Subgroups — #27163121

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, complemented
2	#20263718	C₂	1	2	yes	maximal, complemented
3	#20263716	C₂	1	2	yes	maximal, complemented
4	#20263715	C₂	1	1	yes	maximal, characteristic, complemented
5	#41×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₉₇	—	2	1	no	—
7	D₁₅₈₈	—	2	1	no	—
8	D₁₅₈₈	C₂²	1	2	yes	—
9	#15691787	C₂²	1	2	yes	complemented
10	Q₈×C₃₉₇	C₂²	1	1	yes	characteristic, complemented
11	D₄×C₃₉₇	—	2	1	no	—
12	D₄×C₃₉₇	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₄×C₃₉₇	—	2	1	no	—
14	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
15	C₂²×C₃₉₇	—	2	1	no	—
16	C₂²×C₃₉₇	D₄	1	1	yes	characteristic
17	C₂×D₃₉₇	—	2	2	no	—
18	C₂×D₃₉₇	—	4	2	no	—
19	C₄×C₃₉₇	—	2	1	no	—
20	C₄×C₃₉₇	D₄	1	1	yes	characteristic
21	C₄×C₃₉₇	C₂³	1	1	yes	characteristic
22	C₂×C₃₉₇	—	2	1	no	—
23	C₂×C₃₉₇	—	4	1	no	—
24	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
25	D₃₉₇	—	4	2	no	—
26	C₃₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	397	1	no	maximal, Sylow
28	#41	D₃₉₇	1	1	yes	characteristic, complemented
29	C₂×D₄	—	397	1	no	—
30	SD₁₆	—	397	2	no	—
31	D₈	—	397	2	no	—
32	M₁₆	—	397	1	no	—
33	C₂³	—	794	1	no	—
34	Q₈	C₂×D₃₉₇	1	1	yes	characteristic, complemented
35	D₄	—	2	1	no	—
36	D₄	—	794	1	no	—
37	D₄	—	397	2	no	—
38	D₄	C₂×D₃₉₇	1	1	yes	characteristic, complemented
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
41	C₈	—	397	2	no	—
42	C₂²	—	2	1	no	—
43	C₂²	—	794	2	no	—
44	C₂²	—	1588	2	no	—
45	C₂²	C₂²⋊D₃₉₇	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	C₂²⋊D₃₉₇	1	1	yes	characteristic
48	C₄	C₂²×D₃₉₇	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	4	1	no	—
51	C₂	—	1588	2	no	—
52	C₂	C₂×C₂²⋊D₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₃₉₇
commutator, cosocle	C₄×C₃₉₇	C₂³
Frattini	C₄	C₂²×D₃₉₇
Fitting	#41×C₃₉₇	C₂
socle	C₂×C₃₉₇	C₂×D₄
hypercenter	#41	D₃₉₇
hypocenter	C₃₉₇	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₅₈₈	C₂	semidirect
2. #20263718	C₂	semidirect
3. #20263716	C₂	semidirect
4. #20263715	C₂	semidirect
5. #41×C₃₉₇	C₂	semidirect
9. #15691787	C₂²	semidirect
10. Q₈×C₃₉₇	C₂²	semidirect
12. D₄×C₃₉₇	C₂²	semidirect
26. C₃₉₇	AGL(1,ℤ/8)	semidirect
28. #41	D₃₉₇	semidirect
34. Q₈	C₂×D₃₉₇	semidirect
38. D₄	C₂×D₃₉₇	semidirect

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
#20263716	2
#20263715	2
#41×C₃₉₇	2
AGL(1,ℤ/8)	397