Group #21422775 — subgroups

Subgroups — #21422775

Order 7552.

240 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 75 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⋊Q₈×D₅₉	C₂	1	1	yes	maximal, characteristic
2	#16837454	C₂	1	1	yes	maximal, characteristic
3	#16837424	C₂	1	1	yes	maximal, characteristic
4	#16837151	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837150	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₉	C₄	1	2	yes	—
9	C₂×#11745659	—	2	1	no	—
10	#14273727	C₂²	1	1	yes	characteristic
11	#14273726	—	2	1	no	—
12	#14273726	—	2	1	no	—
13	#14273725	C₄	1	2	yes	—
14	C₄⋊C₄×D₅₉	—	2	1	no	—
15	C₄⋊C₄×D₅₉	—	2	1	no	—
16	C₄²×D₅₉	C₂²	1	1	yes	characteristic
17	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
19	Dic₅₉⋊C₈	—	2	1	no	—
20	(C₂×D₅₉)⋊C₈	—	2	1	no	—
21	C₈×Dic₅₉	—	2	1	no	—
22	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₅₉	C₂×C₄	1	2	yes	—
26	Q₈×D₅₉	—	4	4	no	—
27	C₂×C₄×D₅₉	—	2	2	no	—
28	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
30	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
31	C₂×Dic₁₁₈	—	2	2	no	—
32	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₅₉	—	2	1	no	—
34	C₂×C₈×C₅₉	—	2	1	no	—
35	C₄⋊C₄×C₅₉	—	2	1	no	—
36	C₄⋊C₄×C₅₉	—	2	1	no	—
37	C₄²×C₅₉	C₂³	1	1	yes	characteristic
38	C₄⋊Dic₅₉	—	2	1	no	—
39	C₄⋊Dic₅₉	—	2	1	no	—
40	Dic₅₉⋊C₄	—	2	2	no	—
41	Dic₅₉⋊C₄	—	2	2	no	—
42	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₅₉	D₄	1	1	yes	characteristic
44	C₄×Dic₅₉	D₄	1	1	yes	characteristic
45	C₂×C₅₉⋊C₈	—	2	1	no	—
46	C₂×C₅₉⋊C₈	—	2	1	no	—
47	#11745659	—	4	2	no	—
48	C₂²×D₅₉	C₂²⋊C₄	1	1	yes	characteristic
49	Q₈×C₅₉	—	4	2	no	—
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₅₉	—	2	2	no	—
55	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₄×D₅₉	—	2	1	no	—
59	C₄×D₅₉	—	2	1	no	—
60	C₄×D₅₉	—	2	2	no	—
61	C₄×D₅₉	—	4	4	no	—
62	C₄×D₅₉	SD₁₆	1	2	yes	—
63	C₄×D₅₉	Q₁₆	1	2	yes	—
64	Dic₁₁₈	—	4	4	no	—
65	Dic₁₁₈	—	4	2	no	—
66	C₈×C₅₉	—	4	1	no	—
67	C₈×C₅₉	—	4	1	no	—
68	C₅₉⋊C₈	—	4	1	no	—
69	C₅₉⋊C₈	—	4	1	no	—
70	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×D₅₉	—	2	1	no	—
72	C₂×D₅₉	—	2	1	no	—
73	C₂×D₅₉	#103	1	2	yes	—
74	C₄×C₅₉	—	2	1	no	—
75	C₄×C₅₉	—	4	2	no	—
76	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
78	C₄×C₅₉	#137	1	1	yes	characteristic
79	C₄×C₅₉	#137	1	1	yes	characteristic
80	Dic₅₉	—	2	1	no	—
81	Dic₅₉	—	4	2	no	—
82	Dic₅₉	#100	1	2	yes	—
83	Dic₅₉	#103	1	2	yes	—
84	#1548	—	59	1	no	maximal, Sylow
85	C₂×C₅₉	C₂×#100	1	1	yes	characteristic
86	C₂×C₅₉	C₂×#103	1	1	yes	characteristic
87	C₂×C₅₉	#419	1	1	yes	characteristic
88	D₅₉	—	2	2	no	—
89	C₂×C₄⋊Q₈	—	59	1	no	—
90	#431	—	59	1	no	—
91	#423	—	59	1	no	—
92	#333	—	59	1	no	—
93	#333	—	59	1	no	—
94	#333	—	59	1	no	—
95	#333	D₅₉	1	1	yes	characteristic, complemented
96	C₅₉	#1548	1	1	yes	characteristic, complemented, Sylow
97	C₂²×Q₈	—	59	2	no	—
98	C₂×M₁₆	—	118	1	no	—
99	C₄⋊Q₈	—	59	2	no	—
100	C₄⋊Q₈	—	59	1	no	—
101	C₄⋊Q₈	—	118	1	no	—
102	C₄⋊Q₈	—	118	1	no	—
103	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	118	1	no	—
105	C₂×C₄⋊C₄	—	118	1	no	—
106	C₂×C₄²	—	59	1	no	—
107	C₄⋊C₈	—	59	1	no	—
108	C₄⋊C₈	—	59	1	no	—
109	C₄⋊C₈	—	118	1	no	—
110	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
111	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
112	C₂²⋊C₈	—	118	1	no	—
113	C₄×C₈	—	118	1	no	—
114	C₂×Q₈	—	59	2	no	—
115	C₂×Q₈	—	118	2	no	—
116	C₂×Q₈	—	236	4	no	—
117	C₂×Q₈	C₄×D₅₉	1	2	yes	—
118	C₂²×C₄	—	59	1	no	—
119	C₂²×C₄	—	59	1	no	—
120	C₂²×C₄	—	59	1	no	—
121	C₂²×C₄	—	118	2	no	—
122	M₁₆	—	236	2	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	118	1	no	—
126	C₂×C₈	—	118	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	118	2	no	—
130	C₄⋊C₄	—	118	2	no	—
131	C₄⋊C₄	—	118	1	no	—
132	C₄⋊C₄	—	118	1	no	—
133	C₄²	—	59	1	no	—
134	C₄²	—	59	1	no	—
135	C₄²	—	59	1	no	—
136	C₄²	C₂²×D₅₉	1	1	yes	characteristic
137	C₂³	—	59	1	no	—
138	Q₈	—	4	2	no	—
139	Q₈	—	236	2	no	—
140	Q₈	—	236	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	59	1	no	—
143	C₂×C₄	—	59	1	no	—
144	C₂×C₄	—	59	2	no	—
145	C₂×C₄	—	59	2	no	—
146	C₂×C₄	—	59	1	no	—
147	C₂×C₄	—	118	2	no	—
148	C₂×C₄	—	118	1	no	—
149	C₂×C₄	—	118	1	no	—
150	C₂×C₄	—	118	2	no	—
151	C₂×C₄	—	236	4	no	—
152	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
153	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
154	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
155	C₈	—	4	1	no	—
156	C₈	—	4	1	no	—
157	C₈	—	236	1	no	—
158	C₈	—	236	1	no	—
159	C₂²	—	59	2	no	—
160	C₂²	—	118	1	no	—
161	C₂²	—	118	1	no	—
162	C₂²	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	2	no	—
165	C₄	—	59	2	no	—
166	C₄	—	59	2	no	—
167	C₄	—	118	1	no	—
168	C₄	—	236	2	no	—
169	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
170	C₄	Q₁₆×D₅₉	1	1	yes	characteristic
171	C₄	#14273756	1	1	yes	characteristic
172	C₄	#14273759	1	1	yes	characteristic
173	C₂	—	118	2	no	—
174	C₂	#100×D₅₉	1	1	yes	characteristic
175	C₂	#103×D₅₉	1	1	yes	characteristic
176	C₂	#16837409	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₅₉

C₂

yes

maximal, characteristic

#16837454

C₂

yes

maximal, characteristic

#16837424

C₂

yes

maximal, characteristic

#16837151

C₂

yes

maximal, characteristic, complemented

#16837150

C₂

yes

maximal, characteristic, complemented

#16837150

C₂

yes

maximal, characteristic, complemented

C₅₉×#333

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₉

C₄

yes

—

C₂×#11745659

—

#14273727

C₂²

yes

—

—

C₄

yes

—

C₄⋊C₄×D₅₉

—

C₄⋊C₄×D₅₉

—

C₄²×D₅₉

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

(C₂×D₅₉)⋊C₈

—

C₈×Dic₅₉

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

C₂×C₄

yes

—

Q₈×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

#11745659

—

C₂²×D₅₉

C₂²⋊C₄

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

SD₁₆

yes

—

C₄×D₅₉

Q₁₆

yes

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

#103

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

#100

yes

—

Dic₅₉

#103

yes

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

C₂×C₄⋊Q₈

—

—

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#1548

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

118

—

C₄⋊Q₈

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

118

—

102

C₄⋊Q₈

—

118

—

103

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

104

C₂×C₄⋊C₄

—

118

—

105

C₂×C₄⋊C₄

—

118

—

106

C₂×C₄²

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

118

—

110

C₄⋊C₈

C₂×D₅₉

yes

characteristic

111

C₄⋊C₈

C₂×D₅₉

yes

characteristic

112

C₂²⋊C₈

—

118

—

113

C₄×C₈

—

118

—

114

C₂×Q₈

—

115

C₂×Q₈

—

118

—

116

C₂×Q₈

—

236

—

117

C₂×Q₈

C₄×D₅₉

yes

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

118

—

122

M₁₆

—

236

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

118

—

126

C₂×C₈

—

118

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

118

—

130

C₄⋊C₄

—

118

—

131

C₄⋊C₄

—

118

—

132

C₄⋊C₄

—

118

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²×D₅₉

yes

characteristic

137

C₂³

—

138

Q₈

—

139

Q₈

—

236

—

140

Q₈

—

236

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

118

—

148

C₂×C₄

—

118

—

149

C₂×C₄

—

118

—

150

C₂×C₄

—

118

—

151

C₂×C₄

—

236

—

152

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

153

C₂×C₄

D₄×D₅₉

yes

characteristic

154

C₂×C₄

D₄×D₅₉

yes

characteristic

155

C₈

—

156

C₈

—

157

C₈

—

236

—

158

C₈

—

236

—

159

C₂²

—

160

C₂²

—

118

—

161

C₂²

—

118

—

162

C₂²

C₂²⋊C₄×D₅₉

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

118

—

168

C₄

—

236

—

169

C₄

SD₁₆×D₅₉

yes

characteristic

170

C₄

Q₁₆×D₅₉

yes

characteristic

171

C₄

#14273756

yes

characteristic

172

C₄

#14273759

yes

characteristic

173

C₂

—

118

—

174

C₂

#100×D₅₉

yes

characteristic

175

C₂

#103×D₅₉

yes

characteristic

176

C₂

#16837409

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#333	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#333	D₅₉
hypocenter	C₅₉	#1548
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#333

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
4. #16837151	C₂	semidirect
5. #16837150	C₂	semidirect
6. #16837150	C₂	semidirect
7. C₅₉×#333	C₂	semidirect
95. #333	D₅₉	semidirect
96. C₅₉	#1548	semidirect

H ≅ G/N

type

4. #16837151

C₂

semidirect

5. #16837150

C₂

semidirect

6. #16837150

C₂

semidirect

7. C₅₉×#333

C₂

semidirect

95. #333

D₅₉

semidirect

96. C₅₉

#1548

semidirect

Subgroups — #21422775

Conjugacy classes of subgroups

Complements