Group #21426148 — subgroups

Subgroups — #21426148

Order 7552.

170 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 51 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×#14273670	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14273668	C₂	1	1	yes	maximal, characteristic
3	C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837206	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837203	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#495	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
9	C₄×D₂₃₆	—	2	1	no	—
10	#14273670	—	2	2	no	—
11	#14273668	—	2	2	no	—
12	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
14	#14273661	C₂²	1	1	yes	characteristic
15	#14273649	—	2	1	no	—
16	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₅₉	—	2	1	no	—
18	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
19	#124×C₅₉	—	2	1	no	—
20	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₉	—	2	1	no	—
22	C₂×D₄×C₅₉	—	2	1	no	—
23	C₂×D₂₃₆	—	2	1	no	—
24	C₂×C₄×D₅₉	—	4	1	no	—
25	Q₁₆×C₅₉	—	2	1	no	—
26	Q₁₆×C₅₉	—	2	1	no	—
27	SD₁₆×C₅₉	—	4	2	no	—
28	D₈×C₅₉	—	2	1	no	—
29	D₈×C₅₉	—	2	1	no	—
30	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₂²⋊C₄×C₅₉	—	4	1	no	—
33	C₂²⋊C₄×C₅₉	—	4	1	no	—
34	C₄²×C₅₉	D₄	1	1	yes	characteristic
35	(C₂×D₅₉)⋊C₄	—	4	1	no	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	D₄₇₂	—	2	1	no	—
38	D₄₇₂	D₄	1	2	yes	—
39	C₅₉⋊C₁₆	—	2	1	no	—
40	C₅₉⋊C₁₆	—	2	1	no	—
41	C₂³×C₅₉	—	4	1	no	—
42	C₂²×D₅₉	—	4	1	no	—
43	Q₈×C₅₉	—	4	1	no	—
44	Q₈×C₅₉	—	4	1	no	—
45	D₄×C₅₉	—	4	1	no	—
46	D₄×C₅₉	—	4	1	no	—
47	C₂×C₄×C₅₉	—	4	1	no	—
48	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
49	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₅₉	—	4	1	no	—
52	D₂₃₆	—	2	2	no	—
53	D₂₃₆	—	4	1	no	—
54	C₄×D₅₉	—	8	2	no	—
55	C₈×C₅₉	—	2	1	no	—
56	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₈×C₅₉	#41	1	1	yes	characteristic
58	C₂²×C₅₉	—	8	1	no	—
59	C₂²×C₅₉	—	8	1	no	—
60	C₂²×C₅₉	—	8	1	no	—
61	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
62	C₂×D₅₉	—	4	2	no	—
63	C₂×D₅₉	—	8	1	no	—
64	C₂×D₅₉	—	8	1	no	—
65	C₄×C₅₉	—	2	1	no	—
66	C₄×C₅₉	—	2	1	no	—
67	C₄×C₅₉	—	8	1	no	—
68	C₄×C₅₉	#121	1	1	yes	characteristic
69	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
70	Dic₅₉	—	8	1	no	—
71	#2210	—	59	1	no	maximal, Sylow
72	C₂×C₅₉	—	16	1	no	—
73	C₂×C₅₉	#459	1	1	yes	characteristic
74	C₂×C₅₉	#508	1	1	yes	characteristic
75	C₂×C₅₉	#509	1	1	yes	characteristic
76	D₅₉	—	8	2	no	—
77	C₂×SD₃₂	—	59	1	no	—
78	C₂×D₁₆	—	59	1	no	—
79	#495	D₅₉	1	1	yes	characteristic, complemented
80	C₄×D₈	—	59	1	no	—
81	C₄⋊C₁₆	—	59	1	no	—
82	#358	—	59	1	no	—
83	#357	—	59	1	no	—
84	C₅₉	#2210	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
86	C₂×SD₁₆	—	2	1	no	—
87	C₂×D₈	—	59	1	no	—
88	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
89	#124	—	2	1	no	—
90	C₄×D₄	—	118	1	no	—
91	SD₃₂	—	118	2	no	—
92	D₁₆	—	118	2	no	—
93	C₂×C₁₆	—	59	1	no	—
94	C₂×C₁₆	—	59	1	no	—
95	#107	—	59	1	no	—
96	#102	—	118	1	no	—
97	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
98	C₂×Q₈	—	2	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	118	1	no	—
101	C₂²×C₄	—	236	1	no	—
102	Q₁₆	—	2	1	no	—
103	Q₁₆	—	2	1	no	—
104	SD₁₆	—	4	2	no	—
105	D₈	—	2	1	no	—
106	D₈	—	2	1	no	—
107	D₈	—	59	2	no	—
108	D₈	—	118	1	no	—
109	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
110	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
111	C₄⋊C₄	—	118	1	no	—
112	C₂²⋊C₄	—	4	1	no	—
113	C₂²⋊C₄	—	4	1	no	—
114	C₂²⋊C₄	—	236	1	no	—
115	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
116	C₁₆	—	118	1	no	—
117	C₁₆	—	118	1	no	—
118	C₂³	—	4	1	no	—
119	C₂³	—	236	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	4	1	no	—
122	D₄	—	4	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	118	2	no	—
125	D₄	—	236	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	472	2	no	—
128	C₂×C₄	—	236	1	no	—
129	C₂×C₄	#11745669	1	1	yes	characteristic
130	C₂×C₄	#11745669	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	D₄×D₅₉	1	1	yes	characteristic
134	C₈	#11745687	1	1	yes	characteristic
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	8	1	no	—
138	C₂²	—	472	1	no	—
139	C₂²	—	472	1	no	—
140	C₂²	—	236	2	no	—
141	C₂²	C₂×#11745669	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	8	1	no	—
145	C₄	—	472	1	no	—
146	C₄	#14273698	1	1	yes	characteristic
147	C₄	#14273753	1	1	yes	characteristic
148	C₂	—	16	1	no	—
149	C₂	—	472	2	no	—
150	C₂	#16837594	1	1	yes	characteristic
151	C₂	#16837832	1	1	yes	characteristic
152	C₂	#16837837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273670

C₂

yes

maximal, characteristic, complemented

C₂×#14273668

C₂

yes

maximal, characteristic

C₄×D₄₇₂

C₂

yes

maximal, characteristic, complemented

C₄⋊(C₅₉⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

#16837206

C₂

yes

maximal, characteristic, complemented

#16837203

C₂

yes

maximal, characteristic, complemented

C₅₉×#495

C₂

yes

maximal, characteristic, complemented

C₂×D₄₇₂

C₂²

yes

characteristic

C₄×D₂₃₆

—

#14273670

—

#14273668

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273649

—

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

C₂×SD₁₆×C₅₉

—

C₂×D₈×C₅₉

C₂²

yes

characteristic

#124×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

SD₁₆×C₅₉

—

D₈×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

D₄

yes

characteristic

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

D₄₇₂

—

D₄₇₂

D₄

yes

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#2210

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

C₂×SD₃₂

—

C₂×D₁₆

—

#495

D₅₉

yes

characteristic, complemented

C₄×D₈

—

C₄⋊C₁₆

—

#358

—

#357

—

C₅₉

#2210

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

C₂×D₅₉

yes

characteristic

#124

—

C₄×D₄

—

118

—

SD₃₂

—

118

—

D₁₆

—

118

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#102

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

118

—

101

C₂²×C₄

—

236

—

102

Q₁₆

—

103

Q₁₆

—

104

SD₁₆

—

105

D₈

—

106

D₈

—

107

D₈

—

108

D₈

—

118

—

109

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

110

C₂×C₈

C₂²×D₅₉

yes

characteristic

111

C₄⋊C₄

—

118

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

236

—

115

C₄²

C₂²⋊D₅₉

yes

characteristic

116

C₁₆

—

118

—

117

C₁₆

—

118

—

118

C₂³

—

119

C₂³

—

236

—

120

Q₈

—

121

Q₈

—

122

D₄

—

123

D₄

—

124

D₄

—

118

—

125

D₄

—

236

—

126

C₂×C₄

—

127

C₂×C₄

—

472

—

128

C₂×C₄

—

236

—

129

C₂×C₄

#11745669

yes

characteristic

130

C₂×C₄

#11745669

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

132

C₈

—

133

C₈

D₄×D₅₉

yes

characteristic

134

C₈

#11745687

yes

characteristic

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

472

—

139

C₂²

—

472

—

140

C₂²

—

236

—

141

C₂²

C₂×#11745669

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

472

—

146

C₄

#14273698

yes

characteristic

147

C₄

#14273753

yes

characteristic

148

C₂

—

149

C₂

—

472

—

150

C₂

#16837594

yes

characteristic

151

C₂

#16837832

yes

characteristic

152

C₂

#16837837

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#495	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#495	D₅₉
hypocenter	C₅₉	#2210
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

subgroup

G/H

center

C₂²

C₂×#11745669

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#495

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

N	H ≅ G/N	type
1. C₂×#14273670	C₂	semidirect
3. C₄×D₄₇₂	C₂	semidirect
4. C₄⋊(C₅₉⋊C₁₆)	C₂	semidirect
5. #16837206	C₂	semidirect
6. #16837203	C₂	semidirect
7. C₅₉×#495	C₂	semidirect
79. #495	D₅₉	semidirect
84. C₅₉	#2210	semidirect

H ≅ G/N

type

1. C₂×#14273670

C₂

semidirect

3. C₄×D₄₇₂

C₂

semidirect

4. C₄⋊(C₅₉⋊C₁₆)

C₂

semidirect

5. #16837206

C₂

semidirect

6. #16837203

C₂

semidirect

7. C₅₉×#495

C₂

semidirect

79. #495

D₅₉

semidirect

84. C₅₉

#2210

semidirect

Subgroups — #21426148

Conjugacy classes of subgroups

Complements