Subgroups — #27139771

Order 12416.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260513	C₂	1	2	yes	maximal
2	#491×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#15690750	—	2	2	no	—
4	C₄⋊Q₈×C₉₇	C₄	1	1	yes	characteristic
5	#125×C₉₇	C₄	1	1	yes	characteristic
6	#103×C₉₇	—	2	2	no	—
7	#97×C₉₇	C₂²	1	1	yes	characteristic
8	#13095486	—	2	4	no	—
9	C₂×Q₈×C₉₇	—	2	1	no	—
10	C₂×C₈×C₉₇	D₄	1	2	yes	—
11	C₄⋊C₄×C₉₇	—	2	1	no	—
12	C₄⋊C₄×C₉₇	—	4	1	no	—
13	C₄⋊C₄×C₉₇	—	4	1	no	—
14	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
15	Q₈×C₉₇	—	4	2	no	—
16	C₂×C₄×C₉₇	—	2	1	no	—
17	C₂×C₄×C₉₇	—	4	1	no	—
18	C₂×C₄×C₉₇	—	4	1	no	—
19	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₉₇	—	2	2	no	—
21	C₂²×C₉₇	#100	1	1	yes	characteristic
22	C₄×C₉₇	—	4	1	no	—
23	C₄×C₉₇	—	8	1	no	—
24	C₄×C₉₇	—	8	1	no	—
25	C₄×C₉₇	#104	1	2	yes	—
26	C₂×C₉₇	—	2	1	no	—
27	C₂×C₉₇	#329	1	1	yes	characteristic
28	#1356	—	97	1	no	maximal, Sylow
29	C₉₇	#1356	1	1	yes	characteristic, complemented, Sylow
30	#491	D₉₇	1	1	yes	characteristic
31	#347	—	97	2	no	—
32	C₄⋊Q₈	Dic₉₇	1	1	yes	characteristic
33	#125	Dic₉₇	1	1	yes	characteristic
34	M₃₂	—	194	2	no	—
35	#103	—	2	2	no	—
36	#97	C₂×D₉₇	1	1	yes	characteristic
37	C₂×Q₈	—	2	1	no	—
38	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	4	1	no	—
41	C₄⋊C₄	—	4	1	no	—
42	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
43	C₁₆	—	194	4	no	—
44	Q₈	—	4	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₂²	#15690727	1	1	yes	characteristic
51	C₄	—	4	1	no	—
52	C₄	—	8	1	no	—
53	C₄	—	8	1	no	—
54	C₄	#15690737	1	2	yes	—
55	C₂	—	2	1	no	—
56	C₂	#20260475	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260475
commutator	C₄²×C₉₇	C₂×C₄
Frattini	#97	C₂×D₉₇
Fitting	#491×C₉₇	C₂
socle	C₂×C₉₇	#329
prime radical	C₂²×C₉₇	#100
cosocle	#97×C₉₇	C₂²
hypercenter	#491	D₉₇
hypocenter	C₉₇	#1356
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #491

Complements

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

N	H ≅ G/N	type
29. C₉₇	#1356	semidirect

Maximal subgroups

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

subgroup	index
#20260513	2
#491×C₉₇	2
#1356	97