Subgroups — C₂×#20263731 #27163139

Order 12704.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263731	C₂	1	8	yes	maximal, direct product factor
2	C₂×Dic₇₉₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₁₅₈₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊D₃₉₇	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₃₉₇	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₉₇	C₂²	1	2	yes	—
8	Dic₇₉₄	C₂²	1	4	yes	complemented
9	D₁₅₈₈	C₂²	1	4	yes	complemented
10	C₂²⋊D₃₉₇	C₂²	1	8	yes	complemented
11	C₂×Dic₃₉₇	C₂²	1	2	yes	—
12	C₄×D₃₉₇	C₂²	1	8	yes	complemented
13	C₂³×C₃₉₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₃₉₇	C₂²	1	4	yes	complemented
17	C₂²×C₃₉₇	—	2	2	no	—
18	C₂²×C₃₉₇	C₂³	1	2	yes	—
19	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
20	C₂×D₃₉₇	—	2	4	no	—
21	C₂×D₃₉₇	C₂³	1	4	yes	—
22	C₄×C₃₉₇	C₂³	1	2	yes	—
23	C₄×C₃₉₇	C₂³	1	2	yes	—
24	Dic₃₉₇	C₂³	1	4	yes	—
25	C₂×C₃₉₇	—	2	2	no	—
26	C₂×C₃₉₇	#41	1	2	yes	complemented
27	C₂×C₃₉₇	C₂⁴	1	1	yes	characteristic
28	D₃₉₇	—	2	4	no	—
29	C₃₉₇	C₂×#41	1	1	yes	characteristic, complemented, Sylow
30	C₂×#41	—	397	1	no	maximal, Sylow
31	#41	—	397	8	no	—
32	C₂×Q₈	—	397	1	no	—
33	C₂×D₄	—	397	2	no	—
34	C₂×D₄	—	397	1	no	—
35	C₂²×C₄	—	397	2	no	—
36	C₂²×C₄	D₃₉₇	1	1	yes	characteristic, complemented
37	C₂³	—	397	2	no	—
38	C₂³	C₂×D₃₉₇	1	1	yes	characteristic
39	Q₈	—	397	4	no	—
40	D₄	—	397	8	no	—
41	D₄	—	397	4	no	—
42	C₂×C₄	—	397	2	no	—
43	C₂×C₄	—	397	8	no	—
44	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
45	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
46	C₂×C₄	C₂×D₃₉₇	1	4	yes	complemented
47	C₂²	—	2	2	no	—
48	C₂²	—	794	4	no	—
49	C₂²	—	397	4	no	—
50	C₂²	C₂²×D₃₉₇	1	2	yes	—
51	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
52	C₄	—	397	4	no	—
53	C₄	C₂²×D₃₉₇	1	2	yes	—
54	C₄	C₂²×D₃₉₇	1	2	yes	—
55	C₂	—	2	2	no	—
56	C₂	—	794	4	no	—
57	C₂	#20263731	1	2	yes	direct product factor
58	C₂	C₂³×D₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₉₇
commutator, cosocle	C₂×C₃₉₇	C₂⁴
Frattini	C₂	C₂³×D₃₉₇
Fitting	C₂²×C₄×C₃₉₇	C₂
socle	C₂²×C₃₉₇	C₂³
hypercenter	C₂²×C₄	D₃₉₇
hypocenter	C₃₉₇	C₂×#41
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
1. #20263731	C₂	direct (N × H)
2. C₂×Dic₇₉₄	C₂	semidirect
3. C₂×D₁₅₈₈	C₂	semidirect
4. C₂×C₂²⋊D₃₉₇	C₂	semidirect
5. C₂×C₄×D₃₉₇	C₂	semidirect
6. C₂²×C₄×C₃₉₇	C₂	semidirect
8. Dic₇₉₄	C₂²	semidirect
9. D₁₅₈₈	C₂²	semidirect
10. C₂²⋊D₃₉₇	C₂²	semidirect
12. C₄×D₃₉₇	C₂²	semidirect
16. C₂×C₄×C₃₉₇	C₂²	semidirect
26. C₂×C₃₉₇	#41	semidirect
29. C₃₉₇	C₂×#41	semidirect
36. C₂²×C₄	D₃₉₇	semidirect
46. C₂×C₄	C₂×D₃₉₇	semidirect
57. C₂	#20263731	direct (N × H)

Maximal subgroups

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

subgroup	index
#20263731	2
C₂×Dic₇₉₄	2
C₂×D₁₅₈₈	2
C₂×C₂²⋊D₃₉₇	2
C₂×C₄×D₃₉₇	2
C₂²×C₄×C₃₉₇	2
C₂×#41	397

Subgroups — C2×#20263731 #27163139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20263731 #27163139