Group #21423316 — subgroups

Subgroups — #21423316

Order 7552.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	#16837754	C₂	1	1	yes	maximal, characteristic
3	#16837609	C₂	1	1	yes	maximal, characteristic
4	#16837150	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#16837132	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#475	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745672	—	2	1	no	—
9	#14273727	—	2	1	no	—
10	#14273727	C₂²	1	1	yes	characteristic
11	#14273726	—	2	2	no	—
12	#14273726	—	2	1	no	—
13	Q₈×Dic₅₉	—	2	1	no	—
14	C₄×Dic₁₁₈	—	2	1	no	—
15	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
16	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
17	#14273651	—	2	1	no	—
18	#14273650	—	2	2	no	—
19	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
22	#106×C₅₉	—	2	1	no	—
23	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
24	#103×C₅₉	—	2	1	no	—
25	C₂×Q₈×C₅₉	—	2	1	no	—
26	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂×Dic₁₁₈	—	2	2	no	—
28	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
29	C₂×Dic₁₁₈	D₄	1	2	yes	—
30	C₂×C₈×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₄⋊C₄×C₅₉	—	2	1	no	—
33	C₄⋊C₄×C₅₉	—	2	1	no	—
34	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
35	C₄²×C₅₉	—	2	1	no	—
36	C₄²×C₅₉	C₂³	1	1	yes	characteristic
37	#11745672	—	4	2	no	—
38	C₄⋊Dic₅₉	—	2	2	no	—
39	C₄⋊Dic₅₉	—	2	1	no	—
40	C₄⋊Dic₅₉	—	2	1	no	—
41	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
42	Dic₅₉⋊C₄	—	2	2	no	—
43	Dic₅₉⋊C₄	—	2	1	no	—
44	C₄×Dic₅₉	—	2	2	no	—
45	C₄×Dic₅₉	—	2	1	no	—
46	C₂×C₅₉⋊C₈	—	2	1	no	—
47	C₂×C₅₉⋊C₈	—	2	1	no	—
48	Q₈×C₅₉	—	2	1	no	—
49	Q₈×C₅₉	—	4	1	no	—
50	Q₈×C₅₉	—	4	1	no	—
51	Q₈×C₅₉	Q₁₆	1	2	yes	complemented
52	C₂×C₄×C₅₉	—	2	1	no	—
53	C₂×C₄×C₅₉	—	2	1	no	—
54	C₂×C₄×C₅₉	—	4	1	no	—
55	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₅₉	—	2	1	no	—
59	C₂×Dic₅₉	—	2	2	no	—
60	C₂×Dic₅₉	—	2	1	no	—
61	Dic₁₁₈	—	2	1	no	—
62	Dic₁₁₈	—	4	2	no	—
63	Dic₁₁₈	—	8	2	no	—
64	Dic₁₁₈	SD₁₆	1	2	yes	—
65	C₈×C₅₉	—	4	1	no	—
66	C₅₉⋊C₈	—	4	1	no	—
67	C₅₉⋊C₈	—	4	1	no	—
68	C₂²×C₅₉	#120	1	1	yes	characteristic
69	C₄×C₅₉	—	2	2	no	—
70	C₄×C₅₉	—	2	1	no	—
71	C₄×C₅₉	—	4	1	no	—
72	C₄×C₅₉	—	8	1	no	—
73	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
74	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
75	C₄×C₅₉	#137	1	1	yes	characteristic
76	C₄×C₅₉	#137	1	1	yes	characteristic
77	Dic₅₉	—	2	2	no	—
78	Dic₅₉	—	4	2	no	—
79	Dic₅₉	—	4	1	no	—
80	#1645	—	59	1	no	maximal, Sylow
81	C₂×C₅₉	#448	1	1	yes	characteristic
82	C₂×C₅₉	#451	1	1	yes	characteristic
83	C₂×C₅₉	#455	1	1	yes	characteristic
84	Q₈²	—	59	1	no	—
85	#487	—	59	1	no	—
86	#475	D₅₉	1	1	yes	characteristic
87	#462	—	59	1	no	—
88	#333	—	59	1	no	—
89	Q₈⋊C₈	—	59	1	no	—
90	Q₈⋊C₈	—	59	1	no	—
91	C₅₉	#1645	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	118	1	no	—
93	C₄⋊Q₈	—	59	1	no	—
94	C₄⋊Q₈	—	118	1	no	—
95	C₄⋊Q₈	—	118	2	no	—
96	C₄⋊Q₈	—	118	1	no	—
97	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
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	#106	—	2	1	no	—
103	C₄⋊C₈	—	59	1	no	—
104	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
105	#103	—	2	1	no	—
106	#103	—	118	2	no	—
107	#103	—	118	1	no	—
108	C₄×C₈	—	59	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	59	1	no	—
111	C₂×Q₈	—	59	2	no	—
112	C₂×Q₈	—	118	2	no	—
113	C₂×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
114	Q₁₆	—	236	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	118	1	no	—
117	C₂×C₈	—	118	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	59	1	no	—
122	C₄⋊C₄	—	118	2	no	—
123	C₄⋊C₄	—	118	2	no	—
124	C₄⋊C₄	—	118	1	no	—
125	C₄⋊C₄	—	118	1	no	—
126	C₄⋊C₄	—	118	1	no	—
127	C₄⋊C₄	C₂²⋊D₅₉	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	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	59	2	no	—
136	Q₈	—	118	1	no	—
137	Q₈	—	472	2	no	—
138	Q₈	—	236	2	no	—
139	Q₈	#11745672	1	2	yes	complemented
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	118	1	no	—
144	C₂×C₄	—	118	1	no	—
145	C₂×C₄	—	118	2	no	—
146	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
147	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	236	1	no	—
151	C₈	—	236	1	no	—
152	C₂²	#14273695	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	8	1	no	—
157	C₄	—	118	2	no	—
158	C₄	—	236	1	no	—
159	C₄	—	236	2	no	—
160	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
161	C₄	C₂×#11745672	1	1	yes	characteristic
162	C₄	#14273756	1	1	yes	characteristic
163	C₄	#14273757	1	1	yes	characteristic
164	C₂	#16837539	1	1	yes	characteristic
165	C₂	#16837557	1	1	yes	characteristic
166	C₂	#16837578	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₁₈

C₂

yes

maximal, characteristic

#16837754

C₂

yes

maximal, characteristic

#16837609

C₂

yes

maximal, characteristic

#16837150

C₂

yes

maximal, characteristic

Q₈⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

#16837132

C₂

yes

maximal, characteristic

C₅₉×#475

C₂

yes

maximal, characteristic

C₂×#11745672

—

#14273727

—

#14273727

C₂²

yes

characteristic

#14273726

—

#14273726

—

Q₈×Dic₅₉

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273651

—

#14273650

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄×Q₈×C₅₉

C₂²

yes

characteristic

#106×C₅₉

—

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

#103×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×Dic₁₁₈

D₄

yes

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×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

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

Q₁₆

yes

complemented

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

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

SD₁₆

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

#120

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

Q₈²

—

#487

—

#475

D₅₉

yes

characteristic

#462

—

#333

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₅₉

#1645

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

—

118

—

C₄⋊Q₈

—

118

—

C₄⋊Q₈

—

118

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

118

—

100

C₄×Q₈

—

118

—

101

C₄×Q₈

C₂×D₅₉

yes

characteristic

102

#106

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₅₉

yes

characteristic

105

#103

—

106

#103

—

118

—

107

#103

—

118

—

108

C₄×C₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

118

—

113

C₂×Q₈

C₂²⋊D₅₉

yes

characteristic

114

Q₁₆

—

236

—

115

C₂×C₈

—

116

C₂×C₈

—

118

—

117

C₂×C₈

—

118

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

118

—

123

C₄⋊C₄

—

118

—

124

C₄⋊C₄

—

118

—

125

C₄⋊C₄

—

118

—

126

C₄⋊C₄

—

118

—

127

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

128

C₄²

—

129

C₄²

—

118

—

130

C₄²

—

118

—

131

C₄²

C₂²×D₅₉

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

118

—

137

Q₈

—

472

—

138

Q₈

—

236

—

139

Q₈

#11745672

yes

complemented

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

118

—

144

C₂×C₄

—

118

—

145

C₂×C₄

—

118

—

146

C₂×C₄

D₄×D₅₉

yes

characteristic

147

C₂×C₄

D₄×D₅₉

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

149

C₈

—

150

C₈

—

236

—

151

C₈

—

236

—

152

C₂²

#14273695

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

118

—

158

C₄

—

236

—

159

C₄

—

236

—

160

C₄

SD₁₆×D₅₉

yes

characteristic

161

C₄

C₂×#11745672

yes

characteristic

162

C₄

#14273756

yes

characteristic

163

C₄

#14273757

yes

characteristic

164

C₂

#16837539

yes

characteristic

165

C₂

#16837557

yes

characteristic

166

C₂

#16837578

yes

characteristic

	subgroup	G/H
center	C₂²	#14273695
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#475	C₂
socle	C₂²×C₅₉	#120
hypercenter	#475	D₅₉
hypocenter	C₅₉	#1645
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#14273695

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#475

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
51. Q₈×C₅₉	Q₁₆	semidirect
91. C₅₉	#1645	semidirect
139. Q₈	#11745672	semidirect

H ≅ G/N

type

51. Q₈×C₅₉

Q₁₆

semidirect

91. C₅₉

#1645

semidirect

139. Q₈

#11745672

semidirect

Subgroups — #21423316

Conjugacy classes of subgroups

Complements