Subgroups — #27144014

Order 12416.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261136	C₂	1	2	yes	maximal
2	#20260563	C₂	1	1	yes	maximal, characteristic
3	#20260560	C₂	1	2	yes	maximal
4	C₁₆×Dic₉₇	C₂	1	1	yes	maximal, characteristic
5	#367×C₉₇	C₂	1	1	yes	maximal, characteristic
6	#15690801	—	2	2	no	—
7	C₂×#13095486	C₂²	1	1	yes	characteristic
8	#15690744	C₂²	1	2	yes	—
9	#15690741	—	2	2	no	—
10	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
12	#107×C₉₇	C₂²	1	2	yes	—
13	C₂×#11715895	D₄	1	1	yes	characteristic
14	C₄⋊Dic₉₇	—	2	2	no	—
15	C₄⋊Dic₉₇	—	4	2	no	—
16	#13095490	—	4	2	no	—
17	C₄×Dic₉₇	D₄	1	1	yes	characteristic
18	#13095486	Q₈	1	2	yes	—
19	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₉₇	—	2	2	no	—
21	C₁₆×C₉₇	Q₈	1	2	yes	—
22	C₂×C₄×C₉₇	—	4	2	no	—
23	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
24	C₂×Dic₉₇	—	4	2	no	—
25	C₂×Dic₉₇	D₈	1	2	yes	—
26	C₈×C₉₇	C₂×Q₈	1	1	yes	characteristic
27	C₈×C₉₇	C₂×Q₈	1	1	yes	characteristic
28	#11715895	Q₁₆	1	2	yes	—
29	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
30	C₄×C₉₇	—	8	2	no	—
31	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
32	C₄×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
33	Dic₉₇	—	2	2	no	—
34	Dic₉₇	—	8	2	no	—
35	C₂×C₉₇	#500	1	1	yes	characteristic
36	C₂×C₉₇	#508	1	2	yes	—
37	#2253	—	97	1	no	maximal, Sylow
38	C₉₇	#2253	1	1	yes	characteristic, complemented, Sylow
39	#499	—	97	2	no	—
40	#367	—	97	1	no	—
41	#367	D₉₇	1	1	yes	characteristic
42	#366	—	97	2	no	—
43	C₄×C₁₆	—	97	1	no	—
44	#125	—	194	2	no	—
45	C₂×C₁₆	—	97	1	no	—
46	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
47	#107	—	97	2	no	—
48	#107	C₂×D₉₇	1	2	yes	—
49	#106	—	194	2	no	—
50	C₄×C₈	—	97	1	no	—
51	C₂×C₈	—	97	1	no	—
52	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	388	2	no	—
55	C₄⋊C₄	—	388	2	no	—
56	C₄⋊C₄	—	194	2	no	—
57	C₄²	—	97	1	no	—
58	C₁₆	—	97	2	no	—
59	C₁₆	Dic₁₉₄	1	2	yes	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	97	2	no	—
62	C₂×C₄	—	388	2	no	—
63	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
64	C₈	—	97	2	no	—
65	C₈	Q₈×D₉₇	1	1	yes	characteristic
66	C₈	C₂×Dic₁₉₄	1	1	yes	characteristic
67	C₂²	D₈×D₉₇	1	1	yes	characteristic
68	C₄	—	8	2	no	—
69	C₄	—	776	2	no	—
70	C₄	—	194	2	no	—
71	C₄	#15690809	1	1	yes	characteristic
72	C₄	Q₁₆×D₉₇	1	1	yes	characteristic
73	C₂	#20261139	1	1	yes	characteristic
74	C₂	#20261166	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₉₇
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#367×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂×D₈
hypercenter	#367	D₉₇
hypocenter	C₉₇	#2253
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
38. C₉₇	#2253	semidirect

Maximal subgroups

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

subgroup	index
#20261136	2
#20260563	2
#20260560	2
C₁₆×Dic₉₇	2
#367×C₉₇	2
#2253	97