Group #21425215 — subgroups

Subgroups — #21425215

Order 7552.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₂×#14273758	C₂	1	2	yes	maximal
2	C₂×#14273646	C₂	1	2	yes	maximal, complemented
3	#16837368	C₂	1	1	yes	maximal, characteristic
4	#16837167	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745685	C₂²	1	1	yes	characteristic
7	#14273758	—	2	4	no	—
8	#14273758	—	2	4	no	—
9	C₂×#11745672	—	2	2	no	—
10	C₂×#11745671	—	2	2	no	—
11	C₂×#11745664	C₂²	1	1	yes	characteristic
12	C₂×#11745664	C₂²	1	2	yes	—
13	#14273646	—	2	4	no	—
14	(C₂×D₅₉)⋊C₈	—	2	2	no	—
15	C₂²×Q₈×C₅₉	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₅₉	—	2	4	no	—
17	C₂×Q₁₆×C₅₉	—	2	2	no	—
18	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
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	—	2	2	no	—
24	C₂×Q₈×C₅₉	D₄	1	4	yes	—
25	C₂²×C₄×C₅₉	—	2	2	no	—
26	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
27	C₂×C₂²⋊D₅₉	—	2	1	no	—
28	#11745685	—	2	2	no	—
29	#11745685	—	2	2	no	—
30	C₂×D₂₃₆	D₄	1	1	yes	characteristic
31	C₂×C₄×D₅₉	—	2	1	no	—
32	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
33	Q₁₆×C₅₉	—	2	8	no	—
34	C₂×C₈×C₅₉	—	2	2	no	—
35	C₂×C₈×C₅₉	D₄	1	2	yes	—
36	#11745672	—	2	8	no	—
37	#11745671	—	2	8	no	—
38	#11745664	—	2	2	no	—
39	#11745664	—	2	2	no	—
40	#11745664	D₄	1	4	yes	—
41	C₂×C₅₉⋊C₈	—	2	2	no	—
42	C₂×C₅₉⋊C₈	—	2	1	no	—
43	C₂³×C₅₉	#41	1	1	yes	characteristic
44	C₂²×D₅₉	—	2	1	no	—
45	Q₈×C₅₉	—	2	4	no	—
46	Q₈×C₅₉	—	2	4	no	—
47	Q₈×C₅₉	—	4	2	no	—
48	Q₈×C₅₉	—	4	2	no	—
49	C₂×C₄×C₅₉	—	2	4	no	—
50	C₂×C₄×C₅₉	—	4	2	no	—
51	C₂×C₄×C₅₉	—	4	2	no	—
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂²⋊D₅₉	—	4	2	no	—
57	C₂×Dic₅₉	—	2	1	no	—
58	D₂₃₆	—	2	2	no	—
59	C₄×D₅₉	—	4	1	no	—
60	C₄×D₅₉	—	4	1	no	—
61	Dic₁₁₈	—	2	2	no	—
62	C₈×C₅₉	—	2	2	no	—
63	C₅₉⋊C₈	—	2	4	no	—
64	C₅₉⋊C₈	—	4	1	no	—
65	C₂²×C₅₉	—	2	2	no	—
66	C₂²×C₅₉	#121	1	1	yes	characteristic
67	C₂²×C₅₉	#121	1	2	yes	—
68	C₂×D₅₉	—	4	1	no	—
69	C₂×D₅₉	—	4	2	no	—
70	C₄×C₅₉	—	4	4	no	—
71	C₄×C₅₉	#120	1	2	yes	—
72	C₄×C₅₉	#120	1	1	yes	characteristic
73	C₄×C₅₉	#127	1	1	yes	characteristic
74	Dic₅₉	—	4	1	no	—
75	#2020	—	59	1	no	maximal, Sylow
76	C₂×C₅₉	—	2	2	no	—
77	C₂×C₅₉	#392	1	1	yes	characteristic
78	C₂×C₅₉	#473	1	2	yes	—
79	D₅₉	—	8	1	no	—
80	C₂×#137	—	59	2	no	—
81	C₂²×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
82	C₂×#101	—	59	2	no	—
83	#408	—	59	1	no	—
84	#341	—	59	1	no	—
85	C₅₉	#2020	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	—	59	1	no	—
87	C₂²×Q₈	C₂×D₅₉	1	2	yes	—
88	#137	—	118	4	no	—
89	#137	—	118	4	no	—
90	C₂×Q₁₆	—	2	4	no	—
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	118	2	no	—
93	C₂×SD₁₆	—	118	2	no	—
94	C₂×M₁₆	—	59	2	no	—
95	C₂×M₁₆	—	59	1	no	—
96	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
97	#101	—	118	4	no	—
98	C₂²⋊C₈	—	118	2	no	—
99	#41	—	118	2	no	—
100	#41	—	118	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	59	1	no	—
107	C₂×Q₈	C₂²⋊D₅₉	1	4	yes	—
108	C₂×D₄	—	59	1	no	—
109	C₂×D₄	—	118	1	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	118	1	no	—
112	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
113	Q₁₆	—	2	8	no	—
114	Q₁₆	—	118	8	no	—
115	SD₁₆	—	118	8	no	—
116	M₁₆	—	59	4	no	—
117	M₁₆	—	118	2	no	—
118	M₁₆	—	118	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	118	2	no	—
121	C₂×C₈	—	118	1	no	—
122	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
123	C₂³	—	118	1	no	—
124	C₂³	#11745687	1	1	yes	characteristic
125	Q₈	—	2	4	no	—
126	Q₈	—	2	4	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	118	2	no	—
130	D₄	—	118	2	no	—
131	D₄	—	236	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	118	1	no	—
136	C₂×C₄	—	236	1	no	—
137	C₂×C₄	—	236	1	no	—
138	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
139	C₂×C₄	D₄×D₅₉	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	118	4	no	—
144	C₈	—	236	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	236	2	no	—
147	C₂²	—	236	1	no	—
148	C₂²	#14273698	1	1	yes	characteristic
149	C₂²	#14273699	1	2	yes	—
150	C₄	—	4	4	no	—
151	C₄	—	236	1	no	—
152	C₄	#14273695	1	2	yes	—
153	C₄	#14273696	1	1	yes	characteristic
154	C₄	#14273723	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	472	1	no	—
157	C₂	#16837310	1	1	yes	characteristic
158	C₂	#16837674	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273758

C₂

yes

maximal

C₂×#14273646

C₂

yes

maximal, complemented

#16837368

C₂

yes

maximal, characteristic

#16837167

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

—

#14273646

—

(C₂×D₅₉)⋊C₈

—

C₂²×Q₈×C₅₉

C₂²

yes

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂²×C₈×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

C₂³

yes

characteristic

C₂×C₂²⋊D₅₉

—

#11745685

—

#11745685

—

C₂×D₂₃₆

D₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

D₄

yes

characteristic

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

—

—

—

—

D₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#41

yes

characteristic

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

#121

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#120

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#2020

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#392

yes

characteristic

C₂×C₅₉

#473

yes

—

D₅₉

—

C₂×#137

—

C₂²×Q₁₆

D₅₉

yes

characteristic, complemented

C₂×#101

—

#408

—

#341

—

C₅₉

#2020

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

C₂×D₅₉

yes

—

#137

—

118

—

#137

—

118

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

118

—

C₂×SD₁₆

—

118

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#101

—

118

—

C₂²⋊C₈

—

118

—

#41

—

118

—

100

#41

—

118

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

C₂²⋊D₅₉

yes

—

108

C₂×D₄

—

109

C₂×D₄

—

118

—

110

C₂²×C₄

—

111

C₂²×C₄

—

118

—

112

C₂²×C₄

C₂²×D₅₉

yes

characteristic

113

Q₁₆

—

114

Q₁₆

—

118

—

115

SD₁₆

—

118

—

116

M₁₆

—

117

M₁₆

—

118

—

118

M₁₆

—

118

—

119

C₂×C₈

—

120

C₂×C₈

—

118

—

121

C₂×C₈

—

118

—

122

C₂×C₈

C₂²⋊D₅₉

yes

—

123

C₂³

—

118

—

124

C₂³

#11745687

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

Q₈

—

118

—

130

D₄

—

118

—

131

D₄

—

236

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

118

—

136

C₂×C₄

—

236

—

137

C₂×C₄

—

236

—

138

C₂×C₄

D₄×D₅₉

yes

characteristic

139

C₂×C₄

D₄×D₅₉

yes

—

140

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

141

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

142

C₈

—

143

C₈

—

118

—

144

C₈

—

236

—

145

C₂²

—

146

C₂²

—

236

—

147

C₂²

—

236

—

148

C₂²

#14273698

yes

characteristic

149

C₂²

#14273699

yes

—

150

C₄

—

151

C₄

—

236

—

152

C₄

#14273695

yes

—

153

C₄

#14273696

yes

characteristic

154

C₄

#14273723

yes

characteristic

155

C₂

—

156

C₂

—

472

—

157

C₂

#16837310

yes

characteristic

158

C₂

#16837674

yes

—

	subgroup	G/H
center	C₂²	#14273698
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂²×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	#121
hypercenter	C₂²×Q₁₆	D₅₉
hypocenter	C₅₉	#2020
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#14273698

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂²×Q₁₆×C₅₉

C₂

socle

C₂²×C₅₉

#121

hypercenter

C₂²×Q₁₆

D₅₉

hypocenter

C₅₉

#2020

LCS

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. C₂×#14273646	C₂	semidirect
4. #16837167	C₂	semidirect
5. C₂²×Q₁₆×C₅₉	C₂	semidirect
81. C₂²×Q₁₆	D₅₉	semidirect
85. C₅₉	#2020	semidirect

H ≅ G/N

type

2. C₂×#14273646

C₂

semidirect

4. #16837167

C₂

semidirect

5. C₂²×Q₁₆×C₅₉

C₂

semidirect

81. C₂²×Q₁₆

D₅₉

semidirect

85. C₅₉

#2020

semidirect

Subgroups — #21425215

Conjugacy classes of subgroups

Complements