Group #28493724 — subgroups

Subgroups — #28493724

Order 15104.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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	#21424250	C₂	1	1	yes	maximal, characteristic
2	#21422868	C₂	1	2	yes	maximal
3	#21422499	C₂	1	2	yes	maximal
4	#21421925	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1995	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	#16837152	—	2	4	no	—
10	Q₈⋊(C₅₉⋊C₈)	—	2	4	no	—
11	C₅₉×#533	C₂²	1	1	yes	characteristic
12	C₅₉×#423	C₂²	1	1	yes	characteristic
13	C₅₉×#376	C₂²	1	1	yes	characteristic
14	C₅₉×#339	—	2	2	no	—
15	C₂²×C₅₉⋊C₈	—	2	2	no	—
16	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
17	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
18	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
19	C₄⋊(C₅₉⋊C₈)	—	4	4	no	—
20	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
21	C₄×C₅₉⋊C₈	D₄	1	4	yes	—
22	C₂×M₁₆×C₅₉	—	2	1	no	—
23	C₄⋊Q₈×C₅₉	Q₈	1	2	yes	—
24	#125×C₅₉	—	2	1	no	—
25	C₂²⋊Q₈×C₅₉	—	2	2	no	—
26	C₄×Q₈×C₅₉	—	2	2	no	—
27	#117×C₅₉	—	2	1	no	—
28	C₂×C₄²×C₅₉	D₄	1	2	yes	—
29	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₉	—	2	2	no	—
31	#95×C₅₉	—	2	2	no	—
32	C₂×Q₈×C₅₉	—	2	2	no	—
33	C₂²×C₄×C₅₉	—	2	2	no	—
34	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	#41	1	2	yes	—
36	M₁₆×C₅₉	—	4	2	no	—
37	C₂×C₈×C₅₉	—	4	1	no	—
38	C₄⋊C₄×C₅₉	—	2	2	no	—
39	C₄⋊C₄×C₅₉	—	2	2	no	—
40	C₄⋊C₄×C₅₉	—	4	2	no	—
41	C₂²⋊C₄×C₅₉	—	4	1	no	—
42	C₄²×C₅₉	—	2	2	no	—
43	C₄²×C₅₉	—	4	1	no	—
44	C₄²×C₅₉	SD₁₆	1	4	yes	—
45	C₄²×C₅₉	C₂×D₄	1	2	yes	—
46	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₅₉	#41	1	1	yes	characteristic
48	C₂×C₅₉⋊C₈	—	2	4	no	—
49	C₂×C₅₉⋊C₈	—	4	2	no	—
50	C₂×C₅₉⋊C₈	—	4	4	no	—
51	C₂³×C₅₉	#123	1	1	yes	characteristic
52	Q₈×C₅₉	—	4	4	no	—
53	C₂×C₄×C₅₉	—	2	4	no	—
54	C₂×C₄×C₅₉	—	4	2	no	—
55	C₂×C₄×C₅₉	—	4	4	no	—
56	C₂×C₄×C₅₉	—	4	2	no	—
57	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₅₉	#123	1	2	yes	—
60	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₅₉	#137	1	2	yes	—
63	C₈×C₅₉	—	8	1	no	—
64	C₅₉⋊C₈	—	4	4	no	—
65	C₅₉⋊C₈	—	8	2	no	—
66	#10670	—	59	1	no	maximal, Sylow
67	C₂²×C₅₉	—	2	2	no	—
68	C₂²×C₅₉	#399	1	1	yes	characteristic
69	C₂²×C₅₉	#483	1	2	yes	—
70	C₄×C₅₉	—	2	2	no	—
71	C₄×C₅₉	—	2	2	no	—
72	C₄×C₅₉	—	4	4	no	—
73	C₄×C₅₉	—	8	2	no	—
74	C₄×C₅₉	#475	1	2	yes	—
75	C₄×C₅₉	#492	1	1	yes	characteristic
76	C₄×C₅₉	#497	1	1	yes	characteristic
77	#1995	D₅₉	1	1	yes	characteristic
78	#1843	—	59	1	no	—
79	#1570	—	59	2	no	—
80	#1481	—	59	2	no	—
81	#1274	—	59	1	no	—
82	C₂×C₅₉	—	2	2	no	—
83	C₂×C₅₉	#2096	1	1	yes	characteristic
84	C₂×C₅₉	#2098	1	2	yes	—
85	#533	C₂×D₅₉	1	1	yes	characteristic
86	#431	—	59	2	no	—
87	#423	C₂×D₅₉	1	1	yes	characteristic
88	C₂×C₄⋊C₈	—	59	2	no	—
89	C₂×C₄⋊C₈	—	118	2	no	—
90	#376	C₂×D₅₉	1	1	yes	characteristic
91	#339	—	2	2	no	—
92	#334	—	118	4	no	—
93	Q₈⋊C₈	—	118	4	no	—
94	C₅₉	#10670	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	2	1	no	—
96	C₂²×C₈	—	118	2	no	—
97	C₄⋊Q₈	Dic₁₁₈	1	2	yes	—
98	#125	—	2	1	no	—
99	C₂²⋊Q₈	—	2	2	no	—
100	C₄×Q₈	—	2	2	no	—
101	#117	—	2	1	no	—
102	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
103	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	118	2	no	—
106	C₄⋊C₈	—	118	2	no	—
107	C₄⋊C₈	—	118	2	no	—
108	C₄⋊C₈	—	236	4	no	—
109	C₂²⋊C₈	—	118	2	no	—
110	C₄×C₈	—	59	4	no	—
111	#95	—	2	2	no	—
112	C₂×Q₈	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	#11745687	1	2	yes	—
115	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
116	M₁₆	—	4	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	118	4	no	—
119	C₂×C₈	—	236	2	no	—
120	C₂×C₈	—	236	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	—	4	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	4	1	no	—
127	C₄²	#11745671	1	4	yes	—
128	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
129	C₄²	#11745685	1	1	yes	characteristic
130	C₄²	D₄×D₅₉	1	2	yes	—
131	C₂³	#14273708	1	1	yes	characteristic
132	Q₈	—	4	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	#14273704	1	2	yes	—
138	C₂×C₄	#14273706	1	1	yes	characteristic
139	C₂×C₄	#14273709	1	2	yes	—
140	C₂×C₄	#14273723	1	1	yes	characteristic
141	C₂×C₄	C₂×#11745671	1	2	yes	—
142	C₂×C₄	#14273757	1	2	yes	—
143	C₈	—	8	1	no	—
144	C₈	—	472	2	no	—
145	C₈	—	236	4	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#16837338	1	1	yes	characteristic
148	C₂²	#16837730	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	8	2	no	—
153	C₄	#16837687	1	2	yes	—
154	C₄	#16837776	1	1	yes	characteristic
155	C₄	#16837797	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#21425639	1	1	yes	characteristic
158	C₂	#21425651	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424250

C₂

yes

maximal, characteristic

#21422868

C₂

yes

maximal

#21422499

C₂

yes

maximal

#21421925

C₂

yes

maximal, characteristic

C₅₉×#1995

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

C₂×C₄⋊(C₅₉⋊C₈)

—

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#16837152

—

Q₈⋊(C₅₉⋊C₈)

—

C₅₉×#533

C₂²

yes

characteristic

C₅₉×#423

C₂²

yes

characteristic

C₅₉×#376

C₂²

yes

characteristic

C₅₉×#339

—

C₂²×C₅₉⋊C₈

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₂×M₁₆×C₅₉

—

C₄⋊Q₈×C₅₉

Q₈

yes

—

#125×C₅₉

—

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

#117×C₅₉

—

C₂×C₄²×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

#95×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

#41

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#123

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#10670

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#399

yes

characteristic

C₂²×C₅₉

#483

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#475

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

D₅₉

yes

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

#2096

yes

characteristic

C₂×C₅₉

#2098

yes

—

#533

C₂×D₅₉

yes

characteristic

#431

—

#423

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

118

—

#376

C₂×D₅₉

yes

characteristic

#339

—

#334

—

118

—

Q₈⋊C₈

—

118

—

C₅₉

#10670

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

118

—

C₄⋊Q₈

Dic₁₁₈

yes

—

#125

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

#117

—

102

C₂×C₄²

C₂²⋊D₅₉

yes

—

103

C₂×C₄²

C₂²×D₅₉

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

118

—

106

C₄⋊C₈

—

118

—

107

C₄⋊C₈

—

118

—

108

C₄⋊C₈

—

236

—

109

C₂²⋊C₈

—

118

—

110

C₄×C₈

—

111

#95

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

#11745687

yes

—

115

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

118

—

119

C₂×C₈

—

236

—

120

C₂×C₈

—

236

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

#11745671

yes

—

128

C₄²

C₂×Dic₁₁₈

yes

characteristic

129

C₄²

#11745685

yes

characteristic

130

C₄²

D₄×D₅₉

yes

—

131

C₂³

#14273708

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#14273704

yes

—

138

C₂×C₄

#14273706

yes

characteristic

139

C₂×C₄

#14273709

yes

—

140

C₂×C₄

#14273723

yes

characteristic

141

C₂×C₄

C₂×#11745671

yes

—

142

C₂×C₄

#14273757

yes

—

143

C₈

—

144

C₈

—

472

—

145

C₈

—

236

—

146

C₂²

—

147

C₂²

#16837338

yes

characteristic

148

C₂²

#16837730

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#16837687

yes

—

154

C₄

#16837776

yes

characteristic

155

C₄

#16837797

yes

characteristic

156

C₂

—

157

C₂

#21425639

yes

characteristic

158

C₂

#21425651

yes

—

	subgroup	G/H
center	C₂×C₄	#14273723
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1995	C₂
socle	C₂²×C₅₉	#399
prime radical	C₂³×C₅₉	#123
hypercenter	#1995	D₅₉
hypocenter	C₅₉	#10670
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

subgroup

G/H

center

C₂×C₄

#14273723

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1995

C₂

C₂²×C₅₉

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

N	H ≅ G/N	type
94. C₅₉	#10670	semidirect

H ≅ G/N

type

94. C₅₉

#10670

semidirect

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

subgroup	index
#21424250	2
#21422868	2
#21422499	2
#21421925	2
C₅₉×#1995	2
#10670	59

subgroup

index

C₅₉×#1995

Subgroups — #28493724

Conjugacy classes of subgroups

Complements

Maximal subgroups