Group #22559724 — subgroups

Subgroups — #22559724

Order 8576.

152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16863603	C₂	1	2	yes	maximal
2	#16863526	C₂	1	1	yes	maximal, characteristic
3	#16863147	C₂	1	1	yes	maximal, characteristic
4	#16863142	C₂	1	2	yes	maximal
5	#500×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761618	—	2	2	no	—
7	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
8	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	Dic₆₇⋊C₈	—	2	1	no	—
11	#14523967	—	2	2	no	—
12	#14523953	—	2	1	no	—
13	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	#107×C₆₇	—	2	1	no	—
15	#107×C₆₇	—	2	1	no	—
16	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
18	#11761618	—	4	4	no	—
19	C₂×C₆₇⋊C₈	—	2	1	no	—
20	C₂×C₆₇⋊C₈	—	2	2	no	—
21	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
22	Dic₆₇⋊C₄	—	2	1	no	—
23	C₄×Dic₆₇	—	2	1	no	—
24	C₂×Q₈×C₆₇	—	2	2	no	—
25	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₆₇	—	2	2	no	—
28	C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₄²×C₆₇	C₂³	1	1	yes	characteristic
30	Q₈×C₆₇	—	4	2	no	—
31	Q₈×C₆₇	—	4	2	no	—
32	C₂×C₄×C₆₇	—	4	2	no	—
33	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
36	C₂×Dic₆₇	—	2	1	no	—
37	C₂×Dic₆₇	—	2	1	no	—
38	Dic₁₃₄	—	2	2	no	—
39	C₈×C₆₇	—	2	1	no	—
40	C₈×C₆₇	Q₁₆	1	2	yes	complemented
41	C₆₇⋊C₈	—	4	2	no	—
42	C₆₇⋊C₈	—	4	1	no	—
43	C₂²×C₆₇	#121	1	1	yes	characteristic
44	C₄×C₆₇	—	2	1	no	—
45	C₄×C₆₇	—	8	2	no	—
46	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
47	C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
48	C₄×C₆₇	#137	1	1	yes	characteristic
49	Dic₆₇	—	4	1	no	—
50	Dic₆₇	—	4	1	no	—
51	C₂×C₆₇	#462	1	1	yes	characteristic
52	C₂×C₆₇	#469	1	1	yes	characteristic
53	C₂×C₆₇	#473	1	1	yes	characteristic
54	#1694	—	67	1	no	maximal, Sylow
55	C₆₇	#1694	1	1	yes	characteristic, complemented, Sylow
56	#500	D₆₇	1	1	yes	characteristic
57	#462	—	67	2	no	—
58	#446	—	67	1	no	—
59	#334	—	67	1	no	—
60	#332	—	67	2	no	—
61	C₂×Q₁₆	—	134	2	no	—
62	C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
63	C₄×Q₈	—	67	1	no	—
64	#107	—	2	1	no	—
65	#107	—	2	1	no	—
66	C₄⋊C₈	—	67	1	no	—
67	C₄⋊C₈	—	67	2	no	—
68	C₄⋊C₈	—	134	1	no	—
69	#103	—	134	2	no	—
70	#97	—	134	1	no	—
71	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
72	C₂×Q₈	—	2	2	no	—
73	C₂×Q₈	—	67	1	no	—
74	Q₁₆	—	268	4	no	—
75	C₂×C₈	—	134	2	no	—
76	C₂×C₈	—	134	1	no	—
77	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	67	1	no	—
82	C₄⋊C₄	—	134	1	no	—
83	C₄²	—	134	1	no	—
84	C₄²	C₂²×D₆₇	1	1	yes	characteristic
85	Q₈	—	4	2	no	—
86	Q₈	—	4	2	no	—
87	Q₈	—	134	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	134	1	no	—
90	C₂×C₄	—	134	1	no	—
91	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
92	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
93	C₂×C₄	#11761627	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	268	2	no	—
96	C₈	—	268	1	no	—
97	C₈	#11761618	1	2	yes	complemented
98	C₂²	#14524014	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	8	2	no	—
101	C₄	—	268	1	no	—
102	C₄	—	268	1	no	—
103	C₄	C₂×#11761618	1	1	yes	characteristic
104	C₄	#14524067	1	2	yes	—
105	C₄	#14524074	1	1	yes	characteristic
106	C₂	#16863603	1	1	yes	characteristic
107	C₂	#16863646	1	1	yes	characteristic
108	C₂	#16863668	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863603

C₂

yes

maximal

#16863526

C₂

yes

maximal, characteristic

#16863147

C₂

yes

maximal, characteristic

#16863142

C₂

yes

maximal

#500×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761618

—

C₄×Dic₁₃₄

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

Dic₆₇⋊C₈

—

#14523967

—

#14523953

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

#107×C₆₇

—

#107×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

D₄

yes

characteristic

#11761618

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

D₄

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

Q₁₆

yes

complemented

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#121

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₄×C₆₇

#137

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#500

D₆₇

yes

—

—

—

—

C₂×Q₁₆

—

134

—

C₄⋊Q₈

C₂×D₆₇

yes

—

C₄×Q₈

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

#103

—

134

—

#97

—

134

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

268

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄²

—

134

—

C₄²

C₂²×D₆₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

134

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

D₄×D₆₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

C₂×C₄

#11761627

yes

characteristic

C₈

—

C₈

—

268

—

C₈

—

268

—

C₈

yes

C₂²

yes

C₄

—

100

C₄

—

101

C₄

—

268

—

102

C₄

—

268

—

103

C₄

C₂×#11761618

yes

characteristic

104

C₄

#14524067

yes

—

105

C₄

#14524074

yes

characteristic

106

C₂

#16863603

yes

characteristic

107

C₂

#16863646

yes

characteristic

108

C₂

#16863668

yes

characteristic

	subgroup	G/H
center	C₂²	#14524014
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#500×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#500	D₆₇
hypocenter	C₆₇	#1694
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#14524014

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#500×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
40. C₈×C₆₇	Q₁₆	semidirect
55. C₆₇	#1694	semidirect
97. C₈	#11761618	semidirect

H ≅ G/N

type

40. C₈×C₆₇

Q₁₆

semidirect

55. C₆₇

#1694

semidirect

97. C₈

#11761618

semidirect

Subgroups — #22559724

Conjugacy classes of subgroups

Complements