Group #21424140 — subgroups

Subgroups — #21424140

Order 7552.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14273734	C₂	1	1	yes	maximal, characteristic
2	#16837438	C₂	1	2	yes	maximal, complemented
3	#16837376	C₂	1	1	yes	maximal, characteristic
4	#16837158	C₂	1	2	yes	maximal, complemented
5	C₅₉×#410	C₂	1	1	yes	maximal, characteristic
6	#14273734	—	2	2	no	—
7	#14273734	—	2	2	no	—
8	C₂×#11745664	C₄	1	2	yes	complemented
9	C₂×#11745664	C₂²	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₈	C₄	1	2	yes	complemented
11	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
12	#14273686	C₂²	1	2	yes	—
13	#14273660	—	2	2	no	—
14	#14273658	—	2	2	no	—
15	#14273642	—	2	2	no	—
16	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
17	#117×C₅₉	C₂²	1	2	yes	—
18	#99×C₅₉	—	2	2	no	—
19	#41×C₅₉	—	2	2	no	—
20	#41×C₅₉	—	2	2	no	—
21	C₂×Q₈×C₅₉	Q₈	1	1	yes	characteristic
22	C₂×D₄×C₅₉	D₄	1	2	yes	—
23	C₂×D₄×C₅₉	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₅₉	—	2	2	no	—
27	C₂²⋊C₄×C₅₉	—	2	2	no	—
28	C₄²×C₅₉	—	2	2	no	—
29	C₂²⋊Dic₅₉	—	2	2	no	—
30	C₄⋊Dic₅₉	—	2	2	no	—
31	C₄×Dic₅₉	—	2	2	no	—
32	#11745664	—	2	2	no	—
33	#11745664	—	2	4	no	—
34	C₂×C₅₉⋊C₈	—	2	2	no	—
35	C₂×C₅₉⋊C₈	—	2	2	no	—
36	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
37	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
40	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
41	C₂³×C₅₉	C₄⋊C₄	1	2	yes	—
42	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₅₉	—	2	2	no	—
44	D₄×C₅₉	—	2	4	no	—
45	D₄×C₅₉	—	2	2	no	—
46	C₂×C₄×C₅₉	—	2	2	no	—
47	C₂×C₄×C₅₉	—	2	2	no	—
48	C₂×C₄×C₅₉	—	4	2	no	—
49	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₅₉	#41	1	2	yes	—
54	C₂×Dic₅₉	—	4	2	no	—
55	C₅₉⋊C₈	—	2	2	no	—
56	C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₅₉	—	2	2	no	—
60	C₂²×C₅₉	—	4	1	no	—
61	C₂²×C₅₉	—	4	2	no	—
62	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₅₉	#121	1	2	yes	—
64	C₄×C₅₉	—	2	2	no	—
65	C₄×C₅₉	—	8	2	no	—
66	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₅₉	#117	1	1	yes	characteristic
68	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
69	Dic₅₉	—	8	2	no	—
70	#1812	—	59	1	no	maximal, Sylow
71	C₂×C₅₉	—	2	2	no	—
72	C₂×C₅₉	—	2	1	no	—
73	C₂×C₅₉	—	4	2	no	—
74	C₂×C₅₉	#380	1	1	yes	characteristic
75	C₂×#131	—	59	1	no	—
76	#427	—	59	2	no	—
77	#410	—	59	1	no	—
78	#410	D₅₉	1	1	yes	characteristic
79	H(3,ℤ/4)	—	59	2	no	—
80	C₅₉	#1812	1	1	yes	characteristic, complemented, Sylow
81	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
82	#131	—	118	2	no	—
83	#131	—	118	2	no	—
84	C₂×M₁₆	—	59	2	no	—
85	C₂×M₁₆	—	59	1	no	—
86	C₂²×C₈	—	59	1	no	—
87	C₂²×C₈	—	59	2	no	—
88	#117	—	59	2	no	—
89	#117	C₂×D₅₉	1	2	yes	—
90	#107	—	118	2	no	—
91	#106	—	118	2	no	—
92	#99	—	2	2	no	—
93	#99	—	118	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	C₂×Q₈	Dic₁₁₈	1	1	yes	characteristic
97	C₂×D₄	Dic₁₁₈	1	1	yes	characteristic
98	C₂×D₄	C₂²⋊D₅₉	1	2	yes	—
99	C₂²×C₄	C₄×D₅₉	1	2	yes	—
100	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
101	M₁₆	—	118	4	no	—
102	M₁₆	—	118	2	no	—
103	C₂×C₈	—	59	1	no	—
104	C₂×C₈	—	59	1	no	—
105	C₂×C₈	—	59	2	no	—
106	C₂×C₈	—	59	2	no	—
107	C₂×C₈	—	59	2	no	—
108	C₂×C₈	—	118	2	no	—
109	C₂×C₈	—	118	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	118	2	no	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₂²⋊C₄	—	118	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	118	2	no	—
116	C₂³	Dic₅₉⋊C₄	1	2	yes	—
117	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
118	Q₈	—	2	2	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	236	2	no	—
125	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
126	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
128	C₂×C₄	D₄×D₅₉	1	2	yes	—
129	C₂×C₄	#11745687	1	2	yes	—
130	C₈	—	59	1	no	—
131	C₈	—	59	1	no	—
132	C₈	—	59	2	no	—
133	C₈	—	118	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
138	C₂²	#14273699	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	2	no	—
141	C₄	—	472	2	no	—
142	C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
143	C₄	#14273683	1	1	yes	characteristic
144	C₄	#14273704	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	#16837260	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273734

C₂

yes

maximal, characteristic

#16837438

C₂

yes

maximal, complemented

#16837376

C₂

yes

maximal, characteristic

#16837158

C₂

yes

maximal, complemented

C₅₉×#410

C₂

yes

maximal, characteristic

#14273734

—

#14273734

—

C₂×#11745664

C₄

yes

complemented

C₂×#11745664

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₄

yes

complemented

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273686

C₂²

yes

—

#14273660

—

#14273658

—

#14273642

—

C₂×#41×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

—

#99×C₅₉

—

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

Q₈

yes

characteristic

C₂×D₄×C₅₉

D₄

yes

—

C₂×D₄×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

C₄⋊C₄

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₅₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

#121

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

#117

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

—

Dic₅₉

—

#1812

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

—

—

—

D₅₉

yes

characteristic

H(3,ℤ/4)

—

C₅₉

#1812

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₉

yes

characteristic

#131

—

118

—

#131

—

118

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₅₉

yes

—

#107

—

118

—

#106

—

118

—

#99

—

#99

—

118

—

#41

—

#41

—

C₂×Q₈

Dic₁₁₈

yes

characteristic

C₂×D₄

Dic₁₁₈

yes

characteristic

C₂×D₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₄×D₅₉

yes

—

100

C₂²×C₄

C₂²×D₅₉

yes

characteristic

101

M₁₆

—

118

—

102

M₁₆

—

118

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

118

—

109

C₂×C₈

—

118

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

118

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

118

—

114

C₄²

—

115

C₄²

—

118

—

116

C₂³

Dic₅₉⋊C₄

yes

—

117

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

118

Q₈

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

236

—

125

C₂×C₄

Dic₅₉⋊C₄

yes

—

126

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

127

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

128

C₂×C₄

D₄×D₅₉

yes

—

129

C₂×C₄

#11745687

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

118

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×Dic₅₉⋊C₄

yes

characteristic

138

C₂²

#14273699

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

472

—

142

C₄

C₂²⋊C₄×D₅₉

yes

characteristic

143

C₄

#14273683

yes

characteristic

144

C₄

#14273704

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#16837260

yes

characteristic

	subgroup	G/H
center	C₄	#14273683
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#410	C₂
socle	C₂×C₅₉	#380
hypercenter	#410	D₅₉
hypocenter	C₅₉	#1812
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#14273683

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#410

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
2. #16837438	C₂	semidirect
4. #16837158	C₂	semidirect
8. C₂×#11745664	C₄	semidirect
10. C₂²×C₅₉⋊C₈	C₄	semidirect
80. C₅₉	#1812	semidirect

H ≅ G/N

type

2. #16837438

C₂

semidirect

4. #16837158

C₂

semidirect

8. C₂×#11745664

C₄

semidirect

10. C₂²×C₅₉⋊C₈

C₄

semidirect

80. C₅₉

#1812

semidirect

Subgroups — #21424140

Conjugacy classes of subgroups

Complements