Group #21425509 — subgroups

Subgroups — #21425509

Order 7552.

258 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 81 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×#14273648	C₂	1	2	yes	maximal
2	#16837268	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837164	C₂	1	2	yes	maximal, complemented
4	#16837157	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#390	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
8	#14273648	—	2	4	no	—
9	C₅₉⋊#95	—	2	2	no	—
10	C₂²×D₄×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
13	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
14	C₂⁴×C₅₉	—	2	1	no	—
15	C₂×D₄×C₅₉	—	2	2	no	—
16	C₂×D₄×C₅₉	—	2	2	no	—
17	C₂×D₄×C₅₉	—	2	1	no	—
18	C₂×D₄×C₅₉	D₄	1	2	yes	—
19	C₂²×C₄×C₅₉	—	2	1	no	—
20	C₂²×C₄×C₅₉	D₄	1	2	yes	—
21	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₉	—	2	2	no	—
23	C₄⋊C₄×C₅₉	—	2	2	no	—
24	C₂²⋊C₄×C₅₉	—	4	4	no	—
25	C₄²×C₅₉	—	2	2	no	—
26	C₄⋊Dic₅₉	—	2	2	no	—
27	C₄⋊Dic₅₉	—	2	2	no	—
28	C₂×C₅₉⋊C₈	—	2	2	no	—
29	C₂×C₅₉⋊C₈	—	2	2	no	—
30	C₂×C₅₉⋊C₈	—	2	2	no	—
31	C₂³×C₅₉	—	2	2	no	—
32	C₂³×C₅₉	—	4	1	no	—
33	C₂³×C₅₉	—	4	1	no	—
34	C₂³×C₅₉	—	4	2	no	—
35	C₂³×C₅₉	—	4	2	no	—
36	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
37	D₄×C₅₉	—	2	4	no	—
38	D₄×C₅₉	—	4	2	no	—
39	C₂×C₄×C₅₉	—	2	2	no	—
40	C₂×C₄×C₅₉	—	2	2	no	—
41	C₂×C₄×C₅₉	—	2	2	no	—
42	C₂×C₄×C₅₉	—	4	1	no	—
43	C₂×C₄×C₅₉	—	4	2	no	—
44	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
46	C₂×C₄×C₅₉	#41	1	2	yes	—
47	C₂×C₄×C₅₉	#41	1	2	yes	—
48	C₂×Dic₅₉	—	4	2	no	—
49	C₂×Dic₅₉	—	4	2	no	—
50	C₂×Dic₅₉	—	4	2	no	—
51	C₅₉⋊C₈	—	4	2	no	—
52	C₂²×C₅₉	—	4	2	no	—
53	C₂²×C₅₉	—	4	4	no	—
54	C₂²×C₅₉	—	8	2	no	—
55	C₂²×C₅₉	—	8	1	no	—
56	C₂²×C₅₉	—	8	1	no	—
57	C₂²×C₅₉	#121	1	1	yes	characteristic
58	C₂²×C₅₉	#123	1	1	yes	characteristic
59	C₂²×C₅₉	#124	1	1	yes	characteristic
60	C₂²×C₅₉	#135	1	2	yes	—
61	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₅₉	—	4	2	no	—
63	C₄×C₅₉	—	8	1	no	—
64	C₄×C₅₉	#123	1	2	yes	—
65	C₄×C₅₉	#126	1	2	yes	—
66	Dic₅₉	—	8	2	no	—
67	#2073	—	59	1	no	maximal, Sylow
68	C₂×C₅₉	—	8	2	no	—
69	C₂×C₅₉	#397	1	1	yes	characteristic
70	C₂×C₅₉	#463	1	2	yes	—
71	C₂×C₅₉	#482	1	2	yes	—
72	C₂×C₅₉	#488	1	1	yes	characteristic
73	C₂×C₅₉	#490	1	1	yes	characteristic
74	C₂×#102	—	59	2	no	—
75	#390	D₅₉	1	1	yes	characteristic
76	#382	—	59	1	no	—
77	#340	—	59	2	no	—
78	#336	—	59	1	no	—
79	C₅₉	#2073	1	1	yes	characteristic, complemented, Sylow
80	C₂²×D₄	C₂×D₅₉	1	1	yes	characteristic
81	C₂²×C₈	—	59	2	no	—
82	C₂×C₄⋊C₄	—	59	2	no	—
83	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
84	C₂×C₂²⋊C₄	—	2	2	no	—
85	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
86	#102	—	118	4	no	—
87	#95	—	118	2	no	—
88	C₂⁴	—	2	1	no	—
89	C₂×D₄	—	2	1	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	D₂₃₆	1	2	yes	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	118	2	no	—
95	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
96	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
97	C₂×C₈	—	118	2	no	—
98	C₂×C₈	—	118	2	no	—
99	C₂×C₈	—	118	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	118	2	no	—
102	C₄⋊C₄	—	118	2	no	—
103	C₂²⋊C₄	—	4	4	no	—
104	C₄²	—	2	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	—	4	1	no	—
107	C₂³	—	4	1	no	—
108	C₂³	—	4	2	no	—
109	C₂³	—	4	2	no	—
110	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
111	D₄	—	2	4	no	—
112	D₄	—	4	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	236	2	no	—
119	C₂×C₄	—	236	2	no	—
120	C₂×C₄	—	236	2	no	—
121	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
122	C₂×C₄	#11745685	1	1	yes	characteristic
123	C₂×C₄	#11745687	1	2	yes	—
124	C₂×C₄	#11745687	1	2	yes	—
125	C₈	—	236	2	no	—
126	C₂²	—	4	2	no	—
127	C₂²	—	4	4	no	—
128	C₂²	—	8	2	no	—
129	C₂²	—	8	1	no	—
130	C₂²	—	8	1	no	—
131	C₂²	#14273701	1	1	yes	characteristic
132	C₂²	#14273708	1	1	yes	characteristic
133	C₂²	#14273714	1	1	yes	characteristic
134	C₂²	#14273747	1	2	yes	—
135	C₂²	#14273753	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	—	8	1	no	—
138	C₄	—	472	2	no	—
139	C₄	#14273711	1	2	yes	—
140	C₄	#14273720	1	2	yes	—
141	C₂	—	8	2	no	—
142	C₂	#16837327	1	1	yes	characteristic
143	C₂	#16837617	1	2	yes	—
144	C₂	#16837723	1	2	yes	—
145	C₂	#16837756	1	1	yes	characteristic
146	C₂	#16837766	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273648

C₂

yes

maximal

#16837268

C₂

yes

maximal, characteristic, complemented

#16837164

C₂

yes

maximal, complemented

#16837157

C₂

yes

maximal, characteristic, complemented

C₅₉×#390

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273648

—

C₅₉⋊#95

—

C₂²×D₄×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂⁴×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

AGL(1,ℤ/8)

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#123

yes

—

C₄×C₅₉

#126

yes

—

Dic₅₉

—

#2073

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#397

yes

characteristic

C₂×C₅₉

#463

yes

—

C₂×C₅₉

#482

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

—

—

—

C₅₉

#2073

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#102

—

118

—

#95

—

118

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₂₃₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

118

—

102

C₄⋊C₄

—

118

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

236

—

119

C₂×C₄

—

236

—

120

C₂×C₄

—

236

—

121

C₂×C₄

C₂×D₂₃₆

yes

characteristic

122

C₂×C₄

#11745685

yes

characteristic

123

C₂×C₄

#11745687

yes

—

124

C₂×C₄

#11745687

yes

—

125

C₈

—

236

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#14273701

yes

characteristic

132

C₂²

#14273708

yes

characteristic

133

C₂²

#14273714

yes

characteristic

134

C₂²

#14273747

yes

—

135

C₂²

#14273753

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

472

—

139

C₄

#14273711

yes

—

140

C₄

#14273720

yes

—

141

C₂

—

142

C₂

#16837327

yes

characteristic

143

C₂

#16837617

yes

—

144

C₂

#16837723

yes

—

145

C₂

#16837756

yes

characteristic

146

C₂

#16837766

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#390	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	#390	D₅₉
hypocenter	C₅₉	#2073
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#390

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
2. #16837268	C₂	semidirect
3. #16837164	C₂	semidirect
4. #16837157	C₂	semidirect
79. C₅₉	#2073	semidirect

H ≅ G/N

type

2. #16837268

C₂

semidirect

3. #16837164

C₂

semidirect

4. #16837157

C₂

semidirect

79. C₅₉

#2073

semidirect

Subgroups — #21425509

Conjugacy classes of subgroups

Complements