Group #21422962 — subgroups

Subgroups — #21422962

Order 7552.

182 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 63 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	C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic
2	#16837481	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837153	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837131	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837131	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₇₂	C₄	1	1	yes	characteristic
9	C₂×#11745659	—	2	1	no	—
10	C₂×C₈×D₅₉	—	2	1	no	—
11	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
12	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
13	#14273661	C₄	1	1	yes	characteristic
14	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
16	#14273649	C₄	1	1	yes	characteristic
17	#14273649	C₄	1	1	yes	characteristic
18	(C₂×D₅₉)⋊C₈	—	2	1	no	—
19	(C₂×D₅₉)⋊C₈	—	2	1	no	—
20	C₄×C₈×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×D₂₃₆	C₂×C₄	1	1	yes	characteristic
24	C₂×D₂₃₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₅₉	—	2	1	no	—
26	C₂×C₄×D₅₉	—	2	1	no	—
27	C₂×C₈×C₅₉	—	2	1	no	—
28	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	(C₂×D₅₉)⋊C₄	—	2	1	no	—
34	(C₂×D₅₉)⋊C₄	—	2	1	no	—
35	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅₉⋊C₈	—	2	1	no	—
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	D₄₇₂	—	2	2	no	—
40	#11745659	—	4	2	no	—
41	C₈×D₅₉	—	4	2	no	—
42	C₂²×D₅₉	—	2	1	no	—
43	C₂²×D₅₉	—	2	1	no	—
44	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
47	C₂×Dic₅₉	—	2	1	no	—
48	C₂×Dic₅₉	—	2	1	no	—
49	D₂₃₆	—	2	1	no	—
50	D₂₃₆	—	2	1	no	—
51	D₂₃₆	—	2	1	no	—
52	D₂₃₆	M₁₆	1	2	yes	complemented
53	C₄×D₅₉	—	4	1	no	—
54	C₄×D₅₉	—	4	1	no	—
55	C₄×D₅₉	—	4	1	no	—
56	C₄×D₅₉	—	4	1	no	—
57	C₈×C₅₉	—	2	2	no	—
58	C₈×C₅₉	—	2	1	no	—
59	C₈×C₅₉	—	2	1	no	—
60	C₈×C₅₉	—	2	1	no	—
61	C₈×C₅₉	—	2	1	no	—
62	C₅₉⋊C₈	—	4	1	no	—
63	C₅₉⋊C₈	—	4	1	no	—
64	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
65	C₂×D₅₉	—	2	2	no	—
66	C₂×D₅₉	—	4	1	no	—
67	C₂×D₅₉	—	4	1	no	—
68	C₂×D₅₉	—	4	1	no	—
69	C₂×D₅₉	—	4	1	no	—
70	C₂×D₅₉	—	4	1	no	—
71	C₄×C₅₉	—	2	1	no	—
72	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
73	C₄×C₅₉	#131	1	1	yes	characteristic
74	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Dic₅₉	—	4	1	no	—
77	Dic₅₉	—	4	1	no	—
78	#1586	—	59	1	no	maximal, Sylow
79	C₂×C₅₉	#435	1	1	yes	characteristic
80	C₂×C₅₉	#442	1	1	yes	characteristic
81	C₂×C₅₉	#443	1	1	yes	characteristic
82	D₅₉	—	4	2	no	—
83	D₅₉	—	8	1	no	—
84	C₄×D₈	—	59	1	no	—
85	#436	—	59	1	no	—
86	C₈×D₄	—	59	1	no	—
87	#334	—	59	1	no	—
88	D₄⋊C₈	—	59	1	no	—
89	D₄⋊C₈	—	59	1	no	—
90	#322	D₅₉	1	1	yes	characteristic, complemented
91	C₅₉	#1586	1	1	yes	characteristic, complemented, Sylow
92	C₂×D₈	—	59	1	no	—
93	C₂×M₁₆	—	118	1	no	—
94	C₂²×C₈	—	118	1	no	—
95	C₄×D₄	—	59	1	no	—
96	C₄×D₄	—	59	1	no	—
97	#107	—	59	1	no	—
98	C₄⋊C₈	—	59	1	no	—
99	C₄⋊C₈	—	59	1	no	—
100	#102	—	59	1	no	—
101	#102	—	59	1	no	—
102	C₂²⋊C₈	—	118	1	no	—
103	C₂²⋊C₈	—	118	1	no	—
104	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
105	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
106	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
107	C₂×D₄	—	59	1	no	—
108	C₂×D₄	—	59	1	no	—
109	C₂²×C₄	—	118	1	no	—
110	C₂²×C₄	—	118	1	no	—
111	D₈	—	118	2	no	—
112	M₁₆	—	236	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	118	1	no	—
115	C₂×C₈	—	118	1	no	—
116	C₂×C₈	—	236	2	no	—
117	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
118	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
119	C₂×C₈	D₂₃₆	1	1	yes	characteristic
120	C₂×C₈	D₂₃₆	1	1	yes	characteristic
121	C₄⋊C₄	—	59	1	no	—
122	C₄⋊C₄	—	59	1	no	—
123	C₂²⋊C₄	—	118	1	no	—
124	C₂²⋊C₄	—	118	1	no	—
125	C₄²	C₂²×D₅₉	1	1	yes	characteristic
126	C₂³	—	118	1	no	—
127	C₂³	—	118	1	no	—
128	D₄	—	59	2	no	—
129	D₄	—	118	1	no	—
130	D₄	—	118	1	no	—
131	D₄	—	118	1	no	—
132	C₂×C₄	—	118	1	no	—
133	C₂×C₄	—	118	1	no	—
134	C₂×C₄	—	236	1	no	—
135	C₂×C₄	—	236	1	no	—
136	C₂×C₄	—	236	1	no	—
137	C₂×C₄	—	236	1	no	—
138	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
139	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
140	C₂×C₄	#11745685	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	236	1	no	—
147	C₈	—	236	1	no	—
148	C₂²	—	118	2	no	—
149	C₂²	—	236	1	no	—
150	C₂²	—	236	1	no	—
151	C₂²	—	236	1	no	—
152	C₂²	—	236	1	no	—
153	C₂²	—	236	1	no	—
154	C₂²	C₄×D₂₃₆	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	236	1	no	—
157	C₄	—	236	1	no	—
158	C₄	M₁₆×D₅₉	1	1	yes	characteristic
159	C₄	#14273735	1	1	yes	characteristic
160	C₄	#14273755	1	1	yes	characteristic
161	C₄	#14273755	1	1	yes	characteristic
162	C₂	—	472	1	no	—
163	C₂	—	236	2	no	—
164	C₂	#16837475	1	1	yes	characteristic
165	C₂	#16837515	1	1	yes	characteristic
166	C₂	#16837520	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄₇₂

C₂

yes

maximal, characteristic

#16837481

C₂

yes

maximal, characteristic, complemented

C₈×D₂₃₆

C₂

yes

maximal, characteristic, complemented

#16837153

C₂

yes

maximal, characteristic, complemented

#16837131

C₂

yes

maximal, characteristic, complemented

#16837131

C₂

yes

maximal, characteristic, complemented

C₅₉×#322

C₂

yes

maximal, characteristic, complemented

C₂×D₄₇₂

C₄

yes

characteristic

C₂×#11745659

—

C₂×C₈×D₅₉

—

C₄×D₂₃₆

C₂²

yes

characteristic

C₄×D₂₃₆

C₂²

yes

characteristic

#14273661

C₄

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273649

C₄

yes

characteristic

#14273649

C₄

yes

characteristic

(C₂×D₅₉)⋊C₈

—

(C₂×D₅₉)⋊C₈

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

C₂×C₄

yes

characteristic

C₂×D₂₃₆

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

C₂×C₄

yes

characteristic

C₄⋊Dic₅₉

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

D₄₇₂

—

#11745659

—

C₈×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

M₁₆

yes

complemented

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×M₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

D₅₉

—

C₄×D₈

—

#436

—

C₈×D₄

—

#334

—

D₄⋊C₈

—

D₄⋊C₈

—

#322

D₅₉

yes

characteristic, complemented

C₅₉

#1586

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

118

—

C₂²×C₈

—

118

—

C₄×D₄

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

100

#102

—

101

#102

—

102

C₂²⋊C₈

—

118

—

103

C₂²⋊C₈

—

118

—

104

C₄×C₈

C₂×D₅₉

yes

characteristic

105

C₄×C₈

C₂×D₅₉

yes

characteristic

106

C₄×C₈

C₂×D₅₉

yes

characteristic

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

118

—

110

C₂²×C₄

—

118

—

111

D₈

—

118

—

112

M₁₆

—

236

—

113

C₂×C₈

—

114

C₂×C₈

—

118

—

115

C₂×C₈

—

118

—

116

C₂×C₈

—

236

—

117

C₂×C₈

C₄×D₅₉

yes

characteristic

118

C₂×C₈

C₄×D₅₉

yes

characteristic

119

C₂×C₈

D₂₃₆

yes

characteristic

120

C₂×C₈

D₂₃₆

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

118

—

124

C₂²⋊C₄

—

118

—

125

C₄²

C₂²×D₅₉

yes

characteristic

126

C₂³

—

118

—

127

C₂³

—

118

—

128

D₄

—

129

D₄

—

118

—

130

D₄

—

118

—

131

D₄

—

118

—

132

C₂×C₄

—

118

—

133

C₂×C₄

—

118

—

134

C₂×C₄

—

236

—

135

C₂×C₄

—

236

—

136

C₂×C₄

—

236

—

137

C₂×C₄

—

236

—

138

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

139

C₂×C₄

C₂×D₂₃₆

yes

characteristic

140

C₂×C₄

#11745685

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

236

—

147

C₈

—

236

—

148

C₂²

—

118

—

149

C₂²

—

236

—

150

C₂²

—

236

—

151

C₂²

—

236

—

152

C₂²

—

236

—

153

C₂²

—

236

—

154

C₂²

C₄×D₂₃₆

yes

characteristic

155

C₄

—

156

C₄

—

236

—

157

C₄

—

236

—

158

C₄

M₁₆×D₅₉

yes

characteristic

159

C₄

#14273735

yes

characteristic

160

C₄

#14273755

yes

characteristic

161

C₄

#14273755

yes

characteristic

162

C₂

—

472

—

163

C₂

—

236

—

164

C₂

#16837475

yes

characteristic

165

C₂

#16837515

yes

characteristic

166

C₂

#16837520

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#322	C₂
socle	C₂²×C₅₉	C₄×D₄
hypercenter	#322	D₅₉
hypocenter	C₅₉	#1586
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#322

C₂

socle

C₂²×C₅₉

C₄×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
2. #16837481	C₂	semidirect
3. C₈×D₂₃₆	C₂	semidirect
4. #16837153	C₂	semidirect
5. #16837131	C₂	semidirect
6. #16837131	C₂	semidirect
7. C₅₉×#322	C₂	semidirect
52. D₂₃₆	M₁₆	semidirect
90. #322	D₅₉	semidirect
91. C₅₉	#1586	semidirect

H ≅ G/N

type

2. #16837481

C₂

semidirect

3. C₈×D₂₃₆

C₂

semidirect

4. #16837153

C₂

semidirect

5. #16837131

C₂

semidirect

6. #16837131

C₂

semidirect

7. C₅₉×#322

C₂

semidirect

52. D₂₃₆

M₁₆

semidirect

90. #322

D₅₉

semidirect

91. C₅₉

#1586

semidirect

Subgroups — #21422962

Conjugacy classes of subgroups

Complements