Group #21424882 — subgroups

Subgroups — #21424882

Order 7552.

196 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16837444	C₂	1	2	yes	maximal
2	C₂×#14273661	C₂	1	1	yes	maximal, characteristic
3	#16837357	C₂	1	1	yes	maximal, characteristic
4	#16837170	C₂	1	2	yes	maximal
5	C₂×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	1	yes	characteristic
7	#14273686	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
9	#14273661	—	2	1	no	—
10	#14273661	—	2	1	no	—
11	#14273661	—	2	2	no	—
12	#14273659	—	2	2	no	—
13	#14273637	—	2	1	no	—
14	C₈×Dic₅₉	—	2	1	no	—
15	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
16	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	#108×C₅₉	C₄	1	4	yes	complemented
18	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₉	—	2	2	no	—
20	M₁₆×C₅₉	—	2	2	no	—
21	M₁₆×C₅₉	C₂×C₄	1	4	yes	—
22	C₂×C₈×C₅₉	—	2	2	no	—
23	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₅₉	Q₈	1	2	yes	—
27	C₂²⋊Dic₅₉	—	2	1	no	—
28	C₄⋊Dic₅₉	—	2	2	no	—
29	C₄⋊Dic₅₉	—	2	2	no	—
30	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
31	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
32	C₄×Dic₅₉	—	2	1	no	—
33	#11745664	—	2	1	no	—
34	#11745664	Q₈	1	2	yes	—
35	C₂×C₅₉⋊C₈	—	2	1	no	—
36	C₂³×C₅₉	#41	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
43	C₂×Dic₅₉	—	2	1	no	—
44	C₂×Dic₅₉	—	2	1	no	—
45	C₂×Dic₅₉	—	4	2	no	—
46	C₂×Dic₅₉	—	8	2	no	—
47	C₈×C₅₉	—	2	4	no	—
48	C₈×C₅₉	C₄⋊C₄	1	2	yes	—
49	C₈×C₅₉	C₄⋊C₄	1	2	yes	—
50	C₅₉⋊C₈	—	2	2	no	—
51	C₂²×C₅₉	—	2	1	no	—
52	C₂²×C₅₉	—	2	1	no	—
53	C₂²×C₅₉	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₅₉	#121	1	1	yes	characteristic
55	C₂²×C₅₉	#125	1	1	yes	characteristic
56	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
58	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
59	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₅₉	—	4	1	no	—
61	Dic₅₉	—	4	1	no	—
62	Dic₅₉	—	8	2	no	—
63	#1952	—	59	1	no	maximal, Sylow
64	C₂×C₅₉	—	2	1	no	—
65	C₂×C₅₉	—	2	1	no	—
66	C₂×C₅₉	#389	1	1	yes	characteristic
67	C₂×C₅₉	#472	1	1	yes	characteristic
68	C₂×C₅₉	#473	1	1	yes	characteristic
69	C₂×#108	D₅₉	1	1	yes	characteristic
70	#428	—	59	2	no	—
71	C₂×#107	—	59	1	no	—
72	#405	—	59	1	no	—
73	#343	—	59	2	no	—
74	C₅₉	#1952	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	—	59	1	no	—
76	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
77	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
78	#117	—	59	1	no	—
79	C₂×C₄⋊C₄	—	59	2	no	—
80	#108	Dic₅₉	1	4	yes	complemented
81	#107	—	118	2	no	—
82	#107	—	118	1	no	—
83	#107	—	118	1	no	—
84	#106	—	118	2	no	—
85	#97	—	118	1	no	—
86	C₄×C₈	—	118	1	no	—
87	C₂²×C₄	—	118	2	no	—
88	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
89	M₁₆	—	2	2	no	—
90	M₁₆	—	59	2	no	—
91	M₁₆	—	118	1	no	—
92	M₁₆	C₂×Dic₅₉	1	4	yes	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	118	1	no	—
95	C₂×C₈	Dic₁₁₈	1	2	yes	—
96	C₂×C₈	D₂₃₆	1	1	yes	characteristic
97	C₂×C₈	D₂₃₆	1	1	yes	characteristic
98	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
99	C₄⋊C₄	—	59	1	no	—
100	C₄⋊C₄	—	59	1	no	—
101	C₄⋊C₄	—	118	2	no	—
102	C₄⋊C₄	—	118	2	no	—
103	C₂²⋊C₄	—	118	1	no	—
104	C₄²	—	118	1	no	—
105	C₂³	#11745689	1	1	yes	characteristic
106	C₂×C₄	—	118	1	no	—
107	C₂×C₄	—	118	1	no	—
108	C₂×C₄	—	472	2	no	—
109	C₂×C₄	—	236	2	no	—
110	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
111	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
112	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
113	C₂×C₄	#11745687	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
115	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
116	C₈	—	2	4	no	—
117	C₈	—	118	2	no	—
118	C₈	C₄⋊Dic₅₉	1	2	yes	—
119	C₈	C₄⋊Dic₅₉	1	2	yes	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	Q₈×Dic₅₉	1	1	yes	characteristic
123	C₂²	#14273702	1	1	yes	characteristic
124	C₂²	#14273718	1	1	yes	characteristic
125	C₄	—	472	2	no	—
126	C₄	—	236	1	no	—
127	C₄	—	236	1	no	—
128	C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
129	C₄	D₄×Dic₅₉	1	1	yes	characteristic
130	C₄	#14273707	1	1	yes	characteristic
131	C₄	#14273726	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#16837301	1	1	yes	characteristic
135	C₂	#16837673	1	1	yes	characteristic
136	C₂	#16837678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837444

C₂

yes

maximal

C₂×#14273661

C₂

yes

maximal, characteristic

#16837357

C₂

yes

maximal, characteristic

#16837170

C₂

yes

maximal

C₂×#108×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

characteristic

#14273686

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

—

—

—

—

—

C₈×Dic₅₉

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#108×C₅₉

C₄

yes

complemented

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

Q₈

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄×Dic₅₉

—

#11745664

—

#11745664

Q₈

yes

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#41

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₄⋊C₄

yes

—

C₈×C₅₉

C₄⋊C₄

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄×Q₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#1952

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

—

—

—

—

C₅₉

#1952

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

#108

Dic₅₉

yes

complemented

#107

—

118

—

#107

—

118

—

#107

—

118

—

#106

—

118

—

#97

—

118

—

C₄×C₈

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

118

—

M₁₆

C₂×Dic₅₉

yes

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

Dic₁₁₈

yes

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂×Dic₅₉

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

118

—

102

C₄⋊C₄

—

118

—

103

C₂²⋊C₄

—

118

—

104

C₄²

—

118

—

105

C₂³

#11745689

yes

characteristic

106

C₂×C₄

—

118

—

107

C₂×C₄

—

118

—

108

C₂×C₄

—

472

—

109

C₂×C₄

—

236

—

110

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

111

C₂×C₄

C₂×D₂₃₆

yes

characteristic

112

C₂×C₄

D₄×D₅₉

yes

characteristic

113

C₂×C₄

#11745687

yes

characteristic

114

C₂×C₄

Q₈×D₅₉

yes

characteristic

115

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

116

C₈

—

117

C₈

—

118

—

118

C₈

C₄⋊Dic₅₉

yes

—

119

C₈

C₄⋊Dic₅₉

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

Q₈×Dic₅₉

yes

characteristic

123

C₂²

#14273702

yes

characteristic

124

C₂²

#14273718

yes

characteristic

125

C₄

—

472

—

126

C₄

—

236

—

127

C₄

—

236

—

128

C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

129

C₄

D₄×Dic₅₉

yes

characteristic

130

C₄

#14273707

yes

characteristic

131

C₄

#14273726

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#16837301

yes

characteristic

135

C₂

#16837673

yes

characteristic

136

C₂

#16837678

yes

characteristic

	subgroup	G/H
center	C₂²	#14273702
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#108×C₅₉	C₂
socle	C₂²×C₅₉	#121
hypercenter	C₂×#108	D₅₉
hypocenter	C₅₉	#1952
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#14273702

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#108×C₅₉

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
17. #108×C₅₉	C₄	semidirect
74. C₅₉	#1952	semidirect
80. #108	Dic₅₉	semidirect

H ≅ G/N

type

17. #108×C₅₉

C₄

semidirect

74. C₅₉

#1952

semidirect

80. #108

Dic₅₉

semidirect

Subgroups — #21424882

Conjugacy classes of subgroups

Complements