Subgroups — C₄×C₂²⋊D₃₉₇ #27163057

Order 12704.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₃₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263710	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊Dic₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#20263708	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×Dic₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₉₇	C₂²	1	1	yes	characteristic
9	C₂²⋊D₃₉₇	C₄	1	4	yes	direct product factor
10	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
13	C₄×D₃₉₇	—	2	2	no	—
14	C₂³×C₃₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₃₉₇	—	2	2	no	—
16	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
18	C₂²×C₃₉₇	—	2	1	no	—
19	C₂²×C₃₉₇	—	2	1	no	—
20	C₂²×C₃₉₇	C₂×C₄	1	2	yes	complemented
21	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
22	C₂×D₃₉₇	—	2	1	no	—
23	C₂×D₃₉₇	—	2	1	no	—
24	C₂×D₃₉₇	C₂×C₄	1	2	yes	complemented
25	C₄×C₃₉₇	—	2	1	no	—
26	C₄×C₃₉₇	D₄	1	2	yes	complemented
27	Dic₃₉₇	—	2	1	no	—
28	Dic₃₉₇	—	2	1	no	—
29	Dic₃₉₇	C₂×C₄	1	2	yes	complemented
30	C₂×C₃₉₇	—	2	2	no	—
31	C₂×C₃₉₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₃₉₇	#41	1	1	yes	characteristic
34	D₃₉₇	—	2	2	no	—
35	C₃₉₇	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
36	C₄×D₄	—	397	1	no	maximal, Sylow
37	C₂×D₄	—	397	1	no	—
38	C₂²×C₄	—	397	1	no	—
39	C₂²×C₄	D₃₉₇	1	1	yes	characteristic, complemented
40	C₄⋊C₄	—	397	1	no	—
41	C₂²⋊C₄	—	397	1	no	—
42	C₂²⋊C₄	—	397	1	no	—
43	C₄²	—	397	1	no	—
44	C₂³	—	397	1	no	—
45	C₂³	C₂×D₃₉₇	1	1	yes	characteristic
46	D₄	—	397	4	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	794	2	no	—
49	C₂×C₄	—	397	1	no	—
50	C₂×C₄	—	397	1	no	—
51	C₂×C₄	—	397	1	no	—
52	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
53	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
54	C₂²	—	2	1	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	794	1	no	—
57	C₂²	—	794	1	no	—
58	C₂²	—	397	2	no	—
59	C₂²	C₄×D₃₉₇	1	2	yes	complemented
60	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
61	C₄	—	2	1	no	—
62	C₄	—	794	1	no	—
63	C₄	—	794	1	no	—
64	C₄	—	397	2	no	—
65	C₄	C₂²⋊D₃₉₇	1	2	yes	direct product factor
66	C₂	—	2	2	no	—
67	C₂	—	794	2	no	—
68	C₂	C₂×C₄×D₃₉₇	1	1	yes	characteristic
69	C₂	C₂×C₂²⋊D₃₉₇	1	1	yes	characteristic
70	C₂	#20263731	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₉₇
commutator	C₂×C₃₉₇	C₂²×C₄
Frattini	C₂²	C₂²×D₃₉₇
Fitting	C₂²×C₄×C₃₉₇	C₂
socle, cosocle	C₂²×C₃₉₇	C₂³
prime radical	C₂×C₂²⋊D₃₉₇	C₂
hypercenter	C₂²×C₄	D₃₉₇
hypocenter	C₃₉₇	C₄×D₄
LCS	G ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₃₉₇	C₂	semidirect
3. #20263710	C₂	semidirect
4. C₂²⋊Dic₃₉₇	C₂	semidirect
5. #20263708	C₂	semidirect
6. C₄×Dic₃₉₇	C₂	semidirect
7. C₂²×C₄×C₃₉₇	C₂	semidirect
9. C₂²⋊D₃₉₇	C₄	direct (N × H)
20. C₂²×C₃₉₇	C₂×C₄	semidirect
24. C₂×D₃₉₇	C₂×C₄	semidirect
26. C₄×C₃₉₇	D₄	semidirect
29. Dic₃₉₇	C₂×C₄	semidirect
35. C₃₉₇	C₄×D₄	semidirect
39. C₂²×C₄	D₃₉₇	semidirect
59. C₂²	C₄×D₃₉₇	semidirect
65. C₄	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
C₂×C₂²⋊D₃₉₇	2
C₂×C₄×D₃₉₇	2
#20263710	2
C₂²⋊Dic₃₉₇	2
#20263708	2
C₄×Dic₃₉₇	2
C₂²×C₄×C₃₉₇	2
C₄×D₄	397

Subgroups — C4×C22⋊D397 #27163057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂²⋊D₃₉₇ #27163057