Subgroups — #27156885

Order 12416.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261666	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261661	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₆×D₉₇	C₂	1	1	yes	maximal, characteristic
4	#15690879	—	2	1	no	—
5	#15690860	—	2	1	no	—
6	D₈×D₉₇	—	2	1	no	—
7	D₁₅₅₂	C₄	1	1	yes	characteristic, complemented
8	#15690751	—	2	1	no	—
9	C₁₆×D₉₇	C₂²	1	1	yes	characteristic
10	D₁₆×C₉₇	C₄	1	1	yes	characteristic, complemented
11	C₄⋊#87604	—	4	1	no	—
12	D₄×D₉₇	—	4	1	no	—
13	D₇₇₆	—	2	1	no	—
14	#13095496	—	4	1	no	—
15	C₈×D₉₇	D₄	1	1	yes	characteristic
16	#13095486	D₄	1	1	yes	characteristic
17	D₈×C₉₇	—	2	1	no	—
18	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×D₉₇	—	8	1	no	—
20	C₂×#87604	—	8	1	no	—
21	D₄×C₉₇	—	4	1	no	—
22	C₂²⋊D₉₇	—	8	1	no	—
23	D₃₈₈	—	4	1	no	—
24	C₄×D₉₇	D₈	1	1	yes	characteristic
25	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	#11715895	SD₁₆	1	1	yes	characteristic
27	C₂²×C₉₇	—	8	1	no	—
28	C₂×D₉₇	—	8	1	no	—
29	C₂×D₉₇	—	16	1	no	—
30	C₂×D₉₇	—	16	1	no	—
31	C₂×D₉₇	D₁₆	1	1	yes	characteristic
32	#87604	—	16	1	no	—
33	C₄×C₉₇	#102	1	1	yes	characteristic
34	Dic₉₇	SD₃₂	1	1	yes	characteristic
35	C₂×C₉₇	—	16	1	no	—
36	C₂×C₉₇	#357	1	1	yes	characteristic
37	D₉₇	—	16	1	no	—
38	D₉₇	D₃₂	1	1	yes	characteristic
39	D₉₇	SD₆₄	1	1	yes	characteristic
40	#1415	—	97	1	no	maximal, Sylow
41	C₉₇	#1415	1	1	yes	characteristic, complemented, Sylow
42	C₂×D₁₆	—	97	1	no	—
43	C₂×C₃₂	—	97	1	no	—
44	#366	—	97	1	no	—
45	C₂×D₈	—	194	1	no	—
46	D₁₆	—	97	1	no	—
47	D₁₆	—	194	1	no	—
48	D₁₆	#87604	1	1	yes	characteristic, complemented
49	C₂×C₁₆	—	97	1	no	—
50	#107	—	194	1	no	—
51	C₃₂	—	194	1	no	—
52	C₂×D₄	—	388	1	no	—
53	D₈	—	2	1	no	—
54	D₈	—	388	1	no	—
55	D₈	—	194	1	no	—
56	C₂×C₈	—	97	1	no	—
57	C₄⋊C₄	—	388	1	no	—
58	C₁₆	—	97	1	no	—
59	C₁₆	C₂×#87604	1	1	yes	characteristic
60	C₂³	—	776	1	no	—
61	D₄	—	4	1	no	—
62	D₄	—	776	1	no	—
63	D₄	—	388	1	no	—
64	C₂×C₄	—	776	1	no	—
65	C₂×C₄	—	97	1	no	—
66	C₈	—	97	1	no	—
67	C₈	C₂²⋊#87604	1	1	yes	characteristic
68	C₂²	—	8	1	no	—
69	C₂²	—	776	1	no	—
70	C₂²	—	1552	1	no	—
71	C₂²	—	1552	1	no	—
72	C₂²	—	97	1	no	—
73	C₄	—	1552	1	no	—
74	C₄	—	97	1	no	—
75	C₄	#15690872	1	1	yes	characteristic
76	C₂	—	16	1	no	—
77	C₂	—	1552	1	no	—
78	C₂	—	97	1	no	—
79	C₂	—	97	1	no	—
80	C₂	#20261647	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261647
commutator	C₁₆×C₉₇	C₂×C₄
Frattini	C₁₆	C₂×#87604
Fitting	D₁₆×C₉₇	C₄
socle	C₂×C₉₇	#357
prime radical	D₁₆×D₉₇	C₂
cosocle	C₁₆×D₉₇	C₂²
hypercenter	D₁₆	#87604
hypocenter	C₉₇	#1415
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
1. #20261666	C₂	semidirect
2. #20261661	C₂	semidirect
7. D₁₅₅₂	C₄	semidirect
10. D₁₆×C₉₇	C₄	semidirect
41. C₉₇	#1415	semidirect
48. D₁₆	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261666	2
#20261661	2
D₁₆×D₉₇	2
#1415	97