Subgroups — #27139937

Order 12416.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690716	C₂	1	1	yes	maximal, characteristic
2	#20260561	C₂	1	1	yes	maximal, characteristic
3	#366×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#15690744	—	2	1	no	—
5	#15690716	C₄	1	2	yes	complemented
6	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
7	#107×C₉₇	—	2	1	no	—
8	C₄⋊Dic₉₇	—	4	1	no	—
9	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
10	C₄⋊C₄×C₉₇	—	4	1	no	—
11	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₉₇	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₉₇	—	8	1	no	—
14	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
15	C₂×Dic₉₇	—	8	1	no	—
16	C₈×C₉₇	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₉₇	Q₁₆	1	1	yes	characteristic
18	C₂²×C₉₇	D₁₆	1	1	yes	characteristic
19	C₄×C₉₇	—	16	1	no	—
20	C₄×C₉₇	#107	1	1	yes	characteristic
21	C₄×C₉₇	Q₃₂	1	1	yes	characteristic
22	Dic₉₇	—	16	1	no	—
23	C₂×C₉₇	#366	1	1	yes	characteristic
24	C₂×C₉₇	SD₆₄	1	1	yes	characteristic
25	C₂×C₉₇	SD₆₄	1	1	yes	characteristic
26	#1424	—	97	1	no	maximal, Sylow
27	C₉₇	#1424	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃₂	—	97	1	no	—
29	#366	—	97	1	no	—
30	#366	D₉₇	1	1	yes	characteristic
31	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
32	#107	—	2	1	no	—
33	#107	—	194	1	no	—
34	C₃₂	—	97	2	no	—
35	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
36	C₄⋊C₄	—	4	1	no	—
37	C₄⋊C₄	—	388	1	no	—
38	C₁₆	Dic₁₉₄	1	1	yes	characteristic
39	C₁₆	C₄×D₉₇	1	1	yes	characteristic
40	C₂×C₄	—	8	1	no	—
41	C₂×C₄	—	776	1	no	—
42	C₂×C₄	#13095496	1	1	yes	characteristic
43	C₈	#13095490	1	1	yes	characteristic
44	C₈	#13095501	1	1	yes	characteristic
45	C₂²	#15690751	1	1	yes	characteristic
46	C₄	—	16	1	no	—
47	C₄	—	1552	1	no	—
48	C₄	#15690743	1	1	yes	characteristic
49	C₄	#15690756	1	1	yes	characteristic
50	C₂	#20260560	1	1	yes	characteristic
51	C₂	#20260572	1	1	yes	characteristic
52	C₂	#20260573	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690751
commutator	C₁₆×C₉₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₉₇
Fitting	#366×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	D₁₆
cosocle	C₂×C₁₆×C₉₇	C₂²
hypercenter	#366	D₉₇
hypocenter	C₉₇	#1424
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
5. #15690716	C₄	semidirect
27. C₉₇	#1424	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690716	2
#20260561	2
#366×C₉₇	2
#1424	97