Subgroups — #27156739

Order 12416.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261627	C₂	1	2	yes	maximal, complemented
2	#20261101	C₂	1	1	yes	maximal, characteristic
3	#15690884	—	2	2	no	—
4	#15690807	C₄	1	1	yes	characteristic
5	#15690730	—	2	2	no	—
6	#15690721	C₂²	1	1	yes	characteristic
7	#125×C₉₇	C₄	1	1	yes	characteristic
8	#13095514	—	2	4	no	—
9	C₂×D₃₈₈	—	2	1	no	—
10	C₂×D₃₈₈	—	4	1	no	—
11	C₂×#11715895	D₄	1	2	yes	—
12	C₄⋊C₄×C₉₇	—	2	1	no	—
13	C₄⋊C₄×C₉₇	—	4	1	no	—
14	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×D₉₇	—	4	1	no	—
16	C₂×C₄×C₉₇	—	2	1	no	—
17	C₂×C₄×C₉₇	—	4	1	no	—
18	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
19	D₃₈₈	—	4	2	no	—
20	D₃₈₈	—	8	2	no	—
21	#11715895	—	2	2	no	—
22	C₂²×C₉₇	#100	1	1	yes	characteristic
23	C₂×D₉₇	—	8	2	no	—
24	C₂×D₉₇	—	8	1	no	—
25	C₄×C₉₇	—	4	1	no	—
26	C₄×C₉₇	—	8	1	no	—
27	C₄×C₉₇	#104	1	2	yes	—
28	C₂×C₉₇	—	2	1	no	—
29	C₂×C₉₇	#329	1	1	yes	characteristic
30	D₉₇	—	16	1	no	—
31	#1355	—	97	1	no	maximal, Sylow
32	C₉₇	#1355	1	1	yes	characteristic, complemented, Sylow
33	#490	—	97	1	no	—
34	#347	—	97	2	no	—
35	#127	—	97	1	no	—
36	#125	#87604	1	1	yes	characteristic
37	M₃₂	—	194	2	no	—
38	#102	—	194	2	no	—
39	#97	—	97	1	no	—
40	C₂×D₄	—	388	1	no	—
41	C₂×D₄	—	194	1	no	—
42	C₂×C₈	—	97	2	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	4	1	no	—
45	C₄²	C₂×#87604	1	1	yes	characteristic
46	C₁₆	—	194	4	no	—
47	C₂³	—	388	1	no	—
48	D₄	—	776	2	no	—
49	D₄	—	388	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	4	1	no	—
52	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
53	C₈	—	194	2	no	—
54	C₂²	—	776	2	no	—
55	C₂²	—	776	1	no	—
56	C₂²	#15690870	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	8	1	no	—
59	C₄	#15690875	1	2	yes	—
60	C₂	—	2	1	no	—
61	C₂	—	1552	1	no	—
62	C₂	#20261591	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261591
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄²	C₂×#87604
Fitting	#125×C₉₇	C₄
socle	C₂×C₉₇	#329
prime radical	#15690807	C₄
cosocle	#15690721	C₂²
hypercenter	#125	#87604
hypocenter	C₉₇	#1355
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #20261627	C₂	semidirect
32. C₉₇	#1355	semidirect

Maximal subgroups

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

subgroup	index
#20261627	2
#20261101	2
#1355	97