Group #21426135 — subgroups

Subgroups — #21426135

Order 7552.

174 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 49 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×#14273672	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14273671	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837777	C₂	1	1	yes	maximal, characteristic
4	#16837496	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837202	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
9	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
10	C₂×#11745669	—	2	1	no	—
11	#14273723	—	2	1	no	—
12	#14273692	—	2	1	no	—
13	#14273672	—	2	2	no	—
14	#14273671	—	2	2	no	—
15	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14273650	—	2	1	no	—
17	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	#107×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₅₉	—	2	1	no	—
22	C₂×C₂²⋊D₅₉	—	4	1	no	—
23	C₂×C₂²⋊D₅₉	—	4	1	no	—
24	C₂×D₂₃₆	—	2	1	no	—
25	C₂×Dic₁₁₈	—	2	1	no	—
26	D₈×C₅₉	—	2	1	no	—
27	D₈×C₅₉	—	2	1	no	—
28	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	#11745669	—	4	2	no	—
31	Dic₅₉⋊C₄	—	4	1	no	—
32	C₄×Dic₅₉	—	4	1	no	—
33	C₄×Dic₅₉	D₄	1	1	yes	characteristic
34	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
35	Dic₂₃₆	—	2	1	no	—
36	Dic₂₃₆	D₄	1	2	yes	complemented
37	D₄₇₂	—	2	1	no	—
38	D₄₇₂	—	2	1	no	—
39	C₁₆×C₅₉	—	2	1	no	—
40	C₅₉⋊C₁₆	—	2	1	no	—
41	C₂³×C₅₉	—	4	1	no	—
42	C₂²×D₅₉	—	4	1	no	—
43	D₄×C₅₉	—	4	1	no	—
44	D₄×C₅₉	—	4	1	no	—
45	C₂×C₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂²⋊D₅₉	—	8	2	no	—
48	C₂²⋊D₅₉	—	8	2	no	—
49	C₂×Dic₅₉	—	4	1	no	—
50	C₂×Dic₅₉	D₈	1	1	yes	characteristic
51	C₂×Dic₅₉	D₈	1	1	yes	characteristic
52	D₂₃₆	—	4	1	no	—
53	D₂₃₆	—	4	1	no	—
54	Dic₁₁₈	—	2	2	no	—
55	Dic₁₁₈	—	4	1	no	—
56	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₈×C₅₉	#41	1	1	yes	characteristic
58	C₅₉⋊C₈	—	2	1	no	—
59	C₂²×C₅₉	—	8	1	no	—
60	C₂²×C₅₉	—	8	1	no	—
61	C₂²×C₅₉	—	8	1	no	—
62	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
63	C₂×D₅₉	—	8	1	no	—
64	C₂×D₅₉	—	8	1	no	—
65	C₂×D₅₉	—	8	1	no	—
66	C₄×C₅₉	—	8	1	no	—
67	C₄×C₅₉	#121	1	1	yes	characteristic
68	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
69	Dic₅₉	—	2	1	no	—
70	Dic₅₉	—	4	2	no	—
71	Dic₅₉	SD₃₂	1	2	yes	complemented
72	#2207	—	59	1	no	maximal, Sylow
73	C₂×C₅₉	—	16	1	no	—
74	C₂×C₅₉	#459	1	1	yes	characteristic
75	C₂×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
76	C₂×C₅₉	#509	1	1	yes	characteristic
77	D₅₉	—	16	1	no	—
78	C₂×SD₃₂	—	59	1	no	—
79	C₂×SD₃₂	—	59	1	no	—
80	#493	—	59	1	no	—
81	C₄×Q₁₆	—	59	1	no	—
82	C₄⋊C₁₆	—	59	1	no	—
83	#357	—	59	1	no	—
84	#357	D₅₉	1	1	yes	characteristic, complemented
85	C₅₉	#2207	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₁₆	—	59	1	no	—
87	C₂×D₈	—	59	1	no	—
88	C₂×D₈	—	118	1	no	—
89	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
90	#127	—	118	1	no	—
91	C₄×Q₈	—	118	1	no	—
92	SD₃₂	—	118	2	no	—
93	SD₃₂	—	118	2	no	—
94	C₂×C₁₆	—	59	1	no	—
95	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
96	#107	C₂×D₅₉	1	1	yes	characteristic
97	#103	—	118	1	no	—
98	C₄×C₈	—	59	1	no	—
99	C₂×Q₈	—	118	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	118	1	no	—
102	C₂×D₄	—	236	1	no	—
103	C₂×D₄	—	236	1	no	—
104	Q₁₆	—	59	2	no	—
105	Q₁₆	—	118	1	no	—
106	D₈	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	118	1	no	—
109	D₈	—	118	1	no	—
110	D₈	—	236	2	no	—
111	C₂×C₈	—	59	1	no	—
112	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	236	1	no	—
115	C₄²	—	59	1	no	—
116	C₄²	—	236	1	no	—
117	C₁₆	—	2	1	no	—
118	C₁₆	—	118	1	no	—
119	C₂³	—	4	1	no	—
120	C₂³	—	236	1	no	—
121	Q₈	—	118	2	no	—
122	Q₈	—	236	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	472	2	no	—
126	D₄	—	472	2	no	—
127	D₄	—	236	1	no	—
128	D₄	—	236	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	59	1	no	—
131	C₂×C₄	—	59	1	no	—
132	C₂×C₄	—	236	1	no	—
133	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
134	C₈	—	118	1	no	—
135	C₈	#11745685	1	1	yes	characteristic
136	C₈	D₄×D₅₉	1	1	yes	characteristic
137	C₂²	—	8	1	no	—
138	C₂²	—	8	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	—	472	1	no	—
141	C₂²	—	472	1	no	—
142	C₂²	—	472	1	no	—
143	C₂²	D₈×D₅₉	1	1	yes	characteristic
144	C₄	—	8	1	no	—
145	C₄	—	59	2	no	—
146	C₄	—	118	1	no	—
147	C₄	—	236	2	no	—
148	C₄	#14273700	1	1	yes	characteristic
149	C₄	#14273754	1	1	yes	characteristic
150	C₂	—	16	1	no	—
151	C₂	—	944	1	no	—
152	C₂	#16837596	1	1	yes	characteristic
153	C₂	SD₃₂×D₅₉	1	1	yes	characteristic
154	C₂	#16837836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273672

C₂

yes

maximal, characteristic, complemented

C₂×#14273671

C₂

yes

maximal, characteristic, complemented

#16837777

C₂

yes

maximal, characteristic

#16837496

C₂

yes

maximal, characteristic, complemented

Dic₅₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#16837202

C₂

yes

maximal, characteristic, complemented

C₅₉×#357

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×D₄₇₂

C₂²

yes

—

—

—

—

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273650

—

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×D₈×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂×C₂²⋊D₅₉

—

C₂×C₂²⋊D₅₉

—

C₂×D₂₃₆

—

C₂×Dic₁₁₈

—

D₈×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

#11745669

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

D₄

yes

complemented

D₄₇₂

—

D₄₇₂

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂²×D₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₂×Dic₅₉

D₈

yes

characteristic

D₂₃₆

—

D₂₃₆

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

SD₃₂

yes

complemented

#2207

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#459

yes

characteristic

C₂×C₅₉

C₂×SD₃₂

yes

characteristic

C₂×C₅₉

#509

yes

characteristic

D₅₉

—

C₂×SD₃₂

—

C₂×SD₃₂

—

#493

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#357

—

#357

D₅₉

yes

characteristic, complemented

C₅₉

#2207

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

118

—

C₂×D₈

C₂×D₅₉

yes

characteristic

#127

—

118

—

C₄×Q₈

—

118

—

SD₃₂

—

118

—

SD₃₂

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#107

C₂×D₅₉

yes

characteristic

#103

—

118

—

C₄×C₈

—

C₂×Q₈

—

118

—

100

C₂×D₄

—

101

C₂×D₄

—

118

—

102

C₂×D₄

—

236

—

103

C₂×D₄

—

236

—

104

Q₁₆

—

105

Q₁₆

—

118

—

106

D₈

—

107

D₈

—

108

D₈

—

118

—

109

D₈

—

118

—

110

D₈

—

236

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₅₉

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

236

—

115

C₄²

—

116

C₄²

—

236

—

117

C₁₆

—

118

C₁₆

—

118

—

119

C₂³

—

120

C₂³

—

236

—

121

Q₈

—

118

—

122

Q₈

—

236

—

123

D₄

—

124

D₄

—

125

D₄

—

472

—

126

D₄

—

472

—

127

D₄

—

236

—

128

D₄

—

236

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

236

—

133

C₂×C₄

D₄×D₅₉

yes

characteristic

134

C₈

—

118

—

135

C₈

#11745685

yes

characteristic

136

C₈

D₄×D₅₉

yes

characteristic

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

472

—

141

C₂²

—

472

—

142

C₂²

—

472

—

143

C₂²

D₈×D₅₉

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

118

—

147

C₄

—

236

—

148

C₄

#14273700

yes

characteristic

149

C₄

#14273754

yes

characteristic

150

C₂

—

151

C₂

—

944

—

152

C₂

#16837596

yes

characteristic

153

C₂

SD₃₂×D₅₉

yes

characteristic

154

C₂

#16837836

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#357	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#357	D₅₉
hypocenter	C₅₉	#2207
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#357

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. C₂×#14273672	C₂	semidirect
2. C₂×#14273671	C₂	semidirect
4. #16837496	C₂	semidirect
5. Dic₅₉⋊C₁₆	C₂	semidirect
6. #16837202	C₂	semidirect
7. C₅₉×#357	C₂	semidirect
36. Dic₂₃₆	D₄	semidirect
71. Dic₅₉	SD₃₂	semidirect
84. #357	D₅₉	semidirect
85. C₅₉	#2207	semidirect

H ≅ G/N

type

1. C₂×#14273672

C₂

semidirect

2. C₂×#14273671

C₂

semidirect

4. #16837496

C₂

semidirect

5. Dic₅₉⋊C₁₆

C₂

semidirect

6. #16837202

C₂

semidirect

7. C₅₉×#357

C₂

semidirect

36. Dic₂₃₆

D₄

semidirect

71. Dic₅₉

SD₃₂

semidirect

84. #357

D₅₉

semidirect

85. C₅₉

#2207

semidirect

Subgroups — #21426135

Conjugacy classes of subgroups

Complements