Group #21426648 — subgroups

Subgroups — #126×Dic₅₉ #21426648

Order 7552.

516 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 229 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	#16837295	C₂	1	3	yes	maximal
2	#16837288	C₂	1	3	yes	maximal, complemented
3	#16837270	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊C₄×Dic₅₉	C₂	1	3	yes	maximal, complemented
5	C₂²⋊C₄×Dic₅₉	C₂	1	3	yes	maximal
6	C₄²×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#126×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₅₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₉	C₂²	1	3	yes	—
10	C₂×C₂²⋊Dic₅₉	C₂²	1	3	yes	—
11	C₂×C₄×Dic₅₉	C₂²	1	3	yes	—
12	C₂×C₄×Dic₅₉	C₂²	1	3	yes	—
13	C₂×C₄×Dic₅₉	C₂²	1	3	yes	—
14	C₅₉⋊#95	C₂²	1	3	yes	—
15	C₅₉⋊#95	C₂²	1	3	yes	—
16	C₅₉⋊#95	C₂²	1	3	yes	—
17	C₅₉⋊#95	C₂²	1	3	yes	—
18	#126×C₅₉	C₄	1	8	yes	complemented
19	C₂×C₄⋊C₄×C₅₉	C₂²	1	3	yes	—
20	C₂×C₂²⋊C₄×C₅₉	C₂²	1	3	yes	—
21	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
22	C₂⁴×C₅₉	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	C₂³	1	3	yes	—
24	C₂²×C₄×C₅₉	C₂³	1	3	yes	—
25	C₂²×Dic₅₉	—	2	6	no	—
26	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅₉	C₂³	1	3	yes	—
28	C₂²×Dic₅₉	C₂³	1	3	yes	—
29	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₅₉	C₂×C₄	1	12	yes	complemented
31	C₂²⋊C₄×C₅₉	C₂×C₄	1	12	yes	—
32	C₄²×C₅₉	C₂×C₄	1	4	yes	complemented
33	C₂²⋊Dic₅₉	—	2	6	no	—
34	C₄⋊Dic₅₉	—	2	6	no	—
35	C₄×Dic₅₉	—	2	6	no	—
36	C₄×Dic₅₉	—	2	6	no	—
37	C₄×Dic₅₉	—	2	6	no	—
38	C₂³×C₅₉	—	2	3	no	—
39	C₂³×C₅₉	—	2	3	no	—
40	C₂³×C₅₉	C₂²×C₄	1	2	yes	—
41	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₅₉	—	2	3	no	—
43	C₂×C₄×C₅₉	—	2	3	no	—
44	C₂×C₄×C₅₉	—	2	3	no	—
45	C₂×C₄×C₅₉	—	2	3	no	—
46	C₂×C₄×C₅₉	C₂²×C₄	1	6	yes	—
47	C₂×C₄×C₅₉	C₂²×C₄	1	6	yes	—
48	C₂×Dic₅₉	—	2	3	no	—
49	C₂×Dic₅₉	—	2	3	no	—
50	C₂×Dic₅₉	—	2	3	no	—
51	C₂×Dic₅₉	—	2	3	no	—
52	C₂×Dic₅₉	—	2	3	no	—
53	C₂×Dic₅₉	—	2	3	no	—
54	C₂×Dic₅₉	—	4	4	no	—
55	C₂×Dic₅₉	#41	1	6	yes	—
56	C₂×Dic₅₉	#41	1	6	yes	—
57	C₂²×C₅₉	—	2	6	no	—
58	C₂²×C₅₉	—	4	1	no	—
59	C₂²×C₅₉	—	4	3	no	—
60	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
61	C₂²×C₅₉	C₂×#41	1	3	yes	—
62	C₂²×C₅₉	C₂×#41	1	3	yes	—
63	C₄×C₅₉	—	2	6	no	—
64	C₄×C₅₉	—	2	6	no	—
65	Dic₅₉	—	2	6	no	—
66	Dic₅₉	—	4	1	no	—
67	Dic₅₉	—	4	3	no	—
68	Dic₅₉	#126	1	4	yes	direct product factor
69	C₄×#126	—	59	1	no	maximal, Sylow
70	C₂×C₅₉	—	4	2	no	—
71	C₂×C₅₉	C₄×#41	1	3	yes	—
72	C₂×C₅₉	C₂×#126	1	1	yes	characteristic
73	C₂×C₅₉	#529	1	3	yes	—
74	C₂×#126	D₅₉	1	1	yes	characteristic
75	#388	—	59	3	no	—
76	#387	—	59	3	no	—
77	#383	—	59	1	no	—
78	C₄×C₄⋊C₄	—	59	3	no	—
79	C₄×C₂²⋊C₄	—	59	3	no	—
80	C₄³	—	59	1	no	—
81	C₅₉	C₄×#126	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₄	—	59	1	no	—
83	#126	Dic₅₉	1	8	yes	direct product factor
84	C₂×C₄⋊C₄	—	59	3	no	—
85	C₂×C₄⋊C₄	C₂×D₅₉	1	3	yes	—
86	C₂×C₂²⋊C₄	—	59	3	no	—
87	C₂×C₂²⋊C₄	C₂×D₅₉	1	3	yes	—
88	C₂×C₄²	—	59	3	no	—
89	C₂×C₄²	—	59	3	no	—
90	C₂×C₄²	—	59	3	no	—
91	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
92	#95	—	59	3	no	—
93	#95	—	59	3	no	—
94	#95	—	59	3	no	—
95	#95	—	59	3	no	—
96	C₂⁴	C₂²×D₅₉	1	1	yes	characteristic
97	C₂²×C₄	—	59	1	no	—
98	C₂²×C₄	—	59	3	no	—
99	C₂²×C₄	—	59	1	no	—
100	C₂²×C₄	—	59	3	no	—
101	C₂²×C₄	—	118	6	no	—
102	C₂²×C₄	C₂²×D₅₉	1	3	yes	—
103	C₂²×C₄	C₂²×D₅₉	1	3	yes	—
104	C₄⋊C₄	—	118	6	no	—
105	C₄⋊C₄	C₂×Dic₅₉	1	12	yes	complemented
106	C₂²⋊C₄	—	118	6	no	—
107	C₂²⋊C₄	C₂×Dic₅₉	1	12	yes	—
108	C₄²	—	118	6	no	—
109	C₄²	—	118	6	no	—
110	C₄²	—	118	6	no	—
111	C₄²	C₂×Dic₅₉	1	4	yes	complemented
112	C₂³	—	2	3	no	—
113	C₂³	—	2	3	no	—
114	C₂³	C₂²×Dic₅₉	1	2	yes	—
115	C₂³	C₂³×D₅₉	1	1	yes	characteristic
116	C₂×C₄	—	2	3	no	—
117	C₂×C₄	—	2	3	no	—
118	C₂×C₄	—	2	3	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	59	6	no	—
121	C₂×C₄	—	59	6	no	—
122	C₂×C₄	—	118	3	no	—
123	C₂×C₄	—	118	3	no	—
124	C₂×C₄	—	118	3	no	—
125	C₂×C₄	—	118	3	no	—
126	C₂×C₄	—	118	3	no	—
127	C₂×C₄	—	118	3	no	—
128	C₂×C₄	—	236	4	no	—
129	C₂×C₄	C₂²×Dic₅₉	1	6	yes	—
130	C₂×C₄	C₂²×Dic₅₉	1	6	yes	—
131	C₂²	—	2	6	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	4	3	no	—
134	C₂²	C₂³×Dic₅₉	1	1	yes	characteristic
135	C₂²	#41×D₅₉	1	3	yes	—
136	C₂²	#41×D₅₉	1	3	yes	—
137	C₄	—	2	6	no	—
138	C₄	—	2	6	no	—
139	C₄	—	59	4	no	—
140	C₄	—	118	6	no	—
141	C₄	—	236	1	no	—
142	C₄	—	236	3	no	—
143	C₂	—	4	2	no	—
144	C₂	#41×Dic₅₉	1	3	yes	—
145	C₂	#126×D₅₉	1	1	yes	characteristic
146	C₂	#16837935	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837295

C₂

yes

maximal

#16837288

C₂

yes

maximal, complemented

#16837270

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×Dic₅₉

C₂

yes

maximal, complemented

C₂²⋊C₄×Dic₅₉

C₂

yes

maximal

C₄²×Dic₅₉

C₂

yes

maximal, characteristic, complemented

C₂×#126×C₅₉

C₂

yes

maximal, characteristic

C₂³×Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₂²⋊Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

#126×C₅₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂⁴×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

C₂×C₄

yes

complemented

C₂²⋊C₄×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

complemented

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂²×C₄

yes

—

C₂³×C₅₉

C₂⁴

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

#41

yes

—

C₂×Dic₅₉

#41

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂³×C₄

yes

characteristic

C₂²×C₅₉

C₂×#41

yes

—

C₂²×C₅₉

C₂×#41

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

#126

yes

direct product factor

C₄×#126

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₄×#41

yes

—

C₂×C₅₉

C₂×#126

yes

characteristic

C₂×C₅₉

#529

yes

—

C₂×#126

D₅₉

yes

—

—

—

C₄×C₄⋊C₄

—

C₄×C₂²⋊C₄

—

C₄³

—

C₅₉

C₄×#126

yes

characteristic, complemented, Sylow

C₂³×C₄

—

#126

Dic₅₉

yes

direct product factor

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

—

—

—

—

C₂⁴

C₂²×D₅₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

118

—

102

C₂²×C₄

C₂²×D₅₉

yes

—

103

C₂²×C₄

C₂²×D₅₉

yes

—

104

C₄⋊C₄

—

118

—

105

C₄⋊C₄

C₂×Dic₅₉

yes

complemented

106

C₂²⋊C₄

—

118

—

107

C₂²⋊C₄

C₂×Dic₅₉

yes

—

108

C₄²

—

118

—

109

C₄²

—

118

—

110

C₄²

—

118

—

111

C₄²

C₂×Dic₅₉

yes

complemented

112

C₂³

—

113

C₂³

—

114

C₂³

C₂²×Dic₅₉

yes

—

115

C₂³

C₂³×D₅₉

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

118

—

123

C₂×C₄

—

118

—

124

C₂×C₄

—

118

—

125

C₂×C₄

—

118

—

126

C₂×C₄

—

118

—

127

C₂×C₄

—

118

—

128

C₂×C₄

—

236

—

129

C₂×C₄

C₂²×Dic₅₉

yes

—

130

C₂×C₄

C₂²×Dic₅₉

yes

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₂³×Dic₅₉

yes

characteristic

135

C₂²

#41×D₅₉

yes

—

136

C₂²

#41×D₅₉

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

118

—

141

C₄

—

236

—

142

C₄

—

236

—

143

C₂

—

144

C₂

#41×Dic₅₉

yes

—

145

C₂

#126×D₅₉

yes

characteristic

146

C₂

#16837935

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator	C₂²×C₅₉	C₂³×C₄
Fitting	C₂×#126×C₅₉	C₂
socle	C₂³×C₅₉	C₂⁴
hypercenter	C₂×#126	D₅₉
hypocenter	C₅₉	C₄×#126
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×#126

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₉

commutator

C₂²×C₅₉

C₂³×C₄

Fitting

C₂×#126×C₅₉

C₂

socle

C₂³×C₅₉

C₂⁴

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂³ ◁ C₂×#126

N	H ≅ G/N	type
2. #16837288	C₂	semidirect
3. #16837270	C₂	semidirect
4. C₄⋊C₄×Dic₅₉	C₂	semidirect
6. C₄²×Dic₅₉	C₂	semidirect
18. #126×C₅₉	C₄	semidirect
30. C₄⋊C₄×C₅₉	C₂×C₄	semidirect
32. C₄²×C₅₉	C₂×C₄	semidirect
68. Dic₅₉	#126	direct (N × H)
81. C₅₉	C₄×#126	semidirect
83. #126	Dic₅₉	direct (N × H)
105. C₄⋊C₄	C₂×Dic₅₉	semidirect
111. C₄²	C₂×Dic₅₉	semidirect

H ≅ G/N

type

2. #16837288

C₂

semidirect

3. #16837270

C₂

semidirect

4. C₄⋊C₄×Dic₅₉

C₂

semidirect

6. C₄²×Dic₅₉

C₂

semidirect

18. #126×C₅₉

C₄

semidirect

30. C₄⋊C₄×C₅₉

C₂×C₄

semidirect

32. C₄²×C₅₉

C₂×C₄

semidirect

68. Dic₅₉

#126

direct (N × H)

81. C₅₉

C₄×#126

semidirect

83. #126

Dic₅₉

direct (N × H)

105. C₄⋊C₄

C₂×Dic₅₉

semidirect

111. C₄²

C₂×Dic₅₉

semidirect

Subgroups — #126×Dic₅₉ #21426648

Conjugacy classes of subgroups

Complements

Subgroups — #126×Dic59 #21426648

Conjugacy classes of subgroups

Complements

Subgroups — #126×Dic₅₉ #21426648