Group #21436989 — subgroups

Subgroups — #21436989

Order 7552.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16838127	C₂	1	2	yes	maximal
2	#16837803	C₂	1	2	yes	maximal
3	#16837708	C₂	1	4	yes	maximal
4	#16837684	C₂	1	2	yes	maximal
5	C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
7	#16837496	C₂	1	2	yes	maximal
8	C₅₉×#500	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
10	#14273725	—	2	2	no	—
11	#14273718	C₂²	1	4	yes	—
12	#14273717	—	2	4	no	—
13	Q₈×Dic₅₉	—	2	2	no	—
14	C₄×Dic₁₁₈	C₂²	1	2	yes	—
15	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
16	#14273692	C₂²	1	4	yes	—
17	#14273661	C₂²	1	1	yes	characteristic
18	#14273658	C₂²	1	4	yes	—
19	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₅₉⋊C₈	C₂²	1	2	yes	—
21	#14273652	C₂²	1	2	yes	—
22	#14273650	C₂²	1	4	yes	—
23	C₈×Dic₅₉	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
25	#107×C₅₉	C₂²	1	2	yes	—
26	#107×C₅₉	C₂²	1	2	yes	—
27	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₅₉	—	2	2	no	—
29	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
30	C₂×Dic₁₁₈	C₂³	1	2	yes	—
31	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₅₉	—	2	2	no	—
34	C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
35	C₄²×C₅₉	C₂³	1	1	yes	characteristic
36	C₄⋊Dic₅₉	—	2	4	no	—
37	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
38	C₄⋊Dic₅₉	C₂³	1	2	yes	—
39	Dic₅₉⋊C₄	—	2	2	no	—
40	Dic₅₉⋊C₄	—	2	4	no	—
41	Dic₅₉⋊C₄	—	2	4	no	—
42	Dic₅₉⋊C₄	D₄	1	2	yes	—
43	C₄×Dic₅₉	—	2	2	no	—
44	C₄×Dic₅₉	—	2	4	no	—
45	C₄×Dic₅₉	C₂³	1	2	yes	—
46	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
47	Dic₂₃₆	Q₈	1	4	yes	complemented
48	Q₈×C₅₉	—	4	4	no	—
49	C₂×C₄×C₅₉	—	2	4	no	—
50	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₅₉	—	2	2	no	—
54	C₂×Dic₅₉	—	2	2	no	—
55	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
56	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
57	Dic₁₁₈	—	2	2	no	—
58	Dic₁₁₈	—	2	2	no	—
59	Dic₁₁₈	C₂×Q₈	1	4	yes	—
60	C₈×C₅₉	—	2	1	no	—
61	C₈×C₅₉	C₂×Q₈	1	2	yes	—
62	C₅₉⋊C₈	—	2	2	no	—
63	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₅₉	—	2	1	no	—
65	C₄×C₅₉	—	4	4	no	—
66	C₄×C₅₉	#135	1	2	yes	—
67	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
69	Dic₅₉	—	2	4	no	—
70	Dic₅₉	—	2	2	no	—
71	Dic₅₉	—	2	2	no	—
72	Dic₅₉	—	4	2	no	—
73	#3383	—	59	1	no	maximal, Sylow
74	C₂×C₅₉	D₄×Q₈	1	1	yes	characteristic
75	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
76	C₂×C₅₉	#578	1	1	yes	characteristic
77	#557	—	59	2	no	—
78	#500	D₅₉	1	1	yes	characteristic
79	#499	—	59	2	no	—
80	#479	—	59	4	no	—
81	#475	—	59	2	no	—
82	C₈×Q₈	—	59	1	no	—
83	C₄×Q₁₆	—	59	2	no	—
84	C₄×Q₁₆	—	59	1	no	—
85	C₅₉	#3383	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₁₆	—	59	1	no	—
87	C₄⋊Q₈	—	118	2	no	—
88	C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
89	#125	—	59	4	no	—
90	#125	—	118	4	no	—
91	C₄×Q₈	—	59	1	no	—
92	C₄×Q₈	—	59	4	no	—
93	C₄×Q₈	—	59	2	no	—
94	C₄×Q₈	—	118	2	no	—
95	#107	—	59	1	no	—
96	#107	C₂×D₅₉	1	2	yes	—
97	#107	C₂×D₅₉	1	2	yes	—
98	#106	—	59	4	no	—
99	C₄⋊C₈	—	59	2	no	—
100	C₄⋊C₈	—	59	1	no	—
101	#103	—	59	2	no	—
102	#103	—	59	4	no	—
103	C₄×C₈	—	59	2	no	—
104	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	59	1	no	—
107	C₂×Q₈	—	59	2	no	—
108	Q₁₆	—	59	4	no	—
109	C₂×C₈	—	59	2	no	—
110	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	59	1	no	—
114	C₄⋊C₄	—	59	2	no	—
115	C₄⋊C₄	—	59	2	no	—
116	C₄⋊C₄	—	118	4	no	—
117	C₄⋊C₄	—	118	4	no	—
118	C₄⋊C₄	—	118	4	no	—
119	C₄⋊C₄	—	118	2	no	—
120	C₄⋊C₄	C₂²×D₅₉	1	4	yes	—
121	C₄²	—	59	2	no	—
122	C₄²	—	118	4	no	—
123	C₄²	—	118	2	no	—
124	C₄²	C₂²×D₅₉	1	1	yes	characteristic
125	Q₈	—	4	4	no	—
126	Q₈	—	59	4	no	—
127	Q₈	—	118	2	no	—
128	Q₈	—	118	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	59	2	no	—
131	C₂×C₄	—	59	2	no	—
132	C₂×C₄	—	118	2	no	—
133	C₂×C₄	—	118	2	no	—
134	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
135	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	118	2	no	—
139	C₈	Q₈×D₅₉	1	2	yes	—
140	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	4	no	—
143	C₄	—	118	2	no	—
144	C₄	—	118	2	no	—
145	C₄	—	118	4	no	—
146	C₄	—	236	2	no	—
147	C₄	#14273746	1	2	yes	—
148	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
149	C₄	#14273775	1	1	yes	characteristic
150	C₂	#16838067	1	1	yes	characteristic
151	C₂	C₂×#14273746	1	1	yes	characteristic
152	C₂	#16838215	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838127

C₂

yes

maximal

#16837803

C₂

yes

maximal

#16837708

C₂

yes

maximal

#16837684

C₂

yes

maximal

C₈×Dic₁₁₈

C₂

yes

maximal, characteristic

C₄×Dic₂₃₆

C₂

yes

maximal, characteristic

#16837496

C₂

yes

maximal

C₅₉×#500

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

characteristic

#14273725

—

#14273718

C₂²

yes

—

#14273717

—

Q₈×Dic₅₉

—

C₄×Dic₁₁₈

C₂²

yes

—

C₄×Dic₁₁₈

C₂²

yes

characteristic

#14273692

C₂²

yes

—

#14273661

C₂²

yes

characteristic

#14273658

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

C₂²

yes

—

#14273652

C₂²

yes

—

#14273650

C₂²

yes

—

C₈×Dic₅₉

C₂²

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

—

#107×C₅₉

C₂²

yes

—

#107×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×Dic₁₁₈

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

D₄

yes

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

Dic₂₃₆

Q₈

yes

complemented

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₄

yes

—

C₂×Dic₅₉

C₂×D₄

yes

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

C₂×Q₈

yes

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₈

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#135

yes

—

C₄×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#3383

—

maximal, Sylow

C₂×C₅₉

D₄×Q₈

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₅₉

#3383

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

118

—

C₄⋊Q₈

C₂×D₅₉

yes

—

#125

—

#125

—

118

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

118

—

#107

—

#107

C₂×D₅₉

yes

—

#107

C₂×D₅₉

yes

—

#106

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#103

—

102

#103

—

103

C₄×C₈

—

104

C₄×C₈

C₂×D₅₉

yes

characteristic

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

Q₁₆

—

109

C₂×C₈

—

110

C₂×C₈

C₂²×D₅₉

yes

characteristic

111

C₂×C₈

C₂²×D₅₉

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

118

—

117

C₄⋊C₄

—

118

—

118

C₄⋊C₄

—

118

—

119

C₄⋊C₄

—

118

—

120

C₄⋊C₄

C₂²×D₅₉

yes

—

121

C₄²

—

122

C₄²

—

118

—

123

C₄²

—

118

—

124

C₄²

C₂²×D₅₉

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

118

—

128

Q₈

—

118

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

118

—

133

C₂×C₄

—

118

—

134

C₂×C₄

D₄×D₅₉

yes

characteristic

135

C₂×C₄

D₄×D₅₉

yes

characteristic

136

C₂×C₄

C₂³×D₅₉

yes

characteristic

137

C₈

—

138

C₈

—

118

—

139

C₈

Q₈×D₅₉

yes

—

140

C₂²

C₂×D₄×D₅₉

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

118

—

144

C₄

—

118

—

145

C₄

—

118

—

146

C₄

—

236

—

147

C₄

#14273746

yes

—

148

C₄

C₂×Q₈×D₅₉

yes

characteristic

149

C₄

#14273775

yes

characteristic

150

C₂

#16838067

yes

characteristic

151

C₂

C₂×#14273746

yes

characteristic

152

C₂

#16838215

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#500	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#500	D₅₉
hypocenter	C₅₉	#3383
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#500

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
47. Dic₂₃₆	Q₈	semidirect
85. C₅₉	#3383	semidirect

H ≅ G/N

type

47. Dic₂₃₆

Q₈

semidirect

85. C₅₉

#3383

semidirect

Subgroups — #21436989

Conjugacy classes of subgroups

Complements