Group #22573215 — subgroups

Subgroups — #22573215

Order 8576.

348 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 141 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	#16864125	C₂	1	2	yes	maximal
2	#16863776	C₂	1	3	yes	maximal
3	#16863607	C₂	1	6	yes	maximal
4	C₄×Dic₂₆₈	C₂	1	3	yes	maximal
5	C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₆₈	C₂²	1	3	yes	—
7	C₂×Dic₂₆₈	C₂²	1	6	yes	—
8	#14524043	C₂²	1	6	yes	—
9	#14524034	—	2	6	no	—
10	Q₈×Dic₆₇	—	2	2	no	—
11	C₄×Dic₁₃₄	C₂²	1	6	yes	—
12	#14523977	C₂²	1	1	yes	characteristic
13	#14523968	C₂²	1	6	yes	—
14	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
16	C₄×C₈×C₆₇	C₂²	1	3	yes	—
17	C₂×Dic₁₃₄	C₂³	1	6	yes	—
18	Dic₂₆₈	—	2	6	no	—
19	Dic₂₆₈	—	4	6	no	—
20	C₄⋊Dic₆₇	—	2	6	no	—
21	C₄⋊Dic₆₇	—	2	6	no	—
22	C₄⋊Dic₆₇	C₂³	1	2	yes	—
23	Dic₆₇⋊C₄	—	2	6	no	—
24	C₄×Dic₆₇	—	2	6	no	—
25	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	C₂³	1	3	yes	—
27	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₆₇	D₄	1	3	yes	—
29	C₄²×C₆₇	C₂³	1	3	yes	—
30	Q₈×C₆₇	Q₁₆	1	4	yes	complemented
31	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
32	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
33	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
34	C₂×Dic₆₇	—	2	2	no	—
35	C₂×Dic₆₇	—	2	6	no	—
36	Dic₁₃₄	—	2	6	no	—
37	Dic₁₃₄	—	2	6	no	—
38	C₈×C₆₇	—	2	3	no	—
39	C₈×C₆₇	#41	1	2	yes	—
40	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
41	C₄×C₆₇	—	2	3	no	—
42	C₄×C₆₇	C₂×Q₁₆	1	6	yes	—
43	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
44	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
45	Dic₆₇	—	4	6	no	—
46	Dic₆₇	—	4	2	no	—
47	C₂×C₆₇	#550	1	1	yes	characteristic
48	C₂×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
49	C₂×C₆₇	#578	1	1	yes	characteristic
50	#3395	—	67	1	no	maximal, Sylow
51	C₆₇	#3395	1	1	yes	characteristic, complemented, Sylow
52	#557	—	67	2	no	—
53	#494	—	67	3	no	—
54	#462	—	67	6	no	—
55	C₈×Q₈	D₆₇	1	1	yes	characteristic
56	C₄×Q₁₆	—	67	3	no	—
57	C₂×Q₁₆	—	67	6	no	—
58	C₂×Q₁₆	—	67	3	no	—
59	C₄⋊Q₈	—	67	6	no	—
60	#125	—	134	6	no	—
61	C₄×Q₈	—	67	6	no	—
62	C₄×Q₈	—	134	2	no	—
63	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
64	#107	—	67	1	no	—
65	C₄⋊C₈	C₂×D₆₇	1	3	yes	—
66	#103	—	67	6	no	—
67	C₄×C₈	C₂×D₆₇	1	3	yes	—
68	C₂×Q₈	—	67	6	no	—
69	C₂×Q₈	D₂₆₈	1	1	yes	characteristic
70	Q₁₆	—	268	6	no	—
71	Q₁₆	—	134	6	no	—
72	C₂×C₈	C₂²×D₆₇	1	3	yes	—
73	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
74	C₄⋊C₄	—	67	2	no	—
75	C₄⋊C₄	—	134	6	no	—
76	C₄⋊C₄	—	134	6	no	—
77	C₄⋊C₄	—	134	6	no	—
78	C₄⋊C₄	D₂₆₈	1	3	yes	—
79	C₄²	—	134	6	no	—
80	C₄²	C₂²×D₆₇	1	3	yes	—
81	Q₈	—	134	6	no	—
82	Q₈	—	134	6	no	—
83	Q₈	Dic₂₆₈	1	4	yes	complemented
84	C₂×C₄	—	134	2	no	—
85	C₂×C₄	—	134	6	no	—
86	C₂×C₄	C₂×D₂₆₈	1	3	yes	—
87	C₂×C₄	C₂×D₂₆₈	1	3	yes	—
88	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
89	C₈	—	2	3	no	—
90	C₈	#11761629	1	2	yes	—
91	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
92	C₄	—	2	3	no	—
93	C₄	—	268	6	no	—
94	C₄	—	268	2	no	—
95	C₄	C₂×Dic₂₆₈	1	6	yes	—
96	C₄	C₂×#11761629	1	1	yes	characteristic
97	C₄	#14524089	1	1	yes	characteristic
98	C₂	#16864068	1	1	yes	characteristic
99	C₂	C₂²×Dic₂₆₈	1	1	yes	characteristic
100	C₂	#16864213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864125

C₂

yes

maximal

#16863776

C₂

yes

maximal

#16863607

C₂

yes

maximal

C₄×Dic₂₆₈

C₂

yes

maximal

C₈×Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

—

C₂×Dic₂₆₈

C₂²

yes

—

#14524043

C₂²

yes

—

#14524034

—

Q₈×Dic₆₇

—

C₄×Dic₁₃₄

C₂²

yes

—

#14523977

C₂²

yes

characteristic

#14523968

C₂²

yes

—

C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

C₂³

yes

—

Dic₂₆₈

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

Q₈×C₆₇

Q₁₆

yes

complemented

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

#41

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₁₆

yes

—

C₄×C₆₇

C₂×#41

yes

characteristic

C₄×C₆₇

H(5,𝔽₂)

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#550

yes

characteristic

C₂×C₆₇

C₂²×Q₁₆

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#557

—

#494

—

#462

—

C₈×Q₈

D₆₇

yes

characteristic

C₄×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

#125

—

134

—

C₄×Q₈

—

C₄×Q₈

—

134

—

C₄×Q₈

C₂×D₆₇

yes

characteristic

#107

—

C₄⋊C₈

C₂×D₆₇

yes

—

#103

—

C₄×C₈

C₂×D₆₇

yes

—

C₂×Q₈

—

C₂×Q₈

D₂₆₈

yes

characteristic

Q₁₆

—

268

—

Q₁₆

—

134

—

C₂×C₈

C₂²×D₆₇

yes

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

D₂₆₈

yes

—

C₄²

—

134

—

C₄²

C₂²×D₆₇

yes

—

Q₈

—

134

—

Q₈

—

134

—

Q₈

Dic₂₆₈

yes

complemented

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

C₂×D₂₆₈

yes

—

C₂×C₄

C₂×D₂₆₈

yes

—

C₂×C₄

C₂³×D₆₇

yes

characteristic

C₈

—

C₈

#11761629

yes

—

C₂²

C₂²×D₂₆₈

yes

characteristic

C₄

—

C₄

—

268

—

C₄

—

268

—

C₄

C₂×Dic₂₆₈

yes

—

C₄

yes

C₄

yes

C₂

yes

C₂

C₂²×Dic₂₆₈

yes

characteristic

100

C₂

#16864213

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₈×Q₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₈×Q₈	D₆₇
hypocenter	C₆₇	#3395
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₈×Q₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

hypercenter

C₈×Q₈

D₆₇

hypocenter

C₆₇

#3395

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
30. Q₈×C₆₇	Q₁₆	semidirect
51. C₆₇	#3395	semidirect
83. Q₈	Dic₂₆₈	semidirect

H ≅ G/N

type

30. Q₈×C₆₇

Q₁₆

semidirect

51. C₆₇

#3395

semidirect

83. Q₈

Dic₂₆₈

semidirect

Subgroups — #22573215

Conjugacy classes of subgroups

Complements