Group #21426364 — subgroups

Subgroups — #21426364

Order 7552.

178 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 53 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×#14273673	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14273672	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273670	C₂	1	1	yes	maximal, characteristic
4	#16837787	C₂	1	1	yes	maximal, characteristic
5	#16837780	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837177	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
9	C₂×#11745672	—	2	1	no	—
10	C₂×#11745671	—	2	1	no	—
11	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
12	#14273725	—	2	1	no	—
13	#14273716	—	2	1	no	—
14	#14273673	—	2	2	no	—
15	#14273672	—	2	2	no	—
16	#14273670	—	2	2	no	—
17	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	Q₃₂×C₅₉	—	2	2	no	—
22	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	C₂×Q₈×C₅₉	—	2	1	no	—
25	C₂×D₂₃₆	—	2	1	no	—
26	C₂×Dic₁₁₈	—	2	1	no	—
27	Q₁₆×C₅₉	—	2	1	no	—
28	Q₁₆×C₅₉	—	2	1	no	—
29	Q₁₆×C₅₉	—	2	1	no	—
30	Q₁₆×C₅₉	—	2	1	no	—
31	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	#11745672	—	4	2	no	—
33	#11745671	—	4	2	no	—
34	(C₂×D₅₉)⋊C₄	—	4	2	no	—
35	Dic₅₉⋊C₄	—	4	2	no	—
36	C₄×Dic₅₉	D₄	1	1	yes	characteristic
37	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
38	Dic₂₃₆	—	2	1	no	—
39	Dic₂₃₆	—	2	1	no	—
40	D₄₇₂	—	2	1	no	—
41	D₄₇₂	—	2	1	no	—
42	C₁₆×C₅₉	D₄	1	2	yes	—
43	C₅₉⋊C₁₆	D₄	1	2	yes	—
44	C₂²×D₅₉	—	4	1	no	—
45	Q₈×C₅₉	—	4	1	no	—
46	Q₈×C₅₉	—	4	1	no	—
47	Q₈×C₅₉	—	4	1	no	—
48	Q₈×C₅₉	—	4	1	no	—
49	C₂×C₄×C₅₉	—	4	1	no	—
50	C₂×C₄×C₅₉	—	4	1	no	—
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₅₉	—	4	1	no	—
53	C₂×Dic₅₉	D₈	1	2	yes	—
54	D₂₃₆	—	4	1	no	—
55	D₂₃₆	—	4	1	no	—
56	Dic₁₁₈	—	4	1	no	—
57	Dic₁₁₈	—	4	1	no	—
58	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₅₉⋊C₈	D₈	1	2	yes	—
61	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
62	C₂×D₅₉	—	8	1	no	—
63	C₂×D₅₉	—	8	2	no	—
64	C₄×C₅₉	—	8	1	no	—
65	C₄×C₅₉	—	8	1	no	—
66	C₄×C₅₉	#127	1	1	yes	characteristic
67	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
68	Dic₅₉	—	2	2	no	—
69	Dic₅₉	—	8	1	no	—
70	#2251	—	59	1	no	maximal, Sylow
71	C₂×C₅₉	#493	1	1	yes	characteristic
72	C₂×C₅₉	#510	1	2	yes	—
73	D₅₉	—	16	1	no	—
74	C₂×Q₃₂	—	59	1	no	—
75	C₂×Q₃₂	D₅₉	1	1	yes	characteristic, complemented
76	C₂×SD₃₂	—	59	1	no	—
77	C₂×SD₃₂	—	59	1	no	—
78	#495	—	59	1	no	—
79	#494	—	59	1	no	—
80	#346	—	59	1	no	—
81	C₅₉	#2251	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₁₆	—	59	1	no	—
83	C₂×Q₁₆	—	118	1	no	—
84	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
85	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
86	C₂×SD₁₆	—	118	1	no	—
87	C₂×D₈	—	59	1	no	—
88	C₄⋊Q₈	—	118	1	no	—
89	#124	—	118	1	no	—
90	Q₃₂	—	2	2	no	—
91	Q₃₂	—	118	2	no	—
92	SD₃₂	—	118	2	no	—
93	SD₃₂	—	118	2	no	—
94	C₂×C₁₆	—	59	1	no	—
95	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
96	C₄×C₈	—	59	1	no	—
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	118	1	no	—
100	C₂×D₄	—	118	1	no	—
101	Q₁₆	—	2	1	no	—
102	Q₁₆	—	2	1	no	—
103	Q₁₆	—	2	1	no	—
104	Q₁₆	—	2	1	no	—
105	Q₁₆	—	118	1	no	—
106	Q₁₆	—	118	1	no	—
107	Q₁₆	—	236	2	no	—
108	SD₁₆	—	236	2	no	—
109	D₈	—	118	1	no	—
110	D₈	—	118	1	no	—
111	C₂×C₈	—	59	1	no	—
112	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
113	C₄⋊C₄	—	236	2	no	—
114	C₂²⋊C₄	—	236	2	no	—
115	C₄²	—	59	1	no	—
116	C₁₆	—	59	2	no	—
117	C₁₆	C₂²⋊D₅₉	1	2	yes	—
118	C₂³	—	236	1	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	236	1	no	—
124	Q₈	—	236	1	no	—
125	D₄	—	236	1	no	—
126	D₄	—	236	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	59	2	no	—
130	C₂×C₄	—	236	1	no	—
131	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
132	C₈	—	59	2	no	—
133	C₈	D₄×D₅₉	1	1	yes	characteristic
134	C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
135	C₂²	—	472	1	no	—
136	C₂²	—	472	2	no	—
137	C₂²	D₈×D₅₉	1	1	yes	characteristic
138	C₄	—	8	1	no	—
139	C₄	—	8	1	no	—
140	C₄	—	118	2	no	—
141	C₄	—	472	1	no	—
142	C₄	#14273723	1	1	yes	characteristic
143	C₄	D₈×D₅₉	1	1	yes	characteristic
144	C₂	—	944	1	no	—
145	C₂	#16837777	1	1	yes	characteristic
146	C₂	#16837844	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273673

C₂

yes

maximal, characteristic, complemented

C₂×#14273672

C₂

yes

maximal, characteristic

C₂×#14273670

C₂

yes

maximal, characteristic

#16837787

C₂

yes

maximal, characteristic

#16837780

C₂

yes

maximal, characteristic, complemented

#16837177

C₂

yes

maximal, characteristic, complemented

C₂×Q₃₂×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×#11745672

—

C₂×#11745671

—

C₂×D₄₇₂

C₂²

yes

—

—

—

—

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

Q₃₂×C₅₉

—

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×D₂₃₆

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

#11745672

—

#11745671

—

(C₂×D₅₉)⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

D₄₇₂

—

D₄₇₂

—

C₁₆×C₅₉

D₄

yes

—

C₅₉⋊C₁₆

D₄

yes

—

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

—

D₂₃₆

—

D₂₃₆

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₅₉⋊C₈

D₈

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#127

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

—

C₂×Q₃₂

—

C₂×Q₃₂

D₅₉

yes

characteristic, complemented

C₂×SD₃₂

—

C₂×SD₃₂

—

#495

—

#494

—

#346

—

C₅₉

#2251

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₂×SD₁₆

—

118

—

C₂×D₈

—

C₄⋊Q₈

—

118

—

#124

—

118

—

Q₃₂

—

Q₃₂

—

118

—

SD₃₂

—

118

—

SD₃₂

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

100

C₂×D₄

—

118

—

101

Q₁₆

—

102

Q₁₆

—

103

Q₁₆

—

104

Q₁₆

—

105

Q₁₆

—

118

—

106

Q₁₆

—

118

—

107

Q₁₆

—

236

—

108

SD₁₆

—

236

—

109

D₈

—

118

—

110

D₈

—

118

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₅₉

yes

characteristic

113

C₄⋊C₄

—

236

—

114

C₂²⋊C₄

—

236

—

115

C₄²

—

116

C₁₆

—

117

C₁₆

C₂²⋊D₅₉

yes

—

118

C₂³

—

236

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

236

—

124

Q₈

—

236

—

125

D₄

—

236

—

126

D₄

—

236

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

236

—

131

C₂×C₄

D₄×D₅₉

yes

characteristic

132

C₈

—

133

C₈

D₄×D₅₉

yes

characteristic

134

C₈

C₂×C₂²⋊D₅₉

yes

characteristic

135

C₂²

—

472

—

136

C₂²

—

472

—

137

C₂²

D₈×D₅₉

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

118

—

141

C₄

—

472

—

142

C₄

#14273723

yes

characteristic

143

C₄

D₈×D₅₉

yes

characteristic

144

C₂

—

944

—

145

C₂

#16837777

yes

characteristic

146

C₂

#16837844

yes

—

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₂×Q₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₂×Q₃₂	D₅₉
hypocenter	C₅₉	#2251
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₂×Q₃₂×C₅₉

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₂×Q₃₂

D₅₉

hypocenter

C₅₉

#2251

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
1. C₂×#14273673	C₂	semidirect
5. #16837780	C₂	semidirect
6. #16837177	C₂	semidirect
7. C₂×Q₃₂×C₅₉	C₂	semidirect
75. C₂×Q₃₂	D₅₉	semidirect
81. C₅₉	#2251	semidirect

H ≅ G/N

type

1. C₂×#14273673

C₂

semidirect

5. #16837780

C₂

semidirect

6. #16837177

C₂

semidirect

7. C₂×Q₃₂×C₅₉

C₂

semidirect

75. C₂×Q₃₂

D₅₉

semidirect

81. C₅₉

#2251

semidirect

Subgroups — #21426364

Conjugacy classes of subgroups

Complements