Group #21434680 — subgroups

Subgroups — #21434680

Order 7552.

368 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16837752	C₂	1	2	yes	maximal, complemented
4	#16837640	C₂	1	4	yes	maximal
5	Q₁₆×Dic₅₉	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×D₅₉	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₅₉	C₂²	1	2	yes	—
9	#14273727	C₂²	1	2	yes	—
10	#14273718	—	2	2	no	—
11	#14273707	—	2	2	no	—
12	#14273703	C₂²	1	4	yes	—
13	Q₈×Dic₅₉	C₂²	1	4	yes	—
14	#14273692	—	2	2	no	—
15	#14273685	—	2	2	no	—
16	C₄²×D₅₉	C₂²	1	1	yes	characteristic
17	#14273661	C₂²	1	4	yes	—
18	#14273651	C₂²	1	8	yes	—
19	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₅₉	C₂²	1	2	yes	—
21	C₂×Q₁₆×C₅₉	C₂²	1	4	yes	—
22	C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
23	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₅₉	C₂³	1	4	yes	—
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	Q₁₆×C₅₉	—	2	8	no	—
30	C₂×C₈×C₅₉	C₂³	1	2	yes	—
31	C₄⋊C₄×C₅₉	—	2	2	no	—
32	C₄⋊C₄×C₅₉	—	2	2	no	—
33	C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	(C₂×D₅₉)⋊C₄	—	2	4	no	—
35	C₄⋊Dic₅₉	—	2	2	no	—
36	C₄⋊Dic₅₉	—	2	4	no	—
37	C₄⋊Dic₅₉	C₂³	1	4	yes	—
38	Dic₅₉⋊C₄	—	2	4	no	—
39	C₄×Dic₅₉	—	2	4	no	—
40	C₄×Dic₅₉	D₄	1	1	yes	characteristic
41	C₄×Dic₅₉	D₄	1	1	yes	characteristic
42	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
43	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
44	C₈×D₅₉	—	2	4	no	—
45	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₅₉	—	2	8	no	—
47	C₂×C₄×C₅₉	—	2	4	no	—
48	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₅₉	—	2	4	no	—
52	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
55	C₄×D₅₉	—	2	2	no	—
56	C₄×D₅₉	—	2	2	no	—
57	C₄×D₅₉	Q₁₆	1	4	yes	complemented
58	Dic₁₁₈	—	4	4	no	—
59	C₈×C₅₉	#41	1	4	yes	—
60	C₅₉⋊C₈	—	2	2	no	—
61	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₅₉	—	2	1	no	—
63	C₂×D₅₉	—	2	1	no	—
64	C₂×D₅₉	C₂×Q₁₆	1	2	yes	—
65	C₄×C₅₉	—	4	4	no	—
66	C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
67	C₄×C₅₉	#135	1	2	yes	—
68	C₄×C₅₉	C₂×#41	1	2	yes	—
69	Dic₅₉	—	2	1	no	—
70	Dic₅₉	—	2	1	no	—
71	Dic₅₉	—	4	4	no	—
72	Dic₅₉	C₂×Q₁₆	1	2	yes	—
73	#3170	—	59	1	no	maximal, Sylow
74	C₂×C₅₉	#532	1	1	yes	characteristic
75	C₂×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
76	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
77	D₅₉	—	2	2	no	—
78	#533	—	59	2	no	—
79	#500	—	59	1	no	—
80	#494	D₅₉	1	1	yes	characteristic, complemented
81	#487	—	59	2	no	—
82	#467	—	59	4	no	—
83	C₄×Q₁₆	—	59	4	no	—
84	C₂×C₄×C₈	—	59	1	no	—
85	C₅₉	#3170	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₁₆	C₂×D₅₉	1	4	yes	—
87	C₂²×C₈	—	59	2	no	—
88	C₄⋊Q₈	—	59	2	no	—
89	C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
90	#125	—	118	2	no	—
91	C₂²⋊Q₈	—	59	4	no	—
92	C₂²⋊Q₈	—	118	2	no	—
93	C₄×Q₈	—	59	4	no	—
94	C₄×Q₈	—	118	2	no	—
95	#117	—	118	2	no	—
96	C₂×C₄²	—	59	1	no	—
97	#107	—	59	4	no	—
98	#103	—	59	8	no	—
99	C₄×C₈	—	59	2	no	—
100	C₄×C₈	—	59	1	no	—
101	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
102	C₂×Q₈	—	118	2	no	—
103	C₂×Q₈	C₂²×D₅₉	1	4	yes	—
104	C₂²×C₄	—	59	1	no	—
105	C₂²×C₄	—	59	1	no	—
106	C₂²×C₄	—	59	1	no	—
107	Q₁₆	—	2	8	no	—
108	C₂×C₈	—	59	2	no	—
109	C₂×C₈	—	118	4	no	—
110	C₂×C₈	C₂²×D₅₉	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	59	4	no	—
114	C₄⋊C₄	—	118	4	no	—
115	C₄⋊C₄	—	118	4	no	—
116	C₄⋊C₄	—	118	2	no	—
117	C₂²⋊C₄	—	118	4	no	—
118	C₄²	—	59	1	no	—
119	C₄²	—	59	1	no	—
120	C₄²	—	59	1	no	—
121	C₄²	—	118	4	no	—
122	C₄²	C₂²×D₅₉	1	1	yes	characteristic
123	C₂³	—	59	1	no	—
124	Q₈	—	2	8	no	—
125	Q₈	—	236	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	59	1	no	—
128	C₂×C₄	—	59	4	no	—
129	C₂×C₄	—	59	1	no	—
130	C₂×C₄	—	59	1	no	—
131	C₂×C₄	—	118	2	no	—
132	C₂×C₄	—	118	2	no	—
133	C₂×C₄	—	118	4	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	—	118	2	no	—
138	C₈	#11745687	1	4	yes	—
139	C₂²	—	59	2	no	—
140	C₂²	—	118	1	no	—
141	C₂²	—	118	1	no	—
142	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
143	C₄	—	4	4	no	—
144	C₄	—	59	2	no	—
145	C₄	—	118	1	no	—
146	C₄	—	118	1	no	—
147	C₄	—	236	4	no	—
148	C₄	Q₁₆×D₅₉	1	2	yes	—
149	C₄	#14273750	1	2	yes	—
150	C₄	C₂×#11745687	1	2	yes	—
151	C₂	—	118	2	no	—
152	C₂	#16837950	1	1	yes	characteristic
153	C₂	C₂×Q₁₆×D₅₉	1	1	yes	characteristic
154	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

#16837752

C₂

yes

maximal, complemented

#16837640

C₂

yes

maximal

Q₁₆×Dic₅₉

C₂

yes

maximal, complemented

C₄×C₈×D₅₉

C₂

yes

maximal, characteristic

C₅₉×#494

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₉

C₂²

yes

—

#14273727

C₂²

yes

—

#14273718

—

#14273707

—

#14273703

C₂²

yes

—

Q₈×Dic₅₉

C₂²

yes

—

#14273692

—

#14273685

—

C₄²×D₅₉

C₂²

yes

characteristic

#14273661

C₂²

yes

—

#14273651

C₂²

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₅₉

C₂²

yes

—

C₂×Q₁₆×C₅₉

C₂²

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

C₂³

yes

—

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×C₄×D₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

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

—

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

Q₁₆

yes

complemented

Dic₁₁₈

—

C₈×C₅₉

#41

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

C₂×Q₁₆

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×Q₁₆

yes

—

C₄×C₅₉

#135

yes

—

C₄×C₅₉

C₂×#41

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

C₂×Q₁₆

yes

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂²×Q₁₆

yes

characteristic

C₂×C₅₉

C₂×#135

yes

characteristic

D₅₉

—

#533

—

#500

—

#494

D₅₉

yes

characteristic, complemented

#487

—

#467

—

C₄×Q₁₆

—

C₂×C₄×C₈

—

C₅₉

#3170

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₉

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₉

yes

—

#125

—

118

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

C₄×Q₈

—

118

—

#117

—

118

—

C₂×C₄²

—

#107

—

#103

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂×D₅₉

yes

characteristic

102

C₂×Q₈

—

118

—

103

C₂×Q₈

C₂²×D₅₉

yes

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

Q₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

118

—

110

C₂×C₈

C₂²×D₅₉

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

118

—

115

C₄⋊C₄

—

118

—

116

C₄⋊C₄

—

118

—

117

C₂²⋊C₄

—

118

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

118

—

122

C₄²

C₂²×D₅₉

yes

characteristic

123

C₂³

—

124

Q₈

—

125

Q₈

—

236

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

118

—

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

—

118

—

138

C₈

#11745687

yes

—

139

C₂²

—

140

C₂²

—

118

—

141

C₂²

—

118

—

142

C₂²

C₂×D₄×D₅₉

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

118

—

146

C₄

—

118

—

147

C₄

—

236

—

148

C₄

Q₁₆×D₅₉

yes

—

149

C₄

#14273750

yes

—

150

C₄

C₂×#11745687

yes

—

151

C₂

—

118

—

152

C₂

#16837950

yes

characteristic

153

C₂

C₂×Q₁₆×D₅₉

yes

characteristic

154

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×#494	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#494	D₅₉
hypocenter	C₅₉	#3170
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#494

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #16837807	C₂	semidirect
3. #16837752	C₂	semidirect
5. Q₁₆×Dic₅₉	C₂	semidirect
7. C₅₉×#494	C₂	semidirect
57. C₄×D₅₉	Q₁₆	semidirect
80. #494	D₅₉	semidirect
85. C₅₉	#3170	semidirect

H ≅ G/N

type

2. #16837807

C₂

semidirect

3. #16837752

C₂

semidirect

5. Q₁₆×Dic₅₉

C₂

semidirect

7. C₅₉×#494

C₂

semidirect

57. C₄×D₅₉

Q₁₆

semidirect

80. #494

D₅₉

semidirect

85. C₅₉

#3170

semidirect

Subgroups — #21434680

Conjugacy classes of subgroups

Complements