Group #21425572 — subgroups

Subgroups — #21425572

Order 7552.

230 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 93 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×#14273658	C₂	1	2	yes	maximal
2	#16837272	C₂	1	1	yes	maximal, characteristic
3	#16837164	C₂	1	2	yes	maximal
4	#16837157	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#389	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	—	2	1	no	—
8	C₂×C₄⋊Dic₅₉	—	2	1	no	—
9	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
10	#14273658	—	2	4	no	—
11	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
12	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	#95×C₅₉	—	2	1	no	—
16	C₂²×C₄×C₅₉	—	2	1	no	—
17	C₂²×C₄×C₅₉	—	2	1	no	—
18	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
21	C₂²×Dic₅₉	—	2	2	no	—
22	C₄⋊C₄×C₅₉	—	2	1	no	—
23	C₄⋊C₄×C₅₉	—	2	2	no	—
24	C₄⋊C₄×C₅₉	—	4	2	no	—
25	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
26	C₄²×C₅₉	—	2	2	no	—
27	C₄⋊Dic₅₉	—	2	2	no	—
28	C₄⋊Dic₅₉	—	2	2	no	—
29	C₄⋊Dic₅₉	—	4	2	no	—
30	C₄⋊Dic₅₉	—	4	2	no	—
31	C₂×C₅₉⋊C₈	—	2	2	no	—
32	C₂×C₅₉⋊C₈	—	2	2	no	—
33	C₂×C₅₉⋊C₈	Q₈	1	4	yes	—
34	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	—	2	1	no	—
36	C₂×C₄×C₅₉	—	2	1	no	—
37	C₂×C₄×C₅₉	—	2	1	no	—
38	C₂×C₄×C₅₉	—	2	1	no	—
39	C₂×C₄×C₅₉	—	2	2	no	—
40	C₂×C₄×C₅₉	—	4	1	no	—
41	C₂×C₄×C₅₉	—	4	1	no	—
42	C₂×C₄×C₅₉	—	4	1	no	—
43	C₂×C₄×C₅₉	—	4	1	no	—
44	C₂×C₄×C₅₉	—	4	1	no	—
45	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
46	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
50	C₂×C₄×C₅₉	#41	1	2	yes	—
51	C₂×Dic₅₉	—	4	2	no	—
52	C₂×Dic₅₉	—	4	2	no	—
53	C₂×Dic₅₉	—	4	2	no	—
54	C₅₉⋊C₈	—	2	4	no	—
55	C₂²×C₅₉	#121	1	1	yes	characteristic
56	C₂²×C₅₉	#123	1	1	yes	characteristic
57	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₅₉	#137	1	1	yes	characteristic
62	C₄×C₅₉	—	2	2	no	—
63	C₄×C₅₉	—	2	2	no	—
64	C₄×C₅₉	—	4	2	no	—
65	C₄×C₅₉	—	8	1	no	—
66	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₅₉	#125	1	2	yes	—
68	Dic₅₉	—	8	2	no	—
69	#2084	—	59	1	no	maximal, Sylow
70	C₂×C₅₉	#398	1	1	yes	characteristic
71	C₂×C₅₉	#460	1	1	yes	characteristic
72	C₂×C₅₉	#461	1	1	yes	characteristic
73	C₂×C₅₉	#481	1	1	yes	characteristic
74	C₂×C₅₉	#483	1	1	yes	characteristic
75	C₂×C₅₉	#498	1	1	yes	characteristic
76	C₂×C₅₉	#501	1	1	yes	characteristic
77	C₂×#106	—	59	2	no	—
78	#389	D₅₉	1	1	yes	characteristic
79	#384	—	59	1	no	—
80	#340	—	59	2	no	—
81	#336	—	59	1	no	—
82	C₅₉	#2084	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	—	59	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	59	2	no	—
86	C₂×C₄⋊C₄	—	118	1	no	—
87	C₂×C₄⋊C₄	—	118	1	no	—
88	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
91	#106	—	118	4	no	—
92	#95	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	118	2	no	—
96	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
99	C₂×C₈	—	59	4	no	—
100	C₂×C₈	—	118	2	no	—
101	C₂×C₈	—	118	2	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	118	2	no	—
106	C₄⋊C₄	—	118	2	no	—
107	C₄⋊C₄	—	236	2	no	—
108	C₄⋊C₄	—	236	2	no	—
109	C₄⋊C₄	D₂₃₆	1	2	yes	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	236	2	no	—
123	C₂×C₄	—	236	2	no	—
124	C₂×C₄	—	236	2	no	—
125	C₂×C₄	#11745670	1	2	yes	—
126	C₂×C₄	#11745671	1	2	yes	—
127	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
128	C₂×C₄	#11745685	1	1	yes	characteristic
129	C₂×C₄	#11745687	1	2	yes	—
130	C₂×C₄	Q₈×D₅₉	1	2	yes	—
131	C₈	—	118	4	no	—
132	C₂²	#14273701	1	1	yes	characteristic
133	C₂²	#14273708	1	1	yes	characteristic
134	C₂²	#14273725	1	1	yes	characteristic
135	C₂²	C₂×#11745671	1	1	yes	characteristic
136	C₂²	C₂×#11745670	1	1	yes	characteristic
137	C₂²	#14273752	1	1	yes	characteristic
138	C₂²	#14273757	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	8	1	no	—
143	C₄	—	472	2	no	—
144	C₄	#14273707	1	2	yes	—
145	C₄	#14273717	1	2	yes	—
146	C₂	#16837332	1	1	yes	characteristic
147	C₂	#16837602	1	1	yes	characteristic
148	C₂	#16837607	1	1	yes	characteristic
149	C₂	#16837718	1	1	yes	characteristic
150	C₂	#16837730	1	1	yes	characteristic
151	C₂	#16837798	1	1	yes	characteristic
152	C₂	#16837812	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273658

C₂

yes

maximal

#16837272

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal

#16837157

C₂

yes

maximal, characteristic

C₅₉×#389

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273658

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

Q₈

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₄⋊Q₈

yes

characteristic

C₂²×C₅₉

C₂×SD₁₆

yes

characteristic

C₂²×C₅₉

C₂×SD₁₆

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

#125

yes

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

—

—

—

C₅₉

#2084

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₂×C₄²

C₂×D₅₉

yes

characteristic

#106

—

118

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

118

—

101

C₂×C₈

—

118

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

118

—

106

C₄⋊C₄

—

118

—

107

C₄⋊C₄

—

236

—

108

C₄⋊C₄

—

236

—

109

C₄⋊C₄

D₂₃₆

yes

—

110

C₄²

—

111

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

236

—

123

C₂×C₄

—

236

—

124

C₂×C₄

—

236

—

125

C₂×C₄

#11745670

yes

—

126

C₂×C₄

#11745671

yes

—

127

C₂×C₄

C₂×D₂₃₆

yes

characteristic

128

C₂×C₄

#11745685

yes

characteristic

129

C₂×C₄

#11745687

yes

—

130

C₂×C₄

Q₈×D₅₉

yes

—

131

C₈

—

118

—

132

C₂²

#14273701

yes

characteristic

133

C₂²

#14273708

yes

characteristic

134

C₂²

#14273725

yes

characteristic

135

C₂²

C₂×#11745671

yes

characteristic

136

C₂²

C₂×#11745670

yes

characteristic

137

C₂²

#14273752

yes

characteristic

138

C₂²

#14273757

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

472

—

144

C₄

#14273707

yes

—

145

C₄

#14273717

yes

—

146

C₂

#16837332

yes

characteristic

147

C₂

#16837602

yes

characteristic

148

C₂

#16837607

yes

characteristic

149

C₂

#16837718

yes

characteristic

150

C₂

#16837730

yes

characteristic

151

C₂

#16837798

yes

characteristic

152

C₂

#16837812

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#389	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	#389	D₅₉
hypocenter	C₅₉	#2084
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#389

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
82. C₅₉	#2084	semidirect

H ≅ G/N

type

82. C₅₉

#2084

semidirect

Subgroups — #21425572

Conjugacy classes of subgroups

Complements