Subgroups — C₃×C₅×#3274 #14276308

Order 1920.

1112 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×D₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#469	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#460	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#449	C₂	1	2	yes	maximal, complemented
9	C₃×C₅×#447	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×C₅×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₅×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#3274	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×#41	C₂²	1	2	yes	complemented
14	C₂²×C₃×C₅×D₄	—	2	1	no	—
15	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
16	C₃×C₅×AGL(1,ℤ/8)	—	2	8	no	—
17	C₂×C₃×C₅×SD₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₅×SD₁₆	C₂²	1	2	yes	complemented
19	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
20	C₂×C₃×C₅×D₈	C₂²	1	2	yes	complemented
21	C₂×C₃×C₅×D₈	C₂²	1	2	yes	complemented
22	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	complemented
23	C₃×C₅×#127	—	2	1	no	—
24	C₃×C₅×#127	C₂²	1	1	yes	characteristic
25	C₃×C₅×#127	C₂²	1	1	yes	characteristic, complemented
26	C₃×C₅×#121	—	2	1	no	—
27	C₃×C₅×#121	—	2	2	no	—
28	C₃×C₅×#121	C₂²	1	2	yes	—
29	C₃×C₅×#121	C₂²	1	2	yes	complemented
30	C₃×C₅×#120	—	2	2	no	—
31	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
32	C₃×C₄×C₅×D₄	—	2	1	no	—
33	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
34	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
35	C₃×C₅×#106	C₂²	1	1	yes	characteristic, complemented
36	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₃×C₅×#103	C₂²	1	1	yes	characteristic, complemented
38	C₃×C₅×#102	C₂²	1	1	yes	characteristic
39	C₃×C₅×#102	C₂²	1	2	yes	complemented
40	C₃×C₅×#102	C₂²	1	2	yes	complemented
41	C₃×C₅×C₂²⋊C₈	C₂²	1	2	yes	complemented
42	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
43	C₃×#3274	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
44	C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	1	2	yes	complemented
45	C₅×#550	C₂×C₃	1	1	yes	characteristic, complemented
46	C₅×D₄²	C₂×C₃	1	1	yes	characteristic, complemented
47	C₅×#493	C₂×C₃	1	1	yes	characteristic, complemented
48	C₅×#469	C₂×C₃	1	2	yes	complemented
49	C₅×#460	C₂×C₃	1	1	yes	characteristic, complemented
50	C₅×#459	C₂×C₃	1	1	yes	characteristic, complemented
51	C₅×#449	C₂×C₃	1	2	yes	complemented
52	C₅×#447	C₂×C₃	1	2	yes	complemented
53	C₄×C₅×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
54	C₅×#436	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂⁴×C₃×C₅	—	2	2	no	—
56	C₃×C₅×#41	—	2	4	no	—
57	C₃×C₅×#41	—	4	2	no	—
58	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₅×D₄	—	2	4	no	—
60	C₂×C₃×C₅×D₄	—	2	2	no	—
61	C₂×C₃×C₅×D₄	—	2	2	no	—
62	C₂×C₃×C₅×D₄	—	2	2	no	—
63	C₂×C₃×C₅×D₄	—	2	1	no	—
64	C₂×C₃×C₅×D₄	—	2	2	no	—
65	C₂×C₃×C₅×D₄	—	2	1	no	—
66	C₂×C₃×C₅×D₄	—	2	1	no	—
67	C₂×C₃×C₅×D₄	—	2	1	no	—
68	C₂×C₃×C₅×D₄	—	4	2	no	—
69	C₂×C₃×C₅×D₄	—	4	4	no	—
70	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
71	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
72	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
73	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
74	C₂²×C₃×C₄×C₅	—	2	2	no	—
75	C₂²×C₃×C₄×C₅	—	2	1	no	—
76	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
77	C₃×C₅×SD₁₆	—	2	4	no	—
78	C₃×C₅×SD₁₆	D₄	1	4	yes	complemented
79	C₃×C₅×D₈	—	2	4	no	—
80	C₃×C₅×D₈	—	2	4	no	—
81	C₃×C₅×D₈	—	4	2	no	—
82	C₃×C₅×M₁₆	—	2	4	no	—
83	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
84	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
85	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
86	C₃×C₅×C₄⋊C₄	—	2	1	no	—
87	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
88	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
91	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
92	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
93	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
94	C₃×C₄²×C₅	—	2	1	no	—
95	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
96	C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	1	2	yes	complemented
97	C₃×#550	C₂×C₅	1	1	yes	characteristic, complemented
98	C₃×D₄²	C₂×C₅	1	1	yes	characteristic, complemented
99	C₃×#493	C₂×C₅	1	1	yes	characteristic, complemented
100	C₃×#469	C₂×C₅	1	2	yes	complemented
101	C₃×#460	C₂×C₅	1	1	yes	characteristic, complemented
102	C₃×#459	C₂×C₅	1	1	yes	characteristic, complemented
103	C₃×#449	C₂×C₅	1	2	yes	complemented
104	C₃×#447	C₂×C₅	1	2	yes	complemented
105	C₃×C₄×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
106	C₃×#436	C₂×C₅	1	1	yes	characteristic, complemented
107	C₂×C₅×#41	C₂²×C₃	1	2	yes	complemented
108	C₂²×C₅×D₄	—	2	1	no	—
109	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
110	C₅×AGL(1,ℤ/8)	—	2	8	no	—
111	C₂×C₅×SD₁₆	C₂²×C₃	1	1	yes	characteristic
112	C₂×C₅×SD₁₆	C₂²×C₃	1	2	yes	complemented
113	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
114	C₂×C₅×D₈	C₂²×C₃	1	2	yes	complemented
115	C₂×C₅×D₈	C₂²×C₃	1	2	yes	complemented
116	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	complemented
117	C₅×#127	—	2	1	no	—
118	C₅×#127	C₂²×C₃	1	1	yes	characteristic
119	C₅×#127	C₂²×C₃	1	1	yes	characteristic, complemented
120	C₅×#121	—	2	1	no	—
121	C₅×#121	—	2	2	no	—
122	C₅×#121	C₂²×C₃	1	2	yes	—
123	C₅×#121	C₂²×C₃	1	2	yes	complemented
124	C₅×#120	—	2	2	no	—
125	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
126	C₄×C₅×D₄	—	2	1	no	—
127	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
128	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
129	C₅×#106	C₂²×C₃	1	1	yes	characteristic, complemented
130	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
131	C₅×#103	C₂²×C₃	1	1	yes	characteristic, complemented
132	C₅×#102	C₂²×C₃	1	1	yes	characteristic
133	C₅×#102	C₂²×C₃	1	2	yes	complemented
134	C₅×#102	C₂²×C₃	1	2	yes	complemented
135	C₅×C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
136	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
137	#3274	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
138	C₂³×C₃×C₅	—	2	1	no	—
139	C₂³×C₃×C₅	—	2	2	no	—
140	C₂³×C₃×C₅	—	2	2	no	—
141	C₂³×C₃×C₅	—	4	4	no	—
142	C₂³×C₃×C₅	—	4	4	no	—
143	C₂³×C₃×C₅	—	8	2	no	—
144	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
145	C₃×C₅×Q₈	—	2	1	no	—
146	C₃×C₅×Q₈	C₂×D₄	1	2	yes	complemented
147	C₃×C₅×D₄	—	2	2	no	—
148	C₃×C₅×D₄	—	2	2	no	—
149	C₃×C₅×D₄	—	2	2	no	—
150	C₃×C₅×D₄	—	2	1	no	—
151	C₃×C₅×D₄	—	2	2	no	—
152	C₃×C₅×D₄	—	2	2	no	—
153	C₃×C₅×D₄	—	4	4	no	—
154	C₃×C₅×D₄	—	4	4	no	—
155	C₃×C₅×D₄	—	4	4	no	—
156	C₃×C₅×D₄	—	4	2	no	—
157	C₃×C₅×D₄	—	4	2	no	—
158	C₃×C₅×D₄	—	8	1	no	—
159	C₃×C₅×D₄	—	8	1	no	—
160	C₃×C₅×D₄	—	8	2	no	—
161	C₃×C₅×D₄	C₂×D₄	1	2	yes	—
162	C₂×C₃×C₄×C₅	—	2	1	no	—
163	C₂×C₃×C₄×C₅	—	2	2	no	—
164	C₂×C₃×C₄×C₅	—	2	2	no	—
165	C₂×C₃×C₄×C₅	—	2	1	no	—
166	C₂×C₃×C₄×C₅	—	2	1	no	—
167	C₂×C₃×C₄×C₅	—	4	4	no	—
168	C₂×C₃×C₄×C₅	—	4	2	no	—
169	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
171	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
172	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
173	C₃×C₅×C₈	—	2	2	no	—
174	C₃×C₅×C₈	—	2	1	no	—
175	C₃×C₅×C₈	C₂×D₄	1	2	yes	complemented
176	C₂×C₃×#41	C₂²×C₅	1	2	yes	complemented
177	C₂²×C₃×D₄	—	2	1	no	—
178	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
179	C₃×AGL(1,ℤ/8)	—	2	8	no	—
180	C₂×C₃×SD₁₆	C₂²×C₅	1	1	yes	characteristic
181	C₂×C₃×SD₁₆	C₂²×C₅	1	2	yes	complemented
182	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
183	C₂×C₃×D₈	C₂²×C₅	1	2	yes	complemented
184	C₂×C₃×D₈	C₂²×C₅	1	2	yes	complemented
185	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	complemented
186	C₃×#127	—	2	1	no	—
187	C₃×#127	C₂²×C₅	1	1	yes	characteristic
188	C₃×#127	C₂²×C₅	1	1	yes	characteristic, complemented
189	C₃×#121	—	2	1	no	—
190	C₃×#121	—	2	2	no	—
191	C₃×#121	C₂²×C₅	1	2	yes	—
192	C₃×#121	C₂²×C₅	1	2	yes	complemented
193	C₃×#120	—	2	2	no	—
194	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
195	C₃×C₄×D₄	—	2	1	no	—
196	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
197	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
198	C₃×#106	C₂²×C₅	1	1	yes	characteristic, complemented
199	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
200	C₃×#103	C₂²×C₅	1	1	yes	characteristic, complemented
201	C₃×#102	C₂²×C₅	1	1	yes	characteristic
202	C₃×#102	C₂²×C₅	1	2	yes	complemented
203	C₃×#102	C₂²×C₅	1	2	yes	complemented
204	C₃×C₂²⋊C₈	C₂²×C₅	1	2	yes	complemented
205	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
206	C₂⁴×C₅	—	2	2	no	—
207	C₅×#41	—	2	4	no	—
208	C₅×#41	—	4	2	no	—
209	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
210	C₂×C₅×D₄	—	2	4	no	—
211	C₂×C₅×D₄	—	2	2	no	—
212	C₂×C₅×D₄	—	2	2	no	—
213	C₂×C₅×D₄	—	2	2	no	—
214	C₂×C₅×D₄	—	2	1	no	—
215	C₂×C₅×D₄	—	2	2	no	—
216	C₂×C₅×D₄	—	2	1	no	—
217	C₂×C₅×D₄	—	2	1	no	—
218	C₂×C₅×D₄	—	2	1	no	—
219	C₂×C₅×D₄	—	4	2	no	—
220	C₂×C₅×D₄	—	4	4	no	—
221	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
222	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
223	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
224	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
225	C₂²×C₄×C₅	—	2	2	no	—
226	C₂²×C₄×C₅	—	2	1	no	—
227	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
228	C₅×SD₁₆	—	2	4	no	—
229	C₅×SD₁₆	C₃×D₄	1	4	yes	complemented
230	C₅×D₈	—	2	4	no	—
231	C₅×D₈	—	2	4	no	—
232	C₅×D₈	—	4	2	no	—
233	C₅×M₁₆	—	2	4	no	—
234	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
235	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
236	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
237	C₅×C₄⋊C₄	—	2	1	no	—
238	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
239	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
240	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
241	C₅×C₂²⋊C₄	—	2	2	no	—
242	C₅×C₂²⋊C₄	—	2	1	no	—
243	C₅×C₂²⋊C₄	—	2	2	no	—
244	C₅×C₂²⋊C₄	C₃×D₄	1	2	yes	—
245	C₄²×C₅	—	2	1	no	—
246	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
247	C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	1	2	yes	complemented
248	#550	C₂×C₃×C₅	1	1	yes	characteristic, complemented
249	D₄²	C₂×C₃×C₅	1	1	yes	characteristic, complemented
250	#493	C₂×C₃×C₅	1	1	yes	characteristic, complemented
251	#469	C₂×C₃×C₅	1	2	yes	complemented
252	#460	C₂×C₃×C₅	1	1	yes	characteristic, complemented
253	#459	C₂×C₃×C₅	1	1	yes	characteristic, complemented
254	#449	C₂×C₃×C₅	1	2	yes	complemented
255	#447	C₂×C₃×C₅	1	2	yes	complemented
256	C₄×SD₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
257	#436	C₂×C₃×C₅	1	1	yes	characteristic, complemented
258	C₂²×C₃×C₅	—	2	2	no	—
259	C₂²×C₃×C₅	—	2	2	no	—
260	C₂²×C₃×C₅	—	2	2	no	—
261	C₂²×C₃×C₅	—	2	2	no	—
262	C₂²×C₃×C₅	—	4	1	no	—
263	C₂²×C₃×C₅	—	4	1	no	—
264	C₂²×C₃×C₅	—	4	2	no	—
265	C₂²×C₃×C₅	—	4	2	no	—
266	C₂²×C₃×C₅	—	4	2	no	—
267	C₂²×C₃×C₅	—	4	2	no	—
268	C₂²×C₃×C₅	—	4	2	no	—
269	C₂²×C₃×C₅	—	4	2	no	—
270	C₂²×C₃×C₅	—	8	8	no	—
271	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
272	C₃×C₄×C₅	—	2	2	no	—
273	C₃×C₄×C₅	—	2	2	no	—
274	C₃×C₄×C₅	—	2	1	no	—
275	C₃×C₄×C₅	—	4	1	no	—
276	C₃×C₄×C₅	—	4	1	no	—
277	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	complemented
278	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
279	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
280	C₂⁴×C₃	—	2	2	no	—
281	C₃×#41	—	2	4	no	—
282	C₃×#41	—	4	2	no	—
283	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
284	C₂×C₃×D₄	—	2	4	no	—
285	C₂×C₃×D₄	—	2	2	no	—
286	C₂×C₃×D₄	—	2	2	no	—
287	C₂×C₃×D₄	—	2	2	no	—
288	C₂×C₃×D₄	—	2	1	no	—
289	C₂×C₃×D₄	—	2	2	no	—
290	C₂×C₃×D₄	—	2	1	no	—
291	C₂×C₃×D₄	—	2	1	no	—
292	C₂×C₃×D₄	—	2	1	no	—
293	C₂×C₃×D₄	—	4	2	no	—
294	C₂×C₃×D₄	—	4	4	no	—
295	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
296	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
297	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
298	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
299	C₂²×C₃×C₄	—	2	2	no	—
300	C₂²×C₃×C₄	—	2	1	no	—
301	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
302	C₃×SD₁₆	—	2	4	no	—
303	C₃×SD₁₆	C₅×D₄	1	4	yes	complemented
304	C₃×D₈	—	2	4	no	—
305	C₃×D₈	—	2	4	no	—
306	C₃×D₈	—	4	2	no	—
307	C₃×M₁₆	—	2	4	no	—
308	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
309	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
310	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
311	C₃×C₄⋊C₄	—	2	1	no	—
312	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
313	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
314	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
315	C₃×C₂²⋊C₄	—	2	2	no	—
316	C₃×C₂²⋊C₄	—	2	1	no	—
317	C₃×C₂²⋊C₄	—	2	2	no	—
318	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
319	C₃×C₄²	—	2	1	no	—
320	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
321	C₂³×C₅	—	2	1	no	—
322	C₂³×C₅	—	2	2	no	—
323	C₂³×C₅	—	2	2	no	—
324	C₂³×C₅	—	4	4	no	—
325	C₂³×C₅	—	4	4	no	—
326	C₂³×C₅	—	8	2	no	—
327	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
328	C₅×Q₈	—	2	1	no	—
329	C₅×Q₈	C₂×C₃×D₄	1	2	yes	complemented
330	C₅×D₄	—	2	2	no	—
331	C₅×D₄	—	2	2	no	—
332	C₅×D₄	—	2	2	no	—
333	C₅×D₄	—	2	1	no	—
334	C₅×D₄	—	2	2	no	—
335	C₅×D₄	—	2	2	no	—
336	C₅×D₄	—	4	4	no	—
337	C₅×D₄	—	4	4	no	—
338	C₅×D₄	—	4	4	no	—
339	C₅×D₄	—	4	2	no	—
340	C₅×D₄	—	4	2	no	—
341	C₅×D₄	—	8	1	no	—
342	C₅×D₄	—	8	1	no	—
343	C₅×D₄	—	8	2	no	—
344	C₅×D₄	C₂×C₃×D₄	1	2	yes	—
345	C₂×C₄×C₅	—	2	1	no	—
346	C₂×C₄×C₅	—	2	2	no	—
347	C₂×C₄×C₅	—	2	2	no	—
348	C₂×C₄×C₅	—	2	1	no	—
349	C₂×C₄×C₅	—	2	1	no	—
350	C₂×C₄×C₅	—	4	4	no	—
351	C₂×C₄×C₅	—	4	2	no	—
352	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
353	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
354	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
355	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
356	C₅×C₈	—	2	2	no	—
357	C₅×C₈	—	2	1	no	—
358	C₅×C₈	C₂×C₃×D₄	1	2	yes	complemented
359	C₂×#41	C₂²×C₃×C₅	1	2	yes	complemented
360	C₂²×D₄	—	2	1	no	—
361	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
362	AGL(1,ℤ/8)	—	2	8	no	—
363	C₂×SD₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
364	C₂×SD₁₆	C₂²×C₃×C₅	1	2	yes	complemented
365	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
366	C₂×D₈	C₂²×C₃×C₅	1	2	yes	complemented
367	C₂×D₈	C₂²×C₃×C₅	1	2	yes	complemented
368	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	complemented
369	#127	—	2	1	no	—
370	#127	C₂²×C₃×C₅	1	1	yes	characteristic
371	#127	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
372	#121	—	2	1	no	—
373	#121	—	2	2	no	—
374	#121	C₂²×C₃×C₅	1	2	yes	—
375	#121	C₂²×C₃×C₅	1	2	yes	complemented
376	#120	—	2	2	no	—
377	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
378	C₄×D₄	—	2	1	no	—
379	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
380	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
381	#106	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
382	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
383	#103	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
384	#102	C₂²×C₃×C₅	1	1	yes	characteristic
385	#102	C₂²×C₃×C₅	1	2	yes	complemented
386	#102	C₂²×C₃×C₅	1	2	yes	complemented
387	C₂²⋊C₈	C₂²×C₃×C₅	1	2	yes	complemented
388	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
389	C₂×C₃×C₅	—	4	2	no	—
390	C₂×C₃×C₅	—	4	2	no	—
391	C₂×C₃×C₅	—	8	2	no	—
392	C₂×C₃×C₅	—	8	2	no	—
393	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
394	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
395	C₂×C₃×C₅	#576	1	1	yes	characteristic
396	C₂³×C₃	—	2	1	no	—
397	C₂³×C₃	—	2	2	no	—
398	C₂³×C₃	—	2	2	no	—
399	C₂³×C₃	—	4	4	no	—
400	C₂³×C₃	—	4	4	no	—
401	C₂³×C₃	—	8	2	no	—
402	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
403	C₃×Q₈	—	2	1	no	—
404	C₃×Q₈	C₂×C₅×D₄	1	2	yes	complemented
405	C₃×D₄	—	2	2	no	—
406	C₃×D₄	—	2	2	no	—
407	C₃×D₄	—	2	2	no	—
408	C₃×D₄	—	2	1	no	—
409	C₃×D₄	—	2	2	no	—
410	C₃×D₄	—	2	2	no	—
411	C₃×D₄	—	4	4	no	—
412	C₃×D₄	—	4	4	no	—
413	C₃×D₄	—	4	4	no	—
414	C₃×D₄	—	4	2	no	—
415	C₃×D₄	—	4	2	no	—
416	C₃×D₄	—	8	1	no	—
417	C₃×D₄	—	8	1	no	—
418	C₃×D₄	—	8	2	no	—
419	C₃×D₄	C₂×C₅×D₄	1	2	yes	—
420	C₂×C₃×C₄	—	2	1	no	—
421	C₂×C₃×C₄	—	2	2	no	—
422	C₂×C₃×C₄	—	2	2	no	—
423	C₂×C₃×C₄	—	2	1	no	—
424	C₂×C₃×C₄	—	2	1	no	—
425	C₂×C₃×C₄	—	4	4	no	—
426	C₂×C₃×C₄	—	4	2	no	—
427	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
428	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
429	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
430	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
431	C₃×C₈	—	2	2	no	—
432	C₃×C₈	—	2	1	no	—
433	C₃×C₈	C₂×C₅×D₄	1	2	yes	complemented
434	C₂²×C₅	—	2	2	no	—
435	C₂²×C₅	—	2	2	no	—
436	C₂²×C₅	—	2	2	no	—
437	C₂²×C₅	—	2	2	no	—
438	C₂²×C₅	—	4	1	no	—
439	C₂²×C₅	—	4	1	no	—
440	C₂²×C₅	—	4	2	no	—
441	C₂²×C₅	—	4	2	no	—
442	C₂²×C₅	—	4	2	no	—
443	C₂²×C₅	—	4	2	no	—
444	C₂²×C₅	—	4	2	no	—
445	C₂²×C₅	—	4	2	no	—
446	C₂²×C₅	—	8	8	no	—
447	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
448	C₄×C₅	—	2	1	no	—
449	C₄×C₅	—	2	2	no	—
450	C₄×C₅	—	2	2	no	—
451	C₄×C₅	—	4	1	no	—
452	C₄×C₅	—	4	1	no	—
453	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	complemented
454	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
455	C₄×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
456	C₂⁴	—	2	2	no	—
457	#41	—	2	4	no	—
458	#41	—	4	2	no	—
459	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
460	C₂×D₄	—	2	4	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	2	1	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	4	2	no	—
470	C₂×D₄	—	4	4	no	—
471	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
472	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
473	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
474	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
478	SD₁₆	—	2	4	no	—
479	SD₁₆	C₃×C₅×D₄	1	4	yes	complemented
480	D₈	—	2	4	no	—
481	D₈	—	2	4	no	—
482	D₈	—	4	2	no	—
483	M₁₆	—	2	4	no	—
484	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
485	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
486	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
489	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
490	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	1	no	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	C₃×C₅×D₄	1	2	yes	—
495	C₄²	—	2	1	no	—
496	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
497	C₃×C₅	#3274	1	1	yes	characteristic, direct product factor, Hall
498	C₂²×C₃	—	2	2	no	—
499	C₂²×C₃	—	2	2	no	—
500	C₂²×C₃	—	2	2	no	—
501	C₂²×C₃	—	2	2	no	—
502	C₂²×C₃	—	4	1	no	—
503	C₂²×C₃	—	4	1	no	—
504	C₂²×C₃	—	4	2	no	—
505	C₂²×C₃	—	4	2	no	—
506	C₂²×C₃	—	4	2	no	—
507	C₂²×C₃	—	4	2	no	—
508	C₂²×C₃	—	4	2	no	—
509	C₂²×C₃	—	4	2	no	—
510	C₂²×C₃	—	8	8	no	—
511	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
512	C₃×C₄	—	2	2	no	—
513	C₃×C₄	—	2	2	no	—
514	C₃×C₄	—	2	1	no	—
515	C₃×C₄	—	4	1	no	—
516	C₃×C₄	—	4	1	no	—
517	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	complemented
518	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
519	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
520	C₂×C₅	—	4	2	no	—
521	C₂×C₅	—	4	2	no	—
522	C₂×C₅	—	8	2	no	—
523	C₂×C₅	—	8	2	no	—
524	C₂×C₅	C₃×D₄²	1	1	yes	characteristic
525	C₂×C₅	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
526	C₂×C₅	C₃×#576	1	1	yes	characteristic
527	C₂³	—	2	1	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	2	no	—
530	C₂³	—	4	4	no	—
531	C₂³	—	4	4	no	—
532	C₂³	—	8	2	no	—
533	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
534	Q₈	—	2	1	no	—
535	Q₈	C₂×C₃×C₅×D₄	1	2	yes	complemented
536	D₄	—	2	2	no	—
537	D₄	—	2	2	no	—
538	D₄	—	2	1	no	—
539	D₄	—	2	2	no	—
540	D₄	—	2	2	no	—
541	D₄	—	2	2	no	—
542	D₄	—	4	4	no	—
543	D₄	—	4	4	no	—
544	D₄	—	4	4	no	—
545	D₄	—	4	2	no	—
546	D₄	—	4	2	no	—
547	D₄	—	8	1	no	—
548	D₄	—	8	1	no	—
549	D₄	—	8	2	no	—
550	D₄	C₂×C₃×C₅×D₄	1	2	yes	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	4	2	no	—
558	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
559	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
560	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
561	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
562	C₈	—	2	2	no	—
563	C₈	—	2	1	no	—
564	C₈	C₂×C₃×C₅×D₄	1	2	yes	complemented
565	C₂×C₃	—	4	2	no	—
566	C₂×C₃	—	4	2	no	—
567	C₂×C₃	—	8	2	no	—
568	C₂×C₃	—	8	2	no	—
569	C₂×C₃	C₅×D₄²	1	1	yes	characteristic
570	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
571	C₂×C₃	C₅×#576	1	1	yes	characteristic
572	C₅	C₃×#3274	1	1	yes	characteristic, direct product factor, Sylow
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	2	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	4	2	no	—
584	C₂²	—	4	2	no	—
585	C₂²	—	8	8	no	—
586	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
587	C₄	—	2	2	no	—
588	C₄	—	2	2	no	—
589	C₄	—	2	1	no	—
590	C₄	—	4	1	no	—
591	C₄	—	4	1	no	—
592	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	complemented
593	C₄	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
594	C₄	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
595	C₃	C₅×#3274	1	1	yes	characteristic, direct product factor, Sylow
596	C₂	—	4	2	no	—
597	C₂	—	4	2	no	—
598	C₂	—	8	2	no	—
599	C₂	—	8	2	no	—
600	C₂	C₃×C₅×D₄²	1	1	yes	characteristic
601	C₂	C₂×C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
602	C₂	C₃×C₅×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄×C₅×D₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	semidirect
2. C₃×C₅×#550	C₂	semidirect
3. C₃×C₅×D₄²	C₂	semidirect
4. C₃×C₅×#493	C₂	semidirect
5. C₃×C₅×#469	C₂	semidirect
6. C₃×C₅×#460	C₂	semidirect
7. C₃×C₅×#459	C₂	semidirect
8. C₃×C₅×#449	C₂	semidirect
9. C₃×C₅×#447	C₂	semidirect
10. C₃×C₄×C₅×SD₁₆	C₂	semidirect
11. C₃×C₅×#436	C₂	semidirect
12. C₅×#3274	C₃	direct (N × H)
13. C₂×C₃×C₅×#41	C₂²	semidirect
18. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
20. C₂×C₃×C₅×D₈	C₂²	semidirect
21. C₂×C₃×C₅×D₈	C₂²	semidirect
22. C₂×C₃×C₅×M₁₆	C₂²	semidirect
25. C₃×C₅×#127	C₂²	semidirect
29. C₃×C₅×#121	C₂²	semidirect
31. C₃×C₄×C₅×Q₈	C₂²	semidirect
35. C₃×C₅×#106	C₂²	semidirect
36. C₃×C₅×C₄⋊C₈	C₂²	semidirect
37. C₃×C₅×#103	C₂²	semidirect
39. C₃×C₅×#102	C₂²	semidirect
40. C₃×C₅×#102	C₂²	semidirect
41. C₃×C₅×C₂²⋊C₈	C₂²	semidirect
42. C₃×C₄×C₅×C₈	C₂²	semidirect
43. C₃×#3274	C₅	direct (N × H)
44. C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	semidirect
45. C₅×#550	C₂×C₃	semidirect
46. C₅×D₄²	C₂×C₃	semidirect
47. C₅×#493	C₂×C₃	semidirect
48. C₅×#469	C₂×C₃	semidirect
49. C₅×#460	C₂×C₃	semidirect
50. C₅×#459	C₂×C₃	semidirect
51. C₅×#449	C₂×C₃	semidirect
52. C₅×#447	C₂×C₃	semidirect
53. C₄×C₅×SD₁₆	C₂×C₃	semidirect
54. C₅×#436	C₂×C₃	semidirect
78. C₃×C₅×SD₁₆	D₄	semidirect
96. C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	semidirect
97. C₃×#550	C₂×C₅	semidirect
98. C₃×D₄²	C₂×C₅	semidirect
99. C₃×#493	C₂×C₅	semidirect
100. C₃×#469	C₂×C₅	semidirect
101. C₃×#460	C₂×C₅	semidirect
102. C₃×#459	C₂×C₅	semidirect
103. C₃×#449	C₂×C₅	semidirect
104. C₃×#447	C₂×C₅	semidirect
105. C₃×C₄×SD₁₆	C₂×C₅	semidirect
106. C₃×#436	C₂×C₅	semidirect
107. C₂×C₅×#41	C₂²×C₃	semidirect
112. C₂×C₅×SD₁₆	C₂²×C₃	semidirect
114. C₂×C₅×D₈	C₂²×C₃	semidirect
115. C₂×C₅×D₈	C₂²×C₃	semidirect
116. C₂×C₅×M₁₆	C₂²×C₃	semidirect
119. C₅×#127	C₂²×C₃	semidirect
123. C₅×#121	C₂²×C₃	semidirect
125. C₄×C₅×Q₈	C₂²×C₃	semidirect
129. C₅×#106	C₂²×C₃	semidirect
130. C₅×C₄⋊C₈	C₂²×C₃	semidirect
131. C₅×#103	C₂²×C₃	semidirect
133. C₅×#102	C₂²×C₃	semidirect
134. C₅×#102	C₂²×C₃	semidirect
135. C₅×C₂²⋊C₈	C₂²×C₃	semidirect
136. C₄×C₅×C₈	C₂²×C₃	semidirect
137. #3274	C₃×C₅	direct (N × H)
146. C₃×C₅×Q₈	C₂×D₄	semidirect
175. C₃×C₅×C₈	C₂×D₄	semidirect
176. C₂×C₃×#41	C₂²×C₅	semidirect
181. C₂×C₃×SD₁₆	C₂²×C₅	semidirect
183. C₂×C₃×D₈	C₂²×C₅	semidirect
184. C₂×C₃×D₈	C₂²×C₅	semidirect
185. C₂×C₃×M₁₆	C₂²×C₅	semidirect
188. C₃×#127	C₂²×C₅	semidirect
192. C₃×#121	C₂²×C₅	semidirect
194. C₃×C₄×Q₈	C₂²×C₅	semidirect
198. C₃×#106	C₂²×C₅	semidirect
199. C₃×C₄⋊C₈	C₂²×C₅	semidirect
200. C₃×#103	C₂²×C₅	semidirect
202. C₃×#102	C₂²×C₅	semidirect
203. C₃×#102	C₂²×C₅	semidirect
204. C₃×C₂²⋊C₈	C₂²×C₅	semidirect
205. C₃×C₄×C₈	C₂²×C₅	semidirect
229. C₅×SD₁₆	C₃×D₄	semidirect
247. C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	semidirect
248. #550	C₂×C₃×C₅	semidirect
249. D₄²	C₂×C₃×C₅	semidirect
250. #493	C₂×C₃×C₅	semidirect
251. #469	C₂×C₃×C₅	semidirect
252. #460	C₂×C₃×C₅	semidirect
253. #459	C₂×C₃×C₅	semidirect
254. #449	C₂×C₃×C₅	semidirect
255. #447	C₂×C₃×C₅	semidirect
256. C₄×SD₁₆	C₂×C₃×C₅	semidirect
257. #436	C₂×C₃×C₅	semidirect
277. C₃×C₄×C₅	AGL(1,ℤ/8)	semidirect
303. C₃×SD₁₆	C₅×D₄	semidirect
329. C₅×Q₈	C₂×C₃×D₄	semidirect
358. C₅×C₈	C₂×C₃×D₄	semidirect
359. C₂×#41	C₂²×C₃×C₅	semidirect
364. C₂×SD₁₆	C₂²×C₃×C₅	semidirect
366. C₂×D₈	C₂²×C₃×C₅	semidirect
367. C₂×D₈	C₂²×C₃×C₅	semidirect
368. C₂×M₁₆	C₂²×C₃×C₅	semidirect
371. #127	C₂²×C₃×C₅	semidirect
375. #121	C₂²×C₃×C₅	semidirect
377. C₄×Q₈	C₂²×C₃×C₅	semidirect
381. #106	C₂²×C₃×C₅	semidirect
382. C₄⋊C₈	C₂²×C₃×C₅	semidirect
383. #103	C₂²×C₃×C₅	semidirect
385. #102	C₂²×C₃×C₅	semidirect
386. #102	C₂²×C₃×C₅	semidirect
387. C₂²⋊C₈	C₂²×C₃×C₅	semidirect
388. C₄×C₈	C₂²×C₃×C₅	semidirect
404. C₃×Q₈	C₂×C₅×D₄	semidirect
433. C₃×C₈	C₂×C₅×D₄	semidirect
453. C₄×C₅	C₃×AGL(1,ℤ/8)	semidirect
479. SD₁₆	C₃×C₅×D₄	semidirect
497. C₃×C₅	#3274	direct (N × H)
517. C₃×C₄	C₅×AGL(1,ℤ/8)	semidirect
535. Q₈	C₂×C₃×C₅×D₄	semidirect
564. C₈	C₂×C₃×C₅×D₄	semidirect
572. C₅	C₃×#3274	direct (N × H)
592. C₄	C₃×C₅×AGL(1,ℤ/8)	semidirect
595. C₃	C₅×#3274	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₅×AGL(1,ℤ/8)	2
C₃×C₅×#550	2
C₃×C₅×D₄²	2
C₃×C₅×#493	2
C₃×C₅×#469	2
C₃×C₅×#460	2
C₃×C₅×#459	2
C₃×C₅×#449	2
C₃×C₅×#447	2
C₃×C₄×C₅×SD₁₆	2
C₃×C₅×#436	2
C₅×#3274	3
C₃×#3274	5

Subgroups — C3×C5×#3274 #14276308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3274 #14276308