Subgroups — #27139872

Order 12416.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260569	C₂	1	2	yes	maximal
2	#346×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#15690716	C₄	1	4	yes	complemented
4	C₂×C₁₆×C₉₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₉₇	C₄	1	1	yes	characteristic
7	C₂×C₈×C₉₇	C₈	1	1	yes	characteristic
8	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
9	C₄²×C₉₇	C₈	1	1	yes	characteristic
10	C₁₆×C₉₇	C₈	1	2	yes	—
11	C₁₆×C₉₇	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₉₇	—	2	1	no	—
13	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
14	C₈×C₉₇	—	2	1	no	—
15	C₈×C₉₇	C₄²	1	1	yes	characteristic
16	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
17	C₂²×C₉₇	M₃₂	1	1	yes	characteristic
18	C₄×C₉₇	—	4	1	no	—
19	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
20	C₄×C₉₇	M₃₂	1	1	yes	characteristic
21	C₂×C₉₇	—	2	1	no	—
22	C₂×C₉₇	#346	1	1	yes	characteristic
23	#1398	—	97	1	no	maximal, Sylow
24	C₉₇	#1398	1	1	yes	characteristic, complemented, Sylow
25	M₆₄	—	97	2	no	—
26	#346	D₉₇	1	1	yes	characteristic
27	C₂×C₁₆	Dic₉₇	1	1	yes	characteristic
28	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
29	C₄×C₈	Dic₉₇	1	1	yes	characteristic
30	C₃₂	—	97	4	no	—
31	C₂×C₈	#11715895	1	1	yes	characteristic
32	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
33	C₄²	#11715895	1	1	yes	characteristic
34	C₁₆	#11715895	1	2	yes	—
35	C₁₆	C₄×D₉₇	1	2	yes	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	C₂×#11715895	1	1	yes	characteristic
38	C₈	—	2	1	no	—
39	C₈	C₄×Dic₉₇	1	1	yes	characteristic
40	C₈	C₂×#11715895	1	1	yes	characteristic
41	C₂²	#15690750	1	1	yes	characteristic
42	C₄	—	4	1	no	—
43	C₄	C₄×#11715895	1	1	yes	characteristic
44	C₄	#15690750	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#20260511	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₉₇
commutator	C₄×C₉₇	C₄×C₈
Frattini	C₂×C₁₆	C₂×D₉₇
Fitting	#346×C₉₇	C₂
socle	C₂×C₉₇	#346
prime radical	C₂²×C₉₇	M₃₂
cosocle	C₂×C₁₆×C₉₇	C₂²
hypercenter	#346	D₉₇
hypocenter	C₉₇	#1398
LCS	G ▷ C₄×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
3. #15690716	C₄	semidirect
24. C₉₇	#1398	semidirect

Maximal subgroups

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

subgroup	index
#20260569	2
#346×C₉₇	2
#1398	97