Group #21424923 — subgroups

Subgroups — #21424923

Order 7552.

360 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16837952	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273653	C₂	1	4	yes	maximal, complemented
3	C₄²×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11745685	C₂²	1	2	yes	—
6	#14273727	C₄	1	1	yes	characteristic, complemented
7	#14273724	C₄	1	1	yes	characteristic, complemented
8	#14273715	C₄	1	2	yes	complemented
9	#14273701	—	2	2	no	—
10	C₄×D₂₃₆	—	2	2	no	—
11	C₂×C₄×Dic₅₉	—	2	2	no	—
12	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
13	#14273653	—	2	8	no	—
14	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₅₉	—	2	2	no	—
17	C₂²×C₄×C₅₉	D₄	1	2	yes	—
18	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
19	C₂×C₂²⋊D₅₉	—	2	2	no	—
20	C₂²×Dic₅₉	—	2	2	no	—
21	#11745685	—	2	4	no	—
22	#11745685	—	2	4	no	—
23	C₂×D₂₃₆	—	2	2	no	—
24	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₅₉	—	2	2	no	—
26	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
27	M₁₆×C₅₉	—	2	4	no	—
28	C₂×C₈×C₅₉	—	2	2	no	—
29	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₅₉	D₄	1	2	yes	—
32	(C₂×D₅₉)⋊C₄	—	2	4	no	—
33	C₄⋊Dic₅₉	—	2	2	no	—
34	C₄×Dic₅₉	—	2	8	no	—
35	C₄×Dic₅₉	—	2	2	no	—
36	C₄×Dic₅₉	D₄	1	4	yes	complemented
37	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×D₅₉	—	2	2	no	—
39	C₂×C₄×C₅₉	—	2	4	no	—
40	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	#41	1	2	yes	—
46	C₂²⋊D₅₉	—	4	4	no	—
47	C₂×Dic₅₉	—	2	8	no	—
48	C₂×Dic₅₉	—	2	4	no	—
49	C₂×Dic₅₉	—	2	2	no	—
50	D₂₃₆	—	2	4	no	—
51	D₂₃₆	—	4	4	no	—
52	C₄×D₅₉	—	4	2	no	—
53	C₄×D₅₉	—	4	2	no	—
54	Dic₁₁₈	—	2	4	no	—
55	C₈×C₅₉	—	4	2	no	—
56	C₂²×C₅₉	—	2	2	no	—
57	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₅₉	#121	1	2	yes	—
59	C₂×D₅₉	—	4	2	no	—
60	C₂×D₅₉	—	4	4	no	—
61	C₄×C₅₉	—	2	2	no	—
62	C₄×C₅₉	#104	1	4	yes	complemented
63	C₄×C₅₉	C₄×D₄	1	2	yes	—
64	C₄×C₅₉	#124	1	1	yes	characteristic
65	C₄×C₅₉	#127	1	1	yes	characteristic
66	Dic₅₉	—	2	8	no	—
67	Dic₅₉	—	4	2	no	—
68	#1962	—	59	1	no	maximal, Sylow
69	C₂×C₅₉	—	2	2	no	—
70	C₂×C₅₉	#390	1	1	yes	characteristic
71	C₂×C₅₉	C₂×#104	1	2	yes	—
72	D₅₉	—	8	2	no	—
73	#532	—	59	1	no	—
74	#423	D₅₉	1	1	yes	characteristic, complemented
75	C₂×#104	—	59	4	no	—
76	C₄³	—	59	1	no	—
77	C₅₉	#1962	1	1	yes	characteristic, complemented, Sylow
78	C₂×#41	—	59	2	no	—
79	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
80	C₄⋊Q₈	—	59	1	no	—
81	#127	—	59	1	no	—
82	#124	—	59	2	no	—
83	#121	—	118	2	no	—
84	C₄×D₄	—	118	2	no	—
85	C₂×C₄²	—	59	2	no	—
86	C₂×C₄²	—	118	2	no	—
87	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
88	C₄⋊C₈	—	2	2	no	—
89	#104	—	118	8	no	—
90	#41	—	118	4	no	—
91	#41	—	118	4	no	—
92	C₂×Q₈	—	59	2	no	—
93	C₂×D₄	—	59	2	no	—
94	C₂×D₄	—	118	2	no	—
95	C₂×D₄	—	118	2	no	—
96	C₂²×C₄	—	118	2	no	—
97	C₂²×C₄	—	118	2	no	—
98	C₂²×C₄	D₂₃₆	1	2	yes	—
99	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
100	M₁₆	—	2	4	no	—
101	C₂×C₈	—	2	2	no	—
102	C₄⋊C₄	—	118	2	no	—
103	C₂²⋊C₄	—	118	4	no	—
104	C₄²	—	59	4	no	—
105	C₄²	—	118	8	no	—
106	C₄²	—	118	2	no	—
107	C₄²	C₄×D₅₉	1	1	yes	characteristic
108	C₄²	C₄×D₅₉	1	1	yes	characteristic
109	C₄²	C₂²⋊D₅₉	1	2	yes	—
110	C₂³	—	118	2	no	—
111	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
112	Q₈	—	118	4	no	—
113	D₄	—	118	4	no	—
114	D₄	—	236	4	no	—
115	D₄	—	236	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	118	8	no	—
118	C₂×C₄	—	118	4	no	—
119	C₂×C₄	—	118	2	no	—
120	C₂×C₄	—	236	2	no	—
121	C₂×C₄	—	236	2	no	—
122	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
123	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
125	C₂×C₄	D₄×D₅₉	1	2	yes	—
126	C₂×C₄	#11745689	1	2	yes	—
127	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
128	C₈	—	4	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	236	4	no	—
131	C₂²	—	236	2	no	—
132	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
133	C₂²	#14273702	1	2	yes	—
134	C₄	—	2	2	no	—
135	C₄	—	118	8	no	—
136	C₄	—	236	2	no	—
137	C₄	#14273653	1	4	yes	complemented
138	C₄	#14273689	1	2	yes	—
139	C₄	#14273716	1	1	yes	characteristic
140	C₄	#14273723	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	—	472	2	no	—
143	C₂	#16837303	1	1	yes	characteristic
144	C₂	C₂×#14273653	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837952

C₂

yes

maximal, characteristic

C₂×#14273653

C₂

yes

maximal, complemented

C₄²×Dic₅₉

C₂

yes

maximal, characteristic, complemented

C₅₉×#423

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₂²

yes

—

#14273727

C₄

yes

characteristic, complemented

#14273724

C₄

yes

characteristic, complemented

#14273715

C₄

yes

complemented

#14273701

—

C₄×D₂₃₆

—

C₂×C₄×Dic₅₉

—

C₂×C₄×Dic₅₉

C₂²

yes

—

#14273653

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₂²⋊D₅₉

—

C₂²×Dic₅₉

—

#11745685

—

#11745685

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

complemented

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

#121

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#104

yes

complemented

C₄×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#1962

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#390

yes

characteristic

C₂×C₅₉

C₂×#104

yes

—

D₅₉

—

#532

—

#423

D₅₉

yes

characteristic, complemented

C₂×#104

—

C₄³

—

C₅₉

#1962

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

118

—

C₄×D₄

—

118

—

C₂×C₄²

—

C₂×C₄²

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

#104

—

118

—

#41

—

118

—

#41

—

118

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

D₂₃₆

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

100

M₁₆

—

101

C₂×C₈

—

102

C₄⋊C₄

—

118

—

103

C₂²⋊C₄

—

118

—

104

C₄²

—

105

C₄²

—

118

—

106

C₄²

—

118

—

107

C₄²

C₄×D₅₉

yes

characteristic

108

C₄²

C₄×D₅₉

yes

characteristic

109

C₄²

C₂²⋊D₅₉

yes

—

110

C₂³

—

118

—

111

C₂³

C₂×D₂₃₆

yes

characteristic

112

Q₈

—

118

—

113

D₄

—

118

—

114

D₄

—

236

—

115

D₄

—

236

—

116

C₂×C₄

—

117

C₂×C₄

—

118

—

118

C₂×C₄

—

118

—

119

C₂×C₄

—

118

—

120

C₂×C₄

—

236

—

121

C₂×C₄

—

236

—

122

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

123

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

—

124

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

125

C₂×C₄

D₄×D₅₉

yes

—

126

C₂×C₄

#11745689

yes

—

127

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

128

C₈

—

129

C₂²

—

130

C₂²

—

236

—

131

C₂²

—

236

—

132

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

133

C₂²

#14273702

yes

—

134

C₄

—

135

C₄

—

118

—

136

C₄

—

236

—

137

C₄

#14273653

yes

complemented

138

C₄

#14273689

yes

—

139

C₄

#14273716

yes

characteristic

140

C₄

#14273723

yes

characteristic

141

C₂

—

142

C₂

—

472

—

143

C₂

#16837303

yes

characteristic

144

C₂

C₂×#14273653

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#423	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#423	D₅₉
hypocenter	C₅₉	#1962
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#423

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#14273653	C₂	semidirect
3. C₄²×Dic₅₉	C₂	semidirect
4. C₅₉×#423	C₂	semidirect
6. #14273727	C₄	semidirect
7. #14273724	C₄	semidirect
8. #14273715	C₄	semidirect
36. C₄×Dic₅₉	D₄	semidirect
62. C₄×C₅₉	#104	semidirect
74. #423	D₅₉	semidirect
77. C₅₉	#1962	semidirect
137. C₄	#14273653	semidirect

H ≅ G/N

type

2. C₂×#14273653

C₂

semidirect

3. C₄²×Dic₅₉

C₂

semidirect

4. C₅₉×#423

C₂

semidirect

6. #14273727

C₄

semidirect

7. #14273724

C₄

semidirect

8. #14273715

C₄

semidirect

36. C₄×Dic₅₉

D₄

semidirect

62. C₄×C₅₉

#104

semidirect

74. #423

D₅₉

semidirect

77. C₅₉

#1962

semidirect

137. C₄

#14273653

semidirect

Subgroups — #21424923

Conjugacy classes of subgroups

Complements