Subgroups — #27139775

Order 12416.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260556	C₂	1	2	yes	maximal
2	#494×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#15690750	—	2	2	no	—
4	C₂×Q₁₆×C₉₇	—	2	2	no	—
5	C₄⋊Q₈×C₉₇	C₄	1	2	yes	—
6	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
7	#13095486	—	4	2	no	—
8	C₂×Q₈×C₉₇	—	2	2	no	—
9	Q₁₆×C₉₇	—	4	2	no	—
10	Q₁₆×C₉₇	—	4	2	no	—
11	C₂×C₈×C₉₇	D₄	1	2	yes	—
12	C₄⋊C₄×C₉₇	—	4	2	no	—
13	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
14	Q₈×C₉₇	—	4	4	no	—
15	C₂×C₄×C₉₇	—	2	1	no	—
16	C₂×C₄×C₉₇	—	4	2	no	—
17	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
18	C₈×C₉₇	D₈	1	2	yes	—
19	C₈×C₉₇	SD₁₆	1	2	yes	—
20	C₂²×C₉₇	#100	1	1	yes	characteristic
21	C₄×C₉₇	—	2	2	no	—
22	C₄×C₉₇	—	8	2	no	—
23	C₄×C₉₇	#102	1	2	yes	—
24	C₂×C₉₇	—	2	1	no	—
25	C₂×C₉₇	#331	1	1	yes	characteristic
26	#1358	—	97	1	no	maximal, Sylow
27	C₉₇	#1358	1	1	yes	characteristic, complemented, Sylow
28	#494	D₉₇	1	1	yes	characteristic
29	#364	—	97	2	no	—
30	C₂×Q₁₆	—	2	2	no	—
31	C₄⋊Q₈	Dic₉₇	1	2	yes	—
32	M₃₂	—	194	2	no	—
33	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
34	C₂×Q₈	—	2	2	no	—
35	Q₁₆	—	4	2	no	—
36	Q₁₆	—	4	2	no	—
37	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
38	C₄⋊C₄	—	4	2	no	—
39	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
40	C₁₆	—	388	2	no	—
41	Q₈	—	4	4	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	4	2	no	—
44	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
45	C₈	#13095496	1	2	yes	—
46	C₈	#13095499	1	2	yes	—
47	C₂²	#15690727	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	8	2	no	—
50	C₄	#15690731	1	2	yes	—
51	C₂	—	2	1	no	—
52	C₂	#20260479	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260479
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#494×C₉₇	C₂
socle	C₂×C₉₇	#331
prime radical	C₂²×C₉₇	#100
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#494	D₉₇
hypocenter	C₉₇	#1358
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
27. C₉₇	#1358	semidirect

Maximal subgroups

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

subgroup	index
#20260556	2
#494×C₉₇	2
#1358	97