Group #2066 — subgroups

Subgroups — #2066

Order 128.

124 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 52 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×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#103	C₂	1	1	yes	maximal, characteristic
4	C₂×#103	C₂	1	1	yes	maximal, characteristic
5	#391	C₂	1	1	yes	maximal, characteristic
6	#389	C₂	1	1	yes	maximal, characteristic
7	#340	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄	—	2	1	no	—
12	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄²	C₂²	1	1	yes	characteristic
16	#106	—	2	2	no	—
17	C₄⋊C₈	—	2	2	no	—
18	#103	—	2	2	no	—
19	#103	—	2	2	no	—
20	#95	—	2	1	no	—
21	#95	—	2	1	no	—
22	#95	—	2	1	no	—
23	C₂×Q₈	—	2	1	no	—
24	C₂×Q₈	—	2	1	no	—
25	C₂×Q₈	—	2	1	no	—
26	C₂×Q₈	—	2	1	no	—
27	C₂×Q₈	—	2	1	no	—
28	C₂×Q₈	Q₈	1	2	yes	—
29	C₂²×C₄	—	2	1	no	—
30	C₂²×C₄	—	2	1	no	—
31	C₂²×C₄	—	2	1	no	—
32	C₂²×C₄	—	2	1	no	—
33	C₂²×C₄	D₄	1	1	yes	characteristic
34	C₂²×C₄	D₄	1	1	yes	characteristic
35	C₂²×C₄	C₂³	1	1	yes	characteristic
36	C₂×C₈	—	2	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	D₄	1	2	yes	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	2	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	4	2	no	—
48	C₄⋊C₄	D₄	1	2	yes	—
49	C₄²	—	2	1	no	—
50	C₄²	—	2	1	no	—
51	C₂³	C₂×D₄	1	1	yes	characteristic
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	Q₈	—	4	1	no	—
55	Q₈	—	4	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	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	—	4	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	4	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	SD₁₆	1	2	yes	—
74	C₂×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₄	#41	1	1	yes	characteristic
78	C₂×C₄	#41	1	1	yes	characteristic
79	C₂×C₄	#41	1	1	yes	characteristic
80	C₈	—	2	2	no	—
81	C₈	—	4	1	no	—
82	C₂²	#121	1	1	yes	characteristic
83	C₂²	#121	1	1	yes	characteristic
84	C₂²	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂²	C₂×SD₁₆	1	1	yes	characteristic
86	C₂²	#135	1	1	yes	characteristic
87	C₂²	#137	1	1	yes	characteristic
88	C₂²	#137	1	1	yes	characteristic
89	C₄	—	2	2	no	—
90	C₄	—	4	2	no	—
91	C₄	—	4	2	no	—
92	C₄	—	4	1	no	—
93	C₄	—	8	1	no	—
94	C₄	—	8	1	no	—
95	C₄	#121	1	1	yes	characteristic
96	C₄	C₂²⋊Q₈	1	1	yes	characteristic
97	C₄	C₂²⋊Q₈	1	1	yes	characteristic
98	C₄	#126	1	1	yes	characteristic
99	C₂	#396	1	1	yes	characteristic
100	C₂	#461	1	1	yes	characteristic
101	C₂	#464	1	1	yes	characteristic
102	C₂	#465	1	1	yes	characteristic
103	C₂	#470	1	1	yes	characteristic
104	C₂	#475	1	1	yes	characteristic
105	C₂	#479	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#106

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×#103

C₂

yes

maximal, characteristic

C₂×#103

C₂

yes

maximal, characteristic

#391

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal, characteristic

#340

C₂

yes

maximal, characteristic

C₂²×Q₈

C₂²

yes

characteristic

C₂²×C₈

C₂²

yes

characteristic

C₂²×C₈

C₂²

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²

C₂²

yes

characteristic

#106

—

C₄⋊C₈

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₈

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

—

C₂×C₈

D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄

yes

—

C₄²

—

C₄²

—

C₂³

C₂×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

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

SD₁₆

yes

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×Q₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₂²

yes

C₂²

yes

C₂²

C₂²⋊Q₈

yes

characteristic

C₂²

C₂×SD₁₆

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#121

yes

characteristic

C₄

C₂²⋊Q₈

yes

characteristic

C₄

C₂²⋊Q₈

yes

C₄

yes

C₂

yes

100

C₂

#461

yes

characteristic

101

C₂

#464

yes

characteristic

102

C₂

#465

yes

characteristic

103

C₂

#470

yes

characteristic

104

C₂

#475

yes

characteristic

105

C₂

#479

yes

characteristic

	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×#106	2
C₂×C₄⋊C₈	2
C₂×#103	2
#391	2
#389	2
#340	2

subgroup

index

C₂×C₄⋊C₈

Subgroups — #2066

Conjugacy classes of subgroups

Maximal subgroups