Group #22558968 — subgroups

Subgroups — #22558968

Order 8576.

184 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 109 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	#16863420	C₂	1	2	yes	maximal
2	#16863149	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#11761612	—	2	2	no	—
5	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
6	C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
7	C₂²×C₈×C₆₇	C₄	1	2	yes	—
8	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
10	C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
11	C₄×C₈×C₆₇	C₂²	1	2	yes	—
12	#11761612	—	4	4	no	—
13	C₂×C₆₇⋊C₈	—	2	4	no	—
14	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₆₇	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₆₇	—	2	4	no	—
18	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
20	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₆₇	D₄	1	1	yes	characteristic
22	C₄²×C₆₇	Q₈	1	1	yes	characteristic
23	C₄²×C₆₇	C₂³	1	1	yes	characteristic
24	C₂³×C₆₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₆₇	—	2	2	no	—
26	C₂×C₄×C₆₇	—	2	2	no	—
27	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₆₇	D₈	1	2	yes	—
31	C₂×C₄×C₆₇	Q₁₆	1	2	yes	—
32	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
35	C₈×C₆₇	—	2	2	no	—
36	C₈×C₆₇	M₁₆	1	4	yes	complemented
37	C₆₇⋊C₈	—	4	4	no	—
38	C₂²×C₆₇	—	2	1	no	—
39	C₂²×C₆₇	—	2	1	no	—
40	C₂²×C₆₇	#107	1	2	yes	—
41	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₆₇	—	2	1	no	—
43	C₄×C₆₇	—	2	1	no	—
44	C₄×C₆₇	#107	1	2	yes	—
45	C₄×C₆₇	#108	1	2	yes	—
46	C₄×C₆₇	C₂×M₁₆	1	2	yes	—
47	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
48	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×C₆₇	—	2	2	no	—
50	C₂×C₆₇	#423	1	1	yes	characteristic
51	C₂×C₆₇	C₂×#107	1	1	yes	characteristic
52	C₂×C₆₇	C₂×#108	1	1	yes	characteristic
53	#1567	—	67	1	no	maximal, Sylow
54	C₆₇	#1567	1	1	yes	characteristic, complemented, Sylow
55	#423	—	67	2	no	—
56	C₂×C₄×C₈	D₆₇	1	1	yes	characteristic
57	#335	—	67	4	no	—
58	C₂×M₁₆	—	134	2	no	—
59	C₂²×C₈	Dic₆₇	1	2	yes	—
60	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
61	C₄⋊C₈	—	67	4	no	—
62	C₄⋊C₈	—	134	2	no	—
63	C₄×C₈	Dic₆₇	1	1	yes	characteristic
64	C₄×C₈	Dic₆₇	1	1	yes	characteristic
65	C₄×C₈	C₂×D₆₇	1	2	yes	—
66	C₂²×C₄	Dic₁₃₄	1	1	yes	characteristic
67	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
68	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
69	M₁₆	—	268	4	no	—
70	C₂×C₈	—	2	4	no	—
71	C₂×C₈	—	134	4	no	—
72	C₂×C₈	C₂×Dic₆₇	1	2	yes	—
73	C₂×C₈	C₂×Dic₆₇	1	2	yes	—
74	C₄²	Dic₁₃₄	1	1	yes	characteristic
75	C₄²	D₂₆₈	1	1	yes	characteristic
76	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
77	C₄²	C₂²×D₆₇	1	1	yes	characteristic
78	C₂³	C₄⋊Dic₆₇	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
82	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
83	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
84	C₂×C₄	D₅₃₆	1	2	yes	—
85	C₂×C₄	Dic₂₆₈	1	2	yes	—
86	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
87	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	268	4	no	—
91	C₈	#11761612	1	4	yes	complemented
92	C₂²	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	#14523977	1	2	yes	—
95	C₂²	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
96	C₄	—	2	1	no	—
97	C₄	—	2	1	no	—
98	C₄	#14523977	1	2	yes	—
99	C₄	#14523979	1	2	yes	—
100	C₄	C₂×#11761612	1	2	yes	—
101	C₄	C₂×D₅₃₆	1	1	yes	characteristic
102	C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
103	C₂	—	2	2	no	—
104	C₂	#16863420	1	1	yes	characteristic
105	C₂	C₂×#14523977	1	1	yes	characteristic
106	C₂	C₂×#14523979	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863420

C₂

yes

maximal

#16863149

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂²×C₈×C₆₇

C₄

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₄

yes

characteristic

C₄×C₈×C₆₇

C₄

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

Q₈

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

Q₈

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

D₈

yes

—

C₂×C₄×C₆₇

Q₁₆

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

M₁₆

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#107

yes

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#107

yes

—

C₄×C₆₇

#108

yes

—

C₄×C₆₇

C₂×M₁₆

yes

—

C₄×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#423

—

C₂×C₄×C₈

D₆₇

yes

characteristic

#335

—

C₂×M₁₆

—

134

—

C₂²×C₈

Dic₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄×C₈

Dic₆₇

yes

characteristic

C₄×C₈

Dic₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

—

C₂²×C₄

Dic₁₃₄

yes

characteristic

C₂²×C₄

D₂₆₈

yes

characteristic

C₂²×C₄

C₂×Dic₆₇

yes

characteristic

M₁₆

—

268

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₂×Dic₆₇

yes

—

C₂×C₈

C₂×Dic₆₇

yes

—

C₄²

Dic₁₃₄

yes

characteristic

C₄²

D₂₆₈

yes

characteristic

C₄²

C₂×Dic₆₇

yes

characteristic

C₄²

C₂²×D₆₇

yes

characteristic

C₂³

C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

D₅₃₆

yes

—

C₂×C₄

Dic₂₆₈

yes

—

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

C₂×C₄

C₂×D₂₆₈

yes

characteristic

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

C₈

—

C₈

—

268

—

C₈

#11761612

yes

complemented

C₂²

—

C₂²

—

C₂²

#14523977

yes

—

C₂²

C₂×C₄⋊Dic₆₇

yes

characteristic

C₄

—

C₄

—

C₄

#14523977

yes

—

C₄

#14523979

yes

—

100

C₄

C₂×#11761612

yes

—

101

C₄

C₂×D₅₃₆

yes

characteristic

102

C₄

C₂×Dic₂₆₈

yes

characteristic

103

C₂

—

104

C₂

#16863420

yes

characteristic

105

C₂

C₂×#14523977

yes

characteristic

106

C₂

C₂×#14523979

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₄²	C₂²×D₆₇
Fitting	C₂×C₄×C₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₄
hypercenter	C₂×C₄×C₈	D₆₇
hypocenter	C₆₇	#1567
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₄²

C₂²×D₆₇

Fitting

C₂×C₄×C₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂×C₄⋊C₄

hypercenter

C₂×C₄×C₈

D₆₇

hypocenter

C₆₇

#1567

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16863149	C₂	semidirect
36. C₈×C₆₇	M₁₆	semidirect
54. C₆₇	#1567	semidirect
91. C₈	#11761612	semidirect

H ≅ G/N

type

2. #16863149

C₂

semidirect

36. C₈×C₆₇

M₁₆

semidirect

54. C₆₇

#1567

semidirect

91. C₈

#11761612

semidirect

Subgroups — #22558968

Conjugacy classes of subgroups

Complements