Subgroups — #104×C₇₂₇ #37551413

Order 23264.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₇₂₇	C₂	1	1	yes	maximal, characteristic
2	M₁₆×C₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×C₇₂₇	C₄	1	1	yes	characteristic, complemented
5	D₄×C₇₂₇	—	2	1	no	—
6	D₄×C₇₂₇	C₄	1	1	yes	characteristic, complemented
7	C₂×C₄×C₇₂₇	—	2	1	no	—
8	C₂×C₄×C₇₂₇	—	2	1	no	—
9	C₂×C₄×C₇₂₇	C₂²	1	1	yes	characteristic
10	C₈×C₇₂₇	—	2	1	no	—
11	C₂²×C₇₂₇	—	2	1	no	—
12	C₂²×C₇₂₇	D₄	1	1	yes	characteristic
13	C₄×C₇₂₇	—	2	2	no	—
14	C₄×C₇₂₇	—	2	1	no	—
15	C₄×C₇₂₇	C₂×C₄	1	1	yes	characteristic
16	C₄×C₇₂₇	D₄	1	1	yes	characteristic
17	C₂×C₇₂₇	—	2	1	no	—
18	C₂×C₇₂₇	—	4	1	no	—
19	C₂×C₇₂₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₇₂₇	#104	1	1	yes	characteristic, direct product factor, Sylow
21	#104	C₇₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	#41	C₂×C₇₂₇	1	1	yes	characteristic
23	M₁₆	C₂×C₇₂₇	1	1	yes	characteristic, complemented
24	C₄²	C₂×C₇₂₇	1	1	yes	characteristic, complemented
25	Q₈	C₄×C₇₂₇	1	1	yes	characteristic, complemented
26	D₄	—	2	1	no	—
27	D₄	C₄×C₇₂₇	1	1	yes	characteristic, complemented
28	C₂×C₄	—	2	1	no	—
29	C₂×C₄	—	2	1	no	—
30	C₂×C₄	C₂²×C₇₂₇	1	1	yes	characteristic
31	C₈	—	2	1	no	—
32	C₂²	—	2	1	no	—
33	C₂²	D₄×C₇₂₇	1	1	yes	characteristic
34	C₄	—	2	2	no	—
35	C₄	—	2	1	no	—
36	C₄	C₂×C₄×C₇₂₇	1	1	yes	characteristic
37	C₄	D₄×C₇₂₇	1	1	yes	characteristic
38	C₂	—	2	1	no	—
39	C₂	—	4	1	no	—
40	C₂	C₂²⋊C₄×C₇₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇₂₇	D₄
commutator	C₄	C₂×C₄×C₇₂₇
Frattini, cosocle	C₂×C₄	C₂²×C₇₂₇
socle	C₂×C₇₂₇	C₂²⋊C₄
prime radical	#41×C₇₂₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₂₇ ◁ C₂×C₄×C₇₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. M₁₆×C₇₂₇	C₂	semidirect
3. C₄²×C₇₂₇	C₂	semidirect
4. Q₈×C₇₂₇	C₄	semidirect
6. D₄×C₇₂₇	C₄	semidirect
20. C₇₂₇	#104	direct (N × H)
21. #104	C₇₂₇	direct (N × H)
23. M₁₆	C₂×C₇₂₇	semidirect
24. C₄²	C₂×C₇₂₇	semidirect
25. Q₈	C₄×C₇₂₇	semidirect
27. D₄	C₄×C₇₂₇	semidirect

Maximal subgroups

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

subgroup	index
#41×C₇₂₇	2
M₁₆×C₇₂₇	2
C₄²×C₇₂₇	2
#104	727

Subgroups — #104×C727 #37551413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #104×C₇₂₇ #37551413