Subgroups — #11756454

Order 960.

476 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×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#91787	C₂	1	2	yes	maximal, complemented
3	#91784	C₂	1	2	yes	maximal, complemented
4	#91784	C₂	1	1	yes	maximal, characteristic, complemented
5	#91784	C₂	1	1	yes	maximal, characteristic, complemented
6	#91783	C₂	1	2	yes	maximal, complemented
7	#91782	C₂	1	2	yes	maximal, complemented
8	#91781	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₄×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#66029	—	3	1	no	maximal, Hall
13	C₂×D₆₀	C₂²	1	1	yes	characteristic
14	C₂×D₆₀	C₂²	1	1	yes	characteristic
15	C₂×D₆₀	C₂²	1	2	yes	—
16	C₂×D₆₀	C₂²	1	2	yes	—
17	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₁₅	C₂²	1	2	yes	—
20	C₂×C₄×D₁₅	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×C₄⋊C₄	C₂²	1	2	yes	—
24	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
25	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
26	#6714	C₂²	1	1	yes	characteristic
27	#6714	C₂²	1	1	yes	characteristic
28	#6714	C₂²	1	2	yes	—
29	#6714	C₂²	1	2	yes	—
30	#6714	C₂²	1	2	yes	—
31	#6714	C₂²	1	2	yes	—
32	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
33	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
34	#6712	C₂²	1	2	yes	—
35	C₄×Dic₁₅	C₂²	1	2	yes	—
36	#5695	—	5	1	no	maximal, Hall
37	C₅×#125	D₃	1	1	yes	characteristic, complemented
38	#4044	—	3	2	no	—
39	#4041	—	3	2	no	—
40	#4041	—	3	1	no	—
41	#4041	—	3	1	no	—
42	#4040	—	3	2	no	—
43	#4039	—	3	2	no	—
44	#4038	—	3	1	no	—
45	D₅×C₄⋊C₄	—	3	1	no	—
46	C₄×D₂₀	—	3	1	no	—
47	C₄×D₂₀	—	3	1	no	—
48	C₂²×D₁₅	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	C₂³	1	2	yes	—
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	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₅	C₂³	1	2	yes	—
58	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
59	D₆₀	—	2	2	no	—
60	D₆₀	—	2	4	no	—
61	D₆₀	—	2	2	no	—
62	D₆₀	—	4	2	no	—
63	C₄×D₁₅	—	2	2	no	—
64	C₄×D₁₅	—	2	2	no	—
65	C₄×D₁₅	—	4	2	no	—
66	C₄×D₁₅	—	4	2	no	—
67	C₃×#125	D₅	1	1	yes	characteristic, complemented
68	#898	—	5	2	no	—
69	#895	—	5	2	no	—
70	#895	—	5	1	no	—
71	#895	—	5	1	no	—
72	#894	—	5	2	no	—
73	#893	—	5	2	no	—
74	#892	—	5	1	no	—
75	D₃×C₄⋊C₄	—	5	1	no	—
76	C₄×D₁₂	—	5	1	no	—
77	C₄×D₁₂	—	5	1	no	—
78	C₂×D₂₀	—	3	1	no	—
79	C₂×D₂₀	—	3	1	no	—
80	C₂×D₂₀	—	3	2	no	—
81	C₂×D₂₀	—	3	2	no	—
82	C₂×C₄×D₅	—	3	1	no	—
83	C₂×C₄×D₅	—	3	1	no	—
84	C₂×C₄×D₅	—	3	2	no	—
85	C₂×C₄×D₅	—	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×C₄⋊C₄	C₂×D₃	1	2	yes	—
89	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
90	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
91	#685	—	3	1	no	—
92	#685	—	3	1	no	—
93	#685	—	3	2	no	—
94	#685	—	3	2	no	—
95	#685	—	3	2	no	—
96	#685	—	3	2	no	—
97	C₄⋊Dic₅	—	3	1	no	—
98	C₄⋊Dic₅	—	3	1	no	—
99	#683	—	3	2	no	—
100	C₄×Dic₅	—	3	2	no	—
101	#553	—	15	1	no	Sylow
102	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
103	C₂×D₁₅	—	2	1	no	—
104	C₂×D₁₅	—	2	1	no	—
105	C₂×D₁₅	—	2	1	no	—
106	C₂×D₁₅	—	2	2	no	—
107	C₂×D₁₅	—	2	1	no	—
108	C₂×D₁₅	—	2	2	no	—
109	C₂×D₁₅	—	2	2	no	—
110	C₂×D₁₅	—	2	1	no	—
111	C₂×D₁₅	#41	1	2	yes	—
112	C₃×C₄×C₅	—	2	2	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	—	2	1	no	—
115	C₃×C₄×C₅	—	2	2	no	—
116	C₃×C₄×C₅	#41	1	2	yes	—
117	Dic₁₅	—	2	1	no	—
118	Dic₁₅	—	2	1	no	—
119	Dic₁₅	—	2	2	no	—
120	C₂×D₁₂	—	5	1	no	—
121	C₂×D₁₂	—	5	1	no	—
122	C₂×D₁₂	—	5	2	no	—
123	C₂×D₁₂	—	5	2	no	—
124	C₂×C₄×D₃	—	5	1	no	—
125	C₂×C₄×D₃	—	5	1	no	—
126	C₂×C₄×D₃	—	5	2	no	—
127	C₂×C₄×D₃	—	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×C₄⋊C₄	C₂×D₅	1	2	yes	—
131	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
132	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
133	(C₂×D₃)⋊C₄	—	5	1	no	—
134	(C₂×D₃)⋊C₄	—	5	1	no	—
135	(C₂×D₃)⋊C₄	—	5	2	no	—
136	(C₂×D₃)⋊C₄	—	5	2	no	—
137	(C₂×D₃)⋊C₄	—	5	2	no	—
138	(C₂×D₃)⋊C₄	—	5	2	no	—
139	C₄⋊Dic₃	—	5	1	no	—
140	C₄⋊Dic₃	—	5	1	no	—
141	Dic₃⋊C₄	—	5	2	no	—
142	C₄×Dic₃	—	5	2	no	—
143	C₂²×D₅	—	3	1	no	—
144	C₂²×D₅	—	3	1	no	—
145	C₂²×D₅	—	3	2	no	—
146	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
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	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₂×Dic₅	—	3	1	no	—
152	C₂×Dic₅	—	3	2	no	—
153	C₂×Dic₅	—	3	1	no	—
154	D₂₀	—	6	2	no	—
155	D₂₀	—	6	4	no	—
156	D₂₀	—	6	2	no	—
157	D₂₀	—	12	2	no	—
158	C₄×D₅	—	6	2	no	—
159	C₄×D₅	—	6	2	no	—
160	C₄×D₅	—	12	2	no	—
161	C₄×D₅	—	12	2	no	—
162	#126	—	15	2	no	—
163	#125	D₁₅	1	1	yes	characteristic, complemented
164	#123	—	15	2	no	—
165	#121	—	15	2	no	—
166	#121	—	15	1	no	—
167	#121	—	15	1	no	—
168	C₄×D₄	—	15	2	no	—
169	C₄×D₄	—	15	1	no	—
170	C₄×D₄	—	15	1	no	—
171	#117	—	15	1	no	—
172	C₂×C₄⋊C₄	—	15	1	no	—
173	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
174	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
175	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
176	D₁₅	—	2	2	no	—
177	D₁₅	—	4	1	no	—
178	D₁₅	—	4	2	no	—
179	C₂²×D₃	—	5	1	no	—
180	C₂²×D₃	—	5	1	no	—
181	C₂²×D₃	—	5	2	no	—
182	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
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	1	yes	characteristic
186	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
187	C₂×Dic₃	—	5	1	no	—
188	C₂×Dic₃	—	5	2	no	—
189	C₂×Dic₃	—	5	1	no	—
190	D₁₂	—	10	2	no	—
191	D₁₂	—	10	4	no	—
192	D₁₂	—	10	2	no	—
193	D₁₂	—	20	2	no	—
194	C₄×D₃	—	10	2	no	—
195	C₄×D₃	—	10	2	no	—
196	C₄×D₃	—	20	2	no	—
197	C₄×D₃	—	20	2	no	—
198	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₂×D₅	—	3	2	no	—
200	C₂×D₅	—	6	1	no	—
201	C₂×D₅	—	6	1	no	—
202	C₂×D₅	—	6	1	no	—
203	C₂×D₅	—	6	2	no	—
204	C₂×D₅	—	6	1	no	—
205	C₂×D₅	—	6	2	no	—
206	C₂×D₅	—	6	2	no	—
207	C₂×D₅	—	6	1	no	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	2	1	no	—
210	C₄×C₅	—	2	1	no	—
211	C₄×C₅	—	2	2	no	—
212	C₄×C₅	#239	1	2	yes	—
213	Dic₅	—	6	1	no	—
214	Dic₅	—	6	1	no	—
215	Dic₅	—	6	2	no	—
216	C₂×D₄	—	15	2	no	—
217	C₂×D₄	—	15	1	no	—
218	C₂×D₄	—	15	1	no	—
219	C₂×D₄	—	15	2	no	—
220	C₂²×C₄	—	15	1	no	—
221	C₂²×C₄	—	15	1	no	—
222	C₂²×C₄	—	15	2	no	—
223	C₂²×C₄	—	15	2	no	—
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	C₄⋊C₄	C₂×D₁₅	1	2	yes	—
230	C₄⋊C₄	C₂×D₁₅	1	2	yes	—
231	C₂²⋊C₄	—	15	2	no	—
232	C₂²⋊C₄	—	15	1	no	—
233	C₂²⋊C₄	—	15	2	no	—
234	C₂²⋊C₄	—	15	2	no	—
235	C₂²⋊C₄	—	15	1	no	—
236	C₂²⋊C₄	—	15	2	no	—
237	C₄²	—	15	2	no	—
238	C₄²	C₂×D₁₅	1	1	yes	characteristic
239	C₃×C₅	#553	1	1	yes	characteristic, complemented, Hall
240	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
241	C₂×D₃	—	5	2	no	—
242	C₂×D₃	—	10	1	no	—
243	C₂×D₃	—	10	1	no	—
244	C₂×D₃	—	10	1	no	—
245	C₂×D₃	—	10	2	no	—
246	C₂×D₃	—	10	1	no	—
247	C₂×D₃	—	10	2	no	—
248	C₂×D₃	—	10	2	no	—
249	C₂×D₃	—	10	1	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	#713	1	2	yes	—
255	Dic₃	—	10	1	no	—
256	Dic₃	—	10	1	no	—
257	Dic₃	—	10	2	no	—
258	C₂×C₅	C₂×#239	1	1	yes	characteristic
259	C₂×C₅	D₃×#41	1	1	yes	characteristic
260	C₂×C₅	#1009	1	1	yes	characteristic
261	D₅	—	6	2	no	—
262	D₅	—	12	1	no	—
263	D₅	—	12	2	no	—
264	C₂³	—	15	1	no	—
265	C₂³	—	15	1	no	—
266	C₂³	—	15	2	no	—
267	D₄	—	30	4	no	—
268	D₄	—	30	2	no	—
269	D₄	—	30	2	no	—
270	D₄	—	60	2	no	—
271	C₂×C₄	—	15	1	no	—
272	C₂×C₄	—	15	2	no	—
273	C₂×C₄	—	15	1	no	—
274	C₂×C₄	—	30	2	no	—
275	C₂×C₄	—	30	2	no	—
276	C₂×C₄	—	60	2	no	—
277	C₂×C₄	—	60	2	no	—
278	C₂×C₄	C₂²×D₁₅	1	2	yes	—
279	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
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	C₂×#713	1	1	yes	characteristic
284	C₂×C₃	D₅×#41	1	1	yes	characteristic
285	C₂×C₃	#4149	1	1	yes	characteristic
286	D₃	—	10	2	no	—
287	D₃	—	20	1	no	—
288	D₃	—	20	2	no	—
289	C₅	#5695	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	15	2	no	—
291	C₂²	—	30	1	no	—
292	C₂²	—	30	1	no	—
293	C₂²	—	30	1	no	—
294	C₂²	—	30	2	no	—
295	C₂²	—	30	1	no	—
296	C₂²	—	30	2	no	—
297	C₂²	—	30	2	no	—
298	C₂²	—	30	1	no	—
299	C₂²	C₂³×D₁₅	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	30	1	no	—
305	C₄	—	30	1	no	—
306	C₄	—	30	2	no	—
307	C₄	#6821	1	2	yes	—
308	C₃	#66029	1	1	yes	characteristic, complemented, Sylow
309	C₂	—	30	2	no	—
310	C₂	—	60	1	no	—
311	C₂	—	60	2	no	—
312	C₂	C₂×#6821	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×#125	C₂
hypercenter	#125	D₁₅
hypocenter	C₃×C₅	#553
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#125	C₂	semidirect
2. #91787	C₂	semidirect
3. #91784	C₂	semidirect
4. #91784	C₂	semidirect
5. #91784	C₂	semidirect
6. #91783	C₂	semidirect
7. #91782	C₂	semidirect
8. #91781	C₂	semidirect
9. C₄⋊C₄×D₁₅	C₂	semidirect
10. C₄×D₆₀	C₂	semidirect
11. C₄×D₆₀	C₂	semidirect
37. C₅×#125	D₃	semidirect
67. C₃×#125	D₅	semidirect
163. #125	D₁₅	semidirect
239. C₃×C₅	#553	semidirect
289. C₅	#5695	semidirect
308. C₃	#66029	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#125	2
#91787	2
#91784	2
#91783	2
#91782	2
#91781	2
C₄⋊C₄×D₁₅	2
C₄×D₆₀	2
#66029	3
#5695	5