Group #2059 — subgroups

Subgroups — #2059

Order 128.

125 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 125 conjugacy classes of subgroups — too many to draw. Only its 54 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×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×#106	C₂	1	1	yes	maximal, characteristic
4	#389	C₂	1	1	yes	maximal, characteristic
5	#340	C₂	1	1	yes	maximal, characteristic
6	#340	C₂	1	1	yes	maximal, characteristic
7	#336	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	#125	—	2	2	no	—
11	#125	—	2	2	no	—
12	C₂×C₄⋊C₄	—	2	1	no	—
13	C₂×C₄⋊C₄	—	2	1	no	—
14	C₂×C₄⋊C₄	—	2	1	no	—
15	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₄²	C₂²	1	1	yes	characteristic
20	#107	—	2	2	no	—
21	#106	—	2	2	no	—
22	#95	—	2	1	no	—
23	C₂²×C₄	—	2	1	no	—
24	C₂²×C₄	—	2	1	no	—
25	C₂²×C₄	—	2	1	no	—
26	C₂²×C₄	—	2	1	no	—
27	C₂²×C₄	D₄	1	1	yes	characteristic
28	C₂²×C₄	D₄	1	1	yes	characteristic
29	C₂²×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	Q₈	1	2	yes	—
35	C₂×C₈	Q₈	1	2	yes	—
36	C₄⋊C₄	—	2	4	no	—
37	C₄⋊C₄	—	2	4	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	4	2	no	—
44	C₄⋊C₄	D₄	1	2	yes	—
45	C₄⋊C₄	D₄	1	2	yes	—
46	C₄⋊C₄	Q₈	1	2	yes	—
47	C₄²	—	2	1	no	—
48	C₄²	—	2	1	no	—
49	C₂³	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	4	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	—	4	1	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	4	1	no	—
68	C₂×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₄	#41	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	—	2	2	no	—
76	C₂²	#120	1	1	yes	characteristic
77	C₂²	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂²	C₄⋊Q₈	1	1	yes	characteristic
79	C₂²	#135	1	1	yes	characteristic
80	C₂²	#135	1	1	yes	characteristic
81	C₂²	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂²	#137	1	1	yes	characteristic
83	C₄	—	4	1	no	—
84	C₄	—	4	2	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	2	no	—
87	C₄	—	4	2	no	—
88	C₄	—	8	1	no	—
89	C₄	C₂²⋊Q₈	1	1	yes	characteristic
90	C₄	C₂²⋊Q₈	1	1	yes	characteristic
91	C₄	C₄⋊Q₈	1	1	yes	characteristic
92	C₄	C₄⋊Q₈	1	1	yes	characteristic
93	C₂	#395	1	1	yes	characteristic
94	C₂	#449	1	1	yes	characteristic
95	C₂	#452	1	1	yes	characteristic
96	C₂	#478	1	1	yes	characteristic
97	C₂	#479	1	1	yes	characteristic
98	C₂	#499	1	1	yes	characteristic
99	C₂	#501	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#125

C₂

yes

maximal, characteristic

C₂×#107

C₂

yes

maximal, characteristic

C₂×#106

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal, characteristic

#340

C₂

yes

maximal, characteristic

#340

C₂

yes

maximal, characteristic

#336

C₂

yes

maximal, characteristic

C₂²×C₈

C₂²

yes

characteristic

C₂²×C₈

C₂²

yes

characteristic

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²

C₂²

yes

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄

yes

characteristic

C₂²×C₄

D₄

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₈

yes

—

C₂×C₈

Q₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄

yes

—

C₄⋊C₄

D₄

yes

—

C₄⋊C₄

Q₈

yes

—

C₄²

—

C₄²

—

C₂³

C₂×D₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×Q₈

yes

characteristic

C₂×C₄

C₂×Q₈

yes

characteristic

C₂×C₄

C₂×Q₈

yes

characteristic

C₂×C₄

#41

yes

characteristic

C₈

—

C₈

—

C₂²

#120

yes

characteristic

C₂²

C₂²⋊Q₈

yes

characteristic

C₂²

C₄⋊Q₈

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₂²⋊Q₈

yes

characteristic

C₄

C₂²⋊Q₈

yes

characteristic

C₄

C₄⋊Q₈

yes

characteristic

C₄

C₄⋊Q₈

yes

C₂

yes

C₂

yes

C₂

yes

C₂

yes

C₂

yes

C₂

yes

C₂

yes

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

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#125	2
C₂×#107	2
C₂×#106	2
#389	2
#340	2
#336	2

subgroup

index

Subgroups — #2059

Conjugacy classes of subgroups

Maximal subgroups