Subgroups — #27139788

Order 12416.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13095486	C₂	1	1	yes	maximal, characteristic
2	#20260486	C₂	1	1	yes	maximal, characteristic
3	#499×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095486	—	2	1	no	—
5	C₄⋊#11715895	—	2	1	no	—
6	#125×C₉₇	—	2	1	no	—
7	#107×C₉₇	C₄	1	1	yes	characteristic
8	#107×C₉₇	C₄	1	1	yes	characteristic
9	#106×C₉₇	—	2	1	no	—
10	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×#11715895	—	4	1	no	—
12	#13095486	—	4	1	no	—
13	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
15	C₄⋊C₄×C₉₇	—	2	1	no	—
16	C₄⋊C₄×C₉₇	—	2	1	no	—
17	C₄⋊C₄×C₉₇	—	4	1	no	—
18	C₄⋊C₄×C₉₇	—	4	1	no	—
19	C₄²×C₉₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₉₇	—	4	1	no	—
21	C₂×C₄×C₉₇	—	4	1	no	—
22	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
24	C₂×C₄×C₉₇	SD₁₆	1	1	yes	characteristic
25	C₈×C₉₇	—	2	1	no	—
26	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
27	C₈×C₉₇	Q₁₆	1	1	yes	characteristic
28	#11715895	—	8	1	no	—
29	C₂²×C₉₇	#102	1	1	yes	characteristic
30	C₄×C₉₇	—	2	1	no	—
31	C₄×C₉₇	—	2	1	no	—
32	C₄×C₉₇	—	8	1	no	—
33	C₄×C₉₇	—	8	1	no	—
34	C₄×C₉₇	#101	1	1	yes	characteristic
35	C₄×C₉₇	#103	1	1	yes	characteristic
36	C₂×C₉₇	#332	1	1	yes	characteristic
37	C₂×C₉₇	#359	1	1	yes	characteristic
38	C₂×C₉₇	#360	1	1	yes	characteristic
39	#1363	—	97	1	no	maximal, Sylow
40	C₉₇	#1363	1	1	yes	characteristic, complemented, Sylow
41	#499	D₉₇	1	1	yes	characteristic
42	C₄⋊C₁₆	—	97	1	no	—
43	#334	—	97	1	no	—
44	#125	—	2	1	no	—
45	C₂×C₁₆	—	194	1	no	—
46	#107	Dic₉₇	1	1	yes	characteristic
47	#107	Dic₉₇	1	1	yes	characteristic
48	#106	—	2	1	no	—
49	C₄⋊C₈	—	194	1	no	—
50	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
51	C₂×C₈	—	388	1	no	—
52	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
53	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	2	1	no	—
56	C₄⋊C₄	—	4	1	no	—
57	C₄⋊C₄	—	4	1	no	—
58	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
59	C₁₆	—	388	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
63	C₂×C₄	#13095496	1	1	yes	characteristic
64	C₂×C₄	#13095499	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	776	1	no	—
67	C₈	#13095498	1	1	yes	characteristic
68	C₈	#13095501	1	1	yes	characteristic
69	C₂²	#15690731	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	2	1	no	—
72	C₄	—	8	1	no	—
73	C₄	—	8	1	no	—
74	C₄	#15690729	1	1	yes	characteristic
75	C₄	#15690734	1	1	yes	characteristic
76	C₂	#20260481	1	1	yes	characteristic
77	C₂	#20260542	1	1	yes	characteristic
78	C₂	#20260545	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690731
commutator	C₂×C₈×C₉₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#499×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	#102
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#499	D₉₇
hypocenter	C₉₇	#1363
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
40. C₉₇	#1363	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13095486	2
#20260486	2
#499×C₉₇	2
#1363	97