Group #21424840 — subgroups

Subgroups — #21424840

Order 7552.

240 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 113 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×#14273661	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273637	C₂	1	1	yes	maximal, characteristic
3	#16837301	C₂	1	2	yes	maximal
4	#16837164	C₂	1	2	yes	maximal
5	C₂×#106×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₅₉	—	2	2	no	—
8	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
9	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
10	#14273661	—	2	2	no	—
11	#14273637	—	2	2	no	—
12	C₅₉⋊#95	—	2	2	no	—
13	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
15	#106×C₅₉	C₄	1	4	yes	complemented
16	C₂²×C₄×C₅₉	—	2	2	no	—
17	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₉	—	2	2	no	—
19	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
20	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₅₉	D₄	1	2	yes	—
23	C₂×C₈×C₅₉	Q₈	1	2	yes	—
24	C₄⋊C₄×C₅₉	—	2	2	no	—
25	C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
26	C₄⋊Dic₅₉	—	2	2	no	—
27	C₄⋊Dic₅₉	—	2	2	no	—
28	C₄⋊Dic₅₉	—	4	4	no	—
29	C₄×Dic₅₉	—	2	1	no	—
30	C₄×Dic₅₉	—	2	1	no	—
31	C₂×C₅₉⋊C₈	—	2	1	no	—
32	C₂×C₅₉⋊C₈	—	2	1	no	—
33	C₂×C₅₉⋊C₈	Q₈	1	2	yes	—
34	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	—	2	4	no	—
36	C₂×C₄×C₅₉	—	4	2	no	—
37	C₂×C₄×C₅₉	—	4	2	no	—
38	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
43	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
44	C₂×Dic₅₉	—	2	1	no	—
45	C₂×Dic₅₉	—	2	1	no	—
46	C₂×Dic₅₉	—	2	2	no	—
47	C₂×Dic₅₉	—	2	2	no	—
48	C₂×Dic₅₉	—	4	2	no	—
49	C₂×Dic₅₉	—	4	2	no	—
50	C₂×Dic₅₉	—	4	2	no	—
51	C₈×C₅₉	C₄⋊C₄	1	4	yes	complemented
52	C₅₉⋊C₈	—	2	2	no	—
53	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₅₉	#121	1	1	yes	characteristic
55	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
56	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
57	C₂²×C₅₉	#137	1	2	yes	—
58	C₄×C₅₉	—	4	4	no	—
59	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
61	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
62	C₄×C₅₉	#125	1	1	yes	characteristic
63	Dic₅₉	—	4	1	no	—
64	Dic₅₉	—	4	1	no	—
65	Dic₅₉	—	8	2	no	—
66	#1944	—	59	1	no	maximal, Sylow
67	C₂×C₅₉	#389	1	1	yes	characteristic
68	C₂×C₅₉	#440	1	2	yes	—
69	C₂×C₅₉	#469	1	1	yes	characteristic
70	C₂×C₅₉	#470	1	1	yes	characteristic
71	C₂×C₅₉	#501	1	2	yes	—
72	C₂×#107	—	59	1	no	—
73	C₂×#106	D₅₉	1	1	yes	characteristic
74	C₂×#97	—	59	1	no	—
75	#389	—	59	2	no	—
76	#340	—	59	2	no	—
77	C₅₉	#1944	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₈	—	59	1	no	—
79	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
80	C₂×C₄⋊C₄	—	59	2	no	—
81	C₂×C₄⋊C₄	—	118	2	no	—
82	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
83	C₂×C₄²	—	59	1	no	—
84	#107	—	118	2	no	—
85	#106	Dic₅₉	1	4	yes	complemented
86	#97	—	118	2	no	—
87	#95	—	118	2	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	59	1	no	—
90	C₂²×C₄	—	59	1	no	—
91	C₂²×C₄	—	118	2	no	—
92	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
93	C₂×C₈	—	59	2	no	—
94	C₂×C₈	—	118	1	no	—
95	C₂×C₈	—	118	1	no	—
96	C₂×C₈	Dic₁₁₈	1	2	yes	—
97	C₂×C₈	D₂₃₆	1	2	yes	—
98	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	118	2	no	—
101	C₄⋊C₄	—	118	2	no	—
102	C₄⋊C₄	—	236	4	no	—
103	C₄⋊C₄	C₂×Dic₅₉	1	4	yes	—
104	C₄²	—	118	1	no	—
105	C₄²	—	118	1	no	—
106	C₂³	D₄×D₅₉	1	1	yes	characteristic
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	118	1	no	—
111	C₂×C₄	—	118	1	no	—
112	C₂×C₄	—	118	2	no	—
113	C₂×C₄	—	118	2	no	—
114	C₂×C₄	—	236	2	no	—
115	C₂×C₄	—	236	2	no	—
116	C₂×C₄	—	236	2	no	—
117	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
118	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
119	C₂×C₄	#11745687	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
121	C₂×C₄	#11745689	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
123	C₈	—	118	2	no	—
124	C₈	C₄⋊Dic₅₉	1	4	yes	complemented
125	C₂²	D₄×Dic₅₉	1	1	yes	characteristic
126	C₂²	#14273702	1	1	yes	characteristic
127	C₂²	#14273726	1	1	yes	characteristic
128	C₂²	#14273754	1	2	yes	—
129	C₂²	#14273756	1	2	yes	—
130	C₄	—	4	4	no	—
131	C₄	—	472	2	no	—
132	C₄	—	236	1	no	—
133	C₄	—	236	1	no	—
134	C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
135	C₄	Q₈×Dic₅₉	1	1	yes	characteristic
136	C₄	#14273707	1	1	yes	characteristic
137	C₄	#14273718	1	1	yes	characteristic
138	C₂	#16837301	1	1	yes	characteristic
139	C₂	#16837503	1	2	yes	—
140	C₂	#16837656	1	1	yes	characteristic
141	C₂	#16837661	1	1	yes	characteristic
142	C₂	#16837809	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273661

C₂

yes

maximal, characteristic

C₂×#14273637

C₂

yes

maximal, characteristic

#16837301

C₂

yes

maximal

#16837164

C₂

yes

maximal

C₂×#106×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273661

—

#14273637

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

#106×C₅₉

C₄

yes

complemented

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

characteristic

C₂²×Dic₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

Q₈

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

Q₈

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

C₄⋊C₄

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

C₄⋊Q₈

yes

characteristic

C₂²×C₅₉

AGL(1,ℤ/8)

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

C₄×Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

#125

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×#107

—

C₂×#106

D₅₉

yes

characteristic

C₂×#97

—

#389

—

#340

—

C₅₉

#1944

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₄²

—

#107

—

118

—

#106

Dic₅₉

yes

complemented

#97

—

118

—

#95

—

118

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

Dic₁₁₈

yes

—

C₂×C₈

D₂₃₆

yes

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

118

—

101

C₄⋊C₄

—

118

—

102

C₄⋊C₄

—

236

—

103

C₄⋊C₄

C₂×Dic₅₉

yes

—

104

C₄²

—

118

—

105

C₄²

—

118

—

106

C₂³

D₄×D₅₉

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

118

—

111

C₂×C₄

—

118

—

112

C₂×C₄

—

118

—

113

C₂×C₄

—

118

—

114

C₂×C₄

—

236

—

115

C₂×C₄

—

236

—

116

C₂×C₄

—

236

—

117

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

118

C₂×C₄

C₂×D₂₃₆

yes

characteristic

119

C₂×C₄

#11745687

yes

characteristic

120

C₂×C₄

Q₈×D₅₉

yes

characteristic

121

C₂×C₄

#11745689

yes

characteristic

122

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

123

C₈

—

118

—

124

C₈

C₄⋊Dic₅₉

yes

complemented

125

C₂²

D₄×Dic₅₉

yes

characteristic

126

C₂²

#14273702

yes

characteristic

127

C₂²

#14273726

yes

characteristic

128

C₂²

#14273754

yes

—

129

C₂²

#14273756

yes

—

130

C₄

—

131

C₄

—

472

—

132

C₄

—

236

—

133

C₄

—

236

—

134

C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

135

C₄

Q₈×Dic₅₉

yes

characteristic

136

C₄

#14273707

yes

characteristic

137

C₄

#14273718

yes

characteristic

138

C₂

#16837301

yes

characteristic

139

C₂

#16837503

yes

—

140

C₂

#16837656

yes

characteristic

141

C₂

#16837661

yes

characteristic

142

C₂

#16837809

yes

—

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#106×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#106	D₅₉
hypocenter	C₅₉	#1944
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#106×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
15. #106×C₅₉	C₄	semidirect
51. C₈×C₅₉	C₄⋊C₄	semidirect
77. C₅₉	#1944	semidirect
85. #106	Dic₅₉	semidirect
124. C₈	C₄⋊Dic₅₉	semidirect

H ≅ G/N

type

15. #106×C₅₉

C₄

semidirect

51. C₈×C₅₉

C₄⋊C₄

semidirect

77. C₅₉

#1944

semidirect

85. #106

Dic₅₉

semidirect

124. C₈

C₄⋊Dic₅₉

semidirect

Subgroups — #21424840

Conjugacy classes of subgroups

Complements