Group #21445546 — subgroups

Subgroups — #21445546

Order 7808.

372 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 189 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	C₂²×#11758194	C₂	1	2	yes	maximal
2	#16839527	C₂	1	4	yes	maximal, complemented
3	C₂²×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758194	—	2	4	no	—
5	C₂×#11758194	—	2	2	no	—
6	C₂×#11758194	C₄	1	8	yes	—
7	C₂×#11758194	C₂²	1	4	yes	—
8	C₂²×#92418	—	2	2	no	—
9	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₆₁	—	2	4	no	—
11	C₂×M₁₆×C₆₁	C₄	1	4	yes	—
12	C₂²×C₈×C₆₁	C₂²	1	2	yes	—
13	C₂⁴×C₆₁	Q₈	1	1	yes	characteristic
14	C₂²×C₄×C₆₁	C₂×C₄	1	4	yes	—
15	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₆₁	D₄	1	2	yes	—
17	C₂²×C₄×C₆₁	D₄	1	4	yes	—
18	C₂²×C₄×C₆₁	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
20	M₁₆×C₆₁	—	2	8	no	—
21	C₂×C₈×C₆₁	—	2	4	no	—
22	C₂×C₈×C₆₁	C₂×C₄	1	4	yes	—
23	#11758194	—	2	8	no	—
24	#11758194	—	2	4	no	—
25	#11758194	C₂×C₄	1	8	yes	—
26	C₂×#92418	—	2	8	no	—
27	C₂×#92418	—	2	4	no	—
28	C₂³×C₆₁	—	4	2	no	—
29	C₂³×C₆₁	C₄⋊C₄	1	2	yes	—
30	C₂³×C₆₁	C₄⋊C₄	1	4	yes	—
31	C₂³×C₆₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₆₁	C₄²	1	4	yes	—
33	C₂×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
36	C₂×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
42	C₈×C₆₁	—	2	4	no	—
43	#92418	—	2	8	no	—
44	C₂²×C₆₁	—	2	4	no	—
45	C₂²×C₆₁	—	2	2	no	—
46	C₂²×C₆₁	—	4	4	no	—
47	C₂²×C₆₁	#95	1	4	yes	—
48	C₂²×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
49	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₆₁	#95	1	4	yes	—
51	C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
52	C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
53	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	#1738	—	61	1	no	maximal, Sylow
55	C₂×C₆₁	—	2	4	no	—
56	C₂×C₆₁	—	2	2	no	—
57	C₂×C₆₁	C₂×#95	1	1	yes	characteristic
58	C₂×C₆₁	#430	1	2	yes	—
59	C₂²×M₁₆	—	61	2	no	—
60	C₂²×M₁₆	D₆₁	1	1	yes	characteristic
61	#341	—	61	4	no	—
62	C₆₁	#1738	1	1	yes	characteristic, complemented, Sylow
63	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
64	C₂×M₁₆	—	2	4	no	—
65	C₂×M₁₆	—	61	8	no	—
66	C₂×M₁₆	—	61	4	no	—
67	C₂×M₁₆	—	122	2	no	—
68	C₂×M₁₆	—	122	4	no	—
69	C₂×M₁₆	Dic₆₁	1	4	yes	—
70	C₂²×C₈	—	122	2	no	—
71	C₂²×C₈	C₂×D₆₁	1	2	yes	—
72	C₂⁴	Dic₁₂₂	1	1	yes	characteristic
73	C₂²×C₄	Dic₁₂₂	1	1	yes	characteristic
74	C₂²×C₄	C₄×D₆₁	1	4	yes	—
75	C₂²×C₄	D₂₄₄	1	2	yes	—
76	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
77	C₂²×C₄	C₂²⋊D₆₁	1	4	yes	—
78	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
79	M₁₆	—	2	8	no	—
80	M₁₆	—	61	8	no	—
81	M₁₆	—	122	8	no	—
82	M₁₆	—	122	4	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	—	122	8	no	—
85	C₂×C₈	—	122	4	no	—
86	C₂×C₈	C₂×Dic₆₁	1	4	yes	—
87	C₂³	—	4	2	no	—
88	C₂³	#11758196	1	4	yes	—
89	C₂³	C₄⋊Dic₆₁	1	2	yes	—
90	C₂³	C₂×Dic₁₂₂	1	1	yes	characteristic
91	C₂×C₄	C₄×Dic₆₁	1	4	yes	—
92	C₂×C₄	#11758196	1	4	yes	—
93	C₂×C₄	C₄⋊Dic₆₁	1	2	yes	—
94	C₂×C₄	#11758198	1	4	yes	—
95	C₂×C₄	#11758198	1	4	yes	—
96	C₂×C₄	C₂²⋊Dic₆₁	1	4	yes	—
97	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
98	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
99	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
101	C₈	—	2	4	no	—
102	C₈	—	122	8	no	—
103	C₂²	—	2	4	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	4	4	no	—
106	C₂²	#14519684	1	4	yes	—
107	C₂²	C₂×#11758196	1	2	yes	—
108	C₂²	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
109	C₄	#14519684	1	4	yes	—
110	C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
111	C₄	C₂×#11758198	1	2	yes	—
112	C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
113	C₂	—	2	4	no	—
114	C₂	—	2	2	no	—
115	C₂	C₂×#14519684	1	1	yes	characteristic
116	C₂	#16839811	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11758194

C₂

yes

maximal

#16839527

C₂

yes

maximal, complemented

C₂²×M₁₆×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

—

C₂×#11758194

—

C₂×#11758194

C₄

yes

—

C₂×#11758194

C₂²

yes

—

C₂²×#92418

—

C₂³×C₄×C₆₁

C₂²

yes

characteristic

C₂×M₁₆×C₆₁

—

C₂×M₁₆×C₆₁

C₄

yes

—

C₂²×C₈×C₆₁

C₂²

yes

—

C₂⁴×C₆₁

Q₈

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

Q₈

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

#11758194

—

#11758194

—

#11758194

C₂×C₄

yes

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

—

C₂³×C₆₁

C₄⋊C₄

yes

—

C₂³×C₆₁

C₄⋊C₄

yes

—

C₂³×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

C₄²

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#95

yes

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

#95

yes

—

C₄×C₆₁

C₂×C₄²

yes

characteristic

C₄×C₆₁

C₂×C₂²⋊C₄

yes

—

C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

#1738

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₂×#95

yes

characteristic

C₂×C₆₁

#430

yes

—

C₂²×M₁₆

—

C₂²×M₁₆

D₆₁

yes

characteristic

#341

—

C₆₁

#1738

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₆₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

122

—

C₂×M₁₆

—

122

—

C₂×M₁₆

Dic₆₁

yes

—

C₂²×C₈

—

122

—

C₂²×C₈

C₂×D₆₁

yes

—

C₂⁴

Dic₁₂₂

yes

characteristic

C₂²×C₄

Dic₁₂₂

yes

characteristic

C₂²×C₄

C₄×D₆₁

yes

—

C₂²×C₄

D₂₄₄

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

122

—

M₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

C₂×Dic₆₁

yes

—

C₂³

—

C₂³

#11758196

yes

—

C₂³

C₄⋊Dic₆₁

yes

—

C₂³

C₂×Dic₁₂₂

yes

characteristic

C₂×C₄

C₄×Dic₆₁

yes

—

C₂×C₄

#11758196

yes

—

C₂×C₄

C₄⋊Dic₆₁

yes

—

C₂×C₄

#11758198

yes

—

C₂×C₄

#11758198

yes

—

C₂×C₄

C₂²⋊Dic₆₁

yes

—

C₂×C₄

C₂×C₄×D₆₁

yes

—

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

101

C₈

—

102

C₈

—

122

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

#14519684

yes

—

107

C₂²

C₂×#11758196

yes

—

108

C₂²

C₂×C₄⋊Dic₆₁

yes

characteristic

109

C₄

#14519684

yes

—

110

C₄

C₂×C₄×Dic₆₁

yes

characteristic

111

C₄

C₂×#11758198

yes

—

112

C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

C₂×#14519684

yes

characteristic

116

C₂

#16839811

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₄₄
commutator	C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂²×M₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₄⋊C₄
hypercenter	C₂²×M₁₆	D₆₁
hypocenter	C₆₁	#1738
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₂₄₄

commutator

C₄×C₆₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂²×M₁₆×C₆₁

C₂

socle

C₂²×C₆₁

C₂×C₄⋊C₄

hypercenter

C₂²×M₁₆

D₆₁

hypocenter

C₆₁

#1738

LCS

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
2. #16839527	C₂	semidirect
62. C₆₁	#1738	semidirect

H ≅ G/N

type

2. #16839527

C₂

semidirect

62. C₆₁

#1738

semidirect

Subgroups — #21445546

Conjugacy classes of subgroups

Complements