Group #21437085 — subgroups

Subgroups — #21437085

Order 7552.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16838125	C₂	1	2	yes	maximal, complemented
2	#16837786	C₂	1	2	yes	maximal
3	#16837782	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837625	C₂	1	4	yes	maximal, complemented
5	#16837608	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₂₃₆	C₂	1	2	yes	maximal, complemented
7	C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₃₆	C₂²	1	2	yes	—
10	C₂×Dic₂₃₆	C₂²	1	2	yes	—
11	C₂×#11745660	C₂²	1	4	yes	—
12	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
13	#14273727	C₂²	1	2	yes	—
14	#14273722	—	2	4	no	—
15	#14273715	C₂²	1	4	yes	—
16	#14273703	—	2	2	no	—
17	C₄×Dic₁₁₈	C₂²	1	4	yes	—
18	C₄×D₂₃₆	C₂²	1	2	yes	—
19	#14273685	—	2	2	no	—
20	#14273661	C₂²	1	1	yes	characteristic
21	#14273652	C₂²	1	4	yes	—
22	#14273649	C₂²	1	2	yes	—
23	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
26	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₅₉	C₂²	1	2	yes	—
28	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
29	C₂×D₂₃₆	C₂³	1	2	yes	—
30	C₂×C₄×D₅₉	—	2	2	no	—
31	C₂×Dic₁₁₈	C₂³	1	4	yes	—
32	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂³	1	2	yes	—
34	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
37	C₄²×C₅₉	C₂³	1	1	yes	characteristic
38	C₄²×C₅₉	C₂³	1	2	yes	—
39	(C₂×D₅₉)⋊C₄	—	2	4	no	—
40	(C₂×D₅₉)⋊C₄	—	2	2	no	—
41	(C₂×D₅₉)⋊C₄	—	2	4	no	—
42	C₄⋊Dic₅₉	—	2	2	no	—
43	C₄⋊Dic₅₉	—	2	2	no	—
44	C₄⋊Dic₅₉	C₂³	1	2	yes	—
45	Dic₅₉⋊C₄	—	2	4	no	—
46	C₄×Dic₅₉	—	2	4	no	—
47	Dic₂₃₆	—	2	4	no	—
48	Dic₂₃₆	—	4	2	no	—
49	D₄₇₂	—	2	2	no	—
50	#11745660	—	4	4	no	—
51	C₂²×D₅₉	—	2	2	no	—
52	Q₈×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₅₉	—	2	2	no	—
59	C₂×Dic₅₉	—	2	4	no	—
60	D₂₃₆	—	2	2	no	—
61	D₂₃₆	—	2	2	no	—
62	C₄×D₅₉	—	4	4	no	—
63	Dic₁₁₈	—	2	4	no	—
64	Dic₁₁₈	—	2	4	no	—
65	C₈×C₅₉	—	2	2	no	—
66	C₈×C₅₉	—	2	1	no	—
67	C₈×C₅₉	#41	1	2	yes	—
68	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
69	C₂×D₅₉	—	4	2	no	—
70	C₂×D₅₉	—	4	2	no	—
71	C₂×D₅₉	—	4	2	no	—
72	C₄×C₅₉	—	2	2	no	—
73	C₄×C₅₉	—	2	2	no	—
74	C₄×C₅₉	—	2	1	no	—
75	C₄×C₅₉	#135	1	2	yes	—
76	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
77	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
78	Dic₅₉	—	4	4	no	—
79	Dic₅₉	—	4	2	no	—
80	#3393	—	59	1	no	maximal, Sylow
81	C₂×C₅₉	#550	1	1	yes	characteristic
82	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
83	C₂×C₅₉	#578	1	1	yes	characteristic
84	D₅₉	—	8	2	no	—
85	#556	—	59	2	no	—
86	#495	—	59	2	no	—
87	#494	—	59	1	no	—
88	#464	—	59	4	no	—
89	#461	—	59	2	no	—
90	C₈×Q₈	D₅₉	1	1	yes	characteristic, complemented
91	C₄×Q₁₆	—	59	2	no	—
92	C₄×D₈	—	59	1	no	—
93	C₅₉	#3393	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	59	2	no	—
95	C₂×Q₁₆	—	59	2	no	—
96	C₂×SD₁₆	—	59	4	no	—
97	C₂×D₈	—	59	1	no	—
98	C₄⋊Q₈	—	59	2	no	—
99	#126	—	118	4	no	—
100	#124	—	59	4	no	—
101	C₂²⋊Q₈	—	118	2	no	—
102	C₄×Q₈	—	59	4	no	—
103	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
104	C₄×D₄	—	59	2	no	—
105	#117	—	118	2	no	—
106	#107	—	59	1	no	—
107	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
108	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
109	#103	—	59	4	no	—
110	#102	—	59	2	no	—
111	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
112	C₄×C₈	C₂×D₅₉	1	2	yes	—
113	C₂×Q₈	—	59	4	no	—
114	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
115	C₂×D₄	—	59	2	no	—
116	C₂²×C₄	—	118	2	no	—
117	Q₁₆	—	118	4	no	—
118	Q₁₆	—	236	2	no	—
119	SD₁₆	—	236	4	no	—
120	D₈	—	118	2	no	—
121	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
122	C₂×C₈	C₂²×D₅₉	1	2	yes	—
123	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
124	C₄⋊C₄	—	59	2	no	—
125	C₄⋊C₄	—	118	2	no	—
126	C₄⋊C₄	—	118	4	no	—
127	C₄⋊C₄	—	118	2	no	—
128	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
129	C₄⋊C₄	D₂₃₆	1	2	yes	—
130	C₂²⋊C₄	—	118	4	no	—
131	C₂²⋊C₄	—	118	2	no	—
132	C₂²⋊C₄	—	118	4	no	—
133	C₄²	—	118	4	no	—
134	C₄²	C₂²×D₅₉	1	1	yes	characteristic
135	C₄²	C₂²×D₅₉	1	2	yes	—
136	C₂³	—	118	2	no	—
137	Q₈	—	2	2	no	—
138	Q₈	—	118	4	no	—
139	Q₈	—	118	4	no	—
140	D₄	—	118	2	no	—
141	D₄	—	118	2	no	—
142	C₂×C₄	—	118	2	no	—
143	C₂×C₄	—	118	4	no	—
144	C₂×C₄	—	236	4	no	—
145	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
146	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
147	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
148	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
149	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	1	no	—
152	C₈	#11745689	1	2	yes	—
153	C₂²	—	236	2	no	—
154	C₂²	—	236	2	no	—
155	C₂²	—	236	2	no	—
156	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	1	no	—
160	C₄	—	236	2	no	—
161	C₄	—	236	4	no	—
162	C₄	#14273749	1	2	yes	—
163	C₄	C₂×#11745689	1	1	yes	characteristic
164	C₄	#14273773	1	1	yes	characteristic
165	C₂	—	472	2	no	—
166	C₂	#16838074	1	1	yes	characteristic
167	C₂	C₂×#14273749	1	1	yes	characteristic
168	C₂	#16838219	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838125

C₂

yes

maximal, complemented

#16837786

C₂

yes

maximal

#16837782

C₂

yes

maximal, characteristic, complemented

#16837625

C₂

yes

maximal, complemented

#16837608

C₂

yes

maximal, complemented

C₄×Dic₂₃₆

C₂

yes

maximal, complemented

C₄×D₄₇₂

C₂

yes

maximal, characteristic

C₈×Q₈×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃₆

C₂²

yes

—

C₂×Dic₂₃₆

C₂²

yes

—

C₂×#11745660

C₂²

yes

—

C₂×D₄₇₂

C₂²

yes

characteristic

#14273727

C₂²

yes

—

#14273722

—

#14273715

C₂²

yes

—

#14273703

—

C₄×Dic₁₁₈

C₂²

yes

—

C₄×D₂₃₆

C₂²

yes

—

#14273685

—

#14273661

C₂²

yes

characteristic

#14273652

C₂²

yes

—

#14273649

C₂²

yes

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×D₂₃₆

C₂³

yes

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

—

D₄₇₂

—

#11745660

—

C₂²×D₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#41

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#135

yes

—

C₄×C₅₉

C₂×#41

yes

characteristic

C₄×C₅₉

H(5,𝔽₂)

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

—

—

—

—

—

C₈×Q₈

D₅₉

yes

characteristic, complemented

C₄×Q₁₆

—

C₄×D₈

—

C₅₉

#3393

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#126

—

118

—

100

#124

—

101

C₂²⋊Q₈

—

118

—

102

C₄×Q₈

—

103

C₄×Q₈

C₂×D₅₉

yes

characteristic

104

C₄×D₄

—

105

#117

—

118

—

106

#107

—

107

C₄⋊C₈

C₂×D₅₉

yes

characteristic

108

C₄⋊C₈

C₂×D₅₉

yes

—

109

#103

—

110

#102

—

111

C₄×C₈

C₂×D₅₉

yes

characteristic

112

C₄×C₈

C₂×D₅₉

yes

—

113

C₂×Q₈

—

114

C₂×Q₈

D₂₃₆

yes

characteristic

115

C₂×D₄

—

116

C₂²×C₄

—

118

—

117

Q₁₆

—

118

—

118

Q₁₆

—

236

—

119

SD₁₆

—

236

—

120

D₈

—

118

—

121

C₂×C₈

C₂²×D₅₉

yes

characteristic

122

C₂×C₈

C₂²×D₅₉

yes

—

123

C₂×C₈

C₂²×D₅₉

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

118

—

126

C₄⋊C₄

—

118

—

127

C₄⋊C₄

—

118

—

128

C₄⋊C₄

D₂₃₆

yes

characteristic

129

C₄⋊C₄

D₂₃₆

yes

—

130

C₂²⋊C₄

—

118

—

131

C₂²⋊C₄

—

118

—

132

C₂²⋊C₄

—

118

—

133

C₄²

—

118

—

134

C₄²

C₂²×D₅₉

yes

characteristic

135

C₄²

C₂²×D₅₉

yes

—

136

C₂³

—

118

—

137

Q₈

—

138

Q₈

—

118

—

139

Q₈

—

118

—

140

D₄

—

118

—

141

D₄

—

118

—

142

C₂×C₄

—

118

—

143

C₂×C₄

—

118

—

144

C₂×C₄

—

236

—

145

C₂×C₄

C₂×D₂₃₆

yes

—

146

C₂×C₄

C₂×D₂₃₆

yes

characteristic

147

C₂×C₄

C₂×D₂₃₆

yes

—

148

C₂×C₄

C₂×D₂₃₆

yes

characteristic

149

C₂×C₄

C₂³×D₅₉

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

#11745689

yes

—

153

C₂²

—

236

—

154

C₂²

—

236

—

155

C₂²

—

236

—

156

C₂²

C₂²×D₂₃₆

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

236

—

161

C₄

—

236

—

162

C₄

#14273749

yes

—

163

C₄

C₂×#11745689

yes

characteristic

164

C₄

#14273773

yes

characteristic

165

C₂

—

472

—

166

C₂

#16838074

yes

characteristic

167

C₂

C₂×#14273749

yes

characteristic

168

C₂

#16838219

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₈×Q₈×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₈×Q₈	D₅₉
hypocenter	C₅₉	#3393
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₈×Q₈×C₅₉

C₂

socle

C₂²×C₅₉

C₂²×D₄

hypercenter

C₈×Q₈

D₅₉

hypocenter

C₅₉

#3393

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #16838125	C₂	semidirect
3. #16837782	C₂	semidirect
4. #16837625	C₂	semidirect
5. #16837608	C₂	semidirect
6. C₄×Dic₂₃₆	C₂	semidirect
8. C₈×Q₈×C₅₉	C₂	semidirect
90. C₈×Q₈	D₅₉	semidirect
93. C₅₉	#3393	semidirect

H ≅ G/N

type

1. #16838125

C₂

semidirect

3. #16837782

C₂

semidirect

4. #16837625

C₂

semidirect

5. #16837608

C₂

semidirect

6. C₄×Dic₂₃₆

C₂

semidirect

8. C₈×Q₈×C₅₉

C₂

semidirect

90. C₈×Q₈

D₅₉

semidirect

93. C₅₉

#3393

semidirect

Subgroups — #21437085

Conjugacy classes of subgroups

Complements