Group #21426164 — subgroups

Subgroups — #21426164

Order 7552.

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

normal subgroups only

This group has 154 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	#16837845	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837838	C₂	1	1	yes	maximal, characteristic
3	#16837785	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	#16837217	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×Dic₂₃₆	C₂²	1	1	yes	characteristic
10	C₂×#11745670	—	2	1	no	—
11	#14273735	—	2	1	no	—
12	#14273714	—	2	1	no	—
13	Dic₄₇₂	—	2	1	no	—
14	#14273672	—	2	1	no	—
15	#14273671	—	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	#11745685	—	2	1	no	—
25	C₂×Dic₁₁₈	—	2	1	no	—
26	D₈×C₅₉	—	2	1	no	—
27	D₈×C₅₉	—	2	1	no	—
28	M₁₆×C₅₉	—	2	1	no	—
29	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₅₉	—	4	2	no	—
31	#11745670	—	4	1	no	—
32	#11745670	—	4	1	no	—
33	C₄×Dic₅₉	D₄	1	1	yes	characteristic
34	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
35	Dic₂₃₆	—	2	1	no	—
36	Dic₂₃₆	—	2	1	no	—
37	Dic₂₃₆	D₄	1	1	yes	characteristic
38	D₄₇₂	D₄	1	1	yes	characteristic
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	D₄×C₅₉	—	4	1	no	—
46	D₄×C₅₉	—	4	1	no	—
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₅₉	—	4	1	no	—
49	C₂×Dic₅₉	—	2	1	no	—
50	C₂×Dic₅₉	—	4	1	no	—
51	D₂₃₆	—	2	1	no	—
52	C₄×D₅₉	—	4	1	no	—
53	Dic₁₁₈	—	2	1	no	—
54	Dic₁₁₈	—	4	1	no	—
55	Dic₁₁₈	—	4	1	no	—
56	C₈×C₅₉	—	4	1	no	—
57	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₈×C₅₉	#41	1	1	yes	characteristic
59	C₅₉⋊C₈	D₈	1	1	yes	characteristic
60	C₅₉⋊C₈	D₈	1	1	yes	characteristic
61	C₂²×C₅₉	—	8	2	no	—
62	C₂²×C₅₉	—	8	1	no	—
63	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×D₅₉	—	4	1	no	—
65	C₄×C₅₉	#121	1	1	yes	characteristic
66	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
67	Dic₅₉	—	4	1	no	—
68	Dic₅₉	—	4	1	no	—
69	Dic₅₉	—	8	1	no	—
70	#2212	—	59	1	no	maximal, Sylow
71	C₂×C₅₉	—	2	1	no	—
72	C₂×C₅₉	—	16	1	no	—
73	C₂×C₅₉	#459	1	1	yes	characteristic
74	D₅₉	—	8	1	no	—
75	#510	—	59	1	no	—
76	#509	—	59	1	no	—
77	#495	—	59	1	no	—
78	#443	—	59	1	no	—
79	#364	—	59	1	no	—
80	#362	—	59	1	no	—
81	#361	D₅₉	1	1	yes	characteristic, complemented
82	C₅₉	#2212	1	1	yes	characteristic, complemented, Sylow
83	#135	—	59	1	no	—
84	C₂×Q₁₆	—	59	1	no	—
85	C₂×SD₁₆	—	118	1	no	—
86	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
87	#131	—	118	1	no	—
88	#124	—	118	1	no	—
89	Q₃₂	—	118	1	no	—
90	SD₃₂	—	118	1	no	—
91	SD₃₂	—	118	1	no	—
92	D₁₆	—	118	1	no	—
93	M₃₂	—	59	1	no	—
94	M₃₂	C₂×D₅₉	1	1	yes	characteristic
95	#108	C₂×D₅₉	1	1	yes	characteristic
96	#104	—	118	1	no	—
97	C₄×C₈	—	59	1	no	—
98	#41	—	118	1	no	—
99	C₂×Q₈	—	118	1	no	—
100	C₂×D₄	—	2	1	no	—
101	Q₁₆	—	59	1	no	—
102	Q₁₆	—	118	1	no	—
103	Q₁₆	—	118	1	no	—
104	SD₁₆	—	118	1	no	—
105	SD₁₆	—	236	1	no	—
106	SD₁₆	—	236	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	2	1	no	—
109	D₈	—	59	1	no	—
110	M₁₆	—	2	1	no	—
111	M₁₆	—	236	1	no	—
112	C₂×C₈	—	59	1	no	—
113	C₂×C₈	—	236	1	no	—
114	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
115	C₂²⋊C₄	—	236	2	no	—
116	C₄²	—	59	1	no	—
117	C₁₆	—	2	1	no	—
118	C₁₆	—	118	1	no	—
119	C₂³	—	4	1	no	—
120	Q₈	—	118	1	no	—
121	Q₈	—	236	1	no	—
122	Q₈	—	236	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	118	1	no	—
126	D₄	—	236	1	no	—
127	C₂×C₄	—	118	1	no	—
128	C₂×C₄	—	236	1	no	—
129	C₂×C₄	—	236	1	no	—
130	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
131	C₈	—	4	1	no	—
132	C₈	—	59	1	no	—
133	C₈	—	59	1	no	—
134	C₈	#11745685	1	1	yes	characteristic
135	C₈	D₄×D₅₉	1	1	yes	characteristic
136	C₂²	—	8	2	no	—
137	C₂²	—	8	1	no	—
138	C₂²	—	236	1	no	—
139	C₂²	#14273754	1	1	yes	characteristic
140	C₄	—	472	1	no	—
141	C₄	—	236	1	no	—
142	C₄	—	236	1	no	—
143	C₄	#14273700	1	1	yes	characteristic
144	C₄	D₈×D₅₉	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	16	1	no	—
147	C₂	—	472	1	no	—
148	C₂	#16837596	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837845

C₂

yes

maximal, characteristic, complemented

#16837838

C₂

yes

maximal, characteristic

#16837785

C₂

yes

maximal, characteristic, complemented

#16837518

C₂

yes

maximal, characteristic, complemented

#16837222

C₂

yes

maximal, characteristic, complemented

#16837217

C₂

yes

maximal, characteristic, complemented

C₅₉×#361

C₂

yes

maximal, characteristic, complemented

#14273749

C₂²

yes

characteristic

C₂×Dic₂₃₆

C₂²

yes

—

—

—

Dic₄₇₂

—

—

—

—

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

—

#11745685

—

C₂×Dic₁₁₈

—

D₈×C₅₉

—

D₈×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

#11745670

—

#11745670

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

D₄

yes

characteristic

D₄₇₂

D₄

yes

characteristic

#11745660

—

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

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

—

C₄×C₅₉

#121

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#2212

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#459

yes

characteristic

D₅₉

—

—

—

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#2212

yes

characteristic, complemented, Sylow

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

118

—

C₂×D₈

C₂×D₅₉

yes

characteristic

#131

—

118

—

#124

—

118

—

Q₃₂

—

118

—

SD₃₂

—

118

—

SD₃₂

—

118

—

D₁₆

—

118

—

M₃₂

—

M₃₂

C₂×D₅₉

yes

characteristic

#108

C₂×D₅₉

yes

characteristic

#104

—

118

—

C₄×C₈

—

#41

—

118

—

C₂×Q₈

—

118

—

100

C₂×D₄

—

101

Q₁₆

—

102

Q₁₆

—

118

—

103

Q₁₆

—

118

—

104

SD₁₆

—

118

—

105

SD₁₆

—

236

—

106

SD₁₆

—

236

—

107

D₈

—

108

D₈

—

109

D₈

—

110

M₁₆

—

111

M₁₆

—

236

—

112

C₂×C₈

—

113

C₂×C₈

—

236

—

114

C₂×C₈

C₂²×D₅₉

yes

characteristic

115

C₂²⋊C₄

—

236

—

116

C₄²

—

117

C₁₆

—

118

C₁₆

—

118

—

119

C₂³

—

120

Q₈

—

118

—

121

Q₈

—

236

—

122

Q₈

—

236

—

123

D₄

—

124

D₄

—

125

D₄

—

118

—

126

D₄

—

236

—

127

C₂×C₄

—

118

—

128

C₂×C₄

—

236

—

129

C₂×C₄

—

236

—

130

C₂×C₄

D₄×D₅₉

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

#11745685

yes

characteristic

135

C₈

D₄×D₅₉

yes

characteristic

136

C₂²

—

137

C₂²

—

138

C₂²

—

236

—

139

C₂²

#14273754

yes

characteristic

140

C₄

—

472

—

141

C₄

—

236

—

142

C₄

—

236

—

143

C₄

#14273700

yes

characteristic

144

C₄

D₈×D₅₉

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

472

—

148

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	#2212
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. #16837845	C₂	semidirect
3. #16837785	C₂	semidirect
4. #16837518	C₂	semidirect
5. #16837222	C₂	semidirect
6. #16837217	C₂	semidirect
7. C₅₉×#361	C₂	semidirect
81. #361	D₅₉	semidirect
82. C₅₉	#2212	semidirect

H ≅ G/N

type

1. #16837845

C₂

semidirect

3. #16837785

C₂

semidirect

4. #16837518

C₂

semidirect

5. #16837222

C₂

semidirect

6. #16837217

C₂

semidirect

7. C₅₉×#361

C₂

semidirect

81. #361

D₅₉

semidirect

82. C₅₉

#2212

semidirect

Subgroups — #21426164

Conjugacy classes of subgroups

Complements