Subgroups — #28392479

Order 14464.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic
2	#20298416	C₂	1	2	yes	maximal
3	#20298373	C₂	1	2	yes	maximal
4	C₄×#14268622	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₁₆×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₂₂₆	C₄	1	1	yes	characteristic
7	C₂×#14268622	C₂²	1	1	yes	characteristic
8	C₂×#14268622	C₂²	1	1	yes	characteristic
9	C₄⋊#11744751	C₄	1	1	yes	characteristic
10	#16813551	C₄	1	2	yes	—
11	C₈×Dic₁₁₃	C₂²	1	2	yes	—
12	C₂×C₁₆×C₁₁₃	C₂²	1	2	yes	—
13	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₂₆	C₈	1	1	yes	characteristic
15	C₂×#11744751	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₁₃	C₈	1	1	yes	characteristic
17	#14268626	C₈	1	2	yes	—
18	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
19	#14268622	—	2	1	no	—
20	#14268622	Q₈	1	2	yes	complemented
21	C₂×C₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
23	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
24	C₁₆×C₁₁₃	—	2	2	no	—
25	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₁₁₃	C₂×C₈	1	2	yes	—
29	C₂×Dic₁₁₃	C₂×C₈	1	2	yes	—
30	Dic₂₂₆	—	2	2	no	—
31	C₈×C₁₁₃	—	2	1	no	—
32	C₈×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
33	C₈×C₁₁₃	#41	1	1	yes	characteristic
34	#11744751	—	2	2	no	—
35	C₂²×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₁₁₃	—	2	1	no	—
37	C₄×C₁₁₃	M₃₂	1	2	yes	—
38	C₄×C₁₁₃	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₁₁₃	#131	1	1	yes	characteristic
40	Dic₁₁₃	—	2	2	no	—
41	Dic₁₁₃	—	2	2	no	—
42	C₂×C₁₁₃	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₁₁₃	C₂×M₃₂	1	1	yes	characteristic
44	C₂×C₁₁₃	#504	1	1	yes	characteristic
45	#2183	—	113	1	no	maximal, Sylow
46	C₁₁₃	#2183	1	1	yes	characteristic, complemented, Sylow
47	C₈×Q₈	—	113	1	no	—
48	C₄⋊C₁₆	—	113	2	no	—
49	C₄⋊C₁₆	D₁₁₃	1	1	yes	characteristic
50	#346	—	113	2	no	—
51	C₄×C₁₆	—	113	1	no	—
52	C₄×Q₈	—	113	1	no	—
53	C₂×C₁₆	—	113	1	no	—
54	C₂×C₁₆	—	113	1	no	—
55	C₂×C₁₆	C₂×D₁₁₃	1	2	yes	—
56	C₄⋊C₈	—	113	2	no	—
57	C₄⋊C₈	—	113	1	no	—
58	C₄×C₈	—	113	2	no	—
59	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
60	C₂×Q₈	—	113	1	no	—
61	C₂×C₈	—	113	2	no	—
62	C₂×C₈	C₄×D₁₁₃	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
64	C₄⋊C₄	—	113	2	no	—
65	C₄⋊C₄	—	113	1	no	—
66	C₄²	—	113	2	no	—
67	C₄²	C₄×D₁₁₃	1	1	yes	characteristic
68	C₁₆	—	2	2	no	—
69	C₁₆	—	113	2	no	—
70	C₁₆	—	226	1	no	—
71	Q₈	—	226	2	no	—
72	C₂×C₄	—	113	2	no	—
73	C₂×C₄	—	113	2	no	—
74	C₂×C₄	C₈×D₁₁₃	1	1	yes	characteristic
75	C₂×C₄	C₈×D₁₁₃	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	226	2	no	—
79	C₈	Q₈×D₁₁₃	1	1	yes	characteristic
80	C₈	#14268646	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₁₁₃	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	226	2	no	—
84	C₄	—	226	2	no	—
85	C₄	#16813561	1	2	yes	—
86	C₄	#16813587	1	1	yes	characteristic
87	C₄	#16813630	1	1	yes	characteristic
88	C₂	#20298722	1	1	yes	characteristic
89	C₂	C₂×#16813561	1	1	yes	characteristic
90	C₂	#20299016	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₁₃
commutator, socle, prime radical	C₂²×C₁₁₃	C₂²×C₈
Fitting	C₄⋊C₁₆×C₁₁₃	C₂
cosocle	C₂×C₈×C₁₁₃	C₂³
hypercenter	C₄⋊C₁₆	D₁₁₃
hypocenter	C₁₁₃	#2183
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
20. #14268622	Q₈	semidirect
46. C₁₁₃	#2183	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₂₂₆	2
#20298416	2
#20298373	2
C₄×#14268622	2
C₄⋊C₁₆×C₁₁₃	2
#2183	113