Group #21434607 — subgroups

Subgroups — #21434607

Order 7552.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#16837957	C₂	1	2	yes	maximal
2	#16837807	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837803	C₂	1	4	yes	maximal, complemented
4	#16837798	C₂	1	2	yes	maximal, complemented
5	#16837439	C₂	1	4	yes	maximal
6	C₄×C₈×D₅₉	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#500	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₅₉	C₂²	1	2	yes	—
9	#14273727	C₂²	1	2	yes	—
10	#14273718	C₂²	1	4	yes	—
11	#14273707	—	2	2	no	—
12	#14273703	—	2	2	no	—
13	Q₈×Dic₅₉	—	2	2	no	—
14	#14273692	—	2	2	no	—
15	#14273685	C₂²	1	4	yes	—
16	C₄²×D₅₉	C₂²	1	1	yes	characteristic
17	#14273661	C₂²	1	4	yes	—
18	#14273658	C₂²	1	8	yes	—
19	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₅₉	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
22	#107×C₅₉	C₂²	1	4	yes	—
23	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₅₉	—	2	2	no	—
25	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
26	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
27	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₁₈	—	2	2	no	—
29	C₂×C₈×C₅₉	C₂³	1	2	yes	—
30	C₄⋊C₄×C₅₉	—	2	2	no	—
31	C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
32	C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	(C₂×D₅₉)⋊C₄	—	2	4	no	—
34	C₄⋊Dic₅₉	—	2	2	no	—
35	C₄⋊Dic₅₉	—	2	4	no	—
36	C₄⋊Dic₅₉	C₂³	1	4	yes	—
37	Dic₅₉⋊C₄	—	2	4	no	—
38	C₄×Dic₅₉	—	2	4	no	—
39	C₄×Dic₅₉	D₄	1	1	yes	characteristic
40	C₄×Dic₅₉	D₄	1	1	yes	characteristic
41	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
42	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
43	C₈×D₅₉	Q₈	1	8	yes	complemented
44	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₅₉	—	4	4	no	—
46	C₂×C₄×C₅₉	—	2	4	no	—
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₉	—	2	4	no	—
51	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
54	C₄×D₅₉	—	2	2	no	—
55	C₄×D₅₉	—	2	2	no	—
56	C₄×D₅₉	C₂×Q₈	1	4	yes	—
57	Dic₁₁₈	—	4	4	no	—
58	C₈×C₅₉	C₂×Q₈	1	4	yes	—
59	C₅₉⋊C₈	C₂×Q₈	1	4	yes	—
60	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
61	C₂×D₅₉	—	2	1	no	—
62	C₂×D₅₉	—	2	1	no	—
63	C₂×D₅₉	C₄⋊Q₈	1	2	yes	—
64	C₄×C₅₉	—	4	4	no	—
65	C₄×C₅₉	#135	1	2	yes	—
66	C₄×C₅₉	#135	1	2	yes	—
67	C₄×C₅₉	C₂²×Q₈	1	2	yes	—
68	Dic₅₉	—	2	1	no	—
69	Dic₅₉	—	2	1	no	—
70	Dic₅₉	—	4	4	no	—
71	Dic₅₉	C₄⋊Q₈	1	2	yes	—
72	#3160	—	59	1	no	maximal, Sylow
73	C₂×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
75	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
76	D₅₉	—	2	2	no	—
77	#533	—	59	2	no	—
78	#500	—	59	1	no	—
79	#500	D₅₉	1	1	yes	characteristic, complemented
80	#499	—	59	4	no	—
81	#498	—	59	2	no	—
82	#427	—	59	4	no	—
83	C₂×C₄×C₈	—	59	1	no	—
84	C₅₉	#3160	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	59	2	no	—
86	C₄⋊Q₈	—	59	2	no	—
87	C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
88	#125	—	59	4	no	—
89	C₂²⋊Q₈	—	118	2	no	—
90	C₂²⋊Q₈	—	118	2	no	—
91	C₄×Q₈	—	118	2	no	—
92	C₄×Q₈	—	118	2	no	—
93	#117	—	59	4	no	—
94	C₂×C₄²	—	59	1	no	—
95	#107	—	59	4	no	—
96	#107	C₂×D₅₉	1	4	yes	—
97	#106	—	59	8	no	—
98	C₄×C₈	—	59	2	no	—
99	C₄×C₈	—	59	1	no	—
100	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	118	2	no	—
103	C₂²×C₄	—	59	1	no	—
104	C₂²×C₄	—	59	1	no	—
105	C₂²×C₄	—	59	1	no	—
106	C₂×C₈	—	59	2	no	—
107	C₂×C₈	—	59	8	no	—
108	C₂×C₈	C₂²×D₅₉	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	59	4	no	—
111	C₄⋊C₄	—	118	4	no	—
112	C₄⋊C₄	—	118	4	no	—
113	C₄⋊C₄	—	118	2	no	—
114	C₄⋊C₄	C₂²×D₅₉	1	4	yes	—
115	C₂²⋊C₄	—	118	4	no	—
116	C₄²	—	59	1	no	—
117	C₄²	—	59	1	no	—
118	C₄²	—	59	1	no	—
119	C₄²	—	118	4	no	—
120	C₄²	C₂²×D₅₉	1	1	yes	characteristic
121	C₂³	—	59	1	no	—
122	Q₈	—	4	4	no	—
123	Q₈	—	236	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	59	1	no	—
126	C₂×C₄	—	59	4	no	—
127	C₂×C₄	—	59	1	no	—
128	C₂×C₄	—	59	1	no	—
129	C₂×C₄	—	118	2	no	—
130	C₂×C₄	—	118	2	no	—
131	C₂×C₄	—	118	4	no	—
132	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
133	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
135	C₈	—	59	4	no	—
136	C₈	Q₈×D₅₉	1	4	yes	—
137	C₂²	—	59	2	no	—
138	C₂²	—	118	1	no	—
139	C₂²	—	118	1	no	—
140	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
141	C₄	—	4	4	no	—
142	C₄	—	59	2	no	—
143	C₄	—	118	1	no	—
144	C₄	—	118	1	no	—
145	C₄	—	236	4	no	—
146	C₄	#14273746	1	2	yes	—
147	C₄	#14273750	1	2	yes	—
148	C₄	C₂×Q₈×D₅₉	1	2	yes	—
149	C₂	—	118	2	no	—
150	C₂	C₄⋊Q₈×D₅₉	1	1	yes	characteristic
151	C₂	C₂×#14273746	1	1	yes	characteristic
152	C₂	C₂×#14273750	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837957

C₂

yes

maximal

#16837807

C₂

yes

maximal, characteristic, complemented

#16837803

C₂

yes

maximal, complemented

#16837798

C₂

yes

maximal, complemented

#16837439

C₂

yes

maximal

C₄×C₈×D₅₉

C₂

yes

maximal, characteristic

C₅₉×#500

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₉

C₂²

yes

—

#14273727

C₂²

yes

—

#14273718

C₂²

yes

—

#14273707

—

#14273703

—

Q₈×Dic₅₉

—

#14273692

—

#14273685

C₂²

yes

—

C₄²×D₅₉

C₂²

yes

characteristic

#14273661

C₂²

yes

—

#14273658

C₂²

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₅₉

C₂²

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

—

#107×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×C₄×D₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

C₄×Dic₅₉

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

C₂³

yes

—

C₈×D₅₉

Q₈

yes

complemented

C₂²×D₅₉

C₂×D₄

yes

characteristic

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×D₄

yes

characteristic

C₂×Dic₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

C₂×D₄

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

C₂×Q₈

yes

—

Dic₁₁₈

—

C₈×C₅₉

C₂×Q₈

yes

—

C₅₉⋊C₈

C₂×Q₈

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

C₄⋊Q₈

yes

—

C₄×C₅₉

—

C₄×C₅₉

#135

yes

—

C₄×C₅₉

#135

yes

—

C₄×C₅₉

C₂²×Q₈

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

C₄⋊Q₈

yes

—

#3160

—

maximal, Sylow

C₂×C₅₉

C₂×C₄⋊Q₈

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#533

—

#500

—

#500

D₅₉

yes

characteristic, complemented

#499

—

#498

—

#427

—

C₂×C₄×C₈

—

C₅₉

#3160

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₉

yes

—

#125

—

C₂²⋊Q₈

—

118

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

118

—

C₄×Q₈

—

118

—

#117

—

C₂×C₄²

—

#107

—

#107

C₂×D₅₉

yes

—

#106

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₅₉

yes

characteristic

101

C₂×Q₈

—

102

C₂×Q₈

—

118

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²×D₅₉

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

118

—

112

C₄⋊C₄

—

118

—

113

C₄⋊C₄

—

118

—

114

C₄⋊C₄

C₂²×D₅₉

yes

—

115

C₂²⋊C₄

—

118

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

118

—

120

C₄²

C₂²×D₅₉

yes

characteristic

121

C₂³

—

122

Q₈

—

123

Q₈

—

236

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

118

—

130

C₂×C₄

—

118

—

131

C₂×C₄

—

118

—

132

C₂×C₄

D₄×D₅₉

yes

characteristic

133

C₂×C₄

D₄×D₅₉

yes

characteristic

134

C₂×C₄

C₂³×D₅₉

yes

characteristic

135

C₈

—

136

C₈

Q₈×D₅₉

yes

—

137

C₂²

—

138

C₂²

—

118

—

139

C₂²

—

118

—

140

C₂²

C₂×D₄×D₅₉

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

118

—

144

C₄

—

118

—

145

C₄

—

236

—

146

C₄

#14273746

yes

—

147

C₄

#14273750

yes

—

148

C₄

C₂×Q₈×D₅₉

yes

—

149

C₂

—

118

—

150

C₂

C₄⋊Q₈×D₅₉

yes

characteristic

151

C₂

C₂×#14273746

yes

characteristic

152

C₂

C₂×#14273750

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	#3160
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
2. #16837807	C₂	semidirect
3. #16837803	C₂	semidirect
4. #16837798	C₂	semidirect
7. C₅₉×#500	C₂	semidirect
43. C₈×D₅₉	Q₈	semidirect
79. #500	D₅₉	semidirect
84. C₅₉	#3160	semidirect

H ≅ G/N

type

2. #16837807

C₂

semidirect

3. #16837803

C₂

semidirect

4. #16837798

C₂

semidirect

7. C₅₉×#500

C₂

semidirect

43. C₈×D₅₉

Q₈

semidirect

79. #500

D₅₉

semidirect

84. C₅₉

#3160

semidirect

Subgroups — #21434607

Conjugacy classes of subgroups

Complements