Group #22556101 — subgroups

Subgroups — C₆₇×#1670 #22556101

Order 8576.

156 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 72 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	#494×C₆₇	C₂	1	1	yes	maximal, characteristic
2	#475×C₆₇	C₂	1	2	yes	maximal
3	C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#335×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#332×C₆₇	C₂	1	2	yes	maximal
6	C₂×Q₁₆×C₆₇	—	2	1	no	—
7	C₂×Q₁₆×C₆₇	—	2	1	no	—
8	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
9	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	#106×C₆₇	—	2	2	no	—
11	C₄⋊C₈×C₆₇	—	2	1	no	—
12	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#103×C₆₇	—	2	2	no	—
15	C₄×C₈×C₆₇	—	2	1	no	—
16	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₆₇	—	2	2	no	—
18	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
19	Q₁₆×C₆₇	—	4	2	no	—
20	Q₁₆×C₆₇	—	4	2	no	—
21	C₂×C₈×C₆₇	—	2	2	no	—
22	C₂×C₈×C₆₇	—	2	1	no	—
23	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₄⋊C₄×C₆₇	—	2	1	no	—
26	C₄⋊C₄×C₆₇	—	2	2	no	—
27	C₄⋊C₄×C₆₇	—	2	2	no	—
28	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
29	C₄²×C₆₇	—	2	1	no	—
30	C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	Q₈×C₆₇	—	2	1	no	—
32	Q₈×C₆₇	—	4	2	no	—
33	Q₈×C₆₇	—	4	2	no	—
34	Q₈×C₆₇	Q₁₆	1	2	yes	complemented
35	C₂×C₄×C₆₇	—	2	1	no	—
36	C₂×C₄×C₆₇	—	2	1	no	—
37	C₂×C₄×C₆₇	—	4	2	no	—
38	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
41	C₈×C₆₇	—	2	1	no	—
42	C₈×C₆₇	—	4	2	no	—
43	C₈×C₆₇	—	4	1	no	—
44	C₈×C₆₇	SD₁₆	1	2	yes	—
45	C₂²×C₆₇	#121	1	1	yes	characteristic
46	C₄×C₆₇	—	2	2	no	—
47	C₄×C₆₇	—	2	1	no	—
48	C₄×C₆₇	—	4	1	no	—
49	C₄×C₆₇	—	8	2	no	—
50	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
52	C₄×C₆₇	#135	1	1	yes	characteristic
53	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₂×C₆₇	#460	1	1	yes	characteristic
55	C₂×C₆₇	#467	1	1	yes	characteristic
56	C₂×C₆₇	#473	1	1	yes	characteristic
57	#1670	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₆₇	#1670	1	1	yes	characteristic, direct product factor, Sylow
59	#494	C₂×C₆₇	1	1	yes	characteristic
60	#475	C₂×C₆₇	1	2	yes	—
61	C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic
62	#335	C₂×C₆₇	1	1	yes	characteristic
63	#332	C₂×C₆₇	1	2	yes	—
64	C₂×Q₁₆	—	2	1	no	—
65	C₂×Q₁₆	—	2	1	no	—
66	C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
67	C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
68	#106	—	2	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
71	C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
72	#103	—	2	2	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
75	C₂×Q₈	—	2	2	no	—
76	C₂×Q₈	D₄×C₆₇	1	1	yes	characteristic
77	Q₁₆	—	4	2	no	—
78	Q₁₆	—	4	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	D₄×C₆₇	1	1	yes	characteristic
82	C₂×C₈	D₄×C₆₇	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂³×C₆₇	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	4	2	no	—
91	Q₈	—	4	2	no	—
92	Q₈	Q₁₆×C₆₇	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
97	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
98	C₂×C₄	#41×C₆₇	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	2	no	—
101	C₈	—	4	1	no	—
102	C₈	SD₁₆×C₆₇	1	2	yes	—
103	C₂²	#121×C₆₇	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	8	2	no	—
108	C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
109	C₄	C₂×Q₁₆×C₆₇	1	1	yes	characteristic
110	C₄	#135×C₆₇	1	1	yes	characteristic
111	C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
112	C₂	#460×C₆₇	1	1	yes	characteristic
113	C₂	#467×C₆₇	1	1	yes	characteristic
114	C₂	#473×C₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#494×C₆₇

C₂

yes

maximal, characteristic

#475×C₆₇

C₂

yes

maximal

C₈×Q₈×C₆₇

C₂

yes

maximal, characteristic

#335×C₆₇

C₂

yes

maximal, characteristic

#332×C₆₇

C₂

yes

maximal

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

C₄×Q₈×C₆₇

C₂²

yes

characteristic

#106×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

C₂²

yes

—

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

#103×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

Q₁₆

yes

complemented

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

SD₁₆

yes

—

C₂²×C₆₇

#121

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₄×C₆₇

C₂×Q₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₆₇

#1670

yes

characteristic, direct product factor, Sylow

#494

C₂×C₆₇

yes

characteristic

#475

C₂×C₆₇

yes

—

C₈×Q₈

C₂×C₆₇

yes

characteristic

#335

C₂×C₆₇

yes

characteristic

#332

C₂×C₆₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂²×C₆₇

yes

—

C₄×Q₈

C₂²×C₆₇

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₆₇

yes

characteristic

C₄⋊C₈

C₂²×C₆₇

yes

—

#103

—

C₄×C₈

—

C₄×C₈

C₂²×C₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄×C₆₇

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₆₇

yes

characteristic

C₂×C₈

D₄×C₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₆₇

yes

characteristic

C₄²

—

C₄²

C₂³×C₆₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

Q₁₆×C₆₇

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₆₇

yes

characteristic

C₂×C₄

C₂×D₄×C₆₇

yes

characteristic

C₂×C₄

#41×C₆₇

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

102

C₈

SD₁₆×C₆₇

yes

—

103

C₂²

#121×C₆₇

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

C₂×SD₁₆×C₆₇

yes

characteristic

109

C₄

C₂×Q₁₆×C₆₇

yes

characteristic

110

C₄

#135×C₆₇

yes

characteristic

111

C₄

AGL(1,ℤ/8)×C₆₇

yes

characteristic

112

C₂

#460×C₆₇

yes

characteristic

113

C₂

#467×C₆₇

yes

characteristic

114

C₂

#473×C₆₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₆₇	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₆₇

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₆₇

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
34. Q₈×C₆₇	Q₁₆	semidirect
57. #1670	C₆₇	direct (N × H)
58. C₆₇	#1670	direct (N × H)
92. Q₈	Q₁₆×C₆₇	semidirect

H ≅ G/N

type

34. Q₈×C₆₇

Q₁₆

semidirect

57. #1670

C₆₇

direct (N × H)

58. C₆₇

#1670

direct (N × H)

92. Q₈

Q₁₆×C₆₇

semidirect

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

subgroup	index
#494×C₆₇	2
#475×C₆₇	2
C₈×Q₈×C₆₇	2
#335×C₆₇	2
#332×C₆₇	2
#1670	67

subgroup

index

#494×C₆₇

#475×C₆₇

C₈×Q₈×C₆₇

#335×C₆₇

#332×C₆₇

#1670

Subgroups — C₆₇×#1670 #22556101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#1670 #22556101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#1670 #22556101