Group #22558228 — subgroups

Subgroups — #22558228

Order 8576.

104 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 35 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	#16863637	C₂	1	1	yes	maximal, characteristic
2	#16863161	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₆₈	C₄	1	1	yes	characteristic
5	C₂×#11761612	—	2	1	no	—
6	#14524022	—	2	1	no	—
7	#14523977	C₄	1	1	yes	characteristic
8	#14523968	—	2	1	no	—
9	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₆₇	—	2	1	no	—
11	C₂×Dic₁₃₄	—	2	1	no	—
12	Dic₂₆₈	—	4	1	no	—
13	#11761612	—	2	2	no	—
14	#11761612	—	4	1	no	—
15	C₂×C₆₇⋊C₈	—	4	1	no	—
16	C₄⋊Dic₆₇	—	2	1	no	—
17	Dic₆₇⋊C₄	—	4	1	no	—
18	C₂²⋊Dic₆₇	—	4	1	no	—
19	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₆₇	—	4	1	no	—
21	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
23	C₁₆×C₆₇	—	4	1	no	—
24	C₂³×C₆₇	SD₁₆	1	1	yes	characteristic
25	C₂×C₄×C₆₇	—	2	1	no	—
26	C₂×C₄×C₆₇	—	2	1	no	—
27	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
29	C₂×Dic₆₇	—	4	1	no	—
30	C₂×Dic₆₇	—	4	1	no	—
31	Dic₁₃₄	—	4	1	no	—
32	Dic₁₃₄	—	4	1	no	—
33	C₈×C₆₇	—	2	1	no	—
34	C₈×C₆₇	—	2	1	no	—
35	C₆₇⋊C₈	—	4	2	no	—
36	C₂²×C₆₇	—	2	1	no	—
37	C₂²×C₆₇	—	2	1	no	—
38	C₂²×C₆₇	—	2	1	no	—
39	C₂²×C₆₇	#102	1	1	yes	characteristic
40	C₄×C₆₇	—	2	1	no	—
41	C₄×C₆₇	#99	1	1	yes	characteristic
42	C₄×C₆₇	#104	1	1	yes	characteristic
43	Dic₆₇	—	8	1	no	—
44	Dic₆₇	—	8	1	no	—
45	C₂×C₆₇	—	4	1	no	—
46	C₂×C₆₇	#327	1	1	yes	characteristic
47	C₂×C₆₇	#359	1	1	yes	characteristic
48	C₂×C₆₇	#362	1	1	yes	characteristic
49	#1350	—	67	1	no	maximal, Sylow
50	C₆₇	#1350	1	1	yes	characteristic, complemented, Sylow
51	#467	—	67	1	no	—
52	C₂²⋊C₁₆	D₆₇	1	1	yes	characteristic
53	#341	—	67	1	no	—
54	C₂×Q₁₆	—	67	1	no	—
55	C₂×M₁₆	—	134	1	no	—
56	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
57	C₂²⋊Q₈	—	134	1	no	—
58	C₂×C₁₆	—	2	1	no	—
59	#107	—	67	1	no	—
60	#103	—	134	1	no	—
61	C₂×Q₈	—	134	1	no	—
62	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
63	Q₁₆	—	268	1	no	—
64	M₁₆	—	268	1	no	—
65	M₁₆	—	134	2	no	—
66	C₂×C₈	—	4	1	no	—
67	C₂×C₈	—	268	1	no	—
68	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
69	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
70	C₄⋊C₄	—	268	1	no	—
71	C₄⋊C₄	—	134	1	no	—
72	C₂²⋊C₄	—	268	1	no	—
73	C₁₆	—	4	1	no	—
74	C₂³	#11761617	1	1	yes	characteristic
75	Q₈	—	268	1	no	—
76	Q₈	—	268	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	268	1	no	—
80	C₂×C₄	—	268	1	no	—
81	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
82	C₂×C₄	D₅₃₆	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	2	1	no	—
85	C₈	—	268	2	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	1	no	—
88	C₂²	—	2	1	no	—
89	C₂²	#14523965	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	536	1	no	—
92	C₄	—	536	1	no	—
93	C₄	#14523957	1	1	yes	characteristic
94	C₄	#14523969	1	1	yes	characteristic
95	C₂	—	4	1	no	—
96	C₂	#16863131	1	1	yes	characteristic
97	C₂	#16863204	1	1	yes	characteristic
98	C₂	#16863213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863637

C₂

yes

maximal, characteristic

#16863161

C₂

yes

maximal, characteristic

C₂²⋊C₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₄

yes

—

—

C₄

yes

characteristic

#14523968

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×Dic₁₃₄

—

Dic₂₆₈

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

D₈

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#102

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#467

—

C₂²⋊C₁₆

D₆₇

yes

characteristic

#341

—

C₂×Q₁₆

—

C₂×M₁₆

—

134

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂²⋊Q₈

—

134

—

C₂×C₁₆

—

#107

—

#103

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

D₂₆₈

yes

characteristic

Q₁₆

—

268

—

M₁₆

—

268

—

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

268

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

268

—

C₁₆

—

C₂³

#11761617

yes

characteristic

Q₈

—

268

—

Q₈

—

268

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

—

268

—

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

C₂×C₄

D₅₃₆

yes

characteristic

C₈

—

C₈

—

C₈

—

268

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#14523965

yes

characteristic

C₄

—

C₄

—

536

—

C₄

—

536

—

C₄

yes

C₄

yes

C₂

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	#14523965
commutator	C₂×C₈×C₆₇	C₂×C₄
Frattini	C₂²×C₈	C₂×D₆₇
Fitting	C₂²⋊C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	#102
hypercenter	C₂²⋊C₁₆	D₆₇
hypocenter	C₆₇	#1350
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

#14523965

commutator

C₂×C₈×C₆₇

C₂×C₄

Frattini

C₂²×C₈

C₂×D₆₇

Fitting

C₂²⋊C₁₆×C₆₇

C₂

socle

C₂²×C₆₇

#102

hypercenter

C₂²⋊C₁₆

D₆₇

hypocenter

C₆₇

#1350

LCS

G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
50. C₆₇	#1350	semidirect

H ≅ G/N

type

50. C₆₇

#1350

semidirect

Subgroups — #22558228

Conjugacy classes of subgroups

Complements