Group #21425814 — subgroups

Subgroups — #21425814

Order 7552.

220 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 69 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×Q₁₆×D₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273666	C₂	1	1	yes	maximal, characteristic
3	#16837439	C₂	1	1	yes	maximal, characteristic
4	#16837207	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837206	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837205	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₉	—	2	1	no	—
9	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
10	C₂×#11745672	—	2	1	no	—
11	Q₁₆×D₅₉	—	2	2	no	—
12	Q₁₆×D₅₉	C₄	1	4	yes	complemented
13	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
14	#14273685	—	2	1	no	—
15	#14273666	—	2	2	no	—
16	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
17	#14273661	C₂²	1	1	yes	characteristic
18	#14273658	—	2	1	no	—
19	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	#107×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×Q₈×C₅₉	—	2	1	no	—
23	Q₈×D₅₉	—	2	4	no	—
24	Q₈×D₅₉	—	4	2	no	—
25	C₂×C₄×D₅₉	—	4	1	no	—
26	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
27	C₂×Dic₁₁₈	—	2	1	no	—
28	C₂×Dic₁₁₈	—	4	1	no	—
29	Q₁₆×C₅₉	—	2	1	no	—
30	Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₅₉	—	2	1	no	—
33	#11745672	—	2	4	no	—
34	(C₂×D₅₉)⋊C₄	—	4	1	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	C₄×Dic₅₉	—	4	1	no	—
37	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
38	Dic₂₃₆	—	2	1	no	—
39	Dic₂₃₆	C₂×C₄	1	2	yes	—
40	C₈×D₅₉	C₂×C₄	1	2	yes	—
41	C₈×D₅₉	D₄	1	2	yes	—
42	C₁₆×C₅₉	—	2	1	no	—
43	C₅₉⋊C₁₆	—	2	1	no	—
44	C₂²×D₅₉	D₈	1	1	yes	characteristic
45	Q₈×C₅₉	—	2	2	no	—
46	Q₈×C₅₉	—	4	1	no	—
47	C₂×C₄×C₅₉	—	4	1	no	—
48	C₂×C₄×C₅₉	—	4	1	no	—
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₅₉	—	4	1	no	—
51	C₂×Dic₅₉	—	4	1	no	—
52	C₂×Dic₅₉	D₈	1	1	yes	characteristic
53	C₄×D₅₉	—	4	4	no	—
54	C₄×D₅₉	C₂²⋊C₄	1	2	yes	—
55	C₄×D₅₉	SD₁₆	1	2	yes	—
56	Dic₁₁₈	—	2	2	no	—
57	Dic₁₁₈	—	4	1	no	—
58	Dic₁₁₈	—	4	4	no	—
59	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
63	C₂×D₅₉	—	2	1	no	—
64	C₂×D₅₉	—	2	1	no	—
65	C₂×D₅₉	#102	1	2	yes	—
66	C₄×C₅₉	—	4	2	no	—
67	C₄×C₅₉	—	8	1	no	—
68	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
70	Dic₅₉	—	4	2	no	—
71	Dic₅₉	—	8	1	no	—
72	Dic₅₉	#102	1	2	yes	—
73	#2141	—	59	1	no	maximal, Sylow
74	C₂×C₅₉	C₂×#102	1	1	yes	characteristic
75	C₂×C₅₉	#510	1	1	yes	characteristic
76	C₂×C₅₉	#510	1	1	yes	characteristic
77	D₅₉	—	2	2	no	—
78	C₂²×Q₁₆	—	59	1	no	—
79	C₂×M₃₂	—	59	1	no	—
80	#427	—	59	1	no	—
81	#358	—	59	1	no	—
82	#358	—	59	1	no	—
83	#358	—	59	1	no	—
84	#358	D₅₉	1	1	yes	characteristic, complemented
85	C₅₉	#2141	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	118	1	no	—
87	C₂×Q₁₆	—	59	4	no	—
88	C₂×Q₁₆	—	59	1	no	—
89	C₂×Q₁₆	—	118	2	no	—
90	C₂×Q₁₆	—	118	1	no	—
91	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
92	C₂²×C₈	—	59	1	no	—
93	#117	—	118	1	no	—
94	M₃₂	—	118	2	no	—
95	C₂×C₁₆	—	59	1	no	—
96	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
97	#107	—	59	1	no	—
98	#107	C₂×D₅₉	1	1	yes	characteristic
99	#106	—	118	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	118	4	no	—
102	C₂×Q₈	—	118	1	no	—
103	C₂×Q₈	—	236	1	no	—
104	C₂×Q₈	—	236	2	no	—
105	C₂²×C₄	—	59	1	no	—
106	C₂²×C₄	—	236	1	no	—
107	Q₁₆	—	2	1	no	—
108	Q₁₆	—	59	2	no	—
109	Q₁₆	—	118	4	no	—
110	Q₁₆	—	118	1	no	—
111	Q₁₆	C₄×D₅₉	1	2	yes	—
112	C₂×C₈	—	59	1	no	—
113	C₂×C₈	—	59	2	no	—
114	C₂×C₈	—	59	2	no	—
115	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	118	1	no	—
118	C₂²⋊C₄	—	236	1	no	—
119	C₄²	—	236	1	no	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	118	1	no	—
122	C₂³	—	59	1	no	—
123	Q₈	—	2	2	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	118	2	no	—
126	Q₈	—	236	4	no	—
127	Q₈	—	236	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	59	1	no	—
131	C₂×C₄	—	59	2	no	—
132	C₂×C₄	—	59	2	no	—
133	C₂×C₄	—	236	4	no	—
134	C₂×C₄	—	236	1	no	—
135	C₂×C₄	—	236	1	no	—
136	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
137	C₈	—	59	2	no	—
138	C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
139	C₈	D₄×D₅₉	1	1	yes	characteristic
140	C₂²	—	59	2	no	—
141	C₂²	—	118	1	no	—
142	C₂²	—	118	1	no	—
143	C₂²	D₈×D₅₉	1	1	yes	characteristic
144	C₄	—	4	2	no	—
145	C₄	—	8	1	no	—
146	C₄	—	59	2	no	—
147	C₄	—	472	1	no	—
148	C₄	—	236	2	no	—
149	C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
150	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
151	C₂	—	118	2	no	—
152	C₂	#102×D₅₉	1	1	yes	characteristic
153	C₂	#16837841	1	1	yes	characteristic
154	C₂	#16837844	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×D₅₉

C₂

yes

maximal, characteristic

C₂×#14273666

C₂

yes

maximal, characteristic

#16837439

C₂

yes

maximal, characteristic

#16837207

C₂

yes

maximal, characteristic, complemented

#16837206

C₂

yes

maximal, characteristic, complemented

#16837205

C₂

yes

maximal, characteristic, complemented

C₅₉×#358

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₉

—

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×#11745672

—

Q₁₆×D₅₉

—

Q₁₆×D₅₉

C₄

yes

complemented

C₂×C₈×D₅₉

C₂²

yes

characteristic

#14273685

—

#14273666

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273658

—

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

Q₈×D₅₉

—

Q₈×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

#11745672

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

C₂×C₄

yes

—

C₈×D₅₉

C₂×C₄

yes

—

C₈×D₅₉

D₄

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂²×D₅₉

D₈

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

C₂²⋊C₄

yes

—

C₄×D₅₉

SD₁₆

yes

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

#102

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

#102

yes

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

C₂²×Q₁₆

—

C₂×M₃₂

—

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#2141

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

118

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

#117

—

118

—

M₃₂

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#107

—

#107

C₂×D₅₉

yes

characteristic

#106

—

118

—

100

C₂×Q₈

—

101

C₂×Q₈

—

118

—

102

C₂×Q₈

—

118

—

103

C₂×Q₈

—

236

—

104

C₂×Q₈

—

236

—

105

C₂²×C₄

—

106

C₂²×C₄

—

236

—

107

Q₁₆

—

108

Q₁₆

—

109

Q₁₆

—

118

—

110

Q₁₆

—

118

—

111

Q₁₆

C₄×D₅₉

yes

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂²×D₅₉

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

—

118

C₂²⋊C₄

—

236

—

119

C₄²

—

236

—

120

C₁₆

—

121

C₁₆

—

118

—

122

C₂³

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

118

—

126

Q₈

—

236

—

127

Q₈

—

236

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

236

—

134

C₂×C₄

—

236

—

135

C₂×C₄

—

236

—

136

C₂×C₄

D₄×D₅₉

yes

characteristic

137

C₈

—

138

C₈

C₂×C₄×D₅₉

yes

characteristic

139

C₈

D₄×D₅₉

yes

characteristic

140

C₂²

—

141

C₂²

—

118

—

142

C₂²

—

118

—

143

C₂²

D₈×D₅₉

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

472

—

148

C₄

—

236

—

149

C₄

C₂²⋊C₄×D₅₉

yes

characteristic

150

C₄

SD₁₆×D₅₉

yes

characteristic

151

C₂

—

118

—

152

C₂

#102×D₅₉

yes

characteristic

153

C₂

#16837841

yes

characteristic

154

C₂

#16837844

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#358	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#358	D₅₉
hypocenter	C₅₉	#2141
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#358

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
4. #16837207	C₂	semidirect
5. #16837206	C₂	semidirect
6. #16837205	C₂	semidirect
7. C₅₉×#358	C₂	semidirect
12. Q₁₆×D₅₉	C₄	semidirect
84. #358	D₅₉	semidirect
85. C₅₉	#2141	semidirect

H ≅ G/N

type

4. #16837207

C₂

semidirect

5. #16837206

C₂

semidirect

6. #16837205

C₂

semidirect

7. C₅₉×#358

C₂

semidirect

12. Q₁₆×D₅₉

C₄

semidirect

84. #358

D₅₉

semidirect

85. C₅₉

#2141

semidirect

Subgroups — #21425814

Conjugacy classes of subgroups

Complements