Subgroups — #28407800

Order 14464.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299701	C₂	1	2	yes	maximal
2	C₄⋊#14268659	C₂	1	1	yes	maximal, characteristic
3	#20299692	C₂	1	2	yes	maximal
4	C₄×#14268659	C₂	1	1	yes	maximal, characteristic
5	Q₈×#11744753	C₂	1	1	yes	maximal, characteristic
6	C₂×#14268659	C₂²	1	1	yes	characteristic
7	C₂×#14268659	C₂²	1	2	yes	—
8	C₂×#14268659	C₂²	1	1	yes	characteristic
9	C₄⋊#11744753	C₄	1	2	yes	—
10	C₄⋊#11744753	C₄	1	1	yes	characteristic
11	C₄×#11744753	C₂²	1	1	yes	characteristic
12	C₄×#11744753	C₂²	1	2	yes	—
13	Q₈×Dic₁₁₃	C₄	1	1	yes	characteristic
14	#14268659	—	2	1	no	—
15	#14268659	—	2	2	no	—
16	#14268659	Q₈	1	2	yes	complemented
17	C₂×#11744753	C₂×C₄	1	1	yes	characteristic
18	C₂×#11744753	C₂×C₄	1	2	yes	—
19	C₂×#11744753	C₂³	1	1	yes	characteristic
20	C₄⋊Dic₁₁₃	C₈	1	2	yes	—
21	C₄⋊Dic₁₁₃	C₈	1	1	yes	characteristic
22	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
23	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₁₁₃	C₈	1	1	yes	characteristic
25	Q₈×C₁₁₃	—	2	2	no	—
26	C₂×C₄×C₁₁₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
28	C₂×Dic₁₁₃	C₂×C₈	1	2	yes	—
29	C₂×Dic₁₁₃	C₂×C₈	1	1	yes	characteristic
30	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
31	#11744753	—	2	1	no	—
32	#11744753	—	2	2	no	—
33	#11744753	C₂×Q₈	1	1	yes	characteristic
34	#11744753	#41	1	1	yes	characteristic
35	C₂²×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₁₁₃	—	2	2	no	—
37	C₄×C₁₁₃	M₃₂	1	2	yes	—
38	Dic₁₁₃	—	2	2	no	—
39	Dic₁₁₃	—	2	1	no	—
40	Dic₁₁₃	C₄×Q₈	1	1	yes	characteristic
41	Dic₁₁₃	#131	1	1	yes	characteristic
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	1	no	—
49	C₄⋊C₁₆	—	113	2	no	—
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	—	113	2	no	—
56	C₄⋊C₈	—	113	2	no	—
57	C₄⋊C₈	—	113	1	no	—
58	C₄×C₈	—	113	1	no	—
59	C₄×C₈	—	113	2	no	—
60	C₂×Q₈	#11744762	1	1	yes	characteristic
61	C₂×C₈	—	113	1	no	—
62	C₂×C₈	—	113	1	no	—
63	C₂×C₈	—	113	2	no	—
64	C₄⋊C₄	—	113	2	no	—
65	C₄⋊C₄	—	113	1	no	—
66	C₄²	—	113	2	no	—
67	C₄²	—	113	1	no	—
68	C₁₆	—	113	2	no	—
69	C₁₆	—	226	1	no	—
70	C₁₆	—	226	2	no	—
71	Q₈	—	2	2	no	—
72	C₂×C₄	—	113	1	no	—
73	C₂×C₄	—	113	2	no	—
74	C₂×C₄	—	113	1	no	—
75	C₂×C₄	C₂×#11744762	1	2	yes	—
76	C₂×C₄	C₂×#11744762	1	1	yes	characteristic
77	C₈	—	113	1	no	—
78	C₈	—	113	1	no	—
79	C₈	—	226	1	no	—
80	C₈	—	226	2	no	—
81	C₂²	C₂²×#11744762	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	113	1	no	—
84	C₄	—	113	1	no	—
85	C₄	—	226	2	no	—
86	C₄	—	226	1	no	—
87	C₄	#16813718	1	2	yes	—
88	C₂	Q₈×#11744762	1	1	yes	characteristic
89	C₂	C₂×#16813718	1	1	yes	characteristic
90	C₂	#20299720	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11744762
commutator, socle, prime radical	C₂²×C₁₁₃	C₂²×C₈
Fitting	C₂×Q₈×C₁₁₃	C₈
cosocle	C₂×#11744753	C₂³
hypercenter	C₂×Q₈	#11744762
hypocenter	C₁₁₃	#2183
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
16. #14268659	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
#20299701	2
C₄⋊#14268659	2
#20299692	2
C₄×#14268659	2
Q₈×#11744753	2
#2183	113