Group #21423613 — subgroups

Subgroups — #21423613

Order 7552.

152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16837805	C₂	1	1	yes	maximal, characteristic
2	#16837613	C₂	1	1	yes	maximal, characteristic
3	#16837609	C₂	1	1	yes	maximal, characteristic
4	#16837529	C₂	1	1	yes	maximal, characteristic
5	#16837152	C₂	1	1	yes	maximal, characteristic
6	#16837147	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#332	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₃₆	—	2	1	no	—
9	C₂×#11745672	—	2	1	no	—
10	#14273727	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
12	#14273660	—	2	1	no	—
13	#14273660	—	2	1	no	—
14	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₅₉⋊C₈	—	2	1	no	—
16	#14273652	—	2	1	no	—
17	#14273651	—	2	1	no	—
18	#14273637	—	2	1	no	—
19	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	C₂×Dic₁₁₈	—	2	1	no	—
25	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
26	C₂×C₈×C₅₉	—	2	1	no	—
27	C₂×C₈×C₅₉	—	2	1	no	—
28	C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	#11745672	—	4	2	no	—
32	C₄⋊Dic₅₉	—	2	1	no	—
33	C₄⋊Dic₅₉	—	2	1	no	—
34	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
35	Dic₅₉⋊C₄	—	2	1	no	—
36	C₄×Dic₅₉	—	2	1	no	—
37	C₂×C₅₉⋊C₈	—	2	1	no	—
38	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
40	Dic₂₃₆	—	4	2	no	—
41	Q₈×C₅₉	—	4	1	no	—
42	Q₈×C₅₉	—	4	1	no	—
43	C₂×C₄×C₅₉	—	4	1	no	—
44	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
47	C₂×Dic₅₉	—	2	1	no	—
48	C₂×Dic₅₉	—	2	1	no	—
49	C₂×Dic₅₉	—	4	1	no	—
50	Dic₁₁₈	—	2	1	no	—
51	Dic₁₁₈	—	2	1	no	—
52	Dic₁₁₈	—	4	1	no	—
53	Dic₁₁₈	—	4	1	no	—
54	C₈×C₅₉	—	4	1	no	—
55	C₈×C₅₉	—	4	1	no	—
56	C₅₉⋊C₈	—	2	1	no	—
57	C₅₉⋊C₈	—	4	1	no	—
58	C₅₉⋊C₈	Q₁₆	1	2	yes	complemented
59	C₂²×C₅₉	#121	1	1	yes	characteristic
60	C₄×C₅₉	—	2	1	no	—
61	C₄×C₅₉	—	8	1	no	—
62	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₄×C₅₉	#137	1	1	yes	characteristic
66	Dic₅₉	—	4	1	no	—
67	Dic₅₉	—	4	1	no	—
68	Dic₅₉	—	8	1	no	—
69	#1694	—	59	1	no	maximal, Sylow
70	C₂×C₅₉	#462	1	1	yes	characteristic
71	C₂×C₅₉	#469	1	1	yes	characteristic
72	C₂×C₅₉	#473	1	1	yes	characteristic
73	#500	—	59	1	no	—
74	#462	—	59	1	no	—
75	#462	—	59	1	no	—
76	#446	—	59	1	no	—
77	#334	—	59	1	no	—
78	#332	—	59	1	no	—
79	#332	D₅₉	1	1	yes	characteristic
80	C₅₉	#1694	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	118	1	no	—
82	C₂×Q₁₆	—	118	1	no	—
83	C₄⋊Q₈	—	59	1	no	—
84	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
85	C₄×Q₈	—	59	1	no	—
86	#107	—	118	1	no	—
87	#107	—	118	1	no	—
88	C₄⋊C₈	—	59	1	no	—
89	C₄⋊C₈	—	118	1	no	—
90	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
91	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
92	#103	—	118	1	no	—
93	#103	—	118	1	no	—
94	#97	—	118	1	no	—
95	C₄×C₈	—	59	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	59	1	no	—
98	C₂×Q₈	—	118	1	no	—
99	Q₁₆	—	236	2	no	—
100	Q₁₆	—	236	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	59	1	no	—
104	C₂×C₈	—	59	1	no	—
105	C₂×C₈	—	118	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	59	1	no	—
109	C₄⋊C₄	—	118	1	no	—
110	C₄⋊C₄	—	118	1	no	—
111	C₄⋊C₄	—	118	1	no	—
112	C₄²	—	118	1	no	—
113	C₄²	C₂²×D₅₉	1	1	yes	characteristic
114	Q₈	—	4	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	118	1	no	—
117	Q₈	—	118	1	no	—
118	Q₈	—	236	1	no	—
119	Q₈	—	236	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	118	1	no	—
122	C₂×C₄	—	118	1	no	—
123	C₂×C₄	—	236	1	no	—
124	C₂×C₄	#11745685	1	1	yes	characteristic
125	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
126	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	59	2	no	—
130	C₈	—	118	1	no	—
131	C₈	—	236	1	no	—
132	C₂²	#14273700	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	472	1	no	—
136	C₄	—	236	1	no	—
137	C₄	—	236	1	no	—
138	C₄	Q₁₆×D₅₉	1	1	yes	characteristic
139	C₄	#14273751	1	1	yes	characteristic
140	C₄	#14273754	1	1	yes	characteristic
141	C₄	#14273756	1	1	yes	characteristic
142	C₂	#16837611	1	1	yes	characteristic
143	C₂	#16837654	1	1	yes	characteristic
144	C₂	#16837676	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837805

C₂

yes

maximal, characteristic

#16837613

C₂

yes

maximal, characteristic

#16837609

C₂

yes

maximal, characteristic

#16837529

C₂

yes

maximal, characteristic

#16837152

C₂

yes

maximal, characteristic

#16837147

C₂

yes

maximal, characteristic

C₅₉×#332

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

—

C₂×#11745672

—

#14273727

C₂²

yes

characteristic

C₄×Dic₁₁₈

C₂²

yes

characteristic

#14273660

—

#14273660

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

#14273652

—

#14273651

—

#14273637

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745672

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

Q₁₆

yes

complemented

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

—

—

—

D₅₉

yes

characteristic

C₅₉

#1694

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄×Q₈

—

#107

—

118

—

#107

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

118

—

#103

—

118

—

#97

—

118

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

Q₁₆

—

236

—

100

Q₁₆

—

236

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

118

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

118

—

110

C₄⋊C₄

—

118

—

111

C₄⋊C₄

—

118

—

112

C₄²

—

118

—

113

C₄²

C₂²×D₅₉

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

118

—

117

Q₈

—

118

—

118

Q₈

—

236

—

119

Q₈

—

236

—

120

C₂×C₄

—

121

C₂×C₄

—

118

—

122

C₂×C₄

—

118

—

123

C₂×C₄

—

236

—

124

C₂×C₄

#11745685

yes

characteristic

125

C₂×C₄

D₄×D₅₉

yes

characteristic

126

C₂×C₄

D₄×D₅₉

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

118

—

131

C₈

—

236

—

132

C₂²

#14273700

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

472

—

136

C₄

—

236

—

137

C₄

—

236

—

138

C₄

Q₁₆×D₅₉

yes

characteristic

139

C₄

#14273751

yes

characteristic

140

C₄

#14273754

yes

characteristic

141

C₄

#14273756

yes

characteristic

142

C₂

#16837611

yes

characteristic

143

C₂

#16837654

yes

characteristic

144

C₂

#16837676

yes

characteristic

	subgroup	G/H
center	C₂²	#14273700
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#332	C₂
socle	C₂²×C₅₉	#121
hypercenter	#332	D₅₉
hypocenter	C₅₉	#1694
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14273700

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#332

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
58. C₅₉⋊C₈	Q₁₆	semidirect
80. C₅₉	#1694	semidirect

H ≅ G/N

type

58. C₅₉⋊C₈

Q₁₆

semidirect

80. C₅₉

#1694

semidirect

Subgroups — #21423613

Conjugacy classes of subgroups

Complements