Group #28478056 — subgroups

Subgroups — #28478056

Order 15104.

518 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 291 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	#21423841	C₂	1	2	yes	maximal
2	C₄×#14273637	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837124	C₂	1	2	yes	maximal
4	C₂×#16837123	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273637	C₄	1	4	yes	—
7	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
10	#16837156	C₄	1	8	yes	—
11	#16837124	C₄	1	8	yes	complemented
12	#16837123	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
15	C₈²×C₅₉	C₄	1	4	yes	complemented
16	C₂²×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₅₉	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
20	#14273637	—	2	8	no	—
21	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
22	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
23	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
27	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
33	C₂²×Dic₅₉	C₄²	1	4	yes	—
34	C₂×C₈×C₅₉	—	2	8	no	—
35	C₂×C₈×C₅₉	C₄²	1	8	yes	—
36	C₂×C₈×C₅₉	C₄²	1	4	yes	—
37	C₂×C₈×C₅₉	M₁₆	1	8	yes	—
38	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
39	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₄×Dic₅₉	—	2	4	no	—
42	C₄×Dic₅₉	—	2	8	no	—
43	C₂×C₅₉⋊C₈	—	2	4	no	—
44	C₂×C₅₉⋊C₈	—	2	8	no	—
45	C₂×C₅₉⋊C₈	C₄²	1	4	yes	—
46	C₂×C₅₉⋊C₈	C₄²	1	8	yes	—
47	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
50	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
52	C₂×C₄×C₅₉	C₂×M₁₆	1	4	yes	—
53	C₂×C₄×C₅₉	#131	1	8	yes	—
54	C₂×Dic₅₉	—	2	4	no	—
55	C₂×Dic₅₉	—	2	8	no	—
56	C₈×C₅₉	—	2	8	no	—
57	C₈×C₅₉	#97	1	8	yes	—
58	C₅₉⋊C₈	—	2	4	no	—
59	C₅₉⋊C₈	—	2	8	no	—
60	#7844	—	59	1	no	maximal, Sylow
61	C₂²×C₅₉	C₄³	1	1	yes	characteristic
62	C₂²×C₅₉	C₄×M₁₆	1	2	yes	—
63	C₂²×C₅₉	#405	1	4	yes	—
64	C₄×C₅₉	C₂×#97	1	2	yes	—
65	C₄×C₅₉	C₄×M₁₆	1	2	yes	—
66	C₄×C₅₉	#405	1	4	yes	—
67	C₄×C₅₉	#405	1	4	yes	—
68	Dic₅₉	—	4	4	no	—
69	#1744	—	59	2	no	—
70	C₄×#97	—	59	1	no	—
71	C₂×#322	—	59	1	no	—
72	C₂×#322	—	59	2	no	—
73	C₂×C₈²	D₅₉	1	1	yes	characteristic
74	C₂×C₅₉	#1453	1	4	yes	—
75	C₂×C₅₉	C₄×#97	1	1	yes	characteristic
76	C₂×C₅₉	#1726	1	2	yes	—
77	C₂×#97	—	59	4	no	—
78	C₂×C₄×C₈	—	59	1	no	—
79	C₂×C₄×C₈	—	59	2	no	—
80	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
82	C₄³	—	59	1	no	—
83	#336	—	59	8	no	—
84	#322	—	59	4	no	—
85	#322	—	59	8	no	—
86	C₈²	Dic₅₉	1	4	yes	complemented
87	C₅₉	#7844	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	59	4	no	—
89	C₂²×C₈	—	59	2	no	—
90	C₂²×C₈	C₄×D₅₉	1	4	yes	—
91	C₂²×C₈	C₄×D₅₉	1	2	yes	—
92	C₂×C₄²	—	59	4	no	—
93	C₂×C₄²	—	59	2	no	—
94	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
95	#97	—	118	8	no	—
96	C₄×C₈	—	59	4	no	—
97	C₄×C₈	—	59	4	no	—
98	C₄×C₈	—	59	4	no	—
99	C₄×C₈	C₄×D₅₉	1	4	yes	—
100	C₄×C₈	C₄×D₅₉	1	2	yes	—
101	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
102	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
103	C₂²×C₄	—	59	4	no	—
104	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
106	C₂×C₈	—	2	8	no	—
107	C₂×C₈	—	59	4	no	—
108	C₂×C₈	—	59	8	no	—
109	C₂×C₈	—	118	4	no	—
110	C₂×C₈	—	118	8	no	—
111	C₂×C₈	#11745659	1	8	yes	—
112	C₂×C₈	C₄×Dic₅₉	1	8	yes	—
113	C₂×C₈	C₄×Dic₅₉	1	4	yes	—
114	C₄²	—	118	4	no	—
115	C₄²	—	118	8	no	—
116	C₄²	C₂×C₄×D₅₉	1	2	yes	—
117	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
118	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
119	C₂³	C₄²×D₅₉	1	1	yes	characteristic
120	C₂×C₄	—	118	4	no	—
121	C₂×C₄	—	118	8	no	—
122	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
123	C₂×C₄	C₄²×D₅₉	1	2	yes	—
124	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
126	C₂×C₄	C₂×#11745659	1	4	yes	—
127	C₂×C₄	#14273733	1	8	yes	—
128	C₈	—	2	8	no	—
129	C₈	—	118	4	no	—
130	C₈	—	118	8	no	—
131	C₈	#14273637	1	8	yes	—
132	C₂²	C₄²×Dic₅₉	1	1	yes	characteristic
133	C₂²	C₄×#11745659	1	2	yes	—
134	C₂²	#16837356	1	4	yes	—
135	C₄	—	236	4	no	—
136	C₄	C₂×#14273637	1	2	yes	—
137	C₄	C₄×#11745659	1	2	yes	—
138	C₄	#16837356	1	4	yes	—
139	C₄	#16837357	1	4	yes	—
140	C₂	#21422390	1	4	yes	—
141	C₂	C₄×#14273637	1	1	yes	characteristic
142	C₂	#21423782	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21423841

C₂

yes

maximal

C₄×#14273637

C₂

yes

maximal, characteristic

C₂×#16837124

C₂

yes

maximal

C₂×#16837123

C₂

yes

maximal, characteristic

C₂×C₈²×C₅₉

C₂

yes

maximal, characteristic

C₂×#14273637

C₄

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄²×Dic₅₉

C₂²

yes

characteristic

#16837156

C₄

yes

—

#16837124

C₄

yes

complemented

#16837123

C₄

yes

complemented

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₈²×C₅₉

C₄

yes

complemented

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₅₉

C₂×C₄

yes

—

C₂×C₄×Dic₅₉

C₂×C₄

yes

—

#14273637

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

—

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

C₂³

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂²×C₄

yes

—

C₂²×Dic₅₉

C₄²

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄²

yes

—

C₂×C₈×C₅₉

C₄²

yes

—

C₂×C₈×C₅₉

M₁₆

yes

—

C₄²×C₅₉

C₂²×C₄

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₄²

yes

—

C₂×C₅₉⋊C₈

C₄²

yes

—

C₂³×C₅₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄²

yes

—

C₂×C₄×C₅₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄²

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#97

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#7844

—

maximal, Sylow

C₂²×C₅₉

C₄³

yes

characteristic

C₂²×C₅₉

C₄×M₁₆

yes

—

C₂²×C₅₉

#405

yes

—

C₄×C₅₉

C₂×#97

yes

—

C₄×C₅₉

C₄×M₁₆

yes

—

C₄×C₅₉

#405

yes

—

C₄×C₅₉

#405

yes

—

Dic₅₉

—

#1744

—

C₄×#97

—

C₂×#322

—

C₂×#322

—

C₂×C₈²

D₅₉

yes

characteristic

C₂×C₅₉

#1453

yes

—

C₂×C₅₉

C₄×#97

yes

characteristic

C₂×C₅₉

#1726

yes

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₄³

—

#336

—

#322

—

#322

—

C₈²

Dic₅₉

yes

complemented

C₅₉

#7844

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄×D₅₉

yes

—

C₂²×C₈

C₄×D₅₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

#97

—

118

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₅₉

yes

—

100

C₄×C₈

C₄×D₅₉

yes

—

101

C₄×C₈

C₂×Dic₅₉

yes

—

102

C₄×C₈

C₂×Dic₅₉

yes

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

105

C₂²×C₄

C₂×C₄×D₅₉

yes

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

118

—

110

C₂×C₈

—

118

—

111

C₂×C₈

#11745659

yes

—

112

C₂×C₈

C₄×Dic₅₉

yes

—

113

C₂×C₈

C₄×Dic₅₉

yes

—

114

C₄²

—

118

—

115

C₄²

—

118

—

116

C₄²

C₂×C₄×D₅₉

yes

—

117

C₄²

C₂×C₄×D₅₉

yes

characteristic

118

C₄²

C₂²×Dic₅₉

yes

characteristic

119

C₂³

C₄²×D₅₉

yes

characteristic

120

C₂×C₄

—

118

—

121

C₂×C₄

—

118

—

122

C₂×C₄

C₄²×D₅₉

yes

characteristic

123

C₂×C₄

C₄²×D₅₉

yes

—

124

C₂×C₄

C₂×C₄×Dic₅₉

yes

characteristic

125

C₂×C₄

C₂×C₄×Dic₅₉

yes

—

126

C₂×C₄

C₂×#11745659

yes

—

127

C₂×C₄

#14273733

yes

—

128

C₈

—

129

C₈

—

118

—

130

C₈

—

118

—

131

C₈

#14273637

yes

—

132

C₂²

C₄²×Dic₅₉

yes

characteristic

133

C₂²

C₄×#11745659

yes

—

134

C₂²

#16837356

yes

—

135

C₄

—

236

—

136

C₄

C₂×#14273637

yes

—

137

C₄

C₄×#11745659

yes

—

138

C₄

#16837356

yes

—

139

C₄

#16837357

yes

—

140

C₂

#21422390

yes

—

141

C₂

C₄×#14273637

yes

characteristic

142

C₂

#21423782

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₉
commutator	C₂²×C₅₉	C₄³
Fitting	C₂×C₈²×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×C₈²	D₅₉
hypocenter	C₅₉	#7844
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₅₉

commutator

C₂²×C₅₉

C₄³

Fitting

C₂×C₈²×C₅₉

C₂

socle, prime radical

C₂³×C₅₉

C₂×C₄²

cosocle

C₂×C₄²×C₅₉

C₂³

hypercenter

C₂×C₈²

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
11. #16837124	C₄	semidirect
12. #16837123	C₄	semidirect
15. C₈²×C₅₉	C₄	semidirect
86. C₈²	Dic₅₉	semidirect
87. C₅₉	#7844	semidirect

H ≅ G/N

type

11. #16837124

C₄

semidirect

12. #16837123

C₄

semidirect

15. C₈²×C₅₉

C₄

semidirect

86. C₈²

Dic₅₉

semidirect

87. C₅₉

#7844

semidirect

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

subgroup	index
#21423841	2
C₄×#14273637	2
C₂×#16837124	2
C₂×#16837123	2
C₂×C₈²×C₅₉	2
#7844	59

subgroup

index

C₂×C₈²×C₅₉

Subgroups — #28478056

Conjugacy classes of subgroups

Complements

Maximal subgroups