Group #28475772 — subgroups

Subgroups — #28475772

Order 15104.

264 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 83 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	#21424134	C₂	1	2	yes	maximal
2	C₅₉×#2112	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273663	—	2	2	no	—
4	#16837366	—	2	2	no	—
5	#16837167	C₄	1	4	yes	—
6	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₅₉	C₄	1	2	yes	—
8	C₅₉×C₂²⋊C₁₆	—	2	2	no	—
9	C₂×#11745664	—	2	4	no	—
10	#14273663	—	4	4	no	—
11	C₂²⋊(C₅₉⋊C₈)	—	4	2	no	—
12	C₂²⋊(C₅₉⋊C₈)	—	4	2	no	—
13	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₅₉	—	4	2	no	—
15	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₉	D₄	1	2	yes	—
18	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
19	M₃₂×C₅₉	—	4	2	no	—
20	C₂×C₁₆×C₅₉	—	2	2	no	—
21	C₂⁴×C₅₉	Q₁₆	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	—	4	1	no	—
23	C₂²×C₄×C₅₉	—	4	1	no	—
24	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
25	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
27	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
28	C₂×C₈×C₅₉	—	2	2	no	—
29	C₂×C₈×C₅₉	—	2	4	no	—
30	C₂×C₈×C₅₉	—	4	2	no	—
31	C₂×C₈×C₅₉	—	4	1	no	—
32	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
35	#11745664	—	4	4	no	—
36	#11745664	—	4	4	no	—
37	C₂×C₅₉⋊C₈	—	4	4	no	—
38	C₁₆×C₅₉	—	4	2	no	—
39	C₂³×C₅₉	—	4	1	no	—
40	C₂³×C₅₉	—	4	2	no	—
41	C₂³×C₅₉	#103	1	2	yes	—
42	C₂³×C₅₉	#107	1	1	yes	characteristic
43	C₂×C₄×C₅₉	—	2	2	no	—
44	C₂×C₄×C₅₉	—	2	1	no	—
45	C₂×C₄×C₅₉	—	2	2	no	—
46	C₂×C₄×C₅₉	—	2	1	no	—
47	C₂×C₄×C₅₉	—	4	1	no	—
48	C₂×C₄×C₅₉	—	4	2	no	—
49	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
50	C₂×C₄×C₅₉	#102	1	2	yes	—
51	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
52	C₈×C₅₉	—	2	1	no	—
53	C₈×C₅₉	—	2	2	no	—
54	C₈×C₅₉	#99	1	2	yes	—
55	C₅₉⋊C₈	—	8	4	no	—
56	#7234	—	59	1	no	maximal, Sylow
57	C₂²×C₅₉	—	2	2	no	—
58	C₂²×C₅₉	—	2	1	no	—
59	C₂²×C₅₉	—	2	4	no	—
60	C₂²×C₅₉	—	2	2	no	—
61	C₂²×C₅₉	—	4	4	no	—
62	C₂²×C₅₉	#340	1	1	yes	characteristic
63	C₄×C₅₉	—	2	2	no	—
64	C₄×C₅₉	—	2	1	no	—
65	C₄×C₅₉	H(3,ℤ/4)	1	1	yes	characteristic
66	C₄×C₅₉	#342	1	1	yes	characteristic
67	#2112	D₅₉	1	1	yes	characteristic
68	#1810	—	59	2	no	—
69	C₂×C₅₉	—	4	2	no	—
70	C₂×C₅₉	—	4	1	no	—
71	C₂×C₅₉	#1289	1	1	yes	characteristic
72	C₂×C₅₉	#1388	1	2	yes	—
73	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
74	C₂×M₃₂	Dic₅₉	1	2	yes	—
75	C₂×#108	—	118	2	no	—
76	#407	—	118	2	no	—
77	C₂²⋊C₁₆	—	2	2	no	—
78	#341	—	59	4	no	—
79	C₅₉	#7234	1	1	yes	characteristic, complemented, Sylow
80	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
81	C₂×M₁₆	—	118	4	no	—
82	C₂²×C₈	—	4	2	no	—
83	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
84	C₂²×C₈	C₄×D₅₉	1	2	yes	—
85	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
86	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
87	M₃₂	—	4	2	no	—
88	C₂×C₁₆	—	2	2	no	—
89	#108	—	236	4	no	—
90	C₂²⋊C₈	—	236	2	no	—
91	C₂²⋊C₈	—	236	2	no	—
92	C₂⁴	Dic₂₃₆	1	1	yes	characteristic
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	#11745660	1	2	yes	—
96	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
97	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
98	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
99	M₁₆	—	236	4	no	—
100	M₁₆	—	236	4	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	236	4	no	—
106	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
107	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
108	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
109	C₁₆	—	4	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	2	no	—
112	C₂³	#14273652	1	2	yes	—
113	C₂³	#14273661	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
121	C₂×C₄	#14273649	1	2	yes	—
122	C₂×C₄	#14273659	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	2	no	—
125	C₈	—	472	4	no	—
126	C₈	#14273642	1	2	yes	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	4	4	no	—
132	C₂²	#16837165	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	#16837159	1	1	yes	characteristic
136	C₄	#16837169	1	1	yes	characteristic
137	C₂	—	4	2	no	—
138	C₂	—	4	1	no	—
139	C₂	#21421964	1	1	yes	characteristic
140	C₂	#21422208	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424134

C₂

yes

maximal

C₅₉×#2112

C₂

yes

maximal, characteristic

C₂×#14273663

—

#16837366

—

#16837167

C₄

yes

—

C₂³×C₈×C₅₉

C₂²

yes

characteristic

C₂×M₃₂×C₅₉

C₄

yes

—

C₅₉×C₂²⋊C₁₆

—

C₂×#11745664

—

#14273663

—

C₂²⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₂³×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

Q₈

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂⁴×C₅₉

Q₁₆

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₉

D₈

yes

characteristic

C₂²×C₄×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄²

yes

characteristic

C₂×C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

#103

yes

—

C₂³×C₅₉

#107

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#95

yes

characteristic

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

#106

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#99

yes

—

C₅₉⋊C₈

—

#7234

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#340

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

D₅₉

yes

characteristic

#1810

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1289

yes

characteristic

C₂×C₅₉

#1388

yes

—

C₂³×C₈

C₂×D₅₉

yes

characteristic

C₂×M₃₂

Dic₅₉

yes

—

C₂×#108

—

118

—

#407

—

118

—

C₂²⋊C₁₆

—

#341

—

C₅₉

#7234

yes

characteristic, complemented, Sylow

C₂³×C₄

D₂₃₆

yes

characteristic

C₂×M₁₆

—

118

—

C₂²×C₈

—

C₂²×C₈

Dic₁₁₈

yes

characteristic

C₂²×C₈

C₄×D₅₉

yes

—

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

C₂²×C₈

C₂²⋊D₅₉

yes

—

M₃₂

—

C₂×C₁₆

—

#108

—

236

—

C₂²⋊C₈

—

236

—

C₂²⋊C₈

—

236

—

C₂⁴

Dic₂₃₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#11745660

yes

—

C₂²×C₄

D₄₇₂

yes

characteristic

C₂²×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂²×C₄

(C₂×D₅₉)⋊C₄

yes

—

M₁₆

—

236

—

100

M₁₆

—

236

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

236

—

106

C₂×C₈

C₄×Dic₅₉

yes

characteristic

107

C₂×C₈

Dic₅₉⋊C₄

yes

—

108

C₂×C₈

C₂²⋊Dic₅₉

yes

characteristic

109

C₁₆

—

110

C₂³

—

111

C₂³

—

112

C₂³

#14273652

yes

—

113

C₂³

#14273661

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₅₉⋊#95

yes

characteristic

121

C₂×C₄

#14273649

yes

—

122

C₂×C₄

#14273659

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

472

—

126

C₈

#14273642

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#16837165

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

#16837159

yes

characteristic

136

C₄

#16837169

yes

characteristic

137

C₂

—

138

C₂

—

139

C₂

#21421964

yes

characteristic

140

C₂

#21422208

yes

—

	subgroup	G/H
center	C₂²	#16837165
commutator	C₂×C₈×C₅₉	C₄²
Frattini	C₂³×C₈	C₂×D₅₉
Fitting	C₅₉×#2112	C₂
socle	C₂²×C₅₉	#340
prime radical	C₂⁴×C₅₉	Q₁₆
cosocle	C₂³×C₈×C₅₉	C₂²
hypercenter	#2112	D₅₉
hypocenter	C₅₉	#7234
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #2112

subgroup

G/H

center

C₂²

#16837165

commutator

C₂×C₈×C₅₉

C₄²

Frattini

C₂³×C₈

C₂×D₅₉

Fitting

C₅₉×#2112

C₂

socle

C₂²×C₅₉

#340

prime radical

C₂⁴×C₅₉

Q₁₆

cosocle

C₂³×C₈×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #2112

N	H ≅ G/N	type
79. C₅₉	#7234	semidirect

H ≅ G/N

type

79. C₅₉

#7234

semidirect

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

subgroup	index
#21424134	2
C₅₉×#2112	2
#7234	59

subgroup

index

#21424134

C₅₉×#2112

#7234

Subgroups — #28475772

Conjugacy classes of subgroups

Complements

Maximal subgroups