Subgroups — #27156853

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	#20261152	C₂	1	1	yes	maximal, characteristic
4	#15690860	—	2	1	no	—
5	#15690860	—	2	1	no	—
6	#15690750	—	2	1	no	—
7	#15690750	C₄	1	1	yes	characteristic
8	C₂×#13095486	—	2	1	no	—
9	C₂×#13095486	C₂²	1	1	yes	characteristic
10	#131×C₉₇	C₄	1	1	yes	characteristic
11	#13095486	—	2	1	no	—
12	#13095486	D₄	1	1	yes	characteristic
13	#13095486	D₄	1	1	yes	characteristic
14	#41×C₉₇	C₈	1	1	yes	characteristic
15	M₁₆×C₉₇	—	2	1	no	—
16	M₁₆×C₉₇	C₈	1	1	yes	characteristic
17	C₂×C₈×C₉₇	—	2	1	no	—
18	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
19	Q₈×C₉₇	C₁₆	1	1	yes	characteristic
20	D₄×C₉₇	—	2	1	no	—
21	D₄×C₉₇	C₁₆	1	1	yes	characteristic
22	C₂×C₄×C₉₇	—	2	1	no	—
23	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
24	C₈×C₉₇	—	2	1	no	—
25	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₉₇	M₁₆	1	1	yes	characteristic
27	C₂²×C₉₇	—	2	1	no	—
28	C₂²×C₉₇	M₃₂	1	1	yes	characteristic
29	C₄×C₉₇	—	2	1	no	—
30	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
32	C₂×C₉₇	—	2	1	no	—
33	C₂×C₉₇	—	4	1	no	—
34	C₂×C₉₇	C₂²⋊C₁₆	1	1	yes	characteristic
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	#87604	1	1	yes	characteristic
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	#11715897	1	1	yes	characteristic
48	M₁₆	—	2	1	no	—
49	M₁₆	#11715897	1	1	yes	characteristic
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	C₂×#87604	1	1	yes	characteristic
52	C₁₆	—	97	1	no	—
53	C₁₆	—	97	1	no	—
54	C₁₆	—	194	1	no	—
55	Q₈	#13095514	1	1	yes	characteristic
56	D₄	—	2	1	no	—
57	D₄	#13095514	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₂×#11715897	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	C₂²⋊#87604	1	1	yes	characteristic
62	C₈	#13095520	1	1	yes	characteristic
63	C₂²	—	2	1	no	—
64	C₂²	#15690884	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	C₂²⋊#11715897	1	1	yes	characteristic
67	C₄	C₂×#13095514	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	C₂²⋊#13095514	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊#87604
commutator	C₄×C₉₇	C₂×C₁₆
Frattini	C₂×C₈	C₂×#87604
Fitting	#131×C₉₇	C₄
socle	C₂×C₉₇	C₂²⋊C₁₆
prime radical	#41×C₉₇	C₈
cosocle	C₂×#13095486	C₂²
hypercenter	#131	#87604
hypocenter	C₉₇	#1401
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

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
#20261152	2
#1401	97