Subgroups — C₃×C₅×#3036 #14276070

Order 1920.

1096 conjugacy classes of subgroups.

The subgroup lattice has 520 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×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#463	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×#461	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#459	C₂	1	2	yes	maximal, complemented
7	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#417	C₂	1	2	yes	maximal, complemented
9	C₅×#3036	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
11	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄×C₅	—	2	1	no	—
13	C₃×C₅×AGL(1,ℤ/8)	—	2	8	no	—
14	C₂×C₃×C₅×SD₁₆	C₂²	1	4	yes	—
15	C₂×C₃×C₅×D₈	C₂²	1	4	yes	—
16	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
17	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₃×C₅×#127	C₂²	1	1	yes	characteristic, complemented
19	C₃×C₅×#124	C₂²	1	2	yes	complemented
20	C₃×C₅×#121	—	2	1	no	—
21	C₃×C₅×#121	—	2	1	no	—
22	C₃×C₄×C₅×D₄	—	2	1	no	—
23	C₃×C₄×C₅×D₄	—	2	1	no	—
24	C₃×C₄×C₅×D₄	—	2	1	no	—
25	C₃×C₄×C₅×D₄	—	2	1	no	—
26	C₃×C₄×C₅×D₄	C₂²	1	2	yes	—
27	C₃×C₄×C₅×D₄	C₂²	1	2	yes	—
28	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₅×C₂²⋊C₄	—	2	1	no	—
30	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
32	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
33	C₃×C₅×#103	C₂²	1	4	yes	complemented
34	C₃×C₅×#102	C₂²	1	4	yes	complemented
35	C₃×#3036	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
36	C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	1	2	yes	complemented
37	C₅×#532	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×C₄×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
39	C₅×#463	C₂×C₃	1	4	yes	complemented
40	C₅×#461	C₂×C₃	1	2	yes	complemented
41	C₅×#459	C₂×C₃	1	2	yes	complemented
42	C₅×#423	C₂×C₃	1	1	yes	characteristic, complemented
43	C₅×#417	C₂×C₃	1	2	yes	complemented
44	C₂⁴×C₃×C₅	—	2	1	no	—
45	C₃×C₅×#41	—	2	4	no	—
46	C₃×C₅×#41	—	2	4	no	—
47	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
48	C₂×C₃×C₅×D₄	—	2	2	no	—
49	C₂×C₃×C₅×D₄	—	2	1	no	—
50	C₂×C₃×C₅×D₄	—	2	1	no	—
51	C₂×C₃×C₅×D₄	—	2	1	no	—
52	C₂×C₃×C₅×D₄	—	2	2	no	—
53	C₂×C₃×C₅×D₄	D₄	1	4	yes	complemented
54	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
55	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
56	C₂²×C₃×C₄×C₅	—	2	1	no	—
57	C₂²×C₃×C₄×C₅	—	2	2	no	—
58	C₂²×C₃×C₄×C₅	—	2	2	no	—
59	C₂²×C₃×C₄×C₅	—	2	2	no	—
60	C₂²×C₃×C₄×C₅	—	4	4	no	—
61	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
62	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
63	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
64	C₃×C₅×SD₁₆	—	2	8	no	—
65	C₃×C₅×D₈	—	2	8	no	—
66	C₃×C₅×M₁₆	—	2	4	no	—
67	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
68	C₃×C₅×C₄⋊C₄	—	2	1	no	—
69	C₃×C₅×C₄⋊C₄	—	2	1	no	—
70	C₃×C₅×C₄⋊C₄	—	2	1	no	—
71	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
72	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
73	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
74	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
75	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
76	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
77	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
78	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
79	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
80	C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	1	2	yes	complemented
81	C₃×#532	C₂×C₅	1	1	yes	characteristic, complemented
82	C₂×C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic, complemented
83	C₃×#463	C₂×C₅	1	4	yes	complemented
84	C₃×#461	C₂×C₅	1	2	yes	complemented
85	C₃×#459	C₂×C₅	1	2	yes	complemented
86	C₃×#423	C₂×C₅	1	1	yes	characteristic, complemented
87	C₃×#417	C₂×C₅	1	2	yes	complemented
88	C₂×C₅×#41	C₂²×C₃	1	2	yes	—
89	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
90	C₂³×C₄×C₅	—	2	1	no	—
91	C₅×AGL(1,ℤ/8)	—	2	8	no	—
92	C₂×C₅×SD₁₆	C₂²×C₃	1	4	yes	—
93	C₂×C₅×D₈	C₂²×C₃	1	4	yes	—
94	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
95	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
96	C₅×#127	C₂²×C₃	1	1	yes	characteristic, complemented
97	C₅×#124	C₂²×C₃	1	2	yes	complemented
98	C₅×#121	—	2	1	no	—
99	C₅×#121	—	2	1	no	—
100	C₄×C₅×D₄	—	2	1	no	—
101	C₄×C₅×D₄	—	2	1	no	—
102	C₄×C₅×D₄	—	2	1	no	—
103	C₄×C₅×D₄	—	2	1	no	—
104	C₄×C₅×D₄	C₂²×C₃	1	2	yes	—
105	C₄×C₅×D₄	C₂²×C₃	1	2	yes	—
106	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
107	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
108	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
109	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
110	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
111	C₅×#103	C₂²×C₃	1	4	yes	complemented
112	C₅×#102	C₂²×C₃	1	4	yes	complemented
113	#3036	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
114	C₂³×C₃×C₅	—	2	4	no	—
115	C₂³×C₃×C₅	—	2	2	no	—
116	C₂³×C₃×C₅	—	2	2	no	—
117	C₂³×C₃×C₅	—	4	2	no	—
118	C₂³×C₃×C₅	—	4	2	no	—
119	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
120	C₃×C₅×Q₈	—	2	2	no	—
121	C₃×C₅×Q₈	—	2	2	no	—
122	C₃×C₅×D₄	—	2	4	no	—
123	C₃×C₅×D₄	—	2	2	no	—
124	C₃×C₅×D₄	—	2	2	no	—
125	C₃×C₅×D₄	—	2	2	no	—
126	C₃×C₅×D₄	—	4	2	no	—
127	C₃×C₅×D₄	—	4	4	no	—
128	C₃×C₅×D₄	—	4	2	no	—
129	C₃×C₅×D₄	C₂×D₄	1	4	yes	—
130	C₂×C₃×C₄×C₅	—	2	2	no	—
131	C₂×C₃×C₄×C₅	—	2	2	no	—
132	C₂×C₃×C₄×C₅	—	2	2	no	—
133	C₂×C₃×C₄×C₅	—	4	1	no	—
134	C₂×C₃×C₄×C₅	—	4	4	no	—
135	C₂×C₃×C₄×C₅	—	4	2	no	—
136	C₂×C₃×C₄×C₅	—	4	4	no	—
137	C₂×C₃×C₄×C₅	—	4	4	no	—
138	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
143	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
144	C₂×C₃×C₄×C₅	#41	1	2	yes	—
145	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
146	C₃×C₅×C₈	—	2	4	no	—
147	C₂×C₃×#41	C₂²×C₅	1	2	yes	—
148	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
149	C₂³×C₃×C₄	—	2	1	no	—
150	C₃×AGL(1,ℤ/8)	—	2	8	no	—
151	C₂×C₃×SD₁₆	C₂²×C₅	1	4	yes	—
152	C₂×C₃×D₈	C₂²×C₅	1	4	yes	—
153	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
154	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
155	C₃×#127	C₂²×C₅	1	1	yes	characteristic, complemented
156	C₃×#124	C₂²×C₅	1	2	yes	complemented
157	C₃×#121	—	2	1	no	—
158	C₃×#121	—	2	1	no	—
159	C₃×C₄×D₄	—	2	1	no	—
160	C₃×C₄×D₄	—	2	1	no	—
161	C₃×C₄×D₄	—	2	1	no	—
162	C₃×C₄×D₄	—	2	1	no	—
163	C₃×C₄×D₄	C₂²×C₅	1	2	yes	—
164	C₃×C₄×D₄	C₂²×C₅	1	2	yes	—
165	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
166	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
167	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
168	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
169	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
170	C₃×#103	C₂²×C₅	1	4	yes	complemented
171	C₃×#102	C₂²×C₅	1	4	yes	complemented
172	C₂⁴×C₅	—	2	1	no	—
173	C₅×#41	—	2	4	no	—
174	C₅×#41	—	2	4	no	—
175	C₂×C₅×Q₈	C₂³×C₃	1	2	yes	—
176	C₂×C₅×D₄	—	2	2	no	—
177	C₂×C₅×D₄	—	2	1	no	—
178	C₂×C₅×D₄	—	2	1	no	—
179	C₂×C₅×D₄	—	2	1	no	—
180	C₂×C₅×D₄	—	2	2	no	—
181	C₂×C₅×D₄	C₃×D₄	1	4	yes	complemented
182	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
183	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
184	C₂²×C₄×C₅	—	2	1	no	—
185	C₂²×C₄×C₅	—	2	2	no	—
186	C₂²×C₄×C₅	—	2	2	no	—
187	C₂²×C₄×C₅	—	2	2	no	—
188	C₂²×C₄×C₅	—	4	4	no	—
189	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
192	C₅×SD₁₆	—	2	8	no	—
193	C₅×D₈	—	2	8	no	—
194	C₅×M₁₆	—	2	4	no	—
195	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
196	C₅×C₄⋊C₄	—	2	1	no	—
197	C₅×C₄⋊C₄	—	2	1	no	—
198	C₅×C₄⋊C₄	—	2	1	no	—
199	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
200	C₅×C₂²⋊C₄	—	2	2	no	—
201	C₅×C₂²⋊C₄	—	2	2	no	—
202	C₅×C₂²⋊C₄	—	2	2	no	—
203	C₅×C₂²⋊C₄	—	2	2	no	—
204	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
205	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
206	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
207	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
208	C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	1	2	yes	complemented
209	#532	C₂×C₃×C₅	1	1	yes	characteristic, complemented
210	C₂×C₄×D₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
211	#463	C₂×C₃×C₅	1	4	yes	complemented
212	#461	C₂×C₃×C₅	1	2	yes	complemented
213	#459	C₂×C₃×C₅	1	2	yes	complemented
214	#423	C₂×C₃×C₅	1	1	yes	characteristic, complemented
215	#417	C₂×C₃×C₅	1	2	yes	complemented
216	C₂²×C₃×C₅	—	2	1	no	—
217	C₂²×C₃×C₅	—	2	1	no	—
218	C₂²×C₃×C₅	—	2	4	no	—
219	C₂²×C₃×C₅	—	4	8	no	—
220	C₂²×C₃×C₅	—	4	2	no	—
221	C₂²×C₃×C₅	—	4	2	no	—
222	C₂²×C₃×C₅	—	4	2	no	—
223	C₂²×C₃×C₅	—	4	2	no	—
224	C₂²×C₃×C₅	—	4	2	no	—
225	C₂²×C₃×C₅	#121	1	2	yes	—
226	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
227	C₃×C₄×C₅	—	2	1	no	—
228	C₃×C₄×C₅	—	2	1	no	—
229	C₃×C₄×C₅	—	2	1	no	—
230	C₃×C₄×C₅	—	4	2	no	—
231	C₃×C₄×C₅	—	4	2	no	—
232	C₃×C₄×C₅	#121	1	2	yes	—
233	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	complemented
234	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
235	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
236	C₂⁴×C₃	—	2	1	no	—
237	C₃×#41	—	2	4	no	—
238	C₃×#41	—	2	4	no	—
239	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
240	C₂×C₃×D₄	—	2	2	no	—
241	C₂×C₃×D₄	—	2	1	no	—
242	C₂×C₃×D₄	—	2	1	no	—
243	C₂×C₃×D₄	—	2	1	no	—
244	C₂×C₃×D₄	—	2	2	no	—
245	C₂×C₃×D₄	C₅×D₄	1	4	yes	complemented
246	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
247	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
248	C₂²×C₃×C₄	—	2	1	no	—
249	C₂²×C₃×C₄	—	2	2	no	—
250	C₂²×C₃×C₄	—	2	2	no	—
251	C₂²×C₃×C₄	—	2	2	no	—
252	C₂²×C₃×C₄	—	4	4	no	—
253	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
254	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
255	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
256	C₃×SD₁₆	—	2	8	no	—
257	C₃×D₈	—	2	8	no	—
258	C₃×M₁₆	—	2	4	no	—
259	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
260	C₃×C₄⋊C₄	—	2	1	no	—
261	C₃×C₄⋊C₄	—	2	1	no	—
262	C₃×C₄⋊C₄	—	2	1	no	—
263	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
264	C₃×C₂²⋊C₄	—	2	2	no	—
265	C₃×C₂²⋊C₄	—	2	2	no	—
266	C₃×C₂²⋊C₄	—	2	2	no	—
267	C₃×C₂²⋊C₄	—	2	2	no	—
268	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
269	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
270	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
271	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
272	C₂³×C₅	—	2	4	no	—
273	C₂³×C₅	—	2	2	no	—
274	C₂³×C₅	—	2	2	no	—
275	C₂³×C₅	—	4	2	no	—
276	C₂³×C₅	—	4	2	no	—
277	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
278	C₅×Q₈	—	2	2	no	—
279	C₅×Q₈	—	2	2	no	—
280	C₅×D₄	—	2	4	no	—
281	C₅×D₄	—	2	2	no	—
282	C₅×D₄	—	2	2	no	—
283	C₅×D₄	—	2	2	no	—
284	C₅×D₄	—	4	2	no	—
285	C₅×D₄	—	4	4	no	—
286	C₅×D₄	—	4	2	no	—
287	C₅×D₄	C₂×C₃×D₄	1	4	yes	—
288	C₂×C₄×C₅	—	2	2	no	—
289	C₂×C₄×C₅	—	2	2	no	—
290	C₂×C₄×C₅	—	2	2	no	—
291	C₂×C₄×C₅	—	4	1	no	—
292	C₂×C₄×C₅	—	4	4	no	—
293	C₂×C₄×C₅	—	4	2	no	—
294	C₂×C₄×C₅	—	4	4	no	—
295	C₂×C₄×C₅	—	4	4	no	—
296	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
297	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
298	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
299	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
300	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
301	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
302	C₂×C₄×C₅	C₃×#41	1	2	yes	—
303	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
304	C₅×C₈	—	2	4	no	—
305	C₂×#41	C₂²×C₃×C₅	1	2	yes	—
306	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
307	C₂³×C₄	—	2	1	no	—
308	AGL(1,ℤ/8)	—	2	8	no	—
309	C₂×SD₁₆	C₂²×C₃×C₅	1	4	yes	—
310	C₂×D₈	C₂²×C₃×C₅	1	4	yes	—
311	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
312	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
313	#127	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
314	#124	C₂²×C₃×C₅	1	2	yes	complemented
315	#121	—	2	1	no	—
316	#121	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	—	2	1	no	—
320	C₄×D₄	—	2	1	no	—
321	C₄×D₄	C₂²×C₃×C₅	1	2	yes	—
322	C₄×D₄	C₂²×C₃×C₅	1	2	yes	—
323	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
324	C₂×C₂²⋊C₄	—	2	1	no	—
325	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
326	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	complemented
327	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	complemented
328	#103	C₂²×C₃×C₅	1	4	yes	complemented
329	#102	C₂²×C₃×C₅	1	4	yes	complemented
330	C₂×C₃×C₅	—	2	2	no	—
331	C₂×C₃×C₅	—	4	4	no	—
332	C₂×C₃×C₅	—	8	2	no	—
333	C₂×C₃×C₅	C₂×#121	1	1	yes	characteristic
334	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
335	C₂×C₃×C₅	#575	1	1	yes	characteristic
336	C₂³×C₃	—	2	4	no	—
337	C₂³×C₃	—	2	2	no	—
338	C₂³×C₃	—	2	2	no	—
339	C₂³×C₃	—	4	2	no	—
340	C₂³×C₃	—	4	2	no	—
341	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
342	C₃×Q₈	—	2	2	no	—
343	C₃×Q₈	—	2	2	no	—
344	C₃×D₄	—	2	4	no	—
345	C₃×D₄	—	2	2	no	—
346	C₃×D₄	—	2	2	no	—
347	C₃×D₄	—	2	2	no	—
348	C₃×D₄	—	4	2	no	—
349	C₃×D₄	—	4	4	no	—
350	C₃×D₄	—	4	2	no	—
351	C₃×D₄	C₂×C₅×D₄	1	4	yes	—
352	C₂×C₃×C₄	—	2	2	no	—
353	C₂×C₃×C₄	—	2	2	no	—
354	C₂×C₃×C₄	—	2	2	no	—
355	C₂×C₃×C₄	—	4	1	no	—
356	C₂×C₃×C₄	—	4	4	no	—
357	C₂×C₃×C₄	—	4	2	no	—
358	C₂×C₃×C₄	—	4	4	no	—
359	C₂×C₃×C₄	—	4	4	no	—
360	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
361	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
362	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
363	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
364	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
365	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
366	C₂×C₃×C₄	C₅×#41	1	2	yes	—
367	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
368	C₃×C₈	—	2	4	no	—
369	C₂²×C₅	—	2	1	no	—
370	C₂²×C₅	—	2	1	no	—
371	C₂²×C₅	—	2	4	no	—
372	C₂²×C₅	—	4	8	no	—
373	C₂²×C₅	—	4	2	no	—
374	C₂²×C₅	—	4	2	no	—
375	C₂²×C₅	—	4	2	no	—
376	C₂²×C₅	—	4	2	no	—
377	C₂²×C₅	—	4	2	no	—
378	C₂²×C₅	C₃×#121	1	2	yes	—
379	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
380	C₄×C₅	—	2	1	no	—
381	C₄×C₅	—	2	1	no	—
382	C₄×C₅	—	2	1	no	—
383	C₄×C₅	—	4	2	no	—
384	C₄×C₅	—	4	2	no	—
385	C₄×C₅	C₃×#121	1	2	yes	—
386	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	complemented
387	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
388	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
389	C₂⁴	—	2	1	no	—
390	#41	—	2	4	no	—
391	#41	—	2	4	no	—
392	C₂×Q₈	C₂³×C₃×C₅	1	2	yes	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	—	2	1	no	—
396	C₂×D₄	—	2	1	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	C₃×C₅×D₄	1	4	yes	complemented
399	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
400	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	4	4	no	—
406	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
407	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
408	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
409	SD₁₆	—	2	8	no	—
410	D₈	—	2	8	no	—
411	M₁₆	—	2	4	no	—
412	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	C₂³×C₃×C₅	1	2	yes	—
417	C₂²⋊C₄	—	2	2	no	—
418	C₂²⋊C₄	—	2	2	no	—
419	C₂²⋊C₄	—	2	2	no	—
420	C₂²⋊C₄	—	2	2	no	—
421	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
422	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
423	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
424	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
425	C₃×C₅	#3036	1	1	yes	characteristic, direct product factor, Hall
426	C₂²×C₃	—	2	1	no	—
427	C₂²×C₃	—	2	1	no	—
428	C₂²×C₃	—	2	4	no	—
429	C₂²×C₃	—	4	8	no	—
430	C₂²×C₃	—	4	2	no	—
431	C₂²×C₃	—	4	2	no	—
432	C₂²×C₃	—	4	2	no	—
433	C₂²×C₃	—	4	2	no	—
434	C₂²×C₃	—	4	2	no	—
435	C₂²×C₃	C₅×#121	1	2	yes	—
436	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
437	C₃×C₄	—	2	1	no	—
438	C₃×C₄	—	2	1	no	—
439	C₃×C₄	—	2	1	no	—
440	C₃×C₄	—	4	2	no	—
441	C₃×C₄	—	4	2	no	—
442	C₃×C₄	C₅×#121	1	2	yes	—
443	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	complemented
444	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
445	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
446	C₂×C₅	—	2	2	no	—
447	C₂×C₅	—	4	4	no	—
448	C₂×C₅	—	8	2	no	—
449	C₂×C₅	C₂×C₃×#121	1	1	yes	characteristic
450	C₂×C₅	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
451	C₂×C₅	C₃×#575	1	1	yes	characteristic
452	C₂³	—	2	2	no	—
453	C₂³	—	2	4	no	—
454	C₂³	—	2	2	no	—
455	C₂³	—	4	2	no	—
456	C₂³	—	4	2	no	—
457	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
458	Q₈	—	2	2	no	—
459	Q₈	—	2	2	no	—
460	D₄	—	2	2	no	—
461	D₄	—	2	4	no	—
462	D₄	—	2	2	no	—
463	D₄	—	2	2	no	—
464	D₄	—	4	2	no	—
465	D₄	—	4	4	no	—
466	D₄	—	4	2	no	—
467	D₄	C₂×C₃×C₅×D₄	1	4	yes	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	4	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	4	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
477	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
478	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
479	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
480	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
481	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
482	C₂×C₄	C₃×C₅×#41	1	2	yes	—
483	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
484	C₈	—	2	4	no	—
485	C₂×C₃	—	2	2	no	—
486	C₂×C₃	—	4	4	no	—
487	C₂×C₃	—	8	2	no	—
488	C₂×C₃	C₂×C₅×#121	1	1	yes	characteristic
489	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
490	C₂×C₃	C₅×#575	1	1	yes	characteristic
491	C₅	C₃×#3036	1	1	yes	characteristic, direct product factor, Sylow
492	C₂²	—	2	1	no	—
493	C₂²	—	2	1	no	—
494	C₂²	—	2	4	no	—
495	C₂²	—	4	8	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	4	2	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	2	no	—
501	C₂²	C₃×C₅×#121	1	2	yes	—
502	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
503	C₄	—	2	1	no	—
504	C₄	—	2	1	no	—
505	C₄	—	2	1	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	2	no	—
508	C₄	C₃×C₅×#121	1	2	yes	—
509	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	complemented
510	C₄	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
511	C₄	C₂×C₃×C₅×#41	1	1	yes	characteristic
512	C₃	C₅×#3036	1	1	yes	characteristic, direct product factor, Sylow
513	C₂	—	2	2	no	—
514	C₂	—	4	4	no	—
515	C₂	—	8	2	no	—
516	C₂	C₂×C₃×C₅×#121	1	1	yes	characteristic
517	C₂	C₂×C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
518	C₂	C₃×C₅×#575	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 ◁ C₂×C₃×C₄²×C₅ ◁ 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×#532	C₂	semidirect
3. C₂×C₃×C₄×C₅×D₄	C₂	semidirect
4. C₃×C₅×#463	C₂	semidirect
5. C₃×C₅×#461	C₂	semidirect
6. C₃×C₅×#459	C₂	semidirect
7. C₃×C₅×#423	C₂	semidirect
8. C₃×C₅×#417	C₂	semidirect
9. C₅×#3036	C₃	direct (N × H)
17. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
18. C₃×C₅×#127	C₂²	semidirect
19. C₃×C₅×#124	C₂²	semidirect
31. C₃×C₅×C₄⋊C₈	C₂²	semidirect
32. C₃×C₅×C₄⋊C₈	C₂²	semidirect
33. C₃×C₅×#103	C₂²	semidirect
34. C₃×C₅×#102	C₂²	semidirect
35. C₃×#3036	C₅	direct (N × H)
36. C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	semidirect
37. C₅×#532	C₂×C₃	semidirect
38. C₂×C₄×C₅×D₄	C₂×C₃	semidirect
39. C₅×#463	C₂×C₃	semidirect
40. C₅×#461	C₂×C₃	semidirect
41. C₅×#459	C₂×C₃	semidirect
42. C₅×#423	C₂×C₃	semidirect
43. C₅×#417	C₂×C₃	semidirect
53. C₂×C₃×C₅×D₄	D₄	semidirect
80. C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	semidirect
81. C₃×#532	C₂×C₅	semidirect
82. C₂×C₃×C₄×D₄	C₂×C₅	semidirect
83. C₃×#463	C₂×C₅	semidirect
84. C₃×#461	C₂×C₅	semidirect
85. C₃×#459	C₂×C₅	semidirect
86. C₃×#423	C₂×C₅	semidirect
87. C₃×#417	C₂×C₅	semidirect
95. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
96. C₅×#127	C₂²×C₃	semidirect
97. C₅×#124	C₂²×C₃	semidirect
109. C₅×C₄⋊C₈	C₂²×C₃	semidirect
110. C₅×C₄⋊C₈	C₂²×C₃	semidirect
111. C₅×#103	C₂²×C₃	semidirect
112. C₅×#102	C₂²×C₃	semidirect
113. #3036	C₃×C₅	direct (N × H)
154. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
155. C₃×#127	C₂²×C₅	semidirect
156. C₃×#124	C₂²×C₅	semidirect
168. C₃×C₄⋊C₈	C₂²×C₅	semidirect
169. C₃×C₄⋊C₈	C₂²×C₅	semidirect
170. C₃×#103	C₂²×C₅	semidirect
171. C₃×#102	C₂²×C₅	semidirect
181. C₂×C₅×D₄	C₃×D₄	semidirect
208. C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	semidirect
209. #532	C₂×C₃×C₅	semidirect
210. C₂×C₄×D₄	C₂×C₃×C₅	semidirect
211. #463	C₂×C₃×C₅	semidirect
212. #461	C₂×C₃×C₅	semidirect
213. #459	C₂×C₃×C₅	semidirect
214. #423	C₂×C₃×C₅	semidirect
215. #417	C₂×C₃×C₅	semidirect
233. C₃×C₄×C₅	AGL(1,ℤ/8)	semidirect
245. C₂×C₃×D₄	C₅×D₄	semidirect
312. C₄⋊Q₈	C₂²×C₃×C₅	semidirect
313. #127	C₂²×C₃×C₅	semidirect
314. #124	C₂²×C₃×C₅	semidirect
326. C₄⋊C₈	C₂²×C₃×C₅	semidirect
327. C₄⋊C₈	C₂²×C₃×C₅	semidirect
328. #103	C₂²×C₃×C₅	semidirect
329. #102	C₂²×C₃×C₅	semidirect
386. C₄×C₅	C₃×AGL(1,ℤ/8)	semidirect
398. C₂×D₄	C₃×C₅×D₄	semidirect
425. C₃×C₅	#3036	direct (N × H)
443. C₃×C₄	C₅×AGL(1,ℤ/8)	semidirect
491. C₅	C₃×#3036	direct (N × H)
509. C₄	C₃×C₅×AGL(1,ℤ/8)	semidirect
512. C₃	C₅×#3036	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×#532	2
C₂×C₃×C₄×C₅×D₄	2
C₃×C₅×#463	2
C₃×C₅×#461	2
C₃×C₅×#459	2
C₃×C₅×#423	2
C₃×C₅×#417	2
C₅×#3036	3
C₃×#3036	5

Subgroups — C3×C5×#3036 #14276070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3036 #14276070