Group #21423441 — subgroups

Subgroups — #21423441

Order 7552.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16837684	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837599	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837153	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837148	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837146	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#492	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745670	—	2	1	no	—
9	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
10	#14273658	—	2	1	no	—
11	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₅₉⋊C₈	—	2	1	no	—
15	#14273650	—	2	1	no	—
16	#14273648	—	2	1	no	—
17	C₈×Dic₅₉	—	2	1	no	—
18	C₂×SD₁₆×C₅₉	—	2	1	no	—
19	C₂×SD₁₆×C₅₉	—	2	1	no	—
20	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
21	#127×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	C₂×D₄×C₅₉	—	2	1	no	—
25	C₂×D₄×C₅₉	—	2	1	no	—
26	C₂×D₄×C₅₉	—	2	1	no	—
27	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
28	SD₁₆×C₅₉	—	4	2	no	—
29	SD₁₆×C₅₉	—	4	2	no	—
30	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₄⋊C₄×C₅₉	—	2	1	no	—
33	C₄⋊C₄×C₅₉	—	2	1	no	—
34	C₄²×C₅₉	C₂³	1	1	yes	characteristic
35	#11745670	—	4	2	no	—
36	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
37	Dic₅₉⋊C₄	—	2	1	no	—
38	C₄×Dic₅₉	—	2	1	no	—
39	C₂×C₅₉⋊C₈	—	2	1	no	—
40	C₂×C₅₉⋊C₈	—	2	1	no	—
41	C₂×C₅₉⋊C₈	—	2	1	no	—
42	C₂³×C₅₉	—	4	1	no	—
43	Q₈×C₅₉	—	4	1	no	—
44	Q₈×C₅₉	—	4	1	no	—
45	D₄×C₅₉	—	4	1	no	—
46	D₄×C₅₉	—	4	1	no	—
47	D₄×C₅₉	—	4	1	no	—
48	D₄×C₅₉	—	4	1	no	—
49	D₄×C₅₉	—	8	1	no	—
50	C₂×C₄×C₅₉	—	4	1	no	—
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
54	C₂×Dic₅₉	—	2	1	no	—
55	C₂×Dic₅₉	—	2	1	no	—
56	Dic₁₁₈	—	2	1	no	—
57	Dic₁₁₈	SD₁₆	1	2	yes	complemented
58	C₈×C₅₉	—	2	1	no	—
59	C₈×C₅₉	SD₁₆	1	2	yes	complemented
60	C₅₉⋊C₈	—	4	1	no	—
61	C₅₉⋊C₈	—	4	1	no	—
62	C₅₉⋊C₈	—	4	1	no	—
63	C₂²×C₅₉	—	8	1	no	—
64	C₂²×C₅₉	—	8	1	no	—
65	C₂²×C₅₉	—	8	1	no	—
66	C₂²×C₅₉	#121	1	1	yes	characteristic
67	C₄×C₅₉	—	2	1	no	—
68	C₄×C₅₉	—	8	1	no	—
69	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₄×C₅₉	#135	1	1	yes	characteristic
72	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
73	Dic₅₉	—	2	2	no	—
74	Dic₅₉	—	4	1	no	—
75	#1666	—	59	1	no	maximal, Sylow
76	C₂×C₅₉	—	16	1	no	—
77	C₂×C₅₉	#460	1	1	yes	characteristic
78	C₂×C₅₉	#465	1	1	yes	characteristic
79	C₂×C₅₉	#471	1	1	yes	characteristic
80	#492	D₅₉	1	1	yes	characteristic
81	#475	—	59	1	no	—
82	#460	—	59	1	no	—
83	C₈×Q₈	—	59	1	no	—
84	#334	—	59	1	no	—
85	#332	—	59	1	no	—
86	#331	—	59	1	no	—
87	C₅₉	#1666	1	1	yes	characteristic, complemented, Sylow
88	C₂×SD₁₆	—	2	1	no	—
89	C₂×SD₁₆	—	2	1	no	—
90	C₂×SD₁₆	—	118	1	no	—
91	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
92	#127	C₂×D₅₉	1	1	yes	characteristic
93	C₄×Q₈	—	59	1	no	—
94	#106	—	118	1	no	—
95	C₄⋊C₈	—	59	1	no	—
96	C₄⋊C₈	—	59	1	no	—
97	C₄⋊C₈	—	59	1	no	—
98	C₄⋊C₈	—	118	1	no	—
99	#103	—	118	1	no	—
100	#102	—	118	1	no	—
101	C₄×C₈	—	118	1	no	—
102	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	59	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	SD₁₆	—	4	2	no	—
109	SD₁₆	—	4	2	no	—
110	SD₁₆	—	236	2	no	—
111	C₂×C₈	—	118	1	no	—
112	C₂×C₈	—	118	1	no	—
113	C₂×C₈	—	118	1	no	—
114	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	59	1	no	—
119	C₄⋊C₄	—	118	1	no	—
120	C₄²	—	118	1	no	—
121	C₄²	C₂²×D₅₉	1	1	yes	characteristic
122	C₂³	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	59	2	no	—
126	Q₈	—	118	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	8	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	118	1	no	—
134	C₂×C₄	—	118	1	no	—
135	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
136	C₂×C₄	#11745687	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	236	1	no	—
140	C₈	—	236	1	no	—
141	C₈	—	236	1	no	—
142	C₈	#11745670	1	2	yes	complemented
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	#14273698	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	8	1	no	—
149	C₄	—	118	2	no	—
150	C₄	—	236	1	no	—
151	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
152	C₄	C₂×#11745670	1	1	yes	characteristic
153	C₄	#14273748	1	1	yes	characteristic
154	C₄	#14273753	1	1	yes	characteristic
155	C₂	—	16	1	no	—
156	C₂	#16837599	1	1	yes	characteristic
157	C₂	#16837628	1	1	yes	characteristic
158	C₂	#16837662	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837684

C₂

yes

maximal, characteristic, complemented

#16837599

C₂

yes

maximal, characteristic

C₈×Dic₁₁₈

C₂

yes

maximal, characteristic, complemented

#16837153

C₂

yes

maximal, characteristic, complemented

#16837148

C₂

yes

maximal, characteristic, complemented

#16837146

C₂

yes

maximal, characteristic

C₅₉×#492

C₂

yes

maximal, characteristic

C₂×#11745670

—

C₄×Dic₁₁₈

C₂²

yes

characteristic

#14273658

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

#14273650

—

#14273648

—

C₈×Dic₅₉

—

C₂×SD₁₆×C₅₉

—

C₂×SD₁₆×C₅₉

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#127×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×Dic₁₁₈

D₄

yes

characteristic

SD₁₆×C₅₉

—

SD₁₆×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745670

—

C₄⋊Dic₅₉

D₄

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

SD₁₆

yes

complemented

C₈×C₅₉

—

C₈×C₅₉

SD₁₆

yes

complemented

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#1666

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

#475

—

#460

—

C₈×Q₈

—

#334

—

#332

—

#331

—

C₅₉

#1666

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

118

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

#127

C₂×D₅₉

yes

characteristic

C₄×Q₈

—

#106

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

#103

—

118

—

100

#102

—

118

—

101

C₄×C₈

—

118

—

102

C₄×C₈

C₂×D₅₉

yes

characteristic

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

SD₁₆

—

109

SD₁₆

—

110

SD₁₆

—

236

—

111

C₂×C₈

—

118

—

112

C₂×C₈

—

118

—

113

C₂×C₈

—

118

—

114

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

115

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

118

—

120

C₄²

—

118

—

121

C₄²

C₂²×D₅₉

yes

characteristic

122

C₂³

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

118

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

118

—

134

C₂×C₄

—

118

—

135

C₂×C₄

D₄×D₅₉

yes

characteristic

136

C₂×C₄

#11745687

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

138

C₈

—

139

C₈

—

236

—

140

C₈

—

236

—

141

C₈

—

236

—

142

C₈

#11745670

yes

complemented

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#14273698

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

118

—

150

C₄

—

236

—

151

C₄

SD₁₆×D₅₉

yes

characteristic

152

C₄

C₂×#11745670

yes

characteristic

153

C₄

#14273748

yes

characteristic

154

C₄

#14273753

yes

characteristic

155

C₂

—

156

C₂

#16837599

yes

characteristic

157

C₂

#16837628

yes

characteristic

158

C₂

#16837662

yes

characteristic

	subgroup	G/H
center	C₂²	#14273698
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#492	C₂
socle	C₂²×C₅₉	#121
hypercenter	#492	D₅₉
hypocenter	C₅₉	#1666
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#14273698

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#492

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #16837684	C₂	semidirect
3. C₈×Dic₁₁₈	C₂	semidirect
4. #16837153	C₂	semidirect
5. #16837148	C₂	semidirect
57. Dic₁₁₈	SD₁₆	semidirect
59. C₈×C₅₉	SD₁₆	semidirect
87. C₅₉	#1666	semidirect
142. C₈	#11745670	semidirect

H ≅ G/N

type

1. #16837684

C₂

semidirect

3. C₈×Dic₁₁₈

C₂

semidirect

4. #16837153

C₂

semidirect

5. #16837148

C₂

semidirect

57. Dic₁₁₈

SD₁₆

semidirect

59. C₈×C₅₉

SD₁₆

semidirect

87. C₅₉

#1666

semidirect

142. C₈

#11745670

semidirect

Subgroups — #21423441

Conjugacy classes of subgroups

Complements