Group #21436526 — subgroups

Subgroups — #21436526

Order 7552.

340 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16838126	C₂	1	2	yes	maximal
2	#16837745	C₂	1	4	yes	maximal, complemented
3	#16837684	C₂	1	2	yes	maximal, complemented
4	#16837644	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×D₂₃₆	C₂	1	1	yes	maximal, characteristic
7	#16837439	C₂	1	2	yes	maximal, complemented
8	C₅₉×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₅₉	C₂²	1	2	yes	—
11	#14273722	—	2	4	no	—
12	#14273716	—	2	2	no	—
13	#14273707	C₂²	1	4	yes	—
14	Q₈×Dic₅₉	—	2	2	no	—
15	C₄×Dic₁₁₈	C₂²	1	2	yes	—
16	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
17	#14273685	C₂²	1	4	yes	—
18	#14273661	C₂²	1	1	yes	characteristic
19	#14273658	C₂²	1	4	yes	—
20	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
21	#14273652	C₂²	1	2	yes	—
22	#14273650	C₂²	1	4	yes	—
23	(C₂×D₅₉)⋊C₈	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
25	#107×C₅₉	C₂²	1	2	yes	—
26	#107×C₅₉	C₂²	1	2	yes	—
27	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₅₉	—	2	2	no	—
29	C₂×D₂₃₆	D₄	1	1	yes	characteristic
30	C₂×C₄×D₅₉	C₂³	1	2	yes	—
31	C₂×Dic₁₁₈	C₂³	1	2	yes	—
32	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₅₉	—	2	2	no	—
35	C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
36	C₄²×C₅₉	C₂³	1	1	yes	characteristic
37	(C₂×D₅₉)⋊C₄	—	2	4	no	—
38	(C₂×D₅₉)⋊C₄	—	2	4	no	—
39	(C₂×D₅₉)⋊C₄	D₄	1	2	yes	—
40	C₄⋊Dic₅₉	—	2	4	no	—
41	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
42	C₄⋊Dic₅₉	C₂³	1	2	yes	—
43	Dic₅₉⋊C₄	—	2	2	no	—
44	C₄×Dic₅₉	—	2	2	no	—
45	C₄×Dic₅₉	—	2	4	no	—
46	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
47	Dic₂₃₆	—	2	2	no	—
48	C₈×D₅₉	—	2	4	no	—
49	C₂²×D₅₉	C₂×D₄	1	2	yes	—
50	Q₈×C₅₉	—	4	4	no	—
51	C₂×C₄×C₅₉	—	2	4	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	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₅₉	—	2	2	no	—
56	C₂×Dic₅₉	—	2	2	no	—
57	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
58	D₂₃₆	—	2	2	no	—
59	C₄×D₅₉	—	2	2	no	—
60	C₄×D₅₉	—	2	2	no	—
61	Dic₁₁₈	—	2	2	no	—
62	Dic₁₁₈	—	2	2	no	—
63	C₈×C₅₉	—	2	1	no	—
64	C₈×C₅₉	#41	1	2	yes	—
65	C₅₉⋊C₈	—	2	2	no	—
66	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₅₉	—	2	2	no	—
68	C₂×D₅₉	—	2	2	no	—
69	C₂×D₅₉	—	2	2	no	—
70	C₄×C₅₉	—	2	1	no	—
71	C₄×C₅₉	—	4	4	no	—
72	C₄×C₅₉	#135	1	2	yes	—
73	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
74	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
75	Dic₅₉	—	2	2	no	—
76	Dic₅₉	—	4	2	no	—
77	Dic₅₉	—	4	2	no	—
78	#3342	—	59	1	no	maximal, Sylow
79	C₂×C₅₉	#547	1	1	yes	characteristic
80	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
81	C₂×C₅₉	#578	1	1	yes	characteristic
82	D₅₉	—	4	2	no	—
83	#556	—	59	2	no	—
84	#500	D₅₉	1	1	yes	characteristic, complemented
85	#485	—	59	4	no	—
86	#475	—	59	2	no	—
87	#467	—	59	2	no	—
88	C₄×Q₁₆	—	59	1	no	—
89	C₈×D₄	—	59	1	no	—
90	#427	—	59	2	no	—
91	C₅₉	#3342	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	59	1	no	—
93	C₂²×C₈	—	59	2	no	—
94	C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
95	#126	—	118	4	no	—
96	#124	—	118	2	no	—
97	C₂²⋊Q₈	—	59	4	no	—
98	C₄×Q₈	—	59	2	no	—
99	C₄×Q₈	—	118	2	no	—
100	C₄×D₄	—	59	1	no	—
101	#117	—	59	4	no	—
102	#107	—	59	1	no	—
103	#107	C₂×D₅₉	1	2	yes	—
104	#107	C₂×D₅₉	1	2	yes	—
105	#106	—	59	4	no	—
106	C₄⋊C₈	—	59	1	no	—
107	#103	—	59	2	no	—
108	#103	—	59	4	no	—
109	C₂²⋊C₈	—	59	2	no	—
110	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	—	59	2	no	—
113	C₂×D₄	—	59	1	no	—
114	C₂²×C₄	—	59	2	no	—
115	Q₁₆	—	118	2	no	—
116	C₂×C₈	—	59	2	no	—
117	C₂×C₈	—	118	4	no	—
118	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
119	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	59	1	no	—
122	C₄⋊C₄	—	59	2	no	—
123	C₄⋊C₄	—	118	4	no	—
124	C₄⋊C₄	—	118	2	no	—
125	C₄⋊C₄	C₂²×D₅₉	1	4	yes	—
126	C₂²⋊C₄	—	59	2	no	—
127	C₂²⋊C₄	—	118	4	no	—
128	C₂²⋊C₄	—	118	4	no	—
129	C₄²	—	118	4	no	—
130	C₄²	—	118	2	no	—
131	C₄²	C₂²×D₅₉	1	1	yes	characteristic
132	C₂³	—	59	2	no	—
133	Q₈	—	4	4	no	—
134	Q₈	—	118	2	no	—
135	Q₈	—	118	2	no	—
136	D₄	—	118	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	59	2	no	—
139	C₂×C₄	—	118	2	no	—
140	C₂×C₄	—	118	2	no	—
141	C₂×C₄	—	118	2	no	—
142	C₂×C₄	—	118	2	no	—
143	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
144	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
145	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	118	2	no	—
148	C₈	#11745689	1	2	yes	—
149	C₂²	—	118	2	no	—
150	C₂²	—	118	2	no	—
151	C₂²	—	118	2	no	—
152	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	4	no	—
155	C₄	—	118	2	no	—
156	C₄	—	236	2	no	—
157	C₄	—	236	2	no	—
158	C₄	#14273746	1	2	yes	—
159	C₄	C₂×#11745689	1	1	yes	characteristic
160	C₄	#14273775	1	1	yes	characteristic
161	C₂	—	236	2	no	—
162	C₂	#16838055	1	1	yes	characteristic
163	C₂	C₂×#14273746	1	1	yes	characteristic
164	C₂	#16838215	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838126

C₂

yes

maximal

#16837745

C₂

yes

maximal, complemented

#16837684

C₂

yes

maximal, complemented

#16837644

C₂

yes

maximal, complemented

C₄×Dic₂₃₆

C₂

yes

maximal, characteristic, complemented

C₈×D₂₃₆

C₂

yes

maximal, characteristic

#16837439

C₂

yes

maximal, complemented

C₅₉×#500

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×C₈×D₅₉

C₂²

yes

—

#14273722

—

#14273716

—

#14273707

C₂²

yes

—

Q₈×Dic₅₉

—

C₄×Dic₁₁₈

C₂²

yes

—

C₄×D₂₃₆

C₂²

yes

characteristic

#14273685

C₂²

yes

—

#14273661

C₂²

yes

characteristic

#14273658

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273652

C₂²

yes

—

#14273650

C₂²

yes

—

(C₂×D₅₉)⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

—

#107×C₅₉

C₂²

yes

—

#107×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₂₃₆

D₄

yes

characteristic

C₂×C₄×D₅₉

C₂³

yes

—

C₂×Dic₁₁₈

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

C₂³

yes

—

Dic₂₃₆

—

C₈×D₅₉

—

C₂²×D₅₉

C₂×D₄

yes

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₄

yes

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#41

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#135

yes

—

C₄×C₅₉

C₂×#41

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#556

—

#500

D₅₉

yes

characteristic, complemented

#485

—

#475

—

#467

—

C₄×Q₁₆

—

C₈×D₄

—

#427

—

C₅₉

#3342

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₅₉

yes

—

#126

—

118

—

#124

—

118

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

118

—

100

C₄×D₄

—

101

#117

—

102

#107

—

103

#107

C₂×D₅₉

yes

—

104

#107

C₂×D₅₉

yes

—

105

#106

—

106

C₄⋊C₈

—

107

#103

—

108

#103

—

109

C₂²⋊C₈

—

110

C₄×C₈

C₂×D₅₉

yes

characteristic

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

Q₁₆

—

118

—

116

C₂×C₈

—

117

C₂×C₈

—

118

—

118

C₂×C₈

C₂²×D₅₉

yes

characteristic

119

C₂×C₈

C₂²×D₅₉

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

118

—

124

C₄⋊C₄

—

118

—

125

C₄⋊C₄

C₂²×D₅₉

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

118

—

128

C₂²⋊C₄

—

118

—

129

C₄²

—

118

—

130

C₄²

—

118

—

131

C₄²

C₂²×D₅₉

yes

characteristic

132

C₂³

—

133

Q₈

—

134

Q₈

—

118

—

135

Q₈

—

118

—

136

D₄

—

118

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

118

—

140

C₂×C₄

—

118

—

141

C₂×C₄

—

118

—

142

C₂×C₄

—

118

—

143

C₂×C₄

D₄×D₅₉

yes

characteristic

144

C₂×C₄

D₄×D₅₉

yes

characteristic

145

C₂×C₄

C₂³×D₅₉

yes

characteristic

146

C₈

—

147

C₈

—

118

—

148

C₈

#11745689

yes

—

149

C₂²

—

118

—

150

C₂²

—

118

—

151

C₂²

—

118

—

152

C₂²

C₂×D₄×D₅₉

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

118

—

156

C₄

—

236

—

157

C₄

—

236

—

158

C₄

#14273746

yes

—

159

C₄

C₂×#11745689

yes

characteristic

160

C₄

#14273775

yes

characteristic

161

C₂

—

236

—

162

C₂

#16838055

yes

characteristic

163

C₂

C₂×#14273746

yes

characteristic

164

C₂

#16838215

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#500	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#500	D₅₉
hypocenter	C₅₉	#3342
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#500

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
2. #16837745	C₂	semidirect
3. #16837684	C₂	semidirect
4. #16837644	C₂	semidirect
5. C₄×Dic₂₃₆	C₂	semidirect
7. #16837439	C₂	semidirect
8. C₅₉×#500	C₂	semidirect
84. #500	D₅₉	semidirect
91. C₅₉	#3342	semidirect

H ≅ G/N

type

2. #16837745

C₂

semidirect

3. #16837684

C₂

semidirect

4. #16837644

C₂

semidirect

5. C₄×Dic₂₃₆

C₂

semidirect

7. #16837439

C₂

semidirect

8. C₅₉×#500

C₂

semidirect

84. #500

D₅₉

semidirect

91. C₅₉

#3342

semidirect

Subgroups — #21436526

Conjugacy classes of subgroups

Complements