Group #28505050 — subgroups

Subgroups — #28505050

Order 15104.

342 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 163 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	#21425852	C₂	1	2	yes	maximal
2	#21422178	C₂	1	4	yes	maximal
3	C₄²×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273667	C₂²	1	4	yes	—
5	C₄⋊(C₅₉⋊C₁₆)	—	2	4	no	—
6	C₂×C₄×C₈×C₅₉	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
10	C₄³×C₅₉	C₄	1	1	yes	characteristic
11	#14273667	—	2	8	no	—
12	C₂×C₅₉⋊C₁₆	—	2	4	no	—
13	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅₉	—	2	2	no	—
17	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₅₉	—	2	4	no	—
20	C₄×C₈×C₅₉	—	2	4	no	—
21	C₄×C₈×C₅₉	C₈	1	2	yes	—
22	C₄×C₈×C₅₉	C₈	1	2	yes	—
23	C₄×C₈×C₅₉	D₄	1	2	yes	—
24	C₄×C₈×C₅₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₅₉	—	2	2	no	—
26	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
27	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₉	—	2	2	no	—
29	C₂×C₈×C₅₉	—	2	2	no	—
30	C₂×C₈×C₅₉	—	2	2	no	—
31	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₅₉	#41	1	4	yes	—
36	C₄²×C₅₉	—	2	8	no	—
37	C₄²×C₅₉	—	2	2	no	—
38	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
39	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₅₉	M₁₆	1	4	yes	—
42	C₅₉⋊C₁₆	—	4	4	no	—
43	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₅₉	—	2	8	no	—
45	C₂×C₄×C₅₉	—	2	4	no	—
46	C₂×C₄×C₅₉	—	2	4	no	—
47	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
48	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
49	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	#117	1	2	yes	—
51	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
53	C₈×C₅₉	—	2	2	no	—
54	C₈×C₅₉	—	2	4	no	—
55	C₈×C₅₉	#124	1	2	yes	—
56	C₈×C₅₉	#125	1	2	yes	—
57	#12883	—	59	1	no	maximal, Sylow
58	C₂²×C₅₉	—	2	2	no	—
59	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₂²×C₅₉	#432	1	2	yes	—
61	C₄×C₅₉	—	2	8	no	—
62	C₄×C₅₉	—	2	2	no	—
63	C₄×C₅₉	#364	1	4	yes	—
64	C₄×C₅₉	#382	1	1	yes	characteristic
65	C₄×C₅₉	#423	1	1	yes	characteristic
66	C₄×C₅₉	#431	1	2	yes	—
67	#2150	—	59	2	no	—
68	C₄²×C₈	D₅₉	1	1	yes	characteristic
69	#1376	—	59	4	no	—
70	C₂×C₅₉	—	2	2	no	—
71	C₂×C₅₉	#1831	1	1	yes	characteristic
72	C₂×C₅₉	C₂×#364	1	2	yes	—
73	C₂×M₃₂	—	59	4	no	—
74	C₂×C₄×C₈	Dic₅₉	1	2	yes	—
75	C₂×C₄×C₈	Dic₅₉	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
77	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
78	C₄³	Dic₅₉	1	1	yes	characteristic
79	C₄⋊C₁₆	—	118	4	no	—
80	C₅₉	#12883	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
82	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
83	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
84	C₂×C₄²	—	2	2	no	—
85	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
86	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
87	M₃₂	—	118	8	no	—
88	C₂×C₁₆	—	118	4	no	—
89	C₄×C₈	—	2	4	no	—
90	C₄×C₈	—	2	4	no	—
91	C₄×C₈	C₅₉⋊C₈	1	2	yes	—
92	C₄×C₈	C₅₉⋊C₈	1	2	yes	—
93	C₄×C₈	Dic₁₁₈	1	2	yes	—
94	C₄×C₈	D₂₃₆	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#11745664	1	2	yes	—
97	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₂×C₅₉⋊C₈	1	4	yes	—
102	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
103	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
104	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
105	C₂×C₈	#11745685	1	4	yes	—
106	C₄²	—	2	8	no	—
107	C₄²	—	2	2	no	—
108	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
109	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
110	C₄²	#11745664	1	4	yes	—
111	C₄²	C₄⋊Dic₅₉	1	2	yes	—
112	C₁₆	—	236	4	no	—
113	C₂³	C₂×#11745664	1	1	yes	characteristic
114	C₂×C₄	—	2	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
118	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
119	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
120	C₂×C₄	#14273686	1	2	yes	—
121	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
122	C₂×C₄	C₂×#11745664	1	2	yes	—
123	C₈	—	2	2	no	—
124	C₈	—	2	4	no	—
125	C₈	#14273715	1	2	yes	—
126	C₈	#14273719	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
129	C₂²	#16837459	1	2	yes	—
130	C₄	—	2	8	no	—
131	C₄	—	2	2	no	—
132	C₄	#16837223	1	4	yes	—
133	C₄	#16837268	1	1	yes	characteristic
134	C₄	#16837426	1	1	yes	characteristic
135	C₄	#16837455	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#21424213	1	1	yes	characteristic
138	C₂	C₂×#16837223	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425852

C₂

yes

maximal

#21422178

C₂

yes

maximal

C₄²×C₈×C₅₉

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

—

C₄⋊(C₅₉⋊C₁₆)

—

C₂×C₄×C₈×C₅₉

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₄

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄³×C₅₉

C₄

yes

characteristic

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₈

yes

—

C₄×C₈×C₅₉

C₈

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

M₁₆

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×C₈

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₄²×C₅₉

M₁₆

yes

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#124

yes

—

C₈×C₅₉

#125

yes

—

#12883

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₉

#432

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#364

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

#2150

—

C₄²×C₈

D₅₉

yes

characteristic

#1376

—

C₂×C₅₉

—

C₂×C₅₉

#1831

yes

characteristic

C₂×C₅₉

C₂×#364

yes

—

C₂×M₃₂

—

C₂×C₄×C₈

Dic₅₉

yes

—

C₂×C₄×C₈

Dic₅₉

yes

characteristic

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄³

Dic₅₉

yes

characteristic

C₄⋊C₁₆

—

118

—

C₅₉

#12883

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₅₉

yes

—

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₅₉

yes

—

C₂×C₄²

C₂×Dic₅₉

yes

characteristic

M₃₂

—

118

—

C₂×C₁₆

—

118

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₅₉⋊C₈

yes

—

C₄×C₈

C₅₉⋊C₈

yes

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

D₂₃₆

yes

—

C₂²×C₄

—

C₂²×C₄

#11745664

yes

—

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂×C₅₉⋊C₈

yes

—

102

C₂×C₈

C₄⋊Dic₅₉

yes

—

103

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

104

C₂×C₈

C₂×D₂₃₆

yes

characteristic

105

C₂×C₈

#11745685

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₂×C₅₉⋊C₈

yes

characteristic

109

C₄²

C₂×C₅₉⋊C₈

yes

characteristic

110

C₄²

#11745664

yes

—

111

C₄²

C₄⋊Dic₅₉

yes

—

112

C₁₆

—

236

—

113

C₂³

C₂×#11745664

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₄⋊(C₅₉⋊C₈)

yes

—

118

C₂×C₄

C₄⋊(C₅₉⋊C₈)

yes

—

119

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

120

C₂×C₄

#14273686

yes

—

121

C₂×C₄

C₂²×C₅₉⋊C₈

yes

characteristic

122

C₂×C₄

C₂×#11745664

yes

—

123

C₈

—

124

C₈

—

125

C₈

#14273715

yes

—

126

C₈

#14273719

yes

—

127

C₂²

—

128

C₂²

C₂×C₄⋊(C₅₉⋊C₈)

yes

characteristic

129

C₂²

#16837459

yes

—

130

C₄

—

131

C₄

—

132

C₄

#16837223

yes

—

133

C₄

#16837268

yes

characteristic

134

C₄

#16837426

yes

characteristic

135

C₄

#16837455

yes

—

136

C₂

—

137

C₂

#21424213

yes

characteristic

138

C₂

C₂×#16837223

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄²×C₈×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄²×C₈	D₅₉
hypocenter	C₅₉	#12883
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

subgroup

G/H

center

C₂×C₈

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₄²×C₈×C₅₉

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₈

prime radical

C₂³×C₅₉

C₂×M₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

hypercenter

C₄²×C₈

D₅₉

hypocenter

C₅₉

#12883

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
80. C₅₉	#12883	semidirect

H ≅ G/N

type

80. C₅₉

#12883

semidirect

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

subgroup	index
#21425852	2
#21422178	2
C₄²×C₈×C₅₉	2
#12883	59

subgroup

index

#21425852

#21422178

C₄²×C₈×C₅₉

#12883

Subgroups — #28505050

Conjugacy classes of subgroups

Complements

Maximal subgroups