Group #21456159 — subgroups

Subgroups — C₂×#16840131 #21456159

Order 7808.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14519777	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519769	C₂	1	1	yes	maximal, characteristic
3	#16840131	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14519702	C₂	1	4	yes	maximal
5	C₂×#97×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	2	yes	—
7	#14519777	—	2	2	no	—
8	#14519777	C₂²	1	4	yes	—
9	#14519769	—	2	2	no	—
10	#14519769	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₆₁	—	2	2	no	—
12	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
13	C₂×#11758196	—	2	2	no	—
14	#14519702	C₂²	1	16	yes	—
15	C₂²×C₈×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	#97×C₆₁	C₂²	1	4	yes	—
18	C₂²×C₄×C₆₁	D₄	1	2	yes	—
19	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
20	C₂²×Dic₆₁	—	2	2	no	—
21	C₂²×Dic₆₁	—	2	2	no	—
22	C₂×Dic₁₂₂	—	2	8	no	—
23	C₂×Dic₁₂₂	—	2	2	no	—
24	C₂×Dic₁₂₂	C₂³	1	4	yes	—
25	C₂×C₈×C₆₁	—	2	4	no	—
26	C₂×C₈×C₆₁	C₂³	1	4	yes	—
27	C₄²×C₆₁	D₄	1	2	yes	—
28	C₄²×C₆₁	C₂³	1	2	yes	—
29	C₄⋊Dic₆₁	—	2	8	no	—
30	C₄⋊Dic₆₁	—	2	2	no	—
31	C₄⋊Dic₆₁	C₂³	1	4	yes	—
32	#11758196	—	2	8	no	—
33	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	—	2	4	no	—
35	C₂×C₄×C₆₁	C₂×D₄	1	4	yes	—
36	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	#41	1	4	yes	—
38	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
39	C₂×Dic₆₁	—	2	4	no	—
40	C₂×Dic₆₁	—	2	4	no	—
41	C₂×Dic₆₁	—	4	4	no	—
42	C₂×Dic₆₁	—	4	4	no	—
43	Dic₁₂₂	—	2	8	no	—
44	Dic₁₂₂	—	4	4	no	—
45	C₈×C₆₁	—	2	4	no	—
46	C₂²×C₆₁	#124	1	2	yes	—
47	C₂²×C₆₁	#137	1	4	yes	—
48	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₆₁	—	2	4	no	—
50	C₄×C₆₁	#124	1	2	yes	—
51	C₄×C₆₁	C₂×#41	1	2	yes	—
52	Dic₆₁	—	4	4	no	—
53	Dic₆₁	—	4	4	no	—
54	C₂×#491	—	61	1	no	maximal, Sylow
55	C₂×C₆₁	#491	1	4	yes	complemented
56	C₂×C₆₁	C₂×#124	1	1	yes	characteristic
57	C₂×C₆₁	C₂×#137	1	2	yes	—
58	C₂×C₄⋊Q₈	—	61	1	no	—
59	C₂×#125	—	61	1	no	—
60	#491	—	61	8	no	—
61	C₂×#103	—	61	4	no	—
62	C₂×#97	D₆₁	1	1	yes	characteristic
63	C₆₁	C₂×#491	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₈	—	61	2	no	—
65	C₂²×C₈	C₂×D₆₁	1	2	yes	—
66	C₄⋊Q₈	—	61	4	no	—
67	C₄⋊Q₈	—	122	2	no	—
68	#125	—	61	4	no	—
69	#125	—	122	2	no	—
70	C₂×C₄⋊C₄	—	61	2	no	—
71	C₂×C₄⋊C₄	—	122	2	no	—
72	C₂×C₄⋊C₄	—	122	2	no	—
73	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
74	#103	—	61	16	no	—
75	#97	C₂×D₆₁	1	4	yes	—
76	C₂×Q₈	—	61	4	no	—
77	C₂×Q₈	—	122	2	no	—
78	C₂×Q₈	—	122	8	no	—
79	C₂²×C₄	—	122	2	no	—
80	C₂²×C₄	—	122	2	no	—
81	C₂²×C₄	D₂₄₄	1	2	yes	—
82	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	C₂²×D₆₁	1	4	yes	—
85	C₄⋊C₄	—	61	4	no	—
86	C₄⋊C₄	—	122	2	no	—
87	C₄⋊C₄	—	122	8	no	—
88	C₄⋊C₄	—	122	8	no	—
89	C₄²	D₂₄₄	1	2	yes	—
90	C₄²	C₂²×D₆₁	1	2	yes	—
91	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
92	Q₈	—	122	8	no	—
93	Q₈	—	244	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	122	4	no	—
96	C₂×C₄	—	122	4	no	—
97	C₂×C₄	—	244	4	no	—
98	C₂×C₄	—	244	4	no	—
99	C₂×C₄	C₂×D₂₄₄	1	4	yes	—
100	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
101	C₂×C₄	#11758222	1	4	yes	—
102	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂²	#14519765	1	2	yes	—
105	C₂²	#14519810	1	4	yes	—
106	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	244	4	no	—
109	C₄	—	244	4	no	—
110	C₄	#14519765	1	2	yes	—
111	C₄	C₂×#11758222	1	2	yes	—
112	C₂	#16840131	1	4	yes	direct product factor
113	C₂	C₂×#14519765	1	1	yes	characteristic
114	C₂	C₂×#14519810	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519777

C₂

yes

maximal, characteristic

C₂×#14519769

C₂

yes

maximal, characteristic

#16840131

C₂

yes

maximal, direct product factor

C₂×#14519702

C₂

yes

maximal

C₂×#97×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

—

#14519777

—

#14519777

C₂²

yes

—

#14519769

—

#14519769

C₂²

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

—

#14519702

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#97×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂³

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₂²×C₆₁

#124

yes

—

C₂²×C₆₁

#137

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

#124

yes

—

C₄×C₆₁

C₂×#41

yes

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₄⋊Q₈

—

C₂×#125

—

#491

—

C₂×#103

—

C₂×#97

D₆₁

yes

characteristic

C₆₁

C₂×#491

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₆₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

#125

—

#125

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#103

—

#97

C₂×D₆₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

D₂₄₄

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

D₂₄₄

yes

—

C₄²

C₂²×D₆₁

yes

—

C₂³

C₂×D₂₄₄

yes

characteristic

Q₈

—

122

—

Q₈

—

244

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

C₂×C₄

C₂×D₂₄₄

yes

—

100

C₂×C₄

C₂×D₂₄₄

yes

characteristic

101

C₂×C₄

#11758222

yes

—

102

C₂×C₄

C₂³×D₆₁

yes

characteristic

103

C₈

—

104

C₂²

#14519765

yes

—

105

C₂²

#14519810

yes

—

106

C₂²

C₂²×D₂₄₄

yes

characteristic

107

C₄

—

108

C₄

—

244

—

109

C₄

—

244

—

110

C₄

#14519765

yes

—

111

C₄

C₂×#11758222

yes

—

112

C₂

#16840131

yes

direct product factor

113

C₂

C₂×#14519765

yes

characteristic

114

C₂

C₂×#14519810

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×#97×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#97	D₆₁
hypocenter	C₆₁	C₂×#491
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×#97×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂×#97

D₆₁

hypocenter

C₆₁

C₂×#491

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
3. #16840131	C₂	direct (N × H)
55. C₂×C₆₁	#491	semidirect
63. C₆₁	C₂×#491	semidirect
112. C₂	#16840131	direct (N × H)

H ≅ G/N

type

3. #16840131

C₂

direct (N × H)

55. C₂×C₆₁

#491

semidirect

63. C₆₁

C₂×#491

semidirect

112. C₂

#16840131

direct (N × H)

Subgroups — C₂×#16840131 #21456159

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840131 #21456159

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840131 #21456159