Subgroups — #27139689

Order 12416.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260517	C₂	1	2	yes	maximal
2	#424×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#15690750	—	2	2	no	—
4	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
5	C₂²×C₈×C₉₇	C₄	1	1	yes	characteristic
6	#117×C₉₇	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₉₇	—	2	2	no	—
8	#13095486	—	4	2	no	—
9	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
10	M₁₆×C₉₇	—	4	1	no	—
11	C₂×C₈×C₉₇	—	2	1	no	—
12	C₂×C₈×C₉₇	—	4	1	no	—
13	C₂×C₈×C₉₇	D₄	1	2	yes	—
14	C₄⋊C₄×C₉₇	—	2	1	no	—
15	C₂²⋊C₄×C₉₇	C₈	1	2	yes	—
16	C₄²×C₉₇	—	2	1	no	—
17	C₂³×C₉₇	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₉₇	—	2	1	no	—
19	C₂×C₄×C₉₇	—	2	1	no	—
20	C₂×C₄×C₉₇	—	2	2	no	—
21	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
23	C₈×C₉₇	—	2	2	no	—
24	C₈×C₉₇	—	4	1	no	—
25	C₂²×C₉₇	—	2	1	no	—
26	C₂²×C₉₇	—	4	1	no	—
27	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₉₇	—	2	1	no	—
29	C₄×C₉₇	—	4	2	no	—
30	C₄×C₉₇	#99	1	1	yes	characteristic
31	C₄×C₉₇	#101	1	1	yes	characteristic
32	C₂×C₉₇	—	2	1	no	—
33	C₂×C₉₇	—	4	1	no	—
34	C₂×C₉₇	#324	1	1	yes	characteristic
35	#1328	—	97	1	no	maximal, Sylow
36	C₉₇	#1328	1	1	yes	characteristic, complemented, Sylow
37	#424	D₉₇	1	1	yes	characteristic
38	#349	—	97	2	no	—
39	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
40	C₂²×C₈	Dic₉₇	1	1	yes	characteristic
41	#117	Dic₉₇	1	1	yes	characteristic
42	M₃₂	—	194	2	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
45	M₁₆	—	4	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
49	C₄⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	#11715895	1	2	yes	—
51	C₄²	—	2	1	no	—
52	C₁₆	—	388	2	no	—
53	C₂³	C₂×#11715895	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
58	C₂×C₄	#13095495	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	4	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₂²⋊#11715895	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₄	#15690725	1	1	yes	characteristic
67	C₄	#15690729	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	#20260461	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690725
commutator, prime radical	C₂³×C₉₇	C₂×C₈
Frattini	C₂×M₁₆	C₂×D₉₇
Fitting	#424×C₉₇	C₂
socle	C₂×C₉₇	#324
cosocle	C₂×M₁₆×C₉₇	C₂²
hypercenter	#424	D₉₇
hypocenter	C₉₇	#1328
LCS	G ▷ C₂³×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #424

Complements

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

N	H ≅ G/N	type
36. C₉₇	#1328	semidirect

Maximal subgroups

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

subgroup	index
#20260517	2
#424×C₉₇	2
#1328	97