Subgroups — C₂×D₄×D₃₉₇ #27163132

Order 12704.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃₉₇	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₅₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊D₃₉₇	C₂	1	2	yes	maximal, complemented
4	D₄×D₃₉₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×D₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₉₇	—	2	8	no	—
8	C₂²×D₃₉₇	C₂²	1	1	yes	characteristic
9	C₂²×D₃₉₇	C₂²	1	8	yes	complemented
10	C₂²×D₃₉₇	C₂²	1	2	yes	complemented
11	D₁₅₈₈	C₂²	1	4	yes	complemented
12	C₂²⋊D₃₉₇	C₂²	1	8	yes	complemented
13	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic, complemented
14	C₄×D₃₉₇	C₂²	1	4	yes	complemented
15	C₂³×C₃₉₇	C₂²	1	2	yes	complemented
16	D₄×C₃₉₇	C₂²	1	4	yes	complemented
17	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₃₉₇	—	2	4	no	—
19	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
20	C₂²×C₃₉₇	C₂³	1	4	yes	complemented
21	C₂×D₃₉₇	—	2	4	no	—
22	C₂×D₃₉₇	—	2	16	no	—
23	C₂×D₃₉₇	D₄	1	4	yes	direct product factor
24	C₂×D₃₉₇	C₂³	1	2	yes	—
25	C₂×D₃₉₇	C₂³	1	4	yes	complemented
26	C₄×C₃₉₇	C₂³	1	2	yes	complemented
27	Dic₃₉₇	C₂³	1	2	yes	complemented
28	C₂×C₃₉₇	—	2	4	no	—
29	C₂×C₃₉₇	C₂×D₄	1	2	yes	complemented
30	C₂×C₃₉₇	C₂⁴	1	1	yes	characteristic
31	D₃₉₇	—	2	4	no	—
32	D₃₉₇	C₂×D₄	1	4	yes	direct product factor
33	C₃₉₇	C₂²×D₄	1	1	yes	characteristic, complemented, Sylow
34	C₂²×D₄	—	397	1	no	maximal, Sylow
35	C₂⁴	—	397	2	no	—
36	C₂×D₄	—	397	2	no	—
37	C₂×D₄	—	397	8	no	—
38	C₂×D₄	—	397	1	no	—
39	C₂×D₄	D₃₉₇	1	1	yes	characteristic, direct product factor
40	C₂²×C₄	—	397	1	no	—
41	C₂³	—	794	8	no	—
42	C₂³	—	397	2	no	—
43	C₂³	—	397	8	no	—
44	C₂³	—	397	1	no	—
45	C₂³	C₂×D₃₉₇	1	2	yes	complemented
46	D₄	—	397	8	no	—
47	D₄	—	397	4	no	—
48	D₄	C₂×D₃₉₇	1	4	yes	direct product factor
49	C₂×C₄	—	397	1	no	—
50	C₂×C₄	—	397	4	no	—
51	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic, complemented
52	C₂²	—	2	4	no	—
53	C₂²	—	794	16	no	—
54	C₂²	—	794	4	no	—
55	C₂²	—	397	4	no	—
56	C₂²	—	397	4	no	—
57	C₂²	—	397	2	no	—
58	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
59	C₂²	C₂²×D₃₉₇	1	4	yes	complemented
60	C₄	—	397	2	no	—
61	C₄	C₂²×D₃₉₇	1	2	yes	complemented
62	C₂	—	2	4	no	—
63	C₂	—	794	4	no	—
64	C₂	—	397	4	no	—
65	C₂	D₄×D₃₉₇	1	2	yes	direct product factor
66	C₂	C₂³×D₃₉₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂³×D₃₉₇	C₂	semidirect
2. C₂×D₁₅₈₈	C₂	semidirect
3. C₂×C₂²⋊D₃₉₇	C₂	semidirect
4. D₄×D₃₉₇	C₂	direct (N × H)
5. C₂×C₄×D₃₉₇	C₂	semidirect
6. C₂×D₄×C₃₉₇	C₂	semidirect
9. C₂²×D₃₉₇	C₂²	semidirect
10. C₂²×D₃₉₇	C₂²	semidirect
11. D₁₅₈₈	C₂²	semidirect
12. C₂²⋊D₃₉₇	C₂²	semidirect
13. C₂×Dic₃₉₇	C₂²	semidirect
14. C₄×D₃₉₇	C₂²	semidirect
15. C₂³×C₃₉₇	C₂²	semidirect
16. D₄×C₃₉₇	C₂²	semidirect
17. C₂×C₄×C₃₉₇	C₂²	semidirect
20. C₂²×C₃₉₇	C₂³	semidirect
23. C₂×D₃₉₇	D₄	direct (N × H)
25. C₂×D₃₉₇	C₂³	semidirect
26. C₄×C₃₉₇	C₂³	semidirect
27. Dic₃₉₇	C₂³	semidirect
29. C₂×C₃₉₇	C₂×D₄	semidirect
32. D₃₉₇	C₂×D₄	direct (N × H)
33. C₃₉₇	C₂²×D₄	semidirect
39. C₂×D₄	D₃₉₇	direct (N × H)
45. C₂³	C₂×D₃₉₇	semidirect
48. D₄	C₂×D₃₉₇	direct (N × H)
51. C₂×C₄	C₂×D₃₉₇	semidirect
59. C₂²	C₂²×D₃₉₇	semidirect
61. C₄	C₂²×D₃₉₇	semidirect
65. C₂	D₄×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×D₁₅₈₈	2
C₂×C₂²⋊D₃₉₇	2
D₄×D₃₉₇	2
C₂×C₄×D₃₉₇	2
C₂×D₄×C₃₉₇	2
C₂²×D₄	397

Subgroups — C2×D4×D397 #27163132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×D₃₉₇ #27163132