Subgroups — #20261617

Order 6208.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₉₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	#15690889	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095494	C₂	1	1	yes	maximal, characteristic
4	#13095521	—	2	1	no	—
5	#13095520	C₄	1	2	yes	complemented
6	#13095519	—	2	1	no	—
7	C₄⋊#87604	—	2	1	no	—
8	C₂²⋊#87604	C₄	1	2	yes	—
9	C₄×#87604	—	2	1	no	—
10	C₂×C₄×D₉₇	C₂²	1	1	yes	characteristic
11	#13095494	—	2	1	no	—
12	#13095494	—	2	1	no	—
13	C₂×#11715895	C₄	1	1	yes	characteristic, complemented
14	C₂×C₈×C₉₇	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂×#87604	—	2	2	no	—
17	C₂×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₉₇	C₂×C₄	1	1	yes	characteristic
19	C₄×D₉₇	D₄	1	1	yes	characteristic
20	C₄×D₉₇	D₄	1	1	yes	characteristic
21	C₄×D₉₇	D₄	1	1	yes	characteristic
22	C₄×D₉₇	Q₈	1	1	yes	characteristic
23	#11715897	—	2	2	no	—
24	C₈×C₉₇	—	2	1	no	—
25	#11715895	—	2	1	no	—
26	C₂²×C₉₇	C₄²	1	1	yes	characteristic
27	C₂×D₉₇	—	4	1	no	—
28	C₂×D₉₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₉₇	C₄⋊C₄	1	1	yes	characteristic
30	#87604	—	4	2	no	—
31	C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
33	Dic₉₇	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₉₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₉₇	—	2	1	no	—
36	C₂×C₉₇	#95	1	1	yes	characteristic
37	D₉₇	—	2	1	no	—
38	D₉₇	—	2	1	no	—
39	C₉₇	#344	1	1	yes	characteristic, complemented, Sylow
40	#344	—	97	1	no	maximal, Sylow
41	C₂×M₁₆	—	97	1	no	—
42	C₂×M₁₆	—	97	1	no	—
43	#117	—	97	1	no	—
44	C₂²×C₄	—	97	1	no	—
45	M₁₆	—	97	2	no	—
46	M₁₆	—	194	1	no	—
47	M₁₆	—	194	1	no	—
48	M₁₆	—	194	1	no	—
49	C₂×C₈	—	97	1	no	—
50	C₂×C₈	—	194	1	no	—
51	C₂×C₈	#87604	1	1	yes	characteristic, complemented
52	C₄⋊C₄	—	194	1	no	—
53	C₂²⋊C₄	—	97	2	no	—
54	C₄²	—	194	1	no	—
55	C₂³	—	97	1	no	—
56	C₂×C₄	—	97	1	no	—
57	C₂×C₄	—	97	1	no	—
58	C₂×C₄	—	97	1	no	—
59	C₂×C₄	—	97	1	no	—
60	C₂×C₄	—	97	1	no	—
61	C₂×C₄	—	194	2	no	—
62	C₂×C₄	C₂×#87604	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	194	2	no	—
65	C₈	—	194	1	no	—
66	C₂²	—	97	1	no	—
67	C₂²	—	97	1	no	—
68	C₂²	—	388	1	no	—
69	C₂²	C₄×#87604	1	1	yes	characteristic
70	C₄	—	97	1	no	—
71	C₄	—	97	1	no	—
72	C₄	—	388	2	no	—
73	C₄	C₂²⋊#87604	1	1	yes	characteristic
74	C₄	C₄⋊#87604	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	194	1	no	—
77	C₂	—	194	1	no	—
78	C₂	(C₂×#87604)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#87604
commutator	C₂²×C₉₇	C₄²
Frattini	C₂×C₄	C₂×#87604
Fitting	C₂×C₈×C₉₇	C₄
socle	C₂×C₉₇	#95
hypercenter	C₂×C₈	#87604
hypocenter	C₉₇	#344
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. #13095520	C₄	semidirect
13. C₂×#11715895	C₄	semidirect
14. C₂×C₈×C₉₇	C₄	semidirect
39. C₉₇	#344	semidirect
51. C₂×C₈	#87604	semidirect