Group #7043 — subgroups

Subgroups — #7043

Order 256.

142 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 49 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	#1839	C₂	1	1	yes	maximal, characteristic
2	#1767	C₂	1	1	yes	maximal, characteristic
3	#1745	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈	—	2	1	no	—
5	C₂×#97	—	2	1	no	—
6	C₂×#97	C₄	1	2	yes	—
7	C₄³	C₂²	1	1	yes	characteristic
8	#336	—	2	1	no	—
9	#336	—	2	1	no	—
10	#336	—	2	1	no	—
11	#336	—	2	1	no	—
12	#336	C₄	1	2	yes	—
13	#336	C₄	1	2	yes	—
14	C₂²×C₈	—	2	2	no	—
15	C₂²×C₈	—	2	2	no	—
16	C₂²×C₈	—	2	2	no	—
17	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²	D₄	1	1	yes	characteristic
21	C₂×C₄²	D₄	1	1	yes	characteristic
22	C₂×C₄²	D₄	1	1	yes	characteristic
23	C₂×C₄²	Q₈	1	1	yes	characteristic
24	C₄⋊C₈	—	2	4	no	—
25	#97	—	2	4	no	—
26	#97	—	4	2	no	—
27	C₂²×C₄	C₄²	1	1	yes	characteristic
28	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈	—	2	4	no	—
35	C₂×C₈	—	2	4	no	—
36	C₂×C₈	—	4	2	no	—
37	C₂×C₈	—	4	2	no	—
38	C₂×C₈	—	4	2	no	—
39	C₂×C₈	—	4	2	no	—
40	C₂×C₈	—	4	2	no	—
41	C₂×C₈	—	4	2	no	—
42	C₂×C₈	—	4	2	no	—
43	C₄²	—	2	1	no	—
44	C₄²	—	2	1	no	—
45	C₄²	—	2	2	no	—
46	C₄²	—	2	2	no	—
47	C₄²	—	2	2	no	—
48	C₄²	—	4	1	no	—
49	C₄²	—	4	1	no	—
50	C₄²	—	4	1	no	—
51	C₂³	#95	1	1	yes	characteristic
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	1	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	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	#100	1	1	yes	characteristic
70	C₂×C₄	#101	1	1	yes	characteristic
71	C₂×C₄	#104	1	2	yes	—
72	C₂×C₄	#104	1	2	yes	—
73	C₂×C₄	#108	1	2	yes	—
74	C₈	—	4	4	no	—
75	C₈	—	4	4	no	—
76	C₈	—	8	2	no	—
77	C₂²	H(3,ℤ/4)	1	1	yes	characteristic
78	C₂²	#339	1	1	yes	characteristic
79	C₂²	#339	1	1	yes	characteristic
80	C₂²	#341	1	1	yes	characteristic
81	C₂²	#343	1	1	yes	characteristic
82	C₂²	#344	1	1	yes	characteristic
83	C₂²	#344	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	2	2	no	—
87	C₄	—	2	2	no	—
88	C₄	—	2	2	no	—
89	C₄	—	4	1	no	—
90	C₄	—	4	1	no	—
91	C₄	—	4	1	no	—
92	C₂	#1285	1	1	yes	characteristic
93	C₂	#1287	1	1	yes	characteristic
94	C₂	#1288	1	1	yes	characteristic
95	C₂	#1299	1	1	yes	characteristic
96	C₂	#1300	1	1	yes	characteristic
97	C₂	#1302	1	1	yes	characteristic
98	C₂	#1307	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1839

C₂

yes

maximal, characteristic

#1767

C₂

yes

maximal, characteristic

#1745

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×#97

C₄

yes

—

C₄³

C₂²

yes

—

—

—

—

C₄

yes

—

#336

C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

Q₈

yes

characteristic

C₄⋊C₈

—

#97

—

#97

—

C₂²×C₄

C₄²

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

#95

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

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#104

yes

—

C₂×C₄

#108

yes

—

C₈

—

C₈

—

C₈

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

yes

C₂

yes

C₂

yes

C₂

yes

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center, socle, prime radical	C₂³	#95
commutator	C₂²×C₄	C₄²
Frattini, cosocle	C₄³	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#95

commutator

C₂²×C₄

C₄²

Frattini, cosocle

C₄³

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

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

subgroup	index
#1839	2
#1767	2
#1745	2

subgroup

index

#1839

#1767

#1745

Subgroups — #7043

Conjugacy classes of subgroups

Maximal subgroups