Group #21425040 — subgroups

Subgroups — #21425040

Order 7552.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16837959	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837163	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
5	#14273727	C₄	1	1	yes	characteristic, complemented
6	#14273727	C₄	1	1	yes	characteristic, complemented
7	#14273719	C₄	1	2	yes	complemented
8	#14273706	—	2	1	no	—
9	#14273706	—	2	1	no	—
10	C₄×Dic₁₁₈	—	2	1	no	—
11	C₄×Dic₁₁₈	—	2	1	no	—
12	#14273686	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₅₉	—	2	1	no	—
14	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
15	C₅₉⋊#95	—	2	1	no	—
16	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
18	#97×C₅₉	—	2	1	no	—
19	C₄×C₈×C₅₉	—	2	1	no	—
20	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₉	—	2	2	no	—
24	C₂×Dic₁₁₈	—	2	1	no	—
25	C₂×Dic₁₁₈	—	2	1	no	—
26	M₁₆×C₅₉	—	2	2	no	—
27	M₁₆×C₅₉	Q₈	1	4	yes	complemented
28	C₂×C₈×C₅₉	—	2	2	no	—
29	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₅₉	D₄	1	1	yes	characteristic
32	C₄²×C₅₉	D₄	1	1	yes	characteristic
33	C₂²⋊Dic₅₉	—	2	2	no	—
34	C₄⋊Dic₅₉	—	2	4	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
39	Dic₅₉⋊C₄	—	2	2	no	—
40	Dic₅₉⋊C₄	—	2	2	no	—
41	C₄×Dic₅₉	—	2	2	no	—
42	C₄×Dic₅₉	—	2	4	no	—
43	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₉	—	2	2	no	—
45	C₂×C₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₅₉	#41	1	2	yes	—
54	C₂×Dic₅₉	—	2	4	no	—
55	C₂×Dic₅₉	—	2	2	no	—
56	C₂×Dic₅₉	—	2	2	no	—
57	C₂×Dic₅₉	—	4	2	no	—
58	C₂×Dic₅₉	—	4	2	no	—
59	Dic₁₁₈	—	4	2	no	—
60	Dic₁₁₈	—	4	2	no	—
61	C₈×C₅₉	—	2	4	no	—
62	C₂²×C₅₉	—	2	1	no	—
63	C₂²×C₅₉	—	2	1	no	—
64	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₅₉	#124	1	1	yes	characteristic
66	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₄×C₅₉	—	2	1	no	—
68	C₄×C₅₉	—	2	1	no	—
69	C₄×C₅₉	—	2	1	no	—
70	C₄×C₅₉	#104	1	2	yes	—
71	C₄×C₅₉	C₄×Q₈	1	2	yes	—
72	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
73	Dic₅₉	—	4	4	no	—
74	Dic₅₉	—	4	2	no	—
75	Dic₅₉	—	4	2	no	—
76	#1986	—	59	1	no	maximal, Sylow
77	C₂×C₅₉	—	2	1	no	—
78	C₂×C₅₉	—	2	1	no	—
79	C₂×C₅₉	#391	1	1	yes	characteristic
80	C₂×C₅₉	C₂×#104	1	1	yes	characteristic
81	C₂×C₅₉	#421	1	1	yes	characteristic
82	#533	—	59	1	no	—
83	C₄×M₁₆	D₅₉	1	1	yes	characteristic
84	C₄×C₄⋊C₄	—	59	1	no	—
85	#339	—	59	4	no	—
86	C₅₉	#1986	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
88	C₄⋊Q₈	—	59	1	no	—
89	C₄⋊Q₈	—	59	1	no	—
90	#125	—	59	2	no	—
91	C₂²⋊Q₈	—	118	1	no	—
92	C₂²⋊Q₈	—	118	1	no	—
93	C₄×Q₈	—	118	1	no	—
94	C₄×Q₈	—	118	1	no	—
95	#117	—	59	2	no	—
96	C₂×C₄⋊C₄	—	118	1	no	—
97	C₂×C₄²	—	59	2	no	—
98	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
99	#97	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	#95	—	118	1	no	—
102	C₂×Q₈	—	118	1	no	—
103	C₂×Q₈	—	118	1	no	—
104	C₂²×C₄	—	118	2	no	—
105	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
106	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	Dic₁₁₈	1	4	yes	complemented
110	C₂×C₈	—	2	2	no	—
111	C₄⋊C₄	—	59	2	no	—
112	C₄⋊C₄	—	59	2	no	—
113	C₄⋊C₄	—	118	4	no	—
114	C₄⋊C₄	—	118	2	no	—
115	C₄⋊C₄	—	118	2	no	—
116	C₄⋊C₄	—	118	1	no	—
117	C₄⋊C₄	—	118	1	no	—
118	C₂²⋊C₄	—	118	2	no	—
119	C₄²	—	118	2	no	—
120	C₄²	—	118	4	no	—
121	C₄²	C₄×D₅₉	1	1	yes	characteristic
122	C₄²	C₄×D₅₉	1	1	yes	characteristic
123	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
124	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
125	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
126	Q₈	—	236	2	no	—
127	Q₈	—	236	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	118	4	no	—
131	C₂×C₄	—	118	2	no	—
132	C₂×C₄	—	118	2	no	—
133	C₂×C₄	—	236	2	no	—
134	C₂×C₄	—	236	2	no	—
135	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
138	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
139	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
141	C₂×C₄	#11745685	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
143	C₈	—	2	4	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
147	C₂²	#14273715	1	1	yes	characteristic
148	C₂²	#14273727	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	236	4	no	—
153	C₄	—	236	2	no	—
154	C₄	—	236	2	no	—
155	C₄	#14273653	1	2	yes	—
156	C₄	C₄×Dic₁₁₈	1	2	yes	—
157	C₄	#14273706	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#16837308	1	1	yes	characteristic
161	C₂	C₂×#14273653	1	1	yes	characteristic
162	C₂	#16837418	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

#16837163

C₂

yes

maximal

C₄×M₁₆×C₅₉

C₂

yes

maximal, characteristic

#14273727

C₄

yes

characteristic, complemented

#14273727

C₄

yes

characteristic, complemented

#14273719

C₄

yes

complemented

#14273706

—

#14273706

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

#14273686

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₅₉⋊#95

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

M₁₆×C₅₉

—

M₁₆×C₅₉

Q₈

yes

complemented

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

#124

yes

characteristic

C₂²×C₅₉

C₄⋊Q₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#104

yes

—

C₄×C₅₉

C₄×Q₈

yes

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#1986

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₄×M₁₆

D₅₉

yes

characteristic

C₄×C₄⋊C₄

—

#339

—

C₅₉

#1986

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×D₅₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

118

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

118

—

C₄×Q₈

—

118

—

#117

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#97

—

100

C₄×C₈

—

101

#95

—

118

—

102

C₂×Q₈

—

118

—

103

C₂×Q₈

—

118

—

104

C₂²×C₄

—

118

—

105

C₂²×C₄

D₂₃₆

yes

characteristic

106

C₂²×C₄

D₂₃₆

yes

characteristic

107

C₂²×C₄

C₂²×D₅₉

yes

characteristic

108

M₁₆

—

109

M₁₆

Dic₁₁₈

yes

complemented

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

118

—

114

C₄⋊C₄

—

118

—

115

C₄⋊C₄

—

118

—

116

C₄⋊C₄

—

118

—

117

C₄⋊C₄

—

118

—

118

C₂²⋊C₄

—

118

—

119

C₄²

—

118

—

120

C₄²

—

118

—

121

C₄²

C₄×D₅₉

yes

characteristic

122

C₄²

C₄×D₅₉

yes

characteristic

123

C₄²

C₂²⋊D₅₉

yes

characteristic

124

C₄²

C₂²⋊D₅₉

yes

characteristic

125

C₂³

C₂×D₂₃₆

yes

characteristic

126

Q₈

—

236

—

127

Q₈

—

236

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

118

—

131

C₂×C₄

—

118

—

132

C₂×C₄

—

118

—

133

C₂×C₄

—

236

—

134

C₂×C₄

—

236

—

135

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

136

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

137

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

138

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

139

C₂×C₄

C₂×Dic₁₁₈

yes

—

140

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

141

C₂×C₄

#11745685

yes

—

142

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

147

C₂²

#14273715

yes

characteristic

148

C₂²

#14273727

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

236

—

153

C₄

—

236

—

154

C₄

—

236

—

155

C₄

#14273653

yes

—

156

C₄

C₄×Dic₁₁₈

yes

—

157

C₄

#14273706

yes

—

158

C₂

—

159

C₂

—

160

C₂

#16837308

yes

characteristic

161

C₂

C₂×#14273653

yes

characteristic

162

C₂

#16837418

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₄×M₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	C₄×M₁₆	D₅₉
hypocenter	C₅₉	#1986
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₄×M₁₆×C₅₉

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

hypercenter

C₄×M₁₆

D₅₉

hypocenter

C₅₉

#1986

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
5. #14273727	C₄	semidirect
6. #14273727	C₄	semidirect
7. #14273719	C₄	semidirect
27. M₁₆×C₅₉	Q₈	semidirect
86. C₅₉	#1986	semidirect
109. M₁₆	Dic₁₁₈	semidirect

H ≅ G/N

type

5. #14273727

C₄

semidirect

6. #14273727

C₄

semidirect

7. #14273719

C₄

semidirect

27. M₁₆×C₅₉

Q₈

semidirect

86. C₅₉

#1986

semidirect

109. M₁₆

Dic₁₁₈

semidirect

Subgroups — #21425040

Conjugacy classes of subgroups

Complements