Group #21425009 — subgroups

Subgroups — #21425009

Order 7552.

276 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂²×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
2	#16837412	C₂	1	2	yes	maximal
3	C₂×#14273645	C₂	1	2	yes	maximal
4	#16837357	C₂	1	1	yes	maximal, characteristic
5	C₂×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₁₈	C₂²	1	2	yes	—
7	C₂×Dic₂₃₆	—	2	1	no	—
8	C₂×Dic₂₃₆	—	2	1	no	—
9	C₂×Dic₂₃₆	—	2	2	no	—
10	C₂×Dic₂₃₆	C₄	1	4	yes	complemented
11	C₂×#11745664	C₂²	1	1	yes	characteristic
12	#14273686	C₂²	1	1	yes	characteristic
13	#14273652	—	2	4	no	—
14	#14273645	—	2	4	no	—
15	#14273637	—	2	1	no	—
16	C₈×Dic₅₉	—	2	1	no	—
17	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
18	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	#108×C₅₉	—	2	2	no	—
20	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
21	C₂²×Dic₅₉	—	2	2	no	—
22	C₂×Dic₁₁₈	—	2	2	no	—
23	C₂×Dic₁₁₈	—	2	2	no	—
24	C₂×Dic₁₁₈	—	2	2	no	—
25	C₂×Dic₁₁₈	—	2	2	no	—
26	C₂×Dic₁₁₈	—	2	2	no	—
27	C₂×Dic₁₁₈	C₂×C₄	1	4	yes	—
28	M₁₆×C₅₉	—	2	2	no	—
29	M₁₆×C₅₉	—	2	2	no	—
30	C₂×C₈×C₅₉	—	2	2	no	—
31	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
34	C₂×C₈×C₅₉	D₄	1	2	yes	—
35	C₂²⋊Dic₅₉	—	2	1	no	—
36	C₄⋊Dic₅₉	D₄	1	2	yes	—
37	C₄×Dic₅₉	—	2	1	no	—
38	#11745664	—	2	1	no	—
39	#11745664	D₄	1	2	yes	—
40	C₂×C₅₉⋊C₈	—	2	1	no	—
41	Dic₂₃₆	—	2	4	no	—
42	Dic₂₃₆	—	2	4	no	—
43	C₂³×C₅₉	#41	1	1	yes	characteristic
44	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
50	C₂×Dic₅₉	—	2	4	no	—
51	C₂×Dic₅₉	—	2	2	no	—
52	C₂×Dic₅₉	—	4	2	no	—
53	C₂×Dic₅₉	—	4	2	no	—
54	Dic₁₁₈	—	2	4	no	—
55	Dic₁₁₈	—	2	4	no	—
56	Dic₁₁₈	—	4	2	no	—
57	Dic₁₁₈	—	4	2	no	—
58	C₈×C₅₉	—	4	2	no	—
59	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
60	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
61	C₅₉⋊C₈	—	2	2	no	—
62	C₂²×C₅₉	—	2	2	no	—
63	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
64	C₂²×C₅₉	#121	1	1	yes	characteristic
65	C₂²×C₅₉	#124	1	1	yes	characteristic
66	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
68	C₄×C₅₉	#121	1	1	yes	characteristic
69	C₄×C₅₉	#127	1	1	yes	characteristic
70	Dic₅₉	—	4	4	no	—
71	Dic₅₉	—	4	2	no	—
72	#1979	—	59	1	no	maximal, Sylow
73	C₂×C₅₉	—	2	1	no	—
74	C₂×C₅₉	—	2	1	no	—
75	C₂×C₅₉	#390	1	1	yes	characteristic
76	C₂×C₅₉	#473	1	2	yes	—
77	C₂²×Q₁₆	—	59	1	no	—
78	C₂×#108	D₅₉	1	1	yes	characteristic
79	#419	—	59	2	no	—
80	C₂×#101	—	59	2	no	—
81	#405	—	59	1	no	—
82	C₅₉	#1979	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	59	2	no	—
84	C₂×Q₁₆	—	59	4	no	—
85	C₂×Q₁₆	—	118	1	no	—
86	C₂×Q₁₆	—	118	1	no	—
87	C₂×Q₁₆	—	118	2	no	—
88	C₂×M₁₆	—	59	1	no	—
89	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
90	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
91	#117	—	59	1	no	—
92	#108	—	2	2	no	—
93	#103	—	118	4	no	—
94	#101	—	118	4	no	—
95	#97	—	118	1	no	—
96	C₄×C₈	—	118	1	no	—
97	C₂×Q₈	—	59	4	no	—
98	C₂×Q₈	—	118	2	no	—
99	C₂×Q₈	—	118	2	no	—
100	C₂×Q₈	—	118	2	no	—
101	C₂×Q₈	—	118	2	no	—
102	C₂×Q₈	—	118	2	no	—
103	C₂²×C₄	—	118	2	no	—
104	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
105	Q₁₆	—	118	4	no	—
106	Q₁₆	—	118	4	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	59	2	no	—
110	M₁₆	—	118	1	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	118	1	no	—
113	C₂×C₈	C₄×D₅₉	1	2	yes	—
114	C₂×C₈	D₂₃₆	1	1	yes	characteristic
115	C₂×C₈	D₂₃₆	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
117	C₄⋊C₄	—	59	2	no	—
118	C₂²⋊C₄	—	118	1	no	—
119	C₄²	—	118	1	no	—
120	C₂³	#11745689	1	1	yes	characteristic
121	Q₈	—	118	4	no	—
122	Q₈	—	118	4	no	—
123	Q₈	—	236	2	no	—
124	Q₈	—	236	2	no	—
125	C₂×C₄	—	118	4	no	—
126	C₂×C₄	—	118	2	no	—
127	C₂×C₄	—	236	2	no	—
128	C₂×C₄	—	236	2	no	—
129	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
130	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
131	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
132	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
133	C₂×C₄	#11745689	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
135	C₈	—	4	2	no	—
136	C₈	—	118	2	no	—
137	C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
138	C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	#14273689	1	1	yes	characteristic
141	C₂²	#14273702	1	1	yes	characteristic
142	C₂²	#14273716	1	1	yes	characteristic
143	C₄	—	236	2	no	—
144	C₄	—	236	4	no	—
145	C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
146	C₄	#14273689	1	1	yes	characteristic
147	C₄	#14273702	1	1	yes	characteristic
148	C₄	#14273723	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#16837303	1	1	yes	characteristic
152	C₂	#16837678	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₃₆

C₂

yes

maximal, characteristic

#16837412

C₂

yes

maximal

C₂×#14273645

C₂

yes

maximal

#16837357

C₂

yes

maximal, characteristic

C₂×#108×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

C₄

yes

complemented

C₂×#11745664

C₂²

yes

characteristic

#14273686

C₂²

yes

—

—

—

C₈×Dic₅₉

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#108×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

—

C₄×Dic₅₉

—

#11745664

—

#11745664

D₄

yes

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

Dic₂₃₆

—

C₂³×C₅₉

#41

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄×D₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#1979

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#390

yes

characteristic

C₂×C₅₉

#473

yes

—

C₂²×Q₁₆

—

C₂×#108

D₅₉

yes

—

—

—

C₅₉

#1979

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

C₂×D₅₉

yes

—

—

—

118

—

#101

—

118

—

#97

—

118

—

C₄×C₈

—

118

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

100

C₂×Q₈

—

118

—

101

C₂×Q₈

—

118

—

102

C₂×Q₈

—

118

—

103

C₂²×C₄

—

118

—

104

C₂²×C₄

C₂²×D₅₉

yes

characteristic

105

Q₁₆

—

118

—

106

Q₁₆

—

118

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

118

—

111

C₂×C₈

—

112

C₂×C₈

—

118

—

113

C₂×C₈

C₄×D₅₉

yes

—

114

C₂×C₈

D₂₃₆

yes

characteristic

115

C₂×C₈

D₂₃₆

yes

characteristic

116

C₂×C₈

C₂²⋊D₅₉

yes

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

118

—

119

C₄²

—

118

—

120

C₂³

#11745689

yes

characteristic

121

Q₈

—

118

—

122

Q₈

—

118

—

123

Q₈

—

236

—

124

Q₈

—

236

—

125

C₂×C₄

—

118

—

126

C₂×C₄

—

118

—

127

C₂×C₄

—

236

—

128

C₂×C₄

—

236

—

129

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

130

C₂×C₄

C₂×D₂₃₆

yes

characteristic

131

C₂×C₄

D₄×D₅₉

yes

characteristic

132

C₂×C₄

D₄×D₅₉

yes

characteristic

133

C₂×C₄

#11745689

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

135

C₈

—

136

C₈

—

118

—

137

C₈

(C₂×D₅₉)⋊C₄

yes

—

138

C₈

(C₂×D₅₉)⋊C₄

yes

—

139

C₂²

—

140

C₂²

#14273689

yes

characteristic

141

C₂²

#14273702

yes

characteristic

142

C₂²

#14273716

yes

characteristic

143

C₄

—

236

—

144

C₄

—

236

—

145

C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

146

C₄

#14273689

yes

characteristic

147

C₄

#14273702

yes

characteristic

148

C₄

#14273723

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#16837303

yes

characteristic

152

C₂

#16837678

yes

—

	subgroup	G/H
center	C₂²	#14273702
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#108×C₅₉	C₂
socle	C₂²×C₅₉	#121
hypercenter	C₂×#108	D₅₉
hypocenter	C₅₉	#1979
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#14273702

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#108×C₅₉

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
10. C₂×Dic₂₃₆	C₄	semidirect
82. C₅₉	#1979	semidirect

H ≅ G/N

type

10. C₂×Dic₂₃₆

C₄

semidirect

82. C₅₉

#1979

semidirect

Subgroups — #21425009

Conjugacy classes of subgroups

Complements