Group #21434756 — subgroups

Subgroups — #21434756

Order 7552.

338 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16838015	C₂	1	2	yes	maximal
2	#16837782	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837772	C₂	1	2	yes	maximal, complemented
4	#16837661	C₂	1	2	yes	maximal
5	#16837644	C₂	1	2	yes	maximal
6	#16837508	C₂	1	2	yes	maximal, complemented
7	#16837496	C₂	1	2	yes	maximal, complemented
8	#16837356	C₂	1	1	yes	maximal, characteristic
9	C₅₉×#499	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₃₆	C₂²	1	2	yes	—
11	C₂×Dic₂₃₆	C₂²	1	2	yes	—
12	C₂×#11745659	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
14	#14273727	C₂²	1	2	yes	—
15	#14273722	—	2	4	no	—
16	#14273718	—	2	4	no	—
17	#14273707	C₂²	1	4	yes	—
18	#14273692	C₂²	1	4	yes	—
19	#14273684	C₂²	1	1	yes	characteristic
20	#14273650	C₂²	1	4	yes	—
21	#14273650	C₂²	1	4	yes	—
22	#14273638	C₂²	1	1	yes	characteristic
23	#14273637	C₂²	1	1	yes	characteristic
24	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
25	#125×C₅₉	C₂²	1	2	yes	—
26	#107×C₅₉	C₂²	1	2	yes	—
27	#106×C₅₉	C₂²	1	2	yes	—
28	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₁₈	C₂³	1	4	yes	—
31	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₅₉	—	2	4	no	—
34	C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
35	C₄²×C₅₉	C₂³	1	1	yes	characteristic
36	(C₂×D₅₉)⋊C₄	—	2	4	no	—
37	(C₂×D₅₉)⋊C₄	D₄	1	2	yes	—
38	C₄⋊Dic₅₉	—	2	4	no	—
39	C₄⋊Dic₅₉	—	2	4	no	—
40	Dic₅₉⋊C₄	—	2	4	no	—
41	Dic₅₉⋊C₄	D₄	1	2	yes	—
42	C₄×Dic₅₉	—	2	4	no	—
43	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
44	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
46	Dic₂₃₆	—	2	4	no	—
47	Dic₂₃₆	—	2	4	no	—
48	#11745659	—	2	2	no	—
49	C₈×D₅₉	—	2	2	no	—
50	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	—	2	4	no	—
52	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₅₉	—	2	4	no	—
55	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
56	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
57	C₄×D₅₉	—	2	1	no	—
58	C₄×D₅₉	—	2	1	no	—
59	Dic₁₁₈	—	2	4	no	—
60	Dic₁₁₈	—	2	4	no	—
61	C₈×C₅₉	#41	1	2	yes	—
62	C₈×C₅₉	#41	1	2	yes	—
63	C₅₉⋊C₈	—	2	1	no	—
64	C₅₉⋊C₈	—	2	1	no	—
65	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₅₉	—	2	1	no	—
67	C₂×D₅₉	—	2	2	no	—
68	C₄×C₅₉	—	2	2	no	—
69	C₄×C₅₉	—	4	4	no	—
70	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
71	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
72	Dic₅₉	—	2	1	no	—
73	Dic₅₉	—	2	2	no	—
74	Dic₅₉	—	4	4	no	—
75	#3177	—	59	1	no	maximal, Sylow
76	C₂×C₅₉	#532	1	1	yes	characteristic
77	C₂×C₅₉	#578	1	2	yes	—
78	D₅₉	—	4	1	no	—
79	#541	—	59	2	no	—
80	#499	D₅₉	1	1	yes	characteristic, complemented
81	#494	—	59	1	no	—
82	#491	—	59	2	no	—
83	#470	—	59	2	no	—
84	#467	—	59	2	no	—
85	#441	—	59	2	no	—
86	C₄×Q₁₆	—	59	2	no	—
87	#405	—	59	1	no	—
88	C₅₉	#3177	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₁₆	—	59	2	no	—
90	C₂×Q₁₆	—	59	2	no	—
91	C₂×M₁₆	—	59	1	no	—
92	C₂²×C₈	—	59	1	no	—
93	C₄⋊Q₈	—	59	2	no	—
94	#126	—	118	4	no	—
95	#125	—	118	4	no	—
96	#125	C₂×D₅₉	1	2	yes	—
97	C₂²⋊Q₈	—	59	4	no	—
98	C₄×Q₈	—	59	4	no	—
99	#117	—	59	1	no	—
100	#107	C₂×D₅₉	1	2	yes	—
101	#106	C₂×D₅₉	1	2	yes	—
102	#103	—	59	4	no	—
103	#103	—	59	4	no	—
104	#97	—	59	1	no	—
105	#97	—	59	1	no	—
106	C₄×C₈	—	59	1	no	—
107	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
108	C₂×Q₈	—	59	4	no	—
109	C₂²×C₄	—	59	1	no	—
110	Q₁₆	—	118	4	no	—
111	Q₁₆	—	118	4	no	—
112	M₁₆	—	118	2	no	—
113	C₂×C₈	—	59	1	no	—
114	C₂×C₈	—	59	1	no	—
115	C₂×C₈	—	118	2	no	—
116	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
117	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	59	2	no	—
120	C₄⋊C₄	—	118	4	no	—
121	C₄⋊C₄	—	118	4	no	—
122	C₄⋊C₄	—	118	4	no	—
123	C₄⋊C₄	C₂²×D₅₉	1	4	yes	—
124	C₂²⋊C₄	—	59	2	no	—
125	C₂²⋊C₄	—	118	4	no	—
126	C₄²	—	59	1	no	—
127	C₄²	—	118	4	no	—
128	C₄²	C₂²×D₅₉	1	1	yes	characteristic
129	C₂³	—	59	1	no	—
130	Q₈	—	118	4	no	—
131	Q₈	—	118	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	59	2	no	—
134	C₂×C₄	—	59	1	no	—
135	C₂×C₄	—	118	1	no	—
136	C₂×C₄	—	118	1	no	—
137	C₂×C₄	—	118	4	no	—
138	C₂×C₄	D₄×D₅₉	1	2	yes	—
139	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
140	C₈	—	118	1	no	—
141	C₈	—	118	1	no	—
142	C₈	#11745689	1	2	yes	—
143	C₈	#11745689	1	2	yes	—
144	C₂²	—	118	1	no	—
145	C₂²	—	118	2	no	—
146	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	4	no	—
149	C₄	—	118	1	no	—
150	C₄	—	118	2	no	—
151	C₄	—	236	4	no	—
152	C₄	C₂×#11745689	1	1	yes	characteristic
153	C₄	C₂×#11745689	1	1	yes	characteristic
154	C₂	—	236	1	no	—
155	C₂	#16837953	1	1	yes	characteristic
156	C₂	#16838216	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16838015

C₂

yes

maximal

#16837782

C₂

yes

maximal, characteristic, complemented

#16837772

C₂

yes

maximal, complemented

#16837661

C₂

yes

maximal

#16837644

C₂

yes

maximal

#16837508

C₂

yes

maximal, complemented

#16837496

C₂

yes

maximal, complemented

#16837356

C₂

yes

maximal, characteristic

C₅₉×#499

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃₆

C₂²

yes

—

C₂×Dic₂₃₆

C₂²

yes

—

C₂×#11745659

C₂²

yes

characteristic

C₂×C₈×D₅₉

C₂²

yes

characteristic

#14273727

C₂²

yes

—

#14273722

—

#14273718

—

#14273707

C₂²

yes

—

#14273692

C₂²

yes

—

#14273684

C₂²

yes

characteristic

#14273650

C₂²

yes

—

#14273650

C₂²

yes

—

#14273638

C₂²

yes

characteristic

#14273637

C₂²

yes

characteristic

C₈×Dic₅₉

C₂²

yes

characteristic

#125×C₅₉

C₂²

yes

—

#107×C₅₉

C₂²

yes

—

#106×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄×D₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

D₄

yes

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

C₂³

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

#11745659

—

C₈×D₅₉

—

C₂²×D₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₄

yes

—

C₂×Dic₅₉

C₂×D₄

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

#41

yes

—

C₈×C₅₉

#41

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#41

yes

characteristic

C₄×C₅₉

C₂×#41

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

—

#541

—

#499

D₅₉

yes

characteristic, complemented

—

—

—

—

—

C₄×Q₁₆

—

#405

—

C₅₉

#3177

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

#126

—

118

—

#125

—

118

—

#125

C₂×D₅₉

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

100

#107

C₂×D₅₉

yes

—

101

#106

C₂×D₅₉

yes

—

102

#103

—

103

#103

—

104

#97

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

C₂×D₅₉

yes

characteristic

108

C₂×Q₈

—

109

C₂²×C₄

—

110

Q₁₆

—

118

—

111

Q₁₆

—

118

—

112

M₁₆

—

118

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

118

—

116

C₂×C₈

C₂²×D₅₉

yes

characteristic

117

C₂×C₈

C₂²×D₅₉

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

118

—

121

C₄⋊C₄

—

118

—

122

C₄⋊C₄

—

118

—

123

C₄⋊C₄

C₂²×D₅₉

yes

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

118

—

126

C₄²

—

127

C₄²

—

118

—

128

C₄²

C₂²×D₅₉

yes

characteristic

129

C₂³

—

130

Q₈

—

118

—

131

Q₈

—

118

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

118

—

136

C₂×C₄

—

118

—

137

C₂×C₄

—

118

—

138

C₂×C₄

D₄×D₅₉

yes

—

139

C₂×C₄

C₂³×D₅₉

yes

characteristic

140

C₈

—

118

—

141

C₈

—

118

—

142

C₈

#11745689

yes

—

143

C₈

#11745689

yes

—

144

C₂²

—

118

—

145

C₂²

—

118

—

146

C₂²

C₂×D₄×D₅₉

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

118

—

150

C₄

—

118

—

151

C₄

—

236

—

152

C₄

C₂×#11745689

yes

characteristic

153

C₄

C₂×#11745689

yes

characteristic

154

C₂

—

236

—

155

C₂

#16837953

yes

characteristic

156

C₂

#16838216

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#499	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#499	D₅₉
hypocenter	C₅₉	#3177
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#499

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #499

N	H ≅ G/N	type
2. #16837782	C₂	semidirect
3. #16837772	C₂	semidirect
6. #16837508	C₂	semidirect
7. #16837496	C₂	semidirect
9. C₅₉×#499	C₂	semidirect
80. #499	D₅₉	semidirect
88. C₅₉	#3177	semidirect

H ≅ G/N

type

2. #16837782

C₂

semidirect

3. #16837772

C₂

semidirect

6. #16837508

C₂

semidirect

7. #16837496

C₂

semidirect

9. C₅₉×#499

C₂

semidirect

80. #499

D₅₉

semidirect

88. C₅₉

#3177

semidirect

Subgroups — #21434756

Conjugacy classes of subgroups

Complements