Group #21434682 — subgroups

Subgroups — #21434682

Order 7552.

368 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 149 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	2	yes	maximal
2	#16837807	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837782	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837753	C₂	1	2	yes	maximal, complemented
5	#16837643	C₂	1	4	yes	maximal
6	C₄×Dic₂₃₆	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₃₆	C₂²	1	4	yes	—
9	#14273727	C₂²	1	2	yes	—
10	#14273727	C₂²	1	2	yes	—
11	#14273719	—	2	2	no	—
12	#14273706	—	2	2	no	—
13	#14273706	C₂²	1	4	yes	—
14	C₄×Dic₁₁₈	—	2	2	no	—
15	C₄×Dic₁₁₈	C₂²	1	4	yes	—
16	#14273686	—	2	2	no	—
17	#14273661	C₂²	1	4	yes	—
18	#14273652	C₂²	1	8	yes	—
19	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₉	C₂²	1	2	yes	—
23	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₁₈	—	2	2	no	—
28	C₂×Dic₁₁₈	C₂³	1	4	yes	—
29	C₂×C₈×C₅₉	—	2	4	no	—
30	C₂×C₈×C₅₉	C₂³	1	2	yes	—
31	C₂×C₈×C₅₉	C₂³	1	2	yes	—
32	C₄²×C₅₉	D₄	1	1	yes	characteristic
33	C₄²×C₅₉	D₄	1	1	yes	characteristic
34	C₄²×C₅₉	C₂³	1	1	yes	characteristic
35	C₄²×C₅₉	C₂³	1	1	yes	characteristic
36	C₂²⋊Dic₅₉	—	2	4	no	—
37	C₄⋊Dic₅₉	—	2	2	no	—
38	C₄⋊Dic₅₉	—	2	2	no	—
39	C₄⋊Dic₅₉	—	2	2	no	—
40	C₄⋊Dic₅₉	C₂³	1	4	yes	—
41	Dic₅₉⋊C₄	—	2	4	no	—
42	Dic₅₉⋊C₄	—	2	4	no	—
43	C₄×Dic₅₉	—	2	4	no	—
44	Dic₂₃₆	—	2	8	no	—
45	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	—	2	2	no	—
47	C₂×C₄×C₅₉	—	2	2	no	—
48	C₂×C₄×C₅₉	Q₁₆	1	4	yes	complemented
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₅₉	—	2	4	no	—
56	C₂×Dic₅₉	—	2	4	no	—
57	Dic₁₁₈	—	2	8	no	—
58	Dic₁₁₈	—	4	4	no	—
59	C₈×C₅₉	—	2	2	no	—
60	C₈×C₅₉	#41	1	4	yes	—
61	C₂²×C₅₉	—	2	1	no	—
62	C₂²×C₅₉	—	2	1	no	—
63	C₂²×C₅₉	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₅₉	—	2	1	no	—
66	C₄×C₅₉	—	2	1	no	—
67	C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
68	C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
69	C₄×C₅₉	#135	1	2	yes	—
70	C₄×C₅₉	C₂×#41	1	2	yes	—
71	Dic₅₉	—	4	4	no	—
72	Dic₅₉	—	4	4	no	—
73	#3170	—	59	1	no	maximal, Sylow
74	C₂×C₅₉	—	2	2	no	—
75	C₂×C₅₉	#532	1	1	yes	characteristic
76	C₂×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
77	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
78	#533	—	59	2	no	—
79	#500	—	59	1	no	—
80	#494	—	59	1	no	—
81	#487	—	59	2	no	—
82	#467	—	59	4	no	—
83	C₄×Q₁₆	—	59	4	no	—
84	C₂×C₄×C₈	D₅₉	1	1	yes	characteristic
85	C₅₉	#3170	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₁₆	—	59	4	no	—
87	C₂²×C₈	C₂×D₅₉	1	2	yes	—
88	C₄⋊Q₈	—	59	2	no	—
89	C₄⋊Q₈	—	59	2	no	—
90	#125	—	118	2	no	—
91	C₂²⋊Q₈	—	59	4	no	—
92	C₂²⋊Q₈	—	118	2	no	—
93	C₄×Q₈	—	59	4	no	—
94	C₄×Q₈	—	118	2	no	—
95	#117	—	118	2	no	—
96	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
97	#107	—	59	4	no	—
98	#103	—	59	8	no	—
99	C₄×C₈	C₂×D₅₉	1	2	yes	—
100	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
101	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
102	C₂×Q₈	—	59	4	no	—
103	C₂×Q₈	—	118	2	no	—
104	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
105	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
107	Q₁₆	—	118	8	no	—
108	C₂×C₈	—	2	4	no	—
109	C₂×C₈	C₂²×D₅₉	1	2	yes	—
110	C₂×C₈	C₂²×D₅₉	1	2	yes	—
111	C₄⋊C₄	—	59	4	no	—
112	C₄⋊C₄	—	118	2	no	—
113	C₄⋊C₄	—	118	2	no	—
114	C₄⋊C₄	—	118	4	no	—
115	C₄⋊C₄	—	118	4	no	—
116	C₄⋊C₄	—	118	2	no	—
117	C₂²⋊C₄	—	118	4	no	—
118	C₄²	—	118	4	no	—
119	C₄²	D₂₃₆	1	1	yes	characteristic
120	C₄²	D₂₃₆	1	1	yes	characteristic
121	C₄²	C₂²×D₅₉	1	1	yes	characteristic
122	C₄²	C₂²×D₅₉	1	1	yes	characteristic
123	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
124	Q₈	—	118	8	no	—
125	Q₈	—	236	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	118	4	no	—
129	C₂×C₄	—	118	4	no	—
130	C₂×C₄	Dic₂₃₆	1	4	yes	complemented
131	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
132	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
133	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
134	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
135	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	#11745685	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×Dic₂₃₆	1	2	yes	—
142	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	236	4	no	—
146	C₄	—	236	4	no	—
147	C₄	C₂×Dic₂₃₆	1	2	yes	—
148	C₄	C₂×Dic₂₃₆	1	2	yes	—
149	C₄	#14273749	1	2	yes	—
150	C₄	C₂×#11745685	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	#16837952	1	1	yes	characteristic
153	C₂	C₂²×Dic₂₃₆	1	1	yes	characteristic
154	C₂	C₂×#14273749	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal

#16837807

C₂

yes

maximal, characteristic, complemented

#16837782

C₂

yes

maximal, characteristic, complemented

#16837753

C₂

yes

maximal, complemented

#16837643

C₂

yes

maximal

C₄×Dic₂₃₆

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₅₉

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

—

#14273727

C₂²

yes

—

#14273727

C₂²

yes

—

#14273719

—

#14273706

—

#14273706

C₂²

yes

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₂²

yes

—

#14273686

—

#14273661

C₂²

yes

—

#14273652

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂³

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

Q₁₆

yes

complemented

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#41

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×Q₁₆

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×Q₁₆

yes

—

C₄×C₅₉

C₂×Q₁₆

yes

—

C₄×C₅₉

#135

yes

—

C₄×C₅₉

C₂×#41

yes

—

Dic₅₉

—

Dic₅₉

—

#3170

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#532

yes

characteristic

C₂×C₅₉

C₂²×Q₁₆

yes

C₂×C₅₉

yes

—

—

—

—

—

C₄×Q₁₆

—

C₂×C₄×C₈

D₅₉

yes

characteristic

C₅₉

#3170

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂²×C₈

C₂×D₅₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

118

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

C₄×Q₈

—

118

—

#117

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#107

—

#103

—

C₄×C₈

C₂×D₅₉

yes

—

100

C₄×C₈

C₂×D₅₉

yes

characteristic

101

C₄×C₈

C₂×D₅₉

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

118

—

104

C₂²×C₄

D₂₃₆

yes

characteristic

105

C₂²×C₄

D₂₃₆

yes

characteristic

106

C₂²×C₄

C₂²×D₅₉

yes

characteristic

107

Q₁₆

—

118

—

108

C₂×C₈

—

109

C₂×C₈

C₂²×D₅₉

yes

—

110

C₂×C₈

C₂²×D₅₉

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

118

—

113

C₄⋊C₄

—

118

—

114

C₄⋊C₄

—

118

—

115

C₄⋊C₄

—

118

—

116

C₄⋊C₄

—

118

—

117

C₂²⋊C₄

—

118

—

118

C₄²

—

118

—

119

C₄²

D₂₃₆

yes

characteristic

120

C₄²

D₂₃₆

yes

characteristic

121

C₄²

C₂²×D₅₉

yes

characteristic

122

C₄²

C₂²×D₅₉

yes

characteristic

123

C₂³

C₂×D₂₃₆

yes

characteristic

124

Q₈

—

118

—

125

Q₈

—

236

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

118

—

129

C₂×C₄

—

118

—

130

C₂×C₄

Dic₂₃₆

yes

complemented

131

C₂×C₄

C₂×D₂₃₆

yes

characteristic

132

C₂×C₄

C₂×D₂₃₆

yes

characteristic

133

C₂×C₄

C₂×D₂₃₆

yes

characteristic

134

C₂×C₄

C₂×D₂₃₆

yes

characteristic

135

C₂×C₄

C₂×D₂₃₆

yes

characteristic

136

C₂×C₄

C₂³×D₅₉

yes

characteristic

137

C₈

—

138

C₈

#11745685

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×Dic₂₃₆

yes

—

142

C₂²

C₂²×D₂₃₆

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

236

—

146

C₄

—

236

—

147

C₄

C₂×Dic₂₃₆

yes

—

148

C₄

C₂×Dic₂₃₆

yes

—

149

C₄

#14273749

yes

—

150

C₄

C₂×#11745685

yes

—

151

C₂

—

152

C₂

#16837952

yes

characteristic

153

C₂

C₂²×Dic₂₃₆

yes

characteristic

154

C₂

C₂×#14273749

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂×C₄×C₈×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₂×C₄×C₈	D₅₉
hypocenter	C₅₉	#3170
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₂×C₄×C₈×C₅₉

C₂

socle

C₂²×C₅₉

C₂²×D₄

hypercenter

C₂×C₄×C₈

D₅₉

hypocenter

C₅₉

#3170

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16837807	C₂	semidirect
3. #16837782	C₂	semidirect
4. #16837753	C₂	semidirect
6. C₄×Dic₂₃₆	C₂	semidirect
48. C₂×C₄×C₅₉	Q₁₆	semidirect
85. C₅₉	#3170	semidirect
130. C₂×C₄	Dic₂₃₆	semidirect

H ≅ G/N

type

2. #16837807

C₂

semidirect

3. #16837782

C₂

semidirect

4. #16837753

C₂

semidirect

6. C₄×Dic₂₃₆

C₂

semidirect

48. C₂×C₄×C₅₉

Q₁₆

semidirect

85. C₅₉

#3170

semidirect

130. C₂×C₄

Dic₂₃₆

semidirect

Subgroups — #21434682

Conjugacy classes of subgroups

Complements