Group #28508871 — subgroups

Subgroups — #28508871

Order 15104.

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

normal subgroups only

This group has 238 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	C₂²×C₅₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#21425692	C₂	1	1	yes	maximal, characteristic
3	#21422229	C₂	1	2	yes	maximal, complemented
4	C₃₂×Dic₅₉	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₉×M₆₄	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₄	1	1	yes	characteristic
7	#16837357	C₄	1	1	yes	characteristic
8	C₂×C₅₉⋊C₃₂	C₄	1	4	yes	complemented
9	C₂×C₅₉⋊C₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₅₉⋊C₃₂	C₂²	1	1	yes	characteristic
11	#16837177	C₄	1	2	yes	—
12	C₁₆×Dic₅₉	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×M₆₄	C₄	1	4	yes	complemented
15	C₂×C₃₂×C₅₉	C₂²	1	2	yes	—
16	C₂×#11745664	C₈	1	1	yes	characteristic
17	#14273686	C₈	1	1	yes	characteristic
18	#14273667	C₈	1	4	yes	—
19	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
20	#14273637	C₈	1	2	yes	—
21	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
22	C₅₉⋊C₃₂	C₂×C₄	1	2	yes	—
23	C₅₉⋊C₃₂	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
29	C₃₂×C₅₉	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₅₉	C₄²	1	2	yes	—
32	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₂²⋊Dic₅₉	C₁₆	1	2	yes	—
36	C₄⋊Dic₅₉	C₁₆	1	2	yes	—
37	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
38	#11745664	C₁₆	1	4	yes	—
39	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
40	C₁₆×C₅₉	C₄²	1	2	yes	—
41	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₅₉⋊C₁₆	C₂×C₈	1	4	yes	—
44	C₂³×C₅₉	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₅₉	C₂×C₁₆	1	4	yes	—
50	C₈×C₅₉	C₄×C₈	1	2	yes	—
51	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
52	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
53	C₅₉⋊C₈	C₂×C₁₆	1	4	yes	—
54	#13549	—	59	1	no	maximal, Sylow
55	C₂²×C₅₉	—	2	2	no	—
56	C₂²×C₅₉	C₄×C₁₆	1	2	yes	—
57	C₂²×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
58	C₄×C₅₉	C₄×C₁₆	1	2	yes	—
59	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
60	C₄×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
61	Dic₅₉	—	2	4	no	—
62	C₂×M₆₄	D₅₉	1	1	yes	characteristic
63	C₂²×C₃₂	—	59	1	no	—
64	#2108	—	59	1	no	—
65	#1397	—	59	2	no	—
66	C₄×C₃₂	—	59	2	no	—
67	C₂×C₅₉	—	2	2	no	—
68	C₂×C₅₉	C₂×C₄×C₁₆	1	1	yes	characteristic
69	C₂×C₅₉	#2258	1	2	yes	—
70	C₂×M₃₂	—	59	1	no	—
71	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
72	#405	—	59	1	no	—
73	M₆₄	Dic₅₉	1	4	yes	complemented
74	C₂×C₃₂	—	59	4	no	—
75	C₂×C₃₂	—	59	1	no	—
76	C₂×C₃₂	—	59	1	no	—
77	C₂×C₃₂	C₂×D₅₉	1	2	yes	—
78	#346	—	59	2	no	—
79	C₄×C₁₆	—	59	2	no	—
80	C₅₉	#13549	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₁₆	—	59	1	no	—
82	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
83	#117	—	59	1	no	—
84	M₃₂	—	59	4	no	—
85	C₂×C₁₆	—	59	2	no	—
86	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
87	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
88	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
89	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
90	#97	—	59	2	no	—
91	C₄×C₈	—	59	2	no	—
92	C₃₂	—	59	2	no	—
93	C₃₂	—	59	2	no	—
94	C₃₂	C₂×Dic₅₉	1	4	yes	—
95	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
96	M₁₆	—	59	4	no	—
97	C₂×C₈	—	59	2	no	—
98	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
99	C₂×C₈	C₈×D₅₉	1	2	yes	—
100	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
101	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
102	C₄⋊C₄	—	59	2	no	—
103	C₂²⋊C₄	—	59	2	no	—
104	C₄²	—	59	2	no	—
105	C₁₆	—	59	4	no	—
106	C₁₆	C₄×Dic₅₉	1	2	yes	—
107	C₁₆	C₂×C₄×D₅₉	1	1	yes	characteristic
108	C₁₆	C₂²×Dic₅₉	1	1	yes	characteristic
109	C₂³	C₁₆×D₅₉	1	1	yes	characteristic
110	C₂×C₄	—	59	4	no	—
111	C₂×C₄	C₈×Dic₅₉	1	2	yes	—
112	C₂×C₄	C₁₆×D₅₉	1	1	yes	characteristic
113	C₂×C₄	C₁₆×D₅₉	1	2	yes	—
114	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
115	C₈	—	59	4	no	—
116	C₈	C₈×Dic₅₉	1	2	yes	—
117	C₈	C₂×C₄×Dic₅₉	1	1	yes	characteristic
118	C₈	C₂×C₈×D₅₉	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	C₁₆×Dic₅₉	1	2	yes	—
121	C₂²	C₂×C₁₆×D₅₉	1	1	yes	characteristic
122	C₄	—	118	4	no	—
123	C₄	C₁₆×Dic₅₉	1	2	yes	—
124	C₄	C₂×C₈×Dic₅₉	1	1	yes	characteristic
125	C₄	C₂×C₁₆×D₅₉	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₁₆×Dic₅₉	1	1	yes	characteristic
128	C₂	#21426391	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₅₉⋊C₃₂

C₂

yes

maximal, characteristic

#21425692

C₂

yes

maximal, characteristic

#21422229

C₂

yes

maximal, complemented

C₃₂×Dic₅₉

C₂

yes

maximal, complemented

C₂×C₅₉×M₆₄

C₂

yes

maximal, characteristic

C₂×#14273667

C₄

yes

characteristic

#16837357

C₄

yes

characteristic

C₂×C₅₉⋊C₃₂

C₄

yes

complemented

C₂×C₅₉⋊C₃₂

C₂²

yes

characteristic

C₂×C₅₉⋊C₃₂

C₂²

yes

characteristic

#16837177

C₄

yes

—

C₁₆×Dic₅₉

C₂²

yes

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×M₆₄

C₄

yes

complemented

C₂×C₃₂×C₅₉

C₂²

yes

—

C₂×#11745664

C₈

yes

characteristic

#14273686

C₈

yes

characteristic

#14273667

C₈

yes

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

#14273637

C₈

yes

—

C₈×Dic₅₉

C₂×C₄

yes

—

C₅₉⋊C₃₂

C₂×C₄

yes

—

C₅₉⋊C₃₂

C₂×C₄

yes

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

C₃₂×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₄²

yes

—

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₂×C₈

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂²⋊Dic₅₉

C₁₆

yes

—

C₄⋊Dic₅₉

C₁₆

yes

—

C₄×Dic₅₉

C₂×C₈

yes

—

#11745664

C₁₆

yes

—

C₂×C₅₉⋊C₈

C₂×C₈

yes

—

C₁₆×C₅₉

C₄²

yes

—

C₁₆×C₅₉

C₂²×C₄

yes

characteristic

C₁₆×C₅₉

C₂²×C₄

yes

characteristic

C₅₉⋊C₁₆

C₂×C₈

yes

—

C₂³×C₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

C₄×C₈

yes

—

C₂×C₄×C₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×C₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×Dic₅₉

C₂×C₁₆

yes

—

C₈×C₅₉

C₄×C₈

yes

—

C₈×C₅₉

C₂×C₄²

yes

characteristic

C₈×C₅₉

C₂²×C₈

yes

characteristic

C₅₉⋊C₈

C₂×C₁₆

yes

—

#13549

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₄×C₁₆

yes

—

C₂²×C₅₉

C₂²×C₁₆

yes

characteristic

C₄×C₅₉

C₄×C₁₆

yes

—

C₄×C₅₉

C₂×C₄×C₈

yes

characteristic

C₄×C₅₉

C₂²×C₁₆

yes

characteristic

Dic₅₉

—

C₂×M₆₄

D₅₉

yes

characteristic

C₂²×C₃₂

—

#2108

—

#1397

—

C₄×C₃₂

—

C₂×C₅₉

—

C₂×C₅₉

C₂×C₄×C₁₆

yes

characteristic

C₂×C₅₉

#2258

yes

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

#405

—

M₆₄

Dic₅₉

yes

complemented

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₉

yes

—

#346

—

C₄×C₁₆

—

C₅₉

#13549

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₄×D₅₉

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₉

yes

characteristic

C₂×C₁₆

C₄×D₅₉

yes

—

C₂×C₁₆

C₂×Dic₅₉

yes

—

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₃₂

C₂×Dic₅₉

yes

—

C₂²×C₄

C₈×D₅₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₈×D₅₉

yes

characteristic

C₂×C₈

C₈×D₅₉

yes

—

100

C₂×C₈

C₄×Dic₅₉

yes

—

101

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₁₆

—

106

C₁₆

C₄×Dic₅₉

yes

—

107

C₁₆

C₂×C₄×D₅₉

yes

characteristic

108

C₁₆

C₂²×Dic₅₉

yes

characteristic

109

C₂³

C₁₆×D₅₉

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

C₈×Dic₅₉

yes

—

112

C₂×C₄

C₁₆×D₅₉

yes

characteristic

113

C₂×C₄

C₁₆×D₅₉

yes

—

114

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

115

C₈

—

116

C₈

C₈×Dic₅₉

yes

—

117

C₈

C₂×C₄×Dic₅₉

yes

characteristic

118

C₈

C₂×C₈×D₅₉

yes

characteristic

119

C₂²

—

120

C₂²

C₁₆×Dic₅₉

yes

—

121

C₂²

C₂×C₁₆×D₅₉

yes

characteristic

122

C₄

—

118

—

123

C₄

C₁₆×Dic₅₉

yes

—

124

C₄

C₂×C₈×Dic₅₉

yes

characteristic

125

C₄

C₂×C₁₆×D₅₉

yes

characteristic

126

C₂

—

127

C₂

C₂×C₁₆×Dic₅₉

yes

characteristic

128

C₂

#21426391

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₉
commutator	C₂×C₅₉	C₂×C₄×C₁₆
Fitting	C₂×C₅₉×M₆₄	C₂
socle	C₂²×C₅₉	C₂²×C₁₆
prime radical	C₂³×C₅₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	C₂×M₆₄	D₅₉
hypocenter	C₅₉	#13549
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₅₉

commutator

C₂×C₅₉

C₂×C₄×C₁₆

Fitting

C₂×C₅₉×M₆₄

C₂

socle

C₂²×C₅₉

C₂²×C₁₆

prime radical

C₂³×C₅₉

C₂×C₁₆

cosocle

C₂×C₁₆×C₅₉

C₂³

hypercenter

C₂×M₆₄

D₅₉

C₅₉

G ▷ C₂×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
3. #21422229	C₂	semidirect
4. C₃₂×Dic₅₉	C₂	semidirect
8. C₂×C₅₉⋊C₃₂	C₄	semidirect
14. C₅₉×M₆₄	C₄	semidirect
73. M₆₄	Dic₅₉	semidirect
80. C₅₉	#13549	semidirect

H ≅ G/N

type

3. #21422229

C₂

semidirect

4. C₃₂×Dic₅₉

C₂

semidirect

8. C₂×C₅₉⋊C₃₂

C₄

semidirect

14. C₅₉×M₆₄

C₄

semidirect

73. M₆₄

Dic₅₉

semidirect

80. C₅₉

#13549

semidirect

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

subgroup	index
C₂²×C₅₉⋊C₃₂	2
#21425692	2
#21422229	2
C₃₂×Dic₅₉	2
C₂×C₅₉×M₆₄	2
#13549	59

subgroup

index

C₂²×C₅₉⋊C₃₂

#21425692

#21422229

C₃₂×Dic₅₉

C₂×C₅₉×M₆₄

#13549

Subgroups — #28508871

Conjugacy classes of subgroups

Complements

Maximal subgroups