Subgroups — #11756291

Order 960.

472 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	#91819	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#1220	C₂	1	1	yes	maximal, characteristic, complemented
4	#91784	C₂	1	1	yes	maximal, characteristic, complemented
5	#91782	C₂	1	1	yes	maximal, characteristic, complemented
6	#91781	C₂	1	1	yes	maximal, characteristic, complemented
7	#91777	C₂	1	1	yes	maximal, characteristic, complemented
8	#91774	C₂	1	2	yes	maximal, complemented
9	#91773	C₂	1	2	yes	maximal, complemented
10	#91764	C₂	1	2	yes	maximal, complemented
11	#91761	C₂	1	2	yes	maximal, complemented
12	#65866	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂×#1220	C₂²	1	1	yes	characteristic
15	C₂×#1220	C₂²	1	2	yes	—
16	C₂×D₆₀	C₂²	1	1	yes	characteristic
17	C₂×D₆₀	C₂²	1	2	yes	—
18	C₂×C₄×D₁₅	C₂²	1	2	yes	—
19	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
20	C₂×Dic₃₀	C₂²	1	2	yes	—
21	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄²×C₅	C₂²	1	2	yes	—
25	#6719	C₂²	1	1	yes	characteristic
26	#6714	C₂²	1	2	yes	—
27	#6714	C₂²	1	2	yes	—
28	#6714	C₂²	1	1	yes	characteristic
29	#6714	C₂²	1	2	yes	—
30	#6714	C₂²	1	2	yes	—
31	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
32	#6712	C₂²	1	1	yes	characteristic
33	#6712	C₂²	1	2	yes	—
34	C₄×Dic₁₅	C₂²	1	2	yes	—
35	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
36	#5532	—	5	1	no	maximal, Hall
37	C₅×#117	D₃	1	1	yes	characteristic, complemented
38	#4076	—	3	1	no	—
39	C₄×C₂²⋊D₅	—	3	1	no	—
40	#4041	—	3	1	no	—
41	#4039	—	3	1	no	—
42	#4038	—	3	1	no	—
43	#4034	—	3	1	no	—
44	#4031	—	3	2	no	—
45	#4030	—	3	2	no	—
46	#4021	—	3	2	no	—
47	#4018	—	3	2	no	—
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂²×D₁₅	C₂³	1	2	yes	—
50	C₂²×D₁₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	#1220	—	2	2	no	—
57	#1220	—	4	2	no	—
58	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
59	C₂×Dic₁₅	C₂³	1	2	yes	—
60	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
62	D₆₀	—	2	2	no	—
63	D₆₀	—	4	2	no	—
64	C₄×D₁₅	—	2	2	no	—
65	C₄×D₁₅	—	2	4	no	—
66	Dic₃₀	—	4	2	no	—
67	C₃×#117	D₅	1	1	yes	characteristic, complemented
68	#930	—	5	1	no	—
69	C₄×#67	—	5	1	no	—
70	#895	—	5	1	no	—
71	#893	—	5	1	no	—
72	#892	—	5	1	no	—
73	#888	—	5	1	no	—
74	#885	—	5	2	no	—
75	#884	—	5	2	no	—
76	#875	—	5	2	no	—
77	#872	—	5	2	no	—
78	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
79	C₂×C₂²⋊D₅	—	3	1	no	—
80	C₂×C₂²⋊D₅	—	3	2	no	—
81	C₂×D₂₀	—	3	1	no	—
82	C₂×D₂₀	—	3	2	no	—
83	C₂×C₄×D₅	—	3	2	no	—
84	C₂×C₄×D₅	—	3	1	no	—
85	C₂×Dic₁₀	—	3	2	no	—
86	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
87	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
88	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
89	C₄²×C₅	C₂×D₃	1	2	yes	—
90	C₂²⋊Dic₅	—	3	1	no	—
91	#685	—	3	2	no	—
92	#685	—	3	2	no	—
93	#685	—	3	1	no	—
94	#685	—	3	2	no	—
95	#685	—	3	2	no	—
96	C₄⋊Dic₅	—	3	1	no	—
97	#683	—	3	1	no	—
98	#683	—	3	2	no	—
99	C₄×Dic₅	—	3	2	no	—
100	C₄×Dic₅	—	3	1	no	—
101	#555	—	15	1	no	Sylow
102	C₂²×C₃×C₅	—	2	1	no	—
103	C₂²×C₃×C₅	—	2	1	no	—
104	C₂²×C₃×C₅	—	2	1	no	—
105	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
106	C₂×D₁₅	—	2	1	no	—
107	C₂×D₁₅	—	2	2	no	—
108	C₂×D₁₅	—	2	2	no	—
109	C₂×D₁₅	—	2	1	no	—
110	C₂×D₁₅	—	2	1	no	—
111	C₂×D₁₅	—	2	2	no	—
112	C₃×C₄×C₅	—	2	2	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	—	2	2	no	—
115	C₃×C₄×C₅	#41	1	2	yes	—
116	Dic₁₅	—	2	1	no	—
117	Dic₁₅	—	2	1	no	—
118	Dic₁₅	—	2	2	no	—
119	Dic₁₅	#41	1	2	yes	—
120	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
121	C₂×#67	—	5	1	no	—
122	C₂×#67	—	5	2	no	—
123	C₂×D₁₂	—	5	1	no	—
124	C₂×D₁₂	—	5	2	no	—
125	C₂×C₄×D₃	—	5	2	no	—
126	C₂×C₄×D₃	—	5	1	no	—
127	C₂×Dic₆	—	5	2	no	—
128	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
129	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
130	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
131	C₃×C₄²	C₂×D₅	1	2	yes	—
132	#217	—	5	1	no	—
133	(C₂×D₃)⋊C₄	—	5	2	no	—
134	(C₂×D₃)⋊C₄	—	5	2	no	—
135	(C₂×D₃)⋊C₄	—	5	1	no	—
136	(C₂×D₃)⋊C₄	—	5	2	no	—
137	(C₂×D₃)⋊C₄	—	5	2	no	—
138	C₄⋊Dic₃	—	5	1	no	—
139	Dic₃⋊C₄	—	5	1	no	—
140	Dic₃⋊C₄	—	5	2	no	—
141	C₄×Dic₃	—	5	2	no	—
142	C₄×Dic₃	—	5	1	no	—
143	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
144	C₂²×D₅	—	3	1	no	—
145	C₂²×D₅	—	3	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
148	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
150	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₂²⋊D₅	—	6	2	no	—
152	C₂²⋊D₅	—	12	2	no	—
153	C₂×Dic₅	—	3	1	no	—
154	C₂×Dic₅	—	3	1	no	—
155	C₂×Dic₅	—	3	2	no	—
156	C₂×Dic₅	—	3	1	no	—
157	D₂₀	—	6	2	no	—
158	D₂₀	—	12	2	no	—
159	C₄×D₅	—	6	4	no	—
160	C₄×D₅	—	6	2	no	—
161	Dic₁₀	—	12	2	no	—
162	C₄⋊Q₈	—	15	1	no	—
163	#124	—	15	2	no	—
164	#124	—	15	2	no	—
165	#121	—	15	1	no	—
166	#121	—	15	1	no	—
167	#121	—	15	2	no	—
168	C₄×D₄	—	15	1	no	—
169	C₄×D₄	—	15	1	no	—
170	#117	—	15	1	no	—
171	#117	—	15	2	no	—
172	#117	D₁₅	1	1	yes	characteristic, complemented
173	C₂×C₃×C₅	—	4	1	no	—
174	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
175	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
176	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
177	D₁₅	—	4	2	no	—
178	D₁₅	—	4	1	no	—
179	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
180	C₂²×D₃	—	5	1	no	—
181	C₂²×D₃	—	5	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
184	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
186	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
187	#67	—	10	2	no	—
188	#67	—	20	2	no	—
189	C₂×Dic₃	—	5	1	no	—
190	C₂×Dic₃	—	5	1	no	—
191	C₂×Dic₃	—	5	2	no	—
192	C₂×Dic₃	—	5	1	no	—
193	D₁₂	—	10	2	no	—
194	D₁₂	—	20	2	no	—
195	C₄×D₃	—	10	4	no	—
196	C₄×D₃	—	10	2	no	—
197	Dic₆	—	20	2	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
202	C₂×D₅	—	6	1	no	—
203	C₂×D₅	—	6	2	no	—
204	C₂×D₅	—	6	2	no	—
205	C₂×D₅	—	6	1	no	—
206	C₂×D₅	—	6	1	no	—
207	C₂×D₅	—	6	2	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	—	2	2	no	—
211	C₄×C₅	#235	1	2	yes	—
212	Dic₅	—	3	2	no	—
213	Dic₅	—	6	1	no	—
214	Dic₅	—	6	1	no	—
215	Dic₅	—	6	2	no	—
216	C₂×Q₈	—	15	2	no	—
217	C₂×D₄	—	15	1	no	—
218	C₂×D₄	—	15	2	no	—
219	C₂×D₄	—	15	1	no	—
220	C₂×D₄	—	15	2	no	—
221	C₂²×C₄	—	15	2	no	—
222	C₂²×C₄	—	15	1	no	—
223	C₂²×C₄	C₂×D₁₅	1	1	yes	characteristic
224	C₄⋊C₄	—	15	1	no	—
225	C₄⋊C₄	—	15	2	no	—
226	C₄⋊C₄	—	15	1	no	—
227	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
228	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
229	C₂²⋊C₄	—	15	1	no	—
230	C₂²⋊C₄	—	15	2	no	—
231	C₂²⋊C₄	—	15	2	no	—
232	C₂²⋊C₄	—	15	2	no	—
233	C₂²⋊C₄	—	15	2	no	—
234	C₂²⋊C₄	—	15	1	no	—
235	C₂²⋊C₄	C₂×D₁₅	1	2	yes	—
236	C₄²	—	15	2	no	—
237	C₄²	—	15	1	no	—
238	C₄²	C₂×D₁₅	1	2	yes	—
239	C₃×C₅	#555	1	1	yes	characteristic, complemented, Hall
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
244	C₂×D₃	—	10	1	no	—
245	C₂×D₃	—	10	2	no	—
246	C₂×D₃	—	10	2	no	—
247	C₂×D₃	—	10	1	no	—
248	C₂×D₃	—	10	1	no	—
249	C₂×D₃	—	10	2	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	#709	1	2	yes	—
254	Dic₃	—	5	2	no	—
255	Dic₃	—	10	1	no	—
256	Dic₃	—	10	1	no	—
257	Dic₃	—	10	2	no	—
258	C₂×C₅	—	4	1	no	—
259	C₂×C₅	C₂×#235	1	1	yes	characteristic
260	C₂×C₅	D₃×#41	1	1	yes	characteristic
261	C₂×C₅	#1009	1	1	yes	characteristic
262	D₅	—	12	1	no	—
263	D₅	—	12	2	no	—
264	C₂³	—	15	1	no	—
265	C₂³	—	15	2	no	—
266	C₂³	C₂²×D₁₅	1	1	yes	characteristic
267	Q₈	—	60	2	no	—
268	D₄	—	30	2	no	—
269	D₄	—	30	2	no	—
270	D₄	—	60	2	no	—
271	D₄	—	60	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	15	1	no	—
274	C₂×C₄	—	15	1	no	—
275	C₂×C₄	—	15	1	no	—
276	C₂×C₄	—	15	2	no	—
277	C₂×C₄	—	30	4	no	—
278	C₂×C₄	—	30	2	no	—
279	C₂×C₄	C₂²×D₁₅	1	2	yes	—
280	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₁₅	1	2	yes	—
282	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
283	C₂×C₃	—	4	1	no	—
284	C₂×C₃	C₂×#709	1	1	yes	characteristic
285	C₂×C₃	D₅×#41	1	1	yes	characteristic
286	C₂×C₃	#4149	1	1	yes	characteristic
287	D₃	—	20	1	no	—
288	D₃	—	20	2	no	—
289	C₅	#5532	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	30	1	no	—
294	C₂²	—	30	2	no	—
295	C₂²	—	30	2	no	—
296	C₂²	—	30	1	no	—
297	C₂²	—	30	1	no	—
298	C₂²	—	30	2	no	—
299	C₂²	C₂³×D₁₅	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	15	2	no	—
304	C₄	—	30	1	no	—
305	C₄	—	30	2	no	—
306	C₄	—	30	1	no	—
307	C₄	#6817	1	2	yes	—
308	C₃	#65866	1	1	yes	characteristic, complemented, Sylow
309	C₂	—	4	1	no	—
310	C₂	—	60	1	no	—
311	C₂	—	60	2	no	—
312	C₂	C₂×#6817	1	1	yes	characteristic
313	C₂	#41×D₁₅	1	1	yes	characteristic
314	C₂	#92100	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×#117	C₂
hypercenter	#117	D₁₅
hypocenter	C₃×C₅	#555
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#117	C₂	semidirect
2. #91819	C₂	semidirect
3. C₄×#1220	C₂	semidirect
4. #91784	C₂	semidirect
5. #91782	C₂	semidirect
6. #91781	C₂	semidirect
7. #91777	C₂	semidirect
8. #91774	C₂	semidirect
9. #91773	C₂	semidirect
10. #91764	C₂	semidirect
11. #91761	C₂	semidirect
37. C₅×#117	D₃	semidirect
67. C₃×#117	D₅	semidirect
172. #117	D₁₅	semidirect
239. C₃×C₅	#555	semidirect
289. C₅	#5532	semidirect
308. C₃	#65866	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#117	2
#91819	2
C₄×#1220	2
#91784	2
#91782	2
#91781	2
#91777	2
#91774	2
#91773	2
#91764	2
#91761	2
#65866	3
#5532	5