Group #22561972 — subgroups

Subgroups — C₂×#16863217 #22561972

Order 8576.

168 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 109 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×#14523983	C₂	1	2	yes	maximal
2	#16863217	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#14523983	—	2	2	no	—
5	#14523983	C₂²	1	4	yes	—
6	C₂×C₆₇⋊C₁₆	—	2	2	no	—
7	C₂²×C₈×C₆₇	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₆₇	C₄	1	1	yes	characteristic
10	C₄×C₈×C₆₇	C₄	1	2	yes	—
11	C₄×C₈×C₆₇	C₂²	1	2	yes	—
12	C₆₇⋊C₁₆	—	2	4	no	—
13	C₂²×C₄×C₆₇	—	2	1	no	—
14	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₆₇	C₈	1	2	yes	—
16	C₂×C₈×C₆₇	C₈	1	2	yes	—
17	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
23	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
24	C₄²×C₆₇	—	2	1	no	—
25	C₄²×C₆₇	C₂×C₄	1	2	yes	—
26	C₂³×C₆₇	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₆₇	—	2	4	no	—
28	C₂×C₄×C₆₇	—	2	2	no	—
29	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₆₇	M₁₆	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
36	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
37	C₈×C₆₇	C₂×C₈	1	4	yes	—
38	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₆₇	—	2	2	no	—
41	C₂²×C₆₇	C₄⋊C₈	1	1	yes	characteristic
42	C₂²×C₆₇	C₄⋊C₈	1	1	yes	characteristic
43	C₂²×C₆₇	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₆₇	—	2	4	no	—
45	C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
46	C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
47	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₆₇	—	2	1	no	—
50	C₂×C₆₇	—	2	1	no	—
51	C₂×C₆₇	#364	1	2	yes	complemented
52	C₂×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
53	C₂×#364	—	67	1	no	maximal, Sylow
54	C₆₇	C₂×#364	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₃₂	—	67	2	no	—
56	C₂×C₄×C₈	D₆₇	1	1	yes	characteristic
57	#364	—	67	4	no	—
58	C₂²×C₈	Dic₆₇	1	1	yes	characteristic
59	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
60	C₂×C₄²	Dic₆₇	1	1	yes	characteristic
61	M₃₂	—	67	4	no	—
62	M₃₂	—	134	2	no	—
63	C₂×C₁₆	—	134	2	no	—
64	C₄×C₈	Dic₆₇	1	2	yes	—
65	C₄×C₈	C₂×D₆₇	1	2	yes	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
68	C₂×C₈	C₆₇⋊C₈	1	2	yes	—
69	C₂×C₈	C₆₇⋊C₈	1	2	yes	—
70	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
71	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
72	C₂×C₈	D₂₆₈	1	1	yes	characteristic
73	C₂×C₈	D₂₆₈	1	1	yes	characteristic
74	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
75	C₂×C₈	C₂×Dic₆₇	1	2	yes	—
76	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
77	C₄²	—	2	1	no	—
78	C₄²	C₂×Dic₆₇	1	2	yes	—
79	C₁₆	—	134	4	no	—
80	C₂³	#11761612	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
84	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
85	C₂×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
86	C₂×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
87	C₂×C₄	#11761612	1	1	yes	characteristic
88	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
89	C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
90	C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
91	C₈	C₂×C₆₇⋊C₈	1	4	yes	—
92	C₈	C₂×D₂₆₈	1	1	yes	characteristic
93	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
96	C₂²	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
97	C₂²	C₂×#11761612	1	1	yes	characteristic
98	C₄	—	2	4	no	—
99	C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
100	C₄	C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
101	C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
102	C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	#16863217	1	2	yes	direct product factor
106	C₂	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523983

C₂

yes

maximal

#16863217

C₂

yes

maximal, direct product factor

C₂×C₄×C₈×C₆₇

C₂

yes

maximal, characteristic

#14523983

—

#14523983

C₂²

yes

—

C₂×C₆₇⋊C₁₆

—

C₂²×C₈×C₆₇

C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₄

yes

characteristic

C₄×C₈×C₆₇

C₄

yes

—

C₄×C₈×C₆₇

C₂²

yes

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₈

yes

—

C₂×C₈×C₆₇

C₈

yes

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₂×C₄

yes

—

C₂³×C₆₇

M₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

M₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₂×C₈

yes

—

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

C₄⋊C₈

yes

characteristic

C₂²×C₆₇

C₄⋊C₈

yes

characteristic

C₂²×C₆₇

C₂×M₁₆

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₄⋊C₈

yes

characteristic

C₄×C₆₇

C₄⋊C₈

yes

characteristic

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#364

yes

complemented

C₂×C₆₇

C₂×C₄⋊C₈

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×C₄×C₈

D₆₇

yes

characteristic

#364

—

C₂²×C₈

Dic₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄²

Dic₆₇

yes

characteristic

M₃₂

—

M₃₂

—

134

—

C₂×C₁₆

—

134

—

C₄×C₈

Dic₆₇

yes

—

C₄×C₈

C₂×D₆₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₆₇⋊C₈

yes

—

C₂×C₈

C₆₇⋊C₈

yes

—

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

D₂₆₈

yes

characteristic

C₂×C₈

D₂₆₈

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄²

—

C₄²

C₂×Dic₆₇

yes

—

C₁₆

—

134

—

C₂³

#11761612

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

C₂×C₆₇⋊C₈

yes

characteristic

C₂×C₄

C₂×C₆₇⋊C₈

yes

characteristic

C₂×C₄

#11761612

yes

characteristic

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

C₈

C₄⋊Dic₆₇

yes

characteristic

C₈

C₄⋊Dic₆₇

yes

characteristic

C₈

C₂×C₆₇⋊C₈

yes

—

C₈

C₂×D₂₆₈

yes

characteristic

C₈

C₂×Dic₁₃₄

yes

characteristic

C₂²

—

C₂²

C₄⋊(C₆₇⋊C₈)

yes

characteristic

C₂²

C₄⋊(C₆₇⋊C₈)

yes

characteristic

C₂²

C₂×#11761612

yes

characteristic

C₄

—

C₄

C₄⋊(C₆₇⋊C₈)

yes

characteristic

100

C₄

C₄⋊(C₆₇⋊C₈)

yes

characteristic

101

C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

102

C₄

C₂²×C₆₇⋊C₈

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

#16863217

yes

direct product factor

106

C₂

C₂×C₄⋊(C₆₇⋊C₈)

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₂₆₈
commutator	C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	C₂×C₄×C₈×C₆₇	C₂
socle	C₂²×C₆₇	C₄⋊C₈
hypercenter	C₂×C₄×C₈	D₆₇
hypocenter	C₆₇	C₂×#364
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

D₂₆₈

commutator

C₄×C₆₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

C₂×C₄×C₈×C₆₇

C₂

socle

C₂²×C₆₇

C₄⋊C₈

hypercenter

C₂×C₄×C₈

D₆₇

hypocenter

C₆₇

C₂×#364

LCS

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16863217	C₂	direct (N × H)
51. C₂×C₆₇	#364	semidirect
54. C₆₇	C₂×#364	semidirect
105. C₂	#16863217	direct (N × H)

H ≅ G/N

type

2. #16863217

C₂

direct (N × H)

51. C₂×C₆₇

#364

semidirect

54. C₆₇

C₂×#364

semidirect

105. C₂

#16863217

direct (N × H)

Subgroups — C₂×#16863217 #22561972

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863217 #22561972

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863217 #22561972