Group #22555708 — subgroups

Subgroups — C₆₇×#1277 #22555708

Order 8576.

180 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 120 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×C₄⋊C₈×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×C₆₇	—	2	2	no	—
4	C₂²×C₈×C₆₇	C₄	1	2	yes	—
5	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
6	C₄⋊C₈×C₆₇	—	2	2	no	—
7	C₄⋊C₈×C₆₇	C₄	1	4	yes	—
8	C₄×C₈×C₆₇	—	2	2	no	—
9	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
11	C₂²×C₄×C₆₇	Q₈	1	1	yes	characteristic
12	C₂×C₈×C₆₇	—	2	2	no	—
13	C₂×C₈×C₆₇	—	2	2	no	—
14	C₂×C₈×C₆₇	—	2	4	no	—
15	C₂×C₈×C₆₇	—	4	2	no	—
16	C₂×C₈×C₆₇	—	4	2	no	—
17	C₂×C₈×C₆₇	C₈	1	4	yes	complemented
18	C₄²×C₆₇	C₂×C₄	1	2	yes	—
19	C₄²×C₆₇	D₄	1	2	yes	—
20	C₂³×C₆₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₆₇	—	2	2	no	—
22	C₂×C₄×C₆₇	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₆₇	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₆₇	M₁₆	1	2	yes	—
28	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
29	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
30	C₂×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
31	C₈×C₆₇	—	2	4	no	—
32	C₈×C₆₇	—	4	4	no	—
33	C₂²×C₆₇	#95	1	1	yes	characteristic
34	C₂²×C₆₇	C₄⋊C₈	1	2	yes	—
35	C₂²×C₆₇	#106	1	1	yes	characteristic
36	C₂²×C₆₇	#107	1	1	yes	characteristic
37	C₂²×C₆₇	#108	1	2	yes	—
38	C₄×C₆₇	—	2	2	no	—
39	C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
40	C₄×C₆₇	#97	1	1	yes	characteristic
41	C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
42	C₄×C₆₇	#102	1	2	yes	—
43	C₄×C₆₇	#103	1	2	yes	—
44	C₂×C₆₇	#334	1	2	yes	—
45	C₂×C₆₇	#335	1	2	yes	—
46	C₂×C₆₇	#336	1	1	yes	characteristic
47	C₂×C₆₇	#340	1	1	yes	characteristic
48	C₂×C₆₇	#341	1	1	yes	characteristic
49	#1277	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₆₇	#1277	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₄⋊C₈	C₂×C₆₇	1	2	yes	—
52	C₂×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
53	C₂²×C₈	—	2	2	no	—
54	C₂²×C₈	C₄×C₆₇	1	2	yes	—
55	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
56	C₄⋊C₈	—	2	2	no	—
57	C₄⋊C₈	C₄×C₆₇	1	4	yes	—
58	C₄×C₈	—	2	2	no	—
59	C₂²×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
60	C₂²×C₄	D₄×C₆₇	1	1	yes	characteristic
61	C₂²×C₄	Q₈×C₆₇	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	4	2	no	—
66	C₂×C₈	—	4	2	no	—
67	C₂×C₈	C₈×C₆₇	1	4	yes	complemented
68	C₄²	C₂×C₄×C₆₇	1	2	yes	—
69	C₄²	D₄×C₆₇	1	2	yes	—
70	C₂³	C₄⋊C₄×C₆₇	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	C₄²×C₆₇	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
74	C₂×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
75	C₂×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
76	C₂×C₄	C₂×C₈×C₆₇	1	2	yes	—
77	C₂×C₄	M₁₆×C₆₇	1	2	yes	—
78	C₂×C₄	D₈×C₆₇	1	1	yes	characteristic
79	C₂×C₄	SD₁₆×C₆₇	1	2	yes	—
80	C₂×C₄	Q₁₆×C₆₇	1	1	yes	characteristic
81	C₈	—	2	4	no	—
82	C₈	—	4	4	no	—
83	C₂²	#95×C₆₇	1	1	yes	characteristic
84	C₂²	C₄⋊C₈×C₆₇	1	2	yes	—
85	C₂²	#106×C₆₇	1	1	yes	characteristic
86	C₂²	#107×C₆₇	1	1	yes	characteristic
87	C₂²	#108×C₆₇	1	2	yes	—
88	C₄	—	2	2	no	—
89	C₄	C₄×C₈×C₆₇	1	1	yes	characteristic
90	C₄	#97×C₆₇	1	1	yes	characteristic
91	C₄	C₂²⋊C₈×C₆₇	1	2	yes	—
92	C₄	#102×C₆₇	1	2	yes	—
93	C₄	#103×C₆₇	1	2	yes	—
94	C₂	#334×C₆₇	1	2	yes	—
95	C₂	#335×C₆₇	1	2	yes	—
96	C₂	#336×C₆₇	1	1	yes	characteristic
97	C₂	#340×C₆₇	1	1	yes	characteristic
98	C₂	#341×C₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₈×C₆₇

C₂

yes

maximal

C₂×C₄×C₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

C₄

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

C₄

yes

—

C₄×C₈×C₆₇

—

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×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₈

yes

complemented

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄²

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

M₁₆

yes

—

C₂×C₄×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

Q₁₆

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

#95

yes

characteristic

C₂²×C₆₇

C₄⋊C₈

yes

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₄×C₈

yes

characteristic

C₄×C₆₇

#97

yes

characteristic

C₄×C₆₇

C₂²⋊C₈

yes

—

C₄×C₆₇

#102

yes

—

C₄×C₆₇

#103

yes

—

C₂×C₆₇

#334

yes

—

C₂×C₆₇

#335

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₆₇

#1277

yes

characteristic, direct product factor, Sylow

C₂×C₄⋊C₈

C₂×C₆₇

yes

—

C₂×C₄×C₈

C₂×C₆₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₄×C₆₇

yes

—

C₂×C₄²

C₂²×C₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₄×C₆₇

yes

—

C₄×C₈

—

C₂²×C₄

C₂×C₄×C₆₇

yes

characteristic

C₂²×C₄

D₄×C₆₇

yes

characteristic

C₂²×C₄

Q₈×C₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×C₆₇

yes

complemented

C₄²

C₂×C₄×C₆₇

yes

—

C₄²

D₄×C₆₇

yes

—

C₂³

C₄⋊C₄×C₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄²×C₆₇

yes

characteristic

C₂×C₄

C₂²⋊C₄×C₆₇

yes

—

C₂×C₄

C₂²⋊C₄×C₆₇

yes

characteristic

C₂×C₄

C₄⋊C₄×C₆₇

yes

—

C₂×C₄

C₂×C₈×C₆₇

yes

—

C₂×C₄

M₁₆×C₆₇

yes

—

C₂×C₄

D₈×C₆₇

yes

characteristic

C₂×C₄

SD₁₆×C₆₇

yes

—

C₂×C₄

Q₁₆×C₆₇

yes

characteristic

C₈

—

C₈

—

C₂²

#95×C₆₇

yes

characteristic

C₂²

C₄⋊C₈×C₆₇

yes

—

C₂²

#106×C₆₇

yes

characteristic

C₂²

#107×C₆₇

yes

characteristic

C₂²

#108×C₆₇

yes

—

C₄

—

C₄

C₄×C₈×C₆₇

yes

characteristic

C₄

#97×C₆₇

yes

characteristic

C₄

C₂²⋊C₈×C₆₇

yes

—

C₄

#102×C₆₇

yes

—

C₄

#103×C₆₇

yes

—

C₂

#334×C₆₇

yes

—

C₂

#335×C₆₇

yes

—

C₂

#336×C₆₇

yes

characteristic

C₂

#340×C₆₇

yes

characteristic

C₂

#341×C₆₇

yes

characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	D₄
commutator	C₄	C₄×C₈×C₆₇
Frattini, cosocle	C₂×C₄²	C₂²×C₆₇
socle, prime radical	C₂³×C₆₇	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

subgroup

G/H

center

C₂²×C₄×C₆₇

D₄

commutator

C₄

C₄×C₈×C₆₇

Frattini, cosocle

C₂×C₄²

C₂²×C₆₇

socle, prime radical

C₂³×C₆₇

C₄⋊C₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
17. C₂×C₈×C₆₇	C₈	semidirect
49. #1277	C₆₇	direct (N × H)
50. C₆₇	#1277	direct (N × H)
67. C₂×C₈	C₈×C₆₇	semidirect

H ≅ G/N

type

17. C₂×C₈×C₆₇

C₈

semidirect

49. #1277

C₆₇

direct (N × H)

50. C₆₇

#1277

direct (N × H)

67. C₂×C₈

C₈×C₆₇

semidirect

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

subgroup	index
C₂×C₄⋊C₈×C₆₇	2
C₂×C₄×C₈×C₆₇	2
#1277	67

subgroup

index

C₂×C₄⋊C₈×C₆₇

C₂×C₄×C₈×C₆₇

#1277

Subgroups — C₆₇×#1277 #22555708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#1277 #22555708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#1277 #22555708