Group #21424142 — subgroups

Subgroups — #21424142

Order 7552.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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	#16837440	C₂	1	2	yes	maximal, complemented
2	#16837376	C₂	1	2	yes	maximal
3	#16837159	C₂	1	2	yes	maximal, complemented
4	C₂×#131×C₅₉	C₂	1	1	yes	maximal, characteristic
5	#14273686	C₂²	1	4	yes	—
6	#14273661	—	2	2	no	—
7	#14273659	—	2	2	no	—
8	#14273642	—	2	4	no	—
9	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
10	#131×C₅₉	—	2	2	no	—
11	#131×C₅₉	—	2	2	no	—
12	C₂×M₁₆×C₅₉	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₅₉	C₄	1	2	yes	complemented
15	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	#41×C₅₉	—	2	2	no	—
17	#41×C₅₉	—	2	2	no	—
18	C₂×Q₈×C₅₉	Q₈	1	1	yes	characteristic
19	C₂×D₄×C₅₉	D₄	1	2	yes	—
20	C₂×D₄×C₅₉	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
23	M₁₆×C₅₉	—	2	2	no	—
24	M₁₆×C₅₉	—	2	4	no	—
25	C₂×C₈×C₅₉	—	2	2	no	—
26	C₂×C₈×C₅₉	—	2	2	no	—
27	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₉	D₄	1	2	yes	—
31	C₂×C₈×C₅₉	D₄	1	2	yes	—
32	C₂²⋊Dic₅₉	—	2	4	no	—
33	C₄⋊Dic₅₉	—	2	4	no	—
34	C₄×Dic₅₉	—	2	4	no	—
35	C₂³×C₅₉	C₄⋊C₄	1	2	yes	—
36	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₅₉	—	2	2	no	—
38	D₄×C₅₉	—	2	4	no	—
39	D₄×C₅₉	—	2	2	no	—
40	C₂×C₄×C₅₉	—	2	2	no	—
41	C₂×C₄×C₅₉	—	2	2	no	—
42	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₅₉	#41	1	2	yes	—
47	C₂×Dic₅₉	—	4	4	no	—
48	C₈×C₅₉	—	2	2	no	—
49	C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₅₉	—	2	2	no	—
53	C₂²×C₅₉	—	4	1	no	—
54	C₂²×C₅₉	—	4	2	no	—
55	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₅₉	#121	1	2	yes	—
57	C₄×C₅₉	—	2	2	no	—
58	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₅₉	#117	1	1	yes	characteristic
60	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
61	Dic₅₉	—	8	4	no	—
62	#1812	—	59	1	no	maximal, Sylow
63	C₂×C₅₉	—	2	2	no	—
64	C₂×C₅₉	—	2	1	no	—
65	C₂×C₅₉	—	4	2	no	—
66	C₂×C₅₉	#380	1	1	yes	characteristic
67	C₂×#131	D₅₉	1	1	yes	characteristic
68	#427	—	59	2	no	—
69	#410	—	59	2	no	—
70	H(3,ℤ/4)	—	59	2	no	—
71	C₅₉	#1812	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
73	#131	—	2	2	no	—
74	#131	—	2	2	no	—
75	C₂×M₁₆	Dic₅₉	1	2	yes	complemented
76	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
77	C₂²×C₈	Dic₅₉	1	2	yes	complemented
78	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
79	#117	—	59	4	no	—
80	#107	—	118	2	no	—
81	#106	—	118	2	no	—
82	#99	—	118	4	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	Dic₁₁₈	1	1	yes	characteristic
86	C₂×D₄	Dic₁₁₈	1	1	yes	characteristic
87	C₂×D₄	D₂₃₆	1	2	yes	—
88	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
89	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
95	C₂×C₈	C₂×Dic₅₉	1	1	yes	characteristic
96	C₂×C₈	C₂×Dic₅₉	1	1	yes	characteristic
97	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
98	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
99	C₄⋊C₄	—	118	4	no	—
100	C₂²⋊C₄	—	118	4	no	—
101	C₄²	—	118	4	no	—
102	C₂³	C₄⋊Dic₅₉	1	2	yes	—
103	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	236	4	no	—
110	C₂×C₄	C₄⋊Dic₅₉	1	2	yes	—
111	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
112	C₂×C₄	#11745685	1	2	yes	—
113	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
115	C₈	—	2	2	no	—
116	C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
117	C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
118	C₈	C₂²⋊Dic₅₉	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	2	no	—
122	C₂²	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
123	C₂²	#14273701	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	—	472	4	no	—
126	C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
127	C₄	#14273686	1	1	yes	characteristic
128	C₄	#14273706	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	2	1	no	—
131	C₂	—	4	2	no	—
132	C₂	#16837262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837440

C₂

yes

maximal, complemented

#16837376

C₂

yes

maximal

#16837159

C₂

yes

maximal, complemented

C₂×#131×C₅₉

C₂

yes

maximal, characteristic

#14273686

C₂²

yes

—

#14273661

—

#14273659

—

#14273642

—

C₂×#41×C₅₉

C₂²

yes

characteristic

#131×C₅₉

—

#131×C₅₉

—

C₂×M₁₆×C₅₉

C₄

yes

complemented

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₄

yes

complemented

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#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

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

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

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

—

C₈×C₅₉

—

C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₈×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

#121

yes

—

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

#380

yes

characteristic

C₂×#131

D₅₉

yes

—

—

—

C₅₉

#1812

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₉

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₅₉

yes

complemented

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

Dic₅₉

yes

complemented

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

#107

—

118

—

#106

—

118

—

#99

—

118

—

#41

—

#41

—

C₂×Q₈

Dic₁₁₈

yes

characteristic

C₂×D₄

Dic₁₁₈

yes

characteristic

C₂×D₄

D₂₃₆

yes

—

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂×Dic₅₉

yes

characteristic

C₂×C₈

C₂×Dic₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

—

C₂×C₈

C₂²⋊D₅₉

yes

—

C₄⋊C₄

—

118

—

100

C₂²⋊C₄

—

118

—

101

C₄²

—

118

—

102

C₂³

C₄⋊Dic₅₉

yes

—

103

C₂³

C₂×D₂₃₆

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

236

—

110

C₂×C₄

C₄⋊Dic₅₉

yes

—

111

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

112

C₂×C₄

#11745685

yes

—

113

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

115

C₈

—

116

C₈

C₂²⋊Dic₅₉

yes

characteristic

117

C₈

C₂²⋊Dic₅₉

yes

characteristic

118

C₈

C₂²⋊Dic₅₉

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₄⋊Dic₅₉

yes

characteristic

123

C₂²

#14273701

yes

—

124

C₄

—

125

C₄

—

472

—

126

C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

127

C₄

#14273686

yes

characteristic

128

C₄

#14273706

yes

—

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

#16837262

yes

characteristic

	subgroup	G/H
center	C₄	#14273686
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#131×C₅₉	C₂
socle	C₂×C₅₉	#380
hypercenter	C₂×#131	D₅₉
hypocenter	C₅₉	#1812
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#14273686

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#131×C₅₉

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. #16837440	C₂	semidirect
3. #16837159	C₂	semidirect
12. C₂×M₁₆×C₅₉	C₄	semidirect
14. C₂²×C₈×C₅₉	C₄	semidirect
71. C₅₉	#1812	semidirect
75. C₂×M₁₆	Dic₅₉	semidirect
77. C₂²×C₈	Dic₅₉	semidirect

H ≅ G/N

type

1. #16837440

C₂

semidirect

3. #16837159

C₂

semidirect

12. C₂×M₁₆×C₅₉

C₄

semidirect

14. C₂²×C₈×C₅₉

C₄

semidirect

71. C₅₉

#1812

semidirect

75. C₂×M₁₆

Dic₅₉

semidirect

77. C₂²×C₈

Dic₅₉

semidirect

Subgroups — #21424142

Conjugacy classes of subgroups

Complements