Group #21425196 — subgroups

Subgroups — #21425196

Order 7552.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×#11745672	C₂	1	2	yes	maximal
2	C₂×#14273651	C₂	1	2	yes	maximal
3	#16837308	C₂	1	1	yes	maximal, characteristic
4	#16837157	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₂×#11745672	—	2	4	no	—
8	C₂×#11745672	—	2	4	no	—
9	C₂×#11745672	—	2	4	no	—
10	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
12	#14273651	—	2	4	no	—
13	C₅₉⋊#95	—	2	2	no	—
14	C₂²×Q₈×C₅₉	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₅₉	—	2	4	no	—
16	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
17	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₅₉	—	2	2	no	—
19	C₂×Q₈×C₅₉	—	2	2	no	—
20	C₂×Q₈×C₅₉	—	2	2	no	—
21	C₂×Q₈×C₅₉	—	2	2	no	—
22	C₂×Q₈×C₅₉	—	2	2	no	—
23	C₂×Q₈×C₅₉	D₄	1	4	yes	—
24	C₂²×C₄×C₅₉	—	2	2	no	—
25	C₂²×C₄×C₅₉	D₄	1	2	yes	—
26	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅₉	—	2	1	no	—
28	C₂²×Dic₅₉	—	2	1	no	—
29	C₂×Dic₁₁₈	—	2	2	no	—
30	C₂×Dic₁₁₈	—	2	2	no	—
31	C₂×Dic₁₁₈	—	2	1	no	—
32	C₂×Dic₁₁₈	D₄	1	2	yes	—
33	C₄⋊C₄×C₅₉	—	2	8	no	—
34	C₄²×C₅₉	—	2	2	no	—
35	#11745672	—	2	16	no	—
36	C₄⋊Dic₅₉	—	2	2	no	—
37	C₂×C₅₉⋊C₈	—	2	2	no	—
38	C₂×C₅₉⋊C₈	—	2	2	no	—
39	C₂×C₅₉⋊C₈	D₄	1	4	yes	—
40	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₅₉	—	2	4	no	—
42	Q₈×C₅₉	—	2	4	no	—
43	Q₈×C₅₉	—	4	2	no	—
44	Q₈×C₅₉	—	4	2	no	—
45	C₂×C₄×C₅₉	—	2	2	no	—
46	C₂×C₄×C₅₉	—	2	2	no	—
47	C₂×C₄×C₅₉	—	2	4	no	—
48	C₂×C₄×C₅₉	—	4	2	no	—
49	C₂×C₄×C₅₉	—	4	2	no	—
50	C₂×C₄×C₅₉	Q₁₆	1	4	yes	complemented
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
55	C₂×Dic₅₉	—	2	2	no	—
56	C₂×Dic₅₉	—	4	1	no	—
57	C₂×Dic₅₉	—	4	1	no	—
58	C₂×Dic₅₉	—	4	1	no	—
59	C₂×Dic₅₉	—	4	2	no	—
60	Dic₁₁₈	—	2	4	no	—
61	Dic₁₁₈	—	4	2	no	—
62	C₅₉⋊C₈	—	2	4	no	—
63	C₂²×C₅₉	#120	1	1	yes	characteristic
64	C₂²×C₅₉	#121	1	1	yes	characteristic
65	C₂²×C₅₉	#127	1	1	yes	characteristic
66	C₂²×C₅₉	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₅₉	#137	1	2	yes	—
68	C₄×C₅₉	—	2	4	no	—
69	C₄×C₅₉	—	4	4	no	—
70	C₄×C₅₉	#120	1	2	yes	—
71	C₄×C₅₉	#121	1	2	yes	—
72	Dic₅₉	—	4	2	no	—
73	Dic₅₉	—	8	1	no	—
74	#2016	—	59	1	no	maximal, Sylow
75	C₂×C₅₉	#392	1	1	yes	characteristic
76	C₂×C₅₉	#451	1	2	yes	—
77	C₂×C₅₉	#462	1	2	yes	—
78	C₂×C₅₉	#494	1	1	yes	characteristic
79	C₂×C₅₉	#497	1	1	yes	characteristic
80	C₂²×Q₁₆	—	59	2	no	—
81	C₂×C₄⋊Q₈	D₅₉	1	1	yes	characteristic
82	C₂×#103	—	59	2	no	—
83	#391	—	59	1	no	—
84	#336	—	59	1	no	—
85	C₅₉	#2016	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	59	1	no	—
87	C₂²×Q₈	C₂×D₅₉	1	2	yes	—
88	C₂×Q₁₆	—	118	4	no	—
89	C₂×Q₁₆	—	118	4	no	—
90	C₂×Q₁₆	—	118	4	no	—
91	C₂²×C₈	—	59	2	no	—
92	C₄⋊Q₈	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	—	59	1	no	—
95	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
96	#103	—	118	4	no	—
97	#95	—	118	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	59	2	no	—
104	C₂×Q₈	—	118	1	no	—
105	C₂×Q₈	—	118	2	no	—
106	C₂×Q₈	—	118	2	no	—
107	C₂×Q₈	C₂²⋊D₅₉	1	4	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	118	1	no	—
110	C₂²×C₄	—	118	1	no	—
111	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
113	Q₁₆	—	118	16	no	—
114	C₂×C₈	—	59	4	no	—
115	C₂×C₈	—	118	2	no	—
116	C₂×C₈	—	118	2	no	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	118	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	118	4	no	—
126	Q₈	—	236	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	118	2	no	—
133	C₂×C₄	—	236	1	no	—
134	C₂×C₄	—	236	2	no	—
135	C₂×C₄	—	236	1	no	—
136	C₂×C₄	—	236	1	no	—
137	C₂×C₄	#11745672	1	4	yes	complemented
138	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
139	C₂×C₄	D₄×D₅₉	1	2	yes	—
140	C₂×C₄	#11745687	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
142	C₈	—	118	4	no	—
143	C₂²	#14273696	1	1	yes	characteristic
144	C₂²	#14273698	1	1	yes	characteristic
145	C₂²	#14273723	1	1	yes	characteristic
146	C₂²	C₂×#11745672	1	2	yes	—
147	C₂²	#14273758	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	4	4	no	—
150	C₄	—	472	1	no	—
151	C₄	—	236	2	no	—
152	C₄	#14273695	1	2	yes	—
153	C₄	#14273699	1	2	yes	—
154	C₂	#16837310	1	1	yes	characteristic
155	C₂	#16837558	1	2	yes	—
156	C₂	#16837609	1	2	yes	—
157	C₂	#16837781	1	1	yes	characteristic
158	C₂	#16837794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11745672

C₂

yes

maximal

C₂×#14273651

C₂

yes

maximal

#16837308

C₂

yes

maximal, characteristic

#16837157

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

—

—

—

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

#14273651

—

C₅₉⋊#95

—

C₂²×Q₈×C₅₉

C₂²

yes

—

C₄⋊Q₈×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

#11745672

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

Q₁₆

yes

complemented

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₂×Q₁₆

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#120

yes

—

C₄×C₅₉

#121

yes

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#462

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

D₅₉

yes

—

—

—

C₅₉

#2016

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₉

yes

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#103

—

118

—

#95

—

118

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

118

—

105

C₂×Q₈

—

118

—

106

C₂×Q₈

—

118

—

107

C₂×Q₈

C₂²⋊D₅₉

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

118

—

110

C₂²×C₄

—

118

—

111

C₂²×C₄

C₂²⋊D₅₉

yes

—

112

C₂²×C₄

C₂²×D₅₉

yes

characteristic

113

Q₁₆

—

118

—

114

C₂×C₈

—

115

C₂×C₈

—

118

—

116

C₂×C₈

—

118

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

118

—

119

C₄²

—

120

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

118

—

126

Q₈

—

236

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

118

—

133

C₂×C₄

—

236

—

134

C₂×C₄

—

236

—

135

C₂×C₄

—

236

—

136

C₂×C₄

—

236

—

137

C₂×C₄

#11745672

yes

complemented

138

C₂×C₄

D₄×D₅₉

yes

characteristic

139

C₂×C₄

D₄×D₅₉

yes

—

140

C₂×C₄

#11745687

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

142

C₈

—

118

—

143

C₂²

#14273696

yes

characteristic

144

C₂²

#14273698

yes

characteristic

145

C₂²

#14273723

yes

characteristic

146

C₂²

C₂×#11745672

yes

—

147

C₂²

#14273758

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

472

—

151

C₄

—

236

—

152

C₄

#14273695

yes

—

153

C₄

#14273699

yes

—

154

C₂

#16837310

yes

characteristic

155

C₂

#16837558

yes

—

156

C₂

#16837609

yes

—

157

C₂

#16837781

yes

characteristic

158

C₂

#16837794

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×C₄⋊Q₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₉
hypocenter	C₅₉	#2016
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×C₄⋊Q₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₉

hypocenter

C₅₉

#2016

LCS

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
50. C₂×C₄×C₅₉	Q₁₆	semidirect
85. C₅₉	#2016	semidirect
137. C₂×C₄	#11745672	semidirect

H ≅ G/N

type

50. C₂×C₄×C₅₉

Q₁₆

semidirect

85. C₅₉

#2016

semidirect

137. C₂×C₄

#11745672

semidirect

Subgroups — #21425196

Conjugacy classes of subgroups

Complements