Subgroups — #27158159

Order 12416.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261667	C₂	1	3	yes	maximal, complemented
2	C₂×#15690860	C₂	1	3	yes	maximal, complemented
3	#20261152	C₂	1	1	yes	maximal, characteristic
4	#15690860	C₂²	1	1	yes	characteristic
5	#15690860	C₂²	1	3	yes	—
6	#15690750	C₄	1	3	yes	—
7	C₂×#13095486	C₂²	1	3	yes	—
8	#131×C₉₇	C₄	1	1	yes	characteristic
9	#13095486	C₂×C₄	1	3	yes	—
10	#13095486	C₂³	1	1	yes	characteristic
11	#41×C₉₇	C₈	1	1	yes	characteristic
12	M₁₆×C₉₇	C₈	1	3	yes	—
13	C₂×C₈×C₉₇	C₂×C₄	1	3	yes	—
14	Q₈×C₉₇	C₁₆	1	1	yes	characteristic
15	D₄×C₉₇	C₁₆	1	3	yes	—
16	C₂×C₄×C₉₇	C₂×C₈	1	3	yes	—
17	C₈×C₉₇	C₂×C₈	1	3	yes	—
18	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₉₇	C₂×C₁₆	1	3	yes	—
20	C₄×C₉₇	C₂×C₁₆	1	3	yes	—
21	C₄×C₉₇	C₂²×C₈	1	1	yes	characteristic
22	C₂×C₉₇	—	2	3	no	—
23	C₂×C₉₇	C₂²×C₁₆	1	1	yes	characteristic
24	#2258	—	97	1	no	maximal, Sylow
25	C₉₇	#2258	1	1	yes	characteristic, complemented, Sylow
26	#504	—	97	1	no	—
27	M₆₄	—	97	3	no	—
28	C₂×C₃₂	—	97	3	no	—
29	#131	#87604	1	1	yes	characteristic
30	M₃₂	—	97	3	no	—
31	C₂×C₁₆	—	97	3	no	—
32	C₃₂	—	97	1	no	—
33	C₃₂	—	97	3	no	—
34	#41	#11715897	1	1	yes	characteristic
35	M₁₆	#11715897	1	3	yes	—
36	C₂×C₈	C₂×#87604	1	3	yes	—
37	C₁₆	—	97	3	no	—
38	C₁₆	—	97	1	no	—
39	Q₈	#13095514	1	1	yes	characteristic
40	D₄	#13095514	1	3	yes	—
41	C₂×C₄	C₂×#11715897	1	3	yes	—
42	C₈	C₂×#11715897	1	3	yes	—
43	C₈	C₂²×#87604	1	1	yes	characteristic
44	C₂²	C₂×#13095514	1	3	yes	—
45	C₄	C₂×#13095514	1	3	yes	—
46	C₄	C₂²×#11715897	1	1	yes	characteristic
47	C₂	—	2	3	no	—
48	C₂	C₂²×#13095514	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×#87604
commutator, socle	C₂×C₉₇	C₂²×C₁₆
Fitting	#131×C₉₇	C₄
prime radical	#41×C₉₇	C₈
cosocle	#13095486	C₂³
hypercenter	#131	#87604
hypocenter	C₉₇	#2258
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ 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
25. C₉₇	#2258	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
#2258	97