Group #21422596 — subgroups

Subgroups — #21422596

Order 7552.

240 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16837140	C₂	1	4	yes	maximal
4	C₅₉×#431	C₂	1	1	yes	maximal, characteristic
5	#14273727	C₄	1	1	yes	characteristic, complemented
6	#14273727	C₄	1	1	yes	characteristic, complemented
7	#14273719	—	2	1	no	—
8	#14273706	—	2	1	no	—
9	#14273706	C₂²	1	2	yes	—
10	C₄×Dic₁₁₈	—	2	1	no	—
11	C₄×Dic₁₁₈	C₄	1	2	yes	complemented
12	#14273686	—	2	1	no	—
13	C₂×C₄⋊Dic₅₉	—	2	1	no	—
14	C₂×C₄×Dic₅₉	—	2	1	no	—
15	C₅₉⋊#95	C₂²	1	2	yes	—
16	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₅₉	—	2	1	no	—
18	C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
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	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₅₉	—	2	2	no	—
27	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₅₉	D₄	1	1	yes	characteristic
30	C₄²×C₅₉	D₄	1	1	yes	characteristic
31	C₂²⋊Dic₅₉	—	2	2	no	—
32	C₄⋊Dic₅₉	—	2	4	no	—
33	C₄⋊Dic₅₉	—	2	1	no	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
37	Dic₅₉⋊C₄	—	2	2	no	—
38	Dic₅₉⋊C₄	—	2	2	no	—
39	C₄×Dic₅₉	—	2	2	no	—
40	C₄×Dic₅₉	—	4	2	no	—
41	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	—	2	2	no	—
43	C₂×C₄×C₅₉	—	4	1	no	—
44	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
49	C₂×C₄×C₅₉	Q₁₆	1	2	yes	—
50	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₅₉	—	2	4	no	—
53	C₂×Dic₅₉	—	2	2	no	—
54	C₂×Dic₅₉	—	2	2	no	—
55	C₂×Dic₅₉	—	4	2	no	—
56	C₂×Dic₅₉	—	4	2	no	—
57	Dic₁₁₈	—	4	2	no	—
58	Dic₁₁₈	—	4	2	no	—
59	C₈×C₅₉	—	4	2	no	—
60	C₂²×C₅₉	—	2	1	no	—
61	C₂²×C₅₉	—	2	1	no	—
62	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₄×C₅₉	—	2	1	no	—
66	C₄×C₅₉	—	2	1	no	—
67	C₄×C₅₉	—	2	1	no	—
68	C₄×C₅₉	#103	1	2	yes	—
69	C₄×C₅₉	#103	1	2	yes	—
70	C₄×C₅₉	#104	1	2	yes	—
71	Dic₅₉	—	4	4	no	—
72	Dic₅₉	—	4	2	no	—
73	Dic₅₉	—	4	2	no	—
74	#1503	—	59	1	no	maximal, Sylow
75	C₂×C₅₉	—	2	1	no	—
76	C₂×C₅₉	—	2	1	no	—
77	C₂×C₅₉	#410	1	1	yes	characteristic
78	C₂×C₅₉	C₂×#103	1	1	yes	characteristic
79	C₂×C₅₉	C₂×#104	1	1	yes	characteristic
80	#533	—	59	1	no	—
81	#431	D₅₉	1	1	yes	characteristic
82	C₄×C₄⋊C₄	—	59	1	no	—
83	#328	—	59	4	no	—
84	C₅₉	#1503	1	1	yes	characteristic, complemented, Sylow
85	C₄⋊Q₈	—	59	1	no	—
86	C₄⋊Q₈	—	59	1	no	—
87	#125	—	118	1	no	—
88	C₂²⋊Q₈	—	59	2	no	—
89	C₂²⋊Q₈	—	118	1	no	—
90	C₄×Q₈	—	59	2	no	—
91	C₄×Q₈	—	118	1	no	—
92	#117	—	118	1	no	—
93	C₂×C₄⋊C₄	—	118	1	no	—
94	C₂×C₄²	—	118	1	no	—
95	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
96	C₄⋊C₈	—	2	1	no	—
97	C₂²⋊C₈	C₂×D₅₉	1	2	yes	—
98	C₄×C₈	—	2	1	no	—
99	#95	—	59	2	no	—
100	C₂×Q₈	—	59	2	no	—
101	C₂×Q₈	—	118	1	no	—
102	C₂²×C₄	—	118	2	no	—
103	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
104	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₄⋊C₄	—	59	2	no	—
108	C₄⋊C₄	—	118	4	no	—
109	C₄⋊C₄	—	118	1	no	—
110	C₄⋊C₄	—	118	1	no	—
111	C₄⋊C₄	—	118	2	no	—
112	C₄⋊C₄	—	118	1	no	—
113	C₄⋊C₄	—	118	2	no	—
114	C₂²⋊C₄	—	118	2	no	—
115	C₄²	—	118	2	no	—
116	C₄²	—	236	2	no	—
117	C₄²	C₄×D₅₉	1	1	yes	characteristic
118	C₄²	C₄×D₅₉	1	1	yes	characteristic
119	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
120	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
121	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
122	Q₈	—	236	2	no	—
123	Q₈	—	236	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	118	4	no	—
127	C₂×C₄	—	118	2	no	—
128	C₂×C₄	—	118	2	no	—
129	C₂×C₄	—	236	2	no	—
130	C₂×C₄	—	236	2	no	—
131	C₂×C₄	#11745660	1	2	yes	—
132	C₂×C₄	Dic₂₃₆	1	2	yes	—
133	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
139	C₈	—	4	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
143	C₂²	C₂×#11745660	1	1	yes	characteristic
144	C₂²	C₂×Dic₂₃₆	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	236	4	no	—
149	C₄	—	236	2	no	—
150	C₄	—	236	2	no	—
151	C₄	#14273652	1	2	yes	—
152	C₄	#14273652	1	2	yes	—
153	C₄	#14273653	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#16837375	1	1	yes	characteristic
157	C₂	C₂×#14273652	1	1	yes	characteristic
158	C₂	C₂×#14273653	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

#16837140

C₂

yes

maximal

C₅₉×#431

C₂

yes

maximal, characteristic

#14273727

C₄

yes

characteristic, complemented

#14273727

C₄

yes

characteristic, complemented

#14273719

—

#14273706

—

#14273706

C₂²

yes

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₄

yes

complemented

#14273686

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄×Dic₅₉

—

C₅₉⋊#95

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

C₂²

yes

—

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

C₂×C₄

yes

—

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

—

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

SD₁₆

yes

—

C₂×C₄×C₅₉

Q₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

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

C₂×SD₁₆

yes

characteristic

C₂²×C₅₉

C₂×Q₁₆

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#103

yes

—

C₄×C₅₉

#103

yes

—

C₄×C₅₉

#104

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#1503

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

characteristic

C₄×C₄⋊C₄

—

#328

—

C₅₉

#1503

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

118

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

C₄×Q₈

—

118

—

#117

—

118

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

C₂×D₅₉

yes

—

C₄×C₈

—

#95

—

100

C₂×Q₈

—

101

C₂×Q₈

—

118

—

102

C₂²×C₄

—

118

—

103

C₂²×C₄

D₂₃₆

yes

characteristic

104

C₂²×C₄

D₂₃₆

yes

characteristic

105

C₂²×C₄

C₂²×D₅₉

yes

characteristic

106

C₂×C₈

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

118

—

109

C₄⋊C₄

—

118

—

110

C₄⋊C₄

—

118

—

111

C₄⋊C₄

—

118

—

112

C₄⋊C₄

—

118

—

113

C₄⋊C₄

—

118

—

114

C₂²⋊C₄

—

118

—

115

C₄²

—

118

—

116

C₄²

—

236

—

117

C₄²

C₄×D₅₉

yes

characteristic

118

C₄²

C₄×D₅₉

yes

characteristic

119

C₄²

C₂²⋊D₅₉

yes

characteristic

120

C₄²

C₂²⋊D₅₉

yes

characteristic

121

C₂³

C₂×D₂₃₆

yes

characteristic

122

Q₈

—

236

—

123

Q₈

—

236

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

118

—

127

C₂×C₄

—

118

—

128

C₂×C₄

—

118

—

129

C₂×C₄

—

236

—

130

C₂×C₄

—

236

—

131

C₂×C₄

#11745660

yes

—

132

C₂×C₄

Dic₂₃₆

yes

—

133

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

134

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

135

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

136

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

137

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

143

C₂²

C₂×#11745660

yes

characteristic

144

C₂²

C₂×Dic₂₃₆

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

236

—

149

C₄

—

236

—

150

C₄

—

236

—

151

C₄

#14273652

yes

—

152

C₄

#14273652

yes

—

153

C₄

#14273653

yes

—

154

C₂

—

155

C₂

—

156

C₂

#16837375

yes

characteristic

157

C₂

C₂×#14273652

yes

characteristic

158

C₂

C₂×#14273653

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₅₉)⋊C₄
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#431	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#431	D₅₉
hypocenter	C₅₉	#1503
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₅₉)⋊C₄

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#431

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
5. #14273727	C₄	semidirect
6. #14273727	C₄	semidirect
11. C₄×Dic₁₁₈	C₄	semidirect
84. C₅₉	#1503	semidirect

H ≅ G/N

type

5. #14273727

C₄

semidirect

6. #14273727

C₄

semidirect

11. C₄×Dic₁₁₈

C₄

semidirect

84. C₅₉

#1503

semidirect

Subgroups — #21422596

Conjugacy classes of subgroups

Complements