Subgroups — #28409384

Order 14528.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20300288	C₂	1	1	yes	maximal, characteristic
2	#20300211	C₂	1	4	yes	maximal
3	C₄×C₂₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₅₄	C₂²	1	2	yes	—
6	C₂×C₂₂₇⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₂₂₇	—	2	1	no	—
8	C₄⋊Dic₂₂₇	—	2	1	no	—
9	C₂×Q₈×C₂₂₇	—	2	1	no	—
10	C₄⋊C₄×C₂₂₇	—	2	1	no	—
11	C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
12	C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
13	Dic₄₅₄	—	2	4	no	—
14	C₂×Dic₂₂₇	—	2	2	no	—
15	C₂₂₇⋊C₈	—	2	2	no	—
16	Q₈×C₂₂₇	—	4	2	no	—
17	C₂×C₄×C₂₂₇	—	2	2	no	—
18	C₂×C₄×C₂₂₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
21	C₂²×C₂₂₇	C₂×D₄	1	1	yes	characteristic
22	C₄×C₂₂₇	—	4	2	no	—
23	C₄×C₂₂₇	SD₁₆	1	2	yes	—
24	C₄×C₂₂₇	Q₁₆	1	2	yes	—
25	C₄×C₂₂₇	#41	1	2	yes	—
26	Dic₂₂₇	—	4	2	no	—
27	C₂×C₂₂₇	#124	1	1	yes	characteristic
28	C₂×C₂₂₇	C₂×SD₁₆	1	1	yes	characteristic
29	C₂×C₂₂₇	C₂×Q₁₆	1	1	yes	characteristic
30	C₂₂₇	#487	1	1	yes	characteristic, complemented, Sylow
31	#487	—	227	1	no	maximal, Sylow
32	C₄⋊Q₈	—	227	1	no	—
33	C₄⋊Q₈	D₂₂₇	1	1	yes	characteristic
34	#103	—	227	4	no	—
35	C₄×C₈	—	227	1	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	—	227	2	no	—
38	C₂×C₈	—	227	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	454	1	no	—
41	C₄⋊C₄	—	454	1	no	—
42	C₄⋊C₄	C₂×D₂₂₇	1	2	yes	—
43	C₄²	C₂×D₂₂₇	1	1	yes	characteristic
44	Q₈	—	4	2	no	—
45	Q₈	—	454	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	454	2	no	—
48	C₂×C₄	C₂²⋊D₂₂₇	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₂₂₇	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
51	C₈	—	454	2	no	—
52	C₂²	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	908	2	no	—
55	C₄	#16813872	1	2	yes	—
56	C₄	#16813874	1	2	yes	—
57	C₄	#16813885	1	2	yes	—
58	C₂	#20300277	1	1	yes	characteristic
59	C₂	C₂×#16813872	1	1	yes	characteristic
60	C₂	C₂×#16813874	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₇
commutator, cosocle	C₂×C₄×C₂₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₇
Fitting	C₄⋊Q₈×C₂₂₇	C₂
socle, prime radical	C₂²×C₂₂₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₂₂₇
hypocenter	C₂₂₇	#487
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
30. C₂₂₇	#487	semidirect

Maximal subgroups

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

subgroup	index
#20300288	2
#20300211	2
C₄×C₂₂₇⋊C₈	2
C₄⋊Q₈×C₂₂₇	2
#487	227