Subgroups — #37551526

Order 23264.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₄₅₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₇₂₇	C₂	1	1	yes	maximal, characteristic
3	Dic₇₂₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₇₂₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₇₂₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	(C₂×D₇₂₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×C₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇₂₇	C₂²	1	1	yes	characteristic
9	Dic₁₄₅₄	—	2	2	no	—
10	C₂×Dic₇₂₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₇₂₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₇₂₇	C₂²	1	1	yes	characteristic
13	C₄×D₇₂₇	—	2	2	no	—
14	C₂×C₄×C₇₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇₂₇	C₂²	1	1	yes	characteristic
17	C₂²×C₇₂₇	C₂³	1	1	yes	characteristic
18	C₂×D₇₂₇	—	2	1	no	—
19	C₂×D₇₂₇	—	2	1	no	—
20	C₂×D₇₂₇	Q₈	1	2	yes	complemented
21	C₄×C₇₂₇	—	2	1	no	—
22	C₄×C₇₂₇	—	2	1	no	—
23	C₄×C₇₂₇	—	2	1	no	—
24	Dic₇₂₇	—	2	1	no	—
25	Dic₇₂₇	—	2	1	no	—
26	Dic₇₂₇	D₄	1	2	yes	—
27	C₂×C₇₂₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₇₂₇	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₇₂₇	#41	1	1	yes	characteristic
30	D₇₂₇	—	2	2	no	—
31	C₇₂₇	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
32	C₂²⋊Q₈	—	727	1	no	maximal, Sylow
33	C₂×Q₈	—	727	1	no	—
34	C₂²×C₄	—	727	1	no	—
35	C₄⋊C₄	—	727	1	no	—
36	C₄⋊C₄	—	727	1	no	—
37	C₄⋊C₄	D₇₂₇	1	1	yes	characteristic, complemented
38	C₂²⋊C₄	—	727	1	no	—
39	C₂²⋊C₄	—	727	1	no	—
40	C₂³	—	727	1	no	—
41	Q₈	—	1454	2	no	—
42	C₂×C₄	—	1454	2	no	—
43	C₂×C₄	—	727	1	no	—
44	C₂×C₄	—	727	1	no	—
45	C₂×C₄	—	727	1	no	—
46	C₂×C₄	C₂×D₇₂₇	1	1	yes	characteristic
47	C₂×C₄	C₂×D₇₂₇	1	1	yes	characteristic
48	C₂×C₄	C₂×D₇₂₇	1	1	yes	characteristic
49	C₂²	—	1454	1	no	—
50	C₂²	—	1454	1	no	—
51	C₂²	—	727	2	no	—
52	C₂²	C₂²×D₇₂₇	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	2	1	no	—
55	C₄	—	2	1	no	—
56	C₄	—	1454	1	no	—
57	C₄	—	1454	1	no	—
58	C₄	—	727	2	no	—
59	C₂	—	1454	2	no	—
60	C₂	D₄×D₇₂₇	1	1	yes	characteristic
61	C₂	Q₈×D₇₂₇	1	1	yes	characteristic
62	C₂	#26042740	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇₂₇
commutator, socle, cosocle	C₂²×C₇₂₇	C₂³
Fitting	C₄⋊C₄×C₇₂₇	C₂
prime radical	C₂²×D₇₂₇	C₂²
hypercenter	C₄⋊C₄	D₇₂₇
hypocenter	C₇₂₇	C₂²⋊Q₈
LCS	G ▷ C₂²×C₇₂₇ ▷ C₇₂₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₄₅₄	C₂	semidirect
3. Dic₇₂₇⋊C₄	C₂	semidirect
4. Dic₇₂₇⋊C₄	C₂	semidirect
7. C₄⋊C₄×C₇₂₇	C₂	semidirect
20. C₂×D₇₂₇	Q₈	semidirect
31. C₇₂₇	C₂²⋊Q₈	semidirect
37. C₄⋊C₄	D₇₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₄₅₄	2
C₂×C₄×D₇₂₇	2
Dic₇₂₇⋊C₄	2
(C₂×D₇₂₇)⋊C₄	2
C₄⋊C₄×C₇₂₇	2
C₂²⋊Q₈	727