Group #28482461 — subgroups

Subgroups — #28482461

Order 15104.

422 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 203 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	#21424253	C₂	1	1	yes	maximal, characteristic
2	#21424223	C₂	1	4	yes	maximal
3	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273659	—	2	2	no	—
6	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
7	#16837272	C₂²	1	2	yes	—
8	#16837164	—	2	8	no	—
9	C₈×C₅₉⋊C₈	C₄	1	4	yes	complemented
10	C₂×#106×C₅₉	—	2	2	no	—
11	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×#384	C₂²	1	2	yes	—
13	C₂²×C₅₉⋊C₈	D₄	1	4	yes	—
14	C₂×C₄⋊Dic₅₉	—	2	4	no	—
15	C₂×C₄⋊Dic₅₉	—	2	2	no	—
16	#14273659	—	4	4	no	—
17	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₄×C₅₉⋊C₈	Q₈	1	4	yes	—
19	C₂²×C₈×C₅₉	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
21	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
22	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
23	#106×C₅₉	—	4	4	no	—
24	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₉	Q₈	1	2	yes	—
26	C₂²×C₄×C₅₉	—	4	2	no	—
27	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
29	C₂²×Dic₅₉	—	4	2	no	—
30	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
32	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
33	C₄⋊C₄×C₅₉	—	4	4	no	—
34	C₄⋊C₄×C₅₉	—	4	8	no	—
35	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
38	C₄⋊Dic₅₉	—	4	4	no	—
39	C₄⋊Dic₅₉	—	4	8	no	—
40	C₂×C₅₉⋊C₈	—	2	8	no	—
41	C₂×C₅₉⋊C₈	C₄⋊C₄	1	8	yes	—
42	C₂³×C₅₉	#127	1	1	yes	characteristic
43	C₂×C₄×C₅₉	—	8	2	no	—
44	C₂×C₄×C₅₉	—	8	4	no	—
45	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
49	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
50	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
51	C₂×C₄×C₅₉	#135	1	8	yes	—
52	C₂×Dic₅₉	—	8	2	no	—
53	C₂×Dic₅₉	—	8	4	no	—
54	C₈×C₅₉	#106	1	8	yes	—
55	C₅₉⋊C₈	—	2	8	no	—
56	#8815	—	59	1	no	maximal, Sylow
57	C₂²×C₅₉	#384	1	1	yes	characteristic
58	C₂²×C₅₉	#492	1	1	yes	characteristic
59	C₂²×C₅₉	#492	1	1	yes	characteristic
60	C₂²×C₅₉	#495	1	4	yes	—
61	C₄×C₅₉	—	16	2	no	—
62	C₄×C₅₉	C₂×#106	1	2	yes	—
63	C₄×C₅₉	#427	1	4	yes	—
64	C₄×C₅₉	#498	1	2	yes	—
65	C₄×C₅₉	#499	1	4	yes	—
66	Dic₅₉	—	16	2	no	—
67	#1844	—	59	1	no	—
68	#1844	D₅₉	1	1	yes	characteristic
69	#1834	—	59	4	no	—
70	C₂×C₈²	—	59	1	no	—
71	C₂×C₅₉	#1708	1	2	yes	—
72	C₂×C₅₉	#1708	1	2	yes	—
73	C₂×C₅₉	#1844	1	1	yes	characteristic
74	C₂×C₅₉	#1846	1	2	yes	—
75	C₂×#106	—	2	2	no	—
76	C₂×#106	—	118	2	no	—
77	C₂×C₄×C₈	—	59	2	no	—
78	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
79	#384	—	59	2	no	—
80	#384	C₂×D₅₉	1	2	yes	—
81	#340	—	118	8	no	—
82	C₈²	—	59	4	no	—
83	C₅₉	#8815	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	59	4	no	—
85	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
86	C₂×C₄⋊C₄	—	2	4	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₄⋊C₄	—	118	4	no	—
89	C₂×C₄⋊C₄	—	118	2	no	—
90	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
91	#106	—	4	4	no	—
92	#106	—	236	4	no	—
93	C₄×C₈	—	59	4	no	—
94	C₄×C₈	—	59	4	no	—
95	C₄×C₈	Dic₁₁₈	1	2	yes	—
96	C₄×C₈	C₄×D₅₉	1	2	yes	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	—	236	2	no	—
99	C₂²×C₄	D₄×D₅₉	1	2	yes	—
100	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
101	C₂×C₈	—	59	8	no	—
102	C₂×C₈	—	118	8	no	—
103	C₂×C₈	Dic₅₉⋊C₄	1	4	yes	—
104	C₂×C₈	#11745670	1	4	yes	—
105	C₂×C₈	#11745671	1	4	yes	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	4	8	no	—
108	C₄⋊C₄	—	236	4	no	—
109	C₄⋊C₄	—	236	8	no	—
110	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
111	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
112	C₄²	Q₈×D₅₉	1	2	yes	—
113	C₂³	#14273723	1	1	yes	characteristic
114	C₂×C₄	—	8	2	no	—
115	C₂×C₄	—	8	4	no	—
116	C₂×C₄	—	472	2	no	—
117	C₂×C₄	—	472	4	no	—
118	C₂×C₄	C₄⋊C₄×D₅₉	1	2	yes	—
119	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
120	C₂×C₄	#14273725	1	1	yes	characteristic
121	C₂×C₄	#14273726	1	2	yes	—
122	C₂×C₄	C₂×#11745671	1	2	yes	—
123	C₂×C₄	C₂×#11745670	1	2	yes	—
124	C₂×C₄	#14273748	1	8	yes	—
125	C₈	—	118	8	no	—
126	C₈	#14273658	1	8	yes	—
127	C₂²	#16837271	1	1	yes	characteristic
128	C₂²	#16837774	1	1	yes	characteristic
129	C₂²	#16837776	1	1	yes	characteristic
130	C₂²	#16837783	1	4	yes	—
131	C₄	—	16	2	no	—
132	C₄	—	944	2	no	—
133	C₄	C₂×#14273658	1	2	yes	—
134	C₄	#16837437	1	4	yes	—
135	C₄	#16837798	1	2	yes	—
136	C₄	#16837802	1	4	yes	—
137	C₂	#21423695	1	2	yes	—
138	C₂	#21423698	1	2	yes	—
139	C₂	#21424252	1	1	yes	characteristic
140	C₂	#21424257	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424253

C₂

yes

maximal, characteristic

#21424223

C₂

yes

maximal

C₂×C₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic

C₅₉×#1844

C₂

yes

maximal, characteristic

C₂×#14273659

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

#16837272

C₂²

yes

—

#16837164

—

C₈×C₅₉⋊C₈

C₄

yes

complemented

C₂×#106×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#384

C₂²

yes

—

C₂²×C₅₉⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273659

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

C₄×C₅₉⋊C₈

Q₈

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#106×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

—

C₂³×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

#135

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

#106

yes

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#106

yes

—

C₄×C₅₉

#427

yes

—

C₄×C₅₉

#498

yes

—

C₄×C₅₉

#499

yes

—

Dic₅₉

—

#1844

—

#1844

D₅₉

yes

characteristic

#1834

—

C₂×C₈²

—

C₂×C₅₉

#1708

yes

—

C₂×C₅₉

#1708

yes

—

C₂×C₅₉

#1844

yes

characteristic

C₂×C₅₉

#1846

yes

—

C₂×#106

—

C₂×#106

—

118

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#384

—

#384

C₂×D₅₉

yes

—

#340

—

118

—

C₈²

—

C₅₉

#8815

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

#106

—

#106

—

236

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

C₄×D₅₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

236

—

C₂²×C₄

D₄×D₅₉

yes

—

100

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

118

—

103

C₂×C₈

Dic₅₉⋊C₄

yes

—

104

C₂×C₈

#11745670

yes

—

105

C₂×C₈

#11745671

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

236

—

109

C₄⋊C₄

—

236

—

110

C₄²

C₂×Dic₁₁₈

yes

characteristic

111

C₄²

C₂×C₄×D₅₉

yes

characteristic

112

C₄²

Q₈×D₅₉

yes

—

113

C₂³

#14273723

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

472

—

117

C₂×C₄

—

472

—

118

C₂×C₄

C₄⋊C₄×D₅₉

yes

—

119

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

120

C₂×C₄

#14273725

yes

characteristic

121

C₂×C₄

#14273726

yes

—

122

C₂×C₄

C₂×#11745671

yes

—

123

C₂×C₄

C₂×#11745670

yes

—

124

C₂×C₄

#14273748

yes

—

125

C₈

—

118

—

126

C₈

#14273658

yes

—

127

C₂²

#16837271

yes

characteristic

128

C₂²

#16837774

yes

characteristic

129

C₂²

#16837776

yes

characteristic

130

C₂²

#16837783

yes

—

131

C₄

—

132

C₄

—

944

—

133

C₄

C₂×#14273658

yes

—

134

C₄

#16837437

yes

—

135

C₄

#16837798

yes

—

136

C₄

#16837802

yes

—

137

C₂

#21423695

yes

—

138

C₂

#21423698

yes

—

139

C₂

#21424252

yes

characteristic

140

C₂

#21424257

yes

—

	subgroup	G/H
center	C₂³	#14273723
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1844	C₂
socle, prime radical	C₂³×C₅₉	#127
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1844	D₅₉
hypocenter	C₅₉	#8815
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#14273723

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1844

C₂

socle, prime radical

C₂³×C₅₉

#127

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
9. C₈×C₅₉⋊C₈	C₄	semidirect
83. C₅₉	#8815	semidirect

H ≅ G/N

type

9. C₈×C₅₉⋊C₈

C₄

semidirect

83. C₅₉

#8815

semidirect

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

subgroup	index
#21424253	2
#21424223	2
C₂×C₈×C₅₉⋊C₈	2
C₅₉×#1844	2
#8815	59

subgroup

index

#21424253

#21424223

C₂×C₈×C₅₉⋊C₈

C₅₉×#1844

#8815

Subgroups — #28482461

Conjugacy classes of subgroups

Complements

Maximal subgroups