Subgroups — #28409316

Order 14528.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20300287	C₂	1	1	yes	maximal, characteristic
2	#20300253	C₂	1	1	yes	maximal, characteristic
3	#20300219	C₂	1	1	yes	maximal, characteristic
4	Dic₂₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#20300213	C₂	1	1	yes	maximal, characteristic
6	#20300211	C₂	1	1	yes	maximal, characteristic
7	#106×C₂₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₅₄	—	2	1	no	—
9	C₂×Dic₄₅₄	C₂²	1	1	yes	characteristic
10	C₂×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
12	Dic₂₂₇⋊C₄	—	2	1	no	—
13	Dic₂₂₇⋊C₄	—	2	1	no	—
14	C₄×Dic₂₂₇	—	2	1	no	—
15	C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
19	Dic₄₅₄	—	2	1	no	—
20	Dic₄₅₄	—	4	2	no	—
21	Dic₄₅₄	Q₈	1	2	yes	complemented
22	C₂×Dic₂₂₇	—	2	1	no	—
23	C₂×Dic₂₂₇	—	2	1	no	—
24	C₂×Dic₂₂₇	D₄	1	1	yes	characteristic
25	C₂×Dic₂₂₇	D₄	1	1	yes	characteristic
26	C₂₂₇⋊C₈	—	2	1	no	—
27	C₂×C₄×C₂₂₇	—	2	1	no	—
28	C₂×C₄×C₂₂₇	—	2	1	no	—
29	C₂×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
30	C₈×C₂₂₇	—	2	1	no	—
31	C₂²×C₂₂₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₂₂₇	—	4	1	no	—
33	C₄×C₂₂₇	—	4	1	no	—
34	C₄×C₂₂₇	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₂₂₇	#41	1	1	yes	characteristic
36	Dic₂₂₇	—	2	2	no	—
37	Dic₂₂₇	—	2	1	no	—
38	Dic₂₂₇	—	4	1	no	—
39	Dic₂₂₇	SD₁₆	1	2	yes	—
40	C₂×C₂₂₇	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₂₂₇	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₂₂₇	#137	1	1	yes	characteristic
43	C₂₂₇	#475	1	1	yes	characteristic, complemented, Sylow
44	#475	—	227	1	no	maximal, Sylow
45	C₄⋊Q₈	—	227	1	no	—
46	C₄×Q₈	—	227	1	no	—
47	#106	—	227	1	no	—
48	#106	D₂₂₇	1	1	yes	characteristic
49	C₄⋊C₈	—	227	1	no	—
50	#103	—	227	1	no	—
51	#103	—	227	1	no	—
52	C₂×Q₈	—	454	1	no	—
53	C₂×Q₈	—	227	1	no	—
54	C₂×C₈	—	227	1	no	—
55	C₂×C₈	C₂×D₂₂₇	1	1	yes	characteristic
56	C₄⋊C₄	—	454	1	no	—
57	C₄⋊C₄	—	454	1	no	—
58	C₄⋊C₄	—	227	1	no	—
59	C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
60	C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
61	C₄²	—	454	1	no	—
62	C₄²	—	227	1	no	—
63	Q₈	—	908	2	no	—
64	Q₈	—	454	1	no	—
65	Q₈	—	227	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	454	1	no	—
69	C₂×C₄	—	454	1	no	—
70	C₂×C₄	—	227	1	no	—
71	C₂×C₄	—	227	1	no	—
72	C₂×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	454	1	no	—
75	C₂²	D₄×D₂₂₇	1	1	yes	characteristic
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	908	1	no	—
79	C₄	—	454	1	no	—
80	C₄	—	454	2	no	—
81	C₄	—	227	2	no	—
82	C₄	Q₈×D₂₂₇	1	1	yes	characteristic
83	C₄	#16813884	1	1	yes	characteristic
84	C₂	#20300266	1	1	yes	characteristic
85	C₂	SD₁₆×D₂₂₇	1	1	yes	characteristic
86	C₂	#20300317	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₂₇
commutator, cosocle	C₂×C₄×C₂₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₇
Fitting	#106×C₂₂₇	C₂
socle, prime radical	C₂²×C₂₂₇	C₂×D₄
hypercenter	#106	D₂₂₇
hypocenter	C₂₂₇	#475
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
21. Dic₄₅₄	Q₈	semidirect
43. C₂₂₇	#475	semidirect

Maximal subgroups

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

subgroup	index
#20300287	2
#20300253	2
#20300219	2
Dic₂₂₇⋊C₈	2
#20300213	2
#20300211	2
#106×C₂₂₇	2
#475	227