Group #21445988 — subgroups

Subgroups — #21445988

Order 7808.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×#14519710	C₂	1	2	yes	maximal
2	C₂×#14519708	C₂	1	2	yes	maximal
3	C₂×#14519688	C₂	1	1	yes	maximal, characteristic
4	#16839632	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#382	C₂	1	1	yes	maximal, characteristic
6	C₂²×#92418	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₁	—	2	2	no	—
8	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
9	#14519710	—	2	4	no	—
10	#14519708	—	2	4	no	—
11	#14519688	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
15	#95×C₆₁	—	2	2	no	—
16	C₂²×C₄×C₆₁	—	2	1	no	—
17	C₂²×C₄×C₆₁	—	2	1	no	—
18	C₂²×C₄×C₆₁	D₄	1	2	yes	—
19	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
20	C₂²×Dic₆₁	—	2	2	no	—
21	C₄⋊C₄×C₆₁	—	2	2	no	—
22	C₄⋊C₄×C₆₁	—	2	2	no	—
23	C₄²×C₆₁	C₂×C₄	1	2	yes	—
24	C₄²×C₆₁	Q₈	1	2	yes	—
25	C₄⋊Dic₆₁	—	2	2	no	—
26	C₄⋊Dic₆₁	—	2	2	no	—
27	C₄⋊Dic₆₁	—	4	4	no	—
28	C₂×#92418	C₂×C₄	1	4	yes	—
29	C₂×#92418	D₄	1	4	yes	—
30	C₂×#92418	Q₈	1	4	yes	—
31	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₆₁	—	2	2	no	—
33	C₂×C₄×C₆₁	—	2	2	no	—
34	C₂×C₄×C₆₁	—	4	1	no	—
35	C₂×C₄×C₆₁	—	4	1	no	—
36	C₂×C₄×C₆₁	—	4	2	no	—
37	C₂×C₄×C₆₁	—	4	2	no	—
38	C₂×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
39	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×Dic₆₁	—	4	2	no	—
44	C₂×Dic₆₁	—	4	4	no	—
45	#92418	C₄⋊C₄	1	8	yes	complemented
46	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₆₁	#127	1	1	yes	characteristic
48	C₂²×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₆₁	#137	1	2	yes	—
51	C₄×C₆₁	—	2	4	no	—
52	C₄×C₆₁	—	8	1	no	—
53	C₄×C₆₁	—	8	1	no	—
54	C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
55	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
56	Dic₆₁	—	8	2	no	—
57	#1847	—	61	1	no	maximal, Sylow
58	C₂×C₆₁	#384	1	1	yes	characteristic
59	C₂×C₆₁	#428	1	2	yes	—
60	C₂×C₆₁	#496	1	1	yes	characteristic
61	C₂×C₆₁	#497	1	1	yes	characteristic
62	C₂×C₆₁	#501	1	2	yes	—
63	C₂×#107	—	61	2	no	—
64	C₂×#106	—	61	2	no	—
65	C₂×#97	—	61	1	no	—
66	#384	—	61	1	no	—
67	#382	D₆₁	1	1	yes	characteristic
68	C₆₁	#1847	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	61	2	no	—
70	C₂×C₄⋊C₄	—	61	2	no	—
71	C₂×C₄⋊C₄	—	122	2	no	—
72	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
75	#107	—	122	4	no	—
76	#106	—	122	4	no	—
77	#97	—	61	4	no	—
78	#95	—	2	2	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	122	2	no	—
82	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
83	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
84	C₂×C₈	—	61	4	no	—
85	C₂×C₈	—	61	4	no	—
86	C₂×C₈	—	61	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	122	2	no	—
90	C₄⋊C₄	—	122	2	no	—
91	C₄⋊C₄	—	244	4	no	—
92	C₄²	Dic₁₂₂	1	2	yes	—
93	C₄²	C₄×D₆₁	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	244	2	no	—
102	C₂×C₄	—	244	4	no	—
103	C₂×C₄	#11758196	1	4	yes	—
104	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
106	C₂×C₄	D₄×D₆₁	1	2	yes	—
107	C₂×C₄	Q₈×D₆₁	1	2	yes	—
108	C₈	—	61	8	no	—
109	C₂²	C₂×#11758196	1	1	yes	characteristic
110	C₂²	#14519773	1	1	yes	characteristic
111	C₂²	#14519775	1	1	yes	characteristic
112	C₂²	#14519802	1	2	yes	—
113	C₂²	#14519807	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	488	2	no	—
118	C₄	C₄⋊C₄×D₆₁	1	2	yes	—
119	C₄	#14519776	1	2	yes	—
120	C₂	#16839631	1	1	yes	characteristic
121	C₂	#16839805	1	2	yes	—
122	C₂	#16840152	1	1	yes	characteristic
123	C₂	#16840157	1	1	yes	characteristic
124	C₂	#16840172	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519710

C₂

yes

maximal

C₂×#14519708

C₂

yes

maximal

C₂×#14519688

C₂

yes

maximal, characteristic

#16839632

C₂

yes

maximal, characteristic

C₆₁×#382

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

#14519710

—

#14519708

—

#14519688

C₄

yes

complemented

C₂×C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

Q₈

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

C₂×C₄

yes

—

C₂×#92418

D₄

yes

—

C₂×#92418

Q₈

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×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

—

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

#92418

C₄⋊C₄

yes

complemented

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

#127

yes

characteristic

C₂²×C₆₁

C₄⋊Q₈

yes

characteristic

C₂²×C₆₁

AGL(1,ℤ/8)

yes

—

C₂²×C₆₁

#137

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₄×C₆₁

C₄⋊Q₈

yes

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×#107

—

C₂×#106

—

C₂×#97

—

#384

—

#382

D₆₁

yes

characteristic

C₆₁

#1847

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₆₁

yes

characteristic

C₂×C₄²

C₂×D₆₁

yes

characteristic

#107

—

122

—

#106

—

122

—

#97

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₄²

Dic₁₂₂

yes

—

C₄²

C₄×D₆₁

yes

—

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

244

—

102

C₂×C₄

—

244

—

103

C₂×C₄

#11758196

yes

—

104

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

105

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

106

C₂×C₄

D₄×D₆₁

yes

—

107

C₂×C₄

Q₈×D₆₁

yes

—

108

C₈

—

109

C₂²

C₂×#11758196

yes

characteristic

110

C₂²

#14519773

yes

characteristic

111

C₂²

#14519775

yes

characteristic

112

C₂²

#14519802

yes

—

113

C₂²

#14519807

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

488

—

118

C₄

C₄⋊C₄×D₆₁

yes

—

119

C₄

#14519776

yes

—

120

C₂

#16839631

yes

characteristic

121

C₂

#16839805

yes

—

122

C₂

#16840152

yes

characteristic

123

C₂

#16840157

yes

characteristic

124

C₂

#16840172

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#382	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	#382	D₆₁
hypocenter	C₆₁	#1847
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#382

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
11. #14519688	C₄	semidirect
45. #92418	C₄⋊C₄	semidirect
68. C₆₁	#1847	semidirect

H ≅ G/N

type

11. #14519688

C₄

semidirect

45. #92418

C₄⋊C₄

semidirect

68. C₆₁

#1847

semidirect

Subgroups — #21445988

Conjugacy classes of subgroups

Complements