Group #21422786 — subgroups

Subgroups — #21422786

Order 7552.

216 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16837426	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#16837151	C₂	1	2	yes	maximal, complemented
4	#16837148	C₂	1	2	yes	maximal, complemented
5	C₅₉×#533	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	—	2	1	no	—
7	C₂²×C₅₉⋊C₈	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
9	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
10	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
11	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	#125×C₅₉	—	2	1	no	—
15	C₂²⋊Q₈×C₅₉	—	2	1	no	—
16	C₂²⋊Q₈×C₅₉	C₄	1	2	yes	—
17	C₄×Q₈×C₅₉	—	2	1	no	—
18	C₄×Q₈×C₅₉	C₄	1	2	yes	—
19	#117×C₅₉	—	2	1	no	—
20	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₉	—	2	1	no	—
22	C₂×Q₈×C₅₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₅₉	—	2	2	no	—
27	C₄⋊C₄×C₅₉	—	2	2	no	—
28	C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
32	C₂²⋊C₄×C₅₉	—	2	2	no	—
33	C₄²×C₅₉	—	2	2	no	—
34	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₅₉	D₄	1	1	yes	characteristic
36	C₄²×C₅₉	D₄	1	1	yes	characteristic
37	C₄²×C₅₉	C₂³	1	1	yes	characteristic
38	#11745664	—	4	2	no	—
39	C₂×C₅₉⋊C₈	—	2	2	no	—
40	C₂×C₅₉⋊C₈	—	2	2	no	—
41	C₂×C₅₉⋊C₈	—	4	2	no	—
42	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
43	Q₈×C₅₉	—	4	2	no	—
44	Q₈×C₅₉	—	4	1	no	—
45	Q₈×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	—	2	1	no	—
47	C₂×C₄×C₅₉	—	2	1	no	—
48	C₂×C₄×C₅₉	—	2	2	no	—
49	C₂×C₄×C₅₉	—	2	2	no	—
50	C₂×C₄×C₅₉	—	4	1	no	—
51	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
55	C₂×C₄×C₅₉	Q₁₆	1	2	yes	—
56	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₅₉⋊C₈	—	4	2	no	—
60	C₅₉⋊C₈	—	4	2	no	—
61	C₂²×C₅₉	—	2	1	no	—
62	C₂²×C₅₉	—	2	1	no	—
63	C₂²×C₅₉	#103	1	2	yes	—
64	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₅₉	—	2	1	no	—
66	C₄×C₅₉	—	2	1	no	—
67	C₄×C₅₉	—	2	1	no	—
68	C₄×C₅₉	—	4	2	no	—
69	C₄×C₅₉	—	4	2	no	—
70	C₄×C₅₉	#103	1	2	yes	—
71	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
72	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
73	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₄×C₅₉	#137	1	1	yes	characteristic
75	#1550	—	59	1	no	maximal, Sylow
76	C₂×C₅₉	—	2	2	no	—
77	C₂×C₅₉	#413	1	1	yes	characteristic
78	C₂×C₅₉	C₂×#103	1	1	yes	characteristic
79	C₂×C₅₉	#417	1	1	yes	characteristic
80	#533	D₅₉	1	1	yes	characteristic
81	#423	—	59	1	no	—
82	C₂×C₄⋊C₈	—	59	1	no	—
83	#333	—	59	2	no	—
84	#332	—	59	2	no	—
85	C₅₉	#1550	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₁₆	—	118	1	no	—
87	C₂²×C₈	—	118	1	no	—
88	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
89	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
90	#125	—	2	1	no	—
91	C₂²⋊Q₈	—	2	1	no	—
92	C₂²⋊Q₈	Dic₅₉	1	2	yes	—
93	C₄×Q₈	—	2	1	no	—
94	C₄×Q₈	Dic₅₉	1	2	yes	—
95	#117	—	2	1	no	—
96	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
97	C₄⋊C₈	—	59	2	no	—
98	C₄⋊C₈	—	59	2	no	—
99	C₄⋊C₈	—	118	1	no	—
100	C₄⋊C₈	—	118	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	C₂×Dic₅₉	1	2	yes	—
103	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
106	M₁₆	—	236	2	no	—
107	C₂×C₈	—	118	2	no	—
108	C₂×C₈	—	118	2	no	—
109	C₂×C₈	—	236	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₄²	—	2	2	no	—
118	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
119	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
120	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
121	C₄²	C₂²×D₅₉	1	1	yes	characteristic
122	C₂³	C₂²⋊Dic₅₉	1	1	yes	characteristic
123	Q₈	—	4	2	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	4	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	#11745671	1	2	yes	—
132	C₂×C₄	#11745672	1	2	yes	—
133	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
135	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
136	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
139	C₈	—	236	2	no	—
140	C₈	—	236	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#14273651	1	2	yes	—
144	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	2	no	—
149	C₄	—	4	2	no	—
150	C₄	#14273651	1	2	yes	—
151	C₄	C₂×#11745671	1	1	yes	characteristic
152	C₄	C₂×#11745672	1	1	yes	characteristic
153	C₄	#14273752	1	1	yes	characteristic
154	C₄	#14273757	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#16837386	1	1	yes	characteristic
157	C₂	C₂×#14273651	1	1	yes	characteristic
158	C₂	#16837403	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837426

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

#16837151

C₂

yes

maximal, complemented

#16837148

C₂

yes

maximal, complemented

C₅₉×#533

C₂

yes

maximal, characteristic

C₂×#11745664

—

C₂²×C₅₉⋊C₈

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#125×C₅₉

—

C₂²⋊Q₈×C₅₉

—

C₂²⋊Q₈×C₅₉

C₄

yes

—

C₄×Q₈×C₅₉

—

C₄×Q₈×C₅₉

C₄

yes

—

#117×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

Q₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#103

yes

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#103

yes

—

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

#333

—

#332

—

C₅₉

#1550

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

—

118

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Dic₅₉

yes

—

C₄×Q₈

—

C₄×Q₈

Dic₅₉

yes

—

#117

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

100

C₄⋊C₈

—

118

—

101

C₂×Q₈

—

102

C₂×Q₈

C₂×Dic₅₉

yes

—

103

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

104

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

105

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

106

M₁₆

—

236

—

107

C₂×C₈

—

118

—

108

C₂×C₈

—

118

—

109

C₂×C₈

—

236

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂×Dic₅₉

yes

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂×Dic₅₉

yes

characteristic

119

C₄²

C₂²⋊D₅₉

yes

characteristic

120

C₄²

C₂²⋊D₅₉

yes

characteristic

121

C₄²

C₂²×D₅₉

yes

characteristic

122

C₂³

C₂²⋊Dic₅₉

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#11745671

yes

—

132

C₂×C₄

#11745672

yes

—

133

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

134

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

135

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

136

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

139

C₈

—

236

—

140

C₈

—

236

—

141

C₂²

—

142

C₂²

—

143

C₂²

#14273651

yes

—

144

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#14273651

yes

—

151

C₄

C₂×#11745671

yes

characteristic

152

C₄

C₂×#11745672

yes

characteristic

153

C₄

#14273752

yes

characteristic

154

C₄

#14273757

yes

characteristic

155

C₂

—

156

C₂

#16837386

yes

characteristic

157

C₂

C₂×#14273651

yes

characteristic

158

C₂

#16837403

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#533	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#533	D₅₉
hypocenter	C₅₉	#1550
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#533

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #16837151	C₂	semidirect
4. #16837148	C₂	semidirect
85. C₅₉	#1550	semidirect

H ≅ G/N

type

3. #16837151

C₂

semidirect

4. #16837148

C₂

semidirect

85. C₅₉

#1550

semidirect

Subgroups — #21422786

Conjugacy classes of subgroups

Complements