Subgroups — C₂×C₄⋊#13097544 #27163174

Order 12704.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13097544	C₂	1	2	yes	maximal
2	C₄⋊#13097544	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×D₃₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13097544	—	2	4	no	—
5	C₂×#13097544	C₂²	1	4	yes	—
6	C₂²×D₃₉₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₃₉₇	C₄	1	1	yes	characteristic, complemented
8	C₄×D₃₉₇	C₄	1	2	yes	complemented
9	C₄×D₃₉₇	C₂²	1	2	yes	—
10	C₂×C₄×C₃₉₇	C₄	1	1	yes	characteristic, complemented
11	C₂²×C₃₉₇	C₂×C₄	1	1	yes	characteristic
12	C₂×D₃₉₇	C₂×C₄	1	1	yes	characteristic
13	C₂×D₃₉₇	D₄	1	2	yes	—
14	C₂×D₃₉₇	Q₈	1	2	yes	—
15	C₂×D₃₉₇	C₂³	1	1	yes	characteristic
16	#13097544	—	2	4	no	—
17	C₄×C₃₉₇	C₂×C₄	1	2	yes	complemented
18	Dic₃₉₇	C₂×C₄	1	2	yes	complemented
19	C₂×C₃₉₇	C₄⋊C₄	1	2	yes	complemented
20	C₂×C₃₉₇	C₂²×C₄	1	1	yes	characteristic
21	D₃₉₇	C₄⋊C₄	1	2	yes	complemented
22	D₃₉₇	C₂×D₄	1	1	yes	characteristic
23	D₃₉₇	C₂×Q₈	1	1	yes	characteristic
24	C₃₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₄⋊C₄	—	397	1	no	maximal, Sylow
26	C₂²×C₄	—	397	2	no	—
27	C₂²×C₄	—	397	1	no	—
28	C₄⋊C₄	—	397	4	no	—
29	C₂³	—	397	1	no	—
30	C₂×C₄	—	794	4	no	—
31	C₂×C₄	—	397	4	no	—
32	C₂×C₄	—	397	1	no	—
33	C₂×C₄	—	397	2	no	—
34	C₂×C₄	—	397	2	no	—
35	C₂×C₄	#13097544	1	1	yes	characteristic, complemented
36	C₂²	—	397	2	no	—
37	C₂²	—	397	2	no	—
38	C₂²	—	397	1	no	—
39	C₂²	—	397	1	no	—
40	C₂²	C₂×#13097544	1	1	yes	characteristic
41	C₄	—	794	4	no	—
42	C₄	—	397	2	no	—
43	C₄	C₂×#13097544	1	2	yes	complemented
44	C₂	—	397	1	no	—
45	C₂	—	397	2	no	—
46	C₂	—	397	1	no	—
47	C₂	C₄⋊#13097544	1	2	yes	direct product factor
48	C₂	C₂²×#13097544	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13097544
commutator	C₂×C₃₉₇	C₂²×C₄
Frattini	C₂	C₂²×#13097544
Fitting	C₂×C₄×C₃₉₇	C₄
socle	C₂²×C₃₉₇	C₂×C₄
prime radical	C₂²×D₃₉₇	C₂²
cosocle	C₂×D₃₉₇	C₂³
hypercenter	C₂×C₄	#13097544
hypocenter	C₃₉₇	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₄⋊#13097544	C₂	direct (N × H)
7. C₂×Dic₃₉₇	C₄	semidirect
8. C₄×D₃₉₇	C₄	semidirect
10. C₂×C₄×C₃₉₇	C₄	semidirect
17. C₄×C₃₉₇	C₂×C₄	semidirect
18. Dic₃₉₇	C₂×C₄	semidirect
19. C₂×C₃₉₇	C₄⋊C₄	semidirect
21. D₃₉₇	C₄⋊C₄	semidirect
24. C₃₉₇	C₂×C₄⋊C₄	semidirect
35. C₂×C₄	#13097544	semidirect
43. C₄	C₂×#13097544	semidirect
47. C₂	C₄⋊#13097544	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13097544	2
C₄⋊#13097544	2
C₂×C₄×D₃₉₇	2
C₂×C₄⋊C₄	397

Subgroups — C2×C4⋊#13097544 #27163174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊#13097544 #27163174