Subgroups — #11754145

Order 960.

524 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6817	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	#91322	C₂	1	2	yes	maximal, complemented
5	#91321	C₂	1	2	yes	maximal, complemented
6	#91305	C₂	1	2	yes	maximal, complemented
7	#91304	C₂	1	2	yes	maximal, complemented
8	#91303	C₂	1	4	yes	maximal, complemented
9	#66156	—	3	1	no	maximal, Hall
10	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×#1220	—	2	1	no	—
12	#6817	—	2	1	no	—
13	#6817	—	2	1	no	—
14	#6817	C₂²	1	2	yes	—
15	#6817	C₂²	1	2	yes	complemented
16	C₂×D₆₀	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₄×D₁₅	—	2	1	no	—
18	C₂×Dic₃₀	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₅×#67	—	2	1	no	—
20	C₅×#235	—	2	1	no	—
21	C₅×#235	—	2	1	no	—
22	C₅×#235	C₂²	1	2	yes	—
23	C₅×#235	C₂²	1	2	yes	complemented
24	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₄×C₅×D₃	—	2	1	no	—
26	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic, complemented
27	C₃×#681	C₂²	1	2	yes	complemented
28	C₃×#681	C₂²	1	2	yes	complemented
29	C₂×C₃×#168	C₂²	1	2	yes	complemented
30	#6663	C₂²	1	4	yes	complemented
31	#6660	C₂²	1	4	yes	complemented
32	#6659	C₂²	1	4	yes	complemented
33	#6654	C₂²	1	4	yes	complemented
34	#5748	—	5	1	no	maximal, Hall
35	C₂×C₅×#41	—	3	1	no	—
36	C₂×#709	—	3	1	no	—
37	#4097	—	3	2	no	—
38	#4096	—	3	4	no	—
39	#4095	—	3	2	no	—
40	#4088	—	3	2	no	—
41	#4078	—	3	2	no	—
42	C₂×#681	D₃	1	1	yes	characteristic, complemented
43	C₂³×C₃×C₅	D₄	1	1	yes	characteristic
44	C₂²×D₁₅	—	2	1	no	—
45	C₂²×C₅×D₃	—	2	1	no	—
46	C₂×C₃×C₄×C₅	D₄	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	D₄	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	D₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	#1220	—	2	2	no	—
53	#1220	—	2	2	no	—
54	C₂×Dic₁₅	—	2	1	no	—
55	D₆₀	—	2	1	no	—
56	D₆₀	C₂³	1	2	yes	—
57	C₄×D₁₅	—	2	2	no	—
58	C₄×D₁₅	—	2	2	no	—
59	Dic₃₀	—	2	1	no	—
60	Dic₃₀	C₂³	1	2	yes	—
61	C₅×#67	—	2	2	no	—
62	C₅×#67	—	2	2	no	—
63	C₂×C₅×Dic₃	—	2	1	no	—
64	C₅×D₁₂	—	2	1	no	—
65	C₅×D₁₂	C₂³	1	2	yes	—
66	C₄×C₅×D₃	—	2	2	no	—
67	C₄×C₅×D₃	—	2	2	no	—
68	C₅×Dic₆	—	2	1	no	—
69	C₅×Dic₆	C₂³	1	2	yes	—
70	C₃×#168	C₂³	1	4	yes	—
71	C₂×#235	—	5	1	no	—
72	C₂×#235	D₅	1	1	yes	characteristic, complemented
73	C₂×C₃×M₁₆	—	5	1	no	—
74	#909	—	5	2	no	—
75	#909	—	5	2	no	—
76	#908	—	5	2	no	—
77	#908	—	5	2	no	—
78	#904	—	5	4	no	—
79	C₅×#41	—	3	2	no	—
80	C₅×#41	—	3	2	no	—
81	C₅×#41	—	6	1	no	—
82	C₅×#41	—	6	1	no	—
83	C₂×C₅×Q₈	—	3	1	no	—
84	C₂×C₅×D₄	—	3	1	no	—
85	C₂×C₅×D₄	—	6	1	no	—
86	C₂²×C₄×C₅	—	6	1	no	—
87	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
88	C₂×C₂²⋊D₅	—	6	1	no	—
89	#709	—	3	2	no	—
90	#709	—	3	2	no	—
91	#709	—	6	1	no	—
92	#709	—	6	1	no	—
93	C₂×D₂₀	—	3	1	no	—
94	C₂×C₄×D₅	—	6	1	no	—
95	C₂×Dic₁₀	—	3	1	no	—
96	#689	—	3	4	no	—
97	#688	—	3	4	no	—
98	#687	—	3	4	no	—
99	#686	—	3	4	no	—
100	#681	C₂×D₃	1	2	yes	complemented
101	#681	C₂×D₃	1	2	yes	complemented
102	C₂×#168	C₂×D₃	1	2	yes	complemented
103	#575	—	15	1	no	Sylow
104	C₂²×C₃×C₅	—	4	1	no	—
105	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
108	C₂×D₁₅	—	2	2	no	—
109	C₂×D₁₅	—	4	2	no	—
110	C₂×C₅×D₃	—	2	2	no	—
111	C₂×C₅×D₃	—	4	2	no	—
112	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
113	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
114	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
115	C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
116	Dic₁₅	—	2	2	no	—
117	C₅×Dic₃	—	2	2	no	—
118	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
119	C₂×#67	—	2	1	no	—
120	C₂×#67	—	10	1	no	—
121	#235	—	2	1	no	—
122	#235	—	2	1	no	—
123	#235	—	5	2	no	—
124	#235	—	5	2	no	—
125	#235	—	10	1	no	—
126	#235	—	10	1	no	—
127	#235	C₂×D₅	1	2	yes	—
128	#235	C₂×D₅	1	2	yes	complemented
129	C₂×D₁₂	—	5	1	no	—
130	C₂×D₁₂	C₂×D₅	1	1	yes	characteristic, complemented
131	C₂×C₄×D₃	—	2	1	no	—
132	C₂×C₄×D₃	—	10	1	no	—
133	C₂×Dic₆	—	5	1	no	—
134	C₂×Dic₆	C₂×D₅	1	1	yes	characteristic, complemented
135	C₃×M₁₆	—	5	2	no	—
136	C₃×M₁₆	—	5	2	no	—
137	C₂×C₃×C₈	—	5	2	no	—
138	Dic₁₂	—	5	4	no	—
139	D₂₄	—	5	4	no	—
140	#204	—	5	4	no	—
141	#204	—	5	4	no	—
142	C₂³×C₅	—	6	1	no	—
143	C₂³×C₅	D₁₂	1	1	yes	characteristic
144	C₂²×D₅	—	6	1	no	—
145	C₅×Q₈	—	3	2	no	—
146	C₅×Q₈	—	6	1	no	—
147	C₅×D₄	—	3	2	no	—
148	C₅×D₄	—	6	2	no	—
149	C₅×D₄	—	6	2	no	—
150	C₅×D₄	—	6	1	no	—
151	C₂×C₄×C₅	—	6	1	no	—
152	C₂×C₄×C₅	—	6	2	no	—
153	C₂×C₄×C₅	—	6	2	no	—
154	C₂×C₄×C₅	D₁₂	1	1	yes	characteristic
155	C₂×C₄×C₅	D₁₂	1	1	yes	characteristic
156	C₂×C₄×C₅	D₁₂	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
160	C₂²⋊D₅	—	6	2	no	—
161	C₂²⋊D₅	—	6	2	no	—
162	C₂×Dic₅	—	6	1	no	—
163	D₂₀	—	3	2	no	—
164	D₂₀	—	6	1	no	—
165	C₄×D₅	—	6	2	no	—
166	C₄×D₅	—	6	2	no	—
167	Dic₁₀	—	3	2	no	—
168	Dic₁₀	—	6	1	no	—
169	#168	C₂²×D₃	1	4	yes	—
170	C₂×#41	—	3	1	no	—
171	C₂×#41	—	15	1	no	—
172	#137	—	15	2	no	—
173	#137	—	15	2	no	—
174	AGL(1,ℤ/8)	—	15	2	no	—
175	AGL(1,ℤ/8)	—	15	2	no	—
176	#135	—	15	4	no	—
177	C₂×M₁₆	—	5	1	no	—
178	C₂×C₃×C₅	—	2	1	no	—
179	C₂×C₃×C₅	—	2	1	no	—
180	C₂×C₃×C₅	—	2	1	no	—
181	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
182	D₁₅	—	4	2	no	—
183	C₅×D₃	—	4	2	no	—
184	C₂³×C₃	C₂²⋊D₅	1	1	yes	characteristic
185	C₂²×D₃	—	2	1	no	—
186	C₂²×D₃	—	10	1	no	—
187	C₂×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
193	#67	—	2	2	no	—
194	#67	—	2	2	no	—
195	#67	—	10	2	no	—
196	#67	—	10	2	no	—
197	C₂×Dic₃	—	2	1	no	—
198	C₂×Dic₃	—	10	1	no	—
199	D₁₂	—	2	1	no	—
200	D₁₂	—	5	2	no	—
201	D₁₂	—	10	1	no	—
202	D₁₂	C₂²×D₅	1	2	yes	—
203	C₄×D₃	—	2	2	no	—
204	C₄×D₃	—	2	2	no	—
205	C₄×D₃	—	10	2	no	—
206	C₄×D₃	—	10	2	no	—
207	Dic₆	—	2	1	no	—
208	Dic₆	—	5	2	no	—
209	Dic₆	—	10	1	no	—
210	Dic₆	C₂²×D₅	1	2	yes	—
211	C₃×C₈	—	5	4	no	—
212	C₂²×C₅	—	4	1	no	—
213	C₂²×C₅	—	6	2	no	—
214	C₂²×C₅	—	12	2	no	—
215	C₂²×C₅	C₂×D₁₂	1	1	yes	characteristic
216	C₂²×C₅	C₂×D₁₂	1	1	yes	characteristic
217	C₂²×C₅	C₂×D₁₂	1	1	yes	characteristic
218	C₂×D₅	—	6	2	no	—
219	C₂×D₅	—	12	2	no	—
220	C₄×C₅	—	6	2	no	—
221	C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
222	C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
223	C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
224	C₄×C₅	C₂³×D₃	1	1	yes	characteristic
225	Dic₅	—	6	2	no	—
226	#41	—	3	2	no	—
227	#41	—	3	2	no	—
228	#41	—	6	1	no	—
229	#41	—	6	1	no	—
230	#41	—	15	2	no	—
231	#41	—	15	2	no	—
232	#41	—	30	1	no	—
233	#41	—	30	1	no	—
234	C₂×Q₈	—	3	1	no	—
235	C₂×Q₈	—	15	1	no	—
236	C₂×D₄	—	3	1	no	—
237	C₂×D₄	—	6	1	no	—
238	C₂×D₄	—	15	1	no	—
239	C₂×D₄	—	30	1	no	—
240	C₂²×C₄	—	6	1	no	—
241	C₂²×C₄	—	30	1	no	—
242	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
243	Q₁₆	—	15	4	no	—
244	SD₁₆	—	15	4	no	—
245	SD₁₆	—	15	4	no	—
246	D₈	—	15	4	no	—
247	M₁₆	—	5	2	no	—
248	M₁₆	—	5	2	no	—
249	C₂×C₈	—	5	2	no	—
250	C₃×C₅	#575	1	1	yes	characteristic, complemented, Hall
251	C₂²×C₃	—	4	1	no	—
252	C₂²×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
253	C₂²×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
254	C₂²×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
255	C₂×D₃	—	2	2	no	—
256	C₂×D₃	—	4	2	no	—
257	C₂×D₃	—	10	2	no	—
258	C₂×D₃	—	20	2	no	—
259	C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
260	C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
261	C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
262	C₃×C₄	C₂³×D₅	1	1	yes	characteristic
263	Dic₃	—	2	2	no	—
264	Dic₃	—	10	2	no	—
265	C₂×C₅	—	2	1	no	—
266	C₂×C₅	—	2	1	no	—
267	C₂×C₅	—	2	1	no	—
268	C₂×C₅	—	12	2	no	—
269	C₂×C₅	C₂²×D₁₂	1	1	yes	characteristic
270	D₅	—	12	2	no	—
271	C₂³	—	6	1	no	—
272	C₂³	—	30	1	no	—
273	C₂³	#1203	1	1	yes	characteristic
274	Q₈	—	3	2	no	—
275	Q₈	—	6	1	no	—
276	Q₈	—	15	2	no	—
277	Q₈	—	30	1	no	—
278	D₄	—	3	2	no	—
279	D₄	—	6	2	no	—
280	D₄	—	6	2	no	—
281	D₄	—	6	1	no	—
282	D₄	—	15	2	no	—
283	D₄	—	30	2	no	—
284	D₄	—	30	2	no	—
285	D₄	—	30	1	no	—
286	C₂×C₄	—	6	1	no	—
287	C₂×C₄	—	6	2	no	—
288	C₂×C₄	—	6	2	no	—
289	C₂×C₄	—	30	1	no	—
290	C₂×C₄	—	30	2	no	—
291	C₂×C₄	—	30	2	no	—
292	C₂×C₄	#1203	1	1	yes	characteristic
293	C₂×C₄	#1203	1	1	yes	characteristic
294	C₂×C₄	#1203	1	1	yes	characteristic
295	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
296	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
297	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
298	C₈	—	5	4	no	—
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
303	D₃	—	4	2	no	—
304	D₃	—	20	2	no	—
305	C₅	#5748	1	1	yes	characteristic, complemented, Sylow
306	C₂²	—	4	1	no	—
307	C₂²	—	6	2	no	—
308	C₂²	—	12	2	no	—
309	C₂²	—	30	2	no	—
310	C₂²	—	60	2	no	—
311	C₂²	C₂×#1203	1	1	yes	characteristic
312	C₂²	C₂×#1203	1	1	yes	characteristic
313	C₂²	C₂×#1203	1	1	yes	characteristic
314	C₄	—	6	2	no	—
315	C₄	—	30	2	no	—
316	C₄	C₂×#1203	1	1	yes	characteristic
317	C₄	C₂×#1203	1	1	yes	characteristic
318	C₄	C₂×#1203	1	1	yes	characteristic
319	C₄	C₂²×D₃×D₅	1	1	yes	characteristic
320	C₃	#66156	1	1	yes	characteristic, complemented, Sylow
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	12	2	no	—
325	C₂	—	60	2	no	—
326	C₂	C₂²×#1203	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#1203
commutator, cosocle	C₃×C₄×C₅	C₂⁴
Frattini	C₄	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₄×C₅	C₂²
socle	C₂×C₃×C₅	C₂²×D₄
hypercenter	C₂²×C₄	D₃×D₅
hypocenter	C₃×C₅	#575
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#6817	C₂	semidirect
2. C₂×C₅×#235	C₂	semidirect
3. C₂×C₃×#681	C₂	semidirect
4. #91322	C₂	semidirect
5. #91321	C₂	semidirect
6. #91305	C₂	semidirect
7. #91304	C₂	semidirect
8. #91303	C₂	semidirect
15. #6817	C₂²	semidirect
16. C₂×D₆₀	C₂²	semidirect
18. C₂×Dic₃₀	C₂²	semidirect
23. C₅×#235	C₂²	semidirect
24. C₂×C₅×D₁₂	C₂²	semidirect
26. C₂×C₅×Dic₆	C₂²	semidirect
27. C₃×#681	C₂²	semidirect
28. C₃×#681	C₂²	semidirect
29. C₂×C₃×#168	C₂²	semidirect
30. #6663	C₂²	semidirect
31. #6660	C₂²	semidirect
32. #6659	C₂²	semidirect
33. #6654	C₂²	semidirect
42. C₂×#681	D₃	semidirect
72. C₂×#235	D₅	semidirect
100. #681	C₂×D₃	semidirect
101. #681	C₂×D₃	semidirect
102. C₂×#168	C₂×D₃	semidirect
128. #235	C₂×D₅	semidirect
130. C₂×D₁₂	C₂×D₅	semidirect
134. C₂×Dic₆	C₂×D₅	semidirect
250. C₃×C₅	#575	semidirect
305. C₅	#5748	semidirect
320. C₃	#66156	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6817	2
C₂×C₅×#235	2
C₂×C₃×#681	2
#91322	2
#91321	2
#91305	2
#91304	2
#91303	2
#66156	3
#5748	5