Subgroups — #37551518

Order 23264.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₉₀₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊D₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₇₂₇)⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₄×C₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₂₇	—	2	2	no	—
7	C₂²×D₇₂₇	—	2	4	no	—
8	C₂²×D₇₂₇	C₂²	1	2	yes	—
9	C₂²×D₇₂₇	C₂²	1	1	yes	characteristic, complemented
10	D₂₉₀₈	—	2	4	no	—
11	C₂²⋊D₇₂₇	—	2	2	no	—
12	C₂×Dic₇₂₇	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₇₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇₂₇	C₂²	1	2	yes	complemented
15	C₂²×C₇₂₇	—	2	2	no	—
16	C₂²×C₇₂₇	D₄	1	2	yes	complemented
17	C₂²×C₇₂₇	C₂³	1	1	yes	characteristic
18	C₂×D₇₂₇	—	2	8	no	—
19	C₂×D₇₂₇	—	2	2	no	—
20	C₂×D₇₂₇	—	2	2	no	—
21	C₂×D₇₂₇	—	2	1	no	—
22	C₂×D₇₂₇	—	2	2	no	—
23	C₂×D₇₂₇	D₄	1	4	yes	complemented
24	C₄×C₇₂₇	—	2	2	no	—
25	Dic₇₂₇	—	2	1	no	—
26	C₂×C₇₂₇	—	2	2	no	—
27	C₂×C₇₂₇	C₂×D₄	1	2	yes	—
28	C₂×C₇₂₇	C₂×D₄	1	1	yes	characteristic
29	D₇₂₇	—	2	4	no	—
30	D₇₂₇	—	4	1	no	—
31	C₇₂₇	#120	1	1	yes	characteristic, complemented, Sylow
32	#120	—	727	1	no	maximal, Sylow
33	C₂⁴	—	727	1	no	—
34	C₂×D₄	—	727	1	no	—
35	C₂×D₄	—	727	2	no	—
36	C₂²⋊C₄	—	727	2	no	—
37	C₂²⋊C₄	D₇₂₇	1	1	yes	characteristic, complemented
38	C₂³	—	1454	4	no	—
39	C₂³	—	1454	2	no	—
40	C₂³	—	727	1	no	—
41	C₂³	—	727	2	no	—
42	C₂³	C₂×D₇₂₇	1	1	yes	characteristic
43	D₄	—	1454	4	no	—
44	D₄	—	1454	2	no	—
45	C₂×C₄	—	727	1	no	—
46	C₂×C₄	C₂×D₇₂₇	1	2	yes	complemented
47	C₂²	—	2	2	no	—
48	C₂²	—	1454	8	no	—
49	C₂²	—	1454	2	no	—
50	C₂²	—	1454	1	no	—
51	C₂²	—	1454	2	no	—
52	C₂²	—	1454	2	no	—
53	C₂²	—	727	4	no	—
54	C₂²	D₂₉₀₈	1	2	yes	complemented
55	C₂²	C₂²×D₇₂₇	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	1454	1	no	—
58	C₂	—	2	2	no	—
59	C₂	—	2908	1	no	—
60	C₂	—	1454	4	no	—
61	C₂	D₄×D₇₂₇	1	2	yes	—
62	C₂	C₂×D₂₉₀₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇₂₇
commutator, socle, cosocle	C₂²×C₇₂₇	C₂³
Fitting	C₂²⋊C₄×C₇₂₇	C₂
hypercenter	C₂²⋊C₄	D₇₂₇
hypocenter	C₇₂₇	#120
LCS	G ▷ C₂²×C₇₂₇ ▷ C₇₂₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₇₂₇	C₂	semidirect
2. C₂×D₂₉₀₈	C₂	semidirect
3. C₂×C₂²⋊D₇₂₇	C₂	semidirect
4. (C₂×D₇₂₇)⋊C₄	C₂	semidirect
5. C₂²⋊C₄×C₇₂₇	C₂	semidirect
9. C₂²×D₇₂₇	C₂²	semidirect
12. C₂×Dic₇₂₇	C₂²	semidirect
14. C₂×C₄×C₇₂₇	C₂²	semidirect
16. C₂²×C₇₂₇	D₄	semidirect
23. C₂×D₇₂₇	D₄	semidirect
31. C₇₂₇	#120	semidirect
37. C₂²⋊C₄	D₇₂₇	semidirect
46. C₂×C₄	C₂×D₇₂₇	semidirect
54. C₂²	D₂₉₀₈	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₇₂₇	2
C₂×D₂₉₀₈	2
C₂×C₂²⋊D₇₂₇	2
(C₂×D₇₂₇)⋊C₄	2
C₂²⋊C₄×C₇₂₇	2
#120	727