Subgroups — #27161429

Order 12608.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263326	C₂	1	1	yes	maximal, characteristic
2	#20263250	C₂	1	4	yes	maximal
3	C₄×#13096802	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₉₄	—	2	1	no	—
6	C₂×#13096802	C₂²	1	2	yes	—
7	C₄⋊Dic₁₉₇	—	2	1	no	—
8	C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₉₇	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₉₇	—	2	1	no	—
11	C₄⋊C₄×C₁₉₇	—	2	1	no	—
12	C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
13	Dic₃₉₄	—	4	2	no	—
14	C₂×Dic₁₉₇	—	2	2	no	—
15	#13096802	—	2	2	no	—
16	Q₈×C₁₉₇	—	2	4	no	—
17	C₂×C₄×C₁₉₇	—	2	2	no	—
18	C₂×C₄×C₁₉₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
21	C₂²×C₁₉₇	C₂×D₄	1	1	yes	characteristic
22	C₄×C₁₉₇	—	4	2	no	—
23	C₄×C₁₉₇	SD₁₆	1	2	yes	—
24	C₄×C₁₉₇	Q₁₆	1	2	yes	—
25	C₄×C₁₉₇	#41	1	2	yes	—
26	Dic₁₉₇	—	4	2	no	—
27	C₂×C₁₉₇	#124	1	1	yes	characteristic
28	C₂×C₁₉₇	C₂×SD₁₆	1	1	yes	characteristic
29	C₂×C₁₉₇	C₂×Q₁₆	1	1	yes	characteristic
30	C₁₉₇	#487	1	1	yes	characteristic, complemented, Sylow
31	#487	—	197	1	no	maximal, Sylow
32	C₄⋊Q₈	—	197	1	no	—
33	C₄⋊Q₈	D₁₉₇	1	1	yes	characteristic
34	#103	—	197	4	no	—
35	C₄×C₈	—	197	1	no	—
36	C₂×Q₈	—	394	1	no	—
37	C₂×Q₈	C₂×D₁₉₇	1	2	yes	—
38	C₂×C₈	—	197	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₄⋊C₄	—	394	1	no	—
42	C₄⋊C₄	—	197	2	no	—
43	C₄²	C₂×D₁₉₇	1	1	yes	characteristic
44	Q₈	—	2	4	no	—
45	Q₈	—	788	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	394	2	no	—
48	C₂×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
51	C₈	—	394	2	no	—
52	C₂²	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	788	2	no	—
55	C₄	#15691345	1	2	yes	—
56	C₄	#15691348	1	2	yes	—
57	C₄	#15691357	1	2	yes	—
58	C₂	#20263313	1	1	yes	characteristic
59	C₂	C₂×#15691345	1	1	yes	characteristic
60	C₂	C₂×#15691348	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₇
Fitting	C₄⋊Q₈×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₉₇
hypocenter	C₁₉₇	#487
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
30. C₁₉₇	#487	semidirect

Maximal subgroups

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

subgroup	index
#20263326	2
#20263250	2
C₄×#13096802	2
C₄⋊Q₈×C₁₉₇	2
#487	197