Group #21423340 — subgroups

Subgroups — #21423340

Order 7552.

194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16838130	C₂	1	1	yes	maximal, characteristic
2	#16837807	C₂	1	1	yes	maximal, characteristic
3	#16837688	C₂	1	1	yes	maximal, characteristic
4	#16837612	C₂	1	1	yes	maximal, characteristic
5	#16837147	C₂	1	1	yes	maximal, characteristic
6	#16837134	C₂	1	1	yes	maximal, characteristic
7	C₅₉×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745672	—	2	1	no	—
9	#14273727	C₂²	1	1	yes	characteristic
10	#14273727	C₂²	1	1	yes	characteristic
11	#14273725	—	2	1	no	—
12	#14273719	—	2	1	no	—
13	#14273717	—	2	2	no	—
14	C₄×Dic₁₁₈	—	2	1	no	—
15	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
16	#14273692	—	2	1	no	—
17	#14273661	—	2	2	no	—
18	#14273659	—	2	1	no	—
19	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
20	#14273652	—	2	1	no	—
21	#14273650	—	2	1	no	—
22	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
26	C₂×Dic₁₁₈	—	2	1	no	—
27	C₂×Dic₁₁₈	—	2	1	no	—
28	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
29	C₂×C₈×C₅₉	—	2	1	no	—
30	C₂×C₈×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
33	C₄²×C₅₉	—	2	1	no	—
34	C₄²×C₅₉	C₂³	1	1	yes	characteristic
35	#11745672	—	4	2	no	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	C₄⋊Dic₅₉	—	2	1	no	—
38	C₄⋊Dic₅₉	—	2	1	no	—
39	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
40	C₄⋊Dic₅₉	D₄	1	2	yes	—
41	Dic₅₉⋊C₄	—	2	2	no	—
42	Dic₅₉⋊C₄	—	2	2	no	—
43	Dic₅₉⋊C₄	—	2	2	no	—
44	Dic₅₉⋊C₄	—	2	1	no	—
45	C₄×Dic₅₉	—	2	2	no	—
46	C₄×Dic₅₉	—	2	1	no	—
47	C₂×C₅₉⋊C₈	—	2	1	no	—
48	Q₈×C₅₉	—	2	1	no	—
49	Q₈×C₅₉	—	2	1	no	—
50	C₂×C₄×C₅₉	—	2	1	no	—
51	C₂×C₄×C₅₉	—	2	1	no	—
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₅₉	—	2	1	no	—
56	C₂×Dic₅₉	—	2	1	no	—
57	C₂×Dic₅₉	—	2	2	no	—
58	C₂×Dic₅₉	—	4	1	no	—
59	Dic₁₁₈	—	2	1	no	—
60	Dic₁₁₈	—	4	1	no	—
61	Dic₁₁₈	—	4	1	no	—
62	Dic₁₁₈	—	4	1	no	—
63	Dic₁₁₈	—	4	1	no	—
64	Dic₁₁₈	Q₁₆	1	2	yes	complemented
65	C₈×C₅₉	—	2	2	no	—
66	C₈×C₅₉	—	4	1	no	—
67	C₅₉⋊C₈	—	4	1	no	—
68	C₂²×C₅₉	#120	1	1	yes	characteristic
69	C₄×C₅₉	—	2	1	no	—
70	C₄×C₅₉	—	4	1	no	—
71	C₄×C₅₉	—	4	1	no	—
72	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
73	C₄×C₅₉	#135	1	1	yes	characteristic
74	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₅₉	#137	1	1	yes	characteristic
76	Dic₅₉	—	2	2	no	—
77	Dic₅₉	—	4	2	no	—
78	Dic₅₉	—	4	1	no	—
79	Dic₅₉	—	8	1	no	—
80	#1648	—	59	1	no	maximal, Sylow
81	C₂×C₅₉	#449	1	1	yes	characteristic
82	C₂×C₅₉	#451	1	1	yes	characteristic
83	C₂×C₅₉	#456	1	1	yes	characteristic
84	#557	—	59	1	no	—
85	#500	—	59	1	no	—
86	#475	—	59	1	no	—
87	#462	—	59	1	no	—
88	#332	—	59	1	no	—
89	Q₈⋊C₈	—	59	1	no	—
90	Q₈⋊C₈	D₅₉	1	1	yes	characteristic
91	C₅₉	#1648	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	118	1	no	—
93	C₄⋊Q₈	—	59	1	no	—
94	C₄⋊Q₈	—	59	1	no	—
95	C₄⋊Q₈	—	118	1	no	—
96	#125	—	118	1	no	—
97	#125	—	118	2	no	—
98	C₄×Q₈	—	59	1	no	—
99	C₄×Q₈	—	118	1	no	—
100	C₄×Q₈	—	118	1	no	—
101	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
102	#107	—	118	2	no	—
103	#106	—	118	1	no	—
104	C₄⋊C₈	—	59	1	no	—
105	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
106	#103	—	118	1	no	—
107	#103	—	118	1	no	—
108	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
109	C₂×Q₈	—	59	1	no	—
110	C₂×Q₈	—	118	1	no	—
111	C₂×Q₈	—	118	1	no	—
112	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
113	Q₁₆	—	236	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	118	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	59	1	no	—
119	C₄⋊C₄	—	59	2	no	—
120	C₄⋊C₄	—	118	2	no	—
121	C₄⋊C₄	—	118	2	no	—
122	C₄⋊C₄	—	118	2	no	—
123	C₄⋊C₄	—	118	1	no	—
124	C₄⋊C₄	—	118	1	no	—
125	C₄⋊C₄	—	118	1	no	—
126	C₄⋊C₄	—	118	1	no	—
127	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	118	2	no	—
130	C₄²	—	118	1	no	—
131	C₄²	C₂²×D₅₉	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	2	1	no	—
134	Q₈	—	59	2	no	—
135	Q₈	—	118	1	no	—
136	Q₈	—	236	1	no	—
137	Q₈	—	236	1	no	—
138	Q₈	—	236	1	no	—
139	Q₈	—	236	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	118	1	no	—
143	C₂×C₄	—	118	1	no	—
144	C₂×C₄	—	118	2	no	—
145	C₂×C₄	—	236	1	no	—
146	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
147	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
148	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	4	1	no	—
151	C₈	—	236	1	no	—
152	C₂²	#14273697	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	118	2	no	—
157	C₄	—	472	1	no	—
158	C₄	—	236	2	no	—
159	C₄	—	236	1	no	—
160	C₄	Q₁₆×D₅₉	1	1	yes	characteristic
161	C₄	#14273749	1	1	yes	characteristic
162	C₄	#14273755	1	1	yes	characteristic
163	C₄	#14273756	1	1	yes	characteristic
164	C₂	#16837549	1	1	yes	characteristic
165	C₂	#16837559	1	1	yes	characteristic
166	C₂	#16837586	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838130

C₂

yes

maximal, characteristic

#16837807

C₂

yes

maximal, characteristic

#16837688

C₂

yes

maximal, characteristic

#16837612

C₂

yes

maximal, characteristic

#16837147

C₂

yes

maximal, characteristic

#16837134

C₂

yes

maximal, characteristic

C₅₉×Q₈⋊C₈

C₂

yes

maximal, characteristic

C₂×#11745672

—

#14273727

C₂²

yes

characteristic

#14273727

C₂²

yes

—

—

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₂²

yes

—

—

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273652

—

#14273650

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745672

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

D₄

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

Q₁₆

yes

complemented

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

#120

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

D₅₉

yes

characteristic

C₅₉

#1648

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

118

—

#125

—

118

—

#125

—

118

—

C₄×Q₈

—

C₄×Q₈

—

118

—

100

C₄×Q₈

—

118

—

101

C₄×Q₈

C₂×D₅₉

yes

characteristic

102

#107

—

118

—

103

#106

—

118

—

104

C₄⋊C₈

—

105

C₄⋊C₈

C₂×D₅₉

yes

characteristic

106

#103

—

118

—

107

#103

—

118

—

108

C₄×C₈

C₂×D₅₉

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

118

—

111

C₂×Q₈

—

118

—

112

C₂×Q₈

D₂₃₆

yes

characteristic

113

Q₁₆

—

236

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

118

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

118

—

121

C₄⋊C₄

—

118

—

122

C₄⋊C₄

—

118

—

123

C₄⋊C₄

—

118

—

124

C₄⋊C₄

—

118

—

125

C₄⋊C₄

—

118

—

126

C₄⋊C₄

—

118

—

127

C₄⋊C₄

D₂₃₆

yes

characteristic

128

C₄²

—

129

C₄²

—

118

—

130

C₄²

—

118

—

131

C₄²

C₂²×D₅₉

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

118

—

136

Q₈

—

236

—

137

Q₈

—

236

—

138

Q₈

—

236

—

139

Q₈

—

236

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

118

—

143

C₂×C₄

—

118

—

144

C₂×C₄

—

118

—

145

C₂×C₄

—

236

—

146

C₂×C₄

C₂×D₂₃₆

yes

characteristic

147

C₂×C₄

D₄×D₅₉

yes

characteristic

148

C₂×C₄

D₄×D₅₉

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

236

—

152

C₂²

#14273697

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

118

—

157

C₄

—

472

—

158

C₄

—

236

—

159

C₄

—

236

—

160

C₄

Q₁₆×D₅₉

yes

characteristic

161

C₄

#14273749

yes

characteristic

162

C₄

#14273755

yes

characteristic

163

C₄

#14273756

yes

characteristic

164

C₂

#16837549

yes

characteristic

165

C₂

#16837559

yes

characteristic

166

C₂

#16837586

yes

characteristic

	subgroup	G/H
center	C₂²	#14273697
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×Q₈⋊C₈	C₂
socle	C₂²×C₅₉	#120
hypercenter	Q₈⋊C₈	D₅₉
hypocenter	C₅₉	#1648
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#14273697

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×Q₈⋊C₈

C₂

socle

C₂²×C₅₉

#120

hypercenter

Q₈⋊C₈

D₅₉

hypocenter

C₅₉

#1648

LCS

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
64. Dic₁₁₈	Q₁₆	semidirect
91. C₅₉	#1648	semidirect

H ≅ G/N

type

64. Dic₁₁₈

Q₁₆

semidirect

91. C₅₉

#1648

semidirect

Subgroups — #21423340

Conjugacy classes of subgroups

Complements