Group #21426220 — subgroups

Subgroups — #21426220

Order 7552.

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

normal subgroups only

This group has 140 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	C₂×#14273673	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837820	C₂	1	1	yes	maximal, characteristic
3	#16837678	C₂	1	1	yes	maximal, characteristic
4	#16837674	C₂	1	1	yes	maximal, characteristic
5	#16837231	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837219	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#362	C₂	1	1	yes	maximal, characteristic, complemented
8	#14273760	—	2	1	no	—
9	#14273758	—	2	1	no	—
10	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
11	#14273733	C₂²	1	1	yes	characteristic
12	#14273673	—	2	1	no	—
13	#14273673	—	2	1	no	—
14	#14273666	—	2	1	no	—
15	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₁₆×D₅₉	—	2	1	no	—
17	#14273663	C₂²	1	1	yes	characteristic
18	#14273646	—	2	1	no	—
19	#14273645	—	2	1	no	—
20	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	M₃₂×C₅₉	C₂²	1	1	yes	characteristic
22	#108×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	#11745685	D₄	1	1	yes	characteristic
25	C₂×Dic₁₁₈	—	2	1	no	—
26	Q₁₆×C₅₉	—	2	1	no	—
27	Q₁₆×C₅₉	—	2	1	no	—
28	M₁₆×C₅₉	—	2	1	no	—
29	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	#11745672	—	4	1	no	—
31	#11745671	—	4	1	no	—
32	#11745664	—	2	1	no	—
33	#11745664	D₄	1	1	yes	characteristic
34	Dic₂₃₆	—	2	1	no	—
35	Dic₂₃₆	—	2	1	no	—
36	Dic₂₃₆	—	4	1	no	—
37	#11745660	—	4	1	no	—
38	#11745659	—	2	1	no	—
39	C₈×D₅₉	—	2	1	no	—
40	C₁₆×C₅₉	—	2	1	no	—
41	C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
42	C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
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	C₂²⋊D₅₉	—	2	1	no	—
48	C₂×Dic₅₉	—	4	1	no	—
49	D₂₃₆	D₈	1	1	yes	characteristic
50	C₄×D₅₉	—	2	1	no	—
51	Dic₁₁₈	—	4	1	no	—
52	Dic₁₁₈	—	4	1	no	—
53	Dic₁₁₈	D₈	1	1	yes	characteristic
54	C₈×C₅₉	—	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	C₅₉⋊C₈	—	4	1	no	—
59	C₂²×C₅₉	#135	1	1	yes	characteristic
60	C₂×D₅₉	—	2	1	no	—
61	C₄×C₅₉	—	8	1	no	—
62	C₄×C₅₉	#121	1	1	yes	characteristic
63	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
64	Dic₅₉	—	2	1	no	—
65	Dic₅₉	—	8	1	no	—
66	#2222	—	59	1	no	maximal, Sylow
67	C₂×C₅₉	—	2	1	no	—
68	C₂×C₅₉	#466	1	1	yes	characteristic
69	D₅₉	—	4	1	no	—
70	C₂×Q₃₂	—	59	1	no	—
71	#504	—	59	1	no	—
72	#473	—	59	1	no	—
73	#473	—	59	1	no	—
74	#368	—	59	1	no	—
75	#362	—	59	1	no	—
76	#362	D₅₉	1	1	yes	characteristic, complemented
77	C₅₉	#2222	1	1	yes	characteristic, complemented, Sylow
78	#137	—	118	1	no	—
79	#137	—	118	1	no	—
80	C₂×Q₁₆	—	59	1	no	—
81	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
82	#131	—	59	1	no	—
83	Q₃₂	—	118	1	no	—
84	Q₃₂	—	118	1	no	—
85	M₃₂	—	118	1	no	—
86	M₃₂	C₂×D₅₉	1	1	yes	characteristic
87	C₂×C₁₆	—	59	1	no	—
88	C₂×C₁₆	—	118	1	no	—
89	#108	—	59	1	no	—
90	#108	C₂×D₅₉	1	1	yes	characteristic
91	#101	—	118	1	no	—
92	#101	—	118	1	no	—
93	#41	—	59	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	118	1	no	—
96	Q₁₆	—	2	1	no	—
97	Q₁₆	—	2	1	no	—
98	Q₁₆	—	118	1	no	—
99	Q₁₆	—	118	1	no	—
100	Q₁₆	—	236	1	no	—
101	Q₁₆	—	236	1	no	—
102	SD₁₆	—	236	1	no	—
103	SD₁₆	—	236	1	no	—
104	M₁₆	—	2	1	no	—
105	M₁₆	—	59	1	no	—
106	M₁₆	—	118	1	no	—
107	M₁₆	—	118	1	no	—
108	C₂×C₈	—	118	1	no	—
109	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	59	1	no	—
112	C₁₆	—	59	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	4	1	no	—
115	Q₈	—	59	1	no	—
116	Q₈	—	236	1	no	—
117	Q₈	—	236	1	no	—
118	D₄	—	59	1	no	—
119	D₄	—	118	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	118	1	no	—
122	C₂×C₄	—	236	1	no	—
123	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
124	C₈	—	4	1	no	—
125	C₈	—	118	1	no	—
126	C₈	—	236	1	no	—
127	C₈	#11745685	1	1	yes	characteristic
128	C₈	D₄×D₅₉	1	1	yes	characteristic
129	C₂²	—	118	1	no	—
130	C₂²	#14273748	1	1	yes	characteristic
131	C₄	—	8	1	no	—
132	C₄	—	118	1	no	—
133	C₄	—	472	1	no	—
134	C₄	#14273700	1	1	yes	characteristic
135	C₄	D₈×D₅₉	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	236	1	no	—
138	C₂	#16837637	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273673

C₂

yes

maximal, characteristic, complemented

#16837820

C₂

yes

maximal, characteristic

#16837678

C₂

yes

maximal, characteristic

#16837674

C₂

yes

maximal, characteristic

#16837231

C₂

yes

maximal, characteristic, complemented

#16837219

C₂

yes

maximal, characteristic, complemented

C₅₉×#362

C₂

yes

maximal, characteristic, complemented

#14273760

—

#14273758

—

C₂×Dic₂₃₆

C₂²

yes

characteristic

#14273733

C₂²

yes

—

—

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₁₆×D₅₉

—

#14273663

C₂²

yes

characteristic

#14273646

—

#14273645

—

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

M₃₂×C₅₉

C₂²

yes

characteristic

#108×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

#11745685

D₄

yes

characteristic

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

—

—

D₄

yes

characteristic

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

#11745660

—

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

D₄

yes

characteristic

C₅₉⋊C₁₆

D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

D₈

yes

characteristic

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

D₈

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

#135

yes

characteristic

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#121

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2222

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#466

yes

characteristic

D₅₉

—

C₂×Q₃₂

—

—

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#2222

yes

characteristic, complemented, Sylow

#137

—

118

—

#137

—

118

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

#131

—

Q₃₂

—

118

—

Q₃₂

—

118

—

M₃₂

—

118

—

M₃₂

C₂×D₅₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

118

—

#108

—

#108

C₂×D₅₉

yes

characteristic

#101

—

118

—

#101

—

118

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

118

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

118

—

Q₁₆

—

118

—

100

Q₁₆

—

236

—

101

Q₁₆

—

236

—

102

SD₁₆

—

236

—

103

SD₁₆

—

236

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

118

—

107

M₁₆

—

118

—

108

C₂×C₈

—

118

—

109

C₂×C₈

C₂²×D₅₉

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

236

—

117

Q₈

—

236

—

118

D₄

—

119

D₄

—

118

—

120

C₂×C₄

—

121

C₂×C₄

—

118

—

122

C₂×C₄

—

236

—

123

C₂×C₄

D₄×D₅₉

yes

characteristic

124

C₈

—

125

C₈

—

118

—

126

C₈

—

236

—

127

C₈

#11745685

yes

characteristic

128

C₈

D₄×D₅₉

yes

characteristic

129

C₂²

—

118

—

130

C₂²

#14273748

yes

characteristic

131

C₄

—

132

C₄

—

118

—

133

C₄

—

472

—

134

C₄

#14273700

yes

characteristic

135

C₄

D₈×D₅₉

yes

characteristic

136

C₂

—

137

C₂

—

236

—

138

C₂

#16837637

yes

characteristic

	subgroup	G/H
center	C₂	#16837637
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#362	C₂
socle	C₂×C₅₉	#466
hypercenter	#362	D₅₉
hypocenter	C₅₉	#2222
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #362

subgroup

G/H

center

C₂

#16837637

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#362

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #362

N	H ≅ G/N	type
1. C₂×#14273673	C₂	semidirect
5. #16837231	C₂	semidirect
6. #16837219	C₂	semidirect
7. C₅₉×#362	C₂	semidirect
76. #362	D₅₉	semidirect
77. C₅₉	#2222	semidirect

H ≅ G/N

type

1. C₂×#14273673

C₂

semidirect

5. #16837231

C₂

semidirect

6. #16837219

C₂

semidirect

7. C₅₉×#362

C₂

semidirect

76. #362

D₅₉

semidirect

77. C₅₉

#2222

semidirect

Subgroups — #21426220

Conjugacy classes of subgroups

Complements