Subgroups — #11754366

Order 960.

560 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#924	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
5	#91576	C₂	1	2	yes	maximal, complemented
6	#91559	C₂	1	2	yes	maximal
7	#91431	C₂	1	2	yes	maximal, complemented
8	#91376	C₂	1	1	yes	maximal, characteristic, complemented
9	#91360	C₂	1	1	yes	maximal, characteristic, complemented
10	#91333	C₂	1	1	yes	maximal, characteristic, complemented
11	#91328	C₂	1	2	yes	maximal, complemented
12	#65966	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅	C₂²	1	2	yes	—
15	C₂×Dic₃₀	—	2	4	no	—
16	C₂×Dic₃₀	C₂²	1	2	yes	—
17	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
19	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₂²⋊D₅	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
22	C₂×#1204	C₂²	1	2	yes	—
23	C₂×D₅×Dic₃	C₂²	1	2	yes	—
24	D₅×Dic₆	—	2	4	no	—
25	C₅×#217	C₂²	1	2	yes	—
26	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
28	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
29	C₃×#685	C₂²	1	1	yes	characteristic
30	C₃×#683	C₂²	1	1	yes	characteristic
31	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
32	Dic₃⋊Dic₅	C₂²	1	2	yes	—
33	#6669	C₂²	1	2	yes	—
34	#6668	C₂²	1	2	yes	—
35	#6668	C₂²	1	2	yes	—
36	#6665	C₂²	1	2	yes	—
37	Dic₃×Dic₅	C₂²	1	2	yes	—
38	D₄×Dic₆	—	5	1	no	maximal, Hall
39	C₂×Q₈×D₅	—	3	1	no	—
40	C₂²×Dic₁₀	—	3	1	no	—
41	C₅×C₂²⋊Q₈	—	3	1	no	—
42	#4090	—	3	1	no	—
43	C₄×C₂²⋊D₅	D₃	1	1	yes	characteristic
44	#4043	—	3	1	no	—
45	#4042	—	3	2	no	—
46	#4034	—	3	2	no	—
47	#4033	—	3	1	no	—
48	#4027	—	3	2	no	—
49	#4024	—	3	2	no	—
50	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₅	—	2	4	no	—
56	C₂×Dic₁₅	C₂³	1	2	yes	—
57	C₂×Dic₁₅	C₂³	1	2	yes	—
58	Dic₃₀	—	2	2	no	—
59	Dic₃₀	—	2	4	no	—
60	Dic₃₀	D₄	1	4	yes	complemented
61	C₂×C₅×Dic₃	C₂³	1	2	yes	—
62	C₂×C₅×Dic₃	C₂³	1	2	yes	—
63	C₅×Dic₆	—	2	2	no	—
64	C₃×C₂²⋊D₅	Q₈	1	4	yes	complemented
65	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
66	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
67	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
68	C₃×C₄×D₅	—	2	2	no	—
69	#1204	—	2	4	no	—
70	D₅×Dic₃	—	2	4	no	—
71	C₂²×Dic₆	—	5	1	no	—
72	C₂²×Dic₆	—	5	1	no	—
73	C₃×C₄×D₄	—	5	1	no	—
74	D₄×Dic₃	—	5	2	no	—
75	#924	—	5	1	no	—
76	#924	D₅	1	1	yes	characteristic, complemented
77	#888	—	5	2	no	—
78	#878	—	5	2	no	—
79	#878	—	5	2	no	—
80	#869	—	5	1	no	—
81	C₄×Dic₆	—	5	1	no	—
82	C₂×C₅×Q₈	—	3	1	no	—
83	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
84	C₂×C₂²⋊D₅	C₂×D₃	1	1	yes	characteristic
85	C₂²×Dic₅	—	3	2	no	—
86	Q₈×D₅	—	6	4	no	—
87	C₂×C₄×D₅	—	3	2	no	—
88	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
89	C₂×Dic₁₀	—	3	1	no	—
90	C₂×Dic₁₀	—	3	2	no	—
91	C₂×Dic₁₀	—	3	1	no	—
92	C₂×Dic₁₀	—	3	2	no	—
93	C₂×Dic₁₀	—	6	4	no	—
94	C₅×C₄⋊C₄	—	3	2	no	—
95	C₅×C₄⋊C₄	—	3	1	no	—
96	C₅×C₂²⋊C₄	—	3	2	no	—
97	C₂²⋊Dic₅	C₂×D₃	1	1	yes	characteristic
98	#685	—	3	2	no	—
99	#685	C₂×D₃	1	1	yes	characteristic
100	C₄⋊Dic₅	—	3	2	no	—
101	#683	—	3	2	no	—
102	#683	—	3	2	no	—
103	#683	—	3	2	no	—
104	#683	C₂×D₃	1	1	yes	characteristic
105	C₄×Dic₅	—	3	2	no	—
106	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
107	D₄×Q₈	—	15	1	no	Sylow
108	C₂²×C₃×C₅	—	2	1	no	—
109	C₂²×C₃×C₅	—	2	1	no	—
110	C₂²×C₃×C₅	C₂×Q₈	1	2	yes	complemented
111	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
112	C₂×C₃×D₅	—	2	1	no	—
113	C₂×C₃×D₅	—	2	1	no	—
114	C₂×C₃×D₅	C₂×Q₈	1	2	yes	complemented
115	C₃×C₄×C₅	—	2	1	no	—
116	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
117	Dic₁₅	—	2	2	no	—
118	Dic₁₅	C₂×D₄	1	4	yes	—
119	C₃×Dic₅	—	2	1	no	—
120	C₃×Dic₅	—	2	1	no	—
121	C₃×Dic₅	C₂×Q₈	1	2	yes	complemented
122	C₅×Dic₃	—	2	2	no	—
123	C₅×Dic₃	—	2	2	no	—
124	C₂×C₃×D₄	—	5	1	no	—
125	C₂²×C₃×C₄	—	5	1	no	—
126	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
127	C₂²×Dic₃	—	5	2	no	—
128	C₂²×Dic₃	—	5	2	no	—
129	C₂×Dic₆	—	5	1	no	—
130	C₂×Dic₆	—	5	1	no	—
131	C₂×Dic₆	—	5	2	no	—
132	C₂×Dic₆	—	5	2	no	—
133	C₂×Dic₆	—	10	4	no	—
134	C₂×Dic₆	—	10	4	no	—
135	C₂×Dic₆	C₂×D₅	1	1	yes	characteristic
136	C₃×C₄⋊C₄	—	5	1	no	—
137	C₃×C₂²⋊C₄	—	5	1	no	—
138	C₃×C₂²⋊C₄	—	5	1	no	—
139	C₃×C₄²	—	5	1	no	—
140	#217	—	5	2	no	—
141	#217	C₂×D₅	1	2	yes	—
142	C₄⋊Dic₃	—	5	2	no	—
143	C₄⋊Dic₃	—	5	2	no	—
144	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
145	Dic₃⋊C₄	—	5	2	no	—
146	Dic₃⋊C₄	—	5	2	no	—
147	Dic₃⋊C₄	C₂×D₅	1	2	yes	—
148	C₄×Dic₃	—	5	2	no	—
149	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
150	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
151	C₅×Q₈	—	6	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	3	2	no	—
154	C₂×C₄×C₅	—	3	2	no	—
155	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
157	C₂²⋊D₅	Dic₆	1	4	yes	complemented
158	C₂×Dic₅	—	3	2	no	—
159	C₂×Dic₅	—	3	2	no	—
160	C₂×Dic₅	—	6	4	no	—
161	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
162	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
163	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
164	C₄×D₅	—	2	2	no	—
165	C₄×D₅	—	6	4	no	—
166	Dic₁₀	—	3	4	no	—
167	Dic₁₀	—	6	4	no	—
168	Dic₁₀	—	6	4	no	—
169	Dic₁₀	—	6	2	no	—
170	C₂²×Q₈	—	15	1	no	—
171	C₂²×Q₈	—	15	1	no	—
172	C₄⋊Q₈	—	15	1	no	—
173	C₄⋊Q₈	—	15	2	no	—
174	C₂²⋊Q₈	—	3	1	no	—
175	C₂²⋊Q₈	—	15	2	no	—
176	C₂²⋊Q₈	—	15	2	no	—
177	C₂²⋊Q₈	—	15	1	no	—
178	C₄×Q₈	—	15	1	no	—
179	C₄×D₄	—	5	1	no	—
180	C₄×D₄	—	15	2	no	—
181	C₂×C₃×C₅	—	2	2	no	—
182	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
184	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
185	C₃×D₅	—	2	2	no	—
186	C₂³×C₃	—	5	1	no	—
187	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
188	C₃×D₄	—	5	4	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	5	1	no	—
191	C₂×C₃×C₄	—	5	1	no	—
192	C₂×C₃×C₄	—	5	1	no	—
193	C₂×C₃×C₄	—	10	2	no	—
194	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
196	C₂×Dic₃	—	5	2	no	—
197	C₂×Dic₃	—	5	2	no	—
198	C₂×Dic₃	—	10	4	no	—
199	C₂×Dic₃	—	10	4	no	—
200	C₂×Dic₃	C₂²×D₅	1	2	yes	—
201	C₂×Dic₃	C₂²×D₅	1	2	yes	—
202	Dic₆	—	2	2	no	—
203	Dic₆	—	5	4	no	—
204	Dic₆	—	10	4	no	—
205	Dic₆	—	10	4	no	—
206	Dic₆	—	10	2	no	—
207	C₂²×C₅	—	2	1	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	C₂×Dic₆	1	2	yes	complemented
210	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
211	C₂×D₅	—	2	1	no	—
212	C₂×D₅	—	2	1	no	—
213	C₂×D₅	C₂×Dic₆	1	2	yes	complemented
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	6	2	no	—
216	C₄×C₅	—	6	2	no	—
217	C₄×C₅	D₃×D₄	1	2	yes	—
218	Dic₅	—	2	1	no	—
219	Dic₅	—	2	1	no	—
220	Dic₅	—	3	4	no	—
221	Dic₅	—	6	2	no	—
222	Dic₅	C₂×Dic₆	1	2	yes	complemented
223	C₂×Q₈	—	3	1	no	—
224	C₂×Q₈	—	15	2	no	—
225	C₂×Q₈	—	15	1	no	—
226	C₂×Q₈	—	15	2	no	—
227	C₂×Q₈	—	15	1	no	—
228	C₂×Q₈	—	30	4	no	—
229	C₂×Q₈	—	30	4	no	—
230	C₂×D₄	—	5	1	no	—
231	C₂²×C₄	—	5	1	no	—
232	C₂²×C₄	—	15	2	no	—
233	C₂²×C₄	—	15	2	no	—
234	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
235	C₄⋊C₄	—	3	1	no	—
236	C₄⋊C₄	—	3	2	no	—
237	C₄⋊C₄	—	5	1	no	—
238	C₄⋊C₄	—	15	2	no	—
239	C₄⋊C₄	—	15	2	no	—
240	C₄⋊C₄	—	15	2	no	—
241	C₄⋊C₄	—	15	2	no	—
242	C₂²⋊C₄	—	3	2	no	—
243	C₂²⋊C₄	—	5	1	no	—
244	C₂²⋊C₄	—	5	1	no	—
245	C₂²⋊C₄	—	15	2	no	—
246	C₄²	—	5	1	no	—
247	C₄²	—	15	2	no	—
248	C₃×C₅	D₄×Q₈	1	1	yes	characteristic, complemented, Hall
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	5	2	no	—
252	C₂²×C₃	—	10	1	no	—
253	C₂²×C₃	—	10	1	no	—
254	C₂²×C₃	Q₈×D₅	1	2	yes	complemented
255	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	5	2	no	—
258	C₃×C₄	—	10	1	no	—
259	C₃×C₄	—	10	1	no	—
260	C₃×C₄	D₄×D₅	1	2	yes	—
261	Dic₃	—	2	2	no	—
262	Dic₃	—	2	2	no	—
263	Dic₃	—	5	4	no	—
264	Dic₃	—	10	2	no	—
265	C₂×C₅	—	2	2	no	—
266	C₂×C₅	C₂²×Dic₆	1	1	yes	characteristic
267	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
268	C₂×C₅	#1010	1	1	yes	characteristic
269	D₅	—	2	2	no	—
270	C₂³	—	5	1	no	—
271	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
272	Q₈	—	6	2	no	—
273	Q₈	—	15	4	no	—
274	Q₈	—	30	4	no	—
275	Q₈	—	30	4	no	—
276	Q₈	—	30	2	no	—
277	D₄	—	5	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	3	2	no	—
280	C₂×C₄	—	3	2	no	—
281	C₂×C₄	—	5	1	no	—
282	C₂×C₄	—	5	1	no	—
283	C₂×C₄	—	5	1	no	—
284	C₂×C₄	—	10	2	no	—
285	C₂×C₄	—	15	2	no	—
286	C₂×C₄	—	15	2	no	—
287	C₂×C₄	—	30	4	no	—
288	C₂×C₄	—	30	4	no	—
289	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
290	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	—	10	2	no	—
293	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
294	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
295	C₂×C₃	#4150	1	1	yes	characteristic
296	C₅	D₄×Dic₆	1	1	yes	characteristic, complemented, Sylow
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	5	2	no	—
300	C₂²	—	10	1	no	—
301	C₂²	—	10	1	no	—
302	C₂²	D₅×Dic₆	1	2	yes	complemented
303	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	5	2	no	—
306	C₄	—	6	2	no	—
307	C₄	—	6	2	no	—
308	C₄	—	10	1	no	—
309	C₄	—	10	1	no	—
310	C₄	—	15	4	no	—
311	C₄	—	30	2	no	—
312	C₄	#6790	1	2	yes	—
313	C₃	#65966	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	2	2	no	—
315	C₂	—	10	2	no	—
316	C₂	C₂×D₅×Dic₆	1	1	yes	characteristic
317	C₂	#92001	1	1	yes	characteristic
318	C₂	C₂×#6790	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₂²×C₃×C₄×C₅	C₂²
prime radical	C₂×C₃×C₂²⋊D₅	C₂²
hypercenter	C₂²×C₄	D₃×D₅
hypocenter	C₃×C₅	D₄×Q₈
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₃₀	C₂	semidirect
2. C₂×D₅×Dic₆	C₂	semidirect
3. C₅×#924	C₂	semidirect
5. #91576	C₂	semidirect
7. #91431	C₂	semidirect
8. #91376	C₂	semidirect
9. #91360	C₂	semidirect
10. #91333	C₂	semidirect
11. #91328	C₂	semidirect
60. Dic₃₀	D₄	semidirect
64. C₃×C₂²⋊D₅	Q₈	semidirect
76. #924	D₅	semidirect
110. C₂²×C₃×C₅	C₂×Q₈	semidirect
114. C₂×C₃×D₅	C₂×Q₈	semidirect
121. C₃×Dic₅	C₂×Q₈	semidirect
157. C₂²⋊D₅	Dic₆	semidirect
209. C₂²×C₅	C₂×Dic₆	semidirect
213. C₂×D₅	C₂×Dic₆	semidirect
222. Dic₅	C₂×Dic₆	semidirect
248. C₃×C₅	D₄×Q₈	semidirect
254. C₂²×C₃	Q₈×D₅	semidirect
296. C₅	D₄×Dic₆	semidirect
302. C₂²	D₅×Dic₆	semidirect
313. C₃	#65966	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₃₀	2
C₂×D₅×Dic₆	2
C₅×#924	2
C₃×C₄×C₂²⋊D₅	2
#91576	2
#91559	2
#91431	2
#91376	2
#91360	2
#91333	2
#91328	2
#65966	3
D₄×Dic₆	5