Subgroups — #27156888

Order 12416.

72 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	#20261663	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261168	C₂	1	1	yes	maximal, characteristic
4	#15690881	—	2	1	no	—
5	#15690860	—	2	1	no	—
6	#15690833	—	2	1	no	—
7	#15690753	—	2	1	no	—
8	D₁₅₅₂	C₄	1	1	yes	characteristic
9	C₁₆×D₉₇	C₂²	1	1	yes	characteristic
10	Q₃₂×C₉₇	C₄	1	1	yes	characteristic
11	#13095521	—	4	1	no	—
12	#13095512	—	4	1	no	—
13	#13095498	—	4	1	no	—
14	D₇₇₆	—	2	1	no	—
15	C₈×D₉₇	D₄	1	1	yes	characteristic
16	#13095486	D₄	1	1	yes	characteristic
17	Q₁₆×C₉₇	—	2	1	no	—
18	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
19	Q₈×C₉₇	—	4	1	no	—
20	D₃₈₈	—	4	1	no	—
21	D₃₈₈	—	8	1	no	—
22	C₄×D₉₇	—	8	1	no	—
23	C₄×D₉₇	D₈	1	1	yes	characteristic
24	#11715897	—	8	1	no	—
25	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	#11715895	SD₁₆	1	1	yes	characteristic
27	C₂×D₉₇	—	8	1	no	—
28	C₂×D₉₇	SD₃₂	1	1	yes	characteristic
29	C₄×C₉₇	—	8	1	no	—
30	C₄×C₉₇	#102	1	1	yes	characteristic
31	Dic₉₇	D₁₆	1	1	yes	characteristic
32	C₂×C₉₇	#357	1	1	yes	characteristic
33	D₉₇	—	2	1	no	—
34	D₉₇	—	16	1	no	—
35	#1417	—	97	1	no	maximal, Sylow
36	C₉₇	#1417	1	1	yes	characteristic, complemented, Sylow
37	#508	—	97	1	no	—
38	C₂×C₃₂	—	97	1	no	—
39	#368	—	97	1	no	—
40	#135	—	194	1	no	—
41	Q₃₂	#87604	1	1	yes	characteristic
42	SD₃₂	—	194	1	no	—
43	D₁₆	—	97	1	no	—
44	C₂×C₁₆	—	97	1	no	—
45	#108	—	194	1	no	—
46	C₃₂	—	194	1	no	—
47	#41	—	388	1	no	—
48	Q₁₆	—	2	1	no	—
49	SD₁₆	—	388	1	no	—
50	D₈	—	194	1	no	—
51	M₁₆	—	388	1	no	—
52	C₂×C₈	—	97	1	no	—
53	C₁₆	—	97	1	no	—
54	C₁₆	C₂×#87604	1	1	yes	characteristic
55	Q₈	—	4	1	no	—
56	D₄	—	776	1	no	—
57	D₄	—	388	1	no	—
58	C₂×C₄	—	776	1	no	—
59	C₂×C₄	—	97	1	no	—
60	C₈	—	776	1	no	—
61	C₈	—	97	1	no	—
62	C₈	C₂²⋊#87604	1	1	yes	characteristic
63	C₂²	—	776	1	no	—
64	C₂²	—	97	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	97	1	no	—
67	C₄	#15690872	1	1	yes	characteristic
68	C₂	—	1552	1	no	—
69	C₂	—	194	1	no	—
70	C₂	#20261647	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261647
commutator	C₁₆×C₉₇	C₂×C₄
Frattini	C₁₆	C₂×#87604
Fitting	Q₃₂×C₉₇	C₄
socle	C₂×C₉₇	#357
prime radical	#20261168	C₂
cosocle	C₁₆×D₉₇	C₂²
hypercenter	Q₃₂	#87604
hypocenter	C₉₇	#1417
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. #20261666	C₂	semidirect
2. #20261663	C₂	semidirect
36. C₉₇	#1417	semidirect

Maximal subgroups

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

subgroup	index
#20261666	2
#20261663	2
#20261168	2
#1417	97