Subgroups — #11756440

Order 960.

472 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	#91836	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#91829	C₂	1	1	yes	maximal, characteristic
5	D₄×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
6	#91775	C₂	1	2	yes	maximal, complemented
7	#91774	C₂	1	2	yes	maximal, complemented
8	#91772	C₂	1	2	yes	maximal, complemented
9	#91770	C₂	1	2	yes	maximal, complemented
10	C₄×D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#66015	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₂×#1220	C₂²	1	2	yes	—
16	C₂×#1220	C₂²	1	2	yes	—
17	C₂²×Dic₁₅	C₂²	1	2	yes	—
18	C₂×D₆₀	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₁₅	C₂²	1	2	yes	—
20	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
24	#6719	C₂²	1	2	yes	—
25	#6714	C₂²	1	2	yes	—
26	#6714	C₂²	1	2	yes	—
27	#6714	C₂²	1	2	yes	—
28	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
29	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
30	C₄⋊Dic₁₅	C₂²	1	2	yes	—
31	#6712	C₂²	1	1	yes	characteristic
32	#6712	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₅	C₂²	1	2	yes	—
36	#5681	—	5	1	no	maximal, Hall
37	C₅×#124	D₃	1	1	yes	characteristic, complemented
38	#4093	—	3	1	no	—
39	Q₈×Dic₅	—	3	1	no	—
40	#4086	—	3	1	no	—
41	D₄×Dic₅	—	3	1	no	—
42	#4032	—	3	2	no	—
43	#4031	—	3	2	no	—
44	#4029	—	3	2	no	—
45	#4027	—	3	2	no	—
46	C₄×D₂₀	—	3	1	no	—
47	C₄×Dic₁₀	—	3	1	no	—
48	C₂³×C₃×C₅	C₂³	1	2	yes	—
49	C₂²×D₁₅	C₂³	1	2	yes	—
50	C₃×C₅×Q₈	—	2	2	no	—
51	C₃×C₅×D₄	—	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	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	#1220	—	2	4	no	—
57	#1220	—	4	2	no	—
58	C₂×Dic₁₅	—	4	2	no	—
59	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₅	C₂³	1	2	yes	—
63	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
64	D₆₀	—	2	2	no	—
65	C₄×D₁₅	—	4	2	no	—
66	Dic₃₀	—	2	2	no	—
67	C₃×#124	D₅	1	1	yes	characteristic, complemented
68	#947	—	5	1	no	—
69	Q₈×Dic₃	—	5	1	no	—
70	#940	—	5	1	no	—
71	D₄×Dic₃	—	5	1	no	—
72	#886	—	5	2	no	—
73	#885	—	5	2	no	—
74	#883	—	5	2	no	—
75	#67⋊C₄	—	5	2	no	—
76	C₄×D₁₂	—	5	1	no	—
77	C₄×Dic₆	—	5	1	no	—
78	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
79	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
80	C₂×C₂²⋊D₅	—	3	2	no	—
81	C₂×C₂²⋊D₅	—	3	2	no	—
82	C₂²×Dic₅	—	3	2	no	—
83	C₂×D₂₀	—	3	1	no	—
84	C₂×C₄×D₅	—	3	2	no	—
85	C₂×Dic₁₀	—	3	1	no	—
86	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
87	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
88	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
89	C₂²⋊Dic₅	—	3	2	no	—
90	#685	—	3	2	no	—
91	#685	—	3	2	no	—
92	#685	—	3	2	no	—
93	C₄⋊Dic₅	—	3	1	no	—
94	C₄⋊Dic₅	—	3	1	no	—
95	C₄⋊Dic₅	—	3	2	no	—
96	#683	—	3	1	no	—
97	#683	—	3	1	no	—
98	C₄×Dic₅	—	3	1	no	—
99	C₄×Dic₅	—	3	1	no	—
100	C₄×Dic₅	—	3	2	no	—
101	#559	—	15	1	no	Sylow
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	—	2	2	no	—
104	C₂²×C₃×C₅	—	2	2	no	—
105	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
106	C₂×D₁₅	—	2	2	no	—
107	C₂×D₁₅	—	2	2	no	—
108	C₂×D₁₅	—	2	2	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	—	2	1	no	—
112	C₃×C₄×C₅	#41	1	2	yes	—
113	Dic₁₅	—	2	1	no	—
114	Dic₁₅	—	2	2	no	—
115	Dic₁₅	—	2	1	no	—
116	Dic₁₅	—	2	1	no	—
117	Dic₁₅	#41	1	2	yes	—
118	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
119	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
120	C₂×#67	—	5	2	no	—
121	C₂×#67	—	5	2	no	—
122	C₂²×Dic₃	—	5	2	no	—
123	C₂×D₁₂	—	5	1	no	—
124	C₂×C₄×D₃	—	5	2	no	—
125	C₂×Dic₆	—	5	1	no	—
126	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
127	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
128	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
129	#217	—	5	2	no	—
130	(C₂×D₃)⋊C₄	—	5	2	no	—
131	(C₂×D₃)⋊C₄	—	5	2	no	—
132	(C₂×D₃)⋊C₄	—	5	2	no	—
133	C₄⋊Dic₃	—	5	1	no	—
134	C₄⋊Dic₃	—	5	1	no	—
135	C₄⋊Dic₃	—	5	2	no	—
136	Dic₃⋊C₄	—	5	1	no	—
137	Dic₃⋊C₄	—	5	1	no	—
138	C₄×Dic₃	—	5	1	no	—
139	C₄×Dic₃	—	5	1	no	—
140	C₄×Dic₃	—	5	2	no	—
141	C₂³×C₅	C₂²×D₃	1	2	yes	—
142	C₂²×D₅	—	3	2	no	—
143	C₅×Q₈	—	2	2	no	—
144	C₅×D₄	—	2	2	no	—
145	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
146	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
149	C₂²⋊D₅	—	6	4	no	—
150	C₂²⋊D₅	—	12	2	no	—
151	C₂×Dic₅	—	3	1	no	—
152	C₂×Dic₅	—	3	1	no	—
153	C₂×Dic₅	—	3	1	no	—
154	C₂×Dic₅	—	3	1	no	—
155	C₂×Dic₅	—	3	2	no	—
156	C₂×Dic₅	—	12	2	no	—
157	D₂₀	—	6	2	no	—
158	C₄×D₅	—	12	2	no	—
159	Dic₁₀	—	6	2	no	—
160	#127	—	15	1	no	—
161	#124	—	15	1	no	—
162	#124	—	15	2	no	—
163	#124	D₁₅	1	1	yes	characteristic, complemented
164	#123	—	15	2	no	—
165	#123	—	15	2	no	—
166	C₄×Q₈	—	15	1	no	—
167	C₄×Q₈	—	15	1	no	—
168	C₄×D₄	—	15	1	no	—
169	C₄×D₄	—	15	2	no	—
170	C₄×D₄	—	15	1	no	—
171	C₂×C₃×C₅	—	4	2	no	—
172	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
173	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
174	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
175	D₁₅	—	4	2	no	—
176	C₂³×C₃	C₂²×D₅	1	2	yes	—
177	C₂²×D₃	—	5	2	no	—
178	C₃×Q₈	—	2	2	no	—
179	C₃×D₄	—	2	2	no	—
180	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
181	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
184	#67	—	10	4	no	—
185	#67	—	20	2	no	—
186	C₂×Dic₃	—	5	1	no	—
187	C₂×Dic₃	—	5	1	no	—
188	C₂×Dic₃	—	5	1	no	—
189	C₂×Dic₃	—	5	1	no	—
190	C₂×Dic₃	—	5	2	no	—
191	C₂×Dic₃	—	20	2	no	—
192	D₁₂	—	10	2	no	—
193	C₄×D₃	—	20	2	no	—
194	Dic₆	—	10	2	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₂×D₅	—	6	2	no	—
200	C₂×D₅	—	6	2	no	—
201	C₂×D₅	—	6	2	no	—
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	—	2	1	no	—
205	C₄×C₅	#237	1	2	yes	—
206	Dic₅	—	3	2	no	—
207	Dic₅	—	6	1	no	—
208	Dic₅	—	6	2	no	—
209	Dic₅	—	6	1	no	—
210	Dic₅	—	6	1	no	—
211	C₂×Q₈	—	15	1	no	—
212	C₂×Q₈	C₂×D₁₅	1	1	yes	characteristic
213	C₂×D₄	—	15	2	no	—
214	C₂×D₄	—	15	2	no	—
215	C₂×D₄	—	15	1	no	—
216	C₂×D₄	C₂×D₁₅	1	1	yes	characteristic
217	C₂²×C₄	—	15	2	no	—
218	C₂²×C₄	—	15	2	no	—
219	C₄⋊C₄	—	15	1	no	—
220	C₄⋊C₄	—	15	1	no	—
221	C₄⋊C₄	—	15	1	no	—
222	C₄⋊C₄	—	15	2	no	—
223	C₄⋊C₄	—	15	1	no	—
224	C₂²⋊C₄	—	15	2	no	—
225	C₂²⋊C₄	—	15	2	no	—
226	C₂²⋊C₄	—	15	2	no	—
227	C₂²⋊C₄	—	15	2	no	—
228	C₂²⋊C₄	C₂×D₁₅	1	2	yes	—
229	C₂²⋊C₄	C₂×D₁₅	1	2	yes	—
230	C₄²	—	15	2	no	—
231	C₄²	—	15	1	no	—
232	C₄²	—	15	1	no	—
233	C₄²	C₂×D₁₅	1	1	yes	characteristic
234	C₃×C₅	#559	1	1	yes	characteristic, complemented, Hall
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
239	C₂×D₃	—	10	2	no	—
240	C₂×D₃	—	10	2	no	—
241	C₂×D₃	—	10	2	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	#711	1	2	yes	—
246	Dic₃	—	5	2	no	—
247	Dic₃	—	10	1	no	—
248	Dic₃	—	10	2	no	—
249	Dic₃	—	10	1	no	—
250	Dic₃	—	10	1	no	—
251	C₂×C₅	—	4	2	no	—
252	C₂×C₅	C₂×#237	1	1	yes	characteristic
253	C₂×C₅	D₃×#41	1	1	yes	characteristic
254	C₂×C₅	#1009	1	1	yes	characteristic
255	D₅	—	12	2	no	—
256	C₂³	—	15	2	no	—
257	C₂³	C₂²×D₁₅	1	2	yes	—
258	Q₈	—	2	2	no	—
259	Q₈	—	30	2	no	—
260	D₄	—	2	2	no	—
261	D₄	—	30	4	no	—
262	D₄	—	30	2	no	—
263	D₄	—	60	2	no	—
264	C₂×C₄	—	15	1	no	—
265	C₂×C₄	—	15	1	no	—
266	C₂×C₄	—	15	2	no	—
267	C₂×C₄	—	15	1	no	—
268	C₂×C₄	—	15	1	no	—
269	C₂×C₄	—	60	2	no	—
270	C₂×C₄	—	60	2	no	—
271	C₂×C₄	C₂²×D₁₅	1	2	yes	—
272	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
275	C₂×C₃	—	4	2	no	—
276	C₂×C₃	C₂×#711	1	1	yes	characteristic
277	C₂×C₃	D₅×#41	1	1	yes	characteristic
278	C₂×C₃	#4149	1	1	yes	characteristic
279	D₃	—	20	2	no	—
280	C₅	#5681	1	1	yes	characteristic, complemented, Sylow
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	30	2	no	—
285	C₂²	—	30	2	no	—
286	C₂²	—	30	2	no	—
287	C₂²	C₂³×D₁₅	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	15	2	no	—
292	C₄	—	30	1	no	—
293	C₄	—	30	2	no	—
294	C₄	—	30	1	no	—
295	C₄	—	30	1	no	—
296	C₄	#6819	1	2	yes	—
297	C₃	#66015	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	4	2	no	—
299	C₂	—	60	2	no	—
300	C₂	C₂×#6819	1	1	yes	characteristic
301	C₂	#41×D₁₅	1	1	yes	characteristic
302	C₂	#92100	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×#124	C₂
prime radical	#91829	C₂
hypercenter	#124	D₁₅
hypocenter	C₃×C₅	#559
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#124	C₂	semidirect
2. #91836	C₂	semidirect
3. Q₈×Dic₁₅	C₂	semidirect
5. D₄×Dic₁₅	C₂	semidirect
6. #91775	C₂	semidirect
7. #91774	C₂	semidirect
8. #91772	C₂	semidirect
9. #91770	C₂	semidirect
10. C₄×D₆₀	C₂	semidirect
11. C₄×Dic₃₀	C₂	semidirect
37. C₅×#124	D₃	semidirect
67. C₃×#124	D₅	semidirect
163. #124	D₁₅	semidirect
234. C₃×C₅	#559	semidirect
280. C₅	#5681	semidirect
297. C₃	#66015	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#124	2
#91836	2
Q₈×Dic₁₅	2
#91829	2
D₄×Dic₁₅	2
#91775	2
#91774	2
#91772	2
#91770	2
C₄×D₆₀	2
C₄×Dic₃₀	2
#66015	3
#5681	5