Group #21437247 — subgroups

Subgroups — #21437247

Order 7552.

350 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 129 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	#16838219	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837843	C₂	1	3	yes	maximal, complemented
3	#16837842	C₂	1	3	yes	maximal, complemented
4	#16837832	C₂	1	3	yes	maximal, complemented
5	C₂×#14273673	C₂	1	3	yes	maximal, complemented
6	#16837819	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#578	C₂	1	1	yes	maximal, characteristic, complemented
8	#14273774	—	2	1	no	—
9	#14273760	—	2	3	no	—
10	#14273749	C₂²	1	3	yes	—
11	C₂×Dic₂₃₆	C₂²	1	3	yes	—
12	#14273673	C₂²	1	3	yes	—
13	#14273673	C₂²	1	6	yes	—
14	#14273671	C₂²	1	3	yes	—
15	#14273670	C₂²	1	3	yes	—
16	#14273668	C₂²	1	1	yes	characteristic
17	#14273667	C₂²	1	3	yes	—
18	C₂×C₅₉⋊C₁₆	C₂²	1	3	yes	—
19	2₋¹⁺⁴×C₅₉	—	2	1	no	—
20	#137×C₅₉	—	2	3	no	—
21	#135×C₅₉	C₂²	1	3	yes	—
22	C₂×Q₁₆×C₅₉	C₂²	1	3	yes	—
23	#131×C₅₉	C₂²	1	1	yes	characteristic
24	#41×C₅₉	—	2	3	no	—
25	#41×C₅₉	—	4	3	no	—
26	#41×C₅₉	D₄	1	1	yes	characteristic
27	C₂×Q₈×C₅₉	—	2	3	no	—
28	C₂×Q₈×C₅₉	—	4	1	no	—
29	Q₈×D₅₉	—	4	1	no	—
30	#11745687	—	4	3	no	—
31	#11745685	—	2	3	no	—
32	C₂×Dic₁₁₈	—	2	3	no	—
33	Q₁₆×C₅₉	—	2	3	no	—
34	Q₁₆×C₅₉	C₂³	1	3	yes	—
35	SD₁₆×C₅₉	—	2	3	no	—
36	D₈×C₅₉	C₂³	1	1	yes	characteristic
37	M₁₆×C₅₉	D₄	1	3	yes	—
38	C₂×C₈×C₅₉	C₂³	1	3	yes	—
39	Dic₂₃₆	—	2	3	no	—
40	Dic₂₃₆	C₂³	1	3	yes	—
41	D₄₇₂	C₂³	1	1	yes	characteristic
42	#11745660	—	2	3	no	—
43	C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
44	C₅₉⋊C₁₆	C₂³	1	3	yes	—
45	Q₈×C₅₉	—	2	3	no	—
46	Q₈×C₅₉	—	4	3	no	—
47	Q₈×C₅₉	D₈	1	1	yes	characteristic
48	D₄×C₅₉	—	2	1	no	—
49	D₄×C₅₉	—	4	3	no	—
50	D₄×C₅₉	D₈	1	3	yes	—
51	C₂×C₄×C₅₉	—	4	3	no	—
52	C₂×C₄×C₅₉	—	4	3	no	—
53	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
54	C₂²⋊D₅₉	—	4	3	no	—
55	C₂×Dic₅₉	—	4	3	no	—
56	D₂₃₆	—	2	1	no	—
57	C₄×D₅₉	—	4	3	no	—
58	Dic₁₁₈	—	2	3	no	—
59	Dic₁₁₈	—	4	3	no	—
60	C₈×C₅₉	C₂×D₄	1	3	yes	—
61	C₈×C₅₉	C₂⁴	1	1	yes	characteristic
62	C₂²×C₅₉	—	4	1	no	—
63	C₂²×C₅₉	C₂×D₈	1	3	yes	—
64	C₂×D₅₉	—	4	1	no	—
65	C₄×C₅₉	—	4	3	no	—
66	C₄×C₅₉	C₂×D₈	1	3	yes	—
67	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
68	Dic₅₉	—	4	3	no	—
69	#3417	—	59	1	no	maximal, Sylow
70	C₂×C₅₉	—	2	3	no	—
71	C₂×C₅₉	—	8	1	no	—
72	C₂×C₅₉	C₂²×D₈	1	1	yes	characteristic
73	D₅₉	—	8	1	no	—
74	#578	—	59	1	no	—
75	#578	D₅₉	1	1	yes	characteristic, complemented
76	#510	—	59	3	no	—
77	#510	—	59	3	no	—
78	#508	—	59	3	no	—
79	C₂×Q₃₂	—	59	3	no	—
80	#504	—	59	1	no	—
81	C₅₉	#3417	1	1	yes	characteristic, complemented, Sylow
82	2₋¹⁺⁴	—	2	1	no	—
83	2₋¹⁺⁴	—	118	1	no	—
84	#137	—	2	3	no	—
85	#137	—	118	3	no	—
86	#135	—	59	3	no	—
87	#135	C₂×D₅₉	1	3	yes	—
88	C₂×Q₁₆	—	59	3	no	—
89	C₂×Q₁₆	C₂×D₅₉	1	3	yes	—
90	#131	C₂×D₅₉	1	1	yes	characteristic
91	Q₃₂	—	59	3	no	—
92	Q₃₂	—	59	6	no	—
93	SD₃₂	—	59	3	no	—
94	SD₃₂	—	59	3	no	—
95	D₁₆	—	59	1	no	—
96	M₃₂	—	59	3	no	—
97	C₂×C₁₆	—	59	3	no	—
98	#41	—	2	3	no	—
99	#41	—	4	3	no	—
100	#41	—	118	3	no	—
101	#41	—	236	3	no	—
102	#41	C₂²⋊D₅₉	1	1	yes	characteristic
103	C₂×Q₈	—	2	3	no	—
104	C₂×Q₈	—	4	1	no	—
105	C₂×Q₈	—	118	3	no	—
106	C₂×Q₈	—	236	1	no	—
107	Q₁₆	—	2	3	no	—
108	Q₁₆	—	59	3	no	—
109	Q₁₆	—	118	3	no	—
110	Q₁₆	C₂²×D₅₉	1	3	yes	—
111	SD₁₆	—	2	3	no	—
112	SD₁₆	—	118	3	no	—
113	D₈	—	59	1	no	—
114	D₈	C₂²×D₅₉	1	1	yes	characteristic
115	M₁₆	C₂²⋊D₅₉	1	3	yes	—
116	C₂×C₈	C₂²×D₅₉	1	3	yes	—
117	C₁₆	—	59	1	no	—
118	C₁₆	—	59	3	no	—
119	Q₈	—	2	3	no	—
120	Q₈	—	4	3	no	—
121	Q₈	—	118	3	no	—
122	Q₈	—	236	3	no	—
123	Q₈	#11745669	1	1	yes	characteristic
124	D₄	—	2	1	no	—
125	D₄	—	4	3	no	—
126	D₄	—	118	1	no	—
127	D₄	—	236	3	no	—
128	D₄	#11745669	1	3	yes	—
129	C₂×C₄	—	4	3	no	—
130	C₂×C₄	—	4	3	no	—
131	C₂×C₄	—	236	3	no	—
132	C₂×C₄	—	236	3	no	—
133	C₂×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
134	C₈	C₂×C₂²⋊D₅₉	1	3	yes	—
135	C₈	C₂³×D₅₉	1	1	yes	characteristic
136	C₂²	—	4	1	no	—
137	C₂²	—	236	1	no	—
138	C₂²	C₂×#11745669	1	3	yes	—
139	C₄	—	4	3	no	—
140	C₄	—	236	3	no	—
141	C₄	C₂×#11745669	1	3	yes	—
142	C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
143	C₂	—	2	3	no	—
144	C₂	—	8	1	no	—
145	C₂	—	472	1	no	—
146	C₂	C₂²×#11745669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838219

C₂

yes

maximal, characteristic, complemented

#16837843

C₂

yes

maximal, complemented

#16837842

C₂

yes

maximal, complemented

#16837832

C₂

yes

maximal, complemented

C₂×#14273673

C₂

yes

maximal, complemented

#16837819

C₂

yes

maximal, characteristic, complemented

C₅₉×#578

C₂

yes

maximal, characteristic, complemented

#14273774

—

#14273760

—

#14273749

C₂²

yes

—

C₂×Dic₂₃₆

C₂²

yes

—

#14273673

C₂²

yes

—

#14273673

C₂²

yes

—

#14273671

C₂²

yes

—

#14273670

C₂²

yes

—

#14273668

C₂²

yes

characteristic

#14273667

C₂²

yes

—

C₂×C₅₉⋊C₁₆

C₂²

yes

—

2₋¹⁺⁴×C₅₉

—

#137×C₅₉

—

#135×C₅₉

C₂²

yes

—

C₂×Q₁₆×C₅₉

C₂²

yes

—

#131×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

#41×C₅₉

—

#41×C₅₉

D₄

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

Q₈×D₅₉

—

#11745687

—

#11745685

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

C₂³

yes

—

SD₁₆×C₅₉

—

D₈×C₅₉

C₂³

yes

characteristic

M₁₆×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

Dic₂₃₆

—

Dic₂₃₆

C₂³

yes

—

D₄₇₂

C₂³

yes

characteristic

#11745660

—

C₅₉⋊C₁₆

C₂³

yes

characteristic

C₅₉⋊C₁₆

C₂³

yes

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

D₈

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

D₈

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂×D₄

yes

—

C₈×C₅₉

C₂⁴

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

C₂×D₈

yes

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

—

C₄×C₅₉

C₂²×D₄

yes

characteristic

Dic₅₉

—

#3417

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₂²×D₈

yes

characteristic

D₅₉

—

#578

—

#578

D₅₉

yes

characteristic, complemented

#510

—

#510

—

#508

—

C₂×Q₃₂

—

#504

—

C₅₉

#3417

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

2₋¹⁺⁴

—

118

—

#137

—

#137

—

118

—

#135

—

#135

C₂×D₅₉

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₅₉

yes

—

#131

C₂×D₅₉

yes

characteristic

Q₃₂

—

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

#41

—

#41

—

100

#41

—

118

—

101

#41

—

236

—

102

#41

C₂²⋊D₅₉

yes

characteristic

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

118

—

106

C₂×Q₈

—

236

—

107

Q₁₆

—

108

Q₁₆

—

109

Q₁₆

—

118

—

110

Q₁₆

C₂²×D₅₉

yes

—

111

SD₁₆

—

112

SD₁₆

—

118

—

113

D₈

—

114

D₈

C₂²×D₅₉

yes

characteristic

115

M₁₆

C₂²⋊D₅₉

yes

—

116

C₂×C₈

C₂²×D₅₉

yes

—

117

C₁₆

—

118

C₁₆

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

118

—

122

Q₈

—

236

—

123

Q₈

#11745669

yes

characteristic

124

D₄

—

125

D₄

—

126

D₄

—

118

—

127

D₄

—

236

—

128

D₄

#11745669

yes

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

236

—

132

C₂×C₄

—

236

—

133

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

134

C₈

C₂×C₂²⋊D₅₉

yes

—

135

C₈

C₂³×D₅₉

yes

characteristic

136

C₂²

—

137

C₂²

—

236

—

138

C₂²

C₂×#11745669

yes

—

139

C₄

—

140

C₄

—

236

—

141

C₄

C₂×#11745669

yes

—

142

C₄

C₂²×C₂²⋊D₅₉

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

472

—

146

C₂

C₂²×#11745669

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#11745669
commutator	C₈×C₅₉	C₂⁴
Frattini	C₈	C₂³×D₅₉
Fitting	C₅₉×#578	C₂
socle	C₂×C₅₉	C₂²×D₈
hypercenter	#578	D₅₉
hypocenter	C₅₉	#3417
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #578

subgroup

G/H

center

C₂

C₂²×#11745669

commutator

C₈×C₅₉

C₂⁴

Frattini

C₈

C₂³×D₅₉

Fitting

C₅₉×#578

C₂

socle

C₂×C₅₉

C₂²×D₈

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #578

N	H ≅ G/N	type
1. #16838219	C₂	semidirect
2. #16837843	C₂	semidirect
3. #16837842	C₂	semidirect
4. #16837832	C₂	semidirect
5. C₂×#14273673	C₂	semidirect
6. #16837819	C₂	semidirect
7. C₅₉×#578	C₂	semidirect
75. #578	D₅₉	semidirect
81. C₅₉	#3417	semidirect

H ≅ G/N

type

1. #16838219

C₂

semidirect

2. #16837843

C₂

semidirect

3. #16837842

C₂

semidirect

4. #16837832

C₂

semidirect

5. C₂×#14273673

C₂

semidirect

6. #16837819

C₂

semidirect

7. C₅₉×#578

C₂

semidirect

75. #578

D₅₉

semidirect

81. C₅₉

#3417

semidirect

Subgroups — #21437247

Conjugacy classes of subgroups

Complements