Subgroups — #27156873

Order 12416.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261646	C₂	1	2	yes	maximal
2	#20261114	C₂	1	1	yes	maximal, characteristic
3	#15690871	—	2	2	no	—
4	C₂×#13095501	—	2	2	no	—
5	#15690810	C₂²	1	1	yes	characteristic
6	C₄×#11715895	C₄	1	1	yes	characteristic
7	C₄⋊Q₈×C₉₇	C₄	1	1	yes	characteristic
8	#13095520	—	4	2	no	—
9	C₂×Dic₁₉₄	D₄	1	2	yes	—
10	#13095501	—	4	4	no	—
11	C₂×#11715895	—	2	1	no	—
12	C₄⋊Dic₉₇	—	4	1	no	—
13	C₂×Q₈×C₉₇	—	2	1	no	—
14	C₄⋊C₄×C₉₇	—	4	1	no	—
15	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
16	Q₈×C₉₇	—	4	2	no	—
17	C₂×C₄×C₉₇	—	2	1	no	—
18	C₂×C₄×C₉₇	—	4	1	no	—
19	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×Dic₉₇	—	2	2	no	—
21	Dic₁₉₄	—	4	2	no	—
22	#11715897	—	8	2	no	—
23	#11715895	—	2	2	no	—
24	C₂²×C₉₇	#99	1	1	yes	characteristic
25	C₄×C₉₇	—	2	2	no	—
26	C₄×C₉₇	—	2	1	no	—
27	C₄×C₉₇	—	8	1	no	—
28	Dic₉₇	—	4	2	no	—
29	C₂×C₉₇	—	2	1	no	—
30	C₂×C₉₇	#353	1	1	yes	characteristic
31	#1411	—	97	1	no	maximal, Sylow
32	C₉₇	#1411	1	1	yes	characteristic, complemented, Sylow
33	#494	—	97	1	no	—
34	#356	—	97	2	no	—
35	C₂×Q₁₆	—	194	2	no	—
36	C₄⋊Q₈	—	97	1	no	—
37	C₄⋊Q₈	#87604	1	1	yes	characteristic
38	#101	—	194	2	no	—
39	C₄×C₈	—	97	1	no	—
40	C₂×Q₈	—	2	1	no	—
41	C₂×Q₈	—	97	2	no	—
42	Q₁₆	—	388	4	no	—
43	M₁₆	—	388	2	no	—
44	C₂×C₈	—	194	1	no	—
45	C₄⋊C₄	—	4	1	no	—
46	C₄⋊C₄	—	388	1	no	—
47	C₄²	C₂×#87604	1	1	yes	characteristic
48	Q₈	—	4	2	no	—
49	Q₈	—	388	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	4	1	no	—
52	C₂×C₄	—	194	2	no	—
53	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
54	C₈	—	776	2	no	—
55	C₈	—	194	2	no	—
56	C₂²	#15690868	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	2	1	no	—
59	C₄	—	8	1	no	—
60	C₄	—	388	2	no	—
61	C₂	—	2	1	no	—
62	C₂	#20261639	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261639
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄²	C₂×#87604
Fitting	C₄⋊Q₈×C₉₇	C₄
socle	C₂×C₉₇	#353
prime radical	C₂²×C₉₇	#99
cosocle	#15690810	C₂²
hypercenter	C₄⋊Q₈	#87604
hypocenter	C₉₇	#1411
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
32. C₉₇	#1411	semidirect

Maximal subgroups

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

subgroup	index
#20261646	2
#20261114	2
#1411	97