Group #24918119 — subgroups

Subgroups — #24918119

Order 10624.

338 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 117 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	#19128312	C₂	1	2	yes	maximal
2	#19128103	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128069	C₂	1	2	yes	maximal, complemented
4	#19127956	C₂	1	2	yes	maximal
5	#19127939	C₂	1	2	yes	maximal
6	#19127808	C₂	1	2	yes	maximal, complemented
7	Q₁₆×Dic₈₃	C₂	1	2	yes	maximal, complemented
8	#19127655	C₂	1	1	yes	maximal, characteristic
9	#494×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13050573	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
12	#14581787	—	2	4	no	—
13	#14581784	C₂²	1	2	yes	—
14	#14581782	—	2	4	no	—
15	#14581768	C₂²	1	4	yes	—
16	Q₈×Dic₈₃	C₂²	1	4	yes	—
17	#14581749	C₂²	1	1	yes	characteristic
18	#14581726	C₂²	1	2	yes	—
19	#14581724	C₂²	1	2	yes	—
20	#14581716	C₂²	1	4	yes	—
21	#14581716	C₂²	1	4	yes	—
22	#14581703	C₂²	1	1	yes	characteristic
23	#14581702	C₂²	1	1	yes	characteristic
24	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
25	C₂×Q₁₆×C₈₃	C₂²	1	2	yes	—
26	C₂×Q₁₆×C₈₃	C₂²	1	2	yes	—
27	C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
28	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
30	#13050573	—	2	2	no	—
31	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
32	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
33	C₈×D₈₃	—	2	2	no	—
34	C₄⋊Dic₈₃	—	2	4	no	—
35	C₄⋊Dic₈₃	C₂³	1	4	yes	—
36	Dic₈₃⋊C₄	—	2	4	no	—
37	Dic₈₃⋊C₄	—	2	4	no	—
38	Dic₈₃⋊C₄	D₄	1	2	yes	—
39	(C₂×D₈₃)⋊C₄	—	2	4	no	—
40	(C₂×D₈₃)⋊C₄	D₄	1	2	yes	—
41	C₄×Dic₈₃	—	2	4	no	—
42	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₈₃	C₂³	1	4	yes	—
44	Q₁₆×C₈₃	—	2	4	no	—
45	Q₁₆×C₈₃	—	2	4	no	—
46	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₈₃	—	2	4	no	—
49	C₄²×C₈₃	C₂³	1	1	yes	characteristic
50	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₈₃	—	2	4	no	—
52	Q₈×C₈₃	—	2	4	no	—
53	C₂×C₄×C₈₃	—	2	4	no	—
54	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₈₃	—	2	4	no	—
57	C₂×Dic₈₃	C₂×D₄	1	2	yes	—
58	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
59	C₄×D₈₃	—	2	1	no	—
60	C₄×D₈₃	—	2	1	no	—
61	C₈×C₈₃	#41	1	2	yes	—
62	C₈×C₈₃	#41	1	2	yes	—
63	C₈₃⋊C₈	—	2	1	no	—
64	C₈₃⋊C₈	—	2	1	no	—
65	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₈₃	—	2	1	no	—
67	C₂×D₈₃	—	2	2	no	—
68	C₄×C₈₃	—	2	2	no	—
69	C₄×C₈₃	—	4	4	no	—
70	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
71	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
72	Dic₈₃	—	2	1	no	—
73	Dic₈₃	—	2	2	no	—
74	Dic₈₃	—	4	4	no	—
75	C₂×C₈₃	#532	1	1	yes	characteristic
76	C₂×C₈₃	#578	1	2	yes	—
77	D₈₃	—	4	1	no	—
78	#3177	—	83	1	no	maximal, Sylow
79	C₈₃	#3177	1	1	yes	characteristic, complemented, Sylow
80	#541	—	83	2	no	—
81	#499	—	83	1	no	—
82	#494	D₈₃	1	1	yes	characteristic, complemented
83	#491	—	83	2	no	—
84	#470	—	83	2	no	—
85	#467	—	83	2	no	—
86	#441	—	83	2	no	—
87	C₄×Q₁₆	—	83	2	no	—
88	#405	—	83	1	no	—
89	C₂×Q₁₆	C₂×D₈₃	1	2	yes	—
90	C₂×Q₁₆	C₂×D₈₃	1	2	yes	—
91	C₂×M₁₆	—	83	1	no	—
92	C₂²×C₈	—	83	1	no	—
93	C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
94	#126	—	166	4	no	—
95	#125	—	83	2	no	—
96	#125	—	166	4	no	—
97	C₂²⋊Q₈	—	83	4	no	—
98	C₄×Q₈	—	83	4	no	—
99	#117	—	83	1	no	—
100	#107	—	83	2	no	—
101	#106	—	83	2	no	—
102	#103	—	83	4	no	—
103	#103	—	83	4	no	—
104	#97	—	83	1	no	—
105	#97	—	83	1	no	—
106	C₄×C₈	—	83	1	no	—
107	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
108	C₂×Q₈	C₂²×D₈₃	1	4	yes	—
109	C₂²×C₄	—	83	1	no	—
110	Q₁₆	—	2	4	no	—
111	Q₁₆	—	2	4	no	—
112	M₁₆	—	166	2	no	—
113	C₂×C₈	—	83	1	no	—
114	C₂×C₈	—	83	1	no	—
115	C₂×C₈	—	166	2	no	—
116	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
117	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	83	2	no	—
120	C₄⋊C₄	—	83	4	no	—
121	C₄⋊C₄	—	166	4	no	—
122	C₄⋊C₄	—	166	4	no	—
123	C₄⋊C₄	—	166	4	no	—
124	C₂²⋊C₄	—	83	2	no	—
125	C₂²⋊C₄	—	166	4	no	—
126	C₄²	—	83	1	no	—
127	C₄²	—	166	4	no	—
128	C₄²	C₂²×D₈₃	1	1	yes	characteristic
129	C₂³	—	83	1	no	—
130	Q₈	—	2	4	no	—
131	Q₈	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	83	2	no	—
134	C₂×C₄	—	83	1	no	—
135	C₂×C₄	—	166	4	no	—
136	C₂×C₄	—	166	1	no	—
137	C₂×C₄	—	166	1	no	—
138	C₂×C₄	D₄×D₈₃	1	2	yes	—
139	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
140	C₈	—	166	1	no	—
141	C₈	—	166	1	no	—
142	C₈	#13050589	1	2	yes	—
143	C₈	#13050589	1	2	yes	—
144	C₂²	—	166	1	no	—
145	C₂²	—	166	2	no	—
146	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	4	no	—
149	C₄	—	332	4	no	—
150	C₄	—	166	1	no	—
151	C₄	—	166	2	no	—
152	C₄	C₂×#13050589	1	1	yes	characteristic
153	C₄	C₂×#13050589	1	1	yes	characteristic
154	C₂	—	332	1	no	—
155	C₂	#19128249	1	1	yes	characteristic
156	C₂	#19128514	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19128312

C₂

yes

maximal

#19128103

C₂

yes

maximal, characteristic, complemented

#19128069

C₂

yes

maximal, complemented

#19127956

C₂

yes

maximal

#19127939

C₂

yes

maximal

#19127808

C₂

yes

maximal, complemented

Q₁₆×Dic₈₃

C₂

yes

maximal, complemented

#19127655

C₂

yes

maximal, characteristic

#494×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050573

C₂²

yes

characteristic

C₂×C₈×D₈₃

C₂²

yes

characteristic

#14581787

—

#14581784

C₂²

yes

—

#14581782

—

#14581768

C₂²

yes

—

Q₈×Dic₈₃

C₂²

yes

—

#14581749

C₂²

yes

characteristic

#14581726

C₂²

yes

—

#14581724

C₂²

yes

—

#14581716

C₂²

yes

—

#14581716

C₂²

yes

—

#14581703

C₂²

yes

characteristic

#14581702

C₂²

yes

characteristic

C₈×Dic₈₃

C₂²

yes

characteristic

C₂×Q₁₆×C₈₃

C₂²

yes

—

C₂×Q₁₆×C₈₃

C₂²

yes

—

C₄⋊Q₈×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₄×D₈₃

C₂³

yes

characteristic

#13050573

—

C₂×C₈₃⋊C₈

C₂³

yes

characteristic

C₂×C₈₃⋊C₈

C₂³

yes

characteristic

C₈×D₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂³

yes

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

D₄

yes

—

(C₂×D₈₃)⋊C₄

—

(C₂×D₈₃)⋊C₄

D₄

yes

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₂³

yes

characteristic

C₂×Q₈×C₈₃

C₂³

yes

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂²×D₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×D₄

yes

—

C₂×Dic₈₃

C₂×D₄

yes

characteristic

C₄×D₈₃

—

C₄×D₈₃

—

C₈×C₈₃

#41

yes

—

C₈×C₈₃

#41

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#41

yes

characteristic

C₄×C₈₃

C₂×#41

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

#532

yes

characteristic

C₂×C₈₃

#578

yes

—

D₈₃

—

#3177

—

maximal, Sylow

C₈₃

#3177

yes

characteristic, complemented, Sylow

#541

—

#499

—

#494

D₈₃

yes

characteristic, complemented

—

—

—

—

C₄×Q₁₆

—

#405

—

C₂×Q₁₆

C₂×D₈₃

yes

—

C₂×Q₁₆

C₂×D₈₃

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₈₃

yes

—

#126

—

166

—

#125

—

#125

—

166

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

100

#107

—

101

#106

—

102

#103

—

103

#103

—

104

#97

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

C₂×D₈₃

yes

characteristic

108

C₂×Q₈

C₂²×D₈₃

yes

—

109

C₂²×C₄

—

110

Q₁₆

—

111

Q₁₆

—

112

M₁₆

—

166

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

166

—

116

C₂×C₈

C₂²×D₈₃

yes

characteristic

117

C₂×C₈

C₂²×D₈₃

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

166

—

122

C₄⋊C₄

—

166

—

123

C₄⋊C₄

—

166

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

166

—

126

C₄²

—

127

C₄²

—

166

—

128

C₄²

C₂²×D₈₃

yes

characteristic

129

C₂³

—

130

Q₈

—

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

166

—

136

C₂×C₄

—

166

—

137

C₂×C₄

—

166

—

138

C₂×C₄

D₄×D₈₃

yes

—

139

C₂×C₄

C₂³×D₈₃

yes

characteristic

140

C₈

—

166

—

141

C₈

—

166

—

142

C₈

#13050589

yes

—

143

C₈

#13050589

yes

—

144

C₂²

—

166

—

145

C₂²

—

166

—

146

C₂²

C₂×D₄×D₈₃

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

332

—

150

C₄

—

166

—

151

C₄

—

166

—

152

C₄

C₂×#13050589

yes

characteristic

153

C₄

C₂×#13050589

yes

characteristic

154

C₂

—

332

—

155

C₂

#19128249

yes

characteristic

156

C₂

#19128514

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#494×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂²×D₈₃	C₂×D₄
hypercenter	#494	D₈₃
hypocenter	C₈₃	#3177
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₈₃

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

#494×C₈₃

C₂

socle

C₂²×C₈₃

C₂²×D₄

prime radical

C₂²×D₈₃

C₂×D₄

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #19128103	C₂	semidirect
3. #19128069	C₂	semidirect
6. #19127808	C₂	semidirect
7. Q₁₆×Dic₈₃	C₂	semidirect
9. #494×C₈₃	C₂	semidirect
79. C₈₃	#3177	semidirect
82. #494	D₈₃	semidirect

H ≅ G/N

type

2. #19128103

C₂

semidirect

3. #19128069

C₂

semidirect

6. #19127808

C₂

semidirect

7. Q₁₆×Dic₈₃

C₂

semidirect

9. #494×C₈₃

C₂

semidirect

79. C₈₃

#3177

semidirect

82. #494

D₈₃

semidirect

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

subgroup	index
#19128312	2
#19128103	2
#19128069	2
#19127956	2
#19127939	2
#19127808	2
Q₁₆×Dic₈₃	2
#19127655	2
#494×C₈₃	2
#3177	83

subgroup

index

Q₁₆×Dic₈₃

#494×C₈₃

Subgroups — #24918119

Conjugacy classes of subgroups

Complements

Maximal subgroups