Group #3060 — subgroups

Subgroups — #3060

Order 128.

242 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 100 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×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	#537	C₂	1	2	yes	maximal, complemented
4	#470	C₂	1	2	yes	maximal, complemented
5	#469	C₂	1	2	yes	maximal, complemented
6	#465	C₂	1	4	yes	maximal, complemented
7	C₂×#106	C₂	1	1	yes	maximal, characteristic, complemented
8	#417	C₂	1	2	yes	maximal, complemented
9	C₂×#41	C₂²	1	2	yes	—
10	C₂×#41	C₂²	1	1	yes	characteristic
11	#135	—	2	4	no	—
12	C₂×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆	C₂²	1	2	yes	—
14	C₂×D₈	C₂²	1	1	yes	characteristic
15	#131	—	2	4	no	—
16	C₂×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×M₁₆	C₂²	1	2	yes	—
18	C₂²×C₈	C₂²	1	2	yes	—
19	C₂²×C₈	C₂²	1	1	yes	characteristic
20	C₂²⋊Q₈	—	2	2	no	—
21	C₂²⋊Q₈	C₂²	1	4	yes	—
22	#121	—	2	2	no	—
23	#121	—	2	4	no	—
24	#121	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄	C₂²	1	2	yes	—
26	#106	C₂²	1	2	yes	—
27	#106	C₂²	1	2	yes	—
28	#103	C₂²	1	4	yes	—
29	#102	C₂²	1	4	yes	—
30	#41	—	2	2	no	—
31	#41	—	2	2	no	—
32	#41	—	2	4	no	—
33	#41	—	2	4	no	—
34	C₂×Q₈	D₄	1	1	yes	characteristic
35	C₂×Q₈	C₂³	1	2	yes	—
36	C₂×D₄	—	2	2	no	—
37	C₂×D₄	—	2	4	no	—
38	C₂×D₄	D₄	1	2	yes	—
39	C₂×D₄	D₄	1	1	yes	characteristic
40	C₂×D₄	C₂³	1	2	yes	—
41	C₂²×C₄	—	2	2	no	—
42	C₂²×C₄	—	2	2	no	—
43	C₂²×C₄	C₂³	1	2	yes	—
44	C₂²×C₄	C₂³	1	1	yes	characteristic
45	Q₁₆	—	2	2	no	—
46	SD₁₆	—	2	4	no	—
47	D₈	—	2	2	no	—
48	M₁₆	—	2	2	no	—
49	M₁₆	—	2	4	no	—
50	C₂×C₈	—	2	4	no	—
51	C₂×C₈	D₄	1	4	yes	complemented
52	C₂×C₈	C₂³	1	2	yes	—
53	C₂×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₈	C₂³	1	1	yes	characteristic
55	C₄⋊C₄	—	2	2	no	—
56	C₄⋊C₄	—	2	4	no	—
57	C₄⋊C₄	C₂³	1	4	yes	—
58	C₂²⋊C₄	—	2	4	no	—
59	C₂²⋊C₄	—	2	4	no	—
60	C₂³	—	2	2	no	—
61	C₂³	C₂×D₄	1	2	yes	—
62	C₂³	C₂×D₄	1	1	yes	characteristic
63	Q₈	—	2	2	no	—
64	Q₈	—	2	2	no	—
65	Q₈	—	2	2	no	—
66	D₄	—	2	4	no	—
67	D₄	—	2	2	no	—
68	D₄	—	2	2	no	—
69	D₄	—	2	2	no	—
70	D₄	—	4	4	no	—
71	D₄	—	8	4	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	4	4	no	—
77	C₂×C₄	—	8	2	no	—
78	C₂×C₄	C₂×D₄	1	2	yes	—
79	C₂×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄	C₂×D₄	1	2	yes	—
81	C₂×C₄	#41	1	2	yes	—
82	C₂×C₄	C₂⁴	1	1	yes	characteristic
83	C₈	—	2	2	no	—
84	C₈	C₂×D₄	1	2	yes	—
85	C₈	C₂×D₄	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	2	4	no	—
89	C₂²	—	4	2	no	—
90	C₂²	—	4	2	no	—
91	C₂²	—	4	2	no	—
92	C₂²	#121	1	2	yes	—
93	C₂²	C₂²×D₄	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	4	4	no	—
97	C₄	#121	1	2	yes	—
98	C₄	C₂²×D₄	1	1	yes	characteristic
99	C₄	C₂×#41	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	—	4	2	no	—
102	C₂	—	8	2	no	—
103	C₂	C₂×#121	1	1	yes	characteristic
104	C₂	#577	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#135

C₂

yes

maximal, characteristic, complemented

C₂×#131

C₂

yes

maximal, characteristic, complemented

#537

C₂

yes

maximal, complemented

#470

C₂

yes

maximal, complemented

#469

C₂

yes

maximal, complemented

#465

C₂

yes

maximal, complemented

C₂×#106

C₂

yes

maximal, characteristic, complemented

#417

C₂

yes

maximal, complemented

C₂×#41

C₂²

yes

—

C₂×#41

C₂²

yes

characteristic

#135

—

C₂×Q₁₆

C₂²

yes

characteristic

C₂×SD₁₆

C₂²

yes

—

C₂×D₈

C₂²

yes

characteristic

#131

—

C₂×M₁₆

C₂²

yes

characteristic

C₂×M₁₆

C₂²

yes

—

C₂²×C₈

C₂²

yes

—

C₂²×C₈

C₂²

yes

characteristic

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂²

yes

—

#121

—

#121

—

#121

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

#106

C₂²

yes

—

#106

C₂²

yes

—

#103

C₂²

yes

—

#102

C₂²

yes

—

—

—

—

—

C₂×Q₈

D₄

yes

characteristic

C₂×Q₈

C₂³

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄

yes

—

C₂×D₄

D₄

yes

characteristic

C₂×D₄

C₂³

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

D₄

yes

complemented

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂³

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂³

—

C₂³

C₂×D₄

yes

—

C₂³

C₂×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

#41

yes

—

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

C₂×D₄

yes

—

C₈

C₂×D₄

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#121

yes

—

C₂²

C₂²×D₄

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#121

yes

—

C₄

C₂²×D₄

yes

characteristic

C₄

C₂×#41

yes

characteristic

100

C₂

—

101

C₂

—

102

C₂

—

103

C₂

C₂×#121

yes

characteristic

104

C₂

#577

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ G

LECS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂×#135	C₂	semidirect
2. C₂×#131	C₂	semidirect
3. #537	C₂	semidirect
4. #470	C₂	semidirect
5. #469	C₂	semidirect
6. #465	C₂	semidirect
7. C₂×#106	C₂	semidirect
8. #417	C₂	semidirect
51. C₂×C₈	D₄	semidirect

H ≅ G/N

type

1. C₂×#135

C₂

semidirect

2. C₂×#131

C₂

semidirect

3. #537

C₂

semidirect

4. #470

C₂

semidirect

5. #469

C₂

semidirect

6. #465

C₂

semidirect

7. C₂×#106

C₂

semidirect

8. #417

C₂

semidirect

51. C₂×C₈

D₄

semidirect

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

subgroup	index
C₂×#135	2
C₂×#131	2
#537	2
#470	2
#469	2
#465	2
C₂×#106	2
#417	2

subgroup

index

Subgroups — #3060

Conjugacy classes of subgroups

Complements

Maximal subgroups