Subgroups — C₂×#20263715 #27163101

Order 12704.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263715	C₂	1	4	yes	maximal, direct product factor
2	C₂×#15691787	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic
4	#15691787	C₂²	1	4	yes	complemented
5	C₂³×C₃₉₇	C₄	1	1	yes	characteristic
6	C₂×C₄×C₃₉₇	C₄	1	2	yes	—
7	C₂×C₄×C₃₉₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₃₉₇	C₂²	1	2	yes	—
9	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
10	C₂²×C₃₉₇	—	2	2	no	—
11	C₂²×C₃₉₇	C₂×C₄	1	2	yes	—
12	C₂²×C₃₉₇	C₂×C₄	1	1	yes	characteristic
13	C₄×C₃₉₇	C₂×C₄	1	2	yes	—
14	C₄×C₃₉₇	C₂×C₄	1	1	yes	characteristic
15	C₄×C₃₉₇	C₂³	1	1	yes	characteristic
16	C₂×C₃₉₇	—	2	2	no	—
17	C₂×C₃₉₇	M₁₆	1	2	yes	complemented
18	C₂×C₃₉₇	C₂²×C₄	1	1	yes	characteristic
19	C₃₉₇	C₂×M₁₆	1	1	yes	characteristic, complemented, Sylow
20	C₂×M₁₆	—	397	1	no	maximal, Sylow
21	C₂²×C₄	D₃₉₇	1	1	yes	characteristic
22	M₁₆	—	397	4	no	—
23	C₂×C₈	—	397	2	no	—
24	C₂³	Dic₃₉₇	1	1	yes	characteristic
25	C₂×C₄	Dic₃₉₇	1	1	yes	characteristic
26	C₂×C₄	Dic₃₉₇	1	2	yes	—
27	C₂×C₄	C₂×D₃₉₇	1	2	yes	—
28	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
29	C₈	—	397	4	no	—
30	C₂²	—	2	2	no	—
31	C₂²	C₂×Dic₃₉₇	1	2	yes	—
32	C₂²	C₂×Dic₃₉₇	1	1	yes	characteristic
33	C₄	C₂×Dic₃₉₇	1	2	yes	—
34	C₄	C₂×Dic₃₉₇	1	1	yes	characteristic
35	C₄	C₂²×D₃₉₇	1	1	yes	characteristic
36	C₂	—	2	2	no	—
37	C₂	#20263715	1	2	yes	direct product factor
38	C₂	C₂²×Dic₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₉₇
commutator	C₂×C₃₉₇	C₂²×C₄
Frattini	C₄	C₂²×D₃₉₇
Fitting	C₂²×C₄×C₃₉₇	C₂
socle	C₂²×C₃₉₇	C₂×C₄
prime radical	C₂³×C₃₉₇	C₄
cosocle	C₄×C₃₉₇	C₂³
hypercenter	C₂²×C₄	D₃₉₇
hypocenter	C₃₉₇	C₂×M₁₆
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. #20263715	C₂	direct (N × H)
2. C₂×#15691787	C₂	semidirect
4. #15691787	C₂²	semidirect
17. C₂×C₃₉₇	M₁₆	semidirect
19. C₃₉₇	C₂×M₁₆	semidirect
37. C₂	#20263715	direct (N × H)

Maximal subgroups

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

subgroup	index
#20263715	2
C₂×#15691787	2
C₂²×C₄×C₃₉₇	2
C₂×M₁₆	397

Subgroups — C2×#20263715 #27163101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20263715 #27163101