Group #29600642 — subgroups

Subgroups — #29600642

Order 15616.

166 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₆×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	#21444014	C₂	1	2	yes	maximal
3	C₄×#14519683	C₂	1	1	yes	maximal, characteristic
4	C₃₂×Dic₆₁	C₂	1	2	yes	maximal
5	C₆₁×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₂₂	C₄	1	1	yes	characteristic
7	C₂×#14519683	C₂²	1	1	yes	characteristic
8	C₂×#14519683	C₂²	1	1	yes	characteristic
9	C₄⋊#11758185	C₄	1	1	yes	characteristic
10	#16839580	C₄	1	2	yes	—
11	C₁₆×Dic₆₁	C₂²	1	2	yes	—
12	C₂×C₃₂×C₆₁	C₂²	1	2	yes	—
13	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₂₂	C₈	1	1	yes	characteristic
15	C₂×#11758185	C₂×C₄	1	2	yes	—
16	C₄⋊#92418	C₈	1	1	yes	characteristic
17	#14519706	C₈	1	2	yes	—
18	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
19	#14519683	—	2	1	no	—
20	#14519683	Q₈	1	2	yes	complemented
21	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₃₂×C₆₁	—	2	2	no	—
25	C₂×Dic₁₂₂	C₁₆	1	1	yes	characteristic
26	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
28	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
29	C₄⋊Dic₆₁	C₁₆	1	1	yes	characteristic
30	#11758196	C₁₆	1	2	yes	—
31	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
32	C₂×#92418	C₂×C₈	1	2	yes	—
33	C₁₆×C₆₁	—	2	1	no	—
34	C₁₆×C₆₁	C₂×Q₈	1	1	yes	characteristic
35	C₁₆×C₆₁	#41	1	1	yes	characteristic
36	#11758185	—	2	2	no	—
37	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
40	C₂×Dic₆₁	C₂×C₁₆	1	2	yes	—
41	C₂×Dic₆₁	C₂×C₁₆	1	2	yes	—
42	Dic₁₂₂	C₃₂	1	4	yes	complemented
43	C₈×C₆₁	—	2	1	no	—
44	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
45	C₈×C₆₁	#131	1	1	yes	characteristic
46	#92418	—	2	2	no	—
47	Q₈×C₃₂	—	61	1	no	maximal, Sylow
48	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
49	C₄×C₆₁	—	2	1	no	—
50	C₄×C₆₁	C₂×C₃₂	1	2	yes	—
51	C₄×C₆₁	C₈×Q₈	1	1	yes	characteristic
52	C₄×C₆₁	#504	1	1	yes	characteristic
53	Dic₆₁	—	2	2	no	—
54	Dic₆₁	C₂×C₃₂	1	4	yes	—
55	Q₈×C₁₆	—	61	1	no	—
56	C₄⋊C₃₂	—	61	2	no	—
57	C₄⋊C₃₂	D₆₁	1	1	yes	characteristic
58	C₄×C₃₂	—	61	2	no	—
59	C₄×C₃₂	—	61	1	no	—
60	C₂×C₆₁	Q₈×C₁₆	1	1	yes	characteristic
61	C₂×C₆₁	C₂²×C₃₂	1	1	yes	characteristic
62	C₂×C₆₁	#2258	1	1	yes	characteristic
63	C₈×Q₈	—	61	1	no	—
64	C₂×C₃₂	—	61	1	no	—
65	C₂×C₃₂	—	61	1	no	—
66	C₂×C₃₂	C₂×D₆₁	1	2	yes	—
67	C₄⋊C₁₆	—	61	2	no	—
68	C₄⋊C₁₆	—	61	1	no	—
69	C₄×C₁₆	—	61	2	no	—
70	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
71	C₆₁	Q₈×C₃₂	1	1	yes	characteristic, complemented, Sylow
72	C₄×Q₈	—	61	1	no	—
73	C₂×C₁₆	—	61	2	no	—
74	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
76	C₄⋊C₈	—	61	2	no	—
77	C₄⋊C₈	—	61	1	no	—
78	C₄×C₈	—	61	2	no	—
79	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
80	C₃₂	—	2	2	no	—
81	C₃₂	—	61	2	no	—
82	C₃₂	—	122	1	no	—
83	C₂×Q₈	—	61	1	no	—
84	C₂×C₈	—	61	2	no	—
85	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
86	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
87	C₄⋊C₄	—	61	2	no	—
88	C₄⋊C₄	—	61	1	no	—
89	C₄²	—	61	2	no	—
90	C₄²	C₈×D₆₁	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	122	2	no	—
93	C₁₆	#11758224	1	1	yes	characteristic
94	C₁₆	Q₈×D₆₁	1	1	yes	characteristic
95	Q₈	—	61	4	no	—
96	C₂×C₄	—	61	2	no	—
97	C₂×C₄	—	61	2	no	—
98	C₂×C₄	C₁₆×D₆₁	1	1	yes	characteristic
99	C₂×C₄	C₁₆×D₆₁	1	1	yes	characteristic
100	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	122	2	no	—
103	C₈	#14519742	1	1	yes	characteristic
104	C₈	#14519785	1	1	yes	characteristic
105	C₂²	C₂×C₁₆×D₆₁	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	61	4	no	—
108	C₄	—	122	2	no	—
109	C₄	C₃₂×D₆₁	1	2	yes	—
110	C₄	#16839886	1	1	yes	characteristic
111	C₄	#16840180	1	1	yes	characteristic
112	C₂	#21447721	1	1	yes	characteristic
113	C₂	C₂×C₃₂×D₆₁	1	1	yes	characteristic
114	C₂	#21448114	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₂₂

C₂

yes

maximal, characteristic

#21444014

C₂

yes

maximal

C₄×#14519683

C₂

yes

maximal, characteristic

C₃₂×Dic₆₁

C₂

yes

maximal

C₆₁×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₈×Dic₁₂₂

C₄

yes

characteristic

C₂×#14519683

C₂²

yes

characteristic

C₂×#14519683

C₂²

yes

characteristic

C₄⋊#11758185

C₄

yes

characteristic

#16839580

C₄

yes

—

C₁₆×Dic₆₁

C₂²

yes

—

C₂×C₃₂×C₆₁

C₂²

yes

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₄×Dic₁₂₂

C₈

yes

characteristic

C₂×#11758185

C₂×C₄

yes

—

C₄⋊#92418

C₈

yes

characteristic

#14519706

C₈

yes

—

C₈×Dic₆₁

C₂×C₄

yes

—

#14519683

—

#14519683

Q₈

yes

complemented

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₃₂×C₆₁

—

C₂×Dic₁₂₂

C₁₆

yes

characteristic

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄⋊Dic₆₁

C₁₆

yes

characteristic

#11758196

C₁₆

yes

—

C₄×Dic₆₁

C₂×C₈

yes

—

C₂×#92418

C₂×C₈

yes

—

C₁₆×C₆₁

—

C₁₆×C₆₁

C₂×Q₈

yes

C₁₆×C₆₁

yes

—

C₂×C₄×C₆₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×Dic₆₁

C₂×C₁₆

yes

—

C₂×Dic₆₁

C₂×C₁₆

yes

—

Dic₁₂₂

C₃₂

yes

complemented

C₈×C₆₁

—

C₈×C₆₁

C₄×Q₈

yes

C₈×C₆₁

yes

—

Q₈×C₃₂

—

maximal, Sylow

C₂²×C₆₁

C₂²×C₁₆

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂×C₃₂

yes

—

C₄×C₆₁

C₈×Q₈

yes

characteristic

C₄×C₆₁

#504

yes

characteristic

Dic₆₁

—

Dic₆₁

C₂×C₃₂

yes

—

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

D₆₁

yes

characteristic

C₄×C₃₂

—

C₄×C₃₂

—

C₂×C₆₁

Q₈×C₁₆

yes

characteristic

C₂×C₆₁

C₂²×C₃₂

yes

characteristic

C₂×C₆₁

#2258

yes

characteristic

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₆₁

yes

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₆₁

Q₈×C₃₂

yes

characteristic, complemented, Sylow

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₆₁

yes

characteristic

C₂×C₁₆

C₂²×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₆₁

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

122

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₈×D₆₁

yes

characteristic

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₈×D₆₁

yes

characteristic

C₁₆

—

C₁₆

—

122

—

C₁₆

#11758224

yes

characteristic

C₁₆

Q₈×D₆₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₁₆×D₆₁

yes

characteristic

C₂×C₄

C₁₆×D₆₁

yes

characteristic

100

C₂×C₄

C₂×C₈×D₆₁

yes

characteristic

101

C₈

—

102

C₈

—

122

—

103

C₈

#14519742

yes

characteristic

104

C₈

#14519785

yes

characteristic

105

C₂²

C₂×C₁₆×D₆₁

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

122

—

109

C₄

C₃₂×D₆₁

yes

—

110

C₄

#16839886

yes

characteristic

111

C₄

#16840180

yes

characteristic

112

C₂

#21447721

yes

characteristic

113

C₂

C₂×C₃₂×D₆₁

yes

characteristic

114

C₂

#21448114

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₆₁
commutator	C₂×C₆₁	C₂²×C₃₂
Fitting	C₆₁×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₆₁	C₂²×C₁₆
cosocle	C₂×C₁₆×C₆₁	C₂³
hypercenter	C₄⋊C₃₂	D₆₁
hypocenter	C₆₁	Q₈×C₃₂
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₆₁

commutator

C₂×C₆₁

C₂²×C₃₂

Fitting

C₆₁×C₄⋊C₃₂

C₂

socle, prime radical

C₂²×C₆₁

C₂²×C₁₆

cosocle

C₂×C₁₆×C₆₁

C₂³

hypercenter

C₄⋊C₃₂

D₆₁

hypocenter

C₆₁

Q₈×C₃₂

LCS

G ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

N	H ≅ G/N	type
20. #14519683	Q₈	semidirect
42. Dic₁₂₂	C₃₂	semidirect
71. C₆₁	Q₈×C₃₂	semidirect

H ≅ G/N

type

20. #14519683

Q₈

semidirect

42. Dic₁₂₂

C₃₂

semidirect

71. C₆₁

Q₈×C₃₂

semidirect

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

subgroup	index
C₁₆×Dic₁₂₂	2
#21444014	2
C₄×#14519683	2
C₃₂×Dic₆₁	2
C₆₁×C₄⋊C₃₂	2
Q₈×C₃₂	61

subgroup

index

C₁₆×Dic₁₂₂

#21444014

C₄×#14519683

C₃₂×Dic₆₁

C₆₁×C₄⋊C₃₂

Q₈×C₃₂

Subgroups — #29600642

Conjugacy classes of subgroups

Complements

Maximal subgroups