Group #21422552 — subgroups

Subgroups — #21422552

Order 7552.

232 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 93 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	#16837959	C₂	1	1	yes	maximal, characteristic
2	#16837426	C₂	1	1	yes	maximal, characteristic
3	#16837134	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#11745664	—	2	1	no	—
6	#14273727	C₄	1	1	yes	characteristic
7	#14273727	C₄	1	1	yes	characteristic
8	#14273719	—	2	1	no	—
9	#14273706	—	2	1	no	—
10	#14273706	C₄	1	2	yes	—
11	C₄×Dic₁₁₈	—	2	1	no	—
12	C₄×Dic₁₁₈	C₂²	1	2	yes	—
13	#14273686	—	2	1	no	—
14	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
15	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
16	C₂²×C₈×C₅₉	—	2	1	no	—
17	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₅₉	—	2	1	no	—
19	C₄×C₈×C₅₉	C₂²	1	2	yes	—
20	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
23	C₂×Dic₁₁₈	—	2	1	no	—
24	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₅₉	—	2	4	no	—
26	C₂×C₈×C₅₉	—	2	2	no	—
27	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₅₉	D₄	1	1	yes	characteristic
29	C₄²×C₅₉	D₄	1	1	yes	characteristic
30	C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₂²⋊Dic₅₉	—	2	2	no	—
32	C₄⋊Dic₅₉	—	2	1	no	—
33	C₄⋊Dic₅₉	—	2	1	no	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
36	Dic₅₉⋊C₄	—	2	2	no	—
37	Dic₅₉⋊C₄	—	2	2	no	—
38	C₄×Dic₅₉	—	2	2	no	—
39	#11745664	—	4	2	no	—
40	C₂×C₅₉⋊C₈	—	2	2	no	—
41	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	—	2	2	no	—
43	C₂×C₄×C₅₉	—	2	2	no	—
44	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
48	C₂×C₄×C₅₉	Q₁₆	1	2	yes	—
49	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₅₉	—	2	2	no	—
53	C₂×Dic₅₉	—	2	2	no	—
54	Dic₁₁₈	—	2	4	no	—
55	Dic₁₁₈	—	4	2	no	—
56	C₈×C₅₉	—	2	4	no	—
57	C₅₉⋊C₈	—	4	2	no	—
58	C₂²×C₅₉	—	2	1	no	—
59	C₂²×C₅₉	—	2	1	no	—
60	C₂²×C₅₉	#103	1	2	yes	—
61	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₅₉	—	2	1	no	—
63	C₄×C₅₉	—	2	1	no	—
64	C₄×C₅₉	#103	1	2	yes	—
65	C₄×C₅₉	#104	1	2	yes	—
66	C₄×C₅₉	#131	1	2	yes	—
67	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
69	Dic₅₉	—	4	2	no	—
70	Dic₅₉	—	4	2	no	—
71	#1493	—	59	1	no	maximal, Sylow
72	C₂×C₅₉	—	2	2	no	—
73	C₂×C₅₉	#408	1	1	yes	characteristic
74	C₂×C₅₉	C₂×#103	1	1	yes	characteristic
75	C₂×C₅₉	C₂×#104	1	1	yes	characteristic
76	#533	—	59	1	no	—
77	#423	—	59	1	no	—
78	C₂×C₄×C₈	D₅₉	1	1	yes	characteristic
79	Q₈⋊C₈	—	59	4	no	—
80	C₅₉	#1493	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₁₆	—	118	1	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₄⋊Q₈	—	59	1	no	—
84	C₄⋊Q₈	—	59	1	no	—
85	#125	—	118	1	no	—
86	C₂²⋊Q₈	—	59	2	no	—
87	C₂²⋊Q₈	—	118	1	no	—
88	C₄×Q₈	—	59	2	no	—
89	C₄×Q₈	—	118	1	no	—
90	#117	—	118	1	no	—
91	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
92	C₄⋊C₈	—	59	2	no	—
93	C₄⋊C₈	—	118	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	C₂×D₅₉	1	2	yes	—
96	C₂×Q₈	—	59	2	no	—
97	C₂×Q₈	—	118	1	no	—
98	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
99	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
100	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
101	M₁₆	—	236	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	118	2	no	—
105	C₄⋊C₄	—	59	2	no	—
106	C₄⋊C₄	—	118	1	no	—
107	C₄⋊C₄	—	118	1	no	—
108	C₄⋊C₄	—	118	2	no	—
109	C₄⋊C₄	—	118	1	no	—
110	C₄⋊C₄	—	118	2	no	—
111	C₂²⋊C₄	—	118	2	no	—
112	C₄²	—	118	2	no	—
113	C₄²	C₄×D₅₉	1	1	yes	characteristic
114	C₄²	D₂₃₆	1	1	yes	characteristic
115	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
116	C₄²	C₂²×D₅₉	1	1	yes	characteristic
117	C₂³	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
118	Q₈	—	118	4	no	—
119	Q₈	—	236	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	118	2	no	—
123	C₂×C₄	—	118	2	no	—
124	C₂×C₄	#11745660	1	2	yes	—
125	C₂×C₄	Dic₂₃₆	1	2	yes	—
126	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
127	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
128	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
130	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
132	C₈	—	2	4	no	—
133	C₈	—	236	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	#14273652	1	2	yes	—
137	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	236	2	no	—
141	C₄	—	236	2	no	—
142	C₄	#14273652	1	2	yes	—
143	C₄	#14273655	1	2	yes	—
144	C₄	#14273733	1	2	yes	—
145	C₄	C₂×#11745660	1	1	yes	characteristic
146	C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
147	C₂	—	2	2	no	—
148	C₂	#16837368	1	1	yes	characteristic
149	C₂	C₂×#14273652	1	1	yes	characteristic
150	C₂	C₂×#14273655	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal, characteristic

#16837426

C₂

yes

maximal, characteristic

#16837134

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

—

#14273727

C₄

yes

characteristic

#14273727

C₄

yes

—

—

C₄

yes

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₂²

yes

—

#14273686

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²×C₈×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

Q₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#103

yes

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#103

yes

—

C₄×C₅₉

#104

yes

—

C₄×C₅₉

#131

yes

—

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#1493

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

C₂×C₄×C₈

D₅₉

yes

characteristic

Q₈⋊C₈

—

C₅₉

#1493

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

118

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

C₄×Q₈

—

118

—

#117

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

D₂₃₆

yes

characteristic

100

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

101

M₁₆

—

236

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

118

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

118

—

107

C₄⋊C₄

—

118

—

108

C₄⋊C₄

—

118

—

109

C₄⋊C₄

—

118

—

110

C₄⋊C₄

—

118

—

111

C₂²⋊C₄

—

118

—

112

C₄²

—

118

—

113

C₄²

C₄×D₅₉

yes

characteristic

114

C₄²

D₂₃₆

yes

characteristic

115

C₄²

C₂²⋊D₅₉

yes

characteristic

116

C₄²

C₂²×D₅₉

yes

characteristic

117

C₂³

(C₂×D₅₉)⋊C₄

yes

characteristic

118

Q₈

—

118

—

119

Q₈

—

236

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

118

—

123

C₂×C₄

—

118

—

124

C₂×C₄

#11745660

yes

—

125

C₂×C₄

Dic₂₃₆

yes

—

126

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

127

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

128

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

129

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

130

C₂×C₄

C₂×D₂₃₆

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

132

C₈

—

133

C₈

—

236

—

134

C₂²

—

135

C₂²

—

136

C₂²

#14273652

yes

—

137

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

236

—

141

C₄

—

236

—

142

C₄

#14273652

yes

—

143

C₄

#14273655

yes

—

144

C₄

#14273733

yes

—

145

C₄

C₂×#11745660

yes

characteristic

146

C₄

C₂×Dic₂₃₆

yes

characteristic

147

C₂

—

148

C₂

#16837368

yes

characteristic

149

C₂

C₂×#14273652

yes

characteristic

150

C₂

C₂×#14273655

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₂×C₄×C₈×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	C₂×C₄×C₈	D₅₉
hypocenter	C₅₉	#1493
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₂×C₄×C₈×C₅₉

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

hypercenter

C₂×C₄×C₈

D₅₉

hypocenter

C₅₉

#1493

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #16837134	C₂	semidirect
80. C₅₉	#1493	semidirect

H ≅ G/N

type

3. #16837134

C₂

semidirect

80. C₅₉

#1493

semidirect

Subgroups — #21422552

Conjugacy classes of subgroups

Complements