Group #21445865 — subgroups

Subgroups — #21445865

Order 7808.

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	#16839800	C₂	1	2	yes	maximal, complemented
2	#16839736	C₂	1	2	yes	maximal
3	#16839519	C₂	1	2	yes	maximal, complemented
4	C₂×#131×C₆₁	C₂	1	1	yes	maximal, characteristic
5	#14519736	C₂²	1	4	yes	—
6	#14519711	—	2	2	no	—
7	#14519709	—	2	2	no	—
8	#14519692	—	2	4	no	—
9	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
10	#131×C₆₁	—	2	2	no	—
11	#131×C₆₁	—	2	2	no	—
12	C₂×M₁₆×C₆₁	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₆₁	C₄	1	2	yes	complemented
15	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	#41×C₆₁	—	2	2	no	—
17	#41×C₆₁	—	2	2	no	—
18	C₂×Q₈×C₆₁	Q₈	1	1	yes	characteristic
19	C₂×D₄×C₆₁	D₄	1	2	yes	—
20	C₂×D₄×C₆₁	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
23	M₁₆×C₆₁	—	2	2	no	—
24	M₁₆×C₆₁	—	2	4	no	—
25	C₂×C₈×C₆₁	—	2	2	no	—
26	C₂×C₈×C₆₁	—	2	2	no	—
27	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₆₁	D₄	1	2	yes	—
31	C₂×C₈×C₆₁	D₄	1	2	yes	—
32	C₂²⋊Dic₆₁	—	2	4	no	—
33	C₄⋊Dic₆₁	—	2	4	no	—
34	C₄×Dic₆₁	—	2	4	no	—
35	C₂³×C₆₁	C₄⋊C₄	1	2	yes	—
36	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₆₁	—	2	2	no	—
38	D₄×C₆₁	—	2	4	no	—
39	D₄×C₆₁	—	2	2	no	—
40	C₂×C₄×C₆₁	—	2	2	no	—
41	C₂×C₄×C₆₁	—	2	2	no	—
42	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₆₁	#41	1	2	yes	—
47	C₂×Dic₆₁	—	4	4	no	—
48	C₈×C₆₁	—	2	2	no	—
49	C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₆₁	—	2	2	no	—
53	C₂²×C₆₁	—	4	1	no	—
54	C₂²×C₆₁	—	4	2	no	—
55	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₆₁	#121	1	2	yes	—
57	C₄×C₆₁	—	2	2	no	—
58	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₆₁	#117	1	1	yes	characteristic
60	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
61	Dic₆₁	—	8	4	no	—
62	#1812	—	61	1	no	maximal, Sylow
63	C₂×C₆₁	—	2	2	no	—
64	C₂×C₆₁	—	2	1	no	—
65	C₂×C₆₁	—	4	2	no	—
66	C₂×C₆₁	#380	1	1	yes	characteristic
67	C₂×#131	D₆₁	1	1	yes	characteristic
68	#427	—	61	2	no	—
69	#410	—	61	2	no	—
70	H(3,ℤ/4)	—	61	2	no	—
71	C₆₁	#1812	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
73	#131	—	2	2	no	—
74	#131	—	2	2	no	—
75	C₂×M₁₆	Dic₆₁	1	2	yes	complemented
76	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
77	C₂²×C₈	Dic₆₁	1	2	yes	complemented
78	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
79	#117	—	61	4	no	—
80	#107	—	122	2	no	—
81	#106	—	122	2	no	—
82	#99	—	122	4	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	Dic₁₂₂	1	1	yes	characteristic
86	C₂×D₄	Dic₁₂₂	1	1	yes	characteristic
87	C₂×D₄	D₂₄₄	1	2	yes	—
88	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
89	C₂²×C₄	C₂²×D₆₁	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×Dic₆₁	1	2	yes	—
95	C₂×C₈	C₂×Dic₆₁	1	1	yes	characteristic
96	C₂×C₈	C₂×Dic₆₁	1	1	yes	characteristic
97	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
98	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
99	C₄⋊C₄	—	122	4	no	—
100	C₂²⋊C₄	—	122	4	no	—
101	C₄²	—	122	4	no	—
102	C₂³	C₄⋊Dic₆₁	1	2	yes	—
103	C₂³	C₂×D₂₄₄	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	—	244	4	no	—
110	C₂×C₄	C₄⋊Dic₆₁	1	2	yes	—
111	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
112	C₂×C₄	#11758222	1	2	yes	—
113	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
115	C₈	—	2	2	no	—
116	C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
117	C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
118	C₈	C₂²⋊Dic₆₁	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	2	no	—
122	C₂²	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
123	C₂²	#14519751	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	—	488	4	no	—
126	C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
127	C₄	#14519736	1	1	yes	characteristic
128	C₄	#14519756	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	2	1	no	—
131	C₂	—	4	2	no	—
132	C₂	#16839622	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839800

C₂

yes

maximal, complemented

#16839736

C₂

yes

maximal

#16839519

C₂

yes

maximal, complemented

C₂×#131×C₆₁

C₂

yes

maximal, characteristic

#14519736

C₂²

yes

—

#14519711

—

#14519709

—

#14519692

—

C₂×#41×C₆₁

C₂²

yes

characteristic

#131×C₆₁

—

#131×C₆₁

—

C₂×M₁₆×C₆₁

C₄

yes

complemented

C₂×M₁₆×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₄

yes

complemented

C₂²×C₈×C₆₁

C₂²

yes

characteristic

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

Q₈

yes

characteristic

C₂×D₄×C₆₁

D₄

yes

—

C₂×D₄×C₆₁

Q₈

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

D₄

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₂³×C₆₁

C₄⋊C₄

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₈×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

#121

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

#117

yes

characteristic

C₄×C₆₁

C₂²⋊Q₈

yes

—

Dic₆₁

—

#1812

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#380

yes

characteristic

C₂×#131

D₆₁

yes

—

—

—

C₆₁

#1812

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₆₁

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₆₁

yes

complemented

C₂×M₁₆

C₂×D₆₁

yes

characteristic

C₂²×C₈

Dic₆₁

yes

complemented

C₂²×C₈

C₂×D₆₁

yes

characteristic

#117

—

#107

—

122

—

#106

—

122

—

#99

—

122

—

#41

—

#41

—

C₂×Q₈

Dic₁₂₂

yes

characteristic

C₂×D₄

Dic₁₂₂

yes

characteristic

C₂×D₄

D₂₄₄

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₆₁

yes

—

C₂×C₈

C₂×Dic₆₁

yes

characteristic

C₂×C₈

C₂×Dic₆₁

yes

characteristic

C₂×C₈

C₂²⋊D₆₁

yes

—

C₂×C₈

C₂²⋊D₆₁

yes

—

C₄⋊C₄

—

122

—

100

C₂²⋊C₄

—

122

—

101

C₄²

—

122

—

102

C₂³

C₄⋊Dic₆₁

yes

—

103

C₂³

C₂×D₂₄₄

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

244

—

110

C₂×C₄

C₄⋊Dic₆₁

yes

—

111

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

112

C₂×C₄

#11758222

yes

—

113

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

115

C₈

—

116

C₈

C₂²⋊Dic₆₁

yes

characteristic

117

C₈

C₂²⋊Dic₆₁

yes

characteristic

118

C₈

C₂²⋊Dic₆₁

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₄⋊Dic₆₁

yes

characteristic

123

C₂²

#14519751

yes

—

124

C₄

—

125

C₄

—

488

—

126

C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

127

C₄

#14519736

yes

characteristic

128

C₄

#14519756

yes

—

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

#16839622

yes

characteristic

	subgroup	G/H
center	C₄	#14519736
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×#131×C₆₁	C₂
socle	C₂×C₆₁	#380
hypercenter	C₂×#131	D₆₁
hypocenter	C₆₁	#1812
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#14519736

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×#131×C₆₁

C₂

C₂×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. #16839800	C₂	semidirect
3. #16839519	C₂	semidirect
12. C₂×M₁₆×C₆₁	C₄	semidirect
14. C₂²×C₈×C₆₁	C₄	semidirect
71. C₆₁	#1812	semidirect
75. C₂×M₁₆	Dic₆₁	semidirect
77. C₂²×C₈	Dic₆₁	semidirect

H ≅ G/N

type

1. #16839800

C₂

semidirect

3. #16839519

C₂

semidirect

12. C₂×M₁₆×C₆₁

C₄

semidirect

14. C₂²×C₈×C₆₁

C₄

semidirect

71. C₆₁

#1812

semidirect

75. C₂×M₁₆

Dic₆₁

semidirect

77. C₂²×C₈

Dic₆₁

semidirect

Subgroups — #21445865

Conjugacy classes of subgroups

Complements