Group #7030 — subgroups

Subgroups — #7030

Order 256.

172 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 69 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	#1831	C₂	1	1	yes	maximal, characteristic
2	#1744	C₂	1	1	yes	maximal, characteristic
3	C₄×#97	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈	C₄	1	2	yes	—
5	C₂×#97	—	2	1	no	—
6	C₂×#97	C₄	1	2	yes	—
7	C₂×C₄×C₈	—	2	1	no	—
8	C₂×C₄×C₈	—	2	1	no	—
9	C₄³	C₂²	1	1	yes	characteristic
10	#336	—	2	1	no	—
11	#336	—	2	1	no	—
12	#336	—	2	1	no	—
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	—	4	2	no	—
25	#97	—	2	4	no	—
26	#97	—	2	2	no	—
27	#97	C₈	1	4	yes	complemented
28	C₄×C₈	—	2	4	no	—
29	C₄×C₈	—	2	4	no	—
30	C₂²×C₄	C₄²	1	1	yes	characteristic
31	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈	—	2	4	no	—
38	C₂×C₈	—	2	4	no	—
39	C₂×C₈	—	2	4	no	—
40	C₂×C₈	—	2	4	no	—
41	C₂×C₈	—	4	2	no	—
42	C₂×C₈	—	4	2	no	—
43	C₂×C₈	—	4	2	no	—
44	C₂×C₈	—	4	2	no	—
45	C₂×C₈	—	4	2	no	—
46	C₄²	—	2	1	no	—
47	C₄²	—	2	1	no	—
48	C₄²	—	2	2	no	—
49	C₄²	—	2	2	no	—
50	C₄²	—	2	2	no	—
51	C₄²	—	2	2	no	—
52	C₄²	—	4	1	no	—
53	C₄²	C₂×C₈	1	2	yes	—
54	C₄²	M₁₆	1	2	yes	—
55	C₂³	#95	1	1	yes	characteristic
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	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄	#97	1	1	yes	characteristic
71	C₂×C₄	C₂²⋊C₈	1	2	yes	—
72	C₂×C₄	#100	1	1	yes	characteristic
73	C₂×C₄	#101	1	1	yes	characteristic
74	C₂×C₄	#104	1	2	yes	—
75	C₂×C₄	#104	1	2	yes	—
76	C₂×C₄	#104	1	2	yes	—
77	C₂×C₄	C₄⋊C₈	1	2	yes	—
78	C₂×C₄	#108	1	2	yes	—
79	C₈	—	2	8	no	—
80	C₈	—	4	4	no	—
81	C₈	—	8	2	no	—
82	C₂²	#336	1	1	yes	characteristic
83	C₂²	#339	1	1	yes	characteristic
84	C₂²	#339	1	1	yes	characteristic
85	C₂²	#339	1	1	yes	characteristic
86	C₂²	#341	1	1	yes	characteristic
87	C₂²	#343	1	1	yes	characteristic
88	C₂²	#344	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	2	1	no	—
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	1	no	—
96	C₄	#329	1	2	yes	—
97	C₄	#330	1	2	yes	—
98	C₂	#1276	1	1	yes	characteristic
99	C₂	#1278	1	1	yes	characteristic
100	C₂	#1283	1	1	yes	characteristic
101	C₂	#1295	1	1	yes	characteristic
102	C₂	#1299	1	1	yes	characteristic
103	C₂	#1300	1	1	yes	characteristic
104	C₂	#1302	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1831

C₂

yes

maximal, characteristic

#1744

C₂

yes

maximal, characteristic

C₄×#97

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈

C₄

yes

—

C₂×#97

—

C₂×#97

C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

C₂²

yes

—

—

—

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

—

#97

C₈

yes

complemented

C₄×C₈

—

C₄×C₈

—

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

C₂×C₈

yes

—

C₄²

M₁₆

yes

—

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

C₄×C₈

yes

characteristic

C₂×C₄

#97

yes

characteristic

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#104

yes

—

C₂×C₄

#104

yes

—

C₂×C₄

C₄⋊C₈

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

#329

yes

—

C₄

#330

yes

—

C₂

yes

C₂

yes

100

C₂

#1283

yes

characteristic

101

C₂

#1295

yes

characteristic

102

C₂

#1299

yes

characteristic

103

C₂

#1300

yes

characteristic

104

C₂

#1302

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈
Frattini, cosocle	C₄³	C₂²
socle, prime radical	C₂³	#95
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂²⋊C₄

commutator

C₂×C₄

C₄×C₈

Frattini, cosocle

C₄³

C₂²

C₂³

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
27. #97	C₈	semidirect

H ≅ G/N

type

27. #97

C₈

semidirect

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

subgroup	index
#1831	2
#1744	2
C₄×#97	2

subgroup

index

#1831

#1744

C₄×#97

Subgroups — #7030

Conjugacy classes of subgroups

Complements

Maximal subgroups