Group #21423571 — subgroups

Subgroups — #21423571

Order 7552.

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

normal subgroups only

This group has 168 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	#16837776	C₂	1	1	yes	maximal, characteristic
2	#16837684	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837603	C₂	1	1	yes	maximal, characteristic
4	#16837529	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837154	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837145	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745660	—	2	1	no	—
9	C₂×#11745671	—	2	1	no	—
10	C₂×#11745671	—	2	1	no	—
11	#14273724	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
13	#14273658	—	2	1	no	—
14	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₅₉⋊C₈	—	2	1	no	—
16	#14273650	—	2	1	no	—
17	#14273649	—	2	1	no	—
18	#14273637	—	2	1	no	—
19	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×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×Q₈×C₅₉	—	2	1	no	—
24	C₂×D₂₃₆	—	2	1	no	—
25	C₂×D₂₃₆	—	2	1	no	—
26	C₂×D₂₃₆	—	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	C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	#11745671	—	4	2	no	—
34	#11745671	—	4	2	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	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
41	#11745660	—	4	2	no	—
42	C₂²×D₅₉	—	4	1	no	—
43	Q₈×C₅₉	—	4	1	no	—
44	Q₈×C₅₉	—	4	1	no	—
45	C₂×C₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
49	C₂×Dic₅₉	—	2	1	no	—
50	C₂×Dic₅₉	—	2	1	no	—
51	D₂₃₆	—	4	1	no	—
52	D₂₃₆	—	4	1	no	—
53	D₂₃₆	—	4	1	no	—
54	D₂₃₆	—	4	1	no	—
55	D₂₃₆	—	8	1	no	—
56	Dic₁₁₈	—	2	1	no	—
57	Dic₁₁₈	—	2	1	no	—
58	C₈×C₅₉	—	4	1	no	—
59	C₈×C₅₉	—	4	1	no	—
60	C₅₉⋊C₈	—	2	1	no	—
61	C₅₉⋊C₈	—	4	1	no	—
62	C₅₉⋊C₈	SD₁₆	1	2	yes	complemented
63	C₂²×C₅₉	#121	1	1	yes	characteristic
64	C₂×D₅₉	—	8	1	no	—
65	C₂×D₅₉	—	8	1	no	—
66	C₂×D₅₉	—	8	1	no	—
67	C₄×C₅₉	—	2	1	no	—
68	C₄×C₅₉	—	8	1	no	—
69	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₅₉	#137	1	1	yes	characteristic
73	Dic₅₉	—	4	1	no	—
74	Dic₅₉	—	4	1	no	—
75	#1686	—	59	1	no	maximal, Sylow
76	C₂×C₅₉	#460	1	1	yes	characteristic
77	C₂×C₅₉	#468	1	1	yes	characteristic
78	C₂×C₅₉	#471	1	1	yes	characteristic
79	D₅₉	—	16	1	no	—
80	#492	—	59	1	no	—
81	#475	—	59	1	no	—
82	#460	—	59	1	no	—
83	#446	—	59	1	no	—
84	#335	—	59	1	no	—
85	#332	D₅₉	1	1	yes	characteristic, complemented
86	#331	—	59	1	no	—
87	C₅₉	#1686	1	1	yes	characteristic, complemented, Sylow
88	C₂×SD₁₆	—	118	1	no	—
89	C₂×SD₁₆	—	118	1	no	—
90	C₂×SD₁₆	—	118	1	no	—
91	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
92	#127	—	59	1	no	—
93	C₄×Q₈	—	59	1	no	—
94	#106	—	118	1	no	—
95	C₄⋊C₈	—	59	1	no	—
96	C₄⋊C₈	—	118	1	no	—
97	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
98	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
99	#103	—	118	1	no	—
100	#102	—	118	1	no	—
101	#97	—	118	1	no	—
102	C₄×C₈	—	59	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	59	1	no	—
105	C₂×D₄	—	118	1	no	—
106	C₂×D₄	—	118	1	no	—
107	C₂×D₄	—	118	1	no	—
108	SD₁₆	—	236	2	no	—
109	SD₁₆	—	236	2	no	—
110	SD₁₆	—	236	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	59	1	no	—
114	C₂×C₈	—	59	1	no	—
115	C₂×C₈	—	118	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	59	1	no	—
119	C₄⋊C₄	—	118	1	no	—
120	C₄²	—	118	1	no	—
121	C₄²	C₂²×D₅₉	1	1	yes	characteristic
122	C₂³	—	236	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	118	1	no	—
126	Q₈	—	118	1	no	—
127	D₄	—	472	1	no	—
128	D₄	—	236	1	no	—
129	D₄	—	236	1	no	—
130	D₄	—	236	1	no	—
131	D₄	—	236	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	118	1	no	—
134	C₂×C₄	—	118	1	no	—
135	C₂×C₄	#11745685	1	1	yes	characteristic
136	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
137	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
138	C₈	—	4	1	no	—
139	C₈	—	4	1	no	—
140	C₈	—	59	2	no	—
141	C₈	—	118	1	no	—
142	C₈	—	236	1	no	—
143	C₂²	—	472	1	no	—
144	C₂²	—	472	1	no	—
145	C₂²	—	472	1	no	—
146	C₂²	#14273700	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	8	1	no	—
149	C₄	—	236	1	no	—
150	C₄	—	236	1	no	—
151	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
152	C₄	#14273751	1	1	yes	characteristic
153	C₄	#14273754	1	1	yes	characteristic
154	C₄	#14273759	1	1	yes	characteristic
155	C₂	—	944	1	no	—
156	C₂	#16837601	1	1	yes	characteristic
157	C₂	#16837651	1	1	yes	characteristic
158	C₂	#16837668	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837776

C₂

yes

maximal, characteristic

#16837684

C₂

yes

maximal, characteristic, complemented

#16837603

C₂

yes

maximal, characteristic

#16837529

C₂

yes

maximal, characteristic, complemented

#16837154

C₂

yes

maximal, characteristic, complemented

#16837145

C₂

yes

maximal, characteristic

C₅₉×#332

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

C₄×Dic₁₁₈

C₂²

yes

characteristic

#14273658

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

#14273650

—

#14273649

—

#14273637

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745671

—

#11745671

—

C₄⋊Dic₅₉

D₄

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

#11745660

—

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

SD₁₆

yes

complemented

C₂²×C₅₉

#121

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

—

—

—

—

—

D₅₉

yes

characteristic, complemented

#331

—

C₅₉

#1686

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂×SD₁₆

—

118

—

C₂×SD₁₆

—

118

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

#127

—

C₄×Q₈

—

#106

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

118

—

100

#102

—

118

—

101

#97

—

118

—

102

C₄×C₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×D₄

—

118

—

106

C₂×D₄

—

118

—

107

C₂×D₄

—

118

—

108

SD₁₆

—

236

—

109

SD₁₆

—

236

—

110

SD₁₆

—

236

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

118

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

118

—

120

C₄²

—

118

—

121

C₄²

C₂²×D₅₉

yes

characteristic

122

C₂³

—

236

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

118

—

126

Q₈

—

118

—

127

D₄

—

472

—

128

D₄

—

236

—

129

D₄

—

236

—

130

D₄

—

236

—

131

D₄

—

236

—

132

C₂×C₄

—

133

C₂×C₄

—

118

—

134

C₂×C₄

—

118

—

135

C₂×C₄

#11745685

yes

characteristic

136

C₂×C₄

D₄×D₅₉

yes

characteristic

137

C₂×C₄

D₄×D₅₉

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

118

—

142

C₈

—

236

—

143

C₂²

—

472

—

144

C₂²

—

472

—

145

C₂²

—

472

—

146

C₂²

#14273700

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

236

—

150

C₄

—

236

—

151

C₄

SD₁₆×D₅₉

yes

characteristic

152

C₄

#14273751

yes

characteristic

153

C₄

#14273754

yes

characteristic

154

C₄

#14273759

yes

characteristic

155

C₂

—

944

—

156

C₂

#16837601

yes

characteristic

157

C₂

#16837651

yes

characteristic

158

C₂

#16837668

yes

characteristic

	subgroup	G/H
center	C₂²	#14273700
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#332	C₂
socle	C₂²×C₅₉	#121
hypercenter	#332	D₅₉
hypocenter	C₅₉	#1686
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14273700

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#332

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
2. #16837684	C₂	semidirect
4. #16837529	C₂	semidirect
5. #16837154	C₂	semidirect
7. C₅₉×#332	C₂	semidirect
62. C₅₉⋊C₈	SD₁₆	semidirect
85. #332	D₅₉	semidirect
87. C₅₉	#1686	semidirect

H ≅ G/N

type

2. #16837684

C₂

semidirect

4. #16837529

C₂

semidirect

5. #16837154

C₂

semidirect

7. C₅₉×#332

C₂

semidirect

62. C₅₉⋊C₈

SD₁₆

semidirect

85. #332

D₅₉

semidirect

87. C₅₉

#1686

semidirect

Subgroups — #21423571

Conjugacy classes of subgroups

Complements