Group #28500444 — subgroups

Subgroups — #28500444

Order 15104.

216 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 75 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	#21425978	C₂	1	1	yes	maximal, characteristic
2	#21425932	C₂	1	2	yes	maximal, complemented
3	#21422068	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837834	C₄	1	1	yes	characteristic
7	#16837820	—	2	2	no	—
8	#16837520	C₂²	1	2	yes	—
9	#16837232	C₄	1	1	yes	characteristic
10	#16837210	C₄	1	2	yes	—
11	#16837187	—	2	2	no	—
12	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₅₉×#364	C₂²	1	2	yes	—
15	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	#14273749	C₂×C₄	1	2	yes	—
17	#14273733	—	2	2	no	—
18	Dic₄₇₂	C₈	1	1	yes	characteristic, complemented
19	#14273672	C₈	1	2	yes	complemented
20	D₉₄₄	C₈	1	1	yes	characteristic, complemented
21	#14273666	—	4	2	no	—
22	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
24	C₁₆×D₅₉	—	4	2	no	—
25	#14273663	C₂×C₄	1	2	yes	—
26	#14273655	—	2	2	no	—
27	C₄×C₅₉⋊C₈	—	2	1	no	—
28	M₃₂×C₅₉	—	2	2	no	—
29	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	#11745685	—	2	2	no	—
33	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₅₉	#41	1	1	yes	characteristic
36	#11745664	—	2	2	no	—
37	C₂×C₅₉⋊C₈	—	2	2	no	—
38	Dic₂₃₆	C₂×C₈	1	2	yes	—
39	D₄₇₂	C₂×C₈	1	2	yes	—
40	#11745660	—	2	2	no	—
41	#11745659	—	4	2	no	—
42	C₈×D₅₉	—	4	2	no	—
43	C₁₆×C₅₉	—	2	1	no	—
44	C₁₆×C₅₉	—	4	2	no	—
45	C₁₆×C₅₉	C₂×C₈	1	1	yes	characteristic
46	C₁₆×C₅₉	C₂×C₈	1	1	yes	characteristic
47	C₅₉⋊C₁₆	—	2	1	no	—
48	C₅₉⋊C₁₆	D₈	1	1	yes	characteristic
49	C₅₉⋊C₁₆	D₈	1	1	yes	characteristic
50	C₂×C₄×C₅₉	—	2	1	no	—
51	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
52	C₂²⋊D₅₉	—	4	2	no	—
53	D₂₃₆	—	2	2	no	—
54	C₄×D₅₉	—	4	2	no	—
55	Dic₁₁₈	—	2	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	—	2	4	no	—
61	C₅₉⋊C₈	—	4	2	no	—
62	#12046	—	59	1	no	maximal, Sylow
63	C₂²×C₅₉	#443	1	1	yes	characteristic
64	C₂×D₅₉	—	4	2	no	—
65	C₄×C₅₉	—	2	2	no	—
66	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
67	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
68	Dic₅₉	—	4	2	no	—
69	#2178	—	59	1	no	—
70	#2170	—	59	2	no	—
71	#1373	D₅₉	1	1	yes	characteristic, complemented
72	#1335	—	59	2	no	—
73	C₈×C₁₆	—	59	1	no	—
74	C₂×C₅₉	—	2	1	no	—
75	C₂×C₅₉	C₈×D₈	1	1	yes	characteristic
76	D₅₉	—	8	2	no	—
77	#508	—	59	1	no	—
78	#504	—	118	2	no	—
79	#443	—	59	2	no	—
80	#368	—	59	1	no	—
81	#364	C₂×D₅₉	1	2	yes	—
82	#359	—	59	2	no	—
83	#350	—	118	2	no	—
84	C₄×C₁₆	—	59	1	no	—
85	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
86	C₈²	—	59	1	no	—
87	C₅₉	#12046	1	1	yes	characteristic, complemented, Sylow
88	#135	—	59	2	no	—
89	#131	—	118	2	no	—
90	Q₃₂	—	59	1	no	—
91	SD₃₂	—	59	2	no	—
92	D₁₆	—	59	1	no	—
93	M₃₂	—	2	2	no	—
94	M₃₂	—	236	2	no	—
95	C₂×C₁₆	—	59	1	no	—
96	C₂×C₁₆	—	59	1	no	—
97	C₂×C₁₆	—	236	2	no	—
98	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
99	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
100	#108	—	59	2	no	—
101	#104	—	118	2	no	—
102	C₄×C₈	—	118	1	no	—
103	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
104	#41	—	118	2	no	—
105	Q₁₆	—	59	2	no	—
106	SD₁₆	—	118	2	no	—
107	D₈	—	59	2	no	—
108	M₁₆	—	118	2	no	—
109	M₁₆	—	236	2	no	—
110	C₂×C₈	—	118	2	no	—
111	C₂×C₈	—	236	2	no	—
112	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
113	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
114	C₄²	#11745689	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	4	2	no	—
117	C₁₆	—	59	1	no	—
118	C₁₆	—	59	1	no	—
119	C₁₆	—	118	1	no	—
120	C₁₆	C₈×D₅₉	1	1	yes	characteristic
121	C₁₆	C₈×D₅₉	1	1	yes	characteristic
122	Q₈	—	118	2	no	—
123	D₄	—	118	2	no	—
124	D₄	—	236	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	236	2	no	—
127	C₂×C₄	#14273689	1	1	yes	characteristic
128	C₈	—	118	4	no	—
129	C₈	—	236	2	no	—
130	C₈	C₂×C₈×D₅₉	1	1	yes	characteristic
131	C₈	#14273735	1	1	yes	characteristic
132	C₈	D₈×D₅₉	1	1	yes	characteristic
133	C₈	#14273750	1	1	yes	characteristic
134	C₂²	—	236	2	no	—
135	C₂²	#16837518	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	236	2	no	—
138	C₄	#16837467	1	1	yes	characteristic
139	C₄	#16837484	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	472	2	no	—
142	C₂	#21422890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425978

C₂

yes

maximal, characteristic

#21425932

C₂

yes

maximal, complemented

#21422068

C₂

yes

maximal, complemented

C₁₆×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₅₉×#1373

C₂

yes

maximal, characteristic, complemented

#16837834

C₄

yes

characteristic

#16837820

—

#16837520

C₂²

yes

—

#16837232

C₄

yes

characteristic

#16837210

C₄

yes

—

#16837187

—

C₄×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₈×C₅₉⋊C₈

C₂²

yes

characteristic

C₅₉×#364

C₂²

yes

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

#14273749

C₂×C₄

yes

—

#14273733

—

Dic₄₇₂

C₈

yes

characteristic, complemented

#14273672

C₈

yes

complemented

D₉₄₄

C₈

yes

characteristic, complemented

#14273666

—

C₂×C₅₉⋊C₁₆

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

D₄

yes

characteristic

C₁₆×D₅₉

—

#14273663

C₂×C₄

yes

—

#14273655

—

C₄×C₅₉⋊C₈

—

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂³

yes

characteristic

#11745685

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

C₄²×C₅₉

yes

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

C₂×C₈

yes

—

D₄₇₂

C₂×C₈

yes

—

#11745660

—

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

C₂×C₈

yes

characteristic

C₁₆×C₅₉

C₂×C₈

yes

characteristic

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

D₈

yes

characteristic

C₅₉⋊C₁₆

D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂²⋊D₅₉

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

C₂²×C₈

yes

characteristic

C₈×C₅₉

#131

yes

characteristic

C₈×C₅₉

C₂×D₈

yes

characteristic

C₈×C₅₉

#135

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₈×D₄

yes

characteristic

C₄×C₅₉

C₄×D₈

yes

characteristic

Dic₅₉

—

#2178

—

#2170

—

#1373

D₅₉

yes

characteristic, complemented

#1335

—

C₈×C₁₆

—

C₂×C₅₉

—

C₂×C₅₉

C₈×D₈

yes

characteristic

D₅₉

—

#508

—

#504

—

118

—

#443

—

#368

—

#364

C₂×D₅₉

yes

—

#359

—

#350

—

118

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₈²

—

C₅₉

#12046

yes

characteristic, complemented, Sylow

#135

—

#131

—

118

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

236

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

236

—

C₂×C₁₆

C₄×D₅₉

yes

characteristic

C₂×C₁₆

C₄×D₅₉

yes

characteristic

100

#108

—

101

#104

—

118

—

102

C₄×C₈

—

118

—

103

C₄×C₈

C₂²×D₅₉

yes

characteristic

104

#41

—

118

—

105

Q₁₆

—

106

SD₁₆

—

118

—

107

D₈

—

108

M₁₆

—

118

—

109

M₁₆

—

236

—

110

C₂×C₈

—

118

—

111

C₂×C₈

—

236

—

112

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

113

C₂×C₈

D₄×D₅₉

yes

characteristic

114

C₄²

#11745689

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

119

C₁₆

—

118

—

120

C₁₆

C₈×D₅₉

yes

characteristic

121

C₁₆

C₈×D₅₉

yes

characteristic

122

Q₈

—

118

—

123

D₄

—

118

—

124

D₄

—

236

—

125

C₂×C₄

—

126

C₂×C₄

—

236

—

127

C₂×C₄

#14273689

yes

characteristic

128

C₈

—

118

—

129

C₈

—

236

—

130

C₈

C₂×C₈×D₅₉

yes

characteristic

131

C₈

#14273735

yes

characteristic

132

C₈

D₈×D₅₉

yes

characteristic

133

C₈

#14273750

yes

characteristic

134

C₂²

—

236

—

135

C₂²

#16837518

yes

characteristic

136

C₄

—

137

C₄

—

236

—

138

C₄

#16837467

yes

characteristic

139

C₄

#16837484

yes

characteristic

140

C₂

—

141

C₂

—

472

—

142

C₂

#21422890

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₅₉
commutator	C₈×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1373	C₂
socle	C₂×C₅₉	C₈×D₈
prime radical	#16837834	C₄
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#1373	D₅₉
hypocenter	C₅₉	#12046
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₈

D₈×D₅₉

commutator

C₈×C₅₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1373

C₂

socle

C₂×C₅₉

C₈×D₈

prime radical

#16837834

C₄

cosocle

C₄×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
2. #21425932	C₂	semidirect
3. #21422068	C₂	semidirect
4. C₁₆×C₅₉⋊C₈	C₂	semidirect
5. C₅₉×#1373	C₂	semidirect
18. Dic₄₇₂	C₈	semidirect
19. #14273672	C₈	semidirect
20. D₉₄₄	C₈	semidirect
71. #1373	D₅₉	semidirect
87. C₅₉	#12046	semidirect

H ≅ G/N

type

2. #21425932

C₂

semidirect

3. #21422068

C₂

semidirect

4. C₁₆×C₅₉⋊C₈

C₂

semidirect

5. C₅₉×#1373

C₂

semidirect

18. Dic₄₇₂

C₈

semidirect

19. #14273672

C₈

semidirect

20. D₉₄₄

C₈

semidirect

71. #1373

D₅₉

semidirect

87. C₅₉

#12046

semidirect

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

subgroup	index
#21425978	2
#21425932	2
#21422068	2
C₁₆×C₅₉⋊C₈	2
C₅₉×#1373	2
#12046	59

subgroup

index

#21425978

#21425932

#21422068

C₁₆×C₅₉⋊C₈

C₅₉×#1373

#12046

Subgroups — #28500444

Conjugacy classes of subgroups

Complements

Maximal subgroups