Group #21425979 — subgroups

Subgroups — #21425979

Order 7552.

144 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16837832	C₂	1	1	yes	maximal, characteristic
2	#16837519	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837518	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837231	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837210	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837177	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#359	C₂	1	1	yes	maximal, characteristic, complemented
8	#14273749	C₂²	1	1	yes	characteristic
9	#14273735	—	2	1	no	—
10	#14273734	—	2	1	no	—
11	#14273673	C₄	1	1	yes	characteristic, complemented
12	#14273671	C₄	1	1	yes	characteristic, complemented
13	#14273670	C₄	1	1	yes	characteristic, complemented
14	#14273668	C₄	1	1	yes	characteristic, complemented
15	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14273663	C₂²	1	1	yes	characteristic
17	#14273654	—	2	1	no	—
18	#14273653	—	2	1	no	—
19	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
20	#135×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
22	#108×C₅₉	C₂²	1	1	yes	characteristic
23	#41×C₅₉	—	2	1	no	—
24	#11745685	—	2	1	no	—
25	Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
26	SD₁₆×C₅₉	—	2	1	no	—
27	D₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	M₁₆×C₅₉	—	2	1	no	—
29	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₄×Dic₅₉	D₄	1	1	yes	characteristic
31	#11745664	—	2	1	no	—
32	#11745664	—	4	1	no	—
33	C₂×C₅₉⋊C₈	—	4	1	no	—
34	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
35	Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
36	D₄₇₂	C₂×C₄	1	1	yes	characteristic
37	#11745660	—	2	1	no	—
38	#11745659	—	4	1	no	—
39	C₈×D₅₉	—	4	1	no	—
40	C₁₆×C₅₉	—	2	1	no	—
41	C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
43	Q₈×C₅₉	—	2	1	no	—
44	D₄×C₅₉	—	2	1	no	—
45	D₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	—	4	1	no	—
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₅₉	—	4	1	no	—
49	C₂×Dic₅₉	—	2	1	no	—
50	D₂₃₆	—	2	1	no	—
51	C₄×D₅₉	—	4	1	no	—
52	Dic₁₁₈	—	2	1	no	—
53	C₈×C₅₉	—	4	1	no	—
54	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₈×C₅₉	#41	1	1	yes	characteristic
56	C₅₉⋊C₈	—	4	1	no	—
57	C₅₉⋊C₈	D₈	1	1	yes	characteristic
58	C₅₉⋊C₈	D₈	1	1	yes	characteristic
59	C₂²×C₅₉	—	4	1	no	—
60	C₂²×C₅₉	#135	1	1	yes	characteristic
61	C₂×D₅₉	—	4	1	no	—
62	C₄×C₅₉	—	4	1	no	—
63	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
64	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
65	Dic₅₉	—	4	1	no	—
66	Dic₅₉	—	4	1	no	—
67	#2179	—	59	1	no	maximal, Sylow
68	C₂×C₅₉	—	2	1	no	—
69	C₂×C₅₉	—	8	1	no	—
70	C₂×C₅₉	C₄×D₈	1	1	yes	characteristic
71	D₅₉	—	8	1	no	—
72	#508	—	59	1	no	—
73	#443	—	59	1	no	—
74	#443	—	59	1	no	—
75	#368	—	59	1	no	—
76	#359	—	59	1	no	—
77	#359	D₅₉	1	1	yes	characteristic, complemented
78	#346	—	59	1	no	—
79	C₅₉	#2179	1	1	yes	characteristic, complemented, Sylow
80	#135	—	59	1	no	—
81	#135	C₂×D₅₉	1	1	yes	characteristic
82	#131	—	118	1	no	—
83	#131	—	118	1	no	—
84	Q₃₂	—	59	1	no	—
85	SD₃₂	—	59	1	no	—
86	SD₃₂	—	59	1	no	—
87	D₁₆	—	59	1	no	—
88	C₂×C₁₆	—	59	1	no	—
89	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
90	#108	—	59	1	no	—
91	#108	C₂×D₅₉	1	1	yes	characteristic
92	#104	—	118	1	no	—
93	#104	—	118	1	no	—
94	C₄×C₈	—	59	1	no	—
95	#41	—	2	1	no	—
96	#41	—	118	1	no	—
97	Q₁₆	—	59	1	no	—
98	Q₁₆	C₄×D₅₉	1	1	yes	characteristic
99	SD₁₆	—	2	1	no	—
100	SD₁₆	—	118	1	no	—
101	D₈	—	59	1	no	—
102	D₈	C₄×D₅₉	1	1	yes	characteristic
103	M₁₆	—	2	1	no	—
104	M₁₆	—	118	1	no	—
105	M₁₆	—	236	1	no	—
106	M₁₆	—	236	1	no	—
107	C₂×C₈	—	59	1	no	—
108	C₂×C₈	—	236	1	no	—
109	C₂×C₈	—	236	1	no	—
110	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
111	C₄²	—	59	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	59	1	no	—
114	C₁₆	—	59	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	118	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	4	1	no	—
119	D₄	—	118	1	no	—
120	D₄	—	236	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	118	1	no	—
123	C₂×C₄	—	236	1	no	—
124	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	59	1	no	—
127	C₈	—	59	1	no	—
128	C₈	—	236	1	no	—
129	C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
130	C₈	#11745687	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	236	1	no	—
133	C₂²	#14273748	1	1	yes	characteristic
134	C₄	—	4	1	no	—
135	C₄	—	236	1	no	—
136	C₄	—	236	1	no	—
137	C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
138	C₄	D₈×D₅₉	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	8	1	no	—
141	C₂	—	472	1	no	—
142	C₂	#16837482	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837832

C₂

yes

maximal, characteristic

#16837519

C₂

yes

maximal, characteristic, complemented

#16837518

C₂

yes

maximal, characteristic, complemented

#16837231

C₂

yes

maximal, characteristic, complemented

#16837210

C₂

yes

maximal, characteristic, complemented

#16837177

C₂

yes

maximal, characteristic, complemented

C₅₉×#359

C₂

yes

maximal, characteristic, complemented

#14273749

C₂²

yes

—

—

C₄

yes

characteristic, complemented

#14273671

C₄

yes

characteristic, complemented

#14273670

C₄

yes

characteristic, complemented

#14273668

C₄

yes

characteristic, complemented

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273663

C₂²

yes

characteristic

#14273654

—

#14273653

—

C₈×Dic₅₉

C₂²

yes

characteristic

#135×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#108×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

#11745685

—

Q₁₆×C₅₉

C₂×C₄

yes

characteristic

SD₁₆×C₅₉

—

D₈×C₅₉

C₂×C₄

yes

characteristic

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

C₂×C₄

yes

characteristic

D₄₇₂

C₂×C₄

yes

characteristic

#11745660

—

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

C₂×C₄

yes

characteristic

C₅₉⋊C₁₆

C₂×C₄

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

D₈

yes

characteristic

C₅₉⋊C₈

D₈

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

#135

yes

characteristic

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2179

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₄×D₈

yes

characteristic

D₅₉

—

—

—

—

—

—

D₅₉

yes

characteristic, complemented

#346

—

C₅₉

#2179

yes

characteristic, complemented, Sylow

#135

—

#135

C₂×D₅₉

yes

characteristic

#131

—

118

—

#131

—

118

—

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#108

—

#108

C₂×D₅₉

yes

characteristic

#104

—

118

—

#104

—

118

—

C₄×C₈

—

#41

—

#41

—

118

—

Q₁₆

—

Q₁₆

C₄×D₅₉

yes

characteristic

SD₁₆

—

100

SD₁₆

—

118

—

101

D₈

—

102

D₈

C₄×D₅₉

yes

characteristic

103

M₁₆

—

104

M₁₆

—

118

—

105

M₁₆

—

236

—

106

M₁₆

—

236

—

107

C₂×C₈

—

108

C₂×C₈

—

236

—

109

C₂×C₈

—

236

—

110

C₂×C₈

C₂²×D₅₉

yes

characteristic

111

C₄²

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

Q₈

—

116

Q₈

—

118

—

117

D₄

—

118

D₄

—

119

D₄

—

118

—

120

D₄

—

236

—

121

C₂×C₄

—

122

C₂×C₄

—

118

—

123

C₂×C₄

—

236

—

124

C₂×C₄

D₄×D₅₉

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

236

—

129

C₈

C₂×C₄×D₅₉

yes

characteristic

130

C₈

#11745687

yes

characteristic

131

C₂²

—

132

C₂²

—

236

—

133

C₂²

#14273748

yes

characteristic

134

C₄

—

135

C₄

—

236

—

136

C₄

—

236

—

137

C₄

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

138

C₄

D₈×D₅₉

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

472

—

142

C₂

#16837482

yes

characteristic

	subgroup	G/H
center	C₄	D₈×D₅₉
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#359	C₂
socle	C₂×C₅₉	C₄×D₈
hypercenter	#359	D₅₉
hypocenter	C₅₉	#2179
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

subgroup

G/H

center

C₄

D₈×D₅₉

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#359

C₂

socle

C₂×C₅₉

C₄×D₈

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

N	H ≅ G/N	type
2. #16837519	C₂	semidirect
3. #16837518	C₂	semidirect
4. #16837231	C₂	semidirect
5. #16837210	C₂	semidirect
6. #16837177	C₂	semidirect
7. C₅₉×#359	C₂	semidirect
11. #14273673	C₄	semidirect
12. #14273671	C₄	semidirect
13. #14273670	C₄	semidirect
14. #14273668	C₄	semidirect
77. #359	D₅₉	semidirect
79. C₅₉	#2179	semidirect

H ≅ G/N

type

2. #16837519

C₂

semidirect

3. #16837518

C₂

semidirect

4. #16837231

C₂

semidirect

5. #16837210

C₂

semidirect

6. #16837177

C₂

semidirect

7. C₅₉×#359

C₂

semidirect

11. #14273673

C₄

semidirect

12. #14273671

C₄

semidirect

13. #14273670

C₄

semidirect

14. #14273668

C₄

semidirect

77. #359

D₅₉

semidirect

79. C₅₉

#2179

semidirect

Subgroups — #21425979

Conjugacy classes of subgroups

Complements