Group #21423538 — subgroups

Subgroups — #21423538

Order 7552.

166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16837748	C₂	1	1	yes	maximal, characteristic
2	#16837612	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837154	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837149	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837146	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#460	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745672	—	2	1	no	—
9	#14273727	C₂²	1	1	yes	characteristic
10	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
11	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	#14273650	—	2	1	no	—
14	#14273648	—	2	1	no	—
15	#14273648	—	2	1	no	—
16	C₄×C₅₉⋊C₈	—	2	1	no	—
17	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₅₉	—	2	1	no	—
19	#127×C₅₉	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	#102×C₅₉	—	2	1	no	—
23	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂×D₄×C₅₉	—	2	1	no	—
25	C₂×D₄×C₅₉	—	2	1	no	—
26	C₂×D₄×C₅₉	—	2	1	no	—
27	C₂×Dic₁₁₈	—	2	1	no	—
28	SD₁₆×C₅₉	—	4	2	no	—
29	C₂×C₈×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
32	C₄²×C₅₉	—	2	1	no	—
33	C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	#11745672	—	4	2	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	C₂×C₅₉⋊C₈	—	2	1	no	—
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
41	C₂³×C₅₉	—	4	1	no	—
42	Q₈×C₅₉	—	2	1	no	—
43	Q₈×C₅₉	D₈	1	2	yes	—
44	D₄×C₅₉	—	4	1	no	—
45	D₄×C₅₉	—	4	1	no	—
46	D₄×C₅₉	—	4	1	no	—
47	D₄×C₅₉	—	4	1	no	—
48	D₄×C₅₉	—	8	1	no	—
49	C₂×C₄×C₅₉	—	2	1	no	—
50	C₂×C₄×C₅₉	—	2	1	no	—
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
54	C₂×Dic₅₉	—	4	1	no	—
55	Dic₁₁₈	—	4	1	no	—
56	Dic₁₁₈	—	4	1	no	—
57	C₈×C₅₉	—	4	1	no	—
58	C₅₉⋊C₈	—	2	1	no	—
59	C₅₉⋊C₈	—	2	1	no	—
60	C₅₉⋊C₈	—	4	1	no	—
61	C₅₉⋊C₈	—	4	1	no	—
62	C₂²×C₅₉	—	8	1	no	—
63	C₂²×C₅₉	—	8	1	no	—
64	C₂²×C₅₉	—	8	1	no	—
65	C₂²×C₅₉	#121	1	1	yes	characteristic
66	C₄×C₅₉	—	2	2	no	—
67	C₄×C₅₉	—	2	1	no	—
68	C₄×C₅₉	—	4	1	no	—
69	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
70	C₄×C₅₉	#135	1	1	yes	characteristic
71	C₄×C₅₉	#135	1	1	yes	characteristic
72	C₄×C₅₉	#137	1	1	yes	characteristic
73	Dic₅₉	—	8	1	no	—
74	#1682	—	59	1	no	maximal, Sylow
75	C₂×C₅₉	—	16	1	no	—
76	C₂×C₅₉	#464	1	1	yes	characteristic
77	C₂×C₅₉	#466	1	1	yes	characteristic
78	C₂×C₅₉	#471	1	1	yes	characteristic
79	#486	—	59	1	no	—
80	#462	—	59	1	no	—
81	#460	D₅₉	1	1	yes	characteristic
82	C₈×Q₈	—	59	1	no	—
83	#335	—	59	1	no	—
84	#332	—	59	1	no	—
85	#331	—	59	1	no	—
86	C₅₉	#1682	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	—	118	1	no	—
88	C₂×SD₁₆	—	2	1	no	—
89	C₄⋊Q₈	—	59	1	no	—
90	#127	C₂×D₅₉	1	1	yes	characteristic
91	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
92	C₄⋊C₈	—	59	1	no	—
93	C₄⋊C₈	—	59	1	no	—
94	C₄⋊C₈	—	118	1	no	—
95	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
96	#103	—	118	1	no	—
97	#102	—	2	1	no	—
98	#102	—	118	1	no	—
99	#102	—	118	1	no	—
100	C₄×C₈	—	59	1	no	—
101	C₄×C₈	—	118	1	no	—
102	C₂×Q₈	—	118	1	no	—
103	C₂×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	Q₁₆	—	236	2	no	—
108	SD₁₆	—	4	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	59	1	no	—
111	C₂×C₈	—	59	1	no	—
112	C₂×C₈	—	118	1	no	—
113	C₂×C₈	—	118	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	118	1	no	—
116	C₄⋊C₄	—	118	1	no	—
117	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
118	C₄²	—	2	1	no	—
119	C₄²	C₂²×D₅₉	1	1	yes	characteristic
120	C₂³	—	4	1	no	—
121	Q₈	—	2	1	no	—
122	Q₈	—	236	1	no	—
123	Q₈	—	236	1	no	—
124	Q₈	#11745669	1	2	yes	—
125	D₄	—	4	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	8	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	236	1	no	—
133	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
134	C₂×C₄	#11745687	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
136	C₈	—	4	1	no	—
137	C₈	—	118	1	no	—
138	C₈	—	118	1	no	—
139	C₈	—	236	1	no	—
140	C₈	—	236	1	no	—
141	C₂²	—	8	1	no	—
142	C₂²	—	8	1	no	—
143	C₂²	—	8	1	no	—
144	C₂²	#14273699	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	1	no	—
147	C₄	—	4	1	no	—
148	C₄	—	472	1	no	—
149	C₄	C₂×#11745669	1	1	yes	characteristic
150	C₄	#14273747	1	1	yes	characteristic
151	C₄	#14273748	1	1	yes	characteristic
152	C₄	#14273756	1	1	yes	characteristic
153	C₂	—	16	1	no	—
154	C₂	#16837622	1	1	yes	characteristic
155	C₂	#16837636	1	1	yes	characteristic
156	C₂	#16837663	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837748

C₂

yes

maximal, characteristic

#16837612

C₂

yes

maximal, characteristic, complemented

Q₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#16837154

C₂

yes

maximal, characteristic, complemented

#16837149

C₂

yes

maximal, characteristic, complemented

#16837146

C₂

yes

maximal, characteristic

C₅₉×#460

C₂

yes

maximal, characteristic

C₂×#11745672

—

#14273727

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

—

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₅₉

—

#127×C₅₉

C₂²

yes

characteristic

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

#102×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×Dic₁₁₈

—

SD₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745672

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂³×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

D₈

yes

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1682

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

D₅₉

yes

characteristic

C₈×Q₈

—

#335

—

#332

—

#331

—

C₅₉

#1682

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₂×SD₁₆

—

C₄⋊Q₈

—

#127

C₂×D₅₉

yes

characteristic

C₄×Q₈

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

118

—

#102

—

#102

—

118

—

#102

—

118

—

100

C₄×C₈

—

101

C₄×C₈

—

118

—

102

C₂×Q₈

—

118

—

103

C₂×Q₈

C₂²⋊D₅₉

yes

characteristic

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

Q₁₆

—

236

—

108

SD₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

118

—

113

C₂×C₈

—

118

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

118

—

116

C₄⋊C₄

—

118

—

117

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

118

C₄²

—

119

C₄²

C₂²×D₅₉

yes

characteristic

120

C₂³

—

121

Q₈

—

122

Q₈

—

236

—

123

Q₈

—

236

—

124

Q₈

#11745669

yes

—

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

236

—

133

C₂×C₄

D₄×D₅₉

yes

characteristic

134

C₂×C₄

#11745687

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

136

C₈

—

137

C₈

—

118

—

138

C₈

—

118

—

139

C₈

—

236

—

140

C₈

—

236

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#14273699

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

472

—

149

C₄

C₂×#11745669

yes

characteristic

150

C₄

#14273747

yes

characteristic

151

C₄

#14273748

yes

characteristic

152

C₄

#14273756

yes

characteristic

153

C₂

—

154

C₂

#16837622

yes

characteristic

155

C₂

#16837636

yes

characteristic

156

C₂

#16837663

yes

characteristic

	subgroup	G/H
center	C₂²	#14273699
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#460	C₂
socle	C₂²×C₅₉	#121
hypercenter	#460	D₅₉
hypocenter	C₅₉	#1682
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #460

subgroup

G/H

center

C₂²

#14273699

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#460

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #460

N	H ≅ G/N	type
2. #16837612	C₂	semidirect
3. Q₈×C₅₉⋊C₈	C₂	semidirect
4. #16837154	C₂	semidirect
5. #16837149	C₂	semidirect
86. C₅₉	#1682	semidirect

H ≅ G/N

type

2. #16837612

C₂

semidirect

3. Q₈×C₅₉⋊C₈

C₂

semidirect

4. #16837154

C₂

semidirect

5. #16837149

C₂

semidirect

86. C₅₉

#1682

semidirect

Subgroups — #21423538

Conjugacy classes of subgroups

Complements