Group #28503066 — subgroups

Subgroups — #28503066

Order 15104.

200 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 67 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
2	#21426004	C₂	1	1	yes	maximal, characteristic
3	#21423491	C₂	1	2	yes	maximal
4	#21422152	C₂	1	2	yes	maximal
5	C₅₉×#1372	C₂	1	1	yes	maximal, characteristic
6	#16837807	C₂²	1	2	yes	—
7	#16837613	—	2	2	no	—
8	C₈×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	#16837228	—	2	1	no	—
10	#16837228	—	2	1	no	—
11	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
12	Dic₅₉⋊C₁₆	—	2	1	no	—
13	#16837177	—	2	1	no	—
14	#16837149	—	2	2	no	—
15	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#335	C₂²	1	2	yes	—
17	C₂×Dic₂₃₆	—	4	2	no	—
18	#14273727	—	2	2	no	—
19	C₄×Dic₁₁₈	D₄	1	1	yes	characteristic
20	C₂×C₅₉⋊C₁₆	—	2	1	no	—
21	#14273661	—	2	2	no	—
22	#14273661	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
24	Dic₅₉⋊C₈	—	2	1	no	—
25	#14273652	—	4	2	no	—
26	C₈×Dic₅₉	—	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	—	2	2	no	—
30	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₁₈	—	4	2	no	—
32	C₂×Dic₁₁₈	D₈	1	1	yes	characteristic
33	C₂×C₈×C₅₉	—	4	2	no	—
34	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
36	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₅₉	—	4	2	no	—
38	C₄⋊Dic₅₉	—	4	2	no	—
39	C₄⋊Dic₅₉	D₈	1	1	yes	characteristic
40	Dic₅₉⋊C₄	—	2	1	no	—
41	C₄×Dic₅₉	—	2	1	no	—
42	C₂×C₅₉⋊C₈	—	2	1	no	—
43	Dic₂₃₆	—	8	4	no	—
44	C₁₆×C₅₉	—	2	1	no	—
45	C₁₆×C₅₉	Q₁₆	1	2	yes	complemented
46	C₅₉⋊C₁₆	—	4	1	no	—
47	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
50	C₂×Dic₅₉	—	2	1	no	—
51	C₂×Dic₅₉	—	2	1	no	—
52	C₂×Dic₅₉	—	8	2	no	—
53	Dic₁₁₈	—	2	2	no	—
54	Dic₁₁₈	—	8	2	no	—
55	Dic₁₁₈	—	8	2	no	—
56	C₈×C₅₉	—	2	1	no	—
57	C₈×C₅₉	—	8	2	no	—
58	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₅₉	#137	1	1	yes	characteristic
60	C₅₉⋊C₈	—	4	1	no	—
61	#12460	—	59	1	no	maximal, Sylow
62	C₂²×C₅₉	#466	1	1	yes	characteristic
63	C₄×C₅₉	—	2	1	no	—
64	C₄×C₅₉	#462	1	1	yes	characteristic
65	C₄×C₅₉	#473	1	1	yes	characteristic
66	C₄×C₅₉	#509	1	2	yes	—
67	Dic₅₉	—	4	1	no	—
68	Dic₅₉	—	4	1	no	—
69	Dic₅₉	—	16	2	no	—
70	#2252	—	59	1	no	—
71	#2183	—	59	1	no	—
72	#1674	—	59	2	no	—
73	#1372	D₅₉	1	1	yes	characteristic
74	#1364	—	59	2	no	—
75	C₂×C₅₉	#1674	1	1	yes	characteristic
76	C₂×C₅₉	#2218	1	1	yes	characteristic
77	C₂×C₅₉	#2222	1	1	yes	characteristic
78	#500	—	59	2	no	—
79	#462	—	118	2	no	—
80	C₈×Q₈	—	59	1	no	—
81	#366	—	118	1	no	—
82	#366	—	118	1	no	—
83	C₄⋊C₁₆	—	59	1	no	—
84	C₄⋊C₁₆	—	118	1	no	—
85	#346	—	118	1	no	—
86	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
87	#335	C₂×D₅₉	1	2	yes	—
88	#332	—	118	2	no	—
89	C₅₉	#12460	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₁₆	—	236	2	no	—
91	C₄⋊Q₈	—	118	2	no	—
92	C₄×Q₈	—	59	1	no	—
93	C₂×C₁₆	—	118	1	no	—
94	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
95	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
96	#107	—	118	2	no	—
97	#107	—	118	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	59	1	no	—
100	C₄⋊C₈	—	118	1	no	—
101	#103	—	236	2	no	—
102	C₄×C₈	—	118	1	no	—
103	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
104	C₂×Q₈	—	59	1	no	—
105	C₂×Q₈	—	236	2	no	—
106	Q₁₆	—	472	4	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	118	1	no	—
109	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
110	C₂×C₈	#11745689	1	1	yes	characteristic
111	C₄⋊C₄	—	59	1	no	—
112	C₄⋊C₄	—	118	1	no	—
113	C₄⋊C₄	—	236	2	no	—
114	C₄⋊C₄	—	236	2	no	—
115	C₄²	—	118	1	no	—
116	C₄²	D₄×D₅₉	1	1	yes	characteristic
117	C₁₆	—	2	1	no	—
118	C₁₆	—	236	1	no	—
119	C₁₆	Dic₂₃₆	1	2	yes	complemented
120	Q₈	—	118	2	no	—
121	Q₈	—	472	2	no	—
122	Q₈	—	472	2	no	—
123	C₂×C₄	—	118	1	no	—
124	C₂×C₄	—	118	1	no	—
125	C₂×C₄	—	472	2	no	—
126	C₂×C₄	#14273702	1	1	yes	characteristic
127	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
128	C₂×C₄	#14273750	1	1	yes	characteristic
129	C₈	—	2	1	no	—
130	C₈	—	8	2	no	—
131	C₈	—	236	1	no	—
132	C₈	C₂×Dic₂₃₆	1	1	yes	characteristic
133	C₈	#14273760	1	1	yes	characteristic
134	C₂²	#16837639	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	944	2	no	—
137	C₄	—	236	1	no	—
138	C₄	—	236	1	no	—
139	C₄	#16837613	1	1	yes	characteristic
140	C₄	#16837678	1	1	yes	characteristic
141	C₄	#16837839	1	2	yes	—
142	C₂	#21423491	1	1	yes	characteristic
143	C₂	#21426200	1	1	yes	characteristic
144	C₂	#21426222	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426371

C₂

yes

maximal, characteristic

#21426004

C₂

yes

maximal, characteristic

#21423491

C₂

yes

maximal

#21422152

C₂

yes

maximal

C₅₉×#1372

C₂

yes

maximal, characteristic

#16837807

C₂²

yes

—

#16837613

—

C₈×Dic₁₁₈

C₂²

yes

characteristic

#16837228

—

#16837228

—

C₄⋊(C₅₉⋊C₁₆)

C₂²

yes

characteristic

Dic₅₉⋊C₁₆

—

#16837177

—

#16837149

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×#335

C₂²

yes

—

C₂×Dic₂₃₆

—

#14273727

—

C₄×Dic₁₁₈

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

#14273661

—

#14273661

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

characteristic

Dic₅₉⋊C₈

—

#14273652

—

C₈×Dic₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₈

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

C₁₆×C₅₉

—

C₁₆×C₅₉

Q₁₆

yes

complemented

C₅₉⋊C₁₆

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₁₆

yes

characteristic

C₈×C₅₉

#137

yes

characteristic

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic

#1364

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

118

—

C₈×Q₈

—

#366

—

118

—

#366

—

118

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

118

—

#346

—

118

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

#335

C₂×D₅₉

yes

—

#332

—

118

—

C₅₉

#12460

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

236

—

C₄⋊Q₈

—

118

—

C₄×Q₈

—

C₂×C₁₆

—

118

—

C₂×C₁₆

D₂₃₆

yes

characteristic

C₂×C₁₆

D₂₃₆

yes

characteristic

#107

—

118

—

#107

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

118

—

101

#103

—

236

—

102

C₄×C₈

—

118

—

103

C₄×C₈

C₂²×D₅₉

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

236

—

106

Q₁₆

—

472

—

107

C₂×C₈

—

108

C₂×C₈

—

118

—

109

C₂×C₈

C₂×D₂₃₆

yes

characteristic

110

C₂×C₈

#11745689

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

118

—

113

C₄⋊C₄

—

236

—

114

C₄⋊C₄

—

236

—

115

C₄²

—

118

—

116

C₄²

D₄×D₅₉

yes

characteristic

117

C₁₆

—

118

C₁₆

—

236

—

119

C₁₆

Dic₂₃₆

yes

complemented

120

Q₈

—

118

—

121

Q₈

—

472

—

122

Q₈

—

472

—

123

C₂×C₄

—

118

—

124

C₂×C₄

—

118

—

125

C₂×C₄

—

472

—

126

C₂×C₄

#14273702

yes

characteristic

127

C₂×C₄

D₈×D₅₉

yes

characteristic

128

C₂×C₄

#14273750

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

236

—

132

C₈

C₂×Dic₂₃₆

yes

characteristic

133

C₈

#14273760

yes

characteristic

134

C₂²

#16837639

yes

characteristic

135

C₄

—

136

C₄

—

944

—

137

C₄

—

236

—

138

C₄

—

236

—

139

C₄

#16837613

yes

characteristic

140

C₄

#16837678

yes

characteristic

141

C₄

#16837839

yes

—

142

C₂

#21423491

yes

characteristic

143

C₂

#21426200

yes

characteristic

144

C₂

#21426222

yes

characteristic

	subgroup	G/H
center	C₂²	#16837639
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1372	C₂
socle, prime radical	C₂²×C₅₉	#466
hypercenter	#1372	D₅₉
hypocenter	C₅₉	#12460
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂²

#16837639

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1372

C₂

C₂²×C₅₉

D₅₉

C₅₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
45. C₁₆×C₅₉	Q₁₆	semidirect
89. C₅₉	#12460	semidirect
119. C₁₆	Dic₂₃₆	semidirect

H ≅ G/N

type

45. C₁₆×C₅₉

Q₁₆

semidirect

89. C₅₉

#12460

semidirect

119. C₁₆

Dic₂₃₆

semidirect

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

subgroup	index
#21426371	2
#21426004	2
#21423491	2
#21422152	2
C₅₉×#1372	2
#12460	59

subgroup

index

C₅₉×#1372

Subgroups — #28503066

Conjugacy classes of subgroups

Complements

Maximal subgroups