Group #21426210 — subgroups

Subgroups — #21426210

Order 7552.

160 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 49 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×#14273672	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14273666	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837644	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837639	C₂	1	1	yes	maximal, characteristic
5	#16837205	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837202	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#366	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
9	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
11	#14273707	—	2	1	no	—
12	#14273702	—	2	1	no	—
13	#14273672	—	2	2	no	—
14	#14273666	—	2	2	no	—
15	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14273650	—	2	1	no	—
17	#14273647	—	2	1	no	—
18	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
19	#107×C₅₉	C₂²	1	1	yes	characteristic
20	#107×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×D₂₃₆	—	2	1	no	—
22	C₂×D₂₃₆	—	4	1	no	—
23	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
24	C₂×Dic₁₁₈	—	2	1	no	—
25	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₅₉	—	2	1	no	—
27	C₄⋊C₄×C₅₉	—	2	1	no	—
28	(C₂×D₅₉)⋊C₄	—	4	1	no	—
29	(C₂×D₅₉)⋊C₄	—	4	1	no	—
30	C₄⋊Dic₅₉	—	4	1	no	—
31	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
32	Dic₂₃₆	—	2	1	no	—
33	Dic₂₃₆	—	2	1	no	—
34	D₄₇₂	—	2	1	no	—
35	D₄₇₂	—	2	1	no	—
36	C₈×D₅₉	—	2	1	no	—
37	C₈×D₅₉	—	2	1	no	—
38	C₁₆×C₅₉	D₄	1	2	yes	complemented
39	C₅₉⋊C₁₆	—	2	1	no	—
40	C₂²×D₅₉	—	4	1	no	—
41	C₂²×D₅₉	D₈	1	1	yes	characteristic
42	C₂×C₄×C₅₉	—	4	1	no	—
43	C₂×C₄×C₅₉	—	4	1	no	—
44	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₅₉	—	4	1	no	—
46	C₂×Dic₅₉	D₈	1	1	yes	characteristic
47	D₂₃₆	—	4	1	no	—
48	D₂₃₆	—	4	1	no	—
49	D₂₃₆	—	8	2	no	—
50	C₄×D₅₉	—	2	1	no	—
51	C₄×D₅₉	—	2	1	no	—
52	Dic₁₁₈	—	4	1	no	—
53	Dic₁₁₈	—	4	1	no	—
54	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₈×C₅₉	#41	1	1	yes	characteristic
56	C₅₉⋊C₈	—	2	1	no	—
57	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
58	C₂×D₅₉	—	2	1	no	—
59	C₂×D₅₉	—	2	1	no	—
60	C₂×D₅₉	—	2	1	no	—
61	C₂×D₅₉	—	8	1	no	—
62	C₂×D₅₉	—	8	1	no	—
63	C₂×D₅₉	—	8	1	no	—
64	C₄×C₅₉	—	8	1	no	—
65	C₄×C₅₉	—	8	1	no	—
66	C₄×C₅₉	#121	1	1	yes	characteristic
67	C₄×C₅₉	#135	1	1	yes	characteristic
68	Dic₅₉	—	2	1	no	—
69	Dic₅₉	—	8	1	no	—
70	#2220	—	59	1	no	maximal, Sylow
71	C₂×C₅₉	#466	1	1	yes	characteristic
72	C₂×C₅₉	#509	1	1	yes	characteristic
73	C₂×C₅₉	#510	1	1	yes	characteristic
74	D₅₉	—	4	1	no	—
75	D₅₉	—	16	1	no	—
76	C₂×SD₃₂	—	59	1	no	—
77	C₂×M₃₂	—	59	1	no	—
78	#467	—	59	1	no	—
79	#466	—	59	1	no	—
80	#366	D₅₉	1	1	yes	characteristic, complemented
81	#358	—	59	1	no	—
82	#357	—	59	1	no	—
83	C₅₉	#2220	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₁₆	—	59	1	no	—
85	C₂×D₈	—	59	1	no	—
86	C₂²×C₈	—	59	1	no	—
87	C₂²⋊Q₈	—	118	1	no	—
88	#121	—	118	1	no	—
89	SD₃₂	—	118	2	no	—
90	M₃₂	—	118	2	no	—
91	C₂×C₁₆	—	59	1	no	—
92	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
93	#107	C₂×D₅₉	1	1	yes	characteristic
94	#107	C₂×D₅₉	1	1	yes	characteristic
95	#103	—	118	1	no	—
96	#102	—	118	1	no	—
97	C₂×Q₈	—	118	1	no	—
98	C₂×D₄	—	118	1	no	—
99	C₂×D₄	—	236	1	no	—
100	C₂²×C₄	—	59	1	no	—
101	Q₁₆	—	118	1	no	—
102	Q₁₆	—	118	1	no	—
103	D₈	—	118	1	no	—
104	D₈	—	118	1	no	—
105	C₂×C₈	—	59	1	no	—
106	C₂×C₈	—	118	1	no	—
107	C₂×C₈	—	118	1	no	—
108	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	236	1	no	—
112	C₂²⋊C₄	—	236	1	no	—
113	C₂²⋊C₄	—	236	1	no	—
114	C₁₆	—	118	1	no	—
115	C₁₆	D₂₃₆	1	2	yes	complemented
116	C₂³	—	59	1	no	—
117	C₂³	—	236	1	no	—
118	Q₈	—	236	1	no	—
119	Q₈	—	236	1	no	—
120	D₄	—	472	2	no	—
121	D₄	—	236	1	no	—
122	D₄	—	236	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	59	1	no	—
126	C₂×C₄	—	118	1	no	—
127	C₂×C₄	—	118	1	no	—
128	C₂×C₄	—	236	1	no	—
129	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
130	C₈	—	118	1	no	—
131	C₈	C₂×D₂₃₆	1	1	yes	characteristic
132	C₈	#11745689	1	1	yes	characteristic
133	C₂²	—	118	1	no	—
134	C₂²	—	118	1	no	—
135	C₂²	—	118	1	no	—
136	C₂²	—	472	1	no	—
137	C₂²	—	472	1	no	—
138	C₂²	—	472	1	no	—
139	C₂²	D₈×D₅₉	1	1	yes	characteristic
140	C₄	—	8	1	no	—
141	C₄	—	8	1	no	—
142	C₄	—	118	1	no	—
143	C₄	—	472	1	no	—
144	C₄	#14273702	1	1	yes	characteristic
145	C₄	#14273750	1	1	yes	characteristic
146	C₂	—	944	1	no	—
147	C₂	—	236	1	no	—
148	C₂	#16837639	1	1	yes	characteristic
149	C₂	#16837836	1	1	yes	characteristic
150	C₂	#16837844	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273672

C₂

yes

maximal, characteristic, complemented

C₂×#14273666

C₂

yes

maximal, characteristic, complemented

#16837644

C₂

yes

maximal, characteristic, complemented

#16837639

C₂

yes

maximal, characteristic

#16837205

C₂

yes

maximal, characteristic, complemented

#16837202

C₂

yes

maximal, characteristic, complemented

C₅₉×#366

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×D₄₇₂

C₂²

yes

characteristic

C₂×C₈×D₅₉

C₂²

yes

—

—

—

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273650

—

#14273647

—

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

D₄₇₂

—

D₄₇₂

—

C₈×D₅₉

—

C₈×D₅₉

—

C₁₆×C₅₉

D₄

yes

complemented

C₅₉⋊C₁₆

—

C₂²×D₅₉

—

C₂²×D₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

D₅₉

—

C₂×SD₃₂

—

C₂×M₃₂

—

#467

—

#466

—

#366

D₅₉

yes

characteristic, complemented

#358

—

#357

—

C₅₉

#2220

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₂²×C₈

—

C₂²⋊Q₈

—

118

—

#121

—

118

—

SD₃₂

—

118

—

M₃₂

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#107

C₂×D₅₉

yes

characteristic

#107

C₂×D₅₉

yes

characteristic

#103

—

118

—

#102

—

118

—

C₂×Q₈

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

236

—

100

C₂²×C₄

—

101

Q₁₆

—

118

—

102

Q₁₆

—

118

—

103

D₈

—

118

—

104

D₈

—

118

—

105

C₂×C₈

—

106

C₂×C₈

—

118

—

107

C₂×C₈

—

118

—

108

C₂×C₈

C₂²×D₅₉

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

236

—

112

C₂²⋊C₄

—

236

—

113

C₂²⋊C₄

—

236

—

114

C₁₆

—

118

—

115

C₁₆

D₂₃₆

yes

complemented

116

C₂³

—

117

C₂³

—

236

—

118

Q₈

—

236

—

119

Q₈

—

236

—

120

D₄

—

472

—

121

D₄

—

236

—

122

D₄

—

236

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

118

—

127

C₂×C₄

—

118

—

128

C₂×C₄

—

236

—

129

C₂×C₄

D₄×D₅₉

yes

characteristic

130

C₈

—

118

—

131

C₈

C₂×D₂₃₆

yes

characteristic

132

C₈

#11745689

yes

characteristic

133

C₂²

—

118

—

134

C₂²

—

118

—

135

C₂²

—

118

—

136

C₂²

—

472

—

137

C₂²

—

472

—

138

C₂²

—

472

—

139

C₂²

D₈×D₅₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

118

—

143

C₄

—

472

—

144

C₄

#14273702

yes

characteristic

145

C₄

#14273750

yes

characteristic

146

C₂

—

944

—

147

C₂

—

236

—

148

C₂

#16837639

yes

characteristic

149

C₂

#16837836

yes

characteristic

150

C₂

#16837844

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#366	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#366	D₅₉
hypocenter	C₅₉	#2220
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#366

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

N	H ≅ G/N	type
1. C₂×#14273672	C₂	semidirect
2. C₂×#14273666	C₂	semidirect
3. #16837644	C₂	semidirect
5. #16837205	C₂	semidirect
6. #16837202	C₂	semidirect
7. C₅₉×#366	C₂	semidirect
38. C₁₆×C₅₉	D₄	semidirect
80. #366	D₅₉	semidirect
83. C₅₉	#2220	semidirect
115. C₁₆	D₂₃₆	semidirect

H ≅ G/N

type

1. C₂×#14273672

C₂

semidirect

2. C₂×#14273666

C₂

semidirect

3. #16837644

C₂

semidirect

5. #16837205

C₂

semidirect

6. #16837202

C₂

semidirect

7. C₅₉×#366

C₂

semidirect

38. C₁₆×C₅₉

D₄

semidirect

80. #366

D₅₉

semidirect

83. C₅₉

#2220

semidirect

115. C₁₆

D₂₃₆

semidirect

Subgroups — #21426210

Conjugacy classes of subgroups

Complements