Subgroups — #27144015

Order 12416.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261137	C₂	1	2	yes	maximal
2	#20260563	C₂	1	2	yes	maximal
3	#20260561	C₂	1	2	yes	maximal
4	C₄×C₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
5	#15690802	—	2	2	no	—
6	#15690744	C₂²	1	4	yes	—
7	#15690742	—	2	2	no	—
8	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₉₇	—	2	4	no	—
12	#13095490	—	4	4	no	—
13	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
15	C₄²×C₉₇	D₄	1	1	yes	characteristic
16	C₁₆×C₉₇	Q₈	1	2	yes	—
17	C₁₆×C₉₇	Q₈	1	2	yes	—
18	C₂×C₄×C₉₇	D₈	1	2	yes	—
19	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
20	C₂×Dic₉₇	—	4	4	no	—
21	C₈×C₉₇	Q₁₆	1	2	yes	—
22	C₈×C₉₇	C₂×Q₈	1	1	yes	characteristic
23	C₈×C₉₇	C₂×Q₈	1	1	yes	characteristic
24	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
25	C₄×C₉₇	—	2	2	no	—
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	#508	1	2	yes	—
31	#2253	—	97	1	no	maximal, Sylow
32	C₉₇	#2253	1	1	yes	characteristic, complemented, Sylow
33	#499	—	97	2	no	—
34	#367	—	97	2	no	—
35	#366	—	97	2	no	—
36	C₄×C₁₆	D₉₇	1	1	yes	characteristic
37	#125	—	194	2	no	—
38	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
39	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
40	#107	—	97	4	no	—
41	#106	—	194	2	no	—
42	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
43	C₂×C₈	D₃₈₈	1	1	yes	characteristic
44	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
45	C₄⋊C₄	—	388	4	no	—
46	C₄⋊C₄	—	194	4	no	—
47	C₄²	D₃₈₈	1	1	yes	characteristic
48	C₁₆	Dic₁₉₄	1	2	yes	—
49	C₁₆	Dic₁₉₄	1	2	yes	—
50	C₂×C₄	—	388	4	no	—
51	C₂×C₄	D₇₇₆	1	2	yes	—
52	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
53	C₈	Dic₃₈₈	1	2	yes	—
54	C₈	C₂×Dic₁₉₄	1	1	yes	characteristic
55	C₈	C₂×Dic₁₉₄	1	1	yes	characteristic
56	C₂²	C₂×D₇₇₆	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	776	4	no	—
59	C₄	#15690810	1	1	yes	characteristic
60	C₄	C₂×Dic₃₈₈	1	1	yes	characteristic
61	C₂	#20261140	1	1	yes	characteristic
62	C₂	#20261167	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	#2253
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
32. C₉₇	#2253	semidirect

Maximal subgroups

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

subgroup	index
#20261137	2
#20260563	2
#20260561	2
C₄×C₁₆×C₉₇	2
#2253	97