Group #28504104 — subgroups

Subgroups — #28504104

Order 15104.

306 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16837231	C₂	1	2	yes	maximal
2	C₂×#16837177	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₅₉	C₂	1	2	yes	maximal
4	#21422190	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14273663	C₂²	1	2	yes	—
8	M₁₆×Dic₅₉	—	2	2	no	—
9	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
10	#16837231	C₄	1	8	yes	complemented
11	#16837177	C₄	1	4	yes	—
12	#16837166	—	2	2	no	—
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×#108×C₅₉	C₂²	1	2	yes	—
15	C₂×#11745664	—	2	2	no	—
16	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
18	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
20	#14273663	—	2	2	no	—
21	#14273663	C₂×C₄	1	4	yes	—
22	#14273637	—	4	2	no	—
23	C₈×Dic₅₉	—	4	2	no	—
24	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
25	C₈×Dic₅₉	Q₈	1	2	yes	—
26	C₂×M₁₆×C₅₉	—	2	2	no	—
27	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₅₉	—	2	2	no	—
29	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
30	#108×C₅₉	—	2	2	no	—
31	#108×C₅₉	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₅₉	Q₁₆	1	2	yes	—
34	M₁₆×C₅₉	—	2	4	no	—
35	M₁₆×C₅₉	—	4	2	no	—
36	C₂×C₈×C₅₉	—	4	2	no	—
37	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
39	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₅₉	#41	1	2	yes	—
41	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
42	C₄×Dic₅₉	D₈	1	2	yes	—
43	#11745664	—	2	4	no	—
44	#11745664	—	4	2	no	—
45	C₂×C₅₉⋊C₈	—	2	2	no	—
46	C₂×C₅₉⋊C₈	—	4	2	no	—
47	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
48	C₁₆×C₅₉	—	2	2	no	—
49	C₅₉⋊C₁₆	C₄²	1	4	yes	—
50	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
53	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₅₉	#135	1	2	yes	—
55	C₂×Dic₅₉	—	4	2	no	—
56	C₂×Dic₅₉	#107	1	4	yes	—
57	C₈×C₅₉	—	4	4	no	—
58	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
59	C₈×C₅₉	#117	1	1	yes	characteristic
60	C₈×C₅₉	C₄×Q₈	1	2	yes	—
61	C₅₉⋊C₈	—	2	2	no	—
62	C₅₉⋊C₈	—	4	4	no	—
63	#12676	—	59	1	no	maximal, Sylow
64	C₂²×C₅₉	—	2	2	no	—
65	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
66	C₂²×C₅₉	C₄×Q₁₆	1	2	yes	—
67	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
68	C₄×C₅₉	#427	1	1	yes	characteristic
69	C₄×C₅₉	C₄×D₈	1	2	yes	—
70	Dic₅₉	—	2	4	no	—
71	C₂×#368	—	59	2	no	—
72	C₂×#346	—	59	1	no	—
73	C₄×#108	—	59	2	no	—
74	#1381	—	59	1	no	—
75	#1381	D₅₉	1	1	yes	characteristic
76	C₂×C₅₉	—	2	2	no	—
77	C₂×C₅₉	C₄×#107	1	1	yes	characteristic
78	C₂×C₅₉	#2161	1	2	yes	—
79	C₂²×C₁₆	—	59	1	no	—
80	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
81	C₂×#108	—	59	2	no	—
82	C₂×#108	C₂×D₅₉	1	2	yes	—
83	C₄×M₁₆	—	118	2	no	—
84	C₂×C₄×C₈	—	59	1	no	—
85	#368	—	59	8	no	—
86	#346	—	59	4	no	—
87	#341	—	118	2	no	—
88	C₅₉	#12676	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	118	2	no	—
91	C₂²×C₈	—	59	1	no	—
92	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
93	C₂×C₄²	—	59	1	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	59	2	no	—
96	C₂×C₁₆	—	59	4	no	—
97	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
98	#108	—	2	2	no	—
99	#108	—	59	4	no	—
100	#108	—	118	2	no	—
101	#108	C₄×D₅₉	1	4	yes	—
102	#97	—	236	2	no	—
103	C₄×C₈	—	59	2	no	—
104	C₄×C₈	—	59	2	no	—
105	C₄×C₈	—	236	2	no	—
106	C₂²×C₄	—	59	2	no	—
107	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
108	M₁₆	—	2	4	no	—
109	M₁₆	—	4	2	no	—
110	M₁₆	—	118	4	no	—
111	M₁₆	—	236	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	59	2	no	—
114	C₂×C₈	—	118	2	no	—
115	C₂×C₈	—	236	2	no	—
116	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
117	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
118	C₂×C₈	#11745687	1	2	yes	—
119	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
120	C₄²	—	59	2	no	—
121	C₄²	—	59	2	no	—
122	C₁₆	—	2	2	no	—
123	C₁₆	—	59	4	no	—
124	C₂³	Q₁₆×D₅₉	1	1	yes	characteristic
125	C₂×C₄	—	59	4	no	—
126	C₂×C₄	—	236	2	no	—
127	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
128	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	2	yes	—
129	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
130	C₂×C₄	#14273748	1	2	yes	—
131	C₈	—	4	4	no	—
132	C₈	—	118	2	no	—
133	C₈	—	236	4	no	—
134	C₈	C₄²×D₅₉	1	1	yes	characteristic
135	C₈	#14273683	1	1	yes	characteristic
136	C₈	#14273692	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	#107×D₅₉	1	1	yes	characteristic
139	C₂²	#16837494	1	2	yes	—
140	C₄	—	118	4	no	—
141	C₄	#16837253	1	1	yes	characteristic
142	C₄	#16837437	1	1	yes	characteristic
143	C₄	#16837482	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#21423988	1	1	yes	characteristic
146	C₂	#21425889	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837231

C₂

yes

maximal

C₂×#16837177

C₂

yes

maximal, characteristic

#108×Dic₅₉

C₂

yes

maximal

#21422190

C₂

yes

maximal, characteristic

C₅₉×#1381

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×#14273663

C₂²

yes

—

M₁₆×Dic₅₉

—

C₂×C₈×Dic₅₉

C₂²

yes

characteristic

#16837231

C₄

yes

complemented

#16837177

C₄

yes

—

#16837166

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#108×C₅₉

C₂²

yes

—

C₂×#11745664

—

C₂²×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

#14273663

—

#14273663

C₂×C₄

yes

—

#14273637

—

C₈×Dic₅₉

—

C₈×Dic₅₉

C₂×C₄

yes

—

C₈×Dic₅₉

Q₈

yes

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

#108×C₅₉

—

#108×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

Q₁₆

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₄×Dic₅₉

C₄⋊C₄

yes

—

C₄×Dic₅₉

D₈

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

C₄²

yes

—

C₂³×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

#107

yes

—

C₈×C₅₉

—

C₈×C₅₉

C₂×C₄²

yes

characteristic

C₈×C₅₉

#117

yes

characteristic

C₈×C₅₉

C₄×Q₈

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#12676

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#107

yes

characteristic

C₂²×C₅₉

C₄×Q₁₆

yes

—

C₄×C₅₉

C₄×C₄⋊C₄

yes

characteristic

C₄×C₅₉

#427

yes

characteristic

C₄×C₅₉

C₄×D₈

yes

—

Dic₅₉

—

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

C₄×#107

yes

characteristic

C₂×C₅₉

#2161

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₅₉

yes

—

C₄×M₁₆

—

118

—

C₂×C₄×C₈

—

#368

—

#346

—

#341

—

118

—

C₅₉

#12676

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

118

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₉

yes

—

#108

—

#108

—

100

#108

—

118

—

101

#108

C₄×D₅₉

yes

—

102

#97

—

236

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

236

—

106

C₂²×C₄

—

107

C₂²×C₄

D₄×D₅₉

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

118

—

111

M₁₆

—

236

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

118

—

115

C₂×C₈

—

236

—

116

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

117

C₂×C₈

C₂×C₄×D₅₉

yes

—

118

C₂×C₈

#11745687

yes

—

119

C₂×C₈

Q₈×D₅₉

yes

characteristic

120

C₄²

—

121

C₄²

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

Q₁₆×D₅₉

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

236

—

127

C₂×C₄

C₄⋊C₄×D₅₉

yes

characteristic

128

C₂×C₄

(C₂²⋊D₅₉)⋊C₄

yes

—

129

C₂×C₄

D₈×D₅₉

yes

characteristic

130

C₂×C₄

#14273748

yes

—

131

C₈

—

132

C₈

—

118

—

133

C₈

—

236

—

134

C₈

C₄²×D₅₉

yes

characteristic

135

C₈

#14273683

yes

characteristic

136

C₈

#14273692

yes

—

137

C₂²

—

138

C₂²

#107×D₅₉

yes

characteristic

139

C₂²

#16837494

yes

—

140

C₄

—

118

—

141

C₄

#16837253

yes

characteristic

142

C₄

#16837437

yes

characteristic

143

C₄

#16837482

yes

—

144

C₂

—

145

C₂

#21423988

yes

characteristic

146

C₂

#21425889

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1381	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1381	D₅₉
hypocenter	C₅₉	#12676
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₅₉

commutator

C₈×C₅₉

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1381

C₂

socle

C₂²×C₅₉

C₂×#107

prime radical

C₂³×C₅₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
10. #16837231	C₄	semidirect
88. C₅₉	#12676	semidirect

H ≅ G/N

type

10. #16837231

C₄

semidirect

88. C₅₉

#12676

semidirect

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

subgroup	index
C₂×#16837231	2
C₂×#16837177	2
#108×Dic₅₉	2
#21422190	2
C₅₉×#1381	2
#12676	59

subgroup

index

#108×Dic₅₉

C₅₉×#1381

Subgroups — #28504104

Conjugacy classes of subgroups

Complements

Maximal subgroups