Subgroups — #11756311

Order 960.

580 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×#6817	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#91828	C₂	1	2	yes	maximal, complemented
5	#91819	C₂	1	1	yes	maximal, characteristic, complemented
6	#91813	C₂	1	1	yes	maximal, characteristic, complemented
7	#91785	C₂	1	2	yes	maximal, complemented
8	#91774	C₂	1	2	yes	maximal, complemented
9	#91770	C₂	1	2	yes	maximal, complemented
10	#91764	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#65886	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
16	C₂×#1220	C₂²	1	2	yes	—
17	C₂×#1220	C₂²	1	2	yes	—
18	C₂²×Dic₁₅	C₂²	1	2	yes	—
19	#6817	—	2	4	no	—
20	C₂×D₆₀	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₁₅	C₂²	1	2	yes	—
22	C₂×Dic₃₀	—	2	4	no	—
23	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
24	C₂×Dic₃₀	C₂²	1	2	yes	—
25	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
26	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
30	#6719	C₂²	1	2	yes	—
31	#6714	C₂²	1	2	yes	—
32	#6714	C₂²	1	2	yes	—
33	#6714	C₂²	1	2	yes	—
34	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
35	#6712	C₂²	1	2	yes	—
36	#6712	C₂²	1	2	yes	—
37	C₄×Dic₁₅	C₂²	1	2	yes	—
38	#5552	—	5	1	no	maximal, Hall
39	C₂×#709	—	3	1	no	—
40	C₂²×Dic₁₀	—	3	1	no	—
41	C₄×C₅×D₄	D₃	1	1	yes	characteristic, complemented
42	#4085	—	3	2	no	—
43	#4076	—	3	1	no	—
44	#4070	—	3	1	no	—
45	#4042	—	3	2	no	—
46	#4031	—	3	2	no	—
47	#4027	—	3	2	no	—
48	#4021	—	3	1	no	—
49	C₄×Dic₁₀	—	3	1	no	—
50	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
52	C₂²×D₁₅	C₂³	1	2	yes	—
53	C₃×C₅×D₄	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
61	#1220	—	2	4	no	—
62	#1220	—	2	4	no	—
63	C₂×Dic₁₅	—	2	4	no	—
64	C₂×Dic₁₅	C₂³	1	2	yes	—
65	C₂×Dic₁₅	C₂³	1	2	yes	—
66	C₂×Dic₁₅	C₂³	1	2	yes	—
67	D₆₀	—	2	2	no	—
68	C₄×D₁₅	—	2	4	no	—
69	Dic₃₀	—	2	2	no	—
70	Dic₃₀	—	2	4	no	—
71	Dic₃₀	D₄	1	4	yes	complemented
72	C₂×#235	—	5	1	no	—
73	C₂²×Dic₆	—	5	1	no	—
74	C₃×C₄×D₄	D₅	1	1	yes	characteristic, complemented
75	#939	—	5	2	no	—
76	#930	—	5	1	no	—
77	#924	—	5	1	no	—
78	#896	—	5	2	no	—
79	#885	—	5	2	no	—
80	#67⋊C₄	—	5	2	no	—
81	#875	—	5	1	no	—
82	C₄×Dic₆	—	5	1	no	—
83	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
86	C₂×C₂²⋊D₅	—	3	2	no	—
87	C₂×C₂²⋊D₅	—	3	2	no	—
88	C₂²×Dic₅	—	3	2	no	—
89	#709	—	6	4	no	—
90	C₂×D₂₀	—	3	1	no	—
91	C₂×C₄×D₅	—	3	2	no	—
92	C₂×Dic₁₀	—	3	1	no	—
93	C₂×Dic₁₀	—	3	1	no	—
94	C₂×Dic₁₀	—	3	2	no	—
95	C₂×Dic₁₀	—	6	4	no	—
96	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
97	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
98	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
99	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
100	C₂²⋊Dic₅	—	3	2	no	—
101	#685	—	3	2	no	—
102	#685	—	3	2	no	—
103	#685	—	3	2	no	—
104	C₄⋊Dic₅	—	3	1	no	—
105	#683	—	3	2	no	—
106	#683	—	3	2	no	—
107	C₄×Dic₅	—	3	2	no	—
108	#548	—	15	1	no	Sylow
109	C₂²×C₃×C₅	—	2	1	no	—
110	C₂²×C₃×C₅	—	2	1	no	—
111	C₂²×C₃×C₅	—	2	1	no	—
112	C₂²×C₃×C₅	—	2	1	no	—
113	C₂²×C₃×C₅	—	2	1	no	—
114	C₂²×C₃×C₅	#41	1	2	yes	complemented
115	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
116	C₂×D₁₅	—	2	2	no	—
117	C₂×D₁₅	—	2	2	no	—
118	C₂×D₁₅	—	2	2	no	—
119	C₃×C₄×C₅	—	2	1	no	—
120	C₃×C₄×C₅	—	2	1	no	—
121	C₃×C₄×C₅	—	2	1	no	—
122	C₃×C₄×C₅	—	2	1	no	—
123	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
124	Dic₁₅	—	2	2	no	—
125	Dic₁₅	—	2	2	no	—
126	Dic₁₅	C₂×D₄	1	4	yes	—
127	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
130	C₂×#67	—	5	2	no	—
131	C₂×#67	—	5	2	no	—
132	C₂²×Dic₃	—	5	2	no	—
133	#235	—	10	4	no	—
134	C₂×D₁₂	—	5	1	no	—
135	C₂×C₄×D₃	—	5	2	no	—
136	C₂×Dic₆	—	5	1	no	—
137	C₂×Dic₆	—	5	1	no	—
138	C₂×Dic₆	—	5	2	no	—
139	C₂×Dic₆	—	10	4	no	—
140	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
141	C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
142	C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
143	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
144	#217	—	5	2	no	—
145	(C₂×D₃)⋊C₄	—	5	2	no	—
146	(C₂×D₃)⋊C₄	—	5	2	no	—
147	(C₂×D₃)⋊C₄	—	5	2	no	—
148	C₄⋊Dic₃	—	5	1	no	—
149	Dic₃⋊C₄	—	5	2	no	—
150	Dic₃⋊C₄	—	5	2	no	—
151	C₄×Dic₃	—	5	2	no	—
152	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
153	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
154	C₂²×D₅	—	3	2	no	—
155	C₅×D₄	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
163	C₂²⋊D₅	—	6	4	no	—
164	C₂²⋊D₅	—	6	4	no	—
165	C₂×Dic₅	—	3	2	no	—
166	C₂×Dic₅	—	3	2	no	—
167	C₂×Dic₅	—	3	2	no	—
168	C₂×Dic₅	—	6	4	no	—
169	D₂₀	—	6	2	no	—
170	C₄×D₅	—	6	4	no	—
171	Dic₁₀	—	3	4	no	—
172	Dic₁₀	—	6	4	no	—
173	Dic₁₀	—	6	2	no	—
174	C₂×#41	—	15	1	no	—
175	C₂²×Q₈	—	15	1	no	—
176	#124	—	15	2	no	—
177	#124	—	15	1	no	—
178	C₂²⋊Q₈	—	15	1	no	—
179	C₂²⋊Q₈	—	15	2	no	—
180	#121	—	15	2	no	—
181	#121	—	15	1	no	—
182	C₄×Q₈	—	15	1	no	—
183	C₄×D₄	—	15	2	no	—
184	C₄×D₄	D₁₅	1	1	yes	characteristic, complemented
185	C₂×C₃×C₅	—	2	2	no	—
186	C₂×C₃×C₅	—	4	1	no	—
187	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
189	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
190	D₁₅	—	4	2	no	—
191	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
192	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
193	C₂²×D₃	—	5	2	no	—
194	C₃×D₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
202	#67	—	10	4	no	—
203	#67	—	10	4	no	—
204	C₂×Dic₃	—	5	2	no	—
205	C₂×Dic₃	—	5	2	no	—
206	C₂×Dic₃	—	5	2	no	—
207	C₂×Dic₃	—	10	4	no	—
208	D₁₂	—	10	2	no	—
209	C₄×D₃	—	10	4	no	—
210	Dic₆	—	5	4	no	—
211	Dic₆	—	10	4	no	—
212	Dic₆	—	10	2	no	—
213	C₂²×C₅	—	2	1	no	—
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	—	2	1	no	—
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	#235	1	2	yes	complemented
219	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
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	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	D₃×D₄	1	2	yes	—
228	Dic₅	—	3	4	no	—
229	Dic₅	—	6	2	no	—
230	Dic₅	—	6	2	no	—
231	#41	—	30	4	no	—
232	C₂×Q₈	—	15	1	no	—
233	C₂×Q₈	—	15	2	no	—
234	C₂×Q₈	—	15	1	no	—
235	C₂×Q₈	—	30	4	no	—
236	C₂×D₄	—	15	2	no	—
237	C₂×D₄	—	15	2	no	—
238	C₂×D₄	—	15	1	no	—
239	C₂×D₄	C₂×D₁₅	1	1	yes	characteristic
240	C₂²×C₄	—	15	2	no	—
241	C₂²×C₄	—	15	2	no	—
242	C₂²×C₄	C₂×D₁₅	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₁₅	1	1	yes	characteristic
244	C₄⋊C₄	—	15	2	no	—
245	C₄⋊C₄	—	15	2	no	—
246	C₄⋊C₄	—	15	1	no	—
247	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
248	C₂²⋊C₄	—	15	2	no	—
249	C₂²⋊C₄	—	15	2	no	—
250	C₂²⋊C₄	—	15	2	no	—
251	C₂²⋊C₄	—	15	2	no	—
252	C₂²⋊C₄	C₂×D₁₅	1	1	yes	characteristic
253	C₂²⋊C₄	C₂×D₁₅	1	1	yes	characteristic
254	C₄²	—	15	2	no	—
255	C₄²	C₂×D₁₅	1	1	yes	characteristic
256	C₃×C₅	#548	1	1	yes	characteristic, complemented, Hall
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	#709	1	2	yes	complemented
263	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
264	C₂×D₃	—	10	2	no	—
265	C₂×D₃	—	10	2	no	—
266	C₂×D₃	—	10	2	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	D₄×D₅	1	2	yes	—
272	Dic₃	—	5	4	no	—
273	Dic₃	—	10	2	no	—
274	Dic₃	—	10	2	no	—
275	C₂×C₅	—	2	2	no	—
276	C₂×C₅	—	4	1	no	—
277	C₂×C₅	C₂×#235	1	1	yes	characteristic
278	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
279	C₂×C₅	#1010	1	1	yes	characteristic
280	D₅	—	12	2	no	—
281	C₂³	—	15	2	no	—
282	C₂³	C₂²×D₁₅	1	1	yes	characteristic
283	C₂³	C₂²×D₁₅	1	1	yes	characteristic
284	Q₈	—	15	4	no	—
285	Q₈	—	30	4	no	—
286	Q₈	—	30	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	30	4	no	—
289	D₄	—	30	4	no	—
290	D₄	—	30	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	15	2	no	—
294	C₂×C₄	—	15	2	no	—
295	C₂×C₄	—	15	2	no	—
296	C₂×C₄	—	30	4	no	—
297	C₂×C₄	—	30	4	no	—
298	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
303	C₂×C₃	—	2	2	no	—
304	C₂×C₃	—	4	1	no	—
305	C₂×C₃	C₂×#709	1	1	yes	characteristic
306	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
307	C₂×C₃	#4150	1	1	yes	characteristic
308	D₃	—	20	2	no	—
309	C₅	#5552	1	1	yes	characteristic, complemented, Sylow
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	30	2	no	—
316	C₂²	—	30	2	no	—
317	C₂²	—	30	2	no	—
318	C₂²	#6817	1	2	yes	complemented
319	C₂²	C₂³×D₁₅	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	15	4	no	—
325	C₄	—	30	2	no	—
326	C₄	—	30	2	no	—
327	C₄	D₄×D₁₅	1	2	yes	—
328	C₃	#65886	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	2	2	no	—
330	C₂	—	4	1	no	—
331	C₂	—	60	2	no	—
332	C₂	C₂×#6817	1	1	yes	characteristic
333	C₂	C₂×D₄×D₁₅	1	1	yes	characteristic
334	C₂	#92101	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₄×C₅×D₄	C₂
hypercenter	C₄×D₄	D₁₅
hypocenter	C₃×C₅	#548
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#6817	C₂	semidirect
2. C₂²×Dic₃₀	C₂	semidirect
3. C₃×C₄×C₅×D₄	C₂	semidirect
4. #91828	C₂	semidirect
5. #91819	C₂	semidirect
6. #91813	C₂	semidirect
7. #91785	C₂	semidirect
8. #91774	C₂	semidirect
9. #91770	C₂	semidirect
10. #91764	C₂	semidirect
11. C₄×Dic₃₀	C₂	semidirect
41. C₄×C₅×D₄	D₃	semidirect
71. Dic₃₀	D₄	semidirect
74. C₃×C₄×D₄	D₅	semidirect
114. C₂²×C₃×C₅	#41	semidirect
184. C₄×D₄	D₁₅	semidirect
218. C₂²×C₅	#235	semidirect
256. C₃×C₅	#548	semidirect
262. C₂²×C₃	#709	semidirect
309. C₅	#5552	semidirect
318. C₂²	#6817	semidirect
328. C₃	#65886	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
C₂²×Dic₃₀	2
C₃×C₄×C₅×D₄	2
#91828	2
#91819	2
#91813	2
#91785	2
#91774	2
#91770	2
#91764	2
C₄×Dic₃₀	2
#65886	3
#5552	5