Group #28475286 — subgroups

Subgroups — #28475286

Order 15104.

304 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 107 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	#21423949	C₂	1	2	yes	maximal
2	C₅₉×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
4	C₂×#14273638	C₄	1	4	yes	—
5	#16837157	—	2	2	no	—
6	C₂×C₄⋊C₈×C₅₉	C₄	1	2	yes	—
7	C₄³×C₅₉	C₂²	1	1	yes	characteristic
8	C₅₉×#336	—	2	2	no	—
9	C₂²×C₅₉⋊C₈	—	2	4	no	—
10	C₄⋊(C₅₉⋊C₈)	—	2	8	no	—
11	#14273638	—	4	4	no	—
12	C₂²×C₈×C₅₉	—	2	2	no	—
13	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
16	C₂×C₄²×C₅₉	D₄	1	2	yes	—
17	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
18	C₄⋊C₈×C₅₉	—	4	2	no	—
19	C₂²×C₄×C₅₉	C₄²	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
21	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
23	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₉	—	4	2	no	—
25	C₂×C₈×C₅₉	—	4	4	no	—
26	C₄²×C₅₉	—	2	1	no	—
27	C₄²×C₅₉	—	2	2	no	—
28	C₄²×C₅₉	—	2	1	no	—
29	C₄²×C₅₉	—	2	2	no	—
30	C₄²×C₅₉	—	2	4	no	—
31	C₄²×C₅₉	—	2	2	no	—
32	C₄²×C₅₉	—	4	1	no	—
33	C₂×C₅₉⋊C₈	—	2	8	no	—
34	C₂×C₅₉⋊C₈	—	4	4	no	—
35	C₂×C₅₉⋊C₈	—	4	4	no	—
36	C₂³×C₅₉	#95	1	1	yes	characteristic
37	C₂×C₄×C₅₉	—	2	2	no	—
38	C₂×C₄×C₅₉	—	2	1	no	—
39	C₂×C₄×C₅₉	—	2	2	no	—
40	C₂×C₄×C₅₉	—	2	1	no	—
41	C₂×C₄×C₅₉	—	2	2	no	—
42	C₂×C₄×C₅₉	—	2	2	no	—
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	#100	1	2	yes	—
47	C₂×C₄×C₅₉	#100	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#101	1	2	yes	—
49	C₂×C₄×C₅₉	#101	1	1	yes	characteristic
50	C₂×C₄×C₅₉	#104	1	4	yes	—
51	C₂×C₄×C₅₉	#108	1	2	yes	—
52	C₈×C₅₉	—	8	2	no	—
53	C₅₉⋊C₈	—	4	8	no	—
54	#7042	—	59	1	no	maximal, Sylow
55	C₂²×C₅₉	H(3,ℤ/4)	1	1	yes	characteristic
56	C₂²×C₅₉	#339	1	2	yes	—
57	C₂²×C₅₉	#341	1	1	yes	characteristic
58	C₂²×C₅₉	#343	1	2	yes	—
59	C₂²×C₅₉	#343	1	1	yes	characteristic
60	C₄×C₅₉	—	2	2	no	—
61	C₄×C₅₉	—	2	2	no	—
62	C₄×C₅₉	—	2	2	no	—
63	C₄×C₅₉	—	4	2	no	—
64	C₄×C₅₉	—	4	1	no	—
65	C₄×C₅₉	#329	1	2	yes	—
66	C₄×C₅₉	#330	1	2	yes	—
67	#1831	D₅₉	1	1	yes	characteristic
68	#1767	—	59	2	no	—
69	C₂×C₅₉	#1285	1	2	yes	—
70	C₂×C₅₉	#1290	1	1	yes	characteristic
71	C₂×C₅₉	#1295	1	1	yes	characteristic
72	C₂×C₅₉	#1299	1	2	yes	—
73	C₂×C₅₉	#1309	1	1	yes	characteristic
74	C₂×C₄⋊C₈	—	118	2	no	—
75	C₂×C₄⋊C₈	Dic₅₉	1	2	yes	—
76	C₂×#97	—	59	4	no	—
77	C₄³	C₂×D₅₉	1	1	yes	characteristic
78	#336	—	2	2	no	—
79	#336	—	118	2	no	—
80	C₅₉	#7042	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	2	2	no	—
82	C₂²×C₈	—	118	4	no	—
83	C₂×C₄²	Dic₁₁₈	1	1	yes	characteristic
84	C₂×C₄²	C₄×D₅₉	1	2	yes	—
85	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
86	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
87	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	118	8	no	—
90	#97	—	236	4	no	—
91	C₂²×C₄	C₄×Dic₅₉	1	1	yes	characteristic
92	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
93	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
94	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
95	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	4	4	no	—
98	C₂×C₈	—	118	8	no	—
99	C₂×C₈	—	236	4	no	—
100	C₂×C₈	—	236	4	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	2	4	no	—
106	C₄²	—	2	2	no	—
107	C₄²	—	4	1	no	—
108	C₂³	C₅₉⋊#95	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	#14273643	1	2	yes	—
119	C₂×C₄	#14273644	1	1	yes	characteristic
120	C₂×C₄	#14273645	1	2	yes	—
121	C₂×C₄	#14273646	1	1	yes	characteristic
122	C₂×C₄	#14273654	1	4	yes	—
123	C₂×C₄	#14273663	1	2	yes	—
124	C₈	—	8	2	no	—
125	C₈	—	236	8	no	—
126	C₂²	#16837159	1	1	yes	characteristic
127	C₂²	#16837161	1	2	yes	—
128	C₂²	#16837167	1	1	yes	characteristic
129	C₂²	#16837170	1	2	yes	—
130	C₂²	#16837171	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	2	no	—
135	C₄	—	4	1	no	—
136	C₄	#16837142	1	2	yes	—
137	C₄	#16837144	1	2	yes	—
138	C₂	#21421953	1	2	yes	—
139	C₂	#21421966	1	1	yes	characteristic
140	C₂	#21421977	1	1	yes	characteristic
141	C₂	#21421987	1	2	yes	—
142	C₂	C₅₉⋊#1309	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21423949

C₂

yes

maximal

C₅₉×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

—

C₂×#14273638

C₄

yes

—

#16837157

—

C₂×C₄⋊C₈×C₅₉

C₄

yes

—

C₄³×C₅₉

C₂²

yes

characteristic

C₅₉×#336

—

C₂²×C₅₉⋊C₈

—

C₄⋊(C₅₉⋊C₈)

—

#14273638

—

C₂²×C₈×C₅₉

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

Q₈

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²×C₄×C₅₉

C₄²

yes

characteristic

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₉

C₄⋊C₄

yes

—

C₂²×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#95

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#100

yes

—

C₂×C₄×C₅₉

#100

yes

characteristic

C₂×C₄×C₅₉

#101

yes

—

C₂×C₄×C₅₉

#101

yes

characteristic

C₂×C₄×C₅₉

#104

yes

—

C₂×C₄×C₅₉

#108

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#341

yes

characteristic

C₂²×C₅₉

#343

yes

—

C₂²×C₅₉

#343

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#329

yes

—

C₄×C₅₉

#330

yes

—

#1831

D₅₉

yes

characteristic

#1767

—

C₂×C₅₉

#1285

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#1309

yes

characteristic

C₂×C₄⋊C₈

—

118

—

C₂×C₄⋊C₈

Dic₅₉

yes

—

C₂×#97

—

C₄³

C₂×D₅₉

yes

characteristic

#336

—

#336

—

118

—

C₅₉

#7042

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

118

—

C₂×C₄²

Dic₁₁₈

yes

characteristic

C₂×C₄²

C₄×D₅₉

yes

—

C₂×C₄²

D₂₃₆

yes

characteristic

C₂×C₄²

C₂×Dic₅₉

yes

characteristic

C₂×C₄²

C₂²⋊D₅₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

#97

—

236

—

C₂²×C₄

C₄×Dic₅₉

yes

characteristic

C₂²×C₄

Dic₅₉⋊C₄

yes

—

C₂²×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂²×C₄

(C₂×D₅₉)⋊C₄

yes

—

C₂²×C₄

C₂²⋊Dic₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

236

—

100

C₂×C₈

—

236

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₂³

C₅₉⋊#95

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#14273643

yes

—

119

C₂×C₄

#14273644

yes

characteristic

120

C₂×C₄

#14273645

yes

—

121

C₂×C₄

#14273646

yes

characteristic

122

C₂×C₄

#14273654

yes

—

123

C₂×C₄

#14273663

yes

—

124

C₈

—

125

C₈

—

236

—

126

C₂²

#16837159

yes

characteristic

127

C₂²

#16837161

yes

—

128

C₂²

#16837167

yes

characteristic

129

C₂²

#16837170

yes

—

130

C₂²

#16837171

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#16837142

yes

—

137

C₄

#16837144

yes

—

138

C₂

#21421953

yes

—

139

C₂

#21421966

yes

characteristic

140

C₂

#21421977

yes

characteristic

141

C₂

#21421987

yes

—

142

C₂

C₅₉⋊#1309

yes

characteristic

	subgroup	G/H
center	C₂³	C₅₉⋊#95
commutator	C₂²×C₄×C₅₉	C₄²
Frattini	C₄³	C₂×D₅₉
Fitting	C₅₉×#1831	C₂
socle, prime radical	C₂³×C₅₉	#95
cosocle	C₄³×C₅₉	C₂²
hypercenter	#1831	D₅₉
hypocenter	C₅₉	#7042
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₅₉⋊#95

commutator

C₂²×C₄×C₅₉

C₄²

Frattini

C₄³

C₂×D₅₉

Fitting

C₅₉×#1831

C₂

socle, prime radical

C₂³×C₅₉

#95

cosocle

C₄³×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
80. C₅₉	#7042	semidirect

H ≅ G/N

type

80. C₅₉

#7042

semidirect

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

subgroup	index
#21423949	2
C₅₉×#1831	2
#7042	59

subgroup

index

#21423949

C₅₉×#1831

#7042

Subgroups — #28475286

Conjugacy classes of subgroups

Complements

Maximal subgroups