Subgroups — C₂×#20263714 #27163100

Order 12704.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃₉₇	C₂	1	1	yes	maximal, characteristic
2	#20263714	C₂	1	4	yes	maximal, direct product factor
3	C₂×#15691787	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₉₇	C₄	1	1	yes	characteristic
6	C₂×Dic₃₉₇	C₄	1	1	yes	characteristic
7	C₄×D₃₉₇	C₄	1	2	yes	—
8	C₄×D₃₉₇	C₂²	1	2	yes	—
9	#15691787	C₂²	1	2	yes	complemented
10	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
11	C₈×C₃₉₇	C₂²	1	2	yes	complemented
12	C₂²×C₃₉₇	C₂×C₄	1	1	yes	characteristic
13	C₂×D₃₉₇	—	2	2	no	—
14	C₂×D₃₉₇	C₂×C₄	1	2	yes	—
15	C₄×C₃₉₇	C₂×C₄	1	1	yes	characteristic
16	C₄×C₃₉₇	C₂³	1	1	yes	characteristic
17	Dic₃₉₇	C₂×C₄	1	2	yes	—
18	C₂×C₃₉₇	M₁₆	1	2	yes	complemented
19	C₂×C₃₉₇	C₂²×C₄	1	1	yes	characteristic
20	D₃₉₇	—	2	2	no	—
21	C₃₉₇	C₂×M₁₆	1	1	yes	characteristic, complemented, Sylow
22	C₂×M₁₆	—	397	1	no	maximal, Sylow
23	C₂²×C₄	—	397	1	no	—
24	M₁₆	—	397	4	no	—
25	C₂×C₈	—	397	1	no	—
26	C₂×C₈	D₃₉₇	1	1	yes	characteristic, complemented
27	C₂³	—	397	1	no	—
28	C₂×C₄	—	397	1	no	—
29	C₂×C₄	—	397	2	no	—
30	C₂×C₄	—	397	2	no	—
31	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
32	C₈	—	397	2	no	—
33	C₈	C₂×D₃₉₇	1	2	yes	complemented
34	C₂²	—	794	2	no	—
35	C₂²	—	397	2	no	—
36	C₂²	C₄×D₃₉₇	1	1	yes	characteristic
37	C₄	—	397	2	no	—
38	C₄	C₄×D₃₉₇	1	1	yes	characteristic
39	C₄	C₂²×D₃₉₇	1	1	yes	characteristic
40	C₂	—	794	2	no	—
41	C₂	#20263714	1	2	yes	direct product factor
42	C₂	C₂×C₄×D₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₉₇
commutator	C₂×C₃₉₇	C₂²×C₄
Frattini	C₄	C₂²×D₃₉₇
Fitting	C₂×C₈×C₃₉₇	C₂
socle	C₂²×C₃₉₇	C₂×C₄
prime radical	C₂²×D₃₉₇	C₄
cosocle	C₄×C₃₉₇	C₂³
hypercenter	C₂×C₈	D₃₉₇
hypocenter	C₃₉₇	C₂×M₁₆
LCS	G ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. #20263714	C₂	direct (N × H)
3. C₂×#15691787	C₂	semidirect
4. C₂×C₈×C₃₉₇	C₂	semidirect
9. #15691787	C₂²	semidirect
11. C₈×C₃₉₇	C₂²	semidirect
18. C₂×C₃₉₇	M₁₆	semidirect
21. C₃₉₇	C₂×M₁₆	semidirect
26. C₂×C₈	D₃₉₇	semidirect
33. C₈	C₂×D₃₉₇	semidirect
41. C₂	#20263714	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×C₄×D₃₉₇	2
#20263714	2
C₂×#15691787	2
C₂×C₈×C₃₉₇	2
C₂×M₁₆	397

Subgroups — C2×#20263714 #27163100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20263714 #27163100