Group #22562467 — subgroups

Subgroups — #22562467

Order 8576.

184 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 53 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×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14523987	C₂	1	2	yes	maximal, complemented
3	#16863779	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₆₈	C₂²	1	2	yes	—
7	C₂×#11761616	—	2	2	no	—
8	#14524030	—	2	2	no	—
9	Dic₅₃₆	—	2	2	no	—
10	#14523987	—	2	4	no	—
11	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
13	C₂×D₈×C₆₇	C₂²	1	2	yes	—
14	D₁₆×C₆₇	—	2	2	no	—
15	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₃₄	—	2	2	no	—
17	Dic₂₆₈	—	2	2	no	—
18	Dic₂₆₈	—	2	2	no	—
19	#11761616	—	4	4	no	—
20	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
21	C₂²⋊Dic₆₇	—	4	4	no	—
22	C₄×Dic₆₇	D₄	1	1	yes	characteristic
23	C₆₇⋊C₁₆	D₄	1	2	yes	—
24	C₂×D₄×C₆₇	—	2	2	no	—
25	D₈×C₆₇	—	2	2	no	—
26	D₈×C₆₇	—	2	2	no	—
27	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₁₆×C₆₇	D₄	1	2	yes	—
29	C₂³×C₆₇	—	4	2	no	—
30	D₄×C₆₇	—	4	2	no	—
31	D₄×C₆₇	—	4	2	no	—
32	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₆₇	—	4	2	no	—
34	C₂×Dic₆₇	D₈	1	2	yes	—
35	Dic₁₃₄	—	4	2	no	—
36	Dic₁₃₄	—	4	2	no	—
37	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₆₇⋊C₈	D₈	1	2	yes	—
40	C₂²×C₆₇	—	8	2	no	—
41	C₂²×C₆₇	—	8	4	no	—
42	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
43	C₄×C₆₇	#127	1	1	yes	characteristic
44	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
45	Dic₆₇	—	2	2	no	—
46	Dic₆₇	—	8	2	no	—
47	C₂×C₆₇	—	16	2	no	—
48	C₂×C₆₇	#493	1	1	yes	characteristic
49	C₂×C₆₇	#508	1	2	yes	—
50	#2249	—	67	1	no	maximal, Sylow
51	C₆₇	#2249	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₃₂	—	67	1	no	—
53	C₂×SD₃₂	—	67	2	no	—
54	C₂×D₁₆	D₆₇	1	1	yes	characteristic
55	#495	—	67	2	no	—
56	C₄×C₁₆	—	67	1	no	—
57	C₂×Q₁₆	—	67	2	no	—
58	C₂×SD₁₆	—	134	2	no	—
59	C₂×D₈	C₂×D₆₇	1	2	yes	—
60	#124	—	134	2	no	—
61	Q₃₂	—	134	2	no	—
62	SD₃₂	—	134	4	no	—
63	D₁₆	—	2	2	no	—
64	C₂×C₁₆	—	67	1	no	—
65	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
66	C₄×C₈	—	67	1	no	—
67	C₂×Q₈	—	134	2	no	—
68	C₂×D₄	—	2	2	no	—
69	Q₁₆	—	134	2	no	—
70	Q₁₆	—	134	2	no	—
71	SD₁₆	—	268	4	no	—
72	D₈	—	2	2	no	—
73	D₈	—	2	2	no	—
74	C₂×C₈	—	67	1	no	—
75	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
76	C₂²⋊C₄	—	268	4	no	—
77	C₄²	—	67	1	no	—
78	C₁₆	—	67	2	no	—
79	C₁₆	C₂²⋊D₆₇	1	2	yes	—
80	C₂³	—	4	2	no	—
81	Q₈	—	268	2	no	—
82	Q₈	—	268	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	2	no	—
85	C₂×C₄	—	268	2	no	—
86	C₂×C₄	—	67	2	no	—
87	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
88	C₈	—	67	2	no	—
89	C₈	D₄×D₆₇	1	1	yes	characteristic
90	C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
91	C₂²	—	8	2	no	—
92	C₂²	—	8	4	no	—
93	C₂²	D₈×D₆₇	1	1	yes	characteristic
94	C₄	—	536	2	no	—
95	C₄	—	134	2	no	—
96	C₄	#14524039	1	1	yes	characteristic
97	C₄	D₈×D₆₇	1	1	yes	characteristic
98	C₂	—	16	2	no	—
99	C₂	#16863771	1	1	yes	characteristic
100	C₂	#16863825	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₅₃₆

C₂

yes

maximal, characteristic, complemented

C₂×#14523987

C₂

yes

maximal, complemented

#16863779

C₂

yes

maximal, complemented

C₁₆×Dic₆₇

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

—

C₂×#11761616

—

#14524030

—

Dic₅₃₆

—

#14523987

—

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

C₂×D₈×C₆₇

C₂²

yes

—

D₁₆×C₆₇

—

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761616

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

D₄

yes

characteristic

C₆₇⋊C₁₆

D₄

yes

—

C₂×D₄×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₁₆×C₆₇

D₄

yes

—

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

D₈

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

C₂×D₄

yes

characteristic

C₆₇⋊C₈

D₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

#127

yes

characteristic

C₄×C₆₇

C₂×D₈

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#493

yes

characteristic

C₂×C₆₇

#508

yes

—

#2249

—

maximal, Sylow

C₆₇

#2249

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

D₆₇

yes

characteristic

#495

—

C₄×C₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

134

—

C₂×D₈

C₂×D₆₇

yes

—

#124

—

134

—

Q₃₂

—

134

—

SD₃₂

—

134

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

134

—

C₂×D₄

—

Q₁₆

—

134

—

Q₁₆

—

134

—

SD₁₆

—

268

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₂²⋊C₄

—

268

—

C₄²

—

C₁₆

—

C₁₆

C₂²⋊D₆₇

yes

—

C₂³

—

Q₈

—

268

—

Q₈

—

268

—

D₄

—

D₄

—

C₂×C₄

—

268

—

C₂×C₄

—

C₂×C₄

D₄×D₆₇

yes

characteristic

C₈

—

C₈

D₄×D₆₇

yes

characteristic

C₈

C₂×C₂²⋊D₆₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

D₈×D₆₇

yes

characteristic

C₄

—

536

—

C₄

—

134

—

C₄

#14524039

yes

characteristic

C₄

D₈×D₆₇

yes

characteristic

C₂

—

C₂

#16863771

yes

characteristic

100

C₂

#16863825

yes

—

	subgroup	G/H
center	C₂²	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	C₂×D₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	C₂×D₁₆	D₆₇
hypocenter	C₆₇	#2249
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

subgroup

G/H

center

C₂²

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

C₂×D₁₆×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

hypercenter

C₂×D₁₆

D₆₇

hypocenter

C₆₇

#2249

LCS

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

N	H ≅ G/N	type
1. C₂×Dic₅₃₆	C₂	semidirect
2. C₂×#14523987	C₂	semidirect
3. #16863779	C₂	semidirect
4. C₁₆×Dic₆₇	C₂	semidirect
51. C₆₇	#2249	semidirect

H ≅ G/N

type

1. C₂×Dic₅₃₆

C₂

semidirect

2. C₂×#14523987

C₂

semidirect

3. #16863779

C₂

semidirect

4. C₁₆×Dic₆₇

C₂

semidirect

51. C₆₇

#2249

semidirect

Subgroups — #22562467

Conjugacy classes of subgroups

Complements