Subgroups — #20261114

Order 6208.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13095501	C₂	1	4	yes	maximal
2	#15690810	C₂	1	1	yes	maximal, characteristic
3	C₄×#11715895	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₁₉₄	C₂²	1	2	yes	—
6	#13095501	—	2	8	no	—
7	C₂×#11715895	C₂²	1	2	yes	—
8	C₄⋊Dic₉₇	—	2	2	no	—
9	C₂×Q₈×C₉₇	C₂²	1	2	yes	—
10	C₄⋊C₄×C₉₇	—	2	2	no	—
11	C₄²×C₉₇	C₂²	1	1	yes	characteristic
12	Q₈×C₉₇	—	2	4	no	—
13	C₂×C₄×C₉₇	—	2	2	no	—
14	C₂×C₄×C₉₇	D₄	1	2	yes	—
15	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
16	C₂×Dic₉₇	—	2	2	no	—
17	Dic₁₉₄	—	2	4	no	—
18	#11715895	D₄	1	4	yes	—
19	C₂²×C₉₇	C₂×D₄	1	1	yes	characteristic
20	C₄×C₉₇	—	4	2	no	—
21	C₄×C₉₇	Q₁₆	1	4	yes	complemented
22	C₄×C₉₇	C₂×D₄	1	2	yes	—
23	Dic₉₇	—	4	2	no	—
24	C₂×C₉₇	#127	1	1	yes	characteristic
25	C₂×C₉₇	C₂×Q₁₆	1	2	yes	—
26	C₉₇	#494	1	1	yes	characteristic, complemented, Sylow
27	#494	—	97	1	no	maximal, Sylow
28	C₂×Q₁₆	—	97	4	no	—
29	C₄⋊Q₈	—	97	1	no	—
30	C₄⋊Q₈	D₉₇	1	1	yes	characteristic
31	C₄×C₈	—	97	1	no	—
32	C₂×Q₈	—	97	2	no	—
33	C₂×Q₈	C₂×D₉₇	1	2	yes	—
34	Q₁₆	—	194	8	no	—
35	C₂×C₈	—	97	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	194	2	no	—
38	C₄²	C₂×D₉₇	1	1	yes	characteristic
39	Q₈	—	2	4	no	—
40	Q₈	—	194	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	194	2	no	—
43	C₂×C₄	C₂²⋊D₉₇	1	2	yes	—
44	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
45	C₈	—	97	4	no	—
46	C₂²	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	388	2	no	—
49	C₄	#13095501	1	4	yes	complemented
50	C₄	D₄×D₉₇	1	2	yes	—
51	C₂	#15690806	1	1	yes	characteristic
52	C₂	C₂×#13095501	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₉₇
Fitting	C₄⋊Q₈×C₉₇	C₂
socle	C₂²×C₉₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₉₇
hypocenter	C₉₇	#494
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
21. C₄×C₉₇	Q₁₆	semidirect
26. C₉₇	#494	semidirect
49. C₄	#13095501	semidirect