Subgroups — #11756339

Order 960.

468 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#91836	C₂	1	1	yes	maximal, characteristic, complemented
3	#91835	C₂	1	1	yes	maximal, characteristic, complemented
4	#91787	C₂	1	2	yes	maximal, complemented
5	#91784	C₂	1	1	yes	maximal, characteristic
6	#91783	C₂	1	2	yes	maximal, complemented
7	#91779	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₆₀	C₂	1	2	yes	maximal, complemented
9	C₄×D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
10	#91761	C₂	1	2	yes	maximal, complemented
11	C₄²×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
12	#65914	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂×D₆₀	C₂²	1	2	yes	—
15	C₂×D₆₀	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
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	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
23	C₃×C₄²×C₅	C₂²	1	2	yes	—
24	#6714	C₂²	1	2	yes	—
25	#6714	C₂²	1	2	yes	—
26	#6714	C₂²	1	2	yes	—
27	#6714	C₂²	1	2	yes	—
28	#6714	C₂²	1	2	yes	—
29	C₄⋊Dic₁₅	C₂²	1	2	yes	—
30	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
31	#6712	C₂²	1	2	yes	—
32	#6712	C₂²	1	2	yes	—
33	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
36	#5580	—	5	1	no	maximal, Hall
37	C₄×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
38	#4093	—	3	1	no	—
39	#4092	—	3	1	no	—
40	#4044	—	3	2	no	—
41	#4041	—	3	1	no	—
42	#4040	—	3	2	no	—
43	#4036	—	3	1	no	—
44	C₄×D₂₀	—	3	2	no	—
45	C₄×D₂₀	—	3	1	no	—
46	#4018	—	3	2	no	—
47	C₄²×D₅	—	3	1	no	—
48	C₂²×D₁₅	C₂³	1	2	yes	—
49	C₂²×D₁₅	C₂³	1	1	yes	characteristic
50	C₃×C₅×Q₈	—	2	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
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	1	yes	characteristic
58	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
59	C₂×Dic₁₅	C₂³	1	2	yes	—
60	D₆₀	—	2	4	no	—
61	D₆₀	—	2	2	no	—
62	C₄×D₁₅	—	2	2	no	—
63	C₄×D₁₅	—	2	2	no	—
64	C₄×D₁₅	—	2	2	no	—
65	C₄×D₁₅	—	2	4	no	—
66	C₃×C₄×Q₈	D₅	1	1	yes	characteristic, complemented
67	#947	—	5	1	no	—
68	#946	—	5	1	no	—
69	#898	—	5	2	no	—
70	#895	—	5	1	no	—
71	#894	—	5	2	no	—
72	#890	—	5	1	no	—
73	C₄×D₁₂	—	5	2	no	—
74	C₄×D₁₂	—	5	1	no	—
75	#872	—	5	2	no	—
76	C₄²×D₃	—	5	1	no	—
77	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
78	C₂×D₂₀	—	3	2	no	—
79	C₂×D₂₀	—	3	1	no	—
80	C₂×C₄×D₅	—	3	1	no	—
81	C₂×C₄×D₅	—	3	1	no	—
82	C₂×C₄×D₅	—	3	1	no	—
83	C₂×C₄×D₅	—	3	2	no	—
84	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
85	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
86	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
87	C₄²×C₅	C₂×D₃	1	2	yes	—
88	#685	—	3	2	no	—
89	#685	—	3	2	no	—
90	#685	—	3	2	no	—
91	#685	—	3	2	no	—
92	#685	—	3	2	no	—
93	C₄⋊Dic₅	—	3	2	no	—
94	C₄⋊Dic₅	—	3	1	no	—
95	#683	—	3	2	no	—
96	#683	—	3	2	no	—
97	C₄×Dic₅	—	3	1	no	—
98	C₄×Dic₅	—	3	1	no	—
99	C₄×Dic₅	—	3	1	no	—
100	#529	—	15	1	no	Sylow
101	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
102	C₂×D₁₅	—	2	1	no	—
103	C₂×D₁₅	—	2	1	no	—
104	C₂×D₁₅	—	2	2	no	—
105	C₂×D₁₅	—	2	2	no	—
106	C₂×D₁₅	—	2	2	no	—
107	C₂×D₁₅	#41	1	2	yes	—
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	—	2	2	no	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	#41	1	2	yes	—
112	C₃×C₄×C₅	#41	1	2	yes	—
113	Dic₁₅	—	2	1	no	—
114	Dic₁₅	—	2	1	no	—
115	Dic₁₅	—	2	2	no	—
116	Dic₁₅	—	2	1	no	—
117	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
118	C₂×D₁₂	—	5	2	no	—
119	C₂×D₁₂	—	5	1	no	—
120	C₂×C₄×D₃	—	5	1	no	—
121	C₂×C₄×D₃	—	5	1	no	—
122	C₂×C₄×D₃	—	5	1	no	—
123	C₂×C₄×D₃	—	5	2	no	—
124	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
125	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
126	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
127	C₃×C₄²	C₂×D₅	1	2	yes	—
128	(C₂×D₃)⋊C₄	—	5	2	no	—
129	(C₂×D₃)⋊C₄	—	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	2	no	—
134	C₄⋊Dic₃	—	5	1	no	—
135	Dic₃⋊C₄	—	5	2	no	—
136	Dic₃⋊C₄	—	5	2	no	—
137	C₄×Dic₃	—	5	1	no	—
138	C₄×Dic₃	—	5	1	no	—
139	C₄×Dic₃	—	5	1	no	—
140	C₂²×D₅	—	3	1	no	—
141	C₂²×D₅	—	3	2	no	—
142	C₅×Q₈	—	2	2	no	—
143	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
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×Dic₅	—	3	1	no	—
149	C₂×Dic₅	—	3	1	no	—
150	C₂×Dic₅	—	3	1	no	—
151	C₂×Dic₅	—	3	2	no	—
152	D₂₀	—	6	4	no	—
153	D₂₀	—	6	2	no	—
154	C₄×D₅	—	6	2	no	—
155	C₄×D₅	—	6	2	no	—
156	C₄×D₅	—	6	2	no	—
157	C₄×D₅	—	6	4	no	—
158	#126	—	15	2	no	—
159	#125	—	15	1	no	—
160	#124	—	15	1	no	—
161	#123	—	15	2	no	—
162	C₂²⋊Q₈	—	15	1	no	—
163	#121	—	15	1	no	—
164	C₄×Q₈	D₁₅	1	1	yes	characteristic, complemented
165	C₄×D₄	—	15	1	no	—
166	C₄×D₄	—	15	2	no	—
167	#117	—	15	2	no	—
168	C₂×C₄²	—	15	1	no	—
169	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
170	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
172	D₁₅	—	2	2	no	—
173	D₁₅	—	4	2	no	—
174	C₂²×D₃	—	5	1	no	—
175	C₂²×D₃	—	5	2	no	—
176	C₃×Q₈	—	2	2	no	—
177	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
179	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
180	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
182	C₂×Dic₃	—	5	1	no	—
183	C₂×Dic₃	—	5	1	no	—
184	C₂×Dic₃	—	5	1	no	—
185	C₂×Dic₃	—	5	2	no	—
186	D₁₂	—	10	4	no	—
187	D₁₂	—	10	2	no	—
188	C₄×D₃	—	10	2	no	—
189	C₄×D₃	—	10	2	no	—
190	C₄×D₃	—	10	2	no	—
191	C₄×D₃	—	10	4	no	—
192	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
193	C₂×D₅	—	3	2	no	—
194	C₂×D₅	—	6	1	no	—
195	C₂×D₅	—	6	1	no	—
196	C₂×D₅	—	6	2	no	—
197	C₂×D₅	—	6	2	no	—
198	C₂×D₅	—	6	2	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	2	1	no	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	#235	1	2	yes	—
203	C₄×C₅	#239	1	2	yes	—
204	Dic₅	—	6	1	no	—
205	Dic₅	—	6	1	no	—
206	Dic₅	—	6	2	no	—
207	Dic₅	—	6	1	no	—
208	C₂×Q₈	C₂×D₁₅	1	1	yes	characteristic
209	C₂×D₄	—	15	2	no	—
210	C₂×D₄	—	15	1	no	—
211	C₂²×C₄	—	15	1	no	—
212	C₂²×C₄	—	15	1	no	—
213	C₂²×C₄	—	15	1	no	—
214	C₂²×C₄	—	15	2	no	—
215	C₄⋊C₄	—	15	2	no	—
216	C₄⋊C₄	—	15	2	no	—
217	C₄⋊C₄	—	15	1	no	—
218	C₄⋊C₄	—	15	2	no	—
219	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
220	C₄⋊C₄	C₂×D₁₅	1	2	yes	—
221	C₂²⋊C₄	—	15	2	no	—
222	C₂²⋊C₄	—	15	2	no	—
223	C₂²⋊C₄	—	15	2	no	—
224	C₂²⋊C₄	—	15	2	no	—
225	C₂²⋊C₄	—	15	2	no	—
226	C₄²	—	15	1	no	—
227	C₄²	—	15	1	no	—
228	C₄²	—	15	1	no	—
229	C₄²	C₂×D₁₅	1	1	yes	characteristic
230	C₄²	C₂×D₁₅	1	2	yes	—
231	C₃×C₅	#529	1	1	yes	characteristic, complemented, Hall
232	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
233	C₂×D₃	—	5	2	no	—
234	C₂×D₃	—	10	1	no	—
235	C₂×D₃	—	10	1	no	—
236	C₂×D₃	—	10	2	no	—
237	C₂×D₃	—	10	2	no	—
238	C₂×D₃	—	10	2	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	#709	1	2	yes	—
243	C₃×C₄	#713	1	2	yes	—
244	Dic₃	—	10	1	no	—
245	Dic₃	—	10	1	no	—
246	Dic₃	—	10	2	no	—
247	Dic₃	—	10	1	no	—
248	C₂×C₅	C₂×#235	1	1	yes	characteristic
249	C₂×C₅	C₂×#239	1	1	yes	characteristic
250	C₂×C₅	D₃×#41	1	1	yes	characteristic
251	D₅	—	6	2	no	—
252	D₅	—	12	2	no	—
253	C₂³	—	15	1	no	—
254	C₂³	—	15	2	no	—
255	Q₈	—	2	2	no	—
256	D₄	—	30	4	no	—
257	D₄	—	30	2	no	—
258	C₂×C₄	—	15	1	no	—
259	C₂×C₄	—	15	1	no	—
260	C₂×C₄	—	15	1	no	—
261	C₂×C₄	—	15	2	no	—
262	C₂×C₄	—	30	2	no	—
263	C₂×C₄	—	30	2	no	—
264	C₂×C₄	—	30	2	no	—
265	C₂×C₄	—	30	4	no	—
266	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
267	C₂×C₄	C₂²×D₁₅	1	2	yes	—
268	C₂×C₄	C₂²×D₁₅	1	2	yes	—
269	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
270	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
271	C₂×C₃	C₂×#709	1	1	yes	characteristic
272	C₂×C₃	C₂×#713	1	1	yes	characteristic
273	C₂×C₃	D₅×#41	1	1	yes	characteristic
274	D₃	—	10	2	no	—
275	D₃	—	20	2	no	—
276	C₅	#5580	1	1	yes	characteristic, complemented, Sylow
277	C₂²	—	15	2	no	—
278	C₂²	—	30	1	no	—
279	C₂²	—	30	1	no	—
280	C₂²	—	30	2	no	—
281	C₂²	—	30	2	no	—
282	C₂²	—	30	2	no	—
283	C₂²	C₂³×D₁₅	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	30	1	no	—
288	C₄	—	30	1	no	—
289	C₄	—	30	2	no	—
290	C₄	—	30	1	no	—
291	C₄	#6817	1	2	yes	—
292	C₄	#6821	1	2	yes	—
293	C₃	#65914	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	30	2	no	—
295	C₂	—	60	2	no	—
296	C₂	C₂×#6817	1	1	yes	characteristic
297	C₂	C₂×#6821	1	1	yes	characteristic
298	C₂	#41×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Frattini	C₂²	C₂³×D₁₅
Fitting	C₃×C₄×C₅×Q₈	C₂
prime radical	#91784	C₂
hypercenter	C₄×Q₈	D₁₅
hypocenter	C₃×C₅	#529
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₅×Q₈	C₂	semidirect
2. #91836	C₂	semidirect
3. #91835	C₂	semidirect
4. #91787	C₂	semidirect
6. #91783	C₂	semidirect
7. #91779	C₂	semidirect
8. C₄×D₆₀	C₂	semidirect
9. C₄×D₆₀	C₂	semidirect
10. #91761	C₂	semidirect
11. C₄²×D₁₅	C₂	semidirect
37. C₄×C₅×Q₈	D₃	semidirect
66. C₃×C₄×Q₈	D₅	semidirect
164. C₄×Q₈	D₁₅	semidirect
231. C₃×C₅	#529	semidirect
276. C₅	#5580	semidirect
293. C₃	#65914	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×Q₈	2
#91836	2
#91835	2
#91787	2
#91784	2
#91783	2
#91779	2
C₄×D₆₀	2
#91761	2
C₄²×D₁₅	2
#65914	3
#5580	5