Subgroups — #27158833

Order 12416.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261843	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15690900	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261794	C₂	1	1	yes	maximal, characteristic
4	#15690900	—	2	1	no	—
5	#15690900	—	2	1	no	—
6	#15690885	—	2	1	no	—
7	#15690884	C₄	1	1	yes	characteristic
8	#15690883	—	2	1	no	—
9	C₂×#13095514	C₂²	1	1	yes	characteristic
10	#15690818	C₄	1	1	yes	characteristic
11	#13095514	—	2	1	no	—
12	#13095514	D₄	1	1	yes	characteristic
13	#13095514	D₄	1	1	yes	characteristic
14	#13095511	C₈	1	1	yes	characteristic
15	#13095494	—	2	1	no	—
16	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
17	C₈×D₉₇	—	2	1	no	—
18	M₁₆×C₉₇	C₈	1	1	yes	characteristic
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	—	2	1	no	—
25	#11715895	C₂²⋊C₄	1	1	yes	characteristic
26	#11715895	M₁₆	1	1	yes	characteristic
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	—	97	1	no	—
42	M₃₂	—	194	1	no	—
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	—	194	1	no	—
49	M₁₆	#11715906	1	1	yes	characteristic
50	C₂×C₈	—	97	1	no	—
51	C₂×C₈	—	194	1	no	—
52	C₁₆	—	97	1	no	—
53	C₁₆	—	97	1	no	—
54	C₁₆	—	194	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	C₂×#11715906	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	—	97	1	no	—
62	C₈	—	97	1	no	—
63	C₂²	—	194	1	no	—
64	C₂²	#15690907	1	1	yes	characteristic
65	C₄	—	194	1	no	—
66	C₄	C₂²⋊#11715906	1	1	yes	characteristic
67	C₄	#15690905	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	388	1	no	—
70	C₂	#20261832	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#11715906
commutator	C₄×C₉₇	C₂×C₁₆
Frattini	C₂×C₄	C₂×#11715906
Fitting	M₁₆×C₉₇	C₈
socle	C₂×C₉₇	C₂²⋊C₁₆
prime radical	#13095511	C₈
cosocle	C₂×#13095514	C₂²
hypercenter	M₁₆	#11715906
hypocenter	C₉₇	#1401
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #20261843	C₂	semidirect
2. C₂×#15690900	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
#20261843	2
C₂×#15690900	2
#20261794	2
#1401	97