Group #21426171 — subgroups

Subgroups — #21426171

Order 7552.

170 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 43 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	#16837840	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837838	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837777	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837518	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837222	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837214	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#361	C₂	1	1	yes	maximal, characteristic, complemented
8	#14273749	C₂²	1	1	yes	characteristic
9	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
10	C₂×#11745669	—	2	1	no	—
11	#14273735	—	2	1	no	—
12	#14273723	—	2	1	no	—
13	#14273672	—	2	1	no	—
14	#14273671	—	2	1	no	—
15	D₉₄₄	—	2	1	no	—
16	#14273668	—	2	1	no	—
17	#14273667	C₂²	1	1	yes	characteristic
18	#14273653	—	2	1	no	—
19	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
20	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
21	M₃₂×C₅₉	C₂²	1	1	yes	characteristic
22	#108×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₅₉	—	2	1	no	—
24	C₂×C₂²⋊D₅₉	—	4	2	no	—
25	#11745685	—	2	1	no	—
26	C₂×D₂₃₆	—	2	1	no	—
27	D₈×C₅₉	—	2	1	no	—
28	D₈×C₅₉	—	2	1	no	—
29	M₁₆×C₅₉	—	2	1	no	—
30	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	#11745669	—	4	1	no	—
32	#11745669	—	4	1	no	—
33	C₄×Dic₅₉	D₄	1	1	yes	characteristic
34	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
35	Dic₂₃₆	D₄	1	1	yes	characteristic, complemented
36	D₄₇₂	—	2	1	no	—
37	D₄₇₂	—	2	1	no	—
38	D₄₇₂	D₄	1	1	yes	characteristic, complemented
39	#11745660	—	2	1	no	—
40	#11745659	—	4	1	no	—
41	C₈×D₅₉	—	4	1	no	—
42	C₁₆×C₅₉	—	2	1	no	—
43	C₅₉⋊C₁₆	—	2	1	no	—
44	C₂³×C₅₉	—	4	1	no	—
45	C₂²×D₅₉	—	4	1	no	—
46	D₄×C₅₉	—	4	1	no	—
47	D₄×C₅₉	—	4	1	no	—
48	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂²⋊D₅₉	—	4	1	no	—
50	C₂²⋊D₅₉	—	8	4	no	—
51	C₂×Dic₅₉	—	2	1	no	—
52	D₂₃₆	—	2	1	no	—
53	D₂₃₆	—	4	1	no	—
54	D₂₃₆	—	4	1	no	—
55	C₄×D₅₉	—	4	1	no	—
56	Dic₁₁₈	—	2	1	no	—
57	C₈×C₅₉	—	4	1	no	—
58	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₈×C₅₉	#41	1	1	yes	characteristic
60	C₅₉⋊C₈	D₈	1	1	yes	characteristic
61	C₅₉⋊C₈	D₈	1	1	yes	characteristic
62	C₂²×C₅₉	—	8	2	no	—
63	C₂²×C₅₉	—	8	1	no	—
64	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×D₅₉	—	4	1	no	—
66	C₂×D₅₉	—	8	2	no	—
67	C₂×D₅₉	—	8	1	no	—
68	C₄×C₅₉	#121	1	1	yes	characteristic
69	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
70	Dic₅₉	—	2	2	no	—
71	Dic₅₉	—	4	1	no	—
72	#2213	—	59	1	no	maximal, Sylow
73	C₂×C₅₉	—	2	1	no	—
74	C₂×C₅₉	—	16	1	no	—
75	C₂×C₅₉	#459	1	1	yes	characteristic
76	D₅₉	—	8	1	no	—
77	D₅₉	—	16	1	no	—
78	#509	—	59	1	no	—
79	#509	—	59	1	no	—
80	#493	—	59	1	no	—
81	#443	—	59	1	no	—
82	#364	—	59	1	no	—
83	#361	—	59	1	no	—
84	#361	D₅₉	1	1	yes	characteristic, complemented
85	C₅₉	#2213	1	1	yes	characteristic, complemented, Sylow
86	#135	—	59	1	no	—
87	C₂×D₈	—	59	1	no	—
88	C₂×D₈	—	118	1	no	—
89	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
90	#131	—	118	1	no	—
91	#127	—	118	1	no	—
92	SD₃₂	—	118	1	no	—
93	SD₃₂	—	118	1	no	—
94	D₁₆	—	118	1	no	—
95	D₁₆	—	118	1	no	—
96	M₃₂	—	59	1	no	—
97	M₃₂	C₂×D₅₉	1	1	yes	characteristic
98	#108	C₂×D₅₉	1	1	yes	characteristic
99	#104	—	118	1	no	—
100	C₄×C₈	—	59	1	no	—
101	#41	—	118	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	118	1	no	—
104	C₂×D₄	—	236	2	no	—
105	Q₁₆	—	59	1	no	—
106	SD₁₆	—	118	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	2	1	no	—
109	D₈	—	59	1	no	—
110	D₈	—	118	1	no	—
111	D₈	—	118	1	no	—
112	D₈	—	236	1	no	—
113	D₈	—	236	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	236	1	no	—
116	C₂×C₈	—	59	1	no	—
117	C₂×C₈	—	236	1	no	—
118	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
119	C₄²	—	59	1	no	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	118	1	no	—
122	C₂³	—	4	1	no	—
123	C₂³	—	236	1	no	—
124	Q₈	—	118	1	no	—
125	D₄	—	4	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	118	1	no	—
128	D₄	—	472	4	no	—
129	D₄	—	236	1	no	—
130	D₄	—	236	1	no	—
131	D₄	—	236	1	no	—
132	C₂×C₄	—	118	1	no	—
133	C₂×C₄	—	236	1	no	—
134	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
135	C₈	—	4	1	no	—
136	C₈	—	59	1	no	—
137	C₈	—	59	1	no	—
138	C₈	#11745685	1	1	yes	characteristic
139	C₈	D₄×D₅₉	1	1	yes	characteristic
140	C₂²	—	8	2	no	—
141	C₂²	—	8	1	no	—
142	C₂²	—	472	2	no	—
143	C₂²	—	472	1	no	—
144	C₂²	—	236	1	no	—
145	C₂²	#14273754	1	1	yes	characteristic
146	C₄	—	118	2	no	—
147	C₄	—	236	1	no	—
148	C₄	#14273700	1	1	yes	characteristic
149	C₄	D₈×D₅₉	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	16	1	no	—
152	C₂	—	944	1	no	—
153	C₂	—	472	1	no	—
154	C₂	#16837596	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837840

C₂

yes

maximal, characteristic, complemented

#16837838

C₂

yes

maximal, characteristic, complemented

#16837777

C₂

yes

maximal, characteristic, complemented

#16837518

C₂

yes

maximal, characteristic, complemented

#16837222

C₂

yes

maximal, characteristic, complemented

#16837214

C₂

yes

maximal, characteristic, complemented

C₅₉×#361

C₂

yes

maximal, characteristic, complemented

#14273749

C₂²

yes

characteristic

C₂×D₄₇₂

C₂²

yes

—

—

—

—

—

D₉₄₄

—

#14273668

—

#14273667

C₂²

yes

characteristic

#14273653

—

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×D₈×C₅₉

C₂²

yes

characteristic

M₃₂×C₅₉

C₂²

yes

characteristic

#108×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂×C₂²⋊D₅₉

—

#11745685

—

C₂×D₂₃₆

—

D₈×C₅₉

—

D₈×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

#11745669

—

#11745669

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

D₄

yes

characteristic, complemented

D₄₇₂

—

D₄₇₂

—

D₄₇₂

D₄

yes

characteristic, complemented

#11745660

—

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂²×D₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

D₈

yes

characteristic

C₅₉⋊C₈

D₈

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

#121

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2213

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#459

yes

characteristic

D₅₉

—

D₅₉

—

—

—

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#2213

yes

characteristic, complemented, Sylow

#135

—

C₂×D₈

—

C₂×D₈

—

118

—

C₂×D₈

C₂×D₅₉

yes

characteristic

#131

—

118

—

#127

—

118

—

SD₃₂

—

118

—

SD₃₂

—

118

—

D₁₆

—

118

—

D₁₆

—

118

—

M₃₂

—

M₃₂

C₂×D₅₉

yes

characteristic

#108

C₂×D₅₉

yes

characteristic

#104

—

118

—

100

C₄×C₈

—

101

#41

—

118

—

102

C₂×D₄

—

103

C₂×D₄

—

118

—

104

C₂×D₄

—

236

—

105

Q₁₆

—

106

SD₁₆

—

118

—

107

D₈

—

108

D₈

—

109

D₈

—

110

D₈

—

118

—

111

D₈

—

118

—

112

D₈

—

236

—

113

D₈

—

236

—

114

M₁₆

—

115

M₁₆

—

236

—

116

C₂×C₈

—

117

C₂×C₈

—

236

—

118

C₂×C₈

C₂²×D₅₉

yes

characteristic

119

C₄²

—

120

C₁₆

—

121

C₁₆

—

118

—

122

C₂³

—

123

C₂³

—

236

—

124

Q₈

—

118

—

125

D₄

—

126

D₄

—

127

D₄

—

118

—

128

D₄

—

472

—

129

D₄

—

236

—

130

D₄

—

236

—

131

D₄

—

236

—

132

C₂×C₄

—

118

—

133

C₂×C₄

—

236

—

134

C₂×C₄

D₄×D₅₉

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

#11745685

yes

characteristic

139

C₈

D₄×D₅₉

yes

characteristic

140

C₂²

—

141

C₂²

—

142

C₂²

—

472

—

143

C₂²

—

472

—

144

C₂²

—

236

—

145

C₂²

#14273754

yes

characteristic

146

C₄

—

118

—

147

C₄

—

236

—

148

C₄

#14273700

yes

characteristic

149

C₄

D₈×D₅₉

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

—

944

—

153

C₂

—

472

—

154

C₂

#16837596

yes

characteristic

	subgroup	G/H
center	C₂	#16837596
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#361	C₂
socle	C₂×C₅₉	#459
hypercenter	#361	D₅₉
hypocenter	C₅₉	#2213
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

subgroup

G/H

center

C₂

#16837596

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#361

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

N	H ≅ G/N	type
1. #16837840	C₂	semidirect
2. #16837838	C₂	semidirect
3. #16837777	C₂	semidirect
4. #16837518	C₂	semidirect
5. #16837222	C₂	semidirect
6. #16837214	C₂	semidirect
7. C₅₉×#361	C₂	semidirect
35. Dic₂₃₆	D₄	semidirect
38. D₄₇₂	D₄	semidirect
84. #361	D₅₉	semidirect
85. C₅₉	#2213	semidirect

H ≅ G/N

type

1. #16837840

C₂

semidirect

2. #16837838

C₂

semidirect

3. #16837777

C₂

semidirect

4. #16837518

C₂

semidirect

5. #16837222

C₂

semidirect

6. #16837214

C₂

semidirect

7. C₅₉×#361

C₂

semidirect

35. Dic₂₃₆

D₄

semidirect

38. D₄₇₂

D₄

semidirect

84. #361

D₅₉

semidirect

85. C₅₉

#2213

semidirect

Subgroups — #21426171

Conjugacy classes of subgroups

Complements