Group #21423560 — subgroups

Subgroups — #21423560

Order 7552.

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

normal subgroups only

This group has 182 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	#16837774	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837683	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837594	C₂	1	1	yes	maximal, characteristic
4	#16837479	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837154	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837149	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745670	—	2	1	no	—
9	C₂×#11745670	—	2	1	no	—
10	C₂×#11745669	—	2	1	no	—
11	C₂×#11745659	—	2	1	no	—
12	#14273727	C₂²	1	1	yes	characteristic
13	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
14	#14273661	—	2	1	no	—
15	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
16	#14273649	—	2	1	no	—
17	#14273648	—	2	1	no	—
18	(C₂×D₅₉)⋊C₈	—	2	1	no	—
19	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	#127×C₅₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₅₉	—	2	1	no	—
24	C₂×D₄×C₅₉	—	2	1	no	—
25	C₂×D₄×C₅₉	—	2	1	no	—
26	C₂×D₂₃₆	D₄	1	1	yes	characteristic
27	C₂×C₄×D₅₉	—	2	1	no	—
28	C₂×Dic₁₁₈	—	2	1	no	—
29	C₂×C₈×C₅₉	—	2	1	no	—
30	C₂×C₈×C₅₉	—	2	1	no	—
31	C₄²×C₅₉	C₂³	1	1	yes	characteristic
32	#11745670	—	4	2	no	—
33	#11745670	—	4	2	no	—
34	#11745669	—	4	2	no	—
35	(C₂×D₅₉)⋊C₄	—	2	1	no	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	C₄⋊Dic₅₉	—	2	1	no	—
38	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
39	C₂×C₅₉⋊C₈	—	2	1	no	—
40	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
42	#11745659	—	4	2	no	—
43	C₂³×C₅₉	—	4	1	no	—
44	C₂²×D₅₉	—	2	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	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
53	C₂×Dic₅₉	—	2	1	no	—
54	C₂×Dic₅₉	—	4	1	no	—
55	D₂₃₆	—	2	1	no	—
56	D₂₃₆	—	2	1	no	—
57	C₄×D₅₉	—	4	1	no	—
58	C₄×D₅₉	—	4	1	no	—
59	Dic₁₁₈	—	4	1	no	—
60	Dic₁₁₈	—	4	1	no	—
61	C₈×C₅₉	—	4	1	no	—
62	C₈×C₅₉	—	4	1	no	—
63	C₅₉⋊C₈	—	2	1	no	—
64	C₅₉⋊C₈	—	4	1	no	—
65	C₅₉⋊C₈	D₈	1	2	yes	complemented
66	C₂²×C₅₉	—	8	1	no	—
67	C₂²×C₅₉	—	8	1	no	—
68	C₂²×C₅₉	—	8	1	no	—
69	C₂²×C₅₉	#121	1	1	yes	characteristic
70	C₂×D₅₉	—	4	1	no	—
71	C₂×D₅₉	—	4	1	no	—
72	C₂×D₅₉	—	4	1	no	—
73	C₄×C₅₉	—	2	1	no	—
74	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
75	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄×C₅₉	#137	1	1	yes	characteristic
78	Dic₅₉	—	4	1	no	—
79	Dic₅₉	—	8	1	no	—
80	#1685	—	59	1	no	maximal, Sylow
81	C₂×C₅₉	—	16	1	no	—
82	C₂×C₅₉	#459	1	1	yes	characteristic
83	C₂×C₅₉	#468	1	1	yes	characteristic
84	C₂×C₅₉	#471	1	1	yes	characteristic
85	D₅₉	—	8	1	no	—
86	#492	—	59	1	no	—
87	#474	—	59	1	no	—
88	#459	—	59	1	no	—
89	#436	—	59	1	no	—
90	#335	—	59	1	no	—
91	#332	—	59	1	no	—
92	#331	D₅₉	1	1	yes	characteristic, complemented
93	C₅₉	#1685	1	1	yes	characteristic, complemented, Sylow
94	C₂×SD₁₆	—	118	1	no	—
95	C₂×SD₁₆	—	118	1	no	—
96	C₂×D₈	—	118	1	no	—
97	C₂×M₁₆	—	118	1	no	—
98	C₄⋊Q₈	—	59	1	no	—
99	#127	C₂×D₅₉	1	1	yes	characteristic
100	C₄×D₄	—	59	1	no	—
101	#107	—	118	1	no	—
102	C₄⋊C₈	—	59	1	no	—
103	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
104	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
105	#102	—	118	1	no	—
106	#102	—	118	1	no	—
107	C₂²⋊C₈	—	118	1	no	—
108	C₄×C₈	—	59	1	no	—
109	C₂×Q₈	—	118	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	59	1	no	—
114	C₂²×C₄	—	118	1	no	—
115	SD₁₆	—	236	2	no	—
116	SD₁₆	—	236	2	no	—
117	D₈	—	236	2	no	—
118	M₁₆	—	236	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	59	1	no	—
122	C₂×C₈	—	59	1	no	—
123	C₂×C₈	—	118	1	no	—
124	C₄⋊C₄	—	59	1	no	—
125	C₄⋊C₄	—	118	1	no	—
126	C₄⋊C₄	—	118	1	no	—
127	C₂²⋊C₄	—	118	1	no	—
128	C₄²	C₂²×D₅₉	1	1	yes	characteristic
129	C₂³	—	4	1	no	—
130	C₂³	—	118	1	no	—
131	Q₈	—	236	1	no	—
132	Q₈	—	236	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	8	1	no	—
138	D₄	—	118	1	no	—
139	D₄	—	118	1	no	—
140	C₂×C₄	—	118	1	no	—
141	C₂×C₄	—	236	1	no	—
142	C₂×C₄	—	236	1	no	—
143	C₂×C₄	—	236	1	no	—
144	C₂×C₄	#11745685	1	1	yes	characteristic
145	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
146	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
147	C₈	—	4	1	no	—
148	C₈	—	4	1	no	—
149	C₈	—	59	2	no	—
150	C₈	—	118	1	no	—
151	C₈	—	236	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	—	8	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	236	1	no	—
156	C₂²	—	236	1	no	—
157	C₂²	—	236	1	no	—
158	C₂²	#14273700	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	472	1	no	—
161	C₄	—	236	1	no	—
162	C₄	D₈×D₅₉	1	1	yes	characteristic
163	C₄	#14273751	1	1	yes	characteristic
164	C₄	#14273754	1	1	yes	characteristic
165	C₄	#14273756	1	1	yes	characteristic
166	C₂	—	16	1	no	—
167	C₂	—	472	1	no	—
168	C₂	#16837596	1	1	yes	characteristic
169	C₂	#16837647	1	1	yes	characteristic
170	C₂	#16837664	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837774

C₂

yes

maximal, characteristic, complemented

#16837683

C₂

yes

maximal, characteristic, complemented

#16837594

C₂

yes

maximal, characteristic

#16837479

C₂

yes

maximal, characteristic, complemented

#16837154

C₂

yes

maximal, characteristic, complemented

#16837149

C₂

yes

maximal, characteristic, complemented

C₅₉×#331

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×D₂₃₆

C₂²

yes

characteristic

#14273661

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273649

—

#14273648

—

(C₂×D₅₉)⋊C₈

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

#127×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₂₃₆

D₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂³

yes

—

—

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

#11745659

—

C₂³×C₅₉

—

C₂²×D₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×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

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

D₈

yes

complemented

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#1685

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

—

—

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#1685

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂×SD₁₆

—

118

—

C₂×D₈

—

118

—

C₂×M₁₆

—

118

—

C₄⋊Q₈

—

#127

C₂×D₅₉

yes

characteristic

100

C₄×D₄

—

101

#107

—

118

—

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₅₉

yes

characteristic

104

C₄⋊C₈

C₂×D₅₉

yes

characteristic

105

#102

—

118

—

106

#102

—

118

—

107

C₂²⋊C₈

—

118

—

108

C₄×C₈

—

109

C₂×Q₈

—

118

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

118

—

115

SD₁₆

—

236

—

116

SD₁₆

—

236

—

117

D₈

—

236

—

118

M₁₆

—

236

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

118

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

118

—

126

C₄⋊C₄

—

118

—

127

C₂²⋊C₄

—

118

—

128

C₄²

C₂²×D₅₉

yes

characteristic

129

C₂³

—

130

C₂³

—

118

—

131

Q₈

—

236

—

132

Q₈

—

236

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

118

—

139

D₄

—

118

—

140

C₂×C₄

—

118

—

141

C₂×C₄

—

236

—

142

C₂×C₄

—

236

—

143

C₂×C₄

—

236

—

144

C₂×C₄

#11745685

yes

characteristic

145

C₂×C₄

D₄×D₅₉

yes

characteristic

146

C₂×C₄

D₄×D₅₉

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

118

—

151

C₈

—

236

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

236

—

156

C₂²

—

236

—

157

C₂²

—

236

—

158

C₂²

#14273700

yes

characteristic

159

C₄

—

160

C₄

—

472

—

161

C₄

—

236

—

162

C₄

D₈×D₅₉

yes

characteristic

163

C₄

#14273751

yes

characteristic

164

C₄

#14273754

yes

characteristic

165

C₄

#14273756

yes

characteristic

166

C₂

—

167

C₂

—

472

—

168

C₂

#16837596

yes

characteristic

169

C₂

#16837647

yes

characteristic

170

C₂

#16837664

yes

characteristic

	subgroup	G/H
center	C₂²	#14273700
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#331	C₂
socle	C₂²×C₅₉	#121
hypercenter	#331	D₅₉
hypocenter	C₅₉	#1685
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14273700

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#331

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
1. #16837774	C₂	semidirect
2. #16837683	C₂	semidirect
4. #16837479	C₂	semidirect
5. #16837154	C₂	semidirect
6. #16837149	C₂	semidirect
7. C₅₉×#331	C₂	semidirect
65. C₅₉⋊C₈	D₈	semidirect
92. #331	D₅₉	semidirect
93. C₅₉	#1685	semidirect

H ≅ G/N

type

1. #16837774

C₂

semidirect

2. #16837683

C₂

semidirect

4. #16837479

C₂

semidirect

5. #16837154

C₂

semidirect

6. #16837149

C₂

semidirect

7. C₅₉×#331

C₂

semidirect

65. C₅₉⋊C₈

D₈

semidirect

92. #331

D₅₉

semidirect

93. C₅₉

#1685

semidirect

Subgroups — #21423560

Conjugacy classes of subgroups

Complements