Subgroups — #11754356

Order 960.

620 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6779	C₂	1	2	yes	maximal, complemented
2	#91819	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4076	C₂	1	1	yes	maximal, characteristic, complemented
5	#91569	C₂	1	2	yes	maximal, complemented
6	#91469	C₂	1	1	yes	maximal, characteristic, complemented
7	#91458	C₂	1	1	yes	maximal, characteristic, complemented
8	#91428	C₂	1	1	yes	maximal, characteristic, complemented
9	Dic₃×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
10	#91395	C₂	1	2	yes	maximal, complemented
11	#91353	C₂	1	2	yes	maximal, complemented
12	#65811	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂×#1220	C₂²	1	2	yes	—
15	C₂×D₆₀	C₂²	1	1	yes	characteristic
16	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
19	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
20	C₂×#1204	C₂²	1	2	yes	—
21	C₂×#1202	C₂²	1	2	yes	—
22	C₂×#1202	C₂²	1	2	yes	—
23	C₂×#1200	C₂²	1	2	yes	—
24	#6779	—	2	8	no	—
25	C₂×D₅×Dic₃	C₂²	1	2	yes	—
26	#6714	C₂²	1	2	yes	—
27	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
28	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
29	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
30	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
31	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
32	C₃×#685	C₂²	1	2	yes	—
33	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
34	Dic₃⋊Dic₅	C₂²	1	2	yes	—
35	#6667	C₂²	1	2	yes	—
36	#6665	C₂²	1	2	yes	—
37	#5596	—	5	1	no	maximal, Hall
38	C₂×#709	—	3	2	no	—
39	C₂×C₄²×C₅	—	3	1	no	—
40	#4076	—	3	1	no	—
41	#4076	—	3	2	no	—
42	#4076	D₃	1	1	yes	characteristic, complemented
43	#4021	—	3	2	no	—
44	#4020	—	3	1	no	—
45	C₄×D₂₀	—	3	1	no	—
46	C₄×D₂₀	—	3	2	no	—
47	C₄×D₂₀	—	3	1	no	—
48	#4015	—	3	1	no	—
49	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
50	C₂²×D₁₅	C₂³	1	2	yes	—
51	C₂²×C₃×D₅	C₂³	1	2	yes	—
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	#1220	—	2	4	no	—
56	C₂×Dic₁₅	C₂³	1	2	yes	—
57	D₆₀	—	2	2	no	—
58	C₂×C₅×Dic₃	—	2	2	no	—
59	C₂×C₅×Dic₃	D₄	1	4	yes	complemented
60	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
62	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
63	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
64	C₃×C₂²⋊D₅	—	2	4	no	—
65	C₂×C₃×Dic₅	C₂³	1	2	yes	—
66	C₃×D₂₀	—	2	2	no	—
67	#1204	—	2	4	no	—
68	#1202	—	2	4	no	—
69	#1202	—	2	4	no	—
70	#1200	—	2	4	no	—
71	D₅×Dic₃	—	2	4	no	—
72	C₂×#237	—	5	2	no	—
73	C₃×#121	—	5	1	no	—
74	#940	—	5	1	no	—
75	#939	—	5	2	no	—
76	D₄×Dic₃	—	5	1	no	—
77	#930	—	5	1	no	—
78	C₂×C₄×Dic₃	D₅	1	1	yes	characteristic, complemented
79	#893	—	5	1	no	—
80	#888	—	5	1	no	—
81	#885	—	5	2	no	—
82	#67⋊C₄	—	5	2	no	—
83	C₂²×C₄×C₅	—	3	1	no	—
84	C₂²×C₄×C₅	—	3	1	no	—
85	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
86	C₂×C₂²⋊D₅	—	3	2	no	—
87	C₂×C₂²⋊D₅	C₂×D₃	1	2	yes	—
88	#709	—	6	8	no	—
89	C₂×D₂₀	—	3	2	no	—
90	C₂×D₂₀	—	3	2	no	—
91	C₂×D₂₀	—	3	1	no	—
92	C₂×D₂₀	C₂×D₃	1	1	yes	characteristic
93	C₂×C₄×D₅	—	3	2	no	—
94	C₂×C₄×D₅	—	3	2	no	—
95	C₂×Dic₁₀	—	3	2	no	—
96	C₄²×C₅	—	3	1	no	—
97	C₄²×C₅	—	3	1	no	—
98	C₄²×C₅	—	3	1	no	—
99	C₄²×C₅	—	3	1	no	—
100	#685	—	3	2	no	—
101	#685	—	3	2	no	—
102	#685	—	3	2	no	—
103	#685	C₂×D₃	1	2	yes	—
104	C₄⋊Dic₅	—	3	1	no	—
105	C₄⋊Dic₅	—	3	2	no	—
106	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
107	#532	—	15	1	no	Sylow
108	C₂²×C₃×C₅	—	2	1	no	—
109	C₂²×C₃×C₅	—	2	1	no	—
110	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
111	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
112	C₂×D₁₅	—	2	2	no	—
113	C₂×D₁₅	—	2	2	no	—
114	C₂×D₁₅	—	2	2	no	—
115	C₂×C₃×D₅	—	2	2	no	—
116	C₂×C₃×D₅	—	2	2	no	—
117	C₂×C₃×D₅	—	2	2	no	—
118	C₃×C₄×C₅	—	2	1	no	—
119	C₃×C₄×C₅	#41	1	2	yes	complemented
120	Dic₁₅	—	2	2	no	—
121	C₃×Dic₅	—	2	2	no	—
122	C₅×Dic₃	—	2	1	no	—
123	C₅×Dic₃	C₂×D₄	1	2	yes	—
124	C₅×Dic₃	C₂×D₄	1	2	yes	—
125	C₅×Dic₃	#41	1	2	yes	complemented
126	C₂×C₃×D₄	—	5	2	no	—
127	C₂×C₃×D₄	—	5	1	no	—
128	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
129	C₂×#67	—	5	2	no	—
130	C₂×#67	—	5	2	no	—
131	C₂×#67	—	5	2	no	—
132	C₂²×Dic₃	—	5	2	no	—
133	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
134	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
135	#237	—	10	8	no	—
136	C₂×D₁₂	—	5	1	no	—
137	C₂×C₄×D₃	—	5	2	no	—
138	C₂×Dic₆	—	5	2	no	—
139	C₃×C₄⋊C₄	—	5	1	no	—
140	C₃×C₂²⋊C₄	—	5	2	no	—
141	#217	—	5	2	no	—
142	(C₂×D₃)⋊C₄	—	5	2	no	—
143	(C₂×D₃)⋊C₄	—	5	2	no	—
144	C₄⋊Dic₃	—	5	1	no	—
145	Dic₃⋊C₄	—	5	2	no	—
146	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
147	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
148	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
149	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
150	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₂²×D₅	—	3	2	no	—
152	C₂²×D₅	C₂²×D₃	1	2	yes	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	3	4	no	—
155	C₂×C₄×C₅	—	3	1	no	—
156	C₂×C₄×C₅	—	3	1	no	—
157	C₂×C₄×C₅	—	3	1	no	—
158	C₂×C₄×C₅	—	3	1	no	—
159	C₂×C₄×C₅	—	6	2	no	—
160	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
162	C₂²⋊D₅	—	2	4	no	—
163	C₂²⋊D₅	—	6	4	no	—
164	C₂×Dic₅	—	3	2	no	—
165	C₂×Dic₅	C₂²×D₃	1	2	yes	—
166	D₂₀	—	2	2	no	—
167	D₂₀	—	6	4	no	—
168	D₂₀	—	6	4	no	—
169	D₂₀	—	6	2	no	—
170	C₄×D₅	—	6	4	no	—
171	C₄×D₅	—	6	4	no	—
172	Dic₁₀	—	6	4	no	—
173	C₂×#41	—	15	2	no	—
174	C₄⋊Q₈	—	15	1	no	—
175	#127	—	15	1	no	—
176	#124	—	15	2	no	—
177	#121	—	5	1	no	—
178	#121	—	15	2	no	—
179	#121	—	15	1	no	—
180	C₄×D₄	—	15	2	no	—
181	C₄×D₄	—	15	1	no	—
182	C₄×D₄	—	15	1	no	—
183	C₂×C₄²	—	3	1	no	—
184	C₂×C₃×C₅	—	2	2	no	—
185	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
187	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
188	D₁₅	—	4	2	no	—
189	C₃×D₅	—	4	2	no	—
190	C₂³×C₃	—	5	2	no	—
191	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
192	C₂²×D₃	—	5	2	no	—
193	C₃×D₄	—	10	4	no	—
194	C₃×D₄	—	10	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	5	2	no	—
197	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
199	#67	—	10	4	no	—
200	#67	—	10	4	no	—
201	#67	—	10	4	no	—
202	C₂×Dic₃	—	2	2	no	—
203	C₂×Dic₃	—	5	2	no	—
204	C₂×Dic₃	—	10	4	no	—
205	C₂×Dic₃	D₂₀	1	4	yes	complemented
206	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
207	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
208	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
209	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
210	D₁₂	—	10	2	no	—
211	C₄×D₃	—	10	4	no	—
212	Dic₆	—	10	4	no	—
213	C₂²×C₅	—	2	1	no	—
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	D₃×D₄	1	2	yes	—
216	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
217	C₂×D₅	—	2	2	no	—
218	C₂×D₅	—	2	2	no	—
219	C₂×D₅	—	2	2	no	—
220	C₂×D₅	—	6	2	no	—
221	C₂×D₅	—	6	2	no	—
222	C₂×D₅	—	6	2	no	—
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	3	2	no	—
225	C₄×C₅	—	3	2	no	—
226	C₄×C₅	—	3	2	no	—
227	C₄×C₅	—	6	1	no	—
228	C₄×C₅	#237	1	2	yes	complemented
229	Dic₅	—	2	2	no	—
230	Dic₅	—	6	2	no	—
231	#41	—	30	8	no	—
232	C₂×Q₈	—	15	2	no	—
233	C₂×D₄	—	5	2	no	—
234	C₂×D₄	—	5	1	no	—
235	C₂×D₄	—	15	2	no	—
236	C₂×D₄	—	15	2	no	—
237	C₂×D₄	—	15	2	no	—
238	C₂×D₄	—	15	1	no	—
239	C₂²×C₄	—	3	1	no	—
240	C₂²×C₄	—	3	1	no	—
241	C₂²×C₄	—	15	2	no	—
242	C₂²×C₄	—	15	2	no	—
243	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
244	C₄⋊C₄	—	5	1	no	—
245	C₄⋊C₄	—	15	2	no	—
246	C₄⋊C₄	—	15	1	no	—
247	C₂²⋊C₄	—	5	2	no	—
248	C₂²⋊C₄	—	15	2	no	—
249	C₂²⋊C₄	—	15	2	no	—
250	C₂²⋊C₄	—	15	2	no	—
251	C₄²	—	3	1	no	—
252	C₄²	—	3	1	no	—
253	C₄²	—	3	1	no	—
254	C₄²	—	3	1	no	—
255	C₃×C₅	#532	1	1	yes	characteristic, complemented, Hall
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	10	2	no	—
259	C₂²×C₃	—	10	2	no	—
260	C₂²×C₃	—	10	2	no	—
261	C₂²×C₃	C₂×D₂₀	1	2	yes	—
262	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
263	C₂×D₃	—	10	2	no	—
264	C₂×D₃	—	10	2	no	—
265	C₂×D₃	—	10	2	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	10	2	no	—
268	C₃×C₄	#709	1	2	yes	complemented
269	Dic₃	—	2	1	no	—
270	Dic₃	—	10	2	no	—
271	Dic₃	C₂×D₂₀	1	2	yes	—
272	Dic₃	C₂×D₂₀	1	2	yes	—
273	Dic₃	#709	1	2	yes	complemented
274	C₂×C₅	—	2	2	no	—
275	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
276	C₂×C₅	C₂×#237	1	1	yes	characteristic
277	C₂×C₅	D₃×#41	1	1	yes	characteristic
278	D₅	—	4	2	no	—
279	D₅	—	12	2	no	—
280	C₂³	—	5	2	no	—
281	C₂³	—	15	2	no	—
282	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
283	Q₈	—	30	4	no	—
284	D₄	—	10	4	no	—
285	D₄	—	10	2	no	—
286	D₄	—	30	4	no	—
287	D₄	—	30	4	no	—
288	D₄	—	30	4	no	—
289	D₄	—	30	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	3	4	no	—
292	C₂×C₄	—	3	1	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	3	1	no	—
295	C₂×C₄	—	3	1	no	—
296	C₂×C₄	—	5	2	no	—
297	C₂×C₄	—	6	2	no	—
298	C₂×C₄	—	15	2	no	—
299	C₂×C₄	—	30	4	no	—
300	C₂×C₄	—	30	4	no	—
301	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
302	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
303	C₂×C₃	—	2	2	no	—
304	C₂×C₃	—	20	2	no	—
305	C₂×C₃	C₂²×D₂₀	1	1	yes	characteristic
306	C₂×C₃	C₂×#709	1	1	yes	characteristic
307	C₂×C₃	C₂×#709	1	1	yes	characteristic
308	D₃	—	20	2	no	—
309	C₅	#5596	1	1	yes	characteristic, complemented, Sylow
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	10	2	no	—
313	C₂²	—	10	2	no	—
314	C₂²	—	10	2	no	—
315	C₂²	—	30	2	no	—
316	C₂²	—	30	2	no	—
317	C₂²	—	30	2	no	—
318	C₂²	D₃×D₂₀	1	2	yes	—
319	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	3	2	no	—
322	C₄	—	3	2	no	—
323	C₄	—	3	2	no	—
324	C₄	—	6	1	no	—
325	C₄	—	10	2	no	—
326	C₄	—	30	2	no	—
327	C₄	#6779	1	2	yes	complemented
328	C₃	#65811	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	2	2	no	—
330	C₂	—	20	2	no	—
331	C₂	—	60	2	no	—
332	C₂	C₂×D₃×D₂₀	1	1	yes	characteristic
333	C₂	D₃×#709	1	1	yes	characteristic
334	C₂	C₂×#6779	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
hypercenter	C₂²×C₄	D₃×D₅
hypocenter	C₃×C₅	#532
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. C₂×#6779	C₂	semidirect
2. #91819	C₂	semidirect
3. C₂×C₄×C₅×Dic₃	C₂	semidirect
4. C₃×#4076	C₂	semidirect
5. #91569	C₂	semidirect
6. #91469	C₂	semidirect
7. #91458	C₂	semidirect
8. #91428	C₂	semidirect
9. Dic₃×D₂₀	C₂	semidirect
10. #91395	C₂	semidirect
11. #91353	C₂	semidirect
42. #4076	D₃	semidirect
59. C₂×C₅×Dic₃	D₄	semidirect
78. C₂×C₄×Dic₃	D₅	semidirect
119. C₃×C₄×C₅	#41	semidirect
125. C₅×Dic₃	#41	semidirect
205. C₂×Dic₃	D₂₀	semidirect
228. C₄×C₅	#237	semidirect
255. C₃×C₅	#532	semidirect
268. C₃×C₄	#709	semidirect
273. Dic₃	#709	semidirect
309. C₅	#5596	semidirect
327. C₄	#6779	semidirect
328. C₃	#65811	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6779	2
#91819	2
C₂×C₄×C₅×Dic₃	2
C₃×#4076	2
#91569	2
#91469	2
#91458	2
#91428	2
Dic₃×D₂₀	2
#91395	2
#91353	2
#65811	3
#5596	5