Subgroups — C₂×#20263718 #27163109

Order 12704.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₈₈	C₂	1	1	yes	maximal, characteristic
2	#20263718	C₂	1	4	yes	maximal, direct product factor
3	C₂×#15691787	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Q₈×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₉₇	—	2	1	no	—
6	D₁₅₈₈	—	2	1	no	—
7	D₁₅₈₈	C₂²	1	2	yes	—
8	#15691787	C₂²	1	2	yes	complemented
9	Q₈×C₃₉₇	—	2	1	no	—
10	Q₈×C₃₉₇	C₂²	1	2	yes	complemented
11	C₂×C₄×C₃₉₇	—	2	1	no	—
12	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
13	C₂²×C₃₉₇	D₄	1	1	yes	characteristic
14	C₂×D₃₉₇	—	2	2	no	—
15	C₂×D₃₉₇	—	4	2	no	—
16	C₄×C₃₉₇	—	2	2	no	—
17	C₄×C₃₉₇	D₄	1	1	yes	characteristic
18	C₄×C₃₉₇	C₂³	1	1	yes	characteristic
19	C₂×C₃₉₇	SD₁₆	1	2	yes	complemented
20	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
21	D₃₉₇	—	4	2	no	—
22	C₃₉₇	C₂×SD₁₆	1	1	yes	characteristic, complemented, Sylow
23	C₂×SD₁₆	—	397	1	no	maximal, Sylow
24	C₂×Q₈	D₃₉₇	1	1	yes	characteristic, complemented
25	C₂×D₄	—	397	1	no	—
26	SD₁₆	—	397	4	no	—
27	C₂×C₈	—	397	1	no	—
28	C₂³	—	794	1	no	—
29	Q₈	—	2	1	no	—
30	Q₈	C₂×D₃₉₇	1	2	yes	complemented
31	D₄	—	794	1	no	—
32	D₄	—	397	2	no	—
33	C₂×C₄	—	2	1	no	—
34	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
35	C₈	—	397	2	no	—
36	C₂²	—	794	2	no	—
37	C₂²	—	1588	2	no	—
38	C₂²	C₂²⋊D₃₉₇	1	1	yes	characteristic
39	C₄	—	2	2	no	—
40	C₄	C₂²⋊D₃₉₇	1	1	yes	characteristic
41	C₄	C₂²×D₃₉₇	1	1	yes	characteristic
42	C₂	—	1588	2	no	—
43	C₂	#20263718	1	2	yes	direct product factor
44	C₂	C₂×C₂²⋊D₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊D₃₉₇
commutator, cosocle	C₄×C₃₉₇	C₂³
Frattini	C₄	C₂²×D₃₉₇
Fitting	C₂×Q₈×C₃₉₇	C₂
socle	C₂²×C₃₉₇	D₄
prime radical	C₂×D₁₅₈₈	C₂
hypercenter	C₂×Q₈	D₃₉₇
hypocenter	C₃₉₇	C₂×SD₁₆
LCS	G ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
2. #20263718	C₂	direct (N × H)
3. C₂×#15691787	C₂	semidirect
4. C₂×Q₈×C₃₉₇	C₂	semidirect
8. #15691787	C₂²	semidirect
10. Q₈×C₃₉₇	C₂²	semidirect
19. C₂×C₃₉₇	SD₁₆	semidirect
22. C₃₉₇	C₂×SD₁₆	semidirect
24. C₂×Q₈	D₃₉₇	semidirect
30. Q₈	C₂×D₃₉₇	semidirect
43. C₂	#20263718	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×D₁₅₈₈	2
#20263718	2
C₂×#15691787	2
C₂×Q₈×C₃₉₇	2
C₂×SD₁₆	397

Subgroups — C2×#20263718 #27163109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20263718 #27163109