Group #28479854 — subgroups

Subgroups — #28479854

Order 15104.

390 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 175 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	C₄×#14273638	C₂	1	1	yes	maximal, characteristic
2	#21422842	C₂	1	2	yes	maximal
3	#21422380	C₂	1	2	yes	maximal
4	#21421929	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1276	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×#14273638	C₄	1	2	yes	—
8	C₂×#14273638	C₄	1	2	yes	—
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	#16837154	C₄	1	4	yes	complemented
12	#16837152	C₄	1	4	yes	complemented
13	#16837124	—	2	2	no	—
14	C₈×C₅₉⋊C₈	—	2	2	no	—
15	C₅₉×#431	C₂²	1	2	yes	—
16	C₄³×C₅₉	C₂²	1	1	yes	characteristic
17	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
20	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
21	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
22	#14273638	—	2	4	no	—
23	#14273638	C₈	1	8	yes	complemented
24	C₄×C₅₉⋊C₈	—	2	2	no	—
25	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
28	C₄×C₅₉⋊C₈	Q₈	1	2	yes	—
29	C₂×C₄²×C₅₉	—	2	2	no	—
30	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
33	C₂²⋊C₈×C₅₉	—	2	2	no	—
34	C₄×C₈×C₅₉	—	2	2	no	—
35	C₂²×C₄×C₅₉	—	2	2	no	—
36	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	#41	1	2	yes	—
38	C₂×C₈×C₅₉	—	2	4	no	—
39	C₄²×C₅₉	—	2	2	no	—
40	C₄²×C₅₉	—	4	4	no	—
41	C₄²×C₅₉	C₂×C₈	1	4	yes	—
42	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
43	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₅₉⋊C₈	—	2	2	no	—
46	C₂×C₅₉⋊C₈	—	2	4	no	—
47	C₂×C₅₉⋊C₈	C₄²	1	1	yes	characteristic
48	C₂×C₅₉⋊C₈	C₄²	1	1	yes	characteristic
49	C₂×C₅₉⋊C₈	C₄²	1	2	yes	—
50	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
51	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
52	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
53	C₂×C₅₉⋊C₈	M₁₆	1	4	yes	—
54	C₂³×C₅₉	#117	1	1	yes	characteristic
55	C₂×C₄×C₅₉	—	2	4	no	—
56	C₂×C₄×C₅₉	—	2	4	no	—
57	C₂×C₄×C₅₉	—	4	4	no	—
58	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
61	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
62	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
63	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
64	C₈×C₅₉	—	4	4	no	—
65	C₅₉⋊C₈	—	2	1	no	—
66	C₅₉⋊C₈	—	2	1	no	—
67	C₅₉⋊C₈	—	4	4	no	—
68	C₅₉⋊C₈	C₄×C₈	1	4	yes	—
69	C₅₉⋊C₈	C₄⋊C₈	1	8	yes	—
70	#8268	—	59	1	no	maximal, Sylow
71	C₂²×C₅₉	—	2	2	no	—
72	C₂²×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₅₉	#431	1	2	yes	—
74	C₄×C₅₉	—	2	2	no	—
75	C₄×C₅₉	—	4	4	no	—
76	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
77	C₄×C₅₉	C₄×M₁₆	1	1	yes	characteristic
78	C₄×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
79	C₄×C₅₉	#444	1	4	yes	—
80	C₄×#97	—	59	1	no	—
81	#1564	—	59	2	no	—
82	#1450	—	59	2	no	—
83	#1276	—	59	1	no	—
84	#1276	D₅₉	1	1	yes	characteristic
85	C₂×C₅₉	—	2	2	no	—
86	C₂×C₅₉	C₄×C₄⋊C₈	1	1	yes	characteristic
87	C₂×C₅₉	#1773	1	2	yes	—
88	#431	—	59	2	no	—
89	#431	C₂×D₅₉	1	2	yes	—
90	C₂×#97	—	59	2	no	—
91	C₂×#97	—	59	2	no	—
92	C₂×C₄×C₈	—	59	1	no	—
93	C₂×C₄×C₈	—	59	1	no	—
94	C₄³	C₂×D₅₉	1	1	yes	characteristic
95	#335	—	59	4	no	—
96	#334	—	59	4	no	—
97	#322	—	118	2	no	—
98	C₈²	—	118	2	no	—
99	C₅₉	#8268	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₈	—	59	1	no	—
101	C₂²×C₈	—	59	2	no	—
102	C₂²×C₈	—	59	1	no	—
103	C₂×C₄²	—	2	2	no	—
104	C₂×C₄²	C₄×D₅₉	1	2	yes	—
105	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
106	C₄⋊C₈	—	59	4	no	—
107	C₄⋊C₈	C₄×D₅₉	1	4	yes	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₂²⋊C₈	—	118	2	no	—
110	#97	—	59	8	no	—
111	#97	—	118	4	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	59	2	no	—
114	C₄×C₈	—	59	2	no	—
115	C₄×C₈	—	59	2	no	—
116	C₄×C₈	—	59	2	no	—
117	C₄×C₈	—	118	2	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
120	C₂²×C₄	#11745687	1	2	yes	—
121	C₂×C₈	—	2	4	no	—
122	C₂×C₈	—	59	2	no	—
123	C₂×C₈	—	59	4	no	—
124	C₂×C₈	—	59	4	no	—
125	C₂×C₈	—	59	4	no	—
126	C₂×C₈	—	59	4	no	—
127	C₂×C₈	—	59	1	no	—
128	C₂×C₈	—	59	1	no	—
129	C₂×C₈	—	118	2	no	—
130	C₂×C₈	—	118	4	no	—
131	C₄²	—	2	2	no	—
132	C₄²	—	4	4	no	—
133	C₄²	C₈×D₅₉	1	4	yes	—
134	C₄²	C₂×C₄×D₅₉	1	2	yes	—
135	C₄²	D₄×D₅₉	1	1	yes	characteristic
136	C₄²	Q₈×D₅₉	1	1	yes	characteristic
137	C₂³	#14273683	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	4	4	no	—
141	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
142	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
143	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	2	yes	—
144	C₂×C₄	#14273692	1	2	yes	—
145	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
146	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
147	C₈	—	4	4	no	—
148	C₈	—	59	8	no	—
149	C₈	—	59	4	no	—
150	C₈	—	118	1	no	—
151	C₈	—	118	1	no	—
152	C₈	—	236	4	no	—
153	C₂²	—	2	2	no	—
154	C₂²	#16837253	1	1	yes	characteristic
155	C₂²	#16837452	1	2	yes	—
156	C₄	—	2	2	no	—
157	C₄	—	4	4	no	—
158	C₄	C₄×C₈×D₅₉	1	1	yes	characteristic
159	C₄	#16837354	1	1	yes	characteristic
160	C₄	C₄⋊C₈×D₅₉	1	2	yes	—
161	C₄	#16837521	1	4	yes	—
162	C₂	—	2	2	no	—
163	C₂	#21423944	1	1	yes	characteristic
164	C₂	#21423979	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14273638

C₂

yes

maximal, characteristic

#21422842

C₂

yes

maximal

#21422380

C₂

yes

maximal

#21421929

C₂

yes

maximal, characteristic

C₅₉×#1276

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

C₂×#14273638

C₄

yes

—

C₂×#14273638

C₄

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₅₉⋊C₈

C₂²

yes

characteristic

#16837154

C₄

yes

complemented

#16837152

C₄

yes

complemented

#16837124

—

C₈×C₅₉⋊C₈

—

C₅₉×#431

C₂²

yes

—

C₄³×C₅₉

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

C₄⋊(C₅₉⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

#14273638

C₈

yes

complemented

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₄×C₅₉⋊C₈

Q₈

yes

—

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

C₂×C₄

yes

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×C₈

yes

—

C₄²×C₅₉

C₂²×C₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₄²

yes

characteristic

C₂×C₅₉⋊C₈

C₄²

yes

characteristic

C₂×C₅₉⋊C₈

C₄²

yes

—

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₈

yes

—

C₂×C₅₉⋊C₈

C₂×C₈

yes

—

C₂×C₅₉⋊C₈

M₁₆

yes

—

C₂³×C₅₉

#117

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

C₄×Q₈

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₄×C₈

yes

—

C₅₉⋊C₈

C₄⋊C₈

yes

—

#8268

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

#431

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄×C₈

yes

characteristic

C₄×C₅₉

C₄×M₁₆

yes

characteristic

C₄×C₅₉

C₂×C₄⋊C₈

yes

—

C₄×C₅₉

#444

yes

—

C₄×#97

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

C₄×C₄⋊C₈

yes

characteristic

C₂×C₅₉

#1773

yes

—

#431

—

#431

C₂×D₅₉

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

C₂×D₅₉

yes

—

—

—

118

—

C₈²

—

118

—

C₅₉

#8268

yes

characteristic, complemented, Sylow

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂×C₄²

—

104

C₂×C₄²

C₄×D₅₉

yes

—

105

C₂×C₄²

C₂²×D₅₉

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

C₄×D₅₉

yes

—

108

C₂²⋊C₈

—

109

C₂²⋊C₈

—

118

—

110

#97

—

111

#97

—

118

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

120

C₂²×C₄

#11745687

yes

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

118

—

130

C₂×C₈

—

118

—

131

C₄²

—

132

C₄²

—

133

C₄²

C₈×D₅₉

yes

—

134

C₄²

C₂×C₄×D₅₉

yes

—

135

C₄²

D₄×D₅₉

yes

characteristic

136

C₄²

Q₈×D₅₉

yes

characteristic

137

C₂³

#14273683

yes

characteristic

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

C₄²×D₅₉

yes

characteristic

142

C₂×C₄

C₄⋊C₄×D₅₉

yes

characteristic

143

C₂×C₄

(C₂²⋊D₅₉)⋊C₄

yes

—

144

C₂×C₄

#14273692

yes

—

145

C₂×C₄

C₂×C₈×D₅₉

yes

—

146

C₂×C₄

M₁₆×D₅₉

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

118

—

151

C₈

—

118

—

152

C₈

—

236

—

153

C₂²

—

154

C₂²

#16837253

yes

characteristic

155

C₂²

#16837452

yes

—

156

C₄

—

157

C₄

—

158

C₄

C₄×C₈×D₅₉

yes

characteristic

159

C₄

#16837354

yes

characteristic

160

C₄

C₄⋊C₈×D₅₉

yes

—

161

C₄

#16837521

yes

—

162

C₂

—

163

C₂

#21423944

yes

characteristic

164

C₂

#21423979

yes

—

	subgroup	G/H
center	C₄²	D₄×D₅₉
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1276	C₂
socle	C₂²×C₅₉	C₄×C₄⋊C₄
prime radical	C₂³×C₅₉	#117
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1276	D₅₉
hypocenter	C₅₉	#8268
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₄²

D₄×D₅₉

commutator

C₄×C₅₉

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1276

C₂

socle

C₂²×C₅₉

C₄×C₄⋊C₄

prime radical

C₂³×C₅₉

#117

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
11. #16837154	C₄	semidirect
12. #16837152	C₄	semidirect
23. #14273638	C₈	semidirect
99. C₅₉	#8268	semidirect

H ≅ G/N

type

11. #16837154

C₄

semidirect

12. #16837152

C₄

semidirect

23. #14273638

C₈

semidirect

99. C₅₉

#8268

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#14273638	2
#21422842	2
#21422380	2
#21421929	2
C₅₉×#1276	2
#8268	59

subgroup

index

C₅₉×#1276

Subgroups — #28479854

Conjugacy classes of subgroups

Complements

Maximal subgroups