Group #22559614 — subgroups

Subgroups — #22559614

Order 8576.

144 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16863801	C₂	1	1	yes	maximal, characteristic
2	#16863692	C₂	1	2	yes	maximal
3	C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
4	#16863144	C₂	1	2	yes	maximal
5	#335×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524043	C₂²	1	2	yes	—
7	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
8	#14523977	—	2	2	no	—
9	#14523977	—	2	1	no	—
10	#14523977	—	2	1	no	—
11	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₆₇⋊C₈	—	2	1	no	—
13	#14523968	—	2	2	no	—
14	C₈×Dic₆₇	—	2	1	no	—
15	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₃₄	—	2	2	no	—
18	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
19	C₂×C₆₇⋊C₈	—	2	1	no	—
20	C₄⋊Dic₆₇	—	2	2	no	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
23	Dic₆₇⋊C₄	—	2	1	no	—
24	C₄×Dic₆₇	—	2	1	no	—
25	C₂×C₈×C₆₇	—	2	2	no	—
26	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
32	C₂×Dic₆₇	—	2	1	no	—
33	C₂×Dic₆₇	—	2	1	no	—
34	C₂×Dic₆₇	—	4	2	no	—
35	Dic₁₃₄	—	2	1	no	—
36	Dic₁₃₄	—	8	2	no	—
37	Dic₁₃₄	Q₁₆	1	2	yes	—
38	C₈×C₆₇	—	2	1	no	—
39	C₈×C₆₇	—	4	2	no	—
40	C₈×C₆₇	Q₁₆	1	2	yes	—
41	C₆₇⋊C₈	—	4	1	no	—
42	C₂²×C₆₇	#121	1	1	yes	characteristic
43	C₄×C₆₇	—	2	1	no	—
44	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
45	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₆₇	#135	1	1	yes	characteristic
47	C₄×C₆₇	#137	1	1	yes	characteristic
48	Dic₆₇	—	2	2	no	—
49	Dic₆₇	—	4	1	no	—
50	Dic₆₇	—	8	2	no	—
51	C₂×C₆₇	#462	1	1	yes	characteristic
52	C₂×C₆₇	#467	1	1	yes	characteristic
53	C₂×C₆₇	#472	1	1	yes	characteristic
54	#1676	—	67	1	no	maximal, Sylow
55	C₆₇	#1676	1	1	yes	characteristic, complemented, Sylow
56	#500	—	67	1	no	—
57	#477	—	67	2	no	—
58	C₈×Q₈	—	67	1	no	—
59	#335	D₆₇	1	1	yes	characteristic
60	#333	—	67	2	no	—
61	C₄⋊Q₈	—	67	2	no	—
62	C₄×Q₈	—	67	1	no	—
63	#107	—	134	2	no	—
64	#107	—	134	1	no	—
65	#107	—	134	1	no	—
66	C₄⋊C₈	—	67	1	no	—
67	C₄⋊C₈	—	134	1	no	—
68	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
69	#103	—	134	2	no	—
70	C₄×C₈	—	134	1	no	—
71	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
72	C₂×Q₈	—	67	1	no	—
73	C₂×Q₈	—	134	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	134	1	no	—
76	C₂×C₈	D₂₆₈	1	1	yes	characteristic
77	C₂×C₈	D₂₆₈	1	1	yes	characteristic
78	C₄⋊C₄	—	67	1	no	—
79	C₄⋊C₄	—	134	2	no	—
80	C₄⋊C₄	—	134	2	no	—
81	C₄⋊C₄	—	134	1	no	—
82	C₄²	—	134	1	no	—
83	C₄²	C₂²×D₆₇	1	1	yes	characteristic
84	Q₈	—	536	2	no	—
85	Q₈	—	67	2	no	—
86	Q₈	—	134	1	no	—
87	C₂×C₄	—	268	2	no	—
88	C₂×C₄	—	134	1	no	—
89	C₂×C₄	—	134	1	no	—
90	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
91	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
92	C₂×C₄	#11761629	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	4	2	no	—
95	C₈	—	268	1	no	—
96	C₈	Dic₂₆₈	1	2	yes	—
97	C₂²	#14524018	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	268	1	no	—
100	C₄	—	536	2	no	—
101	C₄	—	134	2	no	—
102	C₄	Q₁₆×D₆₇	1	1	yes	characteristic
103	C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
104	C₄	#14524062	1	1	yes	characteristic
105	C₄	#14524076	1	1	yes	characteristic
106	C₂	#16863607	1	1	yes	characteristic
107	C₂	#16863638	1	1	yes	characteristic
108	C₂	#16863667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863801

C₂

yes

maximal, characteristic

#16863692

C₂

yes

maximal

C₈×Dic₁₃₄

C₂

yes

maximal, characteristic

#16863144

C₂

yes

maximal

#335×C₆₇

C₂

yes

maximal, characteristic

#14524043

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

—

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

—

#14523968

—

C₈×Dic₆₇

—

C₄⋊C₈×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

Q₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

Q₁₆

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

#121

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₄×C₆₇

C₂×Q₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#500

—

#477

—

C₈×Q₈

—

#335

D₆₇

yes

characteristic

#333

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

134

—

#107

—

134

—

#107

—

134

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

—

#103

—

134

—

C₄×C₈

—

134

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

D₂₆₈

yes

characteristic

C₂×C₈

D₂₆₈

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

—

134

—

C₄²

C₂²×D₆₇

yes

characteristic

Q₈

—

536

—

Q₈

—

Q₈

—

134

—

C₂×C₄

—

268

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

D₄×D₆₇

yes

characteristic

C₂×C₄

C₂×D₂₆₈

yes

characteristic

C₂×C₄

#11761629

yes

characteristic

C₈

—

C₈

—

C₈

—

268

—

C₈

Dic₂₆₈

yes

—

C₂²

#14524018

yes

characteristic

C₄

—

C₄

—

268

—

100

C₄

—

536

—

101

C₄

—

134

—

102

C₄

Q₁₆×D₆₇

yes

characteristic

103

C₄

C₂×Dic₂₆₈

yes

characteristic

104

C₄

#14524062

yes

characteristic

105

C₄

#14524076

yes

characteristic

106

C₂

#16863607

yes

characteristic

107

C₂

#16863638

yes

characteristic

108

C₂

#16863667

yes

characteristic

	subgroup	G/H
center	C₂²	#14524018
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#335×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#335	D₆₇
hypocenter	C₆₇	#1676
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#14524018

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#335×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
55. C₆₇	#1676	semidirect

H ≅ G/N

type

55. C₆₇

#1676

semidirect

Subgroups — #22559614

Conjugacy classes of subgroups

Complements