Group #21422566 — subgroups

Subgroups — #21422566

Order 7552.

246 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16837952	C₂	1	1	yes	maximal, characteristic
2	#16837459	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837131	C₂	1	4	yes	maximal, complemented
4	C₅₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11745685	—	2	1	no	—
6	#14273727	C₄	1	1	yes	characteristic
7	#14273724	C₄	1	1	yes	characteristic
8	#14273715	—	2	1	no	—
9	#14273701	—	2	1	no	—
10	#14273701	C₄	1	2	yes	—
11	C₄×D₂₃₆	—	2	1	no	—
12	C₄×D₂₃₆	C₂²	1	2	yes	—
13	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
14	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
15	#14273638	—	2	1	no	—
16	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₅₉	—	2	1	no	—
18	C₂²⋊C₈×C₅₉	—	2	1	no	—
19	C₄×C₈×C₅₉	C₂²	1	2	yes	—
20	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₅₉	—	2	2	no	—
24	#11745685	—	4	4	no	—
25	C₂×D₂₃₆	—	2	1	no	—
26	C₂×D₂₃₆	—	2	1	no	—
27	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₅₉	—	2	2	no	—
29	C₂×Dic₁₁₈	—	2	1	no	—
30	C₂×C₈×C₅₉	—	2	2	no	—
31	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₅₉	D₄	1	1	yes	characteristic
33	C₄²×C₅₉	D₄	1	1	yes	characteristic
34	C₄²×C₅₉	C₂³	1	1	yes	characteristic
35	(C₂×D₅₉)⋊C₄	—	2	2	no	—
36	(C₂×D₅₉)⋊C₄	—	2	2	no	—
37	C₄⋊Dic₅₉	—	2	1	no	—
38	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
39	C₂×C₅₉⋊C₈	—	2	2	no	—
40	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×D₅₉	—	2	2	no	—
42	C₂×C₄×C₅₉	—	2	2	no	—
43	C₂×C₄×C₅₉	—	2	2	no	—
44	C₂×C₄×C₅₉	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂²⋊D₅₉	—	4	4	no	—
52	C₂×Dic₅₉	—	2	2	no	—
53	D₂₃₆	—	2	4	no	—
54	D₂₃₆	—	4	2	no	—
55	D₂₃₆	—	4	2	no	—
56	C₄×D₅₉	—	4	4	no	—
57	Dic₁₁₈	—	4	2	no	—
58	C₈×C₅₉	—	4	2	no	—
59	C₅₉⋊C₈	—	4	2	no	—
60	C₂²×C₅₉	—	2	1	no	—
61	C₂²×C₅₉	—	2	1	no	—
62	C₂²×C₅₉	—	2	1	no	—
63	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×D₅₉	—	4	2	no	—
65	C₂×D₅₉	—	4	2	no	—
66	C₂×D₅₉	—	4	2	no	—
67	C₄×C₅₉	—	2	1	no	—
68	C₄×C₅₉	—	2	1	no	—
69	C₄×C₅₉	—	2	1	no	—
70	C₄×C₅₉	—	2	1	no	—
71	C₄×C₅₉	#131	1	2	yes	—
72	C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
73	Dic₅₉	—	4	2	no	—
74	#1496	—	59	1	no	maximal, Sylow
75	C₂×C₅₉	—	4	1	no	—
76	C₂×C₅₉	#408	1	1	yes	characteristic
77	C₂×C₅₉	#418	1	1	yes	characteristic
78	C₂×C₅₉	#421	1	1	yes	characteristic
79	D₅₉	—	8	2	no	—
80	#532	—	59	1	no	—
81	#432	—	59	1	no	—
82	#431	D₅₉	1	1	yes	characteristic, complemented
83	D₄⋊C₈	—	59	4	no	—
84	C₅₉	#1496	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	—	118	1	no	—
86	C₄⋊Q₈	—	59	1	no	—
87	#127	—	59	1	no	—
88	#124	—	118	1	no	—
89	#121	—	59	2	no	—
90	#121	—	118	1	no	—
91	C₄×D₄	—	59	2	no	—
92	C₄×D₄	—	118	1	no	—
93	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
94	C₄⋊C₈	—	2	1	no	—
95	C₄⋊C₈	—	59	2	no	—
96	C₂²⋊C₈	—	2	1	no	—
97	C₂²⋊C₈	—	118	1	no	—
98	#97	—	118	1	no	—
99	C₄×C₈	C₂×D₅₉	1	2	yes	—
100	#41	—	236	4	no	—
101	C₂×Q₈	—	118	1	no	—
102	C₂×D₄	—	59	2	no	—
103	C₂×D₄	—	118	2	no	—
104	C₂×D₄	—	118	1	no	—
105	C₂×D₄	—	118	1	no	—
106	C₂²×C₄	—	118	2	no	—
107	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
108	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	118	2	no	—
112	C₄⋊C₄	—	59	2	no	—
113	C₄⋊C₄	—	118	1	no	—
114	C₂²⋊C₄	—	118	2	no	—
115	C₂²⋊C₄	—	118	2	no	—
116	C₄²	C₄×D₅₉	1	1	yes	characteristic
117	C₄²	D₂₃₆	1	1	yes	characteristic
118	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
119	C₄²	C₂²×D₅₉	1	1	yes	characteristic
120	C₂³	—	118	2	no	—
121	C₂³	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
122	Q₈	—	236	2	no	—
123	D₄	—	118	4	no	—
124	D₄	—	236	2	no	—
125	D₄	—	236	2	no	—
126	D₄	—	236	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	118	2	no	—
131	C₂×C₄	—	236	4	no	—
132	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
133	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
136	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	236	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	236	2	no	—
144	C₂²	—	236	2	no	—
145	C₂²	—	236	2	no	—
146	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	236	2	no	—
152	C₄	#14273733	1	2	yes	—
153	C₄	#14273755	1	2	yes	—
154	C₂	—	4	1	no	—
155	C₂	—	472	2	no	—
156	C₂	#16837368	1	1	yes	characteristic
157	C₂	#16837408	1	1	yes	characteristic
158	C₂	#16837420	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837952

C₂

yes

maximal, characteristic

#16837459

C₂

yes

maximal, characteristic, complemented

#16837131

C₂

yes

maximal, complemented

C₅₉×#431

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

—

#14273727

C₄

yes

characteristic

#14273724

C₄

yes

—

—

C₄

yes

—

C₄×D₂₃₆

—

C₄×D₂₃₆

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂×C₂²⋊D₅₉

—

#11745685

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

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

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#131

yes

—

C₄×C₅₉

AGL(1,ℤ/8)

yes

—

Dic₅₉

—

#1496

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#532

—

#432

—

#431

D₅₉

yes

characteristic, complemented

D₄⋊C₈

—

C₅₉

#1496

yes

characteristic, complemented, Sylow

C₂×#41

—

118

—

C₄⋊Q₈

—

#127

—

#124

—

118

—

#121

—

#121

—

118

—

C₄×D₄

—

C₄×D₄

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

118

—

#97

—

118

—

C₄×C₈

C₂×D₅₉

yes

—

100

#41

—

236

—

101

C₂×Q₈

—

118

—

102

C₂×D₄

—

103

C₂×D₄

—

118

—

104

C₂×D₄

—

118

—

105

C₂×D₄

—

118

—

106

C₂²×C₄

—

118

—

107

C₂²×C₄

C₄×D₅₉

yes

characteristic

108

C₂²×C₄

D₂₃₆

yes

characteristic

109

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

118

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

118

—

114

C₂²⋊C₄

—

118

—

115

C₂²⋊C₄

—

118

—

116

C₄²

C₄×D₅₉

yes

characteristic

117

C₄²

D₂₃₆

yes

characteristic

118

C₄²

C₂²⋊D₅₉

yes

characteristic

119

C₄²

C₂²×D₅₉

yes

characteristic

120

C₂³

—

118

—

121

C₂³

(C₂×D₅₉)⋊C₄

yes

characteristic

122

Q₈

—

236

—

123

D₄

—

118

—

124

D₄

—

236

—

125

D₄

—

236

—

126

D₄

—

236

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

118

—

131

C₂×C₄

—

236

—

132

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

133

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

134

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

135

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

136

C₂×C₄

C₂×D₂₃₆

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

138

C₈

—

139

C₈

—

236

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

236

—

144

C₂²

—

236

—

145

C₂²

—

236

—

146

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

236

—

152

C₄

#14273733

yes

—

153

C₄

#14273755

yes

—

154

C₂

—

155

C₂

—

472

—

156

C₂

#16837368

yes

characteristic

157

C₂

#16837408

yes

characteristic

158

C₂

#16837420

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#431	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#431	D₅₉
hypocenter	C₅₉	#1496
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#431

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
2. #16837459	C₂	semidirect
3. #16837131	C₂	semidirect
4. C₅₉×#431	C₂	semidirect
82. #431	D₅₉	semidirect
84. C₅₉	#1496	semidirect

H ≅ G/N

type

2. #16837459

C₂

semidirect

3. #16837131

C₂

semidirect

4. C₅₉×#431

C₂

semidirect

82. #431

D₅₉

semidirect

84. C₅₉

#1496

semidirect

Subgroups — #21422566

Conjugacy classes of subgroups

Complements