Group #21426140 — subgroups

Subgroups — #21426140

Order 7552.

166 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 49 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	C₂×#14273672	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273670	C₂	1	1	yes	maximal, characteristic
3	#16837780	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837484	C₂	1	1	yes	maximal, characteristic
5	Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837205	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
9	C₂×#11745672	—	2	1	no	—
10	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
11	#14273725	—	2	1	no	—
12	#14273689	—	2	1	no	—
13	#14273672	—	2	2	no	—
14	#14273670	—	2	2	no	—
15	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14273647	—	2	1	no	—
17	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	#107×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₉	—	2	1	no	—
22	C₂×D₂₃₆	—	2	1	no	—
23	C₂×C₄×D₅₉	—	4	1	no	—
24	C₂×Dic₁₁₈	—	2	1	no	—
25	Q₁₆×C₅₉	—	2	1	no	—
26	Q₁₆×C₅₉	—	2	1	no	—
27	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₅₉	—	2	1	no	—
29	#11745672	—	4	2	no	—
30	(C₂×D₅₉)⋊C₄	—	4	1	no	—
31	Dic₅₉⋊C₄	—	4	1	no	—
32	Dic₅₉⋊C₄	—	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	D₄₇₂	—	2	1	no	—
38	D₄₇₂	D₄	1	2	yes	—
39	C₁₆×C₅₉	—	2	1	no	—
40	C₅₉⋊C₁₆	—	2	1	no	—
41	C₂²×D₅₉	—	4	1	no	—
42	Q₈×C₅₉	—	4	1	no	—
43	Q₈×C₅₉	—	4	1	no	—
44	C₂×C₄×C₅₉	—	4	1	no	—
45	C₂×C₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₅₉	—	4	1	no	—
48	C₂×Dic₅₉	D₈	1	1	yes	characteristic
49	C₂×Dic₅₉	D₈	1	1	yes	characteristic
50	D₂₃₆	—	2	2	no	—
51	D₂₃₆	—	4	1	no	—
52	C₄×D₅₉	—	8	2	no	—
53	Dic₁₁₈	—	4	1	no	—
54	Dic₁₁₈	—	4	1	no	—
55	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₈×C₅₉	#41	1	1	yes	characteristic
57	C₅₉⋊C₈	—	2	1	no	—
58	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
59	C₂×D₅₉	—	4	2	no	—
60	C₂×D₅₉	—	8	1	no	—
61	C₂×D₅₉	—	8	1	no	—
62	C₄×C₅₉	—	8	1	no	—
63	C₄×C₅₉	—	8	1	no	—
64	C₄×C₅₉	#121	1	1	yes	characteristic
65	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
66	Dic₅₉	—	2	1	no	—
67	Dic₅₉	—	8	1	no	—
68	Dic₅₉	SD₃₂	1	2	yes	complemented
69	#2208	—	59	1	no	maximal, Sylow
70	C₂×C₅₉	#459	1	1	yes	characteristic
71	C₂×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
72	C₂×C₅₉	#510	1	1	yes	characteristic
73	D₅₉	—	8	2	no	—
74	C₂×SD₃₂	—	59	1	no	—
75	C₂×SD₃₂	—	59	1	no	—
76	#494	—	59	1	no	—
77	C₄×D₈	—	59	1	no	—
78	C₄⋊C₁₆	—	59	1	no	—
79	#358	—	59	1	no	—
80	#358	D₅₉	1	1	yes	characteristic, complemented
81	C₅₉	#2208	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₁₆	—	59	1	no	—
83	C₂×Q₁₆	—	118	1	no	—
84	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
85	C₂×D₈	—	59	1	no	—
86	C₄⋊Q₈	—	118	1	no	—
87	C₄×D₄	—	118	1	no	—
88	SD₃₂	—	118	2	no	—
89	SD₃₂	—	118	2	no	—
90	C₂×C₁₆	—	59	1	no	—
91	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
92	#107	C₂×D₅₉	1	1	yes	characteristic
93	#102	—	118	1	no	—
94	C₄×C₈	—	59	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	118	1	no	—
97	C₂×D₄	—	118	1	no	—
98	C₂²×C₄	—	236	1	no	—
99	Q₁₆	—	2	1	no	—
100	Q₁₆	—	2	1	no	—
101	Q₁₆	—	118	1	no	—
102	Q₁₆	—	118	1	no	—
103	Q₁₆	—	236	2	no	—
104	D₈	—	59	2	no	—
105	D₈	—	118	1	no	—
106	C₂×C₈	—	59	1	no	—
107	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	236	1	no	—
110	C₄⋊C₄	—	236	1	no	—
111	C₂²⋊C₄	—	236	1	no	—
112	C₄²	—	59	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	118	1	no	—
115	C₂³	—	236	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	236	1	no	—
119	Q₈	—	236	1	no	—
120	D₄	—	118	2	no	—
121	D₄	—	236	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	59	1	no	—
125	C₂×C₄	—	59	1	no	—
126	C₂×C₄	—	472	2	no	—
127	C₂×C₄	—	236	1	no	—
128	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
129	C₈	—	118	1	no	—
130	C₈	#11745685	1	1	yes	characteristic
131	C₈	D₄×D₅₉	1	1	yes	characteristic
132	C₂²	—	472	1	no	—
133	C₂²	—	472	1	no	—
134	C₂²	—	236	2	no	—
135	C₂²	D₈×D₅₉	1	1	yes	characteristic
136	C₄	—	8	1	no	—
137	C₄	—	8	1	no	—
138	C₄	—	59	2	no	—
139	C₄	—	118	1	no	—
140	C₄	—	472	1	no	—
141	C₄	#14273700	1	1	yes	characteristic
142	C₄	#14273754	1	1	yes	characteristic
143	C₂	—	472	2	no	—
144	C₂	#16837596	1	1	yes	characteristic
145	C₂	SD₃₂×D₅₉	1	1	yes	characteristic
146	C₂	#16837844	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273672

C₂

yes

maximal, characteristic

C₂×#14273670

C₂

yes

maximal, characteristic

#16837780

C₂

yes

maximal, characteristic, complemented

#16837484

C₂

yes

maximal, characteristic

Dic₅₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#16837205

C₂

yes

maximal, characteristic, complemented

C₅₉×#358

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×#11745672

—

C₂×D₄₇₂

C₂²

yes

—

—

—

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273647

—

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

#11745672

—

(C₂×D₅₉)⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

D₄₇₂

—

D₄₇₂

D₄

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₂×Dic₅₉

D₈

yes

characteristic

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

SD₃₂

yes

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×SD₃₂

yes

characteristic

C₂×C₅₉

#510

yes

characteristic

D₅₉

—

C₂×SD₃₂

—

C₂×SD₃₂

—

#494

—

C₄×D₈

—

C₄⋊C₁₆

—

#358

—

#358

D₅₉

yes

characteristic, complemented

C₅₉

#2208

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₂×D₈

—

C₄⋊Q₈

—

118

—

C₄×D₄

—

118

—

SD₃₂

—

118

—

SD₃₂

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#107

C₂×D₅₉

yes

characteristic

#102

—

118

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×D₄

—

118

—

C₂²×C₄

—

236

—

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

118

—

102

Q₁₆

—

118

—

103

Q₁₆

—

236

—

104

D₈

—

105

D₈

—

118

—

106

C₂×C₈

—

107

C₂×C₈

C₂²×D₅₉

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

236

—

110

C₄⋊C₄

—

236

—

111

C₂²⋊C₄

—

236

—

112

C₄²

—

113

C₁₆

—

114

C₁₆

—

118

—

115

C₂³

—

236

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

236

—

119

Q₈

—

236

—

120

D₄

—

118

—

121

D₄

—

236

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

472

—

127

C₂×C₄

—

236

—

128

C₂×C₄

D₄×D₅₉

yes

characteristic

129

C₈

—

118

—

130

C₈

#11745685

yes

characteristic

131

C₈

D₄×D₅₉

yes

characteristic

132

C₂²

—

472

—

133

C₂²

—

472

—

134

C₂²

—

236

—

135

C₂²

D₈×D₅₉

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

118

—

140

C₄

—

472

—

141

C₄

#14273700

yes

characteristic

142

C₄

#14273754

yes

characteristic

143

C₂

—

472

—

144

C₂

#16837596

yes

characteristic

145

C₂

SD₃₂×D₅₉

yes

characteristic

146

C₂

#16837844

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#358	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#358	D₅₉
hypocenter	C₅₉	#2208
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#358

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
3. #16837780	C₂	semidirect
5. Dic₅₉⋊C₁₆	C₂	semidirect
6. #16837205	C₂	semidirect
7. C₅₉×#358	C₂	semidirect
68. Dic₅₉	SD₃₂	semidirect
80. #358	D₅₉	semidirect
81. C₅₉	#2208	semidirect

H ≅ G/N

type

3. #16837780

C₂

semidirect

5. Dic₅₉⋊C₁₆

C₂

semidirect

6. #16837205

C₂

semidirect

7. C₅₉×#358

C₂

semidirect

68. Dic₅₉

SD₃₂

semidirect

80. #358

D₅₉

semidirect

81. C₅₉

#2208

semidirect

Subgroups — #21426140

Conjugacy classes of subgroups

Complements