Subgroups — C₃×C₅×#3272 #14276306

Order 1920.

1092 conjugacy classes of subgroups.

The subgroup lattice has 596 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×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×D₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#468	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#461	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#450	C₂	1	2	yes	maximal, complemented
9	C₃×C₅×#449	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₅×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#3272	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	2	yes	—
18	C₂×C₃×C₅×SD₁₆	C₂²	1	2	yes	complemented
19	C₂×C₃×C₅×SD₁₆	C₂²	1	2	yes	complemented
20	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₅×D₈	C₂²	1	2	yes	complemented
22	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	complemented
23	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₃×C₅×#127	C₂²	1	1	yes	characteristic
25	C₃×C₅×#123	—	2	2	no	—
26	C₃×C₅×C₂²⋊Q₈	C₂²	1	2	yes	complemented
27	C₃×C₅×#121	—	2	1	no	—
28	C₃×C₅×#121	—	2	1	no	—
29	C₃×C₅×#121	C₂²	1	2	yes	—
30	C₃×C₅×#120	—	2	2	no	—
31	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
32	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
33	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
35	C₃×C₅×#107	C₂²	1	1	yes	characteristic, complemented
36	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₃×C₅×#103	C₂²	1	2	yes	complemented
38	C₃×C₅×#102	C₂²	1	1	yes	characteristic
39	C₃×C₅×#102	C₂²	1	1	yes	characteristic, 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×#3272	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×#547	C₂×C₃	1	1	yes	characteristic, complemented
46	C₅×D₄²	C₂×C₃	1	1	yes	characteristic, complemented
47	C₅×#492	C₂×C₃	1	1	yes	characteristic, complemented
48	C₅×#468	C₂×C₃	1	2	yes	complemented
49	C₅×#461	C₂×C₃	1	1	yes	characteristic, complemented
50	C₅×#459	C₂×C₃	1	1	yes	characteristic, complemented
51	C₅×#450	C₂×C₃	1	2	yes	complemented
52	C₅×#449	C₂×C₃	1	2	yes	complemented
53	C₄×C₅×D₈	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	2	yes	—
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	1	no	—
65	C₂×C₃×C₅×D₄	—	4	2	no	—
66	C₂×C₃×C₅×D₄	—	4	4	no	—
67	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
68	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
69	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
70	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
71	C₂²×C₃×C₄×C₅	—	2	1	no	—
72	C₂²×C₃×C₄×C₅	—	2	2	no	—
73	C₂²×C₃×C₄×C₅	—	2	1	no	—
74	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
75	C₃×C₅×SD₁₆	—	2	4	no	—
76	C₃×C₅×SD₁₆	—	2	4	no	—
77	C₃×C₅×SD₁₆	—	4	2	no	—
78	C₃×C₅×D₈	—	2	4	no	—
79	C₃×C₅×D₈	D₄	1	4	yes	complemented
80	C₃×C₅×M₁₆	—	2	4	no	—
81	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
82	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
83	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
84	C₃×C₅×C₄⋊C₄	—	2	1	no	—
85	C₃×C₅×C₄⋊C₄	—	2	2	no	—
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	1	no	—
93	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
94	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
95	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
96	C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	1	2	yes	complemented
97	C₃×#547	C₂×C₅	1	1	yes	characteristic, complemented
98	C₃×D₄²	C₂×C₅	1	1	yes	characteristic, complemented
99	C₃×#492	C₂×C₅	1	1	yes	characteristic, complemented
100	C₃×#468	C₂×C₅	1	2	yes	complemented
101	C₃×#461	C₂×C₅	1	1	yes	characteristic, complemented
102	C₃×#459	C₂×C₅	1	1	yes	characteristic, complemented
103	C₃×#450	C₂×C₅	1	2	yes	complemented
104	C₃×#449	C₂×C₅	1	2	yes	complemented
105	C₃×C₄×D₈	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	2	yes	—
112	C₂×C₅×SD₁₆	C₂²×C₃	1	2	yes	complemented
113	C₂×C₅×SD₁₆	C₂²×C₃	1	2	yes	complemented
114	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
115	C₂×C₅×D₈	C₂²×C₃	1	2	yes	complemented
116	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	complemented
117	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
118	C₅×#127	C₂²×C₃	1	1	yes	characteristic
119	C₅×#123	—	2	2	no	—
120	C₅×C₂²⋊Q₈	C₂²×C₃	1	2	yes	complemented
121	C₅×#121	—	2	1	no	—
122	C₅×#121	—	2	1	no	—
123	C₅×#121	C₂²×C₃	1	2	yes	—
124	C₅×#120	—	2	2	no	—
125	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
126	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
127	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
128	C₂×C₅×C₄⋊C₄	—	2	1	no	—
129	C₅×#107	C₂²×C₃	1	1	yes	characteristic, complemented
130	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
131	C₅×#103	C₂²×C₃	1	2	yes	complemented
132	C₅×#102	C₂²×C₃	1	1	yes	characteristic
133	C₅×#102	C₂²×C₃	1	1	yes	characteristic, 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	#3272	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
138	C₂³×C₃×C₅	—	2	1	no	—
139	C₂³×C₃×C₅	—	2	1	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	2	no	—
146	C₃×C₅×Q₈	—	2	2	no	—
147	C₃×C₅×D₄	—	2	2	no	—
148	C₃×C₅×D₄	—	2	2	no	—
149	C₃×C₅×D₄	—	2	1	no	—
150	C₃×C₅×D₄	—	2	1	no	—
151	C₃×C₅×D₄	—	2	2	no	—
152	C₃×C₅×D₄	—	4	4	no	—
153	C₃×C₅×D₄	—	4	4	no	—
154	C₃×C₅×D₄	—	4	4	no	—
155	C₃×C₅×D₄	—	4	2	no	—
156	C₃×C₅×D₄	—	8	1	no	—
157	C₃×C₅×D₄	C₂×D₄	1	2	yes	—
158	C₃×C₅×D₄	C₂×D₄	1	2	yes	complemented
159	C₂×C₃×C₄×C₅	—	2	2	no	—
160	C₂×C₃×C₄×C₅	—	2	2	no	—
161	C₂×C₃×C₄×C₅	—	2	1	no	—
162	C₂×C₃×C₄×C₅	—	2	1	no	—
163	C₂×C₃×C₄×C₅	—	2	2	no	—
164	C₂×C₃×C₄×C₅	—	4	4	no	—
165	C₂×C₃×C₄×C₅	—	4	2	no	—
166	C₂×C₃×C₄×C₅	—	4	2	no	—
167	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
169	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
171	C₃×C₅×C₈	—	2	2	no	—
172	C₃×C₅×C₈	—	2	1	no	—
173	C₃×C₅×C₈	C₂×D₄	1	2	yes	complemented
174	C₂×C₃×#41	C₂²×C₅	1	2	yes	complemented
175	C₂²×C₃×D₄	—	2	1	no	—
176	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
177	C₃×AGL(1,ℤ/8)	—	2	8	no	—
178	C₂×C₃×SD₁₆	C₂²×C₅	1	2	yes	—
179	C₂×C₃×SD₁₆	C₂²×C₅	1	2	yes	complemented
180	C₂×C₃×SD₁₆	C₂²×C₅	1	2	yes	complemented
181	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
182	C₂×C₃×D₈	C₂²×C₅	1	2	yes	complemented
183	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	complemented
184	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
185	C₃×#127	C₂²×C₅	1	1	yes	characteristic
186	C₃×#123	—	2	2	no	—
187	C₃×C₂²⋊Q₈	C₂²×C₅	1	2	yes	complemented
188	C₃×#121	—	2	1	no	—
189	C₃×#121	—	2	1	no	—
190	C₃×#121	C₂²×C₅	1	2	yes	—
191	C₃×#120	—	2	2	no	—
192	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
193	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
194	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
195	C₂×C₃×C₄⋊C₄	—	2	1	no	—
196	C₃×#107	C₂²×C₅	1	1	yes	characteristic, complemented
197	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
198	C₃×#103	C₂²×C₅	1	2	yes	complemented
199	C₃×#102	C₂²×C₅	1	1	yes	characteristic
200	C₃×#102	C₂²×C₅	1	1	yes	characteristic, complemented
201	C₃×#102	C₂²×C₅	1	2	yes	complemented
202	C₃×C₂²⋊C₈	C₂²×C₅	1	2	yes	complemented
203	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
204	C₂⁴×C₅	—	2	2	no	—
205	C₅×#41	—	2	4	no	—
206	C₅×#41	—	4	2	no	—
207	C₂×C₅×Q₈	C₂³×C₃	1	2	yes	—
208	C₂×C₅×D₄	—	2	4	no	—
209	C₂×C₅×D₄	—	2	2	no	—
210	C₂×C₅×D₄	—	2	2	no	—
211	C₂×C₅×D₄	—	2	2	no	—
212	C₂×C₅×D₄	—	2	1	no	—
213	C₂×C₅×D₄	—	2	1	no	—
214	C₂×C₅×D₄	—	4	2	no	—
215	C₂×C₅×D₄	—	4	4	no	—
216	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
217	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
218	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
219	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
220	C₂²×C₄×C₅	—	2	1	no	—
221	C₂²×C₄×C₅	—	2	2	no	—
222	C₂²×C₄×C₅	—	2	1	no	—
223	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
224	C₅×SD₁₆	—	2	4	no	—
225	C₅×SD₁₆	—	2	4	no	—
226	C₅×SD₁₆	—	4	2	no	—
227	C₅×D₈	—	2	4	no	—
228	C₅×D₈	C₃×D₄	1	4	yes	complemented
229	C₅×M₁₆	—	2	4	no	—
230	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
231	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
232	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
233	C₅×C₄⋊C₄	—	2	1	no	—
234	C₅×C₄⋊C₄	—	2	2	no	—
235	C₅×C₄⋊C₄	—	2	1	no	—
236	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
237	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
238	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
239	C₅×C₂²⋊C₄	—	2	2	no	—
240	C₅×C₂²⋊C₄	—	2	1	no	—
241	C₅×C₂²⋊C₄	—	2	1	no	—
242	C₅×C₂²⋊C₄	—	2	2	no	—
243	C₅×C₂²⋊C₄	C₃×D₄	1	2	yes	—
244	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
245	C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	1	2	yes	complemented
246	#547	C₂×C₃×C₅	1	1	yes	characteristic, complemented
247	D₄²	C₂×C₃×C₅	1	1	yes	characteristic, complemented
248	#492	C₂×C₃×C₅	1	1	yes	characteristic, complemented
249	#468	C₂×C₃×C₅	1	2	yes	complemented
250	#461	C₂×C₃×C₅	1	1	yes	characteristic, complemented
251	#459	C₂×C₃×C₅	1	1	yes	characteristic, complemented
252	#450	C₂×C₃×C₅	1	2	yes	complemented
253	#449	C₂×C₃×C₅	1	2	yes	complemented
254	C₄×D₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
255	#436	C₂×C₃×C₅	1	1	yes	characteristic, complemented
256	C₂²×C₃×C₅	—	2	2	no	—
257	C₂²×C₃×C₅	—	2	2	no	—
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	—	4	1	no	—
262	C₂²×C₃×C₅	—	4	1	no	—
263	C₂²×C₃×C₅	—	4	1	no	—
264	C₂²×C₃×C₅	—	4	1	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	—	8	8	no	—
269	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
270	C₃×C₄×C₅	—	2	2	no	—
271	C₃×C₄×C₅	—	2	1	no	—
272	C₃×C₄×C₅	—	4	2	no	—
273	C₃×C₄×C₅	—	4	1	no	—
274	C₃×C₄×C₅	—	4	1	no	—
275	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	complemented
276	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
277	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
278	C₂⁴×C₃	—	2	2	no	—
279	C₃×#41	—	2	4	no	—
280	C₃×#41	—	4	2	no	—
281	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
282	C₂×C₃×D₄	—	2	4	no	—
283	C₂×C₃×D₄	—	2	2	no	—
284	C₂×C₃×D₄	—	2	2	no	—
285	C₂×C₃×D₄	—	2	2	no	—
286	C₂×C₃×D₄	—	2	1	no	—
287	C₂×C₃×D₄	—	2	1	no	—
288	C₂×C₃×D₄	—	4	2	no	—
289	C₂×C₃×D₄	—	4	4	no	—
290	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
291	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
292	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
293	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
294	C₂²×C₃×C₄	—	2	1	no	—
295	C₂²×C₃×C₄	—	2	2	no	—
296	C₂²×C₃×C₄	—	2	1	no	—
297	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
298	C₃×SD₁₆	—	2	4	no	—
299	C₃×SD₁₆	—	2	4	no	—
300	C₃×SD₁₆	—	4	2	no	—
301	C₃×D₈	—	2	4	no	—
302	C₃×D₈	C₅×D₄	1	4	yes	complemented
303	C₃×M₁₆	—	2	4	no	—
304	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
305	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
306	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
307	C₃×C₄⋊C₄	—	2	1	no	—
308	C₃×C₄⋊C₄	—	2	2	no	—
309	C₃×C₄⋊C₄	—	2	1	no	—
310	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
311	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
312	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
313	C₃×C₂²⋊C₄	—	2	2	no	—
314	C₃×C₂²⋊C₄	—	2	1	no	—
315	C₃×C₂²⋊C₄	—	2	1	no	—
316	C₃×C₂²⋊C₄	—	2	2	no	—
317	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
318	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
319	C₂³×C₅	—	2	1	no	—
320	C₂³×C₅	—	2	1	no	—
321	C₂³×C₅	—	2	2	no	—
322	C₂³×C₅	—	4	4	no	—
323	C₂³×C₅	—	4	4	no	—
324	C₂³×C₅	—	8	2	no	—
325	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
326	C₅×Q₈	—	2	2	no	—
327	C₅×Q₈	—	2	2	no	—
328	C₅×D₄	—	2	2	no	—
329	C₅×D₄	—	2	2	no	—
330	C₅×D₄	—	2	1	no	—
331	C₅×D₄	—	2	1	no	—
332	C₅×D₄	—	2	2	no	—
333	C₅×D₄	—	4	4	no	—
334	C₅×D₄	—	4	4	no	—
335	C₅×D₄	—	4	4	no	—
336	C₅×D₄	—	4	2	no	—
337	C₅×D₄	—	8	1	no	—
338	C₅×D₄	C₂×C₃×D₄	1	2	yes	—
339	C₅×D₄	C₂×C₃×D₄	1	2	yes	complemented
340	C₂×C₄×C₅	—	2	2	no	—
341	C₂×C₄×C₅	—	2	2	no	—
342	C₂×C₄×C₅	—	2	1	no	—
343	C₂×C₄×C₅	—	2	1	no	—
344	C₂×C₄×C₅	—	2	2	no	—
345	C₂×C₄×C₅	—	4	4	no	—
346	C₂×C₄×C₅	—	4	2	no	—
347	C₂×C₄×C₅	—	4	2	no	—
348	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
349	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
350	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
351	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
352	C₅×C₈	—	2	2	no	—
353	C₅×C₈	—	2	1	no	—
354	C₅×C₈	C₂×C₃×D₄	1	2	yes	complemented
355	C₂×#41	C₂²×C₃×C₅	1	2	yes	complemented
356	C₂²×D₄	—	2	1	no	—
357	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
358	AGL(1,ℤ/8)	—	2	8	no	—
359	C₂×SD₁₆	C₂²×C₃×C₅	1	2	yes	—
360	C₂×SD₁₆	C₂²×C₃×C₅	1	2	yes	complemented
361	C₂×SD₁₆	C₂²×C₃×C₅	1	2	yes	complemented
362	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
363	C₂×D₈	C₂²×C₃×C₅	1	2	yes	complemented
364	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	complemented
365	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
366	#127	C₂²×C₃×C₅	1	1	yes	characteristic
367	#123	—	2	2	no	—
368	C₂²⋊Q₈	C₂²×C₃×C₅	1	2	yes	complemented
369	#121	—	2	1	no	—
370	#121	—	2	1	no	—
371	#121	C₂²×C₃×C₅	1	2	yes	—
372	#120	—	2	2	no	—
373	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
374	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
375	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
376	C₂×C₄⋊C₄	—	2	1	no	—
377	#107	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
378	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
379	#103	C₂²×C₃×C₅	1	2	yes	complemented
380	#102	C₂²×C₃×C₅	1	1	yes	characteristic
381	#102	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
382	#102	C₂²×C₃×C₅	1	2	yes	complemented
383	C₂²⋊C₈	C₂²×C₃×C₅	1	2	yes	complemented
384	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
385	C₂×C₃×C₅	—	4	2	no	—
386	C₂×C₃×C₅	—	4	2	no	—
387	C₂×C₃×C₅	—	4	2	no	—
388	C₂×C₃×C₅	—	8	2	no	—
389	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
390	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
391	C₂×C₃×C₅	#577	1	1	yes	characteristic
392	C₂³×C₃	—	2	1	no	—
393	C₂³×C₃	—	2	1	no	—
394	C₂³×C₃	—	2	2	no	—
395	C₂³×C₃	—	4	4	no	—
396	C₂³×C₃	—	4	4	no	—
397	C₂³×C₃	—	8	2	no	—
398	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
399	C₃×Q₈	—	2	2	no	—
400	C₃×Q₈	—	2	2	no	—
401	C₃×D₄	—	2	2	no	—
402	C₃×D₄	—	2	2	no	—
403	C₃×D₄	—	2	1	no	—
404	C₃×D₄	—	2	1	no	—
405	C₃×D₄	—	2	2	no	—
406	C₃×D₄	—	4	4	no	—
407	C₃×D₄	—	4	4	no	—
408	C₃×D₄	—	4	4	no	—
409	C₃×D₄	—	4	2	no	—
410	C₃×D₄	—	8	1	no	—
411	C₃×D₄	C₂×C₅×D₄	1	2	yes	—
412	C₃×D₄	C₂×C₅×D₄	1	2	yes	complemented
413	C₂×C₃×C₄	—	2	2	no	—
414	C₂×C₃×C₄	—	2	2	no	—
415	C₂×C₃×C₄	—	2	1	no	—
416	C₂×C₃×C₄	—	2	1	no	—
417	C₂×C₃×C₄	—	2	2	no	—
418	C₂×C₃×C₄	—	4	4	no	—
419	C₂×C₃×C₄	—	4	2	no	—
420	C₂×C₃×C₄	—	4	2	no	—
421	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
422	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
423	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
424	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
425	C₃×C₈	—	2	2	no	—
426	C₃×C₈	—	2	1	no	—
427	C₃×C₈	C₂×C₅×D₄	1	2	yes	complemented
428	C₂²×C₅	—	2	2	no	—
429	C₂²×C₅	—	2	2	no	—
430	C₂²×C₅	—	2	2	no	—
431	C₂²×C₅	—	2	2	no	—
432	C₂²×C₅	—	2	2	no	—
433	C₂²×C₅	—	4	1	no	—
434	C₂²×C₅	—	4	1	no	—
435	C₂²×C₅	—	4	1	no	—
436	C₂²×C₅	—	4	1	no	—
437	C₂²×C₅	—	4	2	no	—
438	C₂²×C₅	—	4	2	no	—
439	C₂²×C₅	—	4	2	no	—
440	C₂²×C₅	—	8	8	no	—
441	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
442	C₄×C₅	—	2	1	no	—
443	C₄×C₅	—	2	2	no	—
444	C₄×C₅	—	4	1	no	—
445	C₄×C₅	—	4	1	no	—
446	C₄×C₅	—	4	2	no	—
447	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	complemented
448	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
449	C₄×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
450	C₂⁴	—	2	2	no	—
451	#41	—	2	4	no	—
452	#41	—	4	2	no	—
453	C₂×Q₈	C₂³×C₃×C₅	1	2	yes	—
454	C₂×D₄	—	2	4	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	4	2	no	—
461	C₂×D₄	—	4	4	no	—
462	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
463	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
464	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
465	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
470	SD₁₆	—	2	4	no	—
471	SD₁₆	—	2	4	no	—
472	SD₁₆	—	4	2	no	—
473	D₈	—	2	4	no	—
474	D₈	C₃×C₅×D₄	1	4	yes	complemented
475	M₁₆	—	2	4	no	—
476	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
477	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
478	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
483	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
484	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
485	C₂²⋊C₄	—	2	2	no	—
486	C₂²⋊C₄	—	2	1	no	—
487	C₂²⋊C₄	—	2	1	no	—
488	C₂²⋊C₄	—	2	2	no	—
489	C₂²⋊C₄	C₃×C₅×D₄	1	2	yes	—
490	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
491	C₃×C₅	#3272	1	1	yes	characteristic, direct product factor, Hall
492	C₂²×C₃	—	2	2	no	—
493	C₂²×C₃	—	2	2	no	—
494	C₂²×C₃	—	2	2	no	—
495	C₂²×C₃	—	2	2	no	—
496	C₂²×C₃	—	2	2	no	—
497	C₂²×C₃	—	4	1	no	—
498	C₂²×C₃	—	4	1	no	—
499	C₂²×C₃	—	4	1	no	—
500	C₂²×C₃	—	4	1	no	—
501	C₂²×C₃	—	4	2	no	—
502	C₂²×C₃	—	4	2	no	—
503	C₂²×C₃	—	4	2	no	—
504	C₂²×C₃	—	8	8	no	—
505	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
506	C₃×C₄	—	2	2	no	—
507	C₃×C₄	—	2	1	no	—
508	C₃×C₄	—	4	2	no	—
509	C₃×C₄	—	4	1	no	—
510	C₃×C₄	—	4	1	no	—
511	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	complemented
512	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
513	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
514	C₂×C₅	—	4	2	no	—
515	C₂×C₅	—	4	2	no	—
516	C₂×C₅	—	4	2	no	—
517	C₂×C₅	—	8	2	no	—
518	C₂×C₅	C₃×D₄²	1	1	yes	characteristic
519	C₂×C₅	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
520	C₂×C₅	C₃×#577	1	1	yes	characteristic
521	C₂³	—	2	1	no	—
522	C₂³	—	2	1	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	4	4	no	—
525	C₂³	—	4	4	no	—
526	C₂³	—	8	2	no	—
527	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
528	Q₈	—	2	2	no	—
529	Q₈	—	2	2	no	—
530	D₄	—	2	2	no	—
531	D₄	—	2	1	no	—
532	D₄	—	2	1	no	—
533	D₄	—	2	2	no	—
534	D₄	—	2	2	no	—
535	D₄	—	4	4	no	—
536	D₄	—	4	4	no	—
537	D₄	—	4	4	no	—
538	D₄	—	4	2	no	—
539	D₄	—	8	1	no	—
540	D₄	C₂×C₃×C₅×D₄	1	2	yes	—
541	D₄	C₂×C₃×C₅×D₄	1	2	yes	complemented
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	4	4	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
551	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
552	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
553	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
554	C₈	—	2	2	no	—
555	C₈	—	2	1	no	—
556	C₈	C₂×C₃×C₅×D₄	1	2	yes	complemented
557	C₂×C₃	—	4	2	no	—
558	C₂×C₃	—	4	2	no	—
559	C₂×C₃	—	4	2	no	—
560	C₂×C₃	—	8	2	no	—
561	C₂×C₃	C₅×D₄²	1	1	yes	characteristic
562	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
563	C₂×C₃	C₅×#577	1	1	yes	characteristic
564	C₅	C₃×#3272	1	1	yes	characteristic, direct product factor, Sylow
565	C₂²	—	2	2	no	—
566	C₂²	—	2	2	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	2	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	4	1	no	—
571	C₂²	—	4	1	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	4	1	no	—
574	C₂²	—	4	2	no	—
575	C₂²	—	4	2	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	8	8	no	—
578	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
579	C₄	—	2	2	no	—
580	C₄	—	2	1	no	—
581	C₄	—	4	2	no	—
582	C₄	—	4	1	no	—
583	C₄	—	4	1	no	—
584	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	complemented
585	C₄	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
586	C₄	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
587	C₃	C₅×#3272	1	1	yes	characteristic, direct product factor, Sylow
588	C₂	—	4	2	no	—
589	C₂	—	4	2	no	—
590	C₂	—	4	2	no	—
591	C₂	—	8	2	no	—
592	C₂	C₃×C₅×D₄²	1	1	yes	characteristic
593	C₂	C₂×C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
594	C₂	C₃×C₅×#577	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×#547	C₂	semidirect
3. C₃×C₅×D₄²	C₂	semidirect
4. C₃×C₅×#492	C₂	semidirect
5. C₃×C₅×#468	C₂	semidirect
6. C₃×C₅×#461	C₂	semidirect
7. C₃×C₅×#459	C₂	semidirect
8. C₃×C₅×#450	C₂	semidirect
9. C₃×C₅×#449	C₂	semidirect
10. C₃×C₄×C₅×D₈	C₂	semidirect
11. C₃×C₅×#436	C₂	semidirect
12. C₅×#3272	C₃	direct (N × H)
13. C₂×C₃×C₅×#41	C₂²	semidirect
18. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
19. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
21. C₂×C₃×C₅×D₈	C₂²	semidirect
22. C₂×C₃×C₅×M₁₆	C₂²	semidirect
23. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
26. C₃×C₅×C₂²⋊Q₈	C₂²	semidirect
33. C₃×C₄×C₅×D₄	C₂²	semidirect
35. C₃×C₅×#107	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×#3272	C₅	direct (N × H)
44. C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	semidirect
45. C₅×#547	C₂×C₃	semidirect
46. C₅×D₄²	C₂×C₃	semidirect
47. C₅×#492	C₂×C₃	semidirect
48. C₅×#468	C₂×C₃	semidirect
49. C₅×#461	C₂×C₃	semidirect
50. C₅×#459	C₂×C₃	semidirect
51. C₅×#450	C₂×C₃	semidirect
52. C₅×#449	C₂×C₃	semidirect
53. C₄×C₅×D₈	C₂×C₃	semidirect
54. C₅×#436	C₂×C₃	semidirect
79. C₃×C₅×D₈	D₄	semidirect
96. C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	semidirect
97. C₃×#547	C₂×C₅	semidirect
98. C₃×D₄²	C₂×C₅	semidirect
99. C₃×#492	C₂×C₅	semidirect
100. C₃×#468	C₂×C₅	semidirect
101. C₃×#461	C₂×C₅	semidirect
102. C₃×#459	C₂×C₅	semidirect
103. C₃×#450	C₂×C₅	semidirect
104. C₃×#449	C₂×C₅	semidirect
105. C₃×C₄×D₈	C₂×C₅	semidirect
106. C₃×#436	C₂×C₅	semidirect
107. C₂×C₅×#41	C₂²×C₃	semidirect
112. C₂×C₅×SD₁₆	C₂²×C₃	semidirect
113. C₂×C₅×SD₁₆	C₂²×C₃	semidirect
115. C₂×C₅×D₈	C₂²×C₃	semidirect
116. C₂×C₅×M₁₆	C₂²×C₃	semidirect
117. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
120. C₅×C₂²⋊Q₈	C₂²×C₃	semidirect
127. C₄×C₅×D₄	C₂²×C₃	semidirect
129. C₅×#107	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. #3272	C₃×C₅	direct (N × H)
158. C₃×C₅×D₄	C₂×D₄	semidirect
173. C₃×C₅×C₈	C₂×D₄	semidirect
174. C₂×C₃×#41	C₂²×C₅	semidirect
179. C₂×C₃×SD₁₆	C₂²×C₅	semidirect
180. C₂×C₃×SD₁₆	C₂²×C₅	semidirect
182. C₂×C₃×D₈	C₂²×C₅	semidirect
183. C₂×C₃×M₁₆	C₂²×C₅	semidirect
184. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
187. C₃×C₂²⋊Q₈	C₂²×C₅	semidirect
194. C₃×C₄×D₄	C₂²×C₅	semidirect
196. C₃×#107	C₂²×C₅	semidirect
197. C₃×C₄⋊C₈	C₂²×C₅	semidirect
198. C₃×#103	C₂²×C₅	semidirect
200. C₃×#102	C₂²×C₅	semidirect
201. C₃×#102	C₂²×C₅	semidirect
202. C₃×C₂²⋊C₈	C₂²×C₅	semidirect
203. C₃×C₄×C₈	C₂²×C₅	semidirect
228. C₅×D₈	C₃×D₄	semidirect
245. C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	semidirect
246. #547	C₂×C₃×C₅	semidirect
247. D₄²	C₂×C₃×C₅	semidirect
248. #492	C₂×C₃×C₅	semidirect
249. #468	C₂×C₃×C₅	semidirect
250. #461	C₂×C₃×C₅	semidirect
251. #459	C₂×C₃×C₅	semidirect
252. #450	C₂×C₃×C₅	semidirect
253. #449	C₂×C₃×C₅	semidirect
254. C₄×D₈	C₂×C₃×C₅	semidirect
255. #436	C₂×C₃×C₅	semidirect
275. C₃×C₄×C₅	AGL(1,ℤ/8)	semidirect
302. C₃×D₈	C₅×D₄	semidirect
339. C₅×D₄	C₂×C₃×D₄	semidirect
354. C₅×C₈	C₂×C₃×D₄	semidirect
355. C₂×#41	C₂²×C₃×C₅	semidirect
360. C₂×SD₁₆	C₂²×C₃×C₅	semidirect
361. C₂×SD₁₆	C₂²×C₃×C₅	semidirect
363. C₂×D₈	C₂²×C₃×C₅	semidirect
364. C₂×M₁₆	C₂²×C₃×C₅	semidirect
365. C₄⋊Q₈	C₂²×C₃×C₅	semidirect
368. C₂²⋊Q₈	C₂²×C₃×C₅	semidirect
375. C₄×D₄	C₂²×C₃×C₅	semidirect
377. #107	C₂²×C₃×C₅	semidirect
378. C₄⋊C₈	C₂²×C₃×C₅	semidirect
379. #103	C₂²×C₃×C₅	semidirect
381. #102	C₂²×C₃×C₅	semidirect
382. #102	C₂²×C₃×C₅	semidirect
383. C₂²⋊C₈	C₂²×C₃×C₅	semidirect
384. C₄×C₈	C₂²×C₃×C₅	semidirect
412. C₃×D₄	C₂×C₅×D₄	semidirect
427. C₃×C₈	C₂×C₅×D₄	semidirect
447. C₄×C₅	C₃×AGL(1,ℤ/8)	semidirect
474. D₈	C₃×C₅×D₄	semidirect
491. C₃×C₅	#3272	direct (N × H)
511. C₃×C₄	C₅×AGL(1,ℤ/8)	semidirect
541. D₄	C₂×C₃×C₅×D₄	semidirect
556. C₈	C₂×C₃×C₅×D₄	semidirect
564. C₅	C₃×#3272	direct (N × H)
584. C₄	C₃×C₅×AGL(1,ℤ/8)	semidirect
587. C₃	C₅×#3272	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×#547	2
C₃×C₅×D₄²	2
C₃×C₅×#492	2
C₃×C₅×#468	2
C₃×C₅×#461	2
C₃×C₅×#459	2
C₃×C₅×#450	2
C₃×C₅×#449	2
C₃×C₄×C₅×D₈	2
C₃×C₅×#436	2
C₅×#3272	3
C₃×#3272	5

Subgroups — C3×C5×#3272 #14276306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3272 #14276306