Subgroups — #27144011

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	#20260561	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	1	yes	characteristic
17	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
18	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
19	C₂×Dic₉₇	—	4	4	no	—
20	Dic₁₉₄	—	8	4	no	—
21	C₈×C₉₇	Q₁₆	1	2	yes	—
22	C₈×C₉₇	C₂×Q₈	1	2	yes	—
23	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
24	C₄×C₉₇	D₁₆	1	2	yes	—
25	C₄×C₉₇	Q₃₂	1	2	yes	—
26	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
27	C₄×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
28	Dic₉₇	—	8	4	no	—
29	C₂×C₉₇	#500	1	1	yes	characteristic
30	C₂×C₉₇	C₂×D₁₆	1	1	yes	characteristic
31	C₂×C₉₇	C₂×Q₃₂	1	1	yes	characteristic
32	#2252	—	97	1	no	maximal, Sylow
33	C₉₇	#2252	1	1	yes	characteristic, complemented, Sylow
34	#500	—	97	2	no	—
35	#366	—	97	4	no	—
36	C₄×C₁₆	D₉₇	1	1	yes	characteristic
37	C₄⋊Q₈	—	194	2	no	—
38	C₂×C₁₆	C₂×D₉₇	1	2	yes	—
39	#107	—	97	4	no	—
40	#107	—	194	2	no	—
41	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
42	C₂×Q₈	—	388	2	no	—
43	C₂×C₈	D₃₈₈	1	1	yes	characteristic
44	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
45	C₄⋊C₄	—	388	2	no	—
46	C₄⋊C₄	—	194	4	no	—
47	C₄²	D₃₈₈	1	1	yes	characteristic
48	C₁₆	Dic₁₉₄	1	4	yes	complemented
49	Q₈	—	776	4	no	—
50	C₂×C₄	—	388	4	no	—
51	C₂×C₄	D₇₇₆	1	1	yes	characteristic
52	C₂×C₄	D₇₇₆	1	1	yes	characteristic
53	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
54	C₈	Dic₃₈₈	1	2	yes	—
55	C₈	C₂×Dic₁₉₄	1	2	yes	—
56	C₂²	C₂×D₇₇₆	1	1	yes	characteristic
57	C₄	—	776	4	no	—
58	C₄	D₁₅₅₂	1	2	yes	—
59	C₄	Dic₇₇₆	1	2	yes	—
60	C₄	#15690810	1	1	yes	characteristic
61	C₄	C₂×Dic₃₈₈	1	1	yes	characteristic
62	C₂	#20261140	1	1	yes	characteristic
63	C₂	C₂×D₁₅₅₂	1	1	yes	characteristic
64	C₂	C₂×Dic₇₇₆	1	1	yes	characteristic

	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	#2252
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
33. C₉₇	#2252	semidirect
48. 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
#20260561	2
C₄×C₁₆×C₉₇	2
#2252	97