Group #21432815 — subgroups

Subgroups — #21432815

Order 7552.

484 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 189 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	C₂×#14273733	C₂	1	2	yes	maximal, complemented
2	#16837952	C₂	1	1	yes	maximal, characteristic
3	#16837475	C₂	1	8	yes	maximal, complemented
4	#16837426	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837368	C₂	1	2	yes	maximal, complemented
6	C₂×#97×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11745685	C₂²	1	2	yes	—
8	#14273733	—	2	8	no	—
9	C₂×#11745664	C₂²	1	2	yes	—
10	C₂×#11745659	C₂²	1	4	yes	—
11	C₂×C₈×D₅₉	C₂²	1	4	yes	—
12	#14273727	C₄	1	1	yes	characteristic
13	#14273724	C₄	1	1	yes	characteristic
14	#14273715	C₄	1	2	yes	—
15	#14273701	C₄	1	4	yes	—
16	C₄×D₂₃₆	C₂²	1	4	yes	—
17	C₄⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
18	(C₂×D₅₉)⋊C₈	C₂²	1	8	yes	—
19	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
22	#97×C₅₉	C₂²	1	4	yes	—
23	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊D₅₉	C₂×C₄	1	4	yes	—
26	#11745685	—	2	4	no	—
27	#11745685	—	2	4	no	—
28	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
29	C₂×D₂₃₆	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₅₉	C₂³	1	4	yes	—
31	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₅₉	—	2	2	no	—
33	C₂×C₈×C₅₉	D₄	1	4	yes	complemented
34	C₂×C₈×C₅₉	C₂³	1	2	yes	—
35	C₂×C₈×C₅₉	C₂³	1	2	yes	—
36	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₅₉	C₂³	1	2	yes	—
39	(C₂×D₅₉)⋊C₄	C₂×C₄	1	8	yes	—
40	C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
41	#11745664	—	2	4	no	—
42	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
43	#11745659	—	2	8	no	—
44	C₈×D₅₉	—	2	8	no	—
45	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₅₉	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₅₉	—	4	2	no	—
48	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₉	#41	1	2	yes	—
53	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
54	C₂²⋊D₅₉	—	2	8	no	—
55	C₂×Dic₅₉	C₂²×C₄	1	4	yes	—
56	D₂₃₆	—	2	4	no	—
57	D₂₃₆	—	4	4	no	—
58	C₄×D₅₉	—	2	4	no	—
59	C₄×D₅₉	—	2	4	no	—
60	Dic₁₁₈	—	2	4	no	—
61	C₈×C₅₉	—	2	2	no	—
62	C₈×C₅₉	C₂×D₄	1	4	yes	—
63	C₅₉⋊C₈	—	2	4	no	—
64	C₂²×C₅₉	—	2	2	no	—
65	C₂²×C₅₉	C₄×D₄	1	2	yes	—
66	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₅₉	—	2	4	no	—
68	C₂×D₅₉	—	2	8	no	—
69	C₄×C₅₉	—	2	2	no	—
70	C₄×C₅₉	—	2	2	no	—
71	C₄×C₅₉	C₄×D₄	1	2	yes	—
72	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
74	Dic₅₉	—	2	4	no	—
75	#2932	—	59	1	no	maximal, Sylow
76	C₂×C₅₉	—	2	2	no	—
77	C₂×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
78	C₂×C₅₉	#568	1	2	yes	—
79	D₅₉	—	4	4	no	—
80	C₂×#131	—	59	2	no	—
81	#532	—	59	1	no	—
82	#435	—	59	8	no	—
83	#423	—	59	1	no	—
84	#408	—	59	2	no	—
85	C₂×#97	D₅₉	1	1	yes	characteristic, complemented
86	C₅₉	#2932	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	—	59	2	no	—
88	#131	—	118	8	no	—
89	C₂×M₁₆	—	59	2	no	—
90	C₂×M₁₆	—	59	4	no	—
91	C₂²×C₈	—	59	4	no	—
92	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
94	C₄⋊Q₈	—	59	1	no	—
95	#127	—	59	1	no	—
96	#124	—	59	2	no	—
97	#121	—	59	4	no	—
98	C₄×D₄	—	59	4	no	—
99	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
100	C₄⋊C₈	—	59	4	no	—
101	C₂²⋊C₈	—	59	8	no	—
102	#97	C₂×D₅₉	1	4	yes	—
103	#41	—	118	4	no	—
104	#41	—	118	4	no	—
105	C₂×Q₈	—	59	2	no	—
106	C₂×D₄	—	59	4	no	—
107	C₂×D₄	—	59	2	no	—
108	C₂×D₄	—	59	4	no	—
109	C₂²×C₄	—	59	4	no	—
110	C₂²×C₄	C₄×D₅₉	1	2	yes	—
111	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
112	M₁₆	—	118	4	no	—
113	M₁₆	—	118	8	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	59	4	no	—
116	C₂×C₈	—	118	8	no	—
117	C₂×C₈	D₂₃₆	1	4	yes	complemented
118	C₂×C₈	C₂²×D₅₉	1	2	yes	—
119	C₂×C₈	C₂²×D₅₉	1	2	yes	—
120	C₄⋊C₄	—	59	4	no	—
121	C₂²⋊C₄	—	59	8	no	—
122	C₄²	C₄×D₅₉	1	1	yes	characteristic
123	C₄²	C₄×D₅₉	1	1	yes	characteristic
124	C₄²	C₂²×D₅₉	1	2	yes	—
125	C₂³	—	59	4	no	—
126	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
127	Q₈	—	118	4	no	—
128	D₄	—	118	8	no	—
129	D₄	—	118	4	no	—
130	D₄	—	236	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	59	4	no	—
133	C₂×C₄	—	118	4	no	—
134	C₂×C₄	—	118	4	no	—
135	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
139	C₂×C₄	#11745685	1	2	yes	—
140	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	118	4	no	—
143	C₈	C₂×D₂₃₆	1	4	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	—	118	8	no	—
146	C₂²	—	118	4	no	—
147	C₂²	C₄×D₂₃₆	1	2	yes	—
148	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	118	4	no	—
152	C₄	C₄×D₂₃₆	1	2	yes	—
153	C₄	C₂²×D₂₃₆	1	1	yes	characteristic
154	C₄	C₂×#11745685	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	236	4	no	—
157	C₂	C₂×C₄×D₂₃₆	1	1	yes	characteristic
158	C₂	#16838168	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273733

C₂

yes

maximal, complemented

#16837952

C₂

yes

maximal, characteristic

#16837475

C₂

yes

maximal, complemented

#16837426

C₂

yes

maximal, characteristic, complemented

#16837368

C₂

yes

maximal, complemented

C₂×#97×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₂²

yes

—

#14273733

—

C₂×#11745664

C₂²

yes

—

C₂×#11745659

C₂²

yes

—

C₂×C₈×D₅₉

C₂²

yes

—

#14273727

C₄

yes

characteristic

#14273724

C₄

yes

characteristic

#14273715

C₄

yes

—

#14273701

C₄

yes

—

C₄×D₂₃₆

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

(C₂×D₅₉)⋊C₈

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#97×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₂²⋊D₅₉

C₂×C₄

yes

—

#11745685

—

#11745685

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×C₄×D₅₉

C₂³

yes

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

complemented

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

—

(C₂×D₅₉)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

#11745664

—

C₂×C₅₉⋊C₈

C₂³

yes

—

#11745659

—

C₈×D₅₉

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

C₂²×D₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

C₂²×C₄

yes

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×D₄

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄×D₄

yes

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

C₂×#41

yes

characteristic

Dic₅₉

—

#2932

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×C₄×D₄

yes

characteristic

C₂×C₅₉

#568

yes

—

D₅₉

—

—

—

—

—

—

C₂×#97

D₅₉

yes

characteristic, complemented

C₅₉

#2932

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

118

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

#97

C₂×D₅₉

yes

—

103

#41

—

118

—

104

#41

—

118

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₅₉

yes

—

111

C₂²×C₄

C₂²×D₅₉

yes

characteristic

112

M₁₆

—

118

—

113

M₁₆

—

118

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

118

—

117

C₂×C₈

D₂₃₆

yes

complemented

118

C₂×C₈

C₂²×D₅₉

yes

—

119

C₂×C₈

C₂²×D₅₉

yes

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

C₄×D₅₉

yes

characteristic

123

C₄²

C₄×D₅₉

yes

characteristic

124

C₄²

C₂²×D₅₉

yes

—

125

C₂³

—

126

C₂³

C₂×C₄×D₅₉

yes

characteristic

127

Q₈

—

118

—

128

D₄

—

118

—

129

D₄

—

118

—

130

D₄

—

236

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

118

—

134

C₂×C₄

—

118

—

135

C₂×C₄

C₂×C₄×D₅₉

yes

—

136

C₂×C₄

C₂×C₄×D₅₉

yes

—

137

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

138

C₂×C₄

C₂×D₂₃₆

yes

—

139

C₂×C₄

#11745685

yes

—

140

C₂×C₄

C₂³×D₅₉

yes

characteristic

141

C₈

—

142

C₈

—

118

—

143

C₈

C₂×D₂₃₆

yes

—

144

C₂²

—

145

C₂²

—

118

—

146

C₂²

—

118

—

147

C₂²

C₄×D₂₃₆

yes

—

148

C₂²

C₂²×C₄×D₅₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

118

—

152

C₄

C₄×D₂₃₆

yes

—

153

C₄

C₂²×D₂₃₆

yes

characteristic

154

C₄

C₂×#11745685

yes

characteristic

155

C₂

—

156

C₂

—

236

—

157

C₂

C₂×C₄×D₂₃₆

yes

characteristic

158

C₂

#16838168

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₂×#97×C₅₉	C₂
hypercenter	C₂×#97	D₅₉
hypocenter	C₅₉	#2932
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₉

commutator, socle

C₂²×C₅₉

C₂³×C₄

Fitting

C₂×#97×C₅₉

C₂

hypercenter

C₂×#97

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
1. C₂×#14273733	C₂	semidirect
3. #16837475	C₂	semidirect
4. #16837426	C₂	semidirect
5. #16837368	C₂	semidirect
6. C₂×#97×C₅₉	C₂	semidirect
33. C₂×C₈×C₅₉	D₄	semidirect
85. C₂×#97	D₅₉	semidirect
86. C₅₉	#2932	semidirect
117. C₂×C₈	D₂₃₆	semidirect

H ≅ G/N

type

1. C₂×#14273733

C₂

semidirect

3. #16837475

C₂

semidirect

4. #16837426

C₂

semidirect

5. #16837368

C₂

semidirect

6. C₂×#97×C₅₉

C₂

semidirect

33. C₂×C₈×C₅₉

D₄

semidirect

85. C₂×#97

D₅₉

semidirect

86. C₅₉

#2932

semidirect

117. C₂×C₈

D₂₃₆

semidirect

Subgroups — #21432815

Conjugacy classes of subgroups

Complements