Subgroups — #27161173

Order 12608.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₈₈	C₂	1	1	yes	maximal, characteristic
2	#20263291	C₂	1	2	yes	maximal
3	#20263249	C₂	1	2	yes	maximal
4	C₈×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
5	#107×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₄	C₂²	1	2	yes	—
7	Dic₇₈₈	C₄	1	4	yes	complemented
8	C₂×#13096802	C₂²	1	1	yes	characteristic
9	#15691337	—	2	2	no	—
10	C₄×Dic₁₉₇	—	2	2	no	—
11	C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
14	Dic₃₉₄	—	2	2	no	—
15	Dic₃₉₄	C₂×C₄	1	4	yes	—
16	C₂×Dic₁₉₇	—	2	2	no	—
17	C₂×Dic₁₉₇	D₄	1	1	yes	characteristic
18	C₂×Dic₁₉₇	D₄	1	1	yes	characteristic
19	#13096802	—	2	1	no	—
20	C₂×C₄×C₁₉₇	—	2	2	no	—
21	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
22	C₈×C₁₉₇	C₂×C₄	1	2	yes	—
23	C₂²×C₁₉₇	C₂×D₄	1	1	yes	characteristic
24	C₄×C₁₉₇	—	4	2	no	—
25	C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₁₉₇	#41	1	1	yes	characteristic
27	Dic₁₉₇	—	2	4	no	—
28	Dic₁₉₇	—	2	1	no	—
29	Dic₁₉₇	Q₁₆	1	2	yes	complemented
30	C₂×C₁₉₇	C₄×D₄	1	1	yes	characteristic
31	C₂×C₁₉₇	C₂×Q₁₆	1	1	yes	characteristic
32	C₂×C₁₉₇	#135	1	1	yes	characteristic
33	C₁₉₇	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×Q₁₆	—	197	1	no	maximal, Sylow
35	C₂×Q₁₆	—	197	1	no	—
36	C₄×Q₈	—	197	2	no	—
37	#107	D₁₉₇	1	1	yes	characteristic
38	#103	—	197	2	no	—
39	C₄×C₈	—	197	1	no	—
40	C₂×Q₈	—	197	2	no	—
41	Q₁₆	—	197	4	no	—
42	C₂×C₈	—	197	1	no	—
43	C₂×C₈	C₂×D₁₉₇	1	1	yes	characteristic
44	C₄⋊C₄	—	394	2	no	—
45	C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
46	C₄²	—	394	2	no	—
47	C₄²	—	197	1	no	—
48	Q₈	—	394	2	no	—
49	Q₈	—	197	4	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	394	2	no	—
52	C₂×C₄	—	197	1	no	—
53	C₂×C₄	—	197	1	no	—
54	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
55	C₈	—	394	1	no	—
56	C₈	C₄×D₁₉₇	1	2	yes	—
57	C₂²	D₄×D₁₉₇	1	1	yes	characteristic
58	C₄	—	4	2	no	—
59	C₄	—	394	4	no	—
60	C₄	—	394	1	no	—
61	C₄	—	197	2	no	—
62	C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
63	C₄	#15691359	1	1	yes	characteristic
64	C₂	#20263288	1	1	yes	characteristic
65	C₂	Q₁₆×D₁₉₇	1	1	yes	characteristic
66	C₂	#20263345	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₇
commutator	C₄×C₁₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₇
Fitting	#107×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₂×D₄
cosocle	C₂×C₄×C₁₉₇	C₂³
hypercenter	#107	D₁₉₇
hypocenter	C₁₉₇	C₄×Q₁₆
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
7. Dic₇₈₈	C₄	semidirect
29. Dic₁₉₇	Q₁₆	semidirect
33. C₁₉₇	C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₈₈	2
#20263291	2
#20263249	2
C₈×Dic₁₉₇	2
#107×C₁₉₇	2
C₄×Q₁₆	197