Subgroups — #41×D₃₉₇ #27163137

Order 12704.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263732	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263731	C₂	1	3	yes	maximal, complemented
3	#20263730	C₂	1	3	yes	maximal, complemented
4	Q₈×D₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×D₃₉₇	C₂	1	3	yes	maximal, complemented
6	C₂×C₄×D₃₉₇	C₂	1	3	yes	maximal, complemented
7	#41×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₉₇	C₂²	1	3	yes	—
9	Dic₇₉₄	C₂²	1	3	yes	complemented
10	D₁₅₈₈	C₂²	1	3	yes	complemented
11	C₂²⋊D₃₉₇	C₂²	1	6	yes	complemented
12	C₂×Dic₃₉₇	C₂²	1	3	yes	complemented
13	C₄×D₃₉₇	C₂²	1	1	yes	characteristic
14	C₄×D₃₉₇	C₂²	1	3	yes	—
15	C₄×D₃₉₇	C₂²	1	3	yes	complemented
16	C₄×D₃₉₇	C₂²	1	3	yes	complemented
17	Q₈×C₃₉₇	C₂²	1	1	yes	characteristic, complemented
18	D₄×C₃₉₇	C₂²	1	3	yes	complemented
19	C₂×C₄×C₃₉₇	C₂²	1	3	yes	complemented
20	C₂²×C₃₉₇	C₂³	1	3	yes	—
21	C₂×D₃₉₇	—	2	6	no	—
22	C₂×D₃₉₇	C₂³	1	3	yes	—
23	C₂×D₃₉₇	C₂³	1	1	yes	characteristic
24	C₄×C₃₉₇	C₂³	1	3	yes	—
25	C₄×C₃₉₇	C₂³	1	1	yes	characteristic
26	Dic₃₉₇	C₂³	1	1	yes	characteristic
27	Dic₃₉₇	C₂³	1	3	yes	—
28	C₂×C₃₉₇	—	2	3	no	—
29	C₂×C₃₉₇	C₂⁴	1	1	yes	characteristic
30	D₃₉₇	—	2	3	no	—
31	D₃₉₇	#41	1	2	yes	direct product factor
32	C₃₉₇	C₂×#41	1	1	yes	characteristic, complemented, Sylow
33	C₂×#41	—	397	1	no	maximal, Sylow
34	#41	—	397	1	no	—
35	#41	—	397	3	no	—
36	#41	—	397	3	no	—
37	#41	D₃₉₇	1	1	yes	characteristic, direct product factor
38	C₂×Q₈	—	397	1	no	—
39	C₂×D₄	—	397	3	no	—
40	C₂²×C₄	—	397	3	no	—
41	C₂³	—	397	3	no	—
42	Q₈	—	397	3	no	—
43	Q₈	C₂×D₃₉₇	1	1	yes	characteristic, complemented
44	D₄	—	397	6	no	—
45	D₄	—	397	3	no	—
46	D₄	C₂×D₃₉₇	1	3	yes	complemented
47	C₂×C₄	—	397	3	no	—
48	C₂×C₄	—	397	3	no	—
49	C₂×C₄	—	397	3	no	—
50	C₂×C₄	—	397	1	no	—
51	C₂×C₄	—	397	3	no	—
52	C₂×C₄	C₂×D₃₉₇	1	3	yes	complemented
53	C₂²	—	794	6	no	—
54	C₂²	—	397	3	no	—
55	C₂²	—	397	1	no	—
56	C₂²	C₂²×D₃₉₇	1	3	yes	—
57	C₄	—	397	1	no	—
58	C₄	—	397	3	no	—
59	C₄	C₂²×D₃₉₇	1	3	yes	—
60	C₄	C₂²×D₃₉₇	1	1	yes	characteristic
61	C₂	—	2	3	no	—
62	C₂	—	794	3	no	—
63	C₂	—	397	2	no	—
64	C₂	C₂³×D₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₃₉₇
commutator, socle, cosocle	C₂×C₃₉₇	C₂⁴
Frattini	C₂	C₂³×D₃₉₇
Fitting	#41×C₃₉₇	C₂
hypercenter	#41	D₃₉₇
hypocenter	C₃₉₇	C₂×#41
LCS	G ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #20263732	C₂	semidirect
2. #20263731	C₂	semidirect
3. #20263730	C₂	semidirect
4. Q₈×D₃₉₇	C₂	semidirect
5. D₄×D₃₉₇	C₂	semidirect
6. C₂×C₄×D₃₉₇	C₂	semidirect
7. #41×C₃₉₇	C₂	semidirect
9. Dic₇₉₄	C₂²	semidirect
10. D₁₅₈₈	C₂²	semidirect
11. C₂²⋊D₃₉₇	C₂²	semidirect
12. C₂×Dic₃₉₇	C₂²	semidirect
15. C₄×D₃₉₇	C₂²	semidirect
16. C₄×D₃₉₇	C₂²	semidirect
17. Q₈×C₃₉₇	C₂²	semidirect
18. D₄×C₃₉₇	C₂²	semidirect
19. C₂×C₄×C₃₉₇	C₂²	semidirect
31. D₃₉₇	#41	direct (N × H)
32. C₃₉₇	C₂×#41	semidirect
37. #41	D₃₉₇	direct (N × H)
43. Q₈	C₂×D₃₉₇	semidirect
46. D₄	C₂×D₃₉₇	semidirect
52. C₂×C₄	C₂×D₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20263732	2
#20263731	2
#20263730	2
Q₈×D₃₉₇	2
D₄×D₃₉₇	2
C₂×C₄×D₃₉₇	2
#41×C₃₉₇	2
C₂×#41	397

Subgroups — #41×D397 #27163137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×D₃₉₇ #27163137