Group #21425940 — subgroups

Subgroups — #21425940

Order 7552.

174 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 59 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	C₂×#14273668	C₂	1	1	yes	maximal, characteristic
2	D₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837484	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837227	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837204	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
9	D₄×Dic₅₉	—	2	1	no	—
10	#14273689	—	2	1	no	—
11	#14273668	C₄	1	4	yes	complemented
12	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14273661	C₂²	1	1	yes	characteristic
14	#14273648	—	2	1	no	—
15	#14273647	—	2	1	no	—
16	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
19	#107×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×D₄×C₅₉	—	2	1	no	—
21	C₂²×Dic₅₉	—	4	1	no	—
22	C₂×D₂₃₆	—	2	1	no	—
23	C₂×C₄×D₅₉	—	4	1	no	—
24	D₈×C₅₉	—	2	1	no	—
25	D₈×C₅₉	C₂×C₄	1	2	yes	complemented
26	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₅₉	—	2	1	no	—
28	C₂²⋊Dic₅₉	—	4	1	no	—
29	(C₂×D₅₉)⋊C₄	—	4	1	no	—
30	C₄⋊Dic₅₉	—	2	1	no	—
31	C₄×Dic₅₉	D₄	1	1	yes	characteristic
32	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
33	D₄₇₂	—	2	1	no	—
34	D₄₇₂	C₂×C₄	1	2	yes	complemented
35	C₁₆×C₅₉	—	2	1	no	—
36	C₅₉⋊C₁₆	C₂×C₄	1	2	yes	complemented
37	C₂³×C₅₉	—	4	1	no	—
38	C₂²×D₅₉	—	4	1	no	—
39	D₄×C₅₉	—	2	2	no	—
40	D₄×C₅₉	—	4	1	no	—
41	C₂×C₄×C₅₉	—	4	1	no	—
42	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₅₉	—	4	1	no	—
44	C₂×Dic₅₉	—	8	2	no	—
45	C₂×Dic₅₉	D₈	1	1	yes	characteristic
46	C₂×Dic₅₉	D₈	1	1	yes	characteristic
47	D₂₃₆	—	2	2	no	—
48	D₂₃₆	—	4	1	no	—
49	C₄×D₅₉	—	8	2	no	—
50	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₅₉	#41	1	1	yes	characteristic
52	C₅₉⋊C₈	—	2	1	no	—
53	C₂²×C₅₉	—	4	2	no	—
54	C₂²×C₅₉	—	8	1	no	—
55	C₂²×C₅₉	—	8	1	no	—
56	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
57	C₂×D₅₉	—	4	2	no	—
58	C₂×D₅₉	—	8	1	no	—
59	C₂×D₅₉	—	8	1	no	—
60	C₄×C₅₉	—	8	1	no	—
61	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
62	C₄×C₅₉	#135	1	1	yes	characteristic
63	Dic₅₉	—	2	1	no	—
64	Dic₅₉	—	8	1	no	—
65	Dic₅₉	D₁₆	1	2	yes	complemented
66	C₄×D₁₆	—	59	1	no	maximal, Sylow
67	C₂×C₅₉	—	8	2	no	—
68	C₂×C₅₉	C₄×D₈	1	1	yes	characteristic
69	C₂×C₅₉	C₂×D₁₆	1	1	yes	characteristic
70	C₂×C₅₉	#508	1	1	yes	characteristic
71	D₅₉	—	8	2	no	—
72	C₂×D₁₆	—	59	1	no	—
73	C₄×D₈	—	59	1	no	—
74	C₄×D₈	—	59	1	no	—
75	#366	—	59	1	no	—
76	#357	—	59	1	no	—
77	#357	D₅₉	1	1	yes	characteristic, complemented
78	C₄×C₁₆	—	59	1	no	—
79	C₅₉	C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
80	C₂×D₈	—	59	1	no	—
81	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
82	C₄×D₄	—	118	1	no	—
83	C₄×D₄	—	118	1	no	—
84	D₁₆	—	59	4	no	—
85	C₂×C₁₆	—	59	1	no	—
86	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
87	#107	—	59	1	no	—
88	#107	C₂×D₅₉	1	1	yes	characteristic
89	#102	—	118	1	no	—
90	#102	—	118	1	no	—
91	C₄×C₈	—	59	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	118	1	no	—
94	C₂²×C₄	—	236	1	no	—
95	C₂²×C₄	—	236	1	no	—
96	D₈	—	2	1	no	—
97	D₈	—	59	2	no	—
98	D₈	—	118	1	no	—
99	D₈	C₄×D₅₉	1	2	yes	complemented
100	C₂×C₈	—	59	1	no	—
101	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	118	1	no	—
104	C₂²⋊C₄	—	236	1	no	—
105	C₂²⋊C₄	—	236	1	no	—
106	C₄²	—	59	1	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	—	59	2	no	—
109	C₂³	—	4	1	no	—
110	C₂³	—	236	1	no	—
111	D₄	—	2	2	no	—
112	D₄	—	4	1	no	—
113	D₄	—	118	2	no	—
114	D₄	—	236	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	59	1	no	—
117	C₂×C₄	—	59	1	no	—
118	C₂×C₄	—	472	2	no	—
119	C₂×C₄	—	472	2	no	—
120	C₂×C₄	—	236	1	no	—
121	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
122	C₈	—	118	1	no	—
123	C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
124	C₈	#11745687	1	1	yes	characteristic
125	C₂²	—	4	2	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	—	472	1	no	—
129	C₂²	—	472	1	no	—
130	C₂²	—	236	2	no	—
131	C₂²	D₈×D₅₉	1	1	yes	characteristic
132	C₄	—	8	1	no	—
133	C₄	—	59	2	no	—
134	C₄	—	118	1	no	—
135	C₄	—	472	1	no	—
136	C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
137	C₄	#14273748	1	1	yes	characteristic
138	C₂	—	8	2	no	—
139	C₂	—	472	2	no	—
140	C₂	#16837482	1	1	yes	characteristic
141	C₂	D₁₆×D₅₉	1	1	yes	characteristic
142	C₂	#16837833	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273668

C₂

yes

maximal, characteristic

D₈×Dic₅₉

C₂

yes

maximal, characteristic, complemented

#16837484

C₂

yes

maximal, characteristic, complemented

#16837227

C₂

yes

maximal, characteristic, complemented

#16837204

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₅₉

C₂

yes

maximal, characteristic, complemented

C₅₉×#357

C₂

yes

maximal, characteristic, complemented

C₂×D₄₇₂

C₂²

yes

characteristic

D₄×Dic₅₉

—

#14273689

—

#14273668

C₄

yes

complemented

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273648

—

#14273647

—

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×D₈×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂²×Dic₅₉

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

—

D₈×C₅₉

—

D₈×C₅₉

C₂×C₄

yes

complemented

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊Dic₅₉

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

D₄₇₂

—

D₄₇₂

C₂×C₄

yes

complemented

C₁₆×C₅₉

—

C₅₉⋊C₁₆

C₂×C₄

yes

complemented

C₂³×C₅₉

—

C₂²×D₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₂×Dic₅₉

D₈

yes

characteristic

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

#135

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

D₁₆

yes

complemented

C₄×D₁₆

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₄×D₈

yes

characteristic

C₂×C₅₉

C₂×D₁₆

yes

characteristic

C₂×C₅₉

#508

yes

characteristic

D₅₉

—

C₂×D₁₆

—

C₄×D₈

—

C₄×D₈

—

#366

—

#357

—

#357

D₅₉

yes

characteristic, complemented

C₄×C₁₆

—

C₅₉

C₄×D₁₆

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

C₂×D₅₉

yes

characteristic

C₄×D₄

—

118

—

C₄×D₄

—

118

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#107

—

#107

C₂×D₅₉

yes

characteristic

#102

—

118

—

#102

—

118

—

C₄×C₈

—

C₂×D₄

—

C₂×D₄

—

118

—

C₂²×C₄

—

236

—

C₂²×C₄

—

236

—

D₈

—

D₈

—

D₈

—

118

—

D₈

C₄×D₅₉

yes

complemented

100

C₂×C₈

—

101

C₂×C₈

C₂²×D₅₉

yes

characteristic

102

C₄⋊C₄

—

103

C₄⋊C₄

—

118

—

104

C₂²⋊C₄

—

236

—

105

C₂²⋊C₄

—

236

—

106

C₄²

—

107

C₁₆

—

108

C₁₆

—

109

C₂³

—

110

C₂³

—

236

—

111

D₄

—

112

D₄

—

113

D₄

—

118

—

114

D₄

—

236

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

472

—

119

C₂×C₄

—

472

—

120

C₂×C₄

—

236

—

121

C₂×C₄

D₄×D₅₉

yes

characteristic

122

C₈

—

118

—

123

C₈

C₂×C₄×D₅₉

yes

characteristic

124

C₈

#11745687

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

472

—

129

C₂²

—

472

—

130

C₂²

—

236

—

131

C₂²

D₈×D₅₉

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

118

—

135

C₄

—

472

—

136

C₄

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

137

C₄

#14273748

yes

characteristic

138

C₂

—

139

C₂

—

472

—

140

C₂

#16837482

yes

characteristic

141

C₂

D₁₆×D₅₉

yes

characteristic

142

C₂

#16837833

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#357	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#357	D₅₉
hypocenter	C₅₉	C₄×D₁₆
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#357

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

#357

D₅₉

hypocenter

C₅₉

C₄×D₁₆

LCS

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
2. D₈×Dic₅₉	C₂	semidirect
3. #16837484	C₂	semidirect
4. #16837227	C₂	semidirect
5. #16837204	C₂	semidirect
6. C₁₆×Dic₅₉	C₂	semidirect
7. C₅₉×#357	C₂	semidirect
11. #14273668	C₄	semidirect
25. D₈×C₅₉	C₂×C₄	semidirect
34. D₄₇₂	C₂×C₄	semidirect
36. C₅₉⋊C₁₆	C₂×C₄	semidirect
65. Dic₅₉	D₁₆	semidirect
77. #357	D₅₉	semidirect
79. C₅₉	C₄×D₁₆	semidirect
99. D₈	C₄×D₅₉	semidirect

H ≅ G/N

type

2. D₈×Dic₅₉

C₂

semidirect

3. #16837484

C₂

semidirect

4. #16837227

C₂

semidirect

5. #16837204

C₂

semidirect

6. C₁₆×Dic₅₉

C₂

semidirect

7. C₅₉×#357

C₂

semidirect

11. #14273668

C₄

semidirect

25. D₈×C₅₉

C₂×C₄

semidirect

34. D₄₇₂

C₂×C₄

semidirect

36. C₅₉⋊C₁₆

C₂×C₄

semidirect

65. Dic₅₉

D₁₆

semidirect

77. #357

D₅₉

semidirect

79. C₅₉

C₄×D₁₆

semidirect

99. D₈

C₄×D₅₉

semidirect

Subgroups — #21425940

Conjugacy classes of subgroups

Complements