Subgroups — #27156854

Order 12416.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261667	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15690860	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261153	C₂	1	1	yes	maximal, characteristic
4	#15690860	—	2	1	no	—
5	#15690860	—	2	1	no	—
6	#15690816	C₄	1	1	yes	characteristic
7	#15690749	—	2	1	no	—
8	C₂×#13095486	C₂²	1	1	yes	characteristic
9	C₁₆×D₉₇	—	2	1	no	—
10	M₃₂×C₉₇	C₄	1	1	yes	characteristic
11	#13095511	C₈	1	1	yes	characteristic
12	#13095495	C₈	1	1	yes	characteristic
13	#13095494	—	2	1	no	—
14	C₈×D₉₇	—	2	1	no	—
15	#13095486	D₄	1	1	yes	characteristic
16	#13095486	D₄	1	1	yes	characteristic
17	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₉₇	—	2	1	no	—
19	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
20	C₂²⋊D₉₇	—	2	1	no	—
21	D₃₈₈	C₁₆	1	1	yes	characteristic
22	C₄×D₉₇	—	2	1	no	—
23	Dic₁₉₄	C₁₆	1	1	yes	characteristic
24	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₉₇	M₁₆	1	1	yes	characteristic
26	#11715895	—	2	1	no	—
27	C₂²×C₉₇	M₃₂	1	1	yes	characteristic
28	C₂×D₉₇	—	2	1	no	—
29	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
31	Dic₉₇	—	2	1	no	—
32	C₂×C₉₇	—	2	1	no	—
33	C₂×C₉₇	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₉₇	—	4	1	no	—
35	#1401	—	97	1	no	maximal, Sylow
36	C₉₇	#1401	1	1	yes	characteristic, complemented, Sylow
37	#504	—	97	1	no	—
38	M₆₄	—	97	1	no	—
39	C₂×C₃₂	—	97	1	no	—
40	#131	—	97	1	no	—
41	M₃₂	—	194	1	no	—
42	M₃₂	#87604	1	1	yes	characteristic
43	C₂×C₁₆	—	97	1	no	—
44	C₂×C₁₆	—	194	1	no	—
45	C₃₂	—	194	1	no	—
46	C₃₂	—	194	1	no	—
47	#41	—	97	1	no	—
48	M₁₆	—	97	1	no	—
49	M₁₆	—	194	1	no	—
50	C₂×C₈	—	194	1	no	—
51	C₂×C₈	C₂×#87604	1	1	yes	characteristic
52	C₁₆	—	2	1	no	—
53	C₁₆	—	97	1	no	—
54	C₁₆	—	97	1	no	—
55	Q₈	—	97	1	no	—
56	D₄	—	97	1	no	—
57	D₄	—	194	1	no	—
58	C₂×C₄	—	194	1	no	—
59	C₂×C₄	#13095519	1	1	yes	characteristic
60	C₈	—	194	1	no	—
61	C₈	C₂²⋊#87604	1	1	yes	characteristic
62	C₈	#13095521	1	1	yes	characteristic
63	C₂²	—	194	1	no	—
64	C₂²	#15690885	1	1	yes	characteristic
65	C₄	—	194	1	no	—
66	C₄	#15690867	1	1	yes	characteristic
67	C₄	#15690883	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	388	1	no	—
70	C₂	#20261630	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊#87604
commutator	C₄×C₉₇	C₂×C₁₆
Frattini	C₂×C₈	C₂×#87604
Fitting	M₃₂×C₉₇	C₄
socle	C₂×C₉₇	C₂²⋊C₁₆
prime radical	#13095511	C₈
cosocle	C₂×#13095486	C₂²
hypercenter	M₃₂	#87604
hypocenter	C₉₇	#1401
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
1. #20261667	C₂	semidirect
2. C₂×#15690860	C₂	semidirect
36. C₉₇	#1401	semidirect

Maximal subgroups

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

subgroup	index
#20261667	2
C₂×#15690860	2
#20261153	2
#1401	97