Subgroups — #27139909

Order 12416.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260534	C₂	1	2	yes	maximal
2	#491×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#15690729	—	2	2	no	—
4	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
5	#125×C₉₇	C₄	1	1	yes	characteristic
6	#103×C₉₇	—	2	2	no	—
7	#97×C₉₇	C₄	1	1	yes	characteristic
8	#13095495	—	4	2	no	—
9	C₂×Q₈×C₉₇	D₄	1	2	yes	—
10	C₂×C₈×C₉₇	—	2	1	no	—
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	—	2	2	no	—
18	C₂×C₄×C₉₇	—	4	1	no	—
19	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₉₇	—	4	1	no	—
21	#11715895	—	8	2	no	—
22	C₂²×C₉₇	#99	1	1	yes	characteristic
23	C₄×C₉₇	—	2	1	no	—
24	C₄×C₉₇	—	4	1	no	—
25	C₄×C₉₇	—	4	2	no	—
26	C₄×C₉₇	—	8	1	no	—
27	C₂×C₉₇	—	2	1	no	—
28	C₂×C₉₇	#354	1	1	yes	characteristic
29	#1413	—	97	1	no	maximal, Sylow
30	C₉₇	#1413	1	1	yes	characteristic, complemented, Sylow
31	#491	D₉₇	1	1	yes	characteristic
32	#356	—	97	2	no	—
33	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
34	#125	Dic₉₇	1	1	yes	characteristic
35	#103	—	2	2	no	—
36	#101	—	194	2	no	—
37	#97	Dic₉₇	1	1	yes	characteristic
38	C₂×Q₈	C₂²⋊D₉₇	1	2	yes	—
39	M₁₆	—	388	2	no	—
40	C₂×C₈	—	2	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	4	1	no	—
43	C₄⋊C₄	—	4	1	no	—
44	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
45	Q₈	—	4	2	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	4	1	no	—
49	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
50	C₈	—	4	1	no	—
51	C₈	—	776	2	no	—
52	C₂²	#15690725	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	4	1	no	—
55	C₄	—	4	2	no	—
56	C₄	—	8	1	no	—
57	C₂	—	2	1	no	—
58	C₂	#20260530	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260530
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₉₇
Fitting	#491×C₉₇	C₂
socle	C₂×C₉₇	#354
prime radical	C₂²×C₉₇	#99
cosocle	C₄⋊Q₈×C₉₇	C₂²
hypercenter	#491	D₉₇
hypocenter	C₉₇	#1413
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #491

Complements

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

N	H ≅ G/N	type
30. C₉₇	#1413	semidirect

Maximal subgroups

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

subgroup	index
#20260534	2
#491×C₉₇	2
#1413	97