Group #21433176 — subgroups

Subgroups — #21433176

Order 7552.

396 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 217 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	C₈×Dic₁₁₈	C₂	1	4	yes	maximal, complemented
3	#16837526	C₂	1	4	yes	maximal, complemented
4	#16837455	C₂	1	1	yes	maximal, characteristic
5	#16837452	C₂	1	4	yes	maximal
6	C₅₉×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₁₈	C₄	1	1	yes	characteristic
8	C₄×Dic₁₁₈	C₄	1	2	yes	—
9	C₄×Dic₁₁₈	C₄	1	2	yes	—
10	C₄×Dic₁₁₈	C₂²	1	2	yes	—
11	C₄×Dic₁₁₈	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₅₉	C₄	1	1	yes	characteristic
13	C₂×Dic₅₉⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
15	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
16	Dic₅₉⋊C₈	C₂²	1	8	yes	—
17	C₂²⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
18	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
19	C₈×Dic₅₉	C₂²	1	8	yes	—
20	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
22	C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
23	C₄×C₈×C₅₉	C₂²	1	2	yes	—
24	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅₉	C₂×C₄	1	2	yes	—
28	C₂²×Dic₅₉	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₁₈	C₈	1	8	yes	complemented
30	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
31	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₅₉	C₂³	1	4	yes	—
33	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₅₉	C₂³	1	1	yes	characteristic
36	C₄²×C₅₉	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
39	Dic₅₉⋊C₄	C₂×C₄	1	4	yes	—
40	Dic₅₉⋊C₄	C₂×C₄	1	4	yes	—
41	C₄×Dic₅₉	C₂×C₄	1	4	yes	—
42	C₄×Dic₅₉	C₂³	1	4	yes	—
43	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
44	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	—	2	2	no	—
46	C₂×C₄×C₅₉	—	4	1	no	—
47	C₂×C₄×C₅₉	C₂×C₈	1	4	yes	—
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	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₅₉	—	4	2	no	—
55	C₂×Dic₅₉	C₂×C₈	1	8	yes	—
56	C₂×Dic₅₉	C₂²×C₄	1	4	yes	—
57	C₂×Dic₅₉	C₂²×C₄	1	4	yes	—
58	Dic₁₁₈	C₂×C₈	1	8	yes	—
59	Dic₁₁₈	C₂×C₈	1	8	yes	—
60	C₈×C₅₉	—	2	4	no	—
61	C₅₉⋊C₈	—	2	4	no	—
62	C₂²×C₅₉	—	2	1	no	—
63	C₂²×C₅₉	—	2	1	no	—
64	C₂²×C₅₉	C₂²×C₈	1	2	yes	—
65	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₅₉	—	2	1	no	—
67	C₄×C₅₉	—	2	1	no	—
68	C₄×C₅₉	—	2	1	no	—
69	C₄×C₅₉	C₂²×C₈	1	2	yes	—
70	C₄×C₅₉	C₂²×C₈	1	2	yes	—
71	C₄×C₅₉	2₋¹⁺⁴	1	2	yes	—
72	Dic₅₉	—	2	4	no	—
73	Dic₅₉	C₂²×C₈	1	8	yes	—
74	#2982	—	59	1	no	maximal, Sylow
75	C₂×C₅₉	—	2	2	no	—
76	C₂×C₅₉	#519	1	1	yes	characteristic
77	C₂×C₅₉	C₂³×C₈	1	1	yes	characteristic
78	C₂×C₅₉	#568	1	1	yes	characteristic
79	C₂×C₄×Q₈	—	59	1	no	—
80	C₈×Q₈	—	59	4	no	—
81	C₈×Q₈	—	59	4	no	—
82	#431	—	59	1	no	—
83	#431	—	59	4	no	—
84	#431	D₅₉	1	1	yes	characteristic
85	C₅₉	#2982	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	59	1	no	—
87	C₄×Q₈	—	59	2	no	—
88	C₄×Q₈	—	59	2	no	—
89	C₄×Q₈	—	59	2	no	—
90	C₄×Q₈	—	59	2	no	—
91	C₂×C₄⋊C₄	—	59	2	no	—
92	C₂×C₄⋊C₄	—	59	1	no	—
93	C₂×C₄²	—	59	2	no	—
94	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
95	C₄⋊C₈	—	59	8	no	—
96	C₄⋊C₈	—	59	2	no	—
97	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
98	C₂²⋊C₈	—	59	2	no	—
99	C₂²⋊C₈	C₂×D₅₉	1	2	yes	—
100	C₄×C₈	—	59	8	no	—
101	C₄×C₈	—	59	2	no	—
102	C₄×C₈	C₂×D₅₉	1	2	yes	—
103	C₂×Q₈	—	59	8	no	—
104	C₂×Q₈	—	59	2	no	—
105	C₂×Q₈	—	59	2	no	—
106	C₂²×C₄	—	59	2	no	—
107	C₂²×C₄	—	59	2	no	—
108	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
111	C₂×C₈	—	59	4	no	—
112	C₂×C₈	C₂²×D₅₉	1	4	yes	—
113	C₄⋊C₄	—	59	4	no	—
114	C₄⋊C₄	—	59	4	no	—
115	C₄⋊C₄	—	59	2	no	—
116	C₄⋊C₄	—	59	2	no	—
117	C₄²	—	59	4	no	—
118	C₄²	—	59	4	no	—
119	C₄²	C₄×D₅₉	1	1	yes	characteristic
120	C₄²	C₄×D₅₉	1	1	yes	characteristic
121	C₄²	C₂²×D₅₉	1	1	yes	characteristic
122	C₄²	C₂²×D₅₉	1	1	yes	characteristic
123	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
124	Q₈	—	59	8	no	—
125	Q₈	—	59	8	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	59	8	no	—
129	C₂×C₄	—	59	4	no	—
130	C₂×C₄	—	59	4	no	—
131	C₂×C₄	—	236	2	no	—
132	C₂×C₄	C₈×D₅₉	1	4	yes	—
133	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	118	4	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×C₈×D₅₉	1	2	yes	—
144	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	59	8	no	—
149	C₄	—	118	4	no	—
150	C₄	C₂×C₈×D₅₉	1	2	yes	—
151	C₄	C₂×C₈×D₅₉	1	2	yes	—
152	C₄	#14273774	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#16837879	1	1	yes	characteristic
155	C₂	C₂²×C₈×D₅₉	1	1	yes	characteristic
156	C₂	#16838168	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₁₈

C₂

yes

maximal, characteristic

C₈×Dic₁₁₈

C₂

yes

maximal, complemented

#16837526

C₂

yes

maximal, complemented

#16837455

C₂

yes

maximal, characteristic

#16837452

C₂

yes

maximal

C₅₉×#431

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₄

yes

characteristic

C₄×Dic₁₁₈

C₄

yes

—

C₄×Dic₁₁₈

C₄

yes

—

C₄×Dic₁₁₈

C₂²

yes

—

C₄×Dic₁₁₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₄

yes

characteristic

C₂×Dic₅₉⋊C₄

C₄

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

Dic₅₉⋊C₈

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

C₂²

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

—

C₈×Dic₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂×C₄

yes

—

C₂²×Dic₅₉

C₂×C₄

yes

—

C₂×Dic₁₁₈

C₈

yes

complemented

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

C₂×C₄

yes

—

Dic₅₉⋊C₄

C₂×C₄

yes

—

C₄×Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₈

yes

—

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

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×C₈

yes

—

C₂×Dic₅₉

C₂²×C₄

yes

—

C₂×Dic₅₉

C₂²×C₄

yes

—

Dic₁₁₈

C₂×C₈

yes

—

Dic₁₁₈

C₂×C₈

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×C₈

yes

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²×C₈

yes

—

C₄×C₅₉

C₂²×C₈

yes

—

C₄×C₅₉

2₋¹⁺⁴

yes

—

Dic₅₉

—

Dic₅₉

C₂²×C₈

yes

—

#2982

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#519

yes

characteristic

C₂×C₅₉

C₂³×C₈

yes

characteristic

C₂×C₅₉

#568

yes

characteristic

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

#431

—

#431

—

#431

D₅₉

yes

characteristic

C₅₉

#2982

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₉

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₅₉

yes

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

C₂×D₅₉

yes

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₅₉

yes

characteristic

109

C₂²×C₄

C₄×D₅₉

yes

characteristic

110

C₂²×C₄

C₂²×D₅₉

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₅₉

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₄×D₅₉

yes

characteristic

120

C₄²

C₄×D₅₉

yes

characteristic

121

C₄²

C₂²×D₅₉

yes

characteristic

122

C₄²

C₂²×D₅₉

yes

characteristic

123

C₂³

C₂×C₄×D₅₉

yes

characteristic

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

236

—

132

C₂×C₄

C₈×D₅₉

yes

—

133

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

134

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

135

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

136

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

137

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

138

C₂×C₄

C₂³×D₅₉

yes

characteristic

139

C₈

—

140

C₈

—

118

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×C₈×D₅₉

yes

—

144

C₂²

C₂²×C₄×D₅₉

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

118

—

150

C₄

C₂×C₈×D₅₉

yes

—

151

C₄

C₂×C₈×D₅₉

yes

—

152

C₄

#14273774

yes

—

153

C₂

—

154

C₂

#16837879

yes

characteristic

155

C₂

C₂²×C₈×D₅₉

yes

characteristic

156

C₂

#16838168

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator	C₂×C₅₉	C₂³×C₈
Fitting	C₅₉×#431	C₂
socle	C₂²×C₅₉	C₂³×C₄
hypercenter	#431	D₅₉
hypocenter	C₅₉	#2982
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₉

commutator

C₂×C₅₉

C₂³×C₈

Fitting

C₅₉×#431

C₂

socle

C₂²×C₅₉

C₂³×C₄

D₅₉

C₅₉

G ▷ C₂×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. C₈×Dic₁₁₈	C₂	semidirect
3. #16837526	C₂	semidirect
29. C₂×Dic₁₁₈	C₈	semidirect
85. C₅₉	#2982	semidirect

H ≅ G/N

type

2. C₈×Dic₁₁₈

C₂

semidirect

3. #16837526

C₂

semidirect

29. C₂×Dic₁₁₈

C₈

semidirect

85. C₅₉

#2982

semidirect

Subgroups — #21433176

Conjugacy classes of subgroups

Complements