Subgroups — #37551539

Order 23264.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Dic₇₂₇	C₂	1	2	yes	maximal
2	C₄⋊Dic₇₂₇	C₂	1	1	yes	maximal, characteristic
3	Dic₇₂₇⋊C₄	C₂	1	2	yes	maximal
4	C₄×Dic₇₂₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×C₇₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₂₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₇₂₇	C₂²	1	1	yes	characteristic
8	C₂×Dic₇₂₇	C₂²	1	2	yes	—
9	C₂×C₄×C₇₂₇	C₂²	1	2	yes	—
10	C₂×C₄×C₇₂₇	C₂²	1	1	yes	characteristic
11	C₂²×C₇₂₇	C₂³	1	1	yes	characteristic
12	C₄×C₇₂₇	—	2	2	no	—
13	C₄×C₇₂₇	Q₈	1	2	yes	—
14	Dic₇₂₇	—	2	1	no	—
15	Dic₇₂₇	—	2	1	no	—
16	Dic₇₂₇	—	2	2	no	—
17	C₂×C₇₂₇	C₂×Q₈	1	1	yes	characteristic
18	C₂×C₇₂₇	#41	1	1	yes	characteristic
19	C₂×C₇₂₇	#41	1	1	yes	characteristic
20	C₇₂₇	#125	1	1	yes	characteristic, complemented, Sylow
21	#125	—	727	1	no	maximal, Sylow
22	C₄⋊C₄	—	727	2	no	—
23	C₄⋊C₄	—	727	2	no	—
24	C₄⋊C₄	—	727	1	no	—
25	C₄⋊C₄	D₇₂₇	1	1	yes	characteristic
26	C₄²	—	727	1	no	—
27	C₂×C₄	—	727	1	no	—
28	C₂×C₄	—	727	1	no	—
29	C₂×C₄	—	727	2	no	—
30	C₂×C₄	C₂×D₇₂₇	1	2	yes	—
31	C₂×C₄	C₂×D₇₂₇	1	1	yes	characteristic
32	C₂²	C₂²×D₇₂₇	1	1	yes	characteristic
33	C₄	—	2	2	no	—
34	C₄	—	1454	1	no	—
35	C₄	—	1454	1	no	—
36	C₄	—	1454	2	no	—
37	C₄	Dic₁₄₅₄	1	2	yes	—
38	C₂	C₂×Dic₁₄₅₄	1	1	yes	characteristic
39	C₂	#26042739	1	1	yes	characteristic
40	C₂	#26042741	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇₂₇
commutator, socle, prime radical, cosocle	C₂²×C₇₂₇	C₂³
Fitting	C₄⋊C₄×C₇₂₇	C₂
hypercenter	C₄⋊C₄	D₇₂₇
hypocenter	C₇₂₇	#125
LCS	G ▷ C₂²×C₇₂₇ ▷ C₇₂₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

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

N	H ≅ G/N	type
20. C₇₂₇	#125	semidirect

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