Group #22555813 — subgroups

Subgroups — C₆₇×#1382 #22555813

Order 8576.

132 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 80 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	C₂²×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
3	#427×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆×C₆₇	—	2	1	no	—
5	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
6	#117×C₆₇	—	2	1	no	—
7	C₂×C₁₆×C₆₇	—	2	1	no	—
8	C₂×C₁₆×C₆₇	—	2	1	no	—
9	C₂×C₁₆×C₆₇	C₄	1	2	yes	complemented
10	#108×C₆₇	—	2	1	no	—
11	#108×C₆₇	C₄	1	2	yes	complemented
12	#107×C₆₇	C₄	1	2	yes	—
13	#106×C₆₇	—	2	1	no	—
14	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
15	M₁₆×C₆₇	—	2	2	no	—
16	M₁₆×C₆₇	—	4	1	no	—
17	C₂×C₈×C₆₇	—	4	1	no	—
18	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
22	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
24	C₄⋊C₄×C₆₇	—	2	2	no	—
25	C₂²⋊C₄×C₆₇	—	4	1	no	—
26	C₄²×C₆₇	—	4	1	no	—
27	C₁₆×C₆₇	—	2	2	no	—
28	C₂³×C₆₇	SD₁₆	1	1	yes	characteristic
29	C₂×C₄×C₆₇	—	4	2	no	—
30	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
34	C₂×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
36	C₈×C₆₇	—	4	2	no	—
37	C₈×C₆₇	C₄²	1	1	yes	characteristic
38	C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
40	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₆₇	—	2	2	no	—
42	C₂²×C₆₇	#102	1	1	yes	characteristic
43	C₂²×C₆₇	#103	1	1	yes	characteristic
44	C₂²×C₆₇	#106	1	1	yes	characteristic
45	C₄×C₆₇	—	8	2	no	—
46	C₄×C₆₇	#95	1	1	yes	characteristic
47	C₄×C₆₇	#102	1	1	yes	characteristic
48	C₄×C₆₇	#103	1	1	yes	characteristic
49	C₄×C₆₇	#107	1	1	yes	characteristic
50	C₂×C₆₇	—	2	1	no	—
51	C₂×C₆₇	—	2	1	no	—
52	C₂×C₆₇	#340	1	1	yes	characteristic
53	C₂×C₆₇	#359	1	2	yes	—
54	#1382	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₆₇	#1382	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×C₁₆	C₂×C₆₇	1	1	yes	characteristic
57	C₂×#108	C₂×C₆₇	1	1	yes	characteristic
58	#427	C₂×C₆₇	1	1	yes	characteristic
59	C₂×M₁₆	—	2	1	no	—
60	C₂²×C₈	C₂²×C₆₇	1	1	yes	characteristic
61	#117	—	2	1	no	—
62	C₂×C₁₆	—	2	1	no	—
63	C₂×C₁₆	—	2	1	no	—
64	C₂×C₁₆	C₄×C₆₇	1	2	yes	complemented
65	#108	—	2	1	no	—
66	#108	C₄×C₆₇	1	2	yes	complemented
67	#107	C₄×C₆₇	1	2	yes	—
68	#106	—	2	1	no	—
69	C₂²×C₄	D₄×C₆₇	1	1	yes	characteristic
70	M₁₆	—	2	2	no	—
71	M₁₆	—	4	1	no	—
72	C₂×C₈	—	4	1	no	—
73	C₂×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
74	C₂×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
75	C₂×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
76	C₂×C₈	D₄×C₆₇	1	1	yes	characteristic
77	C₂×C₈	D₄×C₆₇	1	1	yes	characteristic
78	C₂×C₈	Q₈×C₆₇	1	1	yes	characteristic
79	C₄⋊C₄	—	2	2	no	—
80	C₂²⋊C₄	—	4	1	no	—
81	C₄²	—	4	1	no	—
82	C₁₆	—	2	2	no	—
83	C₂³	SD₁₆×C₆₇	1	1	yes	characteristic
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
86	C₂×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
87	C₂×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
88	C₂×C₄	D₈×C₆₇	1	1	yes	characteristic
89	C₂×C₄	SD₁₆×C₆₇	1	1	yes	characteristic
90	C₂×C₄	Q₁₆×C₆₇	1	1	yes	characteristic
91	C₈	—	4	2	no	—
92	C₈	C₄²×C₆₇	1	1	yes	characteristic
93	C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
94	C₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
95	C₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
96	C₂²	—	2	2	no	—
97	C₂²	#102×C₆₇	1	1	yes	characteristic
98	C₂²	#103×C₆₇	1	1	yes	characteristic
99	C₂²	#106×C₆₇	1	1	yes	characteristic
100	C₄	—	8	2	no	—
101	C₄	#95×C₆₇	1	1	yes	characteristic
102	C₄	#102×C₆₇	1	1	yes	characteristic
103	C₄	#103×C₆₇	1	1	yes	characteristic
104	C₄	#107×C₆₇	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	2	1	no	—
107	C₂	#340×C₆₇	1	1	yes	characteristic
108	C₂	#359×C₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂×#108×C₆₇

C₂

yes

maximal, characteristic

#427×C₆₇

C₂

yes

maximal, characteristic

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#117×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

C₄

yes

complemented

#108×C₆₇

—

#108×C₆₇

C₄

yes

complemented

#107×C₆₇

C₄

yes

—

#106×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

characteristic

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

Q₁₆

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

C₄²

yes

characteristic

C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#340

yes

characteristic

C₂×C₆₇

#359

yes

—

#1382

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₆₇

#1382

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

C₂×C₆₇

yes

characteristic

C₂×#108

C₂×C₆₇

yes

characteristic

#427

C₂×C₆₇

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₂²×C₆₇

yes

characteristic

#117

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×C₆₇

yes

complemented

#108

—

#108

C₄×C₆₇

yes

complemented

#107

C₄×C₆₇

yes

—

#106

—

C₂²×C₄

D₄×C₆₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₆₇

yes

characteristic

C₂×C₈

C₂×C₄×C₆₇

yes

characteristic

C₂×C₈

C₂×C₄×C₆₇

yes

characteristic

C₂×C₈

D₄×C₆₇

yes

characteristic

C₂×C₈

D₄×C₆₇

yes

characteristic

C₂×C₈

Q₈×C₆₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₂³

SD₁₆×C₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₆₇

yes

characteristic

C₂×C₄

C₂²⋊C₄×C₆₇

yes

characteristic

C₂×C₄

C₄⋊C₄×C₆₇

yes

characteristic

C₂×C₄

D₈×C₆₇

yes

characteristic

C₂×C₄

SD₁₆×C₆₇

yes

characteristic

C₂×C₄

Q₁₆×C₆₇

yes

characteristic

C₈

—

C₈

C₄²×C₆₇

yes

characteristic

C₈

C₂²⋊C₄×C₆₇

yes

characteristic

C₈

C₄⋊C₄×C₆₇

yes

characteristic

C₈

C₄⋊C₄×C₆₇

yes

characteristic

C₂²

—

C₂²

#102×C₆₇

yes

characteristic

C₂²

#103×C₆₇

yes

characteristic

C₂²

#106×C₆₇

yes

characteristic

100

C₄

—

101

C₄

#95×C₆₇

yes

characteristic

102

C₄

#102×C₆₇

yes

characteristic

103

C₄

#103×C₆₇

yes

characteristic

104

C₄

#107×C₆₇

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

#340×C₆₇

yes

characteristic

108

C₂

#359×C₆₇

yes

—

	subgroup	G/H
center	C₂×C₄×C₆₇	D₈
commutator	C₈	C₄²×C₆₇
Frattini, cosocle	C₂²×C₈	C₂²×C₆₇
socle	C₂²×C₆₇	#102
prime radical	C₂³×C₆₇	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₈×C₆₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₆₇

D₈

commutator

C₈

C₄²×C₆₇

Frattini, cosocle

C₂²×C₈

C₂²×C₆₇

socle

C₂²×C₆₇

#102

prime radical

C₂³×C₆₇

SD₁₆

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₈×C₆₇ ◁ G

N	H ≅ G/N	type
9. C₂×C₁₆×C₆₇	C₄	semidirect
11. #108×C₆₇	C₄	semidirect
54. #1382	C₆₇	direct (N × H)
55. C₆₇	#1382	direct (N × H)
64. C₂×C₁₆	C₄×C₆₇	semidirect
66. #108	C₄×C₆₇	semidirect

H ≅ G/N

type

9. C₂×C₁₆×C₆₇

C₄

semidirect

11. #108×C₆₇

C₄

semidirect

54. #1382

C₆₇

direct (N × H)

55. C₆₇

#1382

direct (N × H)

64. C₂×C₁₆

C₄×C₆₇

semidirect

66. #108

C₄×C₆₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₆×C₆₇	2
C₂×#108×C₆₇	2
#427×C₆₇	2
#1382	67

subgroup

index

C₂²×C₁₆×C₆₇

C₂×#108×C₆₇

#427×C₆₇

#1382

Subgroups — C₆₇×#1382 #22555813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#1382 #22555813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#1382 #22555813