Group #22561801 — subgroups

Subgroups — M₃₂×Dic₆₇ #22561801

Order 8576.

196 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#16863171	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₆₇	C₄	1	1	yes	characteristic
8	#14523983	C₄	1	4	yes	complemented
9	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	C₈×Dic₆₇	C₄	1	1	yes	characteristic
11	C₈×Dic₆₇	C₄	1	1	yes	characteristic
12	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
13	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	M₃₂×C₆₇	C₄	1	4	yes	complemented
16	C₂×C₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂²×Dic₆₇	C₈	1	2	yes	—
18	C₂×C₆₇⋊C₈	C₈	1	4	yes	—
19	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₆₇	C₈	1	2	yes	—
22	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
24	C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
25	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
31	C₂³×C₆₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₆₇	C₄²	1	2	yes	—
33	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₆₇	—	2	4	no	—
37	C₂×Dic₆₇	C₂×C₈	1	2	yes	—
38	C₂×Dic₆₇	C₂×C₈	1	2	yes	—
39	C₈×C₆₇	C₄²	1	2	yes	—
40	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₆₇⋊C₈	C₂×C₈	1	2	yes	—
43	C₆₇⋊C₈	C₂×C₈	1	2	yes	—
44	C₂²×C₆₇	—	2	2	no	—
45	C₂²×C₆₇	C₄×C₈	1	2	yes	—
46	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
47	C₄×C₆₇	C₄×C₈	1	2	yes	—
48	C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
49	C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
50	Dic₆₇	—	2	2	no	—
51	Dic₆₇	M₃₂	1	4	yes	direct product factor
52	C₂×C₆₇	—	2	2	no	—
53	C₂×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
54	C₂×C₆₇	C₂×M₃₂	1	2	yes	—
55	C₄×M₃₂	—	67	1	no	maximal, Sylow
56	C₆₇	C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₃₂	—	67	1	no	—
58	C₂×M₃₂	D₆₇	1	1	yes	characteristic
59	C₂×C₄×C₈	—	67	1	no	—
60	#346	—	67	2	no	—
61	C₄×C₁₆	—	67	2	no	—
62	C₂²×C₈	—	67	1	no	—
63	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
64	C₂×C₄²	—	67	1	no	—
65	M₃₂	—	67	4	no	—
66	M₃₂	Dic₆₇	1	4	yes	direct product factor
67	C₂×C₁₆	—	67	2	no	—
68	C₂×C₁₆	C₂×D₆₇	1	2	yes	—
69	C₄×C₈	—	67	1	no	—
70	C₄×C₈	—	67	1	no	—
71	C₄×C₈	—	67	1	no	—
72	C₄×C₈	—	67	1	no	—
73	C₂²×C₄	—	67	2	no	—
74	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
75	C₂×C₈	—	67	4	no	—
76	C₂×C₈	—	67	1	no	—
77	C₂×C₈	—	67	1	no	—
78	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
79	C₂×C₈	C₄×D₆₇	1	2	yes	—
80	C₂×C₈	C₂×Dic₆₇	1	2	yes	—
81	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
82	C₄²	—	67	2	no	—
83	C₄²	—	67	1	no	—
84	C₄²	—	67	1	no	—
85	C₁₆	—	67	4	no	—
86	C₁₆	C₂×Dic₆₇	1	4	yes	complemented
87	C₂³	C₈×D₆₇	1	1	yes	characteristic
88	C₂×C₄	—	67	2	no	—
89	C₂×C₄	—	67	2	no	—
90	C₂×C₄	—	134	4	no	—
91	C₂×C₄	C₄×Dic₆₇	1	2	yes	—
92	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
93	C₂×C₄	C₈×D₆₇	1	2	yes	—
94	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
95	C₈	—	67	2	no	—
96	C₈	—	67	2	no	—
97	C₈	C₄×Dic₆₇	1	2	yes	—
98	C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
99	C₈	C₂²×Dic₆₇	1	1	yes	characteristic
100	C₂²	—	2	2	no	—
101	C₂²	C₈×Dic₆₇	1	2	yes	—
102	C₂²	C₂×C₈×D₆₇	1	1	yes	characteristic
103	C₄	—	67	4	no	—
104	C₄	—	134	2	no	—
105	C₄	C₈×Dic₆₇	1	2	yes	—
106	C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
107	C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	C₂×C₈×Dic₆₇	1	1	yes	characteristic
110	C₂	M₃₂×D₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523983

C₂

yes

maximal, characteristic

C₂×C₈×Dic₆₇

C₂

yes

maximal, characteristic

#16863171

C₂

yes

maximal, complemented

C₁₆×Dic₆₇

C₂

yes

maximal, complemented

C₂×M₃₂×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₆₇

C₄

yes

characteristic

#14523983

C₄

yes

complemented

C₂×C₆₇⋊C₁₆

C₂²

yes

—

C₈×Dic₆₇

C₄

yes

characteristic

C₈×Dic₆₇

C₄

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

C₄

yes

complemented

C₂×C₁₆×C₆₇

C₂²

yes

—

C₂²×Dic₆₇

C₈

yes

—

C₂×C₆₇⋊C₈

C₈

yes

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄×Dic₆₇

C₈

yes

—

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₆₇⋊C₁₆

C₂×C₄

yes

complemented

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₁₆×C₆₇

C₂×C₄

yes

complemented

C₂³×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₄²

yes

—

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×C₈

yes

—

C₂×Dic₆₇

C₂×C₈

yes

—

C₈×C₆₇

C₄²

yes

—

C₈×C₆₇

C₂²×C₄

yes

characteristic

C₈×C₆₇

C₂²×C₄

yes

characteristic

C₆₇⋊C₈

C₂×C₈

yes

—

C₆₇⋊C₈

C₂×C₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×C₈

yes

—

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₄×C₆₇

C₄×C₈

yes

—

C₄×C₆₇

C₂×C₄²

yes

characteristic

C₄×C₆₇

C₂²×C₈

yes

characteristic

Dic₆₇

—

Dic₆₇

M₃₂

yes

direct product factor

C₂×C₆₇

—

C₂×C₆₇

C₂×C₄×C₈

yes

characteristic

C₂×C₆₇

C₂×M₃₂

yes

—

C₄×M₃₂

—

maximal, Sylow

C₆₇

C₄×M₃₂

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

D₆₇

yes

characteristic

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

Dic₆₇

yes

direct product factor

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

—

C₂×C₈

C₂×Dic₆₇

yes

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

C₂×Dic₆₇

yes

complemented

C₂³

C₈×D₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

C₄×Dic₆₇

yes

—

C₂×C₄

C₈×D₆₇

yes

characteristic

C₂×C₄

C₈×D₆₇

yes

—

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

C₈

—

C₈

—

C₈

C₄×Dic₆₇

yes

—

C₈

C₂×C₄×D₆₇

yes

characteristic

C₈

C₂²×Dic₆₇

yes

characteristic

100

C₂²

—

101

C₂²

C₈×Dic₆₇

yes

—

102

C₂²

C₂×C₈×D₆₇

yes

characteristic

103

C₄

—

104

C₄

—

134

—

105

C₄

C₈×Dic₆₇

yes

—

106

C₄

C₂×C₄×Dic₆₇

yes

characteristic

107

C₄

C₂×C₈×D₆₇

yes

characteristic

108

C₂

—

109

C₂

C₂×C₈×Dic₆₇

yes

characteristic

110

C₂

M₃₂×D₆₇

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₆₇
commutator	C₂×C₆₇	C₂×C₄×C₈
Fitting	C₂×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₈
hypercenter	C₂×M₃₂	D₆₇
hypocenter	C₆₇	C₄×M₃₂
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₆₇

commutator

C₂×C₆₇

C₂×C₄×C₈

Fitting

C₂×M₃₂×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×C₈

hypercenter

C₂×M₃₂

D₆₇

hypocenter

C₆₇

C₄×M₃₂

LCS

G ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #16863171	C₂	semidirect
4. C₁₆×Dic₆₇	C₂	semidirect
8. #14523983	C₄	semidirect
15. M₃₂×C₆₇	C₄	semidirect
24. C₆₇⋊C₁₆	C₂×C₄	semidirect
30. C₁₆×C₆₇	C₂×C₄	semidirect
51. Dic₆₇	M₃₂	direct (N × H)
56. C₆₇	C₄×M₃₂	semidirect
66. M₃₂	Dic₆₇	direct (N × H)
86. C₁₆	C₂×Dic₆₇	semidirect

H ≅ G/N

type

3. #16863171

C₂

semidirect

4. C₁₆×Dic₆₇

C₂

semidirect

8. #14523983

C₄

semidirect

15. M₃₂×C₆₇

C₄

semidirect

24. C₆₇⋊C₁₆

C₂×C₄

semidirect

30. C₁₆×C₆₇

C₂×C₄

semidirect

51. Dic₆₇

M₃₂

direct (N × H)

56. C₆₇

C₄×M₃₂

semidirect

66. M₃₂

Dic₆₇

direct (N × H)

86. C₁₆

C₂×Dic₆₇

semidirect

Subgroups — M₃₂×Dic₆₇ #22561801

Conjugacy classes of subgroups

Complements

Subgroups — M32×Dic67 #22561801

Conjugacy classes of subgroups

Complements

Subgroups — M₃₂×Dic₆₇ #22561801