Subgroups — #27144046

Order 12416.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261168	C₂	1	2	yes	maximal, complemented
2	#20260572	C₂	1	2	yes	maximal, complemented
3	D₃₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₆₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690833	—	2	2	no	—
7	#15690753	—	2	2	no	—
8	D₁₅₅₂	C₂²	1	2	yes	—
9	C₁₆×D₉₇	C₂²	1	1	yes	characteristic
10	#15690716	C₂²	1	1	yes	characteristic
11	Q₃₂×C₉₇	C₂²	1	2	yes	—
12	C₃₂×C₉₇	C₂²	1	1	yes	characteristic
13	#13095512	—	4	2	no	—
14	#13095498	—	4	2	no	—
15	D₇₇₆	—	2	2	no	—
16	C₈×D₉₇	D₄	1	1	yes	characteristic
17	#13095486	D₄	1	1	yes	characteristic
18	Q₁₆×C₉₇	—	2	2	no	—
19	C₁₆×C₉₇	C₂³	1	1	yes	characteristic
20	Q₈×C₉₇	—	4	2	no	—
21	D₃₈₈	—	4	2	no	—
22	D₃₈₈	—	8	2	no	—
23	C₄×D₉₇	—	8	2	no	—
24	C₄×D₉₇	D₈	1	1	yes	characteristic
25	C₈×C₉₇	C₂×D₄	1	1	yes	characteristic
26	#11715895	D₈	1	1	yes	characteristic
27	C₂×D₉₇	—	8	2	no	—
28	C₂×D₉₇	D₁₆	1	1	yes	characteristic
29	C₄×C₉₇	—	8	2	no	—
30	C₄×C₉₇	C₂×D₈	1	1	yes	characteristic
31	Dic₉₇	D₁₆	1	1	yes	characteristic
32	C₂×C₉₇	C₂×D₁₆	1	1	yes	characteristic
33	D₉₇	—	2	1	no	—
34	D₉₇	—	16	2	no	—
35	#2262	—	97	1	no	maximal, Sylow
36	C₉₇	#2262	1	1	yes	characteristic, complemented, Sylow
37	#508	—	97	2	no	—
38	Q₆₄	D₉₇	1	1	yes	characteristic, complemented
39	SD₆₄	—	97	2	no	—
40	D₃₂	—	97	1	no	—
41	C₂×C₃₂	—	97	1	no	—
42	#135	—	194	2	no	—
43	Q₃₂	C₂×D₉₇	1	2	yes	—
44	SD₃₂	—	194	2	no	—
45	D₁₆	—	97	2	no	—
46	C₂×C₁₆	—	97	1	no	—
47	C₃₂	—	97	1	no	—
48	C₃₂	C₂×D₉₇	1	1	yes	characteristic
49	#41	—	388	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	388	2	no	—
52	D₈	—	194	2	no	—
53	C₂×C₈	—	97	1	no	—
54	C₁₆	—	97	1	no	—
55	C₁₆	C₂²×D₉₇	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	776	2	no	—
58	D₄	—	388	2	no	—
59	C₂×C₄	—	776	2	no	—
60	C₂×C₄	—	97	1	no	—
61	C₈	—	97	1	no	—
62	C₈	D₄×D₉₇	1	1	yes	characteristic
63	C₂²	—	776	2	no	—
64	C₂²	—	97	1	no	—
65	C₄	—	8	2	no	—
66	C₄	—	97	1	no	—
67	C₄	D₈×D₉₇	1	1	yes	characteristic
68	C₂	—	1552	2	no	—
69	C₂	—	194	1	no	—
70	C₂	D₁₆×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₉₇
commutator, cosocle	C₁₆×C₉₇	C₂³
Frattini	C₁₆	C₂²×D₉₇
Fitting	Q₆₄×C₉₇	C₂
socle	C₂×C₉₇	C₂×D₁₆
hypercenter	Q₆₄	D₉₇
hypocenter	C₉₇	#2262
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #20261168	C₂	semidirect
2. #20260572	C₂	semidirect
3. D₃₁₀₄	C₂	semidirect
4. C₃₂×D₉₇	C₂	semidirect
5. Q₆₄×C₉₇	C₂	semidirect
36. C₉₇	#2262	semidirect
38. Q₆₄	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261168	2
#20260572	2
D₃₁₀₄	2
C₃₂×D₉₇	2
Q₆₄×C₉₇	2
#2262	97