Subgroups — #27156872

Order 12416.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261646	C₂	1	2	yes	maximal, complemented
2	#20261108	C₂	1	1	yes	maximal, characteristic
3	#15690871	—	2	2	no	—
4	C₂×#13095500	—	2	2	no	—
5	#15690810	C₂²	1	1	yes	characteristic
6	#15690807	C₄	1	1	yes	characteristic
7	C₄×C₈×C₉₇	C₄	1	1	yes	characteristic
8	#13095520	—	4	2	no	—
9	C₂×Dic₁₉₄	D₄	1	2	yes	—
10	C₂×D₃₈₈	—	2	1	no	—
11	C₂×D₃₈₈	—	4	1	no	—
12	#13095500	—	4	4	no	—
13	C₄⋊Dic₉₇	—	4	1	no	—
14	C₂×C₈×C₉₇	—	2	1	no	—
15	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×D₉₇	—	4	1	no	—
17	C₂×C₄×C₉₇	—	2	1	no	—
18	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×Dic₉₇	—	2	2	no	—
20	D₃₈₈	—	4	2	no	—
21	D₃₈₈	—	8	2	no	—
22	Dic₁₉₄	—	4	2	no	—
23	#11715897	—	8	2	no	—
24	C₈×C₉₇	—	2	2	no	—
25	C₂²×C₉₇	#99	1	1	yes	characteristic
26	C₂×D₉₇	—	8	2	no	—
27	C₂×D₉₇	—	8	1	no	—
28	C₄×C₉₇	—	2	2	no	—
29	C₄×C₉₇	—	2	1	no	—
30	Dic₉₇	—	4	2	no	—
31	C₂×C₉₇	—	2	1	no	—
32	C₂×C₉₇	#353	1	1	yes	characteristic
33	D₉₇	—	16	1	no	—
34	#1410	—	97	1	no	maximal, Sylow
35	C₉₇	#1410	1	1	yes	characteristic, complemented, Sylow
36	#492	—	97	1	no	—
37	#356	—	97	2	no	—
38	C₂×SD₁₆	—	194	2	no	—
39	C₄⋊Q₈	—	97	1	no	—
40	#127	—	97	1	no	—
41	#101	—	194	2	no	—
42	C₄×C₈	#87604	1	1	yes	characteristic
43	C₂×Q₈	—	97	2	no	—
44	C₂×D₄	—	388	1	no	—
45	C₂×D₄	—	194	1	no	—
46	SD₁₆	—	388	4	no	—
47	M₁₆	—	388	2	no	—
48	C₂×C₈	—	2	1	no	—
49	C₄⋊C₄	—	388	1	no	—
50	C₄²	C₂×#87604	1	1	yes	characteristic
51	C₂³	—	388	1	no	—
52	Q₈	—	388	2	no	—
53	D₄	—	776	2	no	—
54	D₄	—	388	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	194	2	no	—
57	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	776	2	no	—
60	C₂²	—	776	2	no	—
61	C₂²	—	776	1	no	—
62	C₂²	#15690869	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	388	2	no	—
66	C₂	—	2	1	no	—
67	C₂	—	1552	1	no	—
68	C₂	#20261640	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261640
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄²	C₂×#87604
Fitting	C₄×C₈×C₉₇	C₄
socle	C₂×C₉₇	#353
prime radical	#15690807	C₄
cosocle	#15690810	C₂²
hypercenter	C₄×C₈	#87604
hypocenter	C₉₇	#1410
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #20261646	C₂	semidirect
35. C₉₇	#1410	semidirect

Maximal subgroups

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

subgroup	index
#20261646	2
#20261108	2
#1410	97