Subgroups — #27139656

Order 12416.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260511	C₂	1	1	yes	maximal, characteristic
2	#20260456	C₂	1	1	yes	maximal, characteristic
3	#346×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095486	C₄	1	2	yes	—
5	C₄×#11715895	C₄	1	2	yes	—
6	C₂×C₁₆×C₉₇	C₄	1	2	yes	—
7	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
8	C₂×#11715895	—	2	2	no	—
9	#13095486	—	2	2	no	—
10	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
12	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₉₇	C₈	1	4	yes	complemented
14	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
15	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
16	C₈×C₉₇	C₂×C₈	1	2	yes	—
17	C₈×C₉₇	M₁₆	1	2	yes	—
18	#11715895	—	4	2	no	—
19	C₂²×C₉₇	#97	1	1	yes	characteristic
20	C₄×C₉₇	—	2	2	no	—
21	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
22	C₄×C₉₇	#97	1	1	yes	characteristic
23	C₂×C₉₇	#322	1	1	yes	characteristic
24	C₂×C₉₇	#347	1	2	yes	—
25	#1313	—	97	1	no	maximal, Sylow
26	C₉₇	#1313	1	1	yes	characteristic, complemented, Sylow
27	#346	—	97	1	no	—
28	#346	D₉₇	1	1	yes	characteristic
29	#322	—	97	1	no	—
30	C₂×C₁₆	—	97	2	no	—
31	C₂×C₁₆	Dic₉₇	1	2	yes	—
32	C₄×C₈	—	97	2	no	—
33	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
34	C₂×C₈	—	194	2	no	—
35	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
36	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
37	C₄²	C₄×D₉₇	1	1	yes	characteristic
38	C₁₆	—	194	2	no	—
39	C₁₆	#11715895	1	4	yes	complemented
40	C₂×C₄	C₄×Dic₉₇	1	1	yes	characteristic
41	C₂×C₄	#13095494	1	2	yes	—
42	C₈	—	388	2	no	—
43	C₈	C₂×#11715895	1	2	yes	—
44	C₈	#13095494	1	2	yes	—
45	C₂²	#15690720	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₄×#11715895	1	1	yes	characteristic
48	C₄	#15690720	1	1	yes	characteristic
49	C₂	#20260456	1	1	yes	characteristic
50	C₂	#20260512	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₉₇
commutator	C₄×C₉₇	C₄×C₈
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#346×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	#97
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#346	D₉₇
hypocenter	C₉₇	#1313
LCS	G ▷ C₄×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #346

Complements

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

N	H ≅ G/N	type
13. C₁₆×C₉₇	C₈	semidirect
26. C₉₇	#1313	semidirect
39. C₁₆	#11715895	semidirect

Maximal subgroups

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

subgroup	index
#20260511	2
#20260456	2
#346×C₉₇	2
#1313	97