Group #21424908 — subgroups

Subgroups — #21424908

Order 7552.

336 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 113 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×#14273651	C₂	1	2	yes	maximal, complemented
2	C₂×#14273648	C₂	1	2	yes	maximal
3	C₂×C₄×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837268	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#124×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
8	#14273651	—	2	4	no	—
9	#14273648	—	2	4	no	—
10	C₄×C₅₉⋊C₈	—	2	2	no	—
11	C₅₉⋊#95	—	2	2	no	—
12	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₅₉	C₂²	1	1	yes	characteristic
14	#124×C₅₉	—	2	2	no	—
15	#124×C₅₉	C₄	1	4	yes	complemented
16	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
17	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
18	C₂⁴×C₅₉	—	2	1	no	—
19	C₂×Q₈×C₅₉	—	2	1	no	—
20	C₂×Q₈×C₅₉	—	2	2	no	—
21	C₂×Q₈×C₅₉	—	2	2	no	—
22	C₂×Q₈×C₅₉	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₅₉	—	2	2	no	—
24	C₂×D₄×C₅₉	—	2	1	no	—
25	C₂×D₄×C₅₉	—	2	2	no	—
26	C₂×D₄×C₅₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅₉	—	2	1	no	—
28	C₂²×C₄×C₅₉	D₄	1	2	yes	—
29	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅₉	—	2	2	no	—
31	C₂²⋊C₄×C₅₉	—	2	8	no	—
32	C₄²×C₅₉	C₂×C₄	1	2	yes	—
33	C₄²×C₅₉	D₄	1	2	yes	—
34	C₄⋊Dic₅₉	—	2	2	no	—
35	C₄⋊Dic₅₉	—	2	2	no	—
36	C₂×C₅₉⋊C₈	—	2	2	no	—
37	C₂×C₅₉⋊C₈	—	2	2	no	—
38	C₂×C₅₉⋊C₈	D₄	1	4	yes	—
39	C₂³×C₅₉	—	2	2	no	—
40	C₂³×C₅₉	—	4	1	no	—
41	C₂³×C₅₉	—	4	1	no	—
42	C₂³×C₅₉	—	4	4	no	—
43	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₅₉	—	2	4	no	—
45	Q₈×C₅₉	—	4	2	no	—
46	D₄×C₅₉	—	2	4	no	—
47	D₄×C₅₉	—	4	2	no	—
48	C₂×C₄×C₅₉	—	2	4	no	—
49	C₂×C₄×C₅₉	—	2	2	no	—
50	C₂×C₄×C₅₉	—	4	1	no	—
51	C₂×C₄×C₅₉	—	4	1	no	—
52	C₂×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#41	1	2	yes	—
57	C₂×Dic₅₉	—	4	2	no	—
58	C₂×Dic₅₉	—	4	4	no	—
59	C₅₉⋊C₈	—	2	4	no	—
60	C₂²×C₅₉	—	4	2	no	—
61	C₂²×C₅₉	—	4	4	no	—
62	C₂²×C₅₉	—	8	2	no	—
63	C₂²×C₅₉	—	8	2	no	—
64	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₅₉	#124	1	1	yes	characteristic
66	C₂²×C₅₉	#127	1	1	yes	characteristic
67	C₂²×C₅₉	#135	1	4	yes	—
68	C₄×C₅₉	—	2	4	no	—
69	C₄×C₅₉	—	4	2	no	—
70	C₄×C₅₉	C₄×D₄	1	2	yes	—
71	C₄×C₅₉	#121	1	2	yes	—
72	Dic₅₉	—	8	2	no	—
73	#1958	—	59	1	no	maximal, Sylow
74	C₂×C₅₉	—	8	2	no	—
75	C₂×C₅₉	#390	1	1	yes	characteristic
76	C₂×C₅₉	#416	1	2	yes	—
77	C₂×C₅₉	#488	1	2	yes	—
78	C₂×C₅₉	#495	1	2	yes	—
79	C₂×#124	D₅₉	1	1	yes	characteristic
80	C₂×#103	—	59	2	no	—
81	C₂×#102	—	59	2	no	—
82	C₂×C₄×C₈	—	59	1	no	—
83	#382	—	59	1	no	—
84	C₅₉	#1958	1	1	yes	characteristic, complemented, Sylow
85	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
86	C₂²×D₄	C₂×D₅₉	1	1	yes	characteristic
87	C₂²×C₈	—	59	2	no	—
88	#124	—	2	2	no	—
89	#124	Dic₅₉	1	4	yes	complemented
90	C₂×C₄⋊C₄	—	59	2	no	—
91	C₂×C₂²⋊C₄	—	2	2	no	—
92	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
93	#103	—	118	4	no	—
94	#102	—	118	4	no	—
95	C₄×C₈	—	118	2	no	—
96	#95	—	118	2	no	—
97	C₂⁴	—	2	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×Dic₅₉	1	2	yes	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	C₂×Dic₅₉	1	2	yes	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	118	2	no	—
108	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
109	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
110	C₂×C₈	—	59	4	no	—
111	C₂×C₈	—	118	2	no	—
112	C₂×C₈	—	118	2	no	—
113	C₄⋊C₄	—	118	2	no	—
114	C₄⋊C₄	—	118	2	no	—
115	C₂²⋊C₄	—	2	8	no	—
116	C₄²	C₂×Dic₅₉	1	2	yes	—
117	C₄²	C₂²⋊D₅₉	1	2	yes	—
118	C₂³	—	2	2	no	—
119	C₂³	—	4	1	no	—
120	C₂³	—	4	1	no	—
121	C₂³	—	4	4	no	—
122	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
123	Q₈	—	2	4	no	—
124	Q₈	—	4	2	no	—
125	D₄	—	2	4	no	—
126	D₄	—	4	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	236	2	no	—
132	C₂×C₄	—	236	4	no	—
133	C₂×C₄	C₂²⋊Dic₅₉	1	4	yes	—
134	C₂×C₄	D₄×D₅₉	1	2	yes	—
135	C₂×C₄	#11745687	1	2	yes	—
136	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
138	C₈	—	118	4	no	—
139	C₂²	—	4	4	no	—
140	C₂²	—	4	2	no	—
141	C₂²	—	8	2	no	—
142	C₂²	—	8	2	no	—
143	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
144	C₂²	#14273714	1	1	yes	characteristic
145	C₂²	#14273723	1	1	yes	characteristic
146	C₂²	#14273747	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	4	2	no	—
149	C₄	—	472	2	no	—
150	C₄	D₄×Dic₅₉	1	2	yes	—
151	C₄	#14273698	1	2	yes	—
152	C₂	—	8	2	no	—
153	C₂	#16837305	1	1	yes	characteristic
154	C₂	#16837398	1	2	yes	—
155	C₂	#16837756	1	2	yes	—
156	C₂	#16837784	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273651

C₂

yes

maximal, complemented

C₂×#14273648

C₂

yes

maximal

C₂×C₄×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#16837268

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273651

—

#14273648

—

C₄×C₅₉⋊C₈

—

C₅₉⋊#95

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂²×D₄×C₅₉

C₂²

yes

characteristic

#124×C₅₉

—

#124×C₅₉

C₄

yes

complemented

C₂×C₂²⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂⁴×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₄

yes

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

#121

yes

—

Dic₅₉

—

#1958

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#390

yes

characteristic

C₂×C₅₉

#416

yes

—

C₂×C₅₉

#488

yes

—

C₂×C₅₉

#495

yes

—

C₂×#124

D₅₉

yes

characteristic

C₂×#103

—

C₂×#102

—

C₂×C₄×C₈

—

#382

—

C₅₉

#1958

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₉

yes

characteristic

C₂²×D₄

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

#124

—

#124

Dic₅₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#103

—

118

—

#102

—

118

—

C₄×C₈

—

118

—

#95

—

118

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×Dic₅₉

yes

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

C₂×Dic₅₉

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

118

—

108

C₂²×C₄

C₂²⋊D₅₉

yes

—

109

C₂²×C₄

C₂²×D₅₉

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

118

—

112

C₂×C₈

—

118

—

113

C₄⋊C₄

—

118

—

114

C₄⋊C₄

—

118

—

115

C₂²⋊C₄

—

116

C₄²

C₂×Dic₅₉

yes

—

117

C₄²

C₂²⋊D₅₉

yes

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

236

—

132

C₂×C₄

—

236

—

133

C₂×C₄

C₂²⋊Dic₅₉

yes

—

134

C₂×C₄

D₄×D₅₉

yes

—

135

C₂×C₄

#11745687

yes

—

136

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

138

C₈

—

118

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

144

C₂²

#14273714

yes

characteristic

145

C₂²

#14273723

yes

characteristic

146

C₂²

#14273747

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

472

—

150

C₄

D₄×Dic₅₉

yes

—

151

C₄

#14273698

yes

—

152

C₂

—

153

C₂

#16837305

yes

characteristic

154

C₂

#16837398

yes

—

155

C₂

#16837756

yes

—

156

C₂

#16837784

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#124×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#124	D₅₉
hypocenter	C₅₉	#1958
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#124×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×#14273651	C₂	semidirect
3. C₂×C₄×C₅₉⋊C₈	C₂	semidirect
4. #16837268	C₂	semidirect
15. #124×C₅₉	C₄	semidirect
84. C₅₉	#1958	semidirect
89. #124	Dic₅₉	semidirect

H ≅ G/N

type

1. C₂×#14273651

C₂

semidirect

3. C₂×C₄×C₅₉⋊C₈

C₂

semidirect

4. #16837268

C₂

semidirect

15. #124×C₅₉

C₄

semidirect

84. C₅₉

#1958

semidirect

89. #124

Dic₅₉

semidirect

Subgroups — #21424908

Conjugacy classes of subgroups

Complements