Group #24906968 — subgroups

Subgroups — #24906968

Order 10624.

146 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 51 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	#19128097	C₂	1	1	yes	maximal, characteristic
2	#19127997	C₂	1	1	yes	maximal, characteristic
3	#19127908	C₂	1	1	yes	maximal, characteristic
4	#19127828	C₂	1	1	yes	maximal, characteristic
5	#19127453	C₂	1	1	yes	maximal, characteristic
6	#19127449	C₂	1	1	yes	maximal, characteristic
7	#332×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13050580	—	2	1	no	—
9	#14581792	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
11	#14581726	—	2	1	no	—
12	#14581723	—	2	1	no	—
13	#14581723	—	2	1	no	—
14	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₈₃⋊C₈	—	2	1	no	—
16	#14581717	—	2	1	no	—
17	#14581716	—	2	1	no	—
18	#14581702	—	2	1	no	—
19	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₆₆	—	2	1	no	—
24	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
25	#13050580	—	4	2	no	—
26	C₂×C₈₃⋊C₈	—	2	1	no	—
27	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
29	C₄⋊Dic₈₃	—	2	1	no	—
30	C₄⋊Dic₈₃	—	2	1	no	—
31	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
32	Dic₈₃⋊C₄	—	2	1	no	—
33	C₄×Dic₈₃	—	2	1	no	—
34	C₂×Q₈×C₈₃	—	2	1	no	—
35	C₂×C₈×C₈₃	—	2	1	no	—
36	C₂×C₈×C₈₃	—	2	1	no	—
37	C₄⋊C₄×C₈₃	—	2	1	no	—
38	C₄⋊C₄×C₈₃	—	2	1	no	—
39	C₄²×C₈₃	C₂³	1	1	yes	characteristic
40	Q₈×C₈₃	—	4	1	no	—
41	Q₈×C₈₃	—	4	1	no	—
42	C₂×C₄×C₈₃	—	4	1	no	—
43	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
46	C₂×Dic₈₃	—	2	1	no	—
47	C₂×Dic₈₃	—	2	1	no	—
48	C₂×Dic₈₃	—	4	1	no	—
49	Dic₁₆₆	—	2	1	no	—
50	Dic₁₆₆	—	2	1	no	—
51	Dic₁₆₆	—	8	1	no	—
52	C₈×C₈₃	—	4	1	no	—
53	C₈×C₈₃	—	4	1	no	—
54	C₈₃⋊C₈	—	2	1	no	—
55	C₈₃⋊C₈	—	4	1	no	—
56	C₈₃⋊C₈	Q₁₆	1	2	yes	complemented
57	C₂²×C₈₃	#121	1	1	yes	characteristic
58	C₄×C₈₃	—	2	1	no	—
59	C₄×C₈₃	—	8	1	no	—
60	C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₄×C₈₃	#137	1	1	yes	characteristic
63	C₄×C₈₃	#137	1	1	yes	characteristic
64	Dic₈₃	—	4	1	no	—
65	Dic₈₃	—	4	1	no	—
66	Dic₈₃	—	8	1	no	—
67	C₂×C₈₃	#462	1	1	yes	characteristic
68	C₂×C₈₃	#468	1	1	yes	characteristic
69	C₂×C₈₃	#471	1	1	yes	characteristic
70	#1692	—	83	1	no	maximal, Sylow
71	C₈₃	#1692	1	1	yes	characteristic, complemented, Sylow
72	#498	—	83	1	no	—
73	#477	—	83	1	no	—
74	#462	—	83	1	no	—
75	#446	—	83	1	no	—
76	#335	—	83	1	no	—
77	#333	—	83	1	no	—
78	#332	D₈₃	1	1	yes	characteristic
79	C₂×Q₁₆	—	166	1	no	—
80	C₄⋊Q₈	—	83	1	no	—
81	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
82	C₄×Q₈	—	83	1	no	—
83	#107	—	166	1	no	—
84	#106	—	166	1	no	—
85	#106	—	166	1	no	—
86	C₄⋊C₈	—	83	1	no	—
87	C₄⋊C₈	—	166	1	no	—
88	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
89	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
90	#103	—	166	1	no	—
91	#103	—	166	1	no	—
92	#97	—	166	1	no	—
93	C₄×C₈	—	83	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	83	1	no	—
96	C₂×Q₈	—	166	1	no	—
97	Q₁₆	—	332	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	83	1	no	—
101	C₂×C₈	—	83	1	no	—
102	C₂×C₈	—	166	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	83	1	no	—
106	C₄⋊C₄	—	166	1	no	—
107	C₄⋊C₄	—	166	1	no	—
108	C₄⋊C₄	—	166	1	no	—
109	C₄²	—	166	1	no	—
110	C₄²	C₂²×D₈₃	1	1	yes	characteristic
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	664	1	no	—
114	Q₈	—	166	1	no	—
115	Q₈	—	166	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	332	1	no	—
118	C₂×C₄	—	166	1	no	—
119	C₂×C₄	—	166	1	no	—
120	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
121	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
122	C₂×C₄	#13050590	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	332	1	no	—
126	C₈	—	83	2	no	—
127	C₈	—	166	1	no	—
128	C₂²	#14581765	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	332	1	no	—
132	C₄	—	332	1	no	—
133	C₄	—	664	1	no	—
134	C₄	Q₁₆×D₈₃	1	1	yes	characteristic
135	C₄	#14581816	1	1	yes	characteristic
136	C₄	#14581821	1	1	yes	characteristic
137	C₄	#14581821	1	1	yes	characteristic
138	C₂	#19127910	1	1	yes	characteristic
139	C₂	#19127946	1	1	yes	characteristic
140	C₂	#19127967	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128097

C₂

yes

maximal, characteristic

#19127997

C₂

yes

maximal, characteristic

#19127908

C₂

yes

maximal, characteristic

#19127828

C₂

yes

maximal, characteristic

#19127453

C₂

yes

maximal, characteristic

#19127449

C₂

yes

maximal, characteristic

#332×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050580

—

#14581792

C₂²

yes

characteristic

C₄×Dic₁₆₆

C₂²

yes

—

—

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

Dic₈₃⋊C₈

—

#14581717

—

#14581716

—

#14581702

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

D₄

yes

characteristic

#13050580

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₄

yes

characteristic

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

Q₁₆

yes

complemented

C₂²×C₈₃

#121

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×Q₁₆

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₈₃

yes

characteristic

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₃

yes

characteristic

C₄×Q₈

—

#107

—

166

—

#106

—

166

—

#106

—

166

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄⋊C₈

C₂×D₈₃

yes

characteristic

C₄⋊C₈

C₂×D₈₃

yes

characteristic

#103

—

166

—

#103

—

166

—

#97

—

166

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

Q₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

166

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

166

—

107

C₄⋊C₄

—

166

—

108

C₄⋊C₄

—

166

—

109

C₄²

—

166

—

110

C₄²

C₂²×D₈₃

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

664

—

114

Q₈

—

166

—

115

Q₈

—

166

—

116

C₂×C₄

—

117

C₂×C₄

—

332

—

118

C₂×C₄

—

166

—

119

C₂×C₄

—

166

—

120

C₂×C₄

D₄×D₈₃

yes

characteristic

121

C₂×C₄

D₄×D₈₃

yes

characteristic

122

C₂×C₄

#13050590

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

332

—

126

C₈

—

127

C₈

—

166

—

128

C₂²

#14581765

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

332

—

132

C₄

—

332

—

133

C₄

—

664

—

134

C₄

Q₁₆×D₈₃

yes

characteristic

135

C₄

#14581816

yes

characteristic

136

C₄

#14581821

yes

characteristic

137

C₄

#14581821

yes

characteristic

138

C₂

#19127910

yes

characteristic

139

C₂

#19127946

yes

characteristic

140

C₂

#19127967

yes

characteristic

	subgroup	G/H
center	C₂²	#14581765
commutator, cosocle	C₄²×C₈₃	C₂³
Frattini	C₄²	C₂²×D₈₃
Fitting	#332×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	#121
hypercenter	#332	D₈₃
hypocenter	C₈₃	#1692
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14581765

commutator, cosocle

C₄²×C₈₃

C₂³

Frattini

C₄²

C₂²×D₈₃

Fitting

#332×C₈₃

C₂

C₂²×C₈₃

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
56. C₈₃⋊C₈	Q₁₆	semidirect
71. C₈₃	#1692	semidirect

H ≅ G/N

type

56. C₈₃⋊C₈

Q₁₆

semidirect

71. C₈₃

#1692

semidirect

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

subgroup	index
#19128097	2
#19127997	2
#19127908	2
#19127828	2
#19127453	2
#19127449	2
#332×C₈₃	2
#1692	83

subgroup

index

#332×C₈₃

Subgroups — #24906968

Conjugacy classes of subgroups

Complements

Maximal subgroups