Subgroups — #27156742

Order 12416.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261627	C₂	1	2	yes	maximal
2	#20261105	C₂	1	1	yes	maximal, characteristic
3	#15690884	—	2	2	no	—
4	#15690810	C₄	1	1	yes	characteristic
5	#15690733	—	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×Dic₁₉₄	—	2	1	no	—
10	C₂×#11715895	D₄	1	2	yes	—
11	C₄⋊Dic₉₇	—	4	1	no	—
12	C₄⋊C₄×C₉₇	—	2	1	no	—
13	C₄⋊C₄×C₉₇	—	4	1	no	—
14	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₉₇	—	2	1	no	—
16	C₂×C₄×C₉₇	—	4	1	no	—
17	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
18	C₂×Dic₉₇	—	4	1	no	—
19	Dic₁₉₄	—	4	2	no	—
20	#11715895	—	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	#104	1	2	yes	—
25	Dic₉₇	—	8	1	no	—
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	—	97	1	no	—
31	#347	—	97	2	no	—
32	C₄⋊Q₈	—	97	1	no	—
33	#125	#87604	1	1	yes	characteristic
34	M₃₂	—	194	2	no	—
35	#103	—	194	2	no	—
36	#97	—	97	1	no	—
37	C₂×Q₈	—	194	1	no	—
38	C₂×C₈	—	97	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	4	1	no	—
41	C₄⋊C₄	—	388	1	no	—
42	C₄²	C₂×#87604	1	1	yes	characteristic
43	C₁₆	—	194	4	no	—
44	Q₈	—	388	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	388	1	no	—
48	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
49	C₈	—	194	2	no	—
50	C₂²	#15690870	1	1	yes	characteristic
51	C₄	—	4	1	no	—
52	C₄	—	8	1	no	—
53	C₄	—	776	1	no	—
54	C₄	#15690875	1	2	yes	—
55	C₂	—	2	1	no	—
56	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	C₂²×C₉₇	#100
cosocle	#15690721	C₂²
hypercenter	#125	#87604
hypocenter	C₉₇	#1356
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
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
#20261627	2
#20261105	2
#1356	97