Group #28479146 — subgroups

Subgroups — #28479146

Order 15104.

294 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 131 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	#21422388	C₂	1	1	yes	maximal, characteristic
2	#21422379	C₂	1	1	yes	maximal, characteristic
3	#21421927	C₂	1	2	yes	maximal
4	#21421925	C₂	1	2	yes	maximal
5	C₅₉×#1746	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	#16837455	C₂²	1	2	yes	—
8	#16837124	C₄	1	4	yes	complemented
9	#16837123	C₄	1	2	yes	complemented
10	C₈×C₅₉⋊C₈	C₄	1	2	yes	complemented
11	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#376	C₂²	1	1	yes	characteristic
14	C₅₉×#336	C₄	1	4	yes	—
15	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
16	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
17	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
19	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
20	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
25	#97×C₅₉	—	2	2	no	—
26	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
29	#95×C₅₉	—	2	2	no	—
30	C₂²×C₄×C₅₉	—	2	2	no	—
31	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₉	#41	1	2	yes	—
33	C₂×C₈×C₅₉	—	2	4	no	—
34	C₂×C₈×C₅₉	—	2	2	no	—
35	C₂×C₈×C₅₉	—	2	4	no	—
36	C₂×C₈×C₅₉	C₄²	1	2	yes	—
37	C₂×C₈×C₅₉	C₄²	1	2	yes	—
38	C₄²×C₅₉	—	2	4	no	—
39	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₅₉	#41	1	2	yes	—
42	C₂×C₅₉⋊C₈	—	2	4	no	—
43	C₂×C₅₉⋊C₈	—	2	4	no	—
44	C₂³×C₅₉	#117	1	1	yes	characteristic
45	C₂×C₄×C₅₉	—	4	2	no	—
46	C₂×C₄×C₅₉	—	4	4	no	—
47	C₂×C₄×C₅₉	—	4	2	no	—
48	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₅₉	#117	1	2	yes	—
50	C₂×C₄×C₅₉	#117	1	2	yes	—
51	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#131	1	4	yes	—
53	C₈×C₅₉	—	2	2	no	—
54	C₈×C₅₉	—	2	2	no	—
55	C₈×C₅₉	—	4	2	no	—
56	C₅₉⋊C₈	—	4	4	no	—
57	C₅₉⋊C₈	—	4	4	no	—
58	#8141	—	59	1	no	maximal, Sylow
59	C₂²×C₅₉	—	2	2	no	—
60	C₂²×C₅₉	#376	1	1	yes	characteristic
61	C₂²×C₅₉	#433	1	2	yes	—
62	C₄×C₅₉	—	2	2	no	—
63	C₄×C₅₉	—	2	2	no	—
64	C₄×C₅₉	—	8	2	no	—
65	C₄×C₅₉	#405	1	2	yes	—
66	C₄×C₅₉	#433	1	2	yes	—
67	#1746	D₅₉	1	1	yes	characteristic
68	#1452	—	59	1	no	—
69	#1450	—	59	1	no	—
70	#1275	—	59	2	no	—
71	#1274	—	59	2	no	—
72	C₂×C₅₉	—	2	2	no	—
73	C₂×C₅₉	#1746	1	1	yes	characteristic
74	C₂×C₅₉	#1747	1	2	yes	—
75	#431	—	59	2	no	—
76	#431	—	59	2	no	—
77	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
79	#376	C₂×D₅₉	1	1	yes	characteristic
80	#336	Dic₅₉	1	4	yes	—
81	#322	—	59	2	no	—
82	#322	—	59	4	no	—
83	C₈²	—	59	2	no	—
84	C₅₉	#8141	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
86	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
87	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
88	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
89	C₄⋊C₈	—	118	2	no	—
90	C₄⋊C₈	—	118	2	no	—
91	C₂²⋊C₈	—	118	2	no	—
92	C₂²⋊C₈	—	118	2	no	—
93	#97	—	2	2	no	—
94	C₄×C₈	—	59	4	no	—
95	C₄×C₈	—	59	4	no	—
96	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
97	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
98	C₄×C₈	C₄×D₅₉	1	2	yes	—
99	#95	—	2	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	#11745685	1	2	yes	—
102	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	118	4	no	—
107	C₂×C₈	—	118	4	no	—
108	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
109	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
110	C₄²	—	2	4	no	—
111	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
112	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
113	C₄²	#11745685	1	2	yes	—
114	C₂³	#14273686	1	1	yes	characteristic
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
119	C₂×C₄	#14273684	1	2	yes	—
120	C₂×C₄	#14273684	1	2	yes	—
121	C₂×C₄	#14273686	1	1	yes	characteristic
122	C₂×C₄	#14273734	1	4	yes	—
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	4	2	no	—
126	C₈	—	236	4	no	—
127	C₈	—	236	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	#16837248	1	1	yes	characteristic
130	C₂²	#16837463	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	8	2	no	—
134	C₄	#16837358	1	2	yes	—
135	C₄	#16837463	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#21423848	1	1	yes	characteristic
138	C₂	#21423851	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21422388

C₂

yes

maximal, characteristic

#21422379

C₂

yes

maximal, characteristic

#21421927

C₂

yes

maximal

#21421925

C₂

yes

maximal

C₅₉×#1746

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

#16837455

C₂²

yes

—

#16837124

C₄

yes

complemented

#16837123

C₄

yes

complemented

C₈×C₅₉⋊C₈

C₄

yes

complemented

C₂×#97×C₅₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#376

C₂²

yes

characteristic

C₅₉×#336

C₄

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#97×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄²

yes

—

C₂×C₈×C₅₉

C₄²

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

#41

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#117

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#117

yes

characteristic

C₂×C₄×C₅₉

#131

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#8141

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#376

yes

characteristic

C₂²×C₅₉

#433

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#405

yes

—

C₄×C₅₉

#433

yes

—

#1746

D₅₉

yes

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

#1746

yes

characteristic

C₂×C₅₉

#1747

yes

—

#431

—

#431

—

C₂×#97

C₂×D₅₉

yes

characteristic

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#376

C₂×D₅₉

yes

characteristic

#336

Dic₅₉

yes

—

#322

—

#322

—

C₈²

—

C₅₉

#8141

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₅₉

yes

—

C₂²×C₈

C₂×Dic₅₉

yes

—

C₂×C₄²

C₂×Dic₅₉

yes

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

C₄⋊C₈

—

118

—

C₄⋊C₈

—

118

—

C₂²⋊C₈

—

118

—

C₂²⋊C₈

—

118

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₅₉

yes

characteristic

C₄×C₈

C₄×D₅₉

yes

characteristic

C₄×C₈

C₄×D₅₉

yes

—

#95

—

100

C₂²×C₄

—

101

C₂²×C₄

#11745685

yes

—

102

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

118

—

107

C₂×C₈

—

118

—

108

C₂×C₈

C₄×Dic₅₉

yes

—

109

C₂×C₈

C₄×Dic₅₉

yes

—

110

C₄²

—

111

C₄²

C₂×C₄×D₅₉

yes

characteristic

112

C₄²

C₂×C₄×D₅₉

yes

characteristic

113

C₄²

#11745685

yes

—

114

C₂³

#14273686

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂×C₄×Dic₅₉

yes

characteristic

119

C₂×C₄

#14273684

yes

—

120

C₂×C₄

#14273684

yes

—

121

C₂×C₄

#14273686

yes

characteristic

122

C₂×C₄

#14273734

yes

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

236

—

127

C₈

—

236

—

128

C₂²

—

129

C₂²

#16837248

yes

characteristic

130

C₂²

#16837463

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#16837358

yes

—

135

C₄

#16837463

yes

—

136

C₂

—

137

C₂

#21423848

yes

characteristic

138

C₂

#21423851

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1746	C₂
socle	C₂²×C₅₉	#376
prime radical	C₂³×C₅₉	#117
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1746	D₅₉
hypocenter	C₅₉	#8141
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #1746

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₅₉

commutator

C₂×C₄×C₅₉

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1746

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄ ◁ #1746

N	H ≅ G/N	type
8. #16837124	C₄	semidirect
9. #16837123	C₄	semidirect
10. C₈×C₅₉⋊C₈	C₄	semidirect
84. C₅₉	#8141	semidirect

H ≅ G/N

type

8. #16837124

C₄

semidirect

9. #16837123

C₄

semidirect

10. C₈×C₅₉⋊C₈

C₄

semidirect

84. C₅₉

#8141

semidirect

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

subgroup	index
#21422388	2
#21422379	2
#21421927	2
#21421925	2
C₅₉×#1746	2
#8141	59

subgroup

index

C₅₉×#1746

Subgroups — #28479146

Conjugacy classes of subgroups

Complements

Maximal subgroups