Group #21423927 — subgroups

Subgroups — #21423927

Order 7552.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273652	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837165	C₂	1	2	yes	maximal
5	C₂×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₁₈	—	2	2	no	—
8	C₄×Dic₁₁₈	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₅₉⋊C₄	—	2	1	no	—
12	C₂×C₄×Dic₅₉	—	2	1	no	—
13	C₂×C₄×Dic₅₉	—	2	1	no	—
14	#14273652	C₄	1	8	yes	complemented
15	C₅₉⋊#95	—	2	1	no	—
16	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
18	#97×C₅₉	—	2	2	no	—
19	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₉	—	2	1	no	—
23	C₂²×Dic₅₉	—	2	2	no	—
24	C₂²×Dic₅₉	—	2	1	no	—
25	C₂×Dic₁₁₈	—	2	1	no	—
26	C₂×Dic₁₁₈	—	2	2	no	—
27	C₂×Dic₁₁₈	C₂×C₄	1	4	yes	—
28	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₅₉	—	2	2	no	—
30	C₂×C₈×C₅₉	—	2	2	no	—
31	C₂×C₈×C₅₉	C₂×C₄	1	4	yes	—
32	C₄²×C₅₉	C₂×C₄	1	2	yes	—
33	C₄²×C₅₉	D₄	1	2	yes	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	—	2	2	no	—
36	C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
37	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
38	Dic₅₉⋊C₄	—	2	4	no	—
39	C₄×Dic₅₉	—	2	4	no	—
40	C₄×Dic₅₉	—	4	2	no	—
41	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	—	2	1	no	—
43	C₂×C₄×C₅₉	—	2	1	no	—
44	C₂×C₄×C₅₉	—	2	2	no	—
45	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	#41	1	2	yes	—
51	C₂×Dic₅₉	—	2	2	no	—
52	C₂×Dic₅₉	—	2	4	no	—
53	C₂×Dic₅₉	—	2	2	no	—
54	C₂×Dic₅₉	—	2	4	no	—
55	C₂×Dic₅₉	—	4	2	no	—
56	C₂×Dic₅₉	—	4	1	no	—
57	C₂×Dic₅₉	—	4	1	no	—
58	C₂×Dic₅₉	—	4	2	no	—
59	Dic₁₁₈	—	2	2	no	—
60	Dic₁₁₈	—	2	4	no	—
61	Dic₁₁₈	C₄²	1	4	yes	complemented
62	C₈×C₅₉	—	2	4	no	—
63	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₅₉	C₄×D₄	1	2	yes	—
65	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₅₉	#137	1	1	yes	characteristic
67	C₂²×C₅₉	#137	1	2	yes	—
68	C₄×C₅₉	—	2	2	no	—
69	C₄×C₅₉	—	2	2	no	—
70	C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
71	C₄×C₅₉	#117	1	1	yes	characteristic
72	C₄×C₅₉	C₄×D₄	1	2	yes	—
73	Dic₅₉	—	2	4	no	—
74	Dic₅₉	—	4	2	no	—
75	Dic₅₉	—	4	4	no	—
76	#1763	—	59	1	no	maximal, Sylow
77	C₂×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅₉	#417	1	1	yes	characteristic
79	C₂×C₅₉	#419	1	1	yes	characteristic
80	C₂×C₅₉	#440	1	2	yes	—
81	C₂×C₅₉	#441	1	2	yes	—
82	C₂×C₄×Q₈	—	59	1	no	—
83	C₂×#103	—	59	2	no	—
84	C₂×#97	D₅₉	1	1	yes	characteristic
85	C₄×C₄⋊C₄	—	59	1	no	—
86	#340	—	59	2	no	—
87	C₅₉	#1763	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	—	59	1	no	—
89	C₂²×C₈	C₂×D₅₉	1	2	yes	—
90	C₄×Q₈	—	59	4	no	—
91	C₄×Q₈	—	118	2	no	—
92	C₂×C₄⋊C₄	—	59	1	no	—
93	C₂×C₄⋊C₄	—	59	2	no	—
94	C₂×C₄⋊C₄	—	118	1	no	—
95	C₂×C₄²	—	118	1	no	—
96	C₂×C₄²	—	118	1	no	—
97	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
98	#103	—	59	8	no	—
99	#97	—	2	2	no	—
100	#95	—	118	1	no	—
101	C₂×Q₈	—	59	4	no	—
102	C₂×Q₈	—	59	2	no	—
103	C₂×Q₈	—	118	2	no	—
104	C₂×Q₈	—	118	1	no	—
105	C₂²×C₄	—	118	1	no	—
106	C₂²×C₄	—	118	2	no	—
107	C₂²×C₄	—	118	1	no	—
108	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
109	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	C₄×D₅₉	1	4	yes	—
114	C₄⋊C₄	—	59	4	no	—
115	C₄⋊C₄	—	59	2	no	—
116	C₄⋊C₄	—	118	4	no	—
117	C₄⋊C₄	—	118	2	no	—
118	C₄⋊C₄	—	118	1	no	—
119	C₄²	—	118	4	no	—
120	C₄²	—	236	2	no	—
121	C₄²	C₄×D₅₉	1	2	yes	—
122	C₄²	C₂²⋊D₅₉	1	2	yes	—
123	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
124	Q₈	—	59	4	no	—
125	Q₈	—	118	2	no	—
126	Q₈	—	118	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	118	2	no	—
131	C₂×C₄	—	118	4	no	—
132	C₂×C₄	—	118	2	no	—
133	C₂×C₄	—	118	4	no	—
134	C₂×C₄	—	236	2	no	—
135	C₂×C₄	—	236	1	no	—
136	C₂×C₄	—	236	1	no	—
137	C₂×C₄	—	236	2	no	—
138	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
139	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
142	C₂×C₄	#11745685	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
146	C₂²	C₄×D₂₃₆	1	2	yes	—
147	C₂²	#14273755	1	1	yes	characteristic
148	C₂²	#14273760	1	1	yes	characteristic
149	C₂²	#14273760	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	118	4	no	—
153	C₄	—	236	2	no	—
154	C₄	—	236	4	no	—
155	C₄	C₄²×D₅₉	1	1	yes	characteristic
156	C₄	#14273684	1	1	yes	characteristic
157	C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
158	C₂	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
159	C₂	#16837404	1	1	yes	characteristic
160	C₂	#16837412	1	1	yes	characteristic
161	C₂	#16837504	1	2	yes	—
162	C₂	#16837510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₁₈

C₂

yes

maximal, characteristic

C₂×#14273652

C₂

yes

maximal

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

#16837165

C₂

yes

maximal

C₂×#97×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

characteristic

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₄

yes

complemented

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

—

C₂×C₄×Dic₅₉

—

#14273652

C₄

yes

complemented

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#97×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

C₄²

yes

complemented

C₈×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

C₄×D₄

yes

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄²

yes

characteristic

C₄×C₅₉

#117

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#1763

—

maximal, Sylow

C₂×C₅₉

C₄×C₂²⋊C₄

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#441

yes

—

C₂×C₄×Q₈

—

C₂×#103

—

C₂×#97

D₅₉

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₅₉

#1763

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₅₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

118

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

—

118

—

C₂×C₄²

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#103

—

#97

—

100

#95

—

118

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

118

—

104

C₂×Q₈

—

118

—

105

C₂²×C₄

—

118

—

106

C₂²×C₄

—

118

—

107

C₂²×C₄

—

118

—

108

C₂²×C₄

D₂₃₆

yes

characteristic

109

C₂²×C₄

D₂₃₆

yes

characteristic

110

C₂²×C₄

C₂²×D₅₉

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₄×D₅₉

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

118

—

117

C₄⋊C₄

—

118

—

118

C₄⋊C₄

—

118

—

119

C₄²

—

118

—

120

C₄²

—

236

—

121

C₄²

C₄×D₅₉

yes

—

122

C₄²

C₂²⋊D₅₉

yes

—

123

C₂³

C₂×D₂₃₆

yes

characteristic

124

Q₈

—

125

Q₈

—

118

—

126

Q₈

—

118

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

118

—

131

C₂×C₄

—

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×D₅₉)⋊C₄

yes

—

139

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

140

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

141

C₂×C₄

C₂×C₄×D₅₉

yes

—

142

C₂×C₄

#11745685

yes

—

143

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

144

C₈

—

145

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

146

C₂²

C₄×D₂₃₆

yes

—

147

C₂²

#14273755

yes

characteristic

148

C₂²

#14273760

yes

characteristic

149

C₂²

#14273760

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

118

—

153

C₄

—

236

—

154

C₄

—

236

—

155

C₄

C₄²×D₅₉

yes

characteristic

156

C₄

#14273684

yes

characteristic

157

C₄

C₄×C₂²⋊D₅₉

yes

—

158

C₂

C₄×(C₂×D₅₉)⋊C₄

yes

characteristic

159

C₂

#16837404

yes

characteristic

160

C₂

#16837412

yes

characteristic

161

C₂

#16837504

yes

—

162

C₂

#16837510

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₄×C₅₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#97×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#97	D₅₉
hypocenter	C₅₉	#1763
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₄×C₅₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#97×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×#97

D₅₉

hypocenter

C₅₉

#1763

LCS

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
8. C₄×Dic₁₁₈	C₄	semidirect
14. #14273652	C₄	semidirect
61. Dic₁₁₈	C₄²	semidirect
87. C₅₉	#1763	semidirect

H ≅ G/N

type

8. C₄×Dic₁₁₈

C₄

semidirect

14. #14273652

C₄

semidirect

61. Dic₁₁₈

C₄²

semidirect

87. C₅₉

#1763

semidirect

Subgroups — #21423927

Conjugacy classes of subgroups

Complements