Group #28503036 — subgroups

Subgroups — #28503036

Order 15104.

206 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 71 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	#21426371	C₂	1	1	yes	maximal, characteristic, complemented
2	#21423495	C₂	1	2	yes	maximal
3	#21422170	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422153	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2169	C₂	1	1	yes	maximal, characteristic
6	#16837807	C₂²	1	2	yes	—
7	#16837692	—	2	2	no	—
8	#16837228	—	2	1	no	—
9	#16837228	—	2	1	no	—
10	#16837155	C₂²	1	2	yes	—
11	#16837150	—	2	2	no	—
12	C₂×M₃₂×C₅₉	—	2	1	no	—
13	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₅₉×C₂²⋊C₁₆	—	2	1	no	—
16	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	#14273727	—	2	2	no	—
18	#14273661	—	2	2	no	—
19	#14273661	—	2	2	no	—
20	#14273658	—	4	2	no	—
21	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
22	#14273648	—	4	2	no	—
23	C₂²×C₈×C₅₉	—	2	1	no	—
24	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
25	M₃₂×C₅₉	—	4	2	no	—
26	C₂×C₁₆×C₅₉	—	2	1	no	—
27	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
30	C₂²⋊C₈×C₅₉	—	2	1	no	—
31	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	—	2	1	no	—
34	C₂×Dic₁₁₈	—	4	2	no	—
35	C₂×C₈×C₅₉	—	2	1	no	—
36	C₂×C₈×C₅₉	—	4	1	no	—
37	C₂×C₈×C₅₉	—	4	1	no	—
38	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
41	C₂²⋊C₄×C₅₉	—	2	1	no	—
42	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₅₉	—	4	2	no	—
44	C₄⋊Dic₅₉	—	4	2	no	—
45	C₂×C₅₉⋊C₈	—	4	2	no	—
46	C₁₆×C₅₉	—	2	1	no	—
47	C₁₆×C₅₉	—	4	1	no	—
48	C₁₆×C₅₉	SD₁₆	1	2	yes	—
49	C₂³×C₅₉	—	2	1	no	—
50	D₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	—	2	1	no	—
52	C₂×C₄×C₅₉	—	4	1	no	—
53	C₂×C₄×C₅₉	—	4	1	no	—
54	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
57	C₂×Dic₅₉	—	8	2	no	—
58	Dic₁₁₈	—	16	2	no	—
59	C₈×C₅₉	—	2	1	no	—
60	C₈×C₅₉	—	4	1	no	—
61	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₈×C₅₉	#137	1	1	yes	characteristic
63	C₅₉⋊C₈	—	8	2	no	—
64	#12455	—	59	1	no	maximal, Sylow
65	C₂²×C₅₉	—	4	1	no	—
66	C₂²×C₅₉	—	4	1	no	—
67	C₂²×C₅₉	—	4	1	no	—
68	C₂²×C₅₉	#466	1	1	yes	characteristic
69	C₄×C₅₉	—	2	1	no	—
70	C₄×C₅₉	—	4	1	no	—
71	C₄×C₅₉	#461	1	1	yes	characteristic
72	C₄×C₅₉	#473	1	1	yes	characteristic
73	C₄×C₅₉	#510	1	2	yes	—
74	Dic₅₉	—	16	2	no	—
75	#2252	—	59	1	no	—
76	#2169	D₅₉	1	1	yes	characteristic
77	#1675	—	59	2	no	—
78	#1372	—	59	1	no	—
79	#1365	—	59	2	no	—
80	C₂×C₅₉	—	8	1	no	—
81	C₂×C₅₉	#1673	1	1	yes	characteristic
82	C₂×C₅₉	#2219	1	1	yes	characteristic
83	C₂×C₅₉	#2221	1	1	yes	characteristic
84	C₂×M₃₂	—	2	1	no	—
85	#500	—	59	2	no	—
86	#476	—	118	2	no	—
87	C₈×D₄	C₂×D₅₉	1	1	yes	characteristic
88	#366	—	118	1	no	—
89	#366	—	118	1	no	—
90	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
91	C₂²⋊C₁₆	—	2	1	no	—
92	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
93	#335	—	59	2	no	—
94	#333	—	118	2	no	—
95	C₅₉	#12455	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	2	1	no	—
97	C₄⋊Q₈	—	118	2	no	—
98	C₄×D₄	D₂₃₆	1	1	yes	characteristic
99	M₃₂	—	4	2	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
103	#107	—	118	2	no	—
104	#107	—	118	2	no	—
105	#106	—	236	2	no	—
106	C₄⋊C₈	—	118	2	no	—
107	C₄⋊C₈	D₂₃₆	1	1	yes	characteristic
108	#102	—	236	2	no	—
109	C₂²⋊C₈	—	2	1	no	—
110	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
111	C₂×Q₈	—	236	2	no	—
112	C₂×D₄	D₄₇₂	1	1	yes	characteristic
113	C₂²×C₄	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	236	2	no	—
118	C₂×C₈	#11745685	1	1	yes	characteristic
119	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
120	C₄⋊C₄	—	236	2	no	—
121	C₄⋊C₄	—	236	2	no	—
122	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
123	C₂²⋊C₄	—	2	1	no	—
124	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	4	1	no	—
127	C₁₆	#11745670	1	2	yes	—
128	C₂³	—	2	1	no	—
129	Q₈	—	944	2	no	—
130	D₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	472	2	no	—
135	C₂×C₄	#14273701	1	1	yes	characteristic
136	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
137	C₂×C₄	#14273749	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	4	1	no	—
140	C₈	—	472	2	no	—
141	C₈	C₂×#11745670	1	1	yes	characteristic
142	C₈	#14273757	1	1	yes	characteristic
143	C₂²	—	4	1	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	#16837638	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	944	2	no	—
150	C₄	#16837607	1	1	yes	characteristic
151	C₄	#16837677	1	1	yes	characteristic
152	C₄	#16837845	1	2	yes	—
153	C₂	—	8	1	no	—
154	C₂	#21423485	1	1	yes	characteristic
155	C₂	#21426204	1	1	yes	characteristic
156	C₂	#21426216	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426371

C₂

yes

maximal, characteristic, complemented

#21423495

C₂

yes

maximal

#21422170

C₂

yes

maximal, characteristic, complemented

#21422153

C₂

yes

maximal, complemented

C₅₉×#2169

C₂

yes

maximal, characteristic

#16837807

C₂²

yes

—

—

—

—

C₂²

yes

—

#16837150

—

C₂×M₃₂×C₅₉

—

C₈×D₄×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

C₂²

yes

characteristic

C₅₉×C₂²⋊C₁₆

—

C₄×C₁₆×C₅₉

C₂²

yes

—

—

—

—

C₄⋊(C₅₉⋊C₈)

—

#14273648

—

C₂²×C₈×C₅₉

—

C₄×D₄×C₅₉

D₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₄⋊C₈×C₅₉

D₄

yes

characteristic

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×D₄×C₅₉

D₈

yes

characteristic

C₂²×C₄×C₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄⋊C₄×C₅₉

D₈

yes

characteristic

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

SD₁₆

yes

—

C₂³×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×SD₁₆

yes

characteristic

C₈×C₅₉

#137

yes

characteristic

C₅₉⋊C₈

—

#12455

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#466

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

#2252

—

#2169

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×M₃₂

—

#500

—

#476

—

118

—

C₈×D₄

C₂×D₅₉

yes

characteristic

#366

—

118

—

#366

—

118

—

C₄⋊C₁₆

C₂×D₅₉

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

#335

—

#333

—

118

—

C₅₉

#12455

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

—

118

—

C₄×D₄

D₂₃₆

yes

characteristic

M₃₂

—

100

C₂×C₁₆

—

101

C₂×C₁₆

C₂²⋊D₅₉

yes

characteristic

102

C₂×C₁₆

C₂²⋊D₅₉

yes

characteristic

103

#107

—

118

—

104

#107

—

118

—

105

#106

—

236

—

106

C₄⋊C₈

—

118

—

107

C₄⋊C₈

D₂₃₆

yes

characteristic

108

#102

—

236

—

109

C₂²⋊C₈

—

110

C₄×C₈

C₂²×D₅₉

yes

characteristic

111

C₂×Q₈

—

236

—

112

C₂×D₄

D₄₇₂

yes

characteristic

113

C₂²×C₄

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

236

—

118

C₂×C₈

#11745685

yes

characteristic

119

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

120

C₄⋊C₄

—

236

—

121

C₄⋊C₄

—

236

—

122

C₄⋊C₄

D₄₇₂

yes

characteristic

123

C₂²⋊C₄

—

124

C₄²

C₂×D₂₃₆

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₁₆

#11745670

yes

—

128

C₂³

—

129

Q₈

—

944

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

472

—

135

C₂×C₄

#14273701

yes

characteristic

136

C₂×C₄

C₂×D₄₇₂

yes

characteristic

137

C₂×C₄

#14273749

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

472

—

141

C₈

C₂×#11745670

yes

characteristic

142

C₈

#14273757

yes

characteristic

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#16837638

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

944

—

150

C₄

#16837607

yes

characteristic

151

C₄

#16837677

yes

characteristic

152

C₄

#16837845

yes

—

153

C₂

—

154

C₂

#21423485

yes

characteristic

155

C₂

#21426204

yes

characteristic

156

C₂

#21426216

yes

characteristic

	subgroup	G/H
center	C₂²	#16837638
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#2169	C₂
socle	C₂²×C₅₉	#466
prime radical	C₂×D₄×C₅₉	D₈
hypercenter	#2169	D₅₉
hypocenter	C₅₉	#12455
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169

subgroup

G/H

center

C₂²

#16837638

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#2169

C₂

socle

C₂²×C₅₉

#466

prime radical

C₂×D₄×C₅₉

D₈

D₅₉

C₅₉

G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169

N	H ≅ G/N	type
1. #21426371	C₂	semidirect
3. #21422170	C₂	semidirect
4. #21422153	C₂	semidirect
95. C₅₉	#12455	semidirect

H ≅ G/N

type

1. #21426371

C₂

semidirect

3. #21422170

C₂

semidirect

4. #21422153

C₂

semidirect

95. C₅₉

#12455

semidirect

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

subgroup	index
#21426371	2
#21423495	2
#21422170	2
#21422153	2
C₅₉×#2169	2
#12455	59

subgroup

index

C₅₉×#2169

Subgroups — #28503036

Conjugacy classes of subgroups

Complements

Maximal subgroups