Group #22559119 — subgroups

Subgroups — #22559119

Order 8576.

148 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16863526	C₂	1	2	yes	maximal
2	C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#16863147	C₂	1	1	yes	maximal, characteristic
4	#16863128	C₂	1	2	yes	maximal
5	#322×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₆₈	C₄	1	1	yes	characteristic
7	C₄×Dic₁₃₄	C₂²	1	2	yes	—
8	#14523977	C₄	1	1	yes	characteristic
9	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	Dic₆₇⋊C₈	—	2	2	no	—
11	#14523968	C₄	1	2	yes	—
12	#14523953	—	2	2	no	—
13	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₃₄	C₂×C₄	1	2	yes	—
17	Dic₂₆₈	—	2	2	no	—
18	C₂×C₆₇⋊C₈	—	2	2	no	—
19	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
20	Dic₆₇⋊C₄	—	2	2	no	—
21	C₄×Dic₆₇	—	2	2	no	—
22	C₂×C₈×C₆₇	—	2	1	no	—
23	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	D₄	1	2	yes	—
26	C₄²×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
30	C₂×Dic₆₇	—	2	2	no	—
31	C₂×Dic₆₇	—	2	2	no	—
32	Dic₁₃₄	—	2	2	no	—
33	Dic₁₃₄	—	2	2	no	—
34	C₈×C₆₇	—	2	1	no	—
35	C₈×C₆₇	—	2	2	no	—
36	C₈×C₆₇	—	4	1	no	—
37	C₈×C₆₇	M₁₆	1	2	yes	—
38	C₆₇⋊C₈	—	4	2	no	—
39	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
40	C₄×C₆₇	—	2	1	no	—
41	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
42	C₄×C₆₇	#131	1	1	yes	characteristic
43	C₄×C₆₇	#137	1	2	yes	—
44	Dic₆₇	—	4	2	no	—
45	Dic₆₇	—	4	2	no	—
46	C₂×C₆₇	#436	1	1	yes	characteristic
47	C₂×C₆₇	#441	1	1	yes	characteristic
48	C₂×C₆₇	#444	1	1	yes	characteristic
49	#1593	—	67	1	no	maximal, Sylow
50	C₆₇	#1593	1	1	yes	characteristic, complemented, Sylow
51	#446	—	67	2	no	—
52	C₄×Q₁₆	—	67	1	no	—
53	#334	—	67	1	no	—
54	Q₈⋊C₈	—	67	2	no	—
55	#322	D₆₇	1	1	yes	characteristic
56	C₂×Q₁₆	—	67	1	no	—
57	C₄×Q₈	—	67	2	no	—
58	#107	—	67	1	no	—
59	C₄⋊C₈	—	67	2	no	—
60	C₄⋊C₈	—	134	2	no	—
61	#103	—	67	2	no	—
62	#97	—	134	2	no	—
63	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
64	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
65	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
66	C₂×Q₈	—	67	2	no	—
67	Q₁₆	—	134	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	134	2	no	—
70	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
71	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
72	C₂×C₈	D₂₆₈	1	2	yes	—
73	C₄⋊C₄	—	67	2	no	—
74	C₄⋊C₄	—	134	2	no	—
75	C₄²	—	134	2	no	—
76	C₄²	C₂²×D₆₇	1	1	yes	characteristic
77	Q₈	—	134	2	no	—
78	Q₈	—	134	2	no	—
79	C₂×C₄	—	134	2	no	—
80	C₂×C₄	—	134	2	no	—
81	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
82	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
83	C₂×C₄	#11761628	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	1	no	—
87	C₈	—	268	2	no	—
88	C₈	#11761611	1	2	yes	—
89	C₂²	C₄×D₂₆₈	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	268	2	no	—
92	C₄	—	268	2	no	—
93	C₄	C₂×#11761611	1	1	yes	characteristic
94	C₄	#14524049	1	1	yes	characteristic
95	C₄	#14524076	1	2	yes	—
96	C₂	#16863475	1	1	yes	characteristic
97	C₂	#16863504	1	1	yes	characteristic
98	C₂	#16863519	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863526

C₂

yes

maximal

C₄×Dic₂₆₈

C₂

yes

maximal, characteristic

#16863147

C₂

yes

maximal, characteristic

#16863128

C₂

yes

maximal

#322×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₄

yes

characteristic

C₄×Dic₁₃₄

C₂²

yes

—

#14523977

C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

Dic₆₇⋊C₈

—

#14523968

C₄

yes

—

#14523953

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

C₂×C₄

yes

—

Dic₂₆₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

M₁₆

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂×M₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#446

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

#322

D₆₇

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

#103

—

#97

—

134

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

Q₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

D₂₆₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄²

—

134

—

C₄²

C₂²×D₆₇

yes

characteristic

Q₈

—

134

—

Q₈

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

C₂×C₄

C₂×D₂₆₈

yes

characteristic

C₂×C₄

#11761628

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

268

—

C₈

#11761611

yes

—

C₂²

C₄×D₂₆₈

yes

characteristic

C₄

—

C₄

—

268

—

C₄

—

268

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₄²	C₂²×D₆₇
Fitting	#322×C₆₇	C₂
socle	C₂²×C₆₇	C₄×D₄
hypercenter	#322	D₆₇
hypocenter	C₆₇	#1593
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₄²

C₂²×D₆₇

Fitting

#322×C₆₇

C₂

socle

C₂²×C₆₇

C₄×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
50. C₆₇	#1593	semidirect

H ≅ G/N

type

50. C₆₇

#1593

semidirect

Subgroups — #22559119

Conjugacy classes of subgroups

Complements