Subgroups — C₂×C₄×#13097544 #27163172

Order 12704.

108 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	C₄	1	8	yes	direct product factor
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	C₂×C₄	1	4	yes	—
14	C₂×D₃₉₇	C₂³	1	1	yes	characteristic
15	#13097544	C₂×C₄	1	8	yes	direct product factor
16	C₄×C₃₉₇	C₂×C₄	1	2	yes	complemented
17	Dic₃₉₇	C₂×C₄	1	2	yes	complemented
18	C₂×C₃₉₇	C₄²	1	2	yes	complemented
19	C₂×C₃₉₇	C₂²×C₄	1	1	yes	characteristic
20	D₃₉₇	C₄²	1	2	yes	complemented
21	D₃₉₇	C₂²×C₄	1	2	yes	—
22	C₃₉₇	C₂×C₄²	1	1	yes	characteristic, complemented, Sylow
23	C₂×C₄²	—	397	1	no	maximal, Sylow
24	C₂²×C₄	—	397	2	no	—
25	C₂²×C₄	—	397	1	no	—
26	C₄²	—	397	4	no	—
27	C₂³	—	397	1	no	—
28	C₂×C₄	—	397	8	no	—
29	C₂×C₄	—	397	4	no	—
30	C₂×C₄	—	397	1	no	—
31	C₂×C₄	—	397	2	no	—
32	C₂×C₄	—	397	2	no	—
33	C₂×C₄	#13097544	1	1	yes	characteristic, direct product factor
34	C₂²	—	397	4	no	—
35	C₂²	—	397	1	no	—
36	C₂²	—	397	1	no	—
37	C₂²	C₂×#13097544	1	1	yes	characteristic
38	C₄	—	397	8	no	—
39	C₄	—	397	2	no	—
40	C₄	C₂×#13097544	1	2	yes	direct product factor
41	C₂	—	397	2	no	—
42	C₂	—	397	2	no	—
43	C₂	C₄×#13097544	1	2	yes	direct product factor
44	C₂	C₂²×#13097544	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	#13097544
commutator, hypocenter	C₃₉₇	C₂×C₄²
Frattini	C₂	C₂²×#13097544
Fitting	C₂×C₄×C₃₉₇	C₄
socle	C₂²×C₃₉₇	C₂×C₄
prime radical	C₂²×D₃₉₇	C₂²
cosocle	C₂×D₃₉₇	C₂³
LCS	G ▷ C₃₉₇
UCS	C₁ ◁ 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)
4. C₂×#13097544	C₄	direct (N × H)
7. C₂×Dic₃₉₇	C₄	semidirect
8. C₄×D₃₉₇	C₄	semidirect
10. C₂×C₄×C₃₉₇	C₄	semidirect
15. #13097544	C₂×C₄	direct (N × H)
16. C₄×C₃₉₇	C₂×C₄	semidirect
17. Dic₃₉₇	C₂×C₄	semidirect
18. C₂×C₃₉₇	C₄²	semidirect
20. D₃₉₇	C₄²	semidirect
22. C₃₉₇	C₂×C₄²	semidirect
33. C₂×C₄	#13097544	direct (N × H)
40. C₄	C₂×#13097544	direct (N × H)
43. 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×C₄²	397

Subgroups — C2×C4×#13097544 #27163172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13097544 #27163172