Group #21425802 — subgroups

Subgroups — #21425802

Order 7552.

228 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 69 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×#14273750	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837439	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837206	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837204	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837202	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745689	—	2	1	no	—
9	#14273750	—	2	2	no	—
10	#14273750	C₄	1	4	yes	complemented
11	C₂×#11745671	—	2	1	no	—
12	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
14	#14273685	—	2	1	no	—
15	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₁₆×D₅₉	—	2	2	no	—
17	#14273661	C₂²	1	1	yes	characteristic
18	#14273658	—	2	1	no	—
19	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	#107×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×Q₈×C₅₉	—	2	1	no	—
23	#11745689	—	2	4	no	—
24	#11745689	—	4	2	no	—
25	C₂×D₂₃₆	—	2	1	no	—
26	C₂×D₂₃₆	—	4	1	no	—
27	C₂×C₄×D₅₉	—	4	1	no	—
28	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
29	Q₁₆×C₅₉	—	2	1	no	—
30	Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₅₉	—	2	1	no	—
33	#11745671	—	2	4	no	—
34	(C₂×D₅₉)⋊C₄	—	4	1	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	C₄×Dic₅₉	—	4	1	no	—
37	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
38	D₄₇₂	—	2	1	no	—
39	D₄₇₂	C₂×C₄	1	2	yes	—
40	C₈×D₅₉	C₂×C₄	1	2	yes	—
41	C₈×D₅₉	D₄	1	2	yes	—
42	C₁₆×C₅₉	—	2	1	no	—
43	C₅₉⋊C₁₆	—	2	1	no	—
44	C₂²×D₅₉	—	4	1	no	—
45	C₂²×D₅₉	D₈	1	1	yes	characteristic
46	Q₈×C₅₉	—	2	2	no	—
47	Q₈×C₅₉	—	4	1	no	—
48	C₂×C₄×C₅₉	—	4	1	no	—
49	C₂×C₄×C₅₉	—	4	1	no	—
50	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₅₉	—	4	1	no	—
52	C₂×Dic₅₉	D₈	1	1	yes	characteristic
53	D₂₃₆	—	2	2	no	—
54	D₂₃₆	—	4	4	no	—
55	D₂₃₆	—	4	1	no	—
56	C₄×D₅₉	—	4	4	no	—
57	C₄×D₅₉	C₂²⋊C₄	1	2	yes	—
58	C₄×D₅₉	SD₁₆	1	2	yes	—
59	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
63	C₂×D₅₉	—	2	1	no	—
64	C₂×D₅₉	—	2	1	no	—
65	C₂×D₅₉	—	4	2	no	—
66	C₂×D₅₉	—	8	1	no	—
67	C₂×D₅₉	—	8	1	no	—
68	C₂×D₅₉	#102	1	2	yes	—
69	C₄×C₅₉	—	4	2	no	—
70	C₄×C₅₉	—	8	1	no	—
71	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
73	Dic₅₉	—	8	1	no	—
74	Dic₅₉	#102	1	2	yes	—
75	#2138	—	59	1	no	maximal, Sylow
76	C₂×C₅₉	C₂×#102	1	1	yes	characteristic
77	C₂×C₅₉	#508	1	1	yes	characteristic
78	C₂×C₅₉	#508	1	1	yes	characteristic
79	D₅₉	—	2	2	no	—
80	D₅₉	—	8	2	no	—
81	C₂×#135	—	59	1	no	—
82	C₂²×C₁₆	—	59	1	no	—
83	#427	—	59	1	no	—
84	#358	—	59	1	no	—
85	#358	D₅₉	1	1	yes	characteristic, complemented
86	#357	—	59	1	no	—
87	#357	—	59	1	no	—
88	C₅₉	#2138	1	1	yes	characteristic, complemented, Sylow
89	C₂×#41	—	118	1	no	—
90	#135	—	59	4	no	—
91	#135	—	118	2	no	—
92	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
93	C₂×SD₁₆	—	118	1	no	—
94	C₂×D₈	—	59	1	no	—
95	C₂²×C₈	—	59	1	no	—
96	#117	—	118	1	no	—
97	C₂×C₁₆	—	59	1	no	—
98	C₂×C₁₆	—	118	2	no	—
99	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
100	#107	—	59	1	no	—
101	#107	C₂×D₅₉	1	1	yes	characteristic
102	#106	—	118	1	no	—
103	#41	—	118	4	no	—
104	#41	—	236	2	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×D₄	—	118	1	no	—
107	C₂×D₄	—	236	1	no	—
108	C₂²×C₄	—	59	1	no	—
109	C₂²×C₄	—	236	1	no	—
110	Q₁₆	—	2	1	no	—
111	Q₁₆	C₄×D₅₉	1	2	yes	—
112	SD₁₆	—	118	4	no	—
113	D₈	—	59	2	no	—
114	D₈	—	118	1	no	—
115	C₂×C₈	—	59	1	no	—
116	C₂×C₈	—	59	2	no	—
117	C₂×C₈	—	59	2	no	—
118	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	118	1	no	—
121	C₂²⋊C₄	—	236	1	no	—
122	C₄²	—	236	1	no	—
123	C₁₆	—	2	1	no	—
124	C₁₆	—	118	1	no	—
125	C₂³	—	59	1	no	—
126	C₂³	—	236	1	no	—
127	Q₈	—	2	2	no	—
128	Q₈	—	4	1	no	—
129	D₄	—	118	2	no	—
130	D₄	—	236	4	no	—
131	D₄	—	236	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	59	1	no	—
135	C₂×C₄	—	59	2	no	—
136	C₂×C₄	—	59	2	no	—
137	C₂×C₄	—	236	4	no	—
138	C₂×C₄	—	236	1	no	—
139	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
140	C₈	—	59	2	no	—
141	C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
142	C₈	D₄×D₅₉	1	1	yes	characteristic
143	C₂²	—	59	2	no	—
144	C₂²	—	118	1	no	—
145	C₂²	—	118	1	no	—
146	C₂²	—	472	1	no	—
147	C₂²	—	472	1	no	—
148	C₂²	—	236	2	no	—
149	C₂²	D₈×D₅₉	1	1	yes	characteristic
150	C₄	—	4	2	no	—
151	C₄	—	8	1	no	—
152	C₄	—	59	2	no	—
153	C₄	—	472	1	no	—
154	C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
155	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
156	C₂	—	118	2	no	—
157	C₂	—	472	2	no	—
158	C₂	#102×D₅₉	1	1	yes	characteristic
159	C₂	#16837833	1	1	yes	characteristic
160	C₂	#16837835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273750

C₂

yes

maximal, characteristic

C₂×C₁₆×D₅₉

C₂

yes

maximal, characteristic, complemented

#16837439

C₂

yes

maximal, characteristic, complemented

#16837206

C₂

yes

maximal, characteristic, complemented

#16837204

C₂

yes

maximal, characteristic, complemented

#16837202

C₂

yes

maximal, characteristic, complemented

C₅₉×#358

C₂

yes

maximal, characteristic, complemented

C₂×#11745689

—

#14273750

—

#14273750

C₄

yes

complemented

C₂×#11745671

—

C₂×D₄₇₂

C₂²

yes

characteristic

C₂×C₈×D₅₉

C₂²

yes

characteristic

#14273685

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₁₆×D₅₉

—

#14273661

C₂²

yes

characteristic

#14273658

—

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

#11745689

—

#11745689

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

D₄

yes

characteristic

Q₁₆×C₅₉

—

Q₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

#11745671

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

D₄₇₂

—

D₄₇₂

C₂×C₄

yes

—

C₈×D₅₉

C₂×C₄

yes

—

C₈×D₅₉

D₄

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂²×D₅₉

—

C₂²×D₅₉

D₈

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

C₂²⋊C₄

yes

—

C₄×D₅₉

SD₁₆

yes

—

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

#102

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

#102

yes

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

D₅₉

—

C₂×#135

—

C₂²×C₁₆

—

#427

—

#358

—

#358

D₅₉

yes

characteristic, complemented

#357

—

#357

—

C₅₉

#2138

yes

characteristic, complemented, Sylow

C₂×#41

—

118

—

#135

—

#135

—

118

—

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₂×SD₁₆

—

118

—

C₂×D₈

—

C₂²×C₈

—

#117

—

118

—

C₂×C₁₆

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

100

#107

—

101

#107

C₂×D₅₉

yes

characteristic

102

#106

—

118

—

103

#41

—

118

—

104

#41

—

236

—

105

C₂×Q₈

—

106

C₂×D₄

—

118

—

107

C₂×D₄

—

236

—

108

C₂²×C₄

—

109

C₂²×C₄

—

236

—

110

Q₁₆

—

111

Q₁₆

C₄×D₅₉

yes

—

112

SD₁₆

—

118

—

113

D₈

—

114

D₈

—

118

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂²×D₅₉

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

118

—

121

C₂²⋊C₄

—

236

—

122

C₄²

—

236

—

123

C₁₆

—

124

C₁₆

—

118

—

125

C₂³

—

126

C₂³

—

236

—

127

Q₈

—

128

Q₈

—

129

D₄

—

118

—

130

D₄

—

236

—

131

D₄

—

236

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

236

—

138

C₂×C₄

—

236

—

139

C₂×C₄

D₄×D₅₉

yes

characteristic

140

C₈

—

141

C₈

C₂×C₄×D₅₉

yes

characteristic

142

C₈

D₄×D₅₉

yes

characteristic

143

C₂²

—

144

C₂²

—

118

—

145

C₂²

—

118

—

146

C₂²

—

472

—

147

C₂²

—

472

—

148

C₂²

—

236

—

149

C₂²

D₈×D₅₉

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

472

—

154

C₄

C₂²⋊C₄×D₅₉

yes

characteristic

155

C₄

SD₁₆×D₅₉

yes

characteristic

156

C₂

—

118

—

157

C₂

—

472

—

158

C₂

#102×D₅₉

yes

characteristic

159

C₂

#16837833

yes

characteristic

160

C₂

#16837835

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#358	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#358	D₅₉
hypocenter	C₅₉	#2138
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#358

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
2. C₂×C₁₆×D₅₉	C₂	semidirect
3. #16837439	C₂	semidirect
4. #16837206	C₂	semidirect
5. #16837204	C₂	semidirect
6. #16837202	C₂	semidirect
7. C₅₉×#358	C₂	semidirect
10. #14273750	C₄	semidirect
85. #358	D₅₉	semidirect
88. C₅₉	#2138	semidirect

H ≅ G/N

type

2. C₂×C₁₆×D₅₉

C₂

semidirect

3. #16837439

C₂

semidirect

4. #16837206

C₂

semidirect

5. #16837204

C₂

semidirect

6. #16837202

C₂

semidirect

7. C₅₉×#358

C₂

semidirect

10. #14273750

C₄

semidirect

85. #358

D₅₉

semidirect

88. C₅₉

#2138

semidirect

Subgroups — #21425802

Conjugacy classes of subgroups

Complements