Subgroups — #28407467

Order 14464.

396 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11744753	C₂	1	8	yes	maximal, complemented
2	#20299614	C₂	1	6	yes	maximal
3	C₂×Q₈×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11744753	C₂²	1	12	yes	—
5	C₂²⋊#11744753	C₂²	1	4	yes	—
6	C₄×#11744753	C₂²	1	12	yes	—
7	Q₈×Dic₁₁₃	C₄	1	4	yes	—
8	Q₈×Dic₁₁₃	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₁₁₃	C₄	1	3	yes	—
10	C₂×C₄×Dic₁₁₃	C₂²	1	3	yes	—
11	C₂²×Q₈×C₁₁₃	C₄	1	1	yes	characteristic
12	C₂×#11744753	C₂³	1	8	yes	—
13	C₂²×Dic₁₁₃	C₂×C₄	1	3	yes	—
14	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
15	C₄⋊Dic₁₁₃	C₂×C₄	1	12	yes	—
16	C₄×Dic₁₁₃	C₂×C₄	1	6	yes	—
17	C₄×Dic₁₁₃	C₂³	1	6	yes	—
18	C₂×Q₈×C₁₁₃	C₈	1	8	yes	complemented
19	C₂×Q₈×C₁₁₃	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₁₁₃	C₂×C₄	1	3	yes	—
21	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
22	Q₈×C₁₁₃	C₂×C₈	1	16	yes	—
23	C₂×C₄×C₁₁₃	C₂×C₈	1	12	yes	—
24	C₂×C₄×C₁₁₃	C₂²×C₄	1	6	yes	—
25	C₂×Dic₁₁₃	—	2	2	no	—
26	C₂×Dic₁₁₃	—	4	3	no	—
27	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₁₁₃	C₂²×C₄	1	6	yes	—
29	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
30	#11744753	—	2	8	no	—
31	C₂²×C₁₁₃	—	2	1	no	—
32	C₂²×C₁₁₃	—	2	1	no	—
33	C₂²×C₁₁₃	C₂²×C₈	1	2	yes	—
34	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
35	C₄×C₁₁₃	C₂²×C₈	1	12	yes	—
36	Dic₁₁₃	—	2	1	no	—
37	Dic₁₁₃	—	2	6	no	—
38	Dic₁₁₃	2₋¹⁺⁴	1	2	yes	—
39	C₂×C₁₁₃	—	2	2	no	—
40	C₂×C₁₁₃	#519	1	1	yes	characteristic
41	C₂×C₁₁₃	C₂³×C₈	1	1	yes	characteristic
42	C₂×C₁₁₃	#568	1	1	yes	characteristic
43	#2982	—	113	1	no	maximal, Sylow
44	C₁₁₃	#2982	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄×Q₈	—	113	1	no	—
46	C₈×Q₈	—	113	8	no	—
47	#431	—	113	6	no	—
48	C₂²×Q₈	#90673	1	1	yes	characteristic
49	C₄×Q₈	—	113	4	no	—
50	C₄×Q₈	—	113	4	no	—
51	C₂×C₄⋊C₄	—	113	3	no	—
52	C₂×C₄²	—	113	3	no	—
53	C₄⋊C₈	—	113	12	no	—
54	C₂²⋊C₈	—	113	4	no	—
55	C₄×C₈	—	113	12	no	—
56	C₂×Q₈	#11744753	1	8	yes	complemented
57	C₂×Q₈	C₂×#90673	1	4	yes	—
58	C₂²×C₄	—	113	3	no	—
59	C₂²×C₄	—	113	1	no	—
60	C₂²×C₄	C₂×#90673	1	3	yes	—
61	C₂×C₈	—	113	8	no	—
62	C₄⋊C₄	—	113	12	no	—
63	C₄²	—	113	6	no	—
64	C₄²	—	113	6	no	—
65	C₂³	C₂²×#90673	1	1	yes	characteristic
66	Q₈	C₂×#11744753	1	16	yes	—
67	C₂×C₄	—	113	1	no	—
68	C₂×C₄	—	113	1	no	—
69	C₂×C₄	—	113	6	no	—
70	C₂×C₄	—	452	3	no	—
71	C₂×C₄	—	226	2	no	—
72	C₂×C₄	C₂×#11744753	1	12	yes	—
73	C₂×C₄	C₂²×#90673	1	6	yes	—
74	C₈	—	226	8	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	C₂²×#11744753	1	2	yes	—
78	C₂²	C₂³×#90673	1	1	yes	characteristic
79	C₄	—	113	2	no	—
80	C₄	—	226	1	no	—
81	C₄	—	226	6	no	—
82	C₄	C₂²×#11744753	1	12	yes	—
83	C₂	—	2	2	no	—
84	C₂	#20299666	1	1	yes	characteristic
85	C₂	C₂³×#11744753	1	1	yes	characteristic
86	C₂	#20299678	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#90673
commutator	C₂×C₁₁₃	C₂³×C₈
Fitting	C₂²×Q₈×C₁₁₃	C₄
socle	C₂²×C₁₁₃	C₂³×C₄
prime radical	C₂³×C₁₁₃	C₂²×C₄
cosocle	C₂×Dic₁₁₃	C₂⁴
hypercenter	C₂²×Q₈	#90673
hypocenter	C₁₁₃	#2982
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#11744753	C₂	semidirect
18. C₂×Q₈×C₁₁₃	C₈	semidirect
44. C₁₁₃	#2982	semidirect
56. C₂×Q₈	#11744753	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11744753	2
#20299614	2
C₂×Q₈×Dic₁₁₃	2
#2982	113