Group #21434418 — subgroups

Subgroups — #21434418

Order 7552.

400 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 141 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	#16837959	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837756	C₂	1	4	yes	maximal, complemented
3	#16837752	C₂	1	2	yes	maximal, complemented
4	#16837748	C₂	1	2	yes	maximal, complemented
5	#16837398	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#532	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
9	#14273727	C₂²	1	2	yes	—
10	#14273719	C₂²	1	2	yes	—
11	#14273706	—	2	2	no	—
12	C₄×Dic₁₁₈	—	2	2	no	—
13	#14273686	C₂²	1	2	yes	—
14	#14273651	C₂²	1	8	yes	—
15	#14273648	C₂²	1	8	yes	—
16	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
17	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
18	C₂×#41×C₅₉	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
20	#127×C₅₉	C₂²	1	1	yes	characteristic
21	#124×C₅₉	C₂²	1	2	yes	—
22	#121×C₅₉	—	2	2	no	—
23	C₄×D₄×C₅₉	—	2	2	no	—
24	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
25	#41×C₅₉	—	2	8	no	—
26	C₂×Q₈×C₅₉	C₂³	1	2	yes	—
27	C₂×D₄×C₅₉	—	2	2	no	—
28	C₂×D₄×C₅₉	—	2	2	no	—
29	C₂×D₄×C₅₉	C₂³	1	2	yes	—
30	C₂²×C₄×C₅₉	—	2	2	no	—
31	C₂²×C₄×C₅₉	D₄	1	2	yes	—
32	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₁₈	—	2	2	no	—
34	C₄⋊C₄×C₅₉	—	2	2	no	—
35	C₂²⋊C₄×C₅₉	—	2	4	no	—
36	C₄²×C₅₉	D₄	1	2	yes	—
37	C₄²×C₅₉	C₂³	1	1	yes	characteristic
38	C₄²×C₅₉	C₂³	1	1	yes	characteristic
39	C₂²⋊Dic₅₉	—	2	2	no	—
40	C₄⋊Dic₅₉	—	2	2	no	—
41	C₄⋊Dic₅₉	C₂³	1	4	yes	—
42	Dic₅₉⋊C₄	—	2	4	no	—
43	C₄×Dic₅₉	—	2	2	no	—
44	C₂×C₅₉⋊C₈	—	2	4	no	—
45	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
46	C₂³×C₅₉	—	2	2	no	—
47	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₅₉	—	2	4	no	—
49	D₄×C₅₉	—	2	4	no	—
50	D₄×C₅₉	—	4	4	no	—
51	D₄×C₅₉	—	4	4	no	—
52	C₂×C₄×C₅₉	—	2	4	no	—
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	—	4	4	no	—
55	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#41	1	4	yes	—
59	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₅₉	—	2	4	no	—
61	Dic₁₁₈	—	4	4	no	—
62	C₅₉⋊C₈	—	2	4	no	—
63	C₂²×C₅₉	—	2	2	no	—
64	C₂²×C₅₉	—	4	2	no	—
65	C₂²×C₅₉	—	4	4	no	—
66	C₂²×C₅₉	#124	1	2	yes	—
67	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₅₉	—	2	2	no	—
69	C₄×C₅₉	—	4	2	no	—
70	C₄×C₅₉	#124	1	2	yes	—
71	C₄×C₅₉	#135	1	4	yes	—
72	C₄×C₅₉	C₂×#41	1	2	yes	—
73	Dic₅₉	—	4	4	no	—
74	#3131	—	59	1	no	maximal, Sylow
75	C₂×C₅₉	—	2	2	no	—
76	C₂×C₅₉	—	8	2	no	—
77	C₂×C₅₉	C₂×#124	1	1	yes	characteristic
78	C₂×C₅₉	C₂×#135	1	2	yes	—
79	#533	—	59	1	no	—
80	#532	D₅₉	1	1	yes	characteristic
81	#488	—	59	4	no	—
82	#487	—	59	2	no	—
83	#486	—	59	2	no	—
84	#416	—	59	4	no	—
85	C₂×C₄×C₈	—	59	1	no	—
86	C₅₉	#3131	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	C₂×D₅₉	1	2	yes	—
88	C₂²×C₈	—	59	2	no	—
89	C₄⋊Q₈	—	59	2	no	—
90	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
91	#127	C₂×D₅₉	1	1	yes	characteristic
92	#125	—	59	2	no	—
93	#124	C₂×D₅₉	1	2	yes	—
94	C₂²⋊Q₈	—	118	2	no	—
95	#121	—	2	2	no	—
96	C₄×Q₈	—	118	2	no	—
97	C₄×D₄	—	2	2	no	—
98	#117	—	59	2	no	—
99	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
100	#103	—	59	8	no	—
101	#102	—	59	8	no	—
102	C₄×C₈	—	59	2	no	—
103	C₄×C₈	—	59	2	no	—
104	#41	—	2	8	no	—
105	C₂×Q₈	—	118	2	no	—
106	C₂×Q₈	C₂²×D₅₉	1	2	yes	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	C₂²×D₅₉	1	2	yes	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
113	C₂×C₈	—	59	4	no	—
114	C₂×C₈	—	118	4	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	59	4	no	—
117	C₄⋊C₄	—	118	4	no	—
118	C₄⋊C₄	—	118	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	118	2	no	—
121	C₄²	—	118	2	no	—
122	C₄²	C₂²⋊D₅₉	1	2	yes	—
123	C₄²	C₂²×D₅₉	1	1	yes	characteristic
124	C₄²	C₂²×D₅₉	1	1	yes	characteristic
125	C₂³	—	2	2	no	—
126	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	Q₈	—	236	4	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	D₄	—	4	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	118	4	no	—
136	C₂×C₄	#11745687	1	4	yes	—
137	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
139	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
140	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
141	C₈	—	118	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	4	no	—
145	C₂²	#14273714	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	2	no	—
149	C₄	—	236	4	no	—
150	C₄	#14273714	1	2	yes	—
151	C₄	#14273747	1	4	yes	—
152	C₄	C₂×#11745687	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	8	2	no	—
155	C₂	C₂×#14273714	1	1	yes	characteristic
156	C₂	C₂×#14273747	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal, characteristic, complemented

#16837756

C₂

yes

maximal, complemented

#16837752

C₂

yes

maximal, complemented

#16837748

C₂

yes

maximal, complemented

#16837398

C₂

yes

maximal, complemented

C₂×C₄×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₅₉×#532

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

#14273727

C₂²

yes

—

#14273719

C₂²

yes

—

#14273706

—

C₄×Dic₁₁₈

—

#14273686

C₂²

yes

—

#14273651

C₂²

yes

—

#14273648

C₂²

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

—

C₂×#41×C₅₉

C₂²

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#127×C₅₉

C₂²

yes

characteristic

#124×C₅₉

C₂²

yes

—

#121×C₅₉

—

C₄×D₄×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

C₂×Q₈×C₅₉

C₂³

yes

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#124

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#124

yes

—

C₄×C₅₉

#135

yes

—

C₄×C₅₉

C₂×#41

yes

—

Dic₅₉

—

#3131

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₂×#124

yes

characteristic

C₂×C₅₉

C₂×#135

yes

—

#533

—

#532

D₅₉

yes

—

—

—

—

C₂×C₄×C₈

—

C₅₉

#3131

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₉

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

#127

C₂×D₅₉

yes

characteristic

#125

—

#124

C₂×D₅₉

yes

—

C₂²⋊Q₈

—

118

—

#121

—

C₄×Q₈

—

118

—

C₄×D₄

—

#117

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

100

#103

—

101

#102

—

102

C₄×C₈

—

103

C₄×C₈

—

104

#41

—

105

C₂×Q₈

—

118

—

106

C₂×Q₈

C₂²×D₅₉

yes

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂²×D₅₉

yes

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₅₉

yes

—

112

C₂²×C₄

C₂²×D₅₉

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

118

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

—

118

C₄⋊C₄

—

118

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

118

—

121

C₄²

—

118

—

122

C₄²

C₂²⋊D₅₉

yes

—

123

C₄²

C₂²×D₅₉

yes

characteristic

124

C₄²

C₂²×D₅₉

yes

characteristic

125

C₂³

—

126

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

127

Q₈

—

128

Q₈

—

236

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

118

—

136

C₂×C₄

#11745687

yes

—

137

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

138

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

139

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

140

C₂×C₄

C₂³×D₅₉

yes

characteristic

141

C₈

—

118

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#14273714

yes

—

146

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

236

—

150

C₄

#14273714

yes

—

151

C₄

#14273747

yes

—

152

C₄

C₂×#11745687

yes

—

153

C₂

—

154

C₂

—

155

C₂

C₂×#14273714

yes

characteristic

156

C₂

C₂×#14273747

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#532	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#532	D₅₉
hypocenter	C₅₉	#3131
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#532

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16837959	C₂	semidirect
2. #16837756	C₂	semidirect
3. #16837752	C₂	semidirect
4. #16837748	C₂	semidirect
5. #16837398	C₂	semidirect
6. C₂×C₄×C₅₉⋊C₈	C₂	semidirect
86. C₅₉	#3131	semidirect

H ≅ G/N

type

1. #16837959

C₂

semidirect

2. #16837756

C₂

semidirect

3. #16837752

C₂

semidirect

4. #16837748

C₂

semidirect

5. #16837398

C₂

semidirect

6. C₂×C₄×C₅₉⋊C₈

C₂

semidirect

86. C₅₉

#3131

semidirect

Subgroups — #21434418

Conjugacy classes of subgroups

Complements