Group #21423607 — subgroups

Subgroups — #21423607

Order 7552.

166 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16837805	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837607	C₂	1	1	yes	maximal, characteristic
3	#16837480	C₂	1	1	yes	maximal, characteristic
4	#16837152	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837149	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837148	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#461	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745670	—	2	1	no	—
9	C₂×#11745664	—	2	1	no	—
10	#14273727	C₂²	1	1	yes	characteristic
11	#14273660	—	2	1	no	—
12	#14273660	—	2	1	no	—
13	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	#14273650	—	2	1	no	—
16	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
17	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₅₉	—	2	1	no	—
19	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	#103×C₅₉	—	2	1	no	—
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	C₂×D₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	—	2	1	no	—
26	C₂×Dic₁₁₈	—	2	1	no	—
27	SD₁₆×C₅₉	—	4	2	no	—
28	C₂×C₈×C₅₉	—	2	1	no	—
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
32	C₂²⋊C₄×C₅₉	—	2	1	no	—
33	C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	#11745670	—	4	2	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	#11745664	—	4	2	no	—
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	C₂×C₅₉⋊C₈	—	2	1	no	—
40	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
42	C₂³×C₅₉	—	2	1	no	—
43	Q₈×C₅₉	—	4	1	no	—
44	Q₈×C₅₉	—	4	1	no	—
45	D₄×C₅₉	—	2	1	no	—
46	D₄×C₅₉	—	2	1	no	—
47	C₂×C₄×C₅₉	—	2	1	no	—
48	C₂×C₄×C₅₉	—	4	1	no	—
49	C₂×C₄×C₅₉	—	4	1	no	—
50	C₂×C₄×C₅₉	—	4	1	no	—
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
54	C₂×Dic₅₉	—	4	1	no	—
55	Dic₁₁₈	—	4	1	no	—
56	Dic₁₁₈	—	4	1	no	—
57	C₈×C₅₉	—	4	1	no	—
58	C₅₉⋊C₈	—	2	1	no	—
59	C₅₉⋊C₈	—	4	1	no	—
60	C₅₉⋊C₈	—	4	1	no	—
61	C₅₉⋊C₈	SD₁₆	1	2	yes	complemented
62	C₂²×C₅₉	—	4	1	no	—
63	C₂²×C₅₉	—	4	1	no	—
64	C₂²×C₅₉	—	4	1	no	—
65	C₂²×C₅₉	#121	1	1	yes	characteristic
66	C₄×C₅₉	—	2	1	no	—
67	C₄×C₅₉	—	4	1	no	—
68	C₄×C₅₉	—	8	1	no	—
69	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₅₉	#137	1	1	yes	characteristic
73	Dic₅₉	—	8	1	no	—
74	#1693	—	59	1	no	maximal, Sylow
75	C₂×C₅₉	—	8	1	no	—
76	C₂×C₅₉	#461	1	1	yes	characteristic
77	C₂×C₅₉	#469	1	1	yes	characteristic
78	C₂×C₅₉	#473	1	1	yes	characteristic
79	#500	—	59	1	no	—
80	#461	—	59	1	no	—
81	#461	D₅₉	1	1	yes	characteristic
82	#436	—	59	1	no	—
83	#334	—	59	1	no	—
84	#332	—	59	1	no	—
85	#332	—	59	1	no	—
86	C₅₉	#1693	1	1	yes	characteristic, complemented, Sylow
87	C₂×SD₁₆	—	2	1	no	—
88	C₂×SD₁₆	—	118	1	no	—
89	C₂×M₁₆	—	118	1	no	—
90	C₄⋊Q₈	—	59	1	no	—
91	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
92	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
93	#107	—	118	1	no	—
94	#107	—	118	1	no	—
95	C₄⋊C₈	—	59	1	no	—
96	C₄⋊C₈	—	59	1	no	—
97	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
98	#103	—	2	1	no	—
99	#103	—	118	1	no	—
100	C₂²⋊C₈	—	118	1	no	—
101	C₄×C₈	—	59	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	118	1	no	—
104	C₂×D₄	C₂²⋊D₅₉	1	1	yes	characteristic
105	C₂²×C₄	—	2	1	no	—
106	SD₁₆	—	4	2	no	—
107	SD₁₆	—	236	2	no	—
108	M₁₆	—	236	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	59	1	no	—
111	C₂×C₈	—	59	1	no	—
112	C₂×C₈	—	118	1	no	—
113	C₂×C₈	—	118	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	118	1	no	—
117	C₄⋊C₄	—	118	1	no	—
118	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
119	C₂²⋊C₄	—	2	1	no	—
120	C₄²	C₂²×D₅₉	1	1	yes	characteristic
121	C₂³	—	2	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	236	1	no	—
125	Q₈	—	236	1	no	—
126	D₄	—	2	1	no	—
127	D₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	236	1	no	—
133	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
134	C₂×C₄	#11745687	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
136	C₈	—	4	1	no	—
137	C₈	—	59	2	no	—
138	C₈	—	118	1	no	—
139	C₈	—	236	1	no	—
140	C₈	—	236	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	#14273699	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	4	1	no	—
147	C₄	—	8	1	no	—
148	C₄	—	472	1	no	—
149	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
150	C₄	#14273752	1	1	yes	characteristic
151	C₄	#14273753	1	1	yes	characteristic
152	C₄	#14273756	1	1	yes	characteristic
153	C₂	—	8	1	no	—
154	C₂	#16837605	1	1	yes	characteristic
155	C₂	#16837653	1	1	yes	characteristic
156	C₂	#16837675	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837805

C₂

yes

maximal, characteristic, complemented

#16837607

C₂

yes

maximal, characteristic

#16837480

C₂

yes

maximal, characteristic

#16837152

C₂

yes

maximal, characteristic, complemented

#16837149

C₂

yes

maximal, characteristic, complemented

#16837148

C₂

yes

maximal, characteristic, complemented

C₅₉×#461

C₂

yes

maximal, characteristic

C₂×#11745670

—

C₂×#11745664

—

#14273727

C₂²

yes

characteristic

#14273660

—

#14273660

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273650

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₅₉

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄×D₄×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

#103×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

—

C₂×Dic₁₁₈

—

SD₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745670

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂³×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

SD₁₆

yes

complemented

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1693

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

D₅₉

yes

—

—

—

—

C₅₉

#1693

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

118

—

C₂×M₁₆

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄×D₄

C₂×D₅₉

yes

characteristic

#107

—

118

—

#107

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

#103

—

118

—

100

C₂²⋊C₈

—

118

—

101

C₄×C₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

118

—

104

C₂×D₄

C₂²⋊D₅₉

yes

characteristic

105

C₂²×C₄

—

106

SD₁₆

—

107

SD₁₆

—

236

—

108

M₁₆

—

236

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

118

—

113

C₂×C₈

—

118

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

118

—

117

C₄⋊C₄

—

118

—

118

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

119

C₂²⋊C₄

—

120

C₄²

C₂²×D₅₉

yes

characteristic

121

C₂³

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

236

—

125

Q₈

—

236

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

236

—

133

C₂×C₄

D₄×D₅₉

yes

characteristic

134

C₂×C₄

#11745687

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

118

—

139

C₈

—

236

—

140

C₈

—

236

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#14273699

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

472

—

149

C₄

SD₁₆×D₅₉

yes

characteristic

150

C₄

#14273752

yes

characteristic

151

C₄

#14273753

yes

characteristic

152

C₄

#14273756

yes

characteristic

153

C₂

—

154

C₂

#16837605

yes

characteristic

155

C₂

#16837653

yes

characteristic

156

C₂

#16837675

yes

characteristic

	subgroup	G/H
center	C₂²	#14273699
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#461	C₂
socle	C₂²×C₅₉	#121
hypercenter	#461	D₅₉
hypocenter	C₅₉	#1693
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

subgroup

G/H

center

C₂²

#14273699

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#461

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #461

N	H ≅ G/N	type
1. #16837805	C₂	semidirect
4. #16837152	C₂	semidirect
5. #16837149	C₂	semidirect
6. #16837148	C₂	semidirect
61. C₅₉⋊C₈	SD₁₆	semidirect
86. C₅₉	#1693	semidirect

H ≅ G/N

type

1. #16837805

C₂

semidirect

4. #16837152

C₂

semidirect

5. #16837149

C₂

semidirect

6. #16837148

C₂

semidirect

61. C₅₉⋊C₈

SD₁₆

semidirect

86. C₅₉

#1693

semidirect

Subgroups — #21423607

Conjugacy classes of subgroups

Complements