Group #21462159 — subgroups

Subgroups — C₂×#16840685 #21462159

Order 7808.

168 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 82 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×#14519851	C₂	1	2	yes	maximal
2	#16840685	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×#92418	C₂	1	1	yes	maximal, characteristic
4	#14519851	—	2	2	no	—
5	#14519851	C₂²	1	4	yes	—
6	C₂×#11758187	—	2	2	no	—
7	C₂²×#92418	C₄	1	1	yes	characteristic
8	C₂²×#92418	C₂²	1	1	yes	characteristic
9	C₄×#92418	C₄	1	2	yes	—
10	C₄×#92418	C₂²	1	2	yes	—
11	C₂×C₄²×C₆₁	C₄	1	1	yes	characteristic
12	C₂²×C₄×C₆₁	—	2	1	no	—
13	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₄²×C₆₁	—	2	1	no	—
15	C₄²×C₆₁	C₂×C₄	1	2	yes	—
16	C₂×#92418	C₈	1	2	yes	—
17	C₂×#92418	C₈	1	2	yes	—
18	C₂×#92418	C₂×C₄	1	1	yes	characteristic
19	C₂×#92418	C₂×C₄	1	2	yes	—
20	C₂×#92418	D₄	1	1	yes	characteristic
21	C₂×#92418	D₄	1	1	yes	characteristic
22	C₂×#92418	Q₈	1	1	yes	characteristic
23	C₂×#92418	Q₈	1	1	yes	characteristic
24	C₂×#92418	C₂³	1	1	yes	characteristic
25	#11758187	—	2	4	no	—
26	C₂³×C₆₁	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₆₁	—	2	4	no	—
28	C₂×C₄×C₆₁	—	2	2	no	—
29	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₆₁	M₁₆	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	#92418	C₄⋊C₄	1	1	yes	characteristic
36	#92418	C₄⋊C₄	1	1	yes	characteristic
37	#92418	C₂×C₈	1	4	yes	—
38	#92418	C₂×D₄	1	1	yes	characteristic
39	#92418	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₆₁	—	2	2	no	—
41	C₂²×C₆₁	C₄⋊C₈	1	1	yes	characteristic
42	C₂²×C₆₁	C₄⋊C₈	1	1	yes	characteristic
43	C₂²×C₆₁	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₆₁	—	2	4	no	—
45	C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
46	C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
47	C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
49	C₂×#364	—	61	1	no	maximal, Sylow
50	C₂×C₆₁	—	2	1	no	—
51	C₂×C₆₁	—	2	1	no	—
52	C₂×C₆₁	#364	1	2	yes	complemented
53	C₂×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
54	C₂×M₃₂	—	61	2	no	—
55	C₂×C₄×C₈	—	61	1	no	—
56	#364	—	61	4	no	—
57	C₆₁	C₂×#364	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	—	61	1	no	—
59	C₂²×C₈	—	61	1	no	—
60	C₂×C₄²	#6923	1	1	yes	characteristic
61	M₃₂	—	61	4	no	—
62	M₃₂	—	122	2	no	—
63	C₂×C₁₆	—	122	2	no	—
64	C₄×C₈	—	61	2	no	—
65	C₄×C₈	—	61	2	no	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
68	C₂×C₈	—	61	2	no	—
69	C₂×C₈	—	61	2	no	—
70	C₂×C₈	—	61	1	no	—
71	C₂×C₈	—	61	1	no	—
72	C₂×C₈	—	61	1	no	—
73	C₂×C₈	—	61	1	no	—
74	C₂×C₈	—	61	1	no	—
75	C₂×C₈	—	61	1	no	—
76	C₂×C₈	—	61	2	no	—
77	C₄²	—	2	1	no	—
78	C₄²	C₂×#6923	1	2	yes	—
79	C₁₆	—	122	4	no	—
80	C₂³	#11758217	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	D₆₁⋊C₈	1	1	yes	characteristic
84	C₂×C₄	D₆₁⋊C₈	1	1	yes	characteristic
85	C₂×C₄	C₄⋊#6923	1	1	yes	characteristic
86	C₂×C₄	C₄⋊#6923	1	1	yes	characteristic
87	C₂×C₄	#11758217	1	1	yes	characteristic
88	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
89	C₈	—	61	1	no	—
90	C₈	—	61	1	no	—
91	C₈	—	61	1	no	—
92	C₈	—	61	1	no	—
93	C₈	—	61	4	no	—
94	C₂²	—	2	2	no	—
95	C₂²	#14519844	1	1	yes	characteristic
96	C₂²	#14519844	1	1	yes	characteristic
97	C₂²	C₂×#11758217	1	1	yes	characteristic
98	C₄	—	2	4	no	—
99	C₄	#14519844	1	1	yes	characteristic
100	C₄	#14519844	1	1	yes	characteristic
101	C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
102	C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	#16840685	1	2	yes	direct product factor
106	C₂	C₂×#14519844	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519851

C₂

yes

maximal

#16840685

C₂

yes

maximal, direct product factor

C₂×C₄×#92418

C₂

yes

maximal, characteristic

#14519851

—

#14519851

C₂²

yes

—

C₂×#11758187

—

C₂²×#92418

C₄

yes

characteristic

C₂²×#92418

C₂²

yes

characteristic

C₄×#92418

C₄

yes

—

C₄×#92418

C₂²

yes

—

C₂×C₄²×C₆₁

C₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

—

C₂×#92418

C₈

yes

—

C₂×#92418

C₈

yes

—

C₂×#92418

C₂×C₄

yes

characteristic

C₂×#92418

C₂×C₄

yes

—

C₂×#92418

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

C₂×#92418

Q₈

yes

characteristic

C₂×#92418

Q₈

yes

characteristic

C₂×#92418

C₂³

yes

characteristic

#11758187

—

C₂³×C₆₁

M₁₆

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂×C₄×C₆₁

M₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

#92418

C₄⋊C₄

yes

characteristic

#92418

C₄⋊C₄

yes

characteristic

#92418

C₂×C₈

yes

—

#92418

C₂×D₄

yes

characteristic

#92418

C₂×Q₈

yes

characteristic

C₂²×C₆₁

—

C₂²×C₆₁

C₄⋊C₈

yes

characteristic

C₂²×C₆₁

C₄⋊C₈

yes

characteristic

C₂²×C₆₁

C₂×M₁₆

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄⋊C₈

yes

characteristic

C₄×C₆₁

C₄⋊C₈

yes

characteristic

C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×#364

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#364

yes

complemented

C₂×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₂×M₃₂

—

C₂×C₄×C₈

—

#364

—

C₆₁

C₂×#364

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

#6923

yes

characteristic

M₃₂

—

M₃₂

—

122

—

C₂×C₁₆

—

122

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂×#6923

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

C₂×#6923

yes

—

C₁₆

—

122

—

C₂³

#11758217

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₆₁⋊C₈

yes

characteristic

C₂×C₄

D₆₁⋊C₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²×#6923

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

#14519844

yes

characteristic

100

C₄

#14519844

yes

characteristic

101

C₄

C₂×C₄⋊#6923

yes

characteristic

102

C₄

C₂×D₆₁⋊C₈

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

#16840685

yes

direct product factor

106

C₂

C₂×#14519844

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#6923
commutator	C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₂×C₄²×C₆₁	C₄
socle	C₂²×C₆₁	C₄⋊C₈
hypercenter	C₂×C₄²	#6923
hypocenter	C₆₁	C₂×#364
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂×C₄

C₄⋊#6923

commutator

C₄×C₆₁

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₂×C₄²×C₆₁

C₄

socle

C₂²×C₆₁

C₄⋊C₈

C₂×C₄²

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
2. #16840685	C₂	direct (N × H)
52. C₂×C₆₁	#364	semidirect
57. C₆₁	C₂×#364	semidirect
105. C₂	#16840685	direct (N × H)

H ≅ G/N

type

2. #16840685

C₂

direct (N × H)

52. C₂×C₆₁

#364

semidirect

57. C₆₁

C₂×#364

semidirect

105. C₂

#16840685

direct (N × H)

Subgroups — C₂×#16840685 #21462159

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840685 #21462159

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840685 #21462159