Subgroups — #20261614

Order 6208.

98 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	C₂×C₄⋊#87604	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#87604	C₄	1	2	yes	—
5	#13095521	—	2	1	no	—
6	#13095520	C₄	1	2	yes	complemented
7	#13095519	—	2	1	no	—
8	C₄⋊#87604	—	2	1	no	—
9	C₄⋊#87604	—	2	1	no	—
10	C₂×C₄×D₉₇	C₂²	1	1	yes	characteristic
11	#13095495	C₄	1	1	yes	characteristic, complemented
12	#13095494	—	2	1	no	—
13	C₈×D₉₇	—	2	1	no	—
14	M₁₆×C₉₇	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂×#87604	—	2	2	no	—
17	C₂×#87604	—	4	2	no	—
18	C₂×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₉₇	C₂×C₄	1	1	yes	characteristic
20	C₄×D₉₇	D₄	1	1	yes	characteristic
21	C₄×D₉₇	D₄	1	1	yes	characteristic
22	C₄×D₉₇	D₄	1	1	yes	characteristic
23	C₄×D₉₇	Q₈	1	1	yes	characteristic
24	#11715897	—	2	2	no	—
25	C₈×C₉₇	—	2	1	no	—
26	#11715895	—	2	1	no	—
27	C₂²×C₉₇	C₄²	1	1	yes	characteristic
28	C₂×D₉₇	—	2	1	no	—
29	C₂×D₉₇	—	2	1	no	—
30	C₂×D₉₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×D₉₇	C₄⋊C₄	1	1	yes	characteristic
32	#87604	—	4	2	no	—
33	C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
35	Dic₉₇	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₉₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₉₇	—	2	1	no	—
38	C₂×C₉₇	#95	1	1	yes	characteristic
39	D₉₇	—	2	1	no	—
40	D₉₇	#100	1	1	yes	characteristic
41	D₉₇	#101	1	1	yes	characteristic
42	C₉₇	#343	1	1	yes	characteristic, complemented, Sylow
43	#343	—	97	1	no	maximal, Sylow
44	C₂×M₁₆	—	97	1	no	—
45	C₂×M₁₆	—	97	1	no	—
46	C₂×C₄⋊C₄	—	97	1	no	—
47	C₂²×C₄	—	97	1	no	—
48	C₂²×C₄	—	97	2	no	—
49	M₁₆	—	97	2	no	—
50	M₁₆	—	97	1	no	—
51	M₁₆	—	194	1	no	—
52	M₁₆	—	194	1	no	—
53	M₁₆	#87604	1	1	yes	characteristic, complemented
54	C₂×C₈	—	194	1	no	—
55	C₂×C₈	—	194	1	no	—
56	C₄⋊C₄	—	194	1	no	—
57	C₄⋊C₄	—	194	1	no	—
58	C₂³	—	97	1	no	—
59	C₂×C₄	—	97	1	no	—
60	C₂×C₄	—	97	1	no	—
61	C₂×C₄	—	97	1	no	—
62	C₂×C₄	—	97	1	no	—
63	C₂×C₄	—	97	1	no	—
64	C₂×C₄	—	388	2	no	—
65	C₂×C₄	—	194	2	no	—
66	C₂×C₄	C₂×#87604	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	194	2	no	—
69	C₈	—	194	1	no	—
70	C₂²	—	97	1	no	—
71	C₂²	—	97	1	no	—
72	C₂²	—	194	1	no	—
73	C₂²	—	194	1	no	—
74	C₂²	C₄×#87604	1	1	yes	characteristic
75	C₄	—	97	1	no	—
76	C₄	—	97	1	no	—
77	C₄	—	388	2	no	—
78	C₄	C₂²⋊#87604	1	1	yes	characteristic
79	C₄	C₄⋊#87604	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	97	1	no	—
82	C₂	—	97	1	no	—
83	C₂	—	194	1	no	—
84	C₂	(C₂×#87604)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#87604)⋊C₄
commutator	C₂²×C₉₇	C₄²
Frattini	C₂×C₄	C₂×#87604
Fitting	M₁₆×C₉₇	C₄
socle	C₂×C₉₇	#95
hypercenter	M₁₆	#87604
hypocenter	C₉₇	#343
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
6. #13095520	C₄	semidirect
11. #13095495	C₄	semidirect
14. M₁₆×C₉₇	C₄	semidirect
42. C₉₇	#343	semidirect
53. M₁₆	#87604	semidirect