Subgroups — #11756571

Order 960.

532 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₅	C₂	1	8	yes	maximal, complemented
3	#91815	C₂	1	6	yes	maximal
4	#66146	—	3	1	no	maximal, Hall
5	C₂×C₃×C₅×Q₈	C₄	1	8	yes	complemented
6	C₂×C₃×C₅×Q₈	C₂²	1	4	yes	—
7	C₂²×C₃×C₄×C₅	C₂²	1	3	yes	—
8	#6719	C₂²	1	4	yes	—
9	C₄⋊Dic₁₅	C₂²	1	12	yes	—
10	C₄×Dic₁₅	C₂²	1	12	yes	—
11	#5812	—	5	1	no	maximal, Hall
12	C₂²×C₅×Q₈	D₃	1	1	yes	characteristic
13	Q₈×Dic₅	—	3	8	no	—
14	#4072	—	3	6	no	—
15	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
16	C₃×C₅×Q₈	C₂×C₄	1	16	yes	—
17	C₂×C₃×C₄×C₅	C₂×C₄	1	12	yes	—
18	C₂×C₃×C₄×C₅	C₂³	1	6	yes	—
19	C₂×Dic₁₅	C₂³	1	8	yes	—
20	C₂²×C₃×Q₈	D₅	1	1	yes	characteristic
21	Q₈×Dic₃	—	5	8	no	—
22	#926	—	5	6	no	—
23	C₂×C₅×Q₈	Dic₃	1	8	yes	complemented
24	C₂×C₅×Q₈	C₂×D₃	1	4	yes	—
25	C₂²×C₄×C₅	C₂×D₃	1	3	yes	—
26	C₂²⋊Dic₅	—	3	4	no	—
27	C₄⋊Dic₅	—	3	12	no	—
28	C₄×Dic₅	—	3	12	no	—
29	#519	—	15	1	no	Sylow
30	C₂²×C₃×C₅	—	2	2	no	—
31	C₂²×C₃×C₅	C₂²×C₄	1	2	yes	—
32	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
33	C₃×C₄×C₅	C₂²×C₄	1	12	yes	—
34	Dic₁₅	—	2	8	no	—
35	C₂×C₃×Q₈	Dic₅	1	8	yes	complemented
36	C₂×C₃×Q₈	C₂×D₅	1	4	yes	—
37	C₂²×C₃×C₄	C₂×D₅	1	3	yes	—
38	#217	—	5	4	no	—
39	C₄⋊Dic₃	—	5	12	no	—
40	C₄×Dic₃	—	5	12	no	—
41	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
42	C₅×Q₈	C₂×Dic₃	1	16	yes	—
43	C₂×C₄×C₅	C₂×Dic₃	1	12	yes	—
44	C₂×C₄×C₅	C₂²×D₃	1	6	yes	—
45	C₂×Dic₅	—	3	8	no	—
46	C₂²×Q₈	D₁₅	1	1	yes	characteristic
47	C₄×Q₈	—	15	8	no	—
48	#117	—	15	6	no	—
49	C₂×C₃×C₅	—	2	2	no	—
50	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
51	C₂×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
52	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
53	C₃×Q₈	C₂×Dic₅	1	16	yes	—
54	C₂×C₃×C₄	C₂×Dic₅	1	12	yes	—
55	C₂×C₃×C₄	C₂²×D₅	1	6	yes	—
56	C₂×Dic₃	—	5	8	no	—
57	C₂²×C₅	—	2	2	no	—
58	C₂²×C₅	C₂²×Dic₃	1	2	yes	—
59	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
60	C₄×C₅	C₂²×Dic₃	1	12	yes	—
61	Dic₅	—	6	8	no	—
62	C₂×Q₈	Dic₁₅	1	8	yes	complemented
63	C₂×Q₈	C₂×D₁₅	1	4	yes	—
64	C₂²×C₄	C₂×D₁₅	1	3	yes	—
65	C₄⋊C₄	—	15	12	no	—
66	C₂²⋊C₄	—	15	4	no	—
67	C₄²	—	15	12	no	—
68	C₃×C₅	#519	1	1	yes	characteristic, complemented, Hall
69	C₂²×C₃	—	2	2	no	—
70	C₂²×C₃	C₂²×Dic₅	1	2	yes	—
71	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
72	C₃×C₄	C₂²×Dic₅	1	12	yes	—
73	Dic₃	—	10	8	no	—
74	C₂×C₅	—	2	2	no	—
75	C₂×C₅	#1007	1	2	yes	—
76	C₂×C₅	C₂³×Dic₃	1	1	yes	characteristic
77	C₂³	C₂²×D₁₅	1	1	yes	characteristic
78	Q₈	C₂×Dic₁₅	1	16	yes	—
79	C₂×C₄	—	15	8	no	—
80	C₂×C₄	C₂×Dic₁₅	1	12	yes	—
81	C₂×C₄	C₂²×D₁₅	1	6	yes	—
82	C₂×C₃	—	2	2	no	—
83	C₂×C₃	#4147	1	2	yes	—
84	C₂×C₃	C₂³×Dic₅	1	1	yes	characteristic
85	C₅	#5812	1	1	yes	characteristic, complemented, Sylow
86	C₂²	—	2	2	no	—
87	C₂²	C₂²×Dic₁₅	1	2	yes	—
88	C₂²	C₂³×D₁₅	1	1	yes	characteristic
89	C₄	—	30	8	no	—
90	C₄	C₂²×Dic₁₅	1	12	yes	—
91	C₃	#66146	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	2	2	no	—
93	C₂	#92098	1	2	yes	—
94	C₂	C₂³×Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator	C₂×C₃×C₅	C₂³×C₄
Fitting	C₂²×C₃×C₅×Q₈	C₂
socle, cosocle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂³×C₃×C₅	C₂³
hypercenter	C₂²×Q₈	D₁₅
hypocenter	C₃×C₅	#519
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₁₅	C₂	semidirect
5. C₂×C₃×C₅×Q₈	C₄	semidirect
23. C₂×C₅×Q₈	Dic₃	semidirect
35. C₂×C₃×Q₈	Dic₅	semidirect
62. C₂×Q₈	Dic₁₅	semidirect
68. C₃×C₅	#519	semidirect
85. C₅	#5812	semidirect
91. C₃	#66146	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×Q₈	2
Q₈×Dic₁₅	2
#91815	2
#66146	3
#5812	5