Group #21429301 — subgroups

Subgroups — #21429301

Order 7552.

500 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 197 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×#14273727	C₂	1	1	yes	maximal, characteristic
2	#16837322	C₂	1	4	yes	maximal
3	#16837309	C₂	1	2	yes	maximal
4	#16837300	C₂	1	4	yes	maximal
5	#16837271	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₁₈	C₂²	1	2	yes	—
9	#14273727	—	2	4	no	—
10	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
13	C₂×Dic₅₉⋊C₄	C₂²	1	8	yes	—
14	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
15	C₅₉⋊#95	C₂²	1	2	yes	—
16	C₅₉⋊#95	C₂²	1	8	yes	—
17	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₅₉	—	2	4	no	—
19	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₅₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₉	—	2	2	no	—
24	C₂×Q₈×C₅₉	—	2	2	no	—
25	C₂×Q₈×C₅₉	Q₈	1	4	yes	complemented
26	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
27	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
29	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₅₉	C₂³	1	4	yes	—
32	C₂²×Dic₅₉	C₂³	1	4	yes	—
33	C₂×Dic₁₁₈	—	2	4	no	—
34	C₂×Dic₁₁₈	—	4	4	no	—
35	C₄⋊C₄×C₅₉	—	2	4	no	—
36	C₄⋊C₄×C₅₉	—	2	2	no	—
37	C₄²×C₅₉	—	2	4	no	—
38	C₄²×C₅₉	D₄	1	4	yes	—
39	C₄⋊Dic₅₉	—	2	4	no	—
40	C₄⋊Dic₅₉	—	2	4	no	—
41	C₄⋊Dic₅₉	Q₈	1	8	yes	complemented
42	Dic₅₉⋊C₄	—	4	8	no	—
43	C₄×Dic₅₉	—	2	4	no	—
44	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
45	Q₈×C₅₉	—	2	8	no	—
46	C₂×C₄×C₅₉	—	2	2	no	—
47	C₂×C₄×C₅₉	—	2	2	no	—
48	C₂×C₄×C₅₉	—	2	2	no	—
49	C₂×C₄×C₅₉	—	2	2	no	—
50	C₂×C₄×C₅₉	—	2	4	no	—
51	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅₉	C₂×Q₈	1	4	yes	—
55	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₅₉	C₂×Q₈	1	4	yes	—
57	C₂×C₄×C₅₉	#41	1	2	yes	—
58	C₂×Dic₅₉	—	2	4	no	—
59	C₂×Dic₅₉	—	2	4	no	—
60	C₂×Dic₅₉	—	2	4	no	—
61	C₂×Dic₅₉	—	2	8	no	—
62	C₂×Dic₅₉	C₂×Q₈	1	8	yes	—
63	Dic₁₁₈	—	4	8	no	—
64	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
65	C₂²×C₅₉	C₂²×Q₈	1	2	yes	—
66	C₂²×C₅₉	C₂²×Q₈	1	1	yes	characteristic
67	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
68	C₂²×C₅₉	H(5,𝔽₂)	1	2	yes	—
69	C₄×C₅₉	—	2	4	no	—
70	C₄×C₅₉	—	2	2	no	—
71	C₄×C₅₉	—	4	2	no	—
72	C₄×C₅₉	C₂²⋊Q₈	1	4	yes	—
73	C₄×C₅₉	C₄⋊Q₈	1	4	yes	—
74	Dic₅₉	—	2	8	no	—
75	Dic₅₉	—	4	4	no	—
76	#2567	—	59	1	no	maximal, Sylow
77	C₂×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₅₉	#535	1	1	yes	characteristic
80	C₂×C₅₉	#554	1	2	yes	—
81	C₂×C₅₉	Q₈²	1	2	yes	—
82	C₂×C₄⋊Q₈	—	59	1	no	—
83	C₂×C₄×Q₈	D₅₉	1	1	yes	characteristic
84	#395	—	59	4	no	—
85	#391	—	59	2	no	—
86	#389	—	59	4	no	—
87	#384	—	59	2	no	—
88	C₄×C₄⋊C₄	—	59	1	no	—
89	C₅₉	#2567	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	59	2	no	—
91	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
92	C₄⋊Q₈	—	118	4	no	—
93	C₄×Q₈	—	2	4	no	—
94	C₂×C₄⋊C₄	—	59	2	no	—
95	C₂×C₄⋊C₄	—	59	2	no	—
96	C₂×C₄⋊C₄	—	59	2	no	—
97	C₂×C₄⋊C₄	—	59	8	no	—
98	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
100	C₂×C₄²	—	59	2	no	—
101	C₂×C₄²	C₂×D₅₉	1	2	yes	—
102	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
103	#95	—	59	2	no	—
104	#95	—	59	8	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	—	118	4	no	—
108	C₂×Q₈	—	236	4	no	—
109	C₂×Q₈	Dic₁₁₈	1	4	yes	complemented
110	C₂²×C₄	—	59	4	no	—
111	C₂²×C₄	—	59	4	no	—
112	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
113	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
114	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	59	8	no	—
120	C₄⋊C₄	—	118	4	no	—
121	C₄⋊C₄	—	118	4	no	—
122	C₄⋊C₄	—	236	8	no	—
123	C₄²	—	2	4	no	—
124	C₄²	—	118	4	no	—
125	C₄²	C₂²⋊D₅₉	1	4	yes	—
126	C₂³	C₂³×D₅₉	1	1	yes	characteristic
127	Q₈	—	2	8	no	—
128	Q₈	—	236	8	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	59	8	no	—
135	C₂×C₄	—	118	4	no	—
136	C₂×C₄	—	118	4	no	—
137	C₂×C₄	—	118	4	no	—
138	C₂×C₄	—	118	8	no	—
139	C₂×C₄	C₂×Dic₁₁₈	1	4	yes	—
140	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
141	C₂×C₄	#11745685	1	2	yes	—
142	C₂×C₄	Q₈×D₅₉	1	4	yes	—
143	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
144	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
145	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
147	C₂²	C₂×Q₈×D₅₉	1	2	yes	—
148	C₂²	C₂²×Dic₁₁₈	1	1	yes	characteristic
149	C₂²	C₂×#11745685	1	1	yes	characteristic
150	C₂²	#14273773	1	2	yes	—
151	C₄	—	2	4	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	118	8	no	—
155	C₄	—	236	4	no	—
156	C₄	#14273706	1	4	yes	—
157	C₄	#14273725	1	4	yes	—
158	C₂	C₂×#14273706	1	1	yes	characteristic
159	C₂	C₂×#14273725	1	1	yes	characteristic
160	C₂	#16837966	1	1	yes	characteristic
161	C₂	#16838112	1	2	yes	—
162	C₂	Q₈×Dic₁₁₈	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273727

C₂

yes

maximal, characteristic

#16837322

C₂

yes

maximal

#16837309

C₂

yes

maximal

#16837300

C₂

yes

maximal

#16837271

C₂

yes

maximal

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

—

#14273727

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×Dic₅₉⋊C₄

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₄×Q₈×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

Q₈

yes

complemented

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

Q₈

yes

complemented

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂³×C₅₉

C₂⁴

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×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

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×Q₈

yes

—

Dic₁₁₈

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂²×C₅₉

C₂²×Q₈

yes

—

C₂²×C₅₉

C₂²×Q₈

yes

characteristic

C₂²×C₅₉

C₂×#41

yes

characteristic

C₂²×C₅₉

H(5,𝔽₂)

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

C₄⋊Q₈

yes

—

Dic₅₉

—

Dic₅₉

—

#2567

—

maximal, Sylow

C₂×C₅₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₅₉

C₂×C₄⋊Q₈

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

Q₈²

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

D₅₉

yes

—

—

—

—

C₄×C₄⋊C₄

—

C₅₉

#2567

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

—

118

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×D₅₉

yes

—

102

C₂×C₄²

C₂×D₅₉

yes

characteristic

103

#95

—

104

#95

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

118

—

108

C₂×Q₈

—

236

—

109

C₂×Q₈

Dic₁₁₈

yes

complemented

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂²×D₅₉

yes

—

113

C₂²×C₄

C₂²×D₅₉

yes

—

114

C₂²×C₄

C₂²×D₅₉

yes

characteristic

115

C₂²×C₄

C₂²×D₅₉

yes

characteristic

116

C₂²×C₄

C₂²×D₅₉

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

118

—

121

C₄⋊C₄

—

118

—

122

C₄⋊C₄

—

236

—

123

C₄²

—

124

C₄²

—

118

—

125

C₄²

C₂²⋊D₅₉

yes

—

126

C₂³

C₂³×D₅₉

yes

characteristic

127

Q₈

—

128

Q₈

—

236

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

118

—

136

C₂×C₄

—

118

—

137

C₂×C₄

—

118

—

138

C₂×C₄

—

118

—

139

C₂×C₄

C₂×Dic₁₁₈

yes

—

140

C₂×C₄

C₂×Dic₁₁₈

yes

—

141

C₂×C₄

#11745685

yes

—

142

C₂×C₄

Q₈×D₅₉

yes

—

143

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

144

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

145

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

146

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

147

C₂²

C₂×Q₈×D₅₉

yes

—

148

C₂²

C₂²×Dic₁₁₈

yes

characteristic

149

C₂²

C₂×#11745685

yes

characteristic

150

C₂²

#14273773

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

118

—

155

C₄

—

236

—

156

C₄

#14273706

yes

—

157

C₄

#14273725

yes

—

158

C₂

C₂×#14273706

yes

characteristic

159

C₂

C₂×#14273725

yes

characteristic

160

C₂

#16837966

yes

characteristic

161

C₂

#16838112

yes

—

162

C₂

Q₈×Dic₁₁₈

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator, socle	C₂³×C₅₉	C₂⁴
Fitting	C₂×C₄×Q₈×C₅₉	C₂
hypercenter	C₂×C₄×Q₈	D₅₉
hypocenter	C₅₉	#2567
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₉

commutator, socle

C₂³×C₅₉

C₂⁴

Fitting

C₂×C₄×Q₈×C₅₉

C₂

hypercenter

C₂×C₄×Q₈

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₅₉

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
25. C₂×Q₈×C₅₉	Q₈	semidirect
41. C₄⋊Dic₅₉	Q₈	semidirect
89. C₅₉	#2567	semidirect
109. C₂×Q₈	Dic₁₁₈	semidirect

H ≅ G/N

type

25. C₂×Q₈×C₅₉

Q₈

semidirect

41. C₄⋊Dic₅₉

Q₈

semidirect

89. C₅₉

#2567

semidirect

109. C₂×Q₈

Dic₁₁₈

semidirect

Subgroups — #21429301

Conjugacy classes of subgroups

Complements