Group #10623 — subgroups

Subgroups — #10623

Order 256.

175 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 175 conjugacy classes of subgroups — too many to draw. Only its 67 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	#1984	C₂	1	1	yes	maximal, characteristic
2	#1834	C₂	1	1	yes	maximal, characteristic
3	#1803	C₂	1	2	yes	maximal
4	#1296	C₂	1	2	yes	maximal
5	#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#103	—	2	2	no	—
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	#384	C₂²	1	2	yes	—
12	#340	—	2	2	no	—
13	#340	—	2	2	no	—
14	#336	—	2	2	no	—
15	C₂²×Q₈	D₄	1	1	yes	characteristic
16	C₂²×C₈	—	2	1	no	—
17	C₂²×C₈	—	2	2	no	—
18	C₂²×C₈	D₄	1	2	yes	—
19	C₄×Q₈	—	2	2	no	—
20	C₄×Q₈	—	2	2	no	—
21	C₂×C₄⋊C₄	—	2	1	no	—
22	C₂×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄⋊C₄	—	2	2	no	—
25	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
26	C₂×C₄²	—	2	1	no	—
27	C₂×C₄²	C₂³	1	1	yes	characteristic
28	C₄⋊C₈	—	2	2	no	—
29	C₄⋊C₈	—	2	2	no	—
30	C₄⋊C₈	—	2	2	no	—
31	#103	—	4	4	no	—
32	C₄×C₈	—	2	2	no	—
33	C₂×Q₈	—	2	1	no	—
34	C₂×Q₈	—	2	2	no	—
35	C₂×Q₈	—	2	2	no	—
36	C₂×Q₈	D₈	1	2	yes	—
37	C₂²×C₄	—	2	1	no	—
38	C₂²×C₄	—	2	1	no	—
39	C₂²×C₄	—	4	2	no	—
40	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄	#41	1	1	yes	characteristic
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	2	2	no	—
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	—	4	2	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	—	4	2	no	—
49	C₂×C₈	—	4	2	no	—
50	C₂×C₈	—	4	2	no	—
51	C₄⋊C₄	—	2	2	no	—
52	C₄⋊C₄	—	4	2	no	—
53	C₄⋊C₄	—	4	2	no	—
54	C₄⋊C₄	—	4	2	no	—
55	C₄⋊C₄	—	4	2	no	—
56	C₄⋊C₄	—	4	2	no	—
57	C₄⋊C₄	—	8	2	no	—
58	C₄²	—	4	1	no	—
59	C₄²	—	4	1	no	—
60	C₄²	#41	1	2	yes	—
61	C₄²	#41	1	2	yes	—
62	C₂³	#121	1	1	yes	characteristic
63	Q₈	—	2	4	no	—
64	Q₈	—	4	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	4	1	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	8	2	no	—
72	C₂×C₄	—	8	2	no	—
73	C₂×C₄	—	8	2	no	—
74	C₂×C₄	#123	1	2	yes	—
75	C₂×C₄	#123	1	1	yes	characteristic
76	C₂×C₄	#124	1	1	yes	characteristic
77	C₂×C₄	#126	1	2	yes	—
78	C₂×C₄	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄	#135	1	1	yes	characteristic
80	C₂×C₄	#135	1	2	yes	—
81	C₂×C₄	#137	1	2	yes	—
82	C₂×C₄	#137	1	2	yes	—
83	C₈	—	4	2	no	—
84	C₈	—	8	2	no	—
85	C₈	—	8	1	no	—
86	C₂²	#397	1	1	yes	characteristic
87	C₂²	#464	1	2	yes	—
88	C₂²	#466	1	1	yes	characteristic
89	C₂²	#470	1	1	yes	characteristic
90	C₂²	#471	1	2	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	4	2	no	—
93	C₄	—	8	1	no	—
94	C₄	—	16	2	no	—
95	C₄	#480	1	2	yes	—
96	C₄	#485	1	2	yes	—
97	C₄	#485	1	2	yes	—
98	C₄	#488	1	1	yes	characteristic
99	C₄	#491	1	1	yes	characteristic
100	C₂	#1682	1	2	yes	—
101	C₂	#1704	1	2	yes	—
102	C₂	#2074	1	1	yes	characteristic
103	C₂	#2075	1	1	yes	characteristic
104	C₂	#2079	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1984

C₂

yes

maximal, characteristic

#1834

C₂

yes

maximal, characteristic

#1803

C₂

yes

maximal

#1296

C₂

yes

maximal

#1277

C₂

yes

maximal, characteristic

C₂×C₄×Q₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

—

#340

—

#340

—

#336

—

C₂²×Q₈

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

#41

yes

—

C₄²

#41

yes

—

C₂³

#121

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#123

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

C₂×D₈

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#137

yes

—

C₂×C₄

#137

yes

—

C₈

—

C₈

—

C₈

—

C₂²

#397

yes

characteristic

C₂²

#464

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#480

yes

—

C₄

#485

yes

—

C₄

#485

yes

—

C₄

yes

C₄

yes

100

C₂

#1682

yes

—

101

C₂

#1704

yes

—

102

C₂

#2074

yes

characteristic

103

C₂

#2075

yes

characteristic

104

C₂

#2079

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#121

commutator, Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#1984	2
#1834	2
#1803	2
#1296	2
#1277	2

subgroup

index

Subgroups — #10623

Conjugacy classes of subgroups

Maximal subgroups