Subgroups — #27144018

Order 12416.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261140	C₂	1	2	yes	maximal
2	#20260563	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#15690810	—	2	2	no	—
5	#15690744	—	2	2	no	—
6	#15690744	C₂²	1	4	yes	—
7	C₂×C₁₆×C₉₇	C₂²	1	2	yes	—
8	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₉₄	—	4	2	no	—
10	C₄⋊Dic₉₇	—	2	4	no	—
11	C₄⋊Dic₉₇	—	4	2	no	—
12	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
14	C₄²×C₉₇	D₄	1	1	yes	characteristic
15	C₁₆×C₉₇	Q₈	1	4	yes	complemented
16	C₂×C₄×C₉₇	D₈	1	2	yes	—
17	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
18	C₂×Dic₉₇	—	4	4	no	—
19	Dic₁₉₄	—	8	4	no	—
20	C₈×C₉₇	Q₁₆	1	2	yes	—
21	C₈×C₉₇	C₂×Q₈	1	2	yes	—
22	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
23	C₄×C₉₇	SD₃₂	1	4	yes	—
24	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
25	C₄×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
26	Dic₉₇	—	8	4	no	—
27	C₂×C₉₇	#500	1	1	yes	characteristic
28	C₂×C₉₇	C₂×SD₃₂	1	2	yes	—
29	#2254	—	97	1	no	maximal, Sylow
30	C₉₇	#2254	1	1	yes	characteristic, complemented, Sylow
31	#500	—	97	2	no	—
32	#367	—	97	4	no	—
33	C₄×C₁₆	D₉₇	1	1	yes	characteristic
34	C₄⋊Q₈	—	194	2	no	—
35	C₂×C₁₆	C₂×D₉₇	1	2	yes	—
36	#107	—	97	4	no	—
37	#107	—	194	2	no	—
38	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
39	C₂×Q₈	—	388	2	no	—
40	C₂×C₈	D₃₈₈	1	1	yes	characteristic
41	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
42	C₄⋊C₄	—	388	2	no	—
43	C₄⋊C₄	—	194	4	no	—
44	C₄²	D₃₈₈	1	1	yes	characteristic
45	C₁₆	Dic₁₉₄	1	4	yes	complemented
46	Q₈	—	776	4	no	—
47	C₂×C₄	—	388	4	no	—
48	C₂×C₄	D₇₇₆	1	2	yes	—
49	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
50	C₈	Dic₃₈₈	1	2	yes	—
51	C₈	C₂×Dic₁₉₄	1	2	yes	—
52	C₂²	C₂×D₇₇₆	1	1	yes	characteristic
53	C₄	—	776	4	no	—
54	C₄	#15690755	1	4	yes	—
55	C₄	#15690810	1	1	yes	characteristic
56	C₄	C₂×Dic₃₈₈	1	1	yes	characteristic
57	C₂	#20261140	1	1	yes	characteristic
58	C₂	C₂×#15690755	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇₇₆
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	C₄×C₁₆×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂×D₈
hypercenter	C₄×C₁₆	D₉₇
hypocenter	C₉₇	#2254
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₉₇	Q₈	semidirect
30. C₉₇	#2254	semidirect
45. C₁₆	Dic₁₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20261140	2
#20260563	2
C₄×C₁₆×C₉₇	2
#2254	97