Group #21424957 — subgroups

Subgroups — #21424957

Order 7552.

312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×#14273661	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273652	C₂	1	2	yes	maximal
4	#16837308	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₁₈	C₂²	1	2	yes	—
7	C₂×Dic₂₃₆	—	2	2	no	—
8	C₂×Dic₂₃₆	—	2	2	no	—
9	C₂×Dic₂₃₆	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
11	#14273661	—	2	2	no	—
12	#14273652	—	2	4	no	—
13	C₅₉⋊#95	—	2	4	no	—
14	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₅₉	—	2	2	no	—
18	C₂²×C₄×C₅₉	D₄	1	2	yes	—
19	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₉	—	2	2	no	—
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	C₂×C₈×C₅₉	—	2	1	no	—
29	C₂×C₈×C₅₉	—	2	1	no	—
30	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₅₉	D₄	1	2	yes	—
32	C₂×C₈×C₅₉	D₄	1	2	yes	—
33	C₂×C₈×C₅₉	D₄	1	2	yes	—
34	C₄²×C₅₉	—	2	1	no	—
35	C₄²×C₅₉	—	2	1	no	—
36	C₄⋊Dic₅₉	—	2	2	no	—
37	C₄⋊Dic₅₉	—	2	2	no	—
38	Dic₂₃₆	—	2	8	no	—
39	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₉	—	2	2	no	—
41	C₂×C₄×C₅₉	—	2	2	no	—
42	C₂×C₄×C₅₉	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₉	C₂×D₄	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	#41	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
49	C₂×Dic₅₉	—	2	4	no	—
50	C₂×Dic₅₉	—	4	2	no	—
51	C₂×Dic₅₉	—	4	2	no	—
52	C₂×Dic₅₉	—	4	2	no	—
53	C₂×Dic₅₉	—	4	4	no	—
54	Dic₁₁₈	—	2	4	no	—
55	Dic₁₁₈	—	2	4	no	—
56	Dic₁₁₈	—	4	2	no	—
57	Dic₁₁₈	—	4	2	no	—
58	C₈×C₅₉	—	2	2	no	—
59	C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₅₉	#121	1	1	yes	characteristic
62	C₂²×C₅₉	#127	1	1	yes	characteristic
63	C₂²×C₅₉	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₅₉	#135	1	2	yes	—
65	C₄×C₅₉	—	2	4	no	—
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	#124	1	1	yes	characteristic
70	Dic₅₉	—	4	4	no	—
71	Dic₅₉	—	8	2	no	—
72	#1969	—	59	1	no	maximal, Sylow
73	C₂×C₅₉	#390	1	1	yes	characteristic
74	C₂×C₅₉	C₄×Q₁₆	1	2	yes	—
75	C₂×C₅₉	#467	1	2	yes	—
76	C₂×C₅₉	#494	1	1	yes	characteristic
77	C₂×C₅₉	#495	1	1	yes	characteristic
78	C₂²×Q₁₆	—	59	1	no	—
79	C₂×#107	—	59	1	no	—
80	C₂×#103	—	59	2	no	—
81	C₂×C₄×C₈	D₅₉	1	1	yes	characteristic
82	#391	—	59	2	no	—
83	C₅₉	#1969	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₈	—	59	2	no	—
85	C₂×Q₁₆	—	59	4	no	—
86	C₂×Q₁₆	—	118	2	no	—
87	C₂×Q₁₆	—	118	2	no	—
88	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	59	2	no	—
91	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
92	#107	—	118	2	no	—
93	#103	—	118	4	no	—
94	C₄×C₈	—	2	2	no	—
95	#95	—	118	4	no	—
96	C₂×Q₈	—	59	4	no	—
97	C₂×Q₈	—	118	2	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	C₂²×C₄	—	118	2	no	—
103	C₂²×C₄	—	118	2	no	—
104	C₂²×C₄	D₂₃₆	1	2	yes	—
105	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
106	Q₁₆	—	118	8	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₄×D₅₉	1	2	yes	—
110	C₂×C₈	D₂₃₆	1	2	yes	—
111	C₂×C₈	D₂₃₆	1	2	yes	—
112	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
113	C₄⋊C₄	—	118	2	no	—
114	C₄⋊C₄	—	118	2	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
118	Q₈	—	118	4	no	—
119	Q₈	—	118	4	no	—
120	Q₈	—	236	2	no	—
121	Q₈	—	236	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	118	4	no	—
125	C₂×C₄	—	236	2	no	—
126	C₂×C₄	—	236	4	no	—
127	C₂×C₄	—	236	2	no	—
128	C₂×C₄	—	236	2	no	—
129	C₂×C₄	Dic₂₃₆	1	4	yes	complemented
130	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
131	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
132	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
133	C₂×C₄	#11745685	1	1	yes	characteristic
134	C₂×C₄	#11745685	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	(C₂×D₅₉)⋊C₄	1	4	yes	—
138	C₂²	C₄×D₂₃₆	1	1	yes	characteristic
139	C₂²	#14273701	1	1	yes	characteristic
140	C₂²	#14273724	1	1	yes	characteristic
141	C₂²	C₂×Dic₂₃₆	1	2	yes	—
142	C₂²	#14273749	1	2	yes	—
143	C₄	—	2	4	no	—
144	C₄	—	472	2	no	—
145	C₄	—	236	4	no	—
146	C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
147	C₄	C₄×D₂₃₆	1	1	yes	characteristic
148	C₄	#14273701	1	1	yes	characteristic
149	C₄	#14273715	1	1	yes	characteristic
150	C₂	#16837304	1	1	yes	characteristic
151	C₂	C₄×Dic₂₃₆	1	2	yes	—
152	C₂	#16837643	1	2	yes	—
153	C₂	#16837782	1	1	yes	characteristic
154	C₂	#16837786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₃₆

C₂

yes

maximal, characteristic

C₂×#14273661

C₂

yes

maximal, characteristic

C₂×#14273652

C₂

yes

maximal

#16837308

C₂

yes

maximal

C₂×C₄×C₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

C₄

yes

complemented

C₂×C₄⋊Dic₅₉

C₂²

yes

—

#14273661

—

#14273652

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₂₃₆

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

Q₁₆

yes

complemented

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

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₅₉

C₄×D₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₂×Q₁₆

yes

—

C₂²×C₅₉

#135

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₄×Q₁₆

yes

—

C₂×C₅₉

#467

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂²×Q₁₆

—

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

D₅₉

yes

characteristic

#391

—

C₅₉

#1969

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#107

—

118

—

#103

—

118

—

C₄×C₈

—

#95

—

118

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

100

C₂×Q₈

—

118

—

101

C₂×Q₈

—

118

—

102

C₂²×C₄

—

118

—

103

C₂²×C₄

—

118

—

104

C₂²×C₄

D₂₃₆

yes

—

105

C₂²×C₄

C₂²×D₅₉

yes

characteristic

106

Q₁₆

—

118

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₄×D₅₉

yes

—

110

C₂×C₈

D₂₃₆

yes

—

111

C₂×C₈

D₂₃₆

yes

—

112

C₂×C₈

C₂²⋊D₅₉

yes

—

113

C₄⋊C₄

—

118

—

114

C₄⋊C₄

—

118

—

115

C₄²

—

116

C₄²

—

117

C₂³

C₂×D₂₃₆

yes

characteristic

118

Q₈

—

118

—

119

Q₈

—

118

—

120

Q₈

—

236

—

121

Q₈

—

236

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

118

—

125

C₂×C₄

—

236

—

126

C₂×C₄

—

236

—

127

C₂×C₄

—

236

—

128

C₂×C₄

—

236

—

129

C₂×C₄

Dic₂₃₆

yes

complemented

130

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

131

C₂×C₄

C₂×D₂₃₆

yes

characteristic

132

C₂×C₄

C₂×D₂₃₆

yes

characteristic

133

C₂×C₄

#11745685

yes

characteristic

134

C₂×C₄

#11745685

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

136

C₈

—

137

C₈

(C₂×D₅₉)⋊C₄

yes

—

138

C₂²

C₄×D₂₃₆

yes

characteristic

139

C₂²

#14273701

yes

characteristic

140

C₂²

#14273724

yes

characteristic

141

C₂²

C₂×Dic₂₃₆

yes

—

142

C₂²

#14273749

yes

—

143

C₄

—

144

C₄

—

472

—

145

C₄

—

236

—

146

C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

147

C₄

C₄×D₂₃₆

yes

characteristic

148

C₄

#14273701

yes

characteristic

149

C₄

#14273715

yes

characteristic

150

C₂

#16837304

yes

characteristic

151

C₂

C₄×Dic₂₃₆

yes

—

152

C₂

#16837643

yes

—

153

C₂

#16837782

yes

characteristic

154

C₂

#16837786

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×C₄×C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×C₄×C₈	D₅₉
hypocenter	C₅₉	#1969
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×C₄×C₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×C₄×C₈

D₅₉

hypocenter

C₅₉

#1969

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. C₂×Dic₂₃₆	C₄	semidirect
42. C₂×C₄×C₅₉	Q₁₆	semidirect
83. C₅₉	#1969	semidirect
129. C₂×C₄	Dic₂₃₆	semidirect

H ≅ G/N

type

9. C₂×Dic₂₃₆

C₄

semidirect

42. C₂×C₄×C₅₉

Q₁₆

semidirect

83. C₅₉

#1969

semidirect

129. C₂×C₄

Dic₂₃₆

semidirect

Subgroups — #21424957

Conjugacy classes of subgroups

Complements