Group #28500450 — subgroups

Subgroups — #28500450

Order 15104.

212 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 91 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	#21425933	C₂	1	2	yes	maximal, complemented
2	#21422172	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422069	C₂	1	2	yes	maximal, complemented
4	#21422012	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2178	C₂	1	1	yes	maximal, characteristic
6	#16837819	—	2	2	no	—
7	#16837223	C₂²	1	2	yes	—
8	#16837186	—	2	2	no	—
9	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₅₉×#508	C₄	1	1	yes	characteristic
12	C₅₉×#443	C₂²	1	2	yes	—
13	C₅₉×#368	C₄	1	1	yes	characteristic
14	C₅₉×#359	C₄	1	2	yes	—
15	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	#14273667	—	2	2	no	—
17	#14273667	—	4	2	no	—
18	C₂×C₅₉⋊C₁₆	—	4	2	no	—
19	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
20	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
21	C₄×C₅₉⋊C₈	—	2	1	no	—
22	#135×C₅₉	C₂×C₄	1	2	yes	—
23	#131×C₅₉	—	2	2	no	—
24	Q₃₂×C₅₉	C₈	1	1	yes	characteristic, complemented
25	SD₃₂×C₅₉	C₈	1	2	yes	complemented
26	D₁₆×C₅₉	C₈	1	1	yes	characteristic, complemented
27	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
29	#108×C₅₉	C₂×C₄	1	2	yes	—
30	#104×C₅₉	—	2	2	no	—
31	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	#41×C₅₉	—	2	2	no	—
33	Q₁₆×C₅₉	C₂×C₈	1	2	yes	—
34	SD₁₆×C₅₉	—	2	2	no	—
35	D₈×C₅₉	C₂×C₈	1	2	yes	—
36	M₁₆×C₅₉	—	2	2	no	—
37	M₁₆×C₅₉	—	4	2	no	—
38	C₂×C₈×C₅₉	—	4	2	no	—
39	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₅₉	#41	1	1	yes	characteristic
42	C₂×C₅₉⋊C₈	—	2	2	no	—
43	C₁₆×C₅₉	—	2	1	no	—
44	C₁₆×C₅₉	C₂×C₈	1	1	yes	characteristic
45	C₁₆×C₅₉	C₂×C₈	1	1	yes	characteristic
46	C₅₉⋊C₁₆	—	2	1	no	—
47	C₅₉⋊C₁₆	—	4	2	no	—
48	C₅₉⋊C₁₆	D₈	1	2	yes	—
49	Q₈×C₅₉	—	2	2	no	—
50	D₄×C₅₉	—	2	2	no	—
51	D₄×C₅₉	—	4	2	no	—
52	C₂×C₄×C₅₉	—	2	1	no	—
53	C₂×C₄×C₅₉	—	4	2	no	—
54	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
55	C₈×C₅₉	—	4	2	no	—
56	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₅₉	#131	1	1	yes	characteristic
58	C₈×C₅₉	C₂×D₈	1	1	yes	characteristic
59	C₈×C₅₉	#135	1	1	yes	characteristic
60	C₅₉⋊C₈	—	4	2	no	—
61	#12047	—	59	1	no	maximal, Sylow
62	C₂²×C₅₉	—	4	2	no	—
63	C₂²×C₅₉	#443	1	1	yes	characteristic
64	C₄×C₅₉	—	2	2	no	—
65	C₄×C₅₉	—	4	2	no	—
66	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
67	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
68	#2178	D₅₉	1	1	yes	characteristic
69	#2170	—	59	2	no	—
70	#1373	—	59	1	no	—
71	#1335	—	59	2	no	—
72	#1311	—	59	1	no	—
73	C₂×C₅₉	—	2	1	no	—
74	C₂×C₅₉	—	8	2	no	—
75	C₂×C₅₉	C₈×D₈	1	1	yes	characteristic
76	#508	Dic₅₉	1	1	yes	characteristic
77	#504	—	118	2	no	—
78	#443	C₂×D₅₉	1	2	yes	—
79	#368	Dic₅₉	1	1	yes	characteristic
80	#364	—	59	2	no	—
81	#359	Dic₅₉	1	2	yes	—
82	#350	—	118	2	no	—
83	C₄×C₁₆	—	59	1	no	—
84	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
85	C₈²	—	59	1	no	—
86	C₅₉	#12047	1	1	yes	characteristic, complemented, Sylow
87	#135	C₂×Dic₅₉	1	2	yes	—
88	#131	—	2	2	no	—
89	Q₃₂	C₅₉⋊C₈	1	1	yes	characteristic, complemented
90	SD₃₂	C₅₉⋊C₈	1	2	yes	complemented
91	D₁₆	C₅₉⋊C₈	1	1	yes	characteristic, complemented
92	M₃₂	—	118	2	no	—
93	M₃₂	—	236	2	no	—
94	C₂×C₁₆	—	59	1	no	—
95	C₂×C₁₆	—	59	1	no	—
96	C₂×C₁₆	—	236	2	no	—
97	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
98	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
99	#108	C₂×Dic₅₉	1	2	yes	—
100	#104	—	2	2	no	—
101	C₄×C₈	—	118	1	no	—
102	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
103	#41	—	2	2	no	—
104	Q₁₆	C₂×C₅₉⋊C₈	1	2	yes	—
105	SD₁₆	—	2	2	no	—
106	D₈	C₂×C₅₉⋊C₈	1	2	yes	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	4	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	118	2	no	—
111	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
112	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
113	C₄²	#11745687	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	59	2	no	—
116	C₁₆	—	118	1	no	—
117	C₁₆	—	236	2	no	—
118	C₁₆	C₂×C₅₉⋊C₈	1	1	yes	characteristic
119	C₁₆	C₂×C₅₉⋊C₈	1	1	yes	characteristic
120	Q₈	—	2	2	no	—
121	D₄	—	2	2	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
126	C₈	—	4	2	no	—
127	C₈	—	236	2	no	—
128	C₈	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
129	C₈	#14273734	1	1	yes	characteristic
130	C₈	D₈×D₅₉	1	1	yes	characteristic
131	C₈	#14273746	1	1	yes	characteristic
132	C₂²	—	4	2	no	—
133	C₂²	#16837519	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	4	2	no	—
136	C₄	D₄×C₅₉⋊C₈	1	1	yes	characteristic
137	C₄	D₈×Dic₅₉	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	8	2	no	—
140	C₂	D₈×C₅₉⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425933

C₂

yes

maximal, complemented

#21422172

C₂

yes

maximal, characteristic, complemented

#21422069

C₂

yes

maximal, complemented

#21422012

C₂

yes

maximal, characteristic, complemented

C₅₉×#2178

C₂

yes

maximal, characteristic

#16837819

—

#16837223

C₂²

yes

—

#16837186

—

C₄×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₈×C₅₉⋊C₈

C₂²

yes

characteristic

C₅₉×#508

C₄

yes

characteristic

C₅₉×#443

C₂²

yes

—

C₅₉×#368

C₄

yes

characteristic

C₅₉×#359

C₄

yes

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

#14273667

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

D₄

yes

characteristic

C₄×C₅₉⋊C₈

—

#135×C₅₉

C₂×C₄

yes

—

#131×C₅₉

—

Q₃₂×C₅₉

C₈

yes

characteristic, complemented

SD₃₂×C₅₉

C₈

yes

complemented

D₁₆×C₅₉

C₈

yes

characteristic, complemented

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

#108×C₅₉

C₂×C₄

yes

—

#104×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

#41×C₅₉

—

Q₁₆×C₅₉

C₂×C₈

yes

—

SD₁₆×C₅₉

—

D₈×C₅₉

C₂×C₈

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

#41

yes

characteristic

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

C₂×C₈

yes

characteristic

C₁₆×C₅₉

C₂×C₈

yes

characteristic

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

D₈

yes

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

C₂²×C₈

yes

characteristic

C₈×C₅₉

#131

yes

characteristic

C₈×C₅₉

C₂×D₈

yes

characteristic

C₈×C₅₉

#135

yes

characteristic

C₅₉⋊C₈

—

#12047

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#443

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₈×D₄

yes

characteristic

C₄×C₅₉

C₄×D₈

yes

characteristic

#2178

D₅₉

yes

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₈×D₈

yes

characteristic

#508

Dic₅₉

yes

characteristic

#504

—

118

—

#443

C₂×D₅₉

yes

—

#368

Dic₅₉

yes

characteristic

#364

—

#359

Dic₅₉

yes

—

#350

—

118

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₈²

—

C₅₉

#12047

yes

characteristic, complemented, Sylow

#135

C₂×Dic₅₉

yes

—

#131

—

Q₃₂

C₅₉⋊C₈

yes

characteristic, complemented

SD₃₂

C₅₉⋊C₈

yes

complemented

D₁₆

C₅₉⋊C₈

yes

characteristic, complemented

M₃₂

—

118

—

M₃₂

—

236

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

236

—

C₂×C₁₆

C₂×Dic₅₉

yes

characteristic

C₂×C₁₆

C₂×Dic₅₉

yes

characteristic

#108

C₂×Dic₅₉

yes

—

100

#104

—

101

C₄×C₈

—

118

—

102

C₄×C₈

C₂²×D₅₉

yes

characteristic

103

#41

—

104

Q₁₆

C₂×C₅₉⋊C₈

yes

—

105

SD₁₆

—

106

D₈

C₂×C₅₉⋊C₈

yes

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

118

—

111

C₂×C₈

D₄×D₅₉

yes

characteristic

112

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

113

C₄²

#11745687

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

118

—

117

C₁₆

—

236

—

118

C₁₆

C₂×C₅₉⋊C₈

yes

characteristic

119

C₁₆

C₂×C₅₉⋊C₈

yes

characteristic

120

Q₈

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

D₄×Dic₅₉

yes

characteristic

126

C₈

—

127

C₈

—

236

—

128

C₈

C₂²×C₅₉⋊C₈

yes

characteristic

129

C₈

#14273734

yes

characteristic

130

C₈

D₈×D₅₉

yes

characteristic

131

C₈

#14273746

yes

characteristic

132

C₂²

—

133

C₂²

#16837519

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

D₄×C₅₉⋊C₈

yes

characteristic

137

C₄

D₈×Dic₅₉

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

D₈×C₅₉⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₅₉
commutator	C₈×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#2178	C₂
socle	C₂×C₅₉	C₈×D₈
prime radical	C₅₉×#508	C₄
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#2178	D₅₉
hypocenter	C₅₉	#12047
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

subgroup

G/H

center

C₈

D₈×D₅₉

commutator

C₈×C₅₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#2178

C₂

socle

C₂×C₅₉

C₈×D₈

prime radical

C₅₉×#508

C₄

cosocle

C₄×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

N	H ≅ G/N	type
1. #21425933	C₂	semidirect
2. #21422172	C₂	semidirect
3. #21422069	C₂	semidirect
4. #21422012	C₂	semidirect
24. Q₃₂×C₅₉	C₈	semidirect
25. SD₃₂×C₅₉	C₈	semidirect
26. D₁₆×C₅₉	C₈	semidirect
86. C₅₉	#12047	semidirect
89. Q₃₂	C₅₉⋊C₈	semidirect
90. SD₃₂	C₅₉⋊C₈	semidirect
91. D₁₆	C₅₉⋊C₈	semidirect

H ≅ G/N

type

1. #21425933

C₂

semidirect

2. #21422172

C₂

semidirect

3. #21422069

C₂

semidirect

4. #21422012

C₂

semidirect

24. Q₃₂×C₅₉

C₈

semidirect

25. SD₃₂×C₅₉

C₈

semidirect

26. D₁₆×C₅₉

C₈

semidirect

86. C₅₉

#12047

semidirect

89. Q₃₂

C₅₉⋊C₈

semidirect

90. SD₃₂

C₅₉⋊C₈

semidirect

91. D₁₆

C₅₉⋊C₈

semidirect

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

subgroup	index
#21425933	2
#21422172	2
#21422069	2
#21422012	2
C₅₉×#2178	2
#12047	59

subgroup

index

C₅₉×#2178

Subgroups — #28500450

Conjugacy classes of subgroups

Complements

Maximal subgroups