Subgroups — #27161203

Order 12608.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₃₉₄	C₂	1	1	yes	maximal, characteristic
2	#20263255	C₂	1	2	yes	maximal
3	C₄×#13096802	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₉₇	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₄	C₄	1	1	yes	characteristic
7	C₂×#13096802	C₂²	1	1	yes	characteristic
8	C₂×#13096802	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₉₇	C₄	1	1	yes	characteristic
10	#15691337	C₄	1	2	yes	—
11	C₄×Dic₁₉₇	C₂²	1	2	yes	—
12	C₂×C₈×C₁₉₇	C₂²	1	2	yes	—
13	C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
14	Dic₃₉₄	C₈	1	4	yes	complemented
15	C₂×Dic₁₉₇	C₂×C₄	1	2	yes	—
16	C₂×Dic₁₉₇	C₂×C₄	1	2	yes	—
17	#13096802	—	2	1	no	—
18	#13096802	Q₈	1	2	yes	complemented
19	C₂×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
22	C₈×C₁₉₇	—	2	2	no	—
23	C₂²×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₁₉₇	—	2	1	no	—
25	C₄×C₁₉₇	C₂×C₈	1	2	yes	—
26	C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
27	C₄×C₁₉₇	#41	1	1	yes	characteristic
28	Dic₁₉₇	—	2	2	no	—
29	Dic₁₉₇	C₂×C₈	1	4	yes	—
30	C₂×C₁₉₇	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₁₉₇	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₁₉₇	#131	1	1	yes	characteristic
33	C₁₉₇	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₈×Q₈	—	197	1	no	maximal, Sylow
35	C₄×Q₈	—	197	1	no	—
36	C₄⋊C₈	—	197	2	no	—
37	C₄⋊C₈	D₁₉₇	1	1	yes	characteristic
38	C₄×C₈	—	197	2	no	—
39	C₄×C₈	—	197	1	no	—
40	C₂×Q₈	—	197	1	no	—
41	C₂×C₈	—	197	1	no	—
42	C₂×C₈	—	197	1	no	—
43	C₂×C₈	C₂×D₁₉₇	1	2	yes	—
44	C₄⋊C₄	—	197	2	no	—
45	C₄⋊C₄	—	197	1	no	—
46	C₄²	—	197	2	no	—
47	C₄²	C₂×D₁₉₇	1	1	yes	characteristic
48	Q₈	—	197	4	no	—
49	C₂×C₄	—	197	2	no	—
50	C₂×C₄	—	197	2	no	—
51	C₂×C₄	C₄×D₁₉₇	1	1	yes	characteristic
52	C₂×C₄	C₄×D₁₉₇	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	394	1	no	—
56	C₈	—	197	2	no	—
57	C₂²	C₂×C₄×D₁₉₇	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	394	2	no	—
60	C₄	—	197	4	no	—
61	C₄	C₈×D₁₉₇	1	2	yes	—
62	C₄	Q₈×D₁₉₇	1	1	yes	characteristic
63	C₄	#15691357	1	1	yes	characteristic
64	C₂	#20263291	1	1	yes	characteristic
65	C₂	C₂×C₈×D₁₉₇	1	1	yes	characteristic
66	C₂	#20263334	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₉₇
commutator	C₂×C₁₉₇	C₂²×C₈
Fitting	C₄⋊C₈×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₂²×C₄
cosocle	C₂×C₄×C₁₉₇	C₂³
hypercenter	C₄⋊C₈	D₁₉₇
hypocenter	C₁₉₇	C₈×Q₈
LCS	G ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
14. Dic₃₉₄	C₈	semidirect
18. #13096802	Q₈	semidirect
33. C₁₉₇	C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₃₉₄	2
#20263255	2
C₄×#13096802	2
C₈×Dic₁₉₇	2
C₄⋊C₈×C₁₉₇	2
C₈×Q₈	197