Group #21423637 — subgroups

Subgroups — #21423637

Order 7552.

156 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16837749	C₂	1	1	yes	maximal, characteristic
2	#16837698	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837603	C₂	1	1	yes	maximal, characteristic
4	#16837532	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837150	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837145	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#335	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745660	—	2	1	no	—
9	#14273727	C₂²	1	1	yes	characteristic
10	#14273724	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
12	#14273661	—	2	1	no	—
13	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₅₉⋊C₈	—	2	1	no	—
15	#14273652	—	2	1	no	—
16	#14273649	—	2	1	no	—
17	#14273649	—	2	1	no	—
18	#14273649	—	2	1	no	—
19	#14273637	—	2	1	no	—
20	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×D₂₃₆	—	2	1	no	—
24	C₂×D₂₃₆	—	2	1	no	—
25	C₂×D₂₃₆	—	2	1	no	—
26	C₂×Dic₁₁₈	—	2	1	no	—
27	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
28	C₂×C₈×C₅₉	—	2	1	no	—
29	C₂×C₈×C₅₉	—	2	1	no	—
30	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₅₉	—	2	1	no	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
36	Dic₅₉⋊C₄	—	2	1	no	—
37	C₄×Dic₅₉	—	2	1	no	—
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	#11745660	—	4	2	no	—
40	C₂²×D₅₉	—	4	1	no	—
41	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
44	C₂×Dic₅₉	—	2	1	no	—
45	C₂×Dic₅₉	—	2	1	no	—
46	C₂×Dic₅₉	—	4	1	no	—
47	D₂₃₆	—	4	1	no	—
48	D₂₃₆	—	4	1	no	—
49	D₂₃₆	—	4	1	no	—
50	D₂₃₆	—	4	1	no	—
51	D₂₃₆	—	8	1	no	—
52	Dic₁₁₈	—	2	1	no	—
53	Dic₁₁₈	—	2	1	no	—
54	Dic₁₁₈	—	8	1	no	—
55	C₈×C₅₉	—	2	1	no	—
56	C₈×C₅₉	—	2	1	no	—
57	C₈×C₅₉	—	4	1	no	—
58	C₈×C₅₉	—	4	1	no	—
59	C₅₉⋊C₈	—	4	1	no	—
60	C₂²×C₅₉	#121	1	1	yes	characteristic
61	C₂×D₅₉	—	8	1	no	—
62	C₂×D₅₉	—	8	1	no	—
63	C₂×D₅₉	—	8	1	no	—
64	C₄×C₅₉	—	2	1	no	—
65	C₄×C₅₉	#135	1	1	yes	characteristic
66	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₄×C₅₉	#137	1	1	yes	characteristic
68	C₄×C₅₉	#137	1	1	yes	characteristic
69	Dic₅₉	—	4	1	no	—
70	Dic₅₉	—	4	1	no	—
71	Dic₅₉	—	8	1	no	—
72	#1698	—	59	1	no	maximal, Sylow
73	C₂×C₅₉	#464	1	1	yes	characteristic
74	C₂×C₅₉	#468	1	1	yes	characteristic
75	C₂×C₅₉	#472	1	1	yes	characteristic
76	D₅₉	—	16	1	no	—
77	#486	—	59	1	no	—
78	#477	—	59	1	no	—
79	#460	—	59	1	no	—
80	#446	—	59	1	no	—
81	#335	D₅₉	1	1	yes	characteristic, complemented
82	#333	—	59	1	no	—
83	#331	—	59	1	no	—
84	C₅₉	#1698	1	1	yes	characteristic, complemented, Sylow
85	C₂×SD₁₆	—	118	1	no	—
86	C₄⋊Q₈	—	59	1	no	—
87	#127	—	59	1	no	—
88	C₄×Q₈	—	59	1	no	—
89	#107	—	118	1	no	—
90	C₄⋊C₈	—	59	1	no	—
91	C₄⋊C₈	—	118	1	no	—
92	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
93	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
94	#103	—	118	1	no	—
95	#102	—	118	1	no	—
96	#102	—	118	1	no	—
97	#102	—	118	1	no	—
98	#97	—	118	1	no	—
99	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
100	C₂×Q₈	—	59	1	no	—
101	C₂×Q₈	—	118	1	no	—
102	C₂×D₄	—	118	1	no	—
103	C₂×D₄	—	118	1	no	—
104	C₂×D₄	—	118	1	no	—
105	SD₁₆	—	236	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	118	1	no	—
109	C₂×C₈	D₂₃₆	1	1	yes	characteristic
110	C₂×C₈	D₂₃₆	1	1	yes	characteristic
111	C₄⋊C₄	—	59	1	no	—
112	C₄⋊C₄	—	118	1	no	—
113	C₄⋊C₄	—	118	1	no	—
114	C₄⋊C₄	—	118	1	no	—
115	C₄²	—	118	1	no	—
116	C₄²	C₂²×D₅₉	1	1	yes	characteristic
117	C₂³	—	236	1	no	—
118	Q₈	—	118	1	no	—
119	Q₈	—	118	1	no	—
120	Q₈	—	472	1	no	—
121	D₄	—	472	1	no	—
122	D₄	—	236	1	no	—
123	D₄	—	236	1	no	—
124	D₄	—	236	1	no	—
125	D₄	—	236	1	no	—
126	C₂×C₄	—	118	1	no	—
127	C₂×C₄	—	118	1	no	—
128	C₂×C₄	—	236	1	no	—
129	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
130	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
131	C₂×C₄	#11745689	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	4	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	236	1	no	—
137	C₂²	—	472	1	no	—
138	C₂²	—	472	1	no	—
139	C₂²	—	472	1	no	—
140	C₂²	#14273702	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	472	1	no	—
143	C₄	—	236	1	no	—
144	C₄	—	236	1	no	—
145	C₄	#14273749	1	1	yes	characteristic
146	C₄	#14273751	1	1	yes	characteristic
147	C₄	#14273759	1	1	yes	characteristic
148	C₄	#14273760	1	1	yes	characteristic
149	C₂	—	944	1	no	—
150	C₂	#16837625	1	1	yes	characteristic
151	C₂	#16837649	1	1	yes	characteristic
152	C₂	#16837673	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837749

C₂

yes

maximal, characteristic

#16837698

C₂

yes

maximal, characteristic, complemented

#16837603

C₂

yes

maximal, characteristic

#16837532

C₂

yes

maximal, characteristic, complemented

#16837150

C₂

yes

maximal, characteristic, complemented

#16837145

C₂

yes

maximal, characteristic

C₅₉×#335

C₂

yes

maximal, characteristic, complemented

C₂×#11745660

—

#14273727

C₂²

yes

characteristic

#14273724

C₂²

yes

characteristic

C₄×Dic₁₁₈

C₂²

yes

characteristic

#14273661

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

—

—

—

—

—

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

#11745660

—

C₂²×D₅₉

—

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

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

—

—

—

—

D₅₉

yes

characteristic, complemented

#333

—

#331

—

C₅₉

#1698

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₄⋊Q₈

—

#127

—

C₄×Q₈

—

#107

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

118

—

#102

—

118

—

#102

—

118

—

#102

—

118

—

#97

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

100

C₂×Q₈

—

101

C₂×Q₈

—

118

—

102

C₂×D₄

—

118

—

103

C₂×D₄

—

118

—

104

C₂×D₄

—

118

—

105

SD₁₆

—

236

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

118

—

109

C₂×C₈

D₂₃₆

yes

characteristic

110

C₂×C₈

D₂₃₆

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

118

—

113

C₄⋊C₄

—

118

—

114

C₄⋊C₄

—

118

—

115

C₄²

—

118

—

116

C₄²

C₂²×D₅₉

yes

characteristic

117

C₂³

—

236

—

118

Q₈

—

118

—

119

Q₈

—

118

—

120

Q₈

—

472

—

121

D₄

—

472

—

122

D₄

—

236

—

123

D₄

—

236

—

124

D₄

—

236

—

125

D₄

—

236

—

126

C₂×C₄

—

118

—

127

C₂×C₄

—

118

—

128

C₂×C₄

—

236

—

129

C₂×C₄

C₂×D₂₃₆

yes

characteristic

130

C₂×C₄

D₄×D₅₉

yes

characteristic

131

C₂×C₄

#11745689

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

236

—

137

C₂²

—

472

—

138

C₂²

—

472

—

139

C₂²

—

472

—

140

C₂²

#14273702

yes

characteristic

141

C₄

—

142

C₄

—

472

—

143

C₄

—

236

—

144

C₄

—

236

—

145

C₄

#14273749

yes

characteristic

146

C₄

#14273751

yes

characteristic

147

C₄

#14273759

yes

characteristic

148

C₄

#14273760

yes

characteristic

149

C₂

—

944

—

150

C₂

#16837625

yes

characteristic

151

C₂

#16837649

yes

characteristic

152

C₂

#16837673

yes

characteristic

	subgroup	G/H
center	C₂²	#14273702
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#335	C₂
socle	C₂²×C₅₉	#121
hypercenter	#335	D₅₉
hypocenter	C₅₉	#1698
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#14273702

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#335

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
2. #16837698	C₂	semidirect
4. #16837532	C₂	semidirect
5. #16837150	C₂	semidirect
7. C₅₉×#335	C₂	semidirect
81. #335	D₅₉	semidirect
84. C₅₉	#1698	semidirect

H ≅ G/N

type

2. #16837698

C₂

semidirect

4. #16837532

C₂

semidirect

5. #16837150

C₂

semidirect

7. C₅₉×#335

C₂

semidirect

81. #335

D₅₉

semidirect

84. C₅₉

#1698

semidirect

Subgroups — #21423637

Conjugacy classes of subgroups

Complements