Subgroups — C₂²×C₂²⋊D₃₉₇ #27163133

Order 12704.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₃₉₇	C₂	1	12	yes	maximal, direct product factor
3	C₂²×Dic₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₉₇	—	2	4	no	—
6	C₂²×D₃₉₇	C₂²	1	6	yes	complemented
7	C₂²⋊D₃₉₇	C₂²	1	16	yes	direct product factor
8	C₂×Dic₃₉₇	C₂²	1	6	yes	complemented
9	C₂³×C₃₉₇	—	2	4	no	—
10	C₂³×C₃₉₇	C₂²	1	1	yes	characteristic
11	C₂³×C₃₉₇	C₂²	1	6	yes	complemented
12	C₂²×C₃₉₇	—	2	12	no	—
13	C₂²×C₃₉₇	D₄	1	4	yes	complemented
14	C₂²×C₃₉₇	C₂³	1	3	yes	—
15	C₂²×C₃₉₇	C₂³	1	4	yes	complemented
16	C₂×D₃₉₇	—	2	12	no	—
17	C₂×D₃₉₇	C₂³	1	4	yes	complemented
18	Dic₃₉₇	C₂³	1	4	yes	complemented
19	C₂×C₃₉₇	—	2	4	no	—
20	C₂×C₃₉₇	C₂×D₄	1	6	yes	complemented
21	C₂×C₃₉₇	C₂⁴	1	1	yes	characteristic
22	D₃₉₇	—	2	4	no	—
23	C₃₉₇	C₂²×D₄	1	1	yes	characteristic, complemented, Sylow
24	C₂²×D₄	—	397	1	no	maximal, Sylow
25	C₂⁴	—	397	1	no	—
26	C₂⁴	D₃₉₇	1	1	yes	characteristic, complemented
27	C₂×D₄	—	397	12	no	—
28	C₂²×C₄	—	397	1	no	—
29	C₂³	—	2	4	no	—
30	C₂³	—	794	4	no	—
31	C₂³	—	397	6	no	—
32	C₂³	C₂×D₃₉₇	1	1	yes	characteristic
33	C₂³	C₂×D₃₉₇	1	6	yes	complemented
34	D₄	—	397	16	no	—
35	C₂×C₄	—	397	6	no	—
36	C₂²	—	2	12	no	—
37	C₂²	—	794	12	no	—
38	C₂²	—	397	4	no	—
39	C₂²	C₂²⋊D₃₉₇	1	4	yes	direct product factor
40	C₂²	C₂²×D₃₉₇	1	3	yes	—
41	C₂²	C₂²×D₃₉₇	1	4	yes	complemented
42	C₄	—	397	4	no	—
43	C₂	—	2	4	no	—
44	C₂	—	794	4	no	—
45	C₂	C₂×C₂²⋊D₃₉₇	1	6	yes	direct product factor
46	C₂	C₂³×D₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₉₇
commutator, cosocle	C₂×C₃₉₇	C₂⁴
Frattini	C₂	C₂³×D₃₉₇
Fitting	C₂⁴×C₃₉₇	C₂
socle	C₂³×C₃₉₇	C₂²
hypercenter	C₂⁴	D₃₉₇
hypocenter	C₃₉₇	C₂²×D₄
LCS	G ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂³×D₃₉₇	C₂	semidirect
2. C₂×C₂²⋊D₃₉₇	C₂	direct (N × H)
3. C₂²×Dic₃₉₇	C₂	semidirect
4. C₂⁴×C₃₉₇	C₂	semidirect
6. C₂²×D₃₉₇	C₂²	semidirect
7. C₂²⋊D₃₉₇	C₂²	direct (N × H)
8. C₂×Dic₃₉₇	C₂²	semidirect
11. C₂³×C₃₉₇	C₂²	semidirect
13. C₂²×C₃₉₇	D₄	semidirect
15. C₂²×C₃₉₇	C₂³	semidirect
17. C₂×D₃₉₇	C₂³	semidirect
18. Dic₃₉₇	C₂³	semidirect
20. C₂×C₃₉₇	C₂×D₄	semidirect
23. C₃₉₇	C₂²×D₄	semidirect
26. C₂⁴	D₃₉₇	semidirect
33. C₂³	C₂×D₃₉₇	semidirect
39. C₂²	C₂²⋊D₃₉₇	direct (N × H)
41. C₂²	C₂²×D₃₉₇	semidirect
45. C₂	C₂×C₂²⋊D₃₉₇	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×D₃₉₇	2
C₂×C₂²⋊D₃₉₇	2
C₂²×Dic₃₉₇	2
C₂⁴×C₃₉₇	2
C₂²×D₄	397

Subgroups — C22×C22⋊D397 #27163133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊D₃₉₇ #27163133