Subgroups — #27139804

Order 12416.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260556	C₂	1	2	yes	maximal
2	#346×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#15690750	—	2	2	no	—
4	C₂×C₁₆×C₉₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₉₇	C₄	1	1	yes	characteristic
6	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
7	#13095486	—	4	2	no	—
8	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
10	C₄²×C₉₇	Q₈	1	1	yes	characteristic
11	C₁₆×C₉₇	—	2	1	no	—
12	C₁₆×C₉₇	C₈	1	2	yes	—
13	C₂×C₄×C₉₇	—	2	1	no	—
14	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
15	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
16	C₈×C₉₇	M₁₆	1	1	yes	characteristic
17	C₈×C₉₇	SD₁₆	1	2	yes	—
18	C₂²×C₉₇	#108	1	1	yes	characteristic
19	C₄×C₉₇	—	4	1	no	—
20	C₄×C₉₇	C₄⋊C₈	1	1	yes	characteristic
21	C₄×C₉₇	#106	1	1	yes	characteristic
22	C₂×C₉₇	—	2	1	no	—
23	C₂×C₉₇	#334	1	1	yes	characteristic
24	#1369	—	97	1	no	maximal, Sylow
25	C₉₇	#1369	1	1	yes	characteristic, complemented, Sylow
26	#364	—	97	2	no	—
27	#346	D₉₇	1	1	yes	characteristic
28	M₃₂	—	194	2	no	—
29	C₂×C₁₆	Dic₉₇	1	1	yes	characteristic
30	C₂×C₁₆	Dic₉₇	1	1	yes	characteristic
31	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
32	C₂×C₈	D₃₈₈	1	1	yes	characteristic
33	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
34	C₄²	Dic₁₉₄	1	1	yes	characteristic
35	C₁₆	—	2	1	no	—
36	C₁₆	—	388	2	no	—
37	C₁₆	#11715895	1	2	yes	—
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
40	C₈	C₂×#11715895	1	1	yes	characteristic
41	C₈	#13095495	1	1	yes	characteristic
42	C₈	#13095500	1	2	yes	—
43	C₂²	#15690746	1	1	yes	characteristic
44	C₄	—	4	1	no	—
45	C₄	C₄⋊#11715895	1	1	yes	characteristic
46	C₄	#15690742	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#20260486	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690742
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#346×C₉₇	C₂
socle	C₂×C₉₇	#334
prime radical	C₂²×C₉₇	#108
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#346	D₉₇
hypocenter	C₉₇	#1369
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
25. C₉₇	#1369	semidirect

Maximal subgroups

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

subgroup	index
#20260556	2
#346×C₉₇	2
#1369	97