Group #21426130 — subgroups

Subgroups — #21426130

Order 7552.

186 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 49 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	C₂×D₉₄₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14273668	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837777	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837484	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837202	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
9	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
10	C₂×#11745669	—	2	1	no	—
11	#14273723	—	2	1	no	—
12	#14273689	—	2	1	no	—
13	D₉₄₄	—	2	2	no	—
14	#14273668	—	2	2	no	—
15	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14273647	—	2	1	no	—
17	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	#107×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₅₉	—	2	1	no	—
22	C₂×C₂²⋊D₅₉	—	4	1	no	—
23	C₂×C₂²⋊D₅₉	—	4	1	no	—
24	C₂×D₂₃₆	—	2	1	no	—
25	C₂×D₂₃₆	—	2	1	no	—
26	C₂×C₄×D₅₉	—	4	1	no	—
27	D₈×C₅₉	—	2	1	no	—
28	D₈×C₅₉	—	2	1	no	—
29	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₅₉	—	2	1	no	—
31	#11745669	—	4	2	no	—
32	(C₂×D₅₉)⋊C₄	—	4	1	no	—
33	C₄×Dic₅₉	D₄	1	1	yes	characteristic
34	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
35	D₄₇₂	—	2	1	no	—
36	D₄₇₂	—	2	1	no	—
37	D₄₇₂	—	2	1	no	—
38	D₄₇₂	D₄	1	2	yes	complemented
39	C₁₆×C₅₉	—	2	1	no	—
40	C₅₉⋊C₁₆	—	2	1	no	—
41	C₂³×C₅₉	—	4	1	no	—
42	C₂²×D₅₉	—	4	1	no	—
43	C₂²×D₅₉	—	4	1	no	—
44	D₄×C₅₉	—	4	1	no	—
45	D₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	—	4	1	no	—
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₅₉	—	8	2	no	—
49	C₂²⋊D₅₉	—	8	2	no	—
50	C₂×Dic₅₉	D₈	1	1	yes	characteristic
51	C₂×Dic₅₉	D₈	1	1	yes	characteristic
52	D₂₃₆	—	2	2	no	—
53	D₂₃₆	—	4	1	no	—
54	D₂₃₆	—	4	1	no	—
55	D₂₃₆	—	4	1	no	—
56	C₄×D₅₉	—	8	2	no	—
57	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₈×C₅₉	#41	1	1	yes	characteristic
59	C₅₉⋊C₈	—	2	1	no	—
60	C₂²×C₅₉	—	8	1	no	—
61	C₂²×C₅₉	—	8	1	no	—
62	C₂²×C₅₉	—	8	1	no	—
63	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
64	C₂×D₅₉	—	4	2	no	—
65	C₂×D₅₉	—	8	1	no	—
66	C₂×D₅₉	—	8	1	no	—
67	C₂×D₅₉	—	8	1	no	—
68	C₂×D₅₉	—	8	1	no	—
69	C₂×D₅₉	—	8	1	no	—
70	C₄×C₅₉	—	8	1	no	—
71	C₄×C₅₉	#121	1	1	yes	characteristic
72	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
73	Dic₅₉	—	2	1	no	—
74	Dic₅₉	D₁₆	1	2	yes	complemented
75	#2206	—	59	1	no	maximal, Sylow
76	C₂×C₅₉	—	16	1	no	—
77	C₂×C₅₉	#459	1	1	yes	characteristic
78	C₂×C₅₉	C₂×D₁₆	1	1	yes	characteristic
79	C₂×C₅₉	#509	1	1	yes	characteristic
80	D₅₉	—	8	2	no	—
81	D₅₉	—	16	1	no	—
82	C₂×D₁₆	—	59	1	no	—
83	C₂×D₁₆	—	59	1	no	—
84	#493	—	59	1	no	—
85	C₄×D₈	—	59	1	no	—
86	C₄⋊C₁₆	—	59	1	no	—
87	#357	—	59	1	no	—
88	#357	D₅₉	1	1	yes	characteristic, complemented
89	C₅₉	#2206	1	1	yes	characteristic, complemented, Sylow
90	C₂×D₈	—	59	1	no	—
91	C₂×D₈	—	59	1	no	—
92	C₂×D₈	—	118	1	no	—
93	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
94	#127	—	118	1	no	—
95	C₄×D₄	—	118	1	no	—
96	D₁₆	—	118	2	no	—
97	D₁₆	—	118	2	no	—
98	C₂×C₁₆	—	59	1	no	—
99	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
100	#107	C₂×D₅₉	1	1	yes	characteristic
101	#102	—	118	1	no	—
102	C₄×C₈	—	59	1	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	—	118	1	no	—
105	C₂×D₄	—	118	1	no	—
106	C₂×D₄	—	236	1	no	—
107	C₂×D₄	—	236	1	no	—
108	C₂²×C₄	—	236	1	no	—
109	D₈	—	2	1	no	—
110	D₈	—	2	1	no	—
111	D₈	—	59	2	no	—
112	D₈	—	118	1	no	—
113	D₈	—	118	1	no	—
114	D₈	—	118	1	no	—
115	D₈	—	236	2	no	—
116	C₂×C₈	—	59	1	no	—
117	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
118	C₄⋊C₄	—	2	1	no	—
119	C₂²⋊C₄	—	236	1	no	—
120	C₄²	—	59	1	no	—
121	C₁₆	—	2	1	no	—
122	C₁₆	—	118	1	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	236	1	no	—
125	C₂³	—	236	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	118	2	no	—
129	D₄	—	472	2	no	—
130	D₄	—	472	2	no	—
131	D₄	—	236	1	no	—
132	D₄	—	236	1	no	—
133	D₄	—	236	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	59	1	no	—
136	C₂×C₄	—	59	1	no	—
137	C₂×C₄	—	472	2	no	—
138	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
139	C₈	—	118	1	no	—
140	C₈	#11745685	1	1	yes	characteristic
141	C₈	D₄×D₅₉	1	1	yes	characteristic
142	C₂²	—	8	1	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	472	1	no	—
146	C₂²	—	472	1	no	—
147	C₂²	—	472	1	no	—
148	C₂²	—	472	1	no	—
149	C₂²	—	472	1	no	—
150	C₂²	—	236	2	no	—
151	C₂²	D₈×D₅₉	1	1	yes	characteristic
152	C₄	—	8	1	no	—
153	C₄	—	59	2	no	—
154	C₄	—	118	1	no	—
155	C₄	#14273700	1	1	yes	characteristic
156	C₄	#14273754	1	1	yes	characteristic
157	C₂	—	16	1	no	—
158	C₂	—	944	1	no	—
159	C₂	—	472	2	no	—
160	C₂	#16837596	1	1	yes	characteristic
161	C₂	D₁₆×D₅₉	1	1	yes	characteristic
162	C₂	#16837839	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₉₄₄

C₂

yes

maximal, characteristic, complemented

C₂×#14273668

C₂

yes

maximal, characteristic, complemented

#16837777

C₂

yes

maximal, characteristic, complemented

#16837484

C₂

yes

maximal, characteristic, complemented

Dic₅₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#16837202

C₂

yes

maximal, characteristic, complemented

C₅₉×#357

C₂

yes

maximal, characteristic, complemented

C₂×D₄₇₂

C₂²

yes

characteristic

C₂×D₄₇₂

C₂²

yes

—

—

—

D₉₄₄

—

#14273668

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273647

—

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×D₈×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂×C₂²⋊D₅₉

—

C₂×C₂²⋊D₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

—

D₈×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

#11745669

—

(C₂×D₅₉)⋊C₄

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

D₄₇₂

—

D₄₇₂

—

D₄₇₂

—

D₄₇₂

D₄

yes

complemented

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₂×Dic₅₉

D₈

yes

characteristic

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

D₁₆

yes

complemented

#2206

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#459

yes

characteristic

C₂×C₅₉

C₂×D₁₆

yes

characteristic

C₂×C₅₉

#509

yes

characteristic

D₅₉

—

D₅₉

—

C₂×D₁₆

—

C₂×D₁₆

—

#493

—

C₄×D₈

—

C₄⋊C₁₆

—

#357

—

#357

D₅₉

yes

characteristic, complemented

C₅₉

#2206

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

118

—

C₂×D₈

C₂×D₅₉

yes

characteristic

#127

—

118

—

C₄×D₄

—

118

—

D₁₆

—

118

—

D₁₆

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

100

#107

C₂×D₅₉

yes

characteristic

101

#102

—

118

—

102

C₄×C₈

—

103

C₂×D₄

—

104

C₂×D₄

—

118

—

105

C₂×D₄

—

118

—

106

C₂×D₄

—

236

—

107

C₂×D₄

—

236

—

108

C₂²×C₄

—

236

—

109

D₈

—

110

D₈

—

111

D₈

—

112

D₈

—

118

—

113

D₈

—

118

—

114

D₈

—

118

—

115

D₈

—

236

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₅₉

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

236

—

120

C₄²

—

121

C₁₆

—

122

C₁₆

—

118

—

123

C₂³

—

124

C₂³

—

236

—

125

C₂³

—

236

—

126

D₄

—

127

D₄

—

128

D₄

—

118

—

129

D₄

—

472

—

130

D₄

—

472

—

131

D₄

—

236

—

132

D₄

—

236

—

133

D₄

—

236

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

472

—

138

C₂×C₄

D₄×D₅₉

yes

characteristic

139

C₈

—

118

—

140

C₈

#11745685

yes

characteristic

141

C₈

D₄×D₅₉

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

472

—

146

C₂²

—

472

—

147

C₂²

—

472

—

148

C₂²

—

472

—

149

C₂²

—

472

—

150

C₂²

—

236

—

151

C₂²

D₈×D₅₉

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

118

—

155

C₄

#14273700

yes

characteristic

156

C₄

#14273754

yes

characteristic

157

C₂

—

158

C₂

—

944

—

159

C₂

—

472

—

160

C₂

#16837596

yes

characteristic

161

C₂

D₁₆×D₅₉

yes

characteristic

162

C₂

#16837839

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#357	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#357	D₅₉
hypocenter	C₅₉	#2206
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#357

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. C₂×D₉₄₄	C₂	semidirect
2. C₂×#14273668	C₂	semidirect
3. #16837777	C₂	semidirect
4. #16837484	C₂	semidirect
5. Dic₅₉⋊C₁₆	C₂	semidirect
6. #16837202	C₂	semidirect
7. C₅₉×#357	C₂	semidirect
38. D₄₇₂	D₄	semidirect
74. Dic₅₉	D₁₆	semidirect
88. #357	D₅₉	semidirect
89. C₅₉	#2206	semidirect

H ≅ G/N

type

1. C₂×D₉₄₄

C₂

semidirect

2. C₂×#14273668

C₂

semidirect

3. #16837777

C₂

semidirect

4. #16837484

C₂

semidirect

5. Dic₅₉⋊C₁₆

C₂

semidirect

6. #16837202

C₂

semidirect

7. C₅₉×#357

C₂

semidirect

38. D₄₇₂

D₄

semidirect

74. Dic₅₉

D₁₆

semidirect

88. #357

D₅₉

semidirect

89. C₅₉

#2206

semidirect

Subgroups — #21426130

Conjugacy classes of subgroups

Complements