Subgroups — C₂²×C₄×D₇₂₇ #37551582

Order 23264.

236 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
2	C₂²×Dic₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₇₂₇	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₄×C₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₇₂₇	C₄	1	8	yes	direct product factor
6	C₂²×D₇₂₇	C₂²	1	6	yes	—
7	C₂×Dic₇₂₇	C₂²	1	6	yes	complemented
8	C₄×D₇₂₇	C₂²	1	16	yes	direct product factor
9	C₂³×C₇₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇₂₇	C₂²	1	6	yes	complemented
11	C₂²×C₇₂₇	C₂×C₄	1	4	yes	complemented
12	C₂²×C₇₂₇	C₂³	1	3	yes	—
13	C₂×D₇₂₇	C₂×C₄	1	24	yes	direct product factor
14	C₂×D₇₂₇	C₂³	1	4	yes	—
15	C₄×C₇₂₇	C₂³	1	4	yes	complemented
16	Dic₇₂₇	C₂³	1	4	yes	complemented
17	C₂×C₇₂₇	C₂²×C₄	1	6	yes	complemented
18	C₂×C₇₂₇	C₂⁴	1	1	yes	characteristic
19	D₇₂₇	C₂²×C₄	1	8	yes	direct product factor
20	C₇₂₇	C₂³×C₄	1	1	yes	characteristic, complemented, Sylow
21	C₂³×C₄	—	727	1	no	maximal, Sylow
22	C₂⁴	—	727	1	no	—
23	C₂²×C₄	—	727	1	no	—
24	C₂²×C₄	—	727	12	no	—
25	C₂²×C₄	D₇₂₇	1	1	yes	characteristic, direct product factor
26	C₂³	—	727	8	no	—
27	C₂³	—	727	6	no	—
28	C₂³	C₂×D₇₂₇	1	1	yes	characteristic
29	C₂×C₄	—	727	6	no	—
30	C₂×C₄	—	727	16	no	—
31	C₂×C₄	C₂×D₇₂₇	1	6	yes	direct product factor
32	C₂²	—	727	24	no	—
33	C₂²	—	727	4	no	—
34	C₂²	C₄×D₇₂₇	1	4	yes	direct product factor
35	C₂²	C₂²×D₇₂₇	1	3	yes	—
36	C₄	—	727	4	no	—
37	C₄	C₂²×D₇₂₇	1	4	yes	direct product factor
38	C₂	—	727	8	no	—
39	C₂	C₂×C₄×D₇₂₇	1	6	yes	direct product factor
40	C₂	C₂³×D₇₂₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₇₂₇
commutator, hypocenter	C₇₂₇	C₂³×C₄
Frattini	C₂	C₂³×D₇₂₇
Fitting	C₂²×C₄×C₇₂₇	C₂
socle	C₂³×C₇₂₇	C₂²
prime radical	C₂³×D₇₂₇	C₂
cosocle	C₂×C₇₂₇	C₂⁴
LCS	G ▷ C₇₂₇
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×Dic₇₂₇	C₂	semidirect
3. C₂×C₄×D₇₂₇	C₂	direct (N × H)
4. C₂²×C₄×C₇₂₇	C₂	semidirect
5. C₂²×D₇₂₇	C₄	direct (N × H)
7. C₂×Dic₇₂₇	C₂²	semidirect
8. C₄×D₇₂₇	C₂²	direct (N × H)
10. C₂×C₄×C₇₂₇	C₂²	semidirect
11. C₂²×C₇₂₇	C₂×C₄	semidirect
13. C₂×D₇₂₇	C₂×C₄	direct (N × H)
15. C₄×C₇₂₇	C₂³	semidirect
16. Dic₇₂₇	C₂³	semidirect
17. C₂×C₇₂₇	C₂²×C₄	semidirect
19. D₇₂₇	C₂²×C₄	direct (N × H)
20. C₇₂₇	C₂³×C₄	semidirect
25. C₂²×C₄	D₇₂₇	direct (N × H)
31. C₂×C₄	C₂×D₇₂₇	direct (N × H)
34. C₂²	C₄×D₇₂₇	direct (N × H)
37. C₄	C₂²×D₇₂₇	direct (N × H)
39. C₂	C₂×C₄×D₇₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₇₂₇	2
C₂²×Dic₇₂₇	2
C₂×C₄×D₇₂₇	2
C₂²×C₄×C₇₂₇	2
C₂³×C₄	727

Subgroups — C22×C4×D727 #37551582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×D₇₂₇ #37551582