Subgroups — #27139801

Order 12416.

68 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	#20260486	C₂	1	1	yes	maximal, characteristic
3	#334×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095486	C₄	1	1	yes	characteristic
5	C₂×#13095486	C₄	1	1	yes	characteristic
6	C₄⋊#11715895	—	2	1	no	—
7	C₄⋊C₈×C₉₇	—	2	1	no	—
8	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
9	C₂×#11715895	—	4	1	no	—
10	#13095486	—	2	1	no	—
11	#13095486	C₈	1	2	yes	complemented
12	C₂×C₈×C₉₇	—	4	1	no	—
13	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₉₇	Q₈	1	1	yes	characteristic
15	C₄²×C₉₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
17	C₂×C₄×C₉₇	SD₁₆	1	1	yes	characteristic
18	C₂×C₄×C₉₇	SD₁₆	1	1	yes	characteristic
19	C₈×C₉₇	—	2	1	no	—
20	C₈×C₉₇	—	8	1	no	—
21	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
22	C₈×C₉₇	M₁₆	1	1	yes	characteristic
23	#11715895	—	8	1	no	—
24	C₂²×C₉₇	#106	1	1	yes	characteristic
25	C₄×C₉₇	—	2	1	no	—
26	C₄×C₉₇	—	2	1	no	—
27	C₄×C₉₇	C₄⋊C₈	1	1	yes	characteristic
28	C₄×C₉₇	#108	1	1	yes	characteristic
29	C₂×C₉₇	#334	1	1	yes	characteristic
30	C₂×C₉₇	#365	1	1	yes	characteristic
31	C₂×C₉₇	#365	1	1	yes	characteristic
32	#1368	—	97	1	no	maximal, Sylow
33	C₉₇	#1368	1	1	yes	characteristic, complemented, Sylow
34	#346	—	97	1	no	—
35	#334	—	97	1	no	—
36	#334	D₉₇	1	1	yes	characteristic
37	C₂×C₁₆	—	97	1	no	—
38	C₂×C₁₆	—	97	1	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	194	1	no	—
41	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
42	C₂×C₈	—	4	1	no	—
43	C₂×C₈	—	388	1	no	—
44	C₂×C₈	Dic₁₉₄	1	1	yes	characteristic
45	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
46	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
47	C₁₆	—	97	2	no	—
48	C₁₆	—	194	1	no	—
49	C₂×C₄	#13095490	1	1	yes	characteristic
50	C₂×C₄	#13095498	1	1	yes	characteristic
51	C₂×C₄	#13095499	1	1	yes	characteristic
52	C₈	—	2	1	no	—
53	C₈	—	8	1	no	—
54	C₈	—	776	1	no	—
55	C₈	C₈×D₉₇	1	1	yes	characteristic
56	C₈	#13095494	1	1	yes	characteristic
57	C₂²	#15690741	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	#15690739	1	1	yes	characteristic
61	C₄	#15690745	1	1	yes	characteristic
62	C₂	#20260485	1	1	yes	characteristic
63	C₂	#20260557	1	1	yes	characteristic
64	C₂	#20260559	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690741
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#334×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	#106
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#334	D₉₇
hypocenter	C₉₇	#1368
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

Complements

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

N	H ≅ G/N	type
11. #13095486	C₈	semidirect
33. C₉₇	#1368	semidirect

Maximal subgroups

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

subgroup	index
#20260511	2
#20260486	2
#334×C₉₇	2
#1368	97