Group #10363 — subgroups

Subgroups — #10363

Order 256.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	#2862	C₂	1	1	yes	maximal, characteristic
2	#1994	C₂	1	1	yes	maximal, characteristic
3	#1518	C₂	1	2	yes	maximal
4	#1286	C₂	1	2	yes	maximal
5	#1276	C₂	1	1	yes	maximal, characteristic
6	#533	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈	—	2	1	no	—
8	#431	C₂²	1	2	yes	—
9	#423	C₂²	1	1	yes	characteristic
10	#391	—	2	4	no	—
11	#382	—	2	1	no	—
12	#382	C₂²	1	2	yes	—
13	C₄³	C₂²	1	1	yes	characteristic
14	#339	—	2	1	no	—
15	#339	—	2	1	no	—
16	#328	—	2	4	no	—
17	C₂²×Q₈	—	2	2	no	—
18	C₂×M₁₆	—	2	1	no	—
19	C₄⋊Q₈	—	4	2	no	—
20	C₄⋊Q₈	—	4	2	no	—
21	C₄⋊Q₈	Q₈	1	2	yes	complemented
22	#125	—	2	1	no	—
23	C₂²⋊Q₈	—	2	2	no	—
24	C₄×Q₈	—	2	2	no	—
25	#117	—	2	1	no	—
26	C₂×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄²	—	2	2	no	—
29	C₂×C₄²	D₄	1	1	yes	characteristic
30	C₂×C₄²	D₄	1	1	yes	characteristic
31	C₂×C₄²	C₂³	1	1	yes	characteristic
32	C₄⋊C₈	—	2	2	no	—
33	C₄⋊C₈	—	2	2	no	—
34	C₂²⋊C₈	—	2	2	no	—
35	C₄×C₈	—	2	2	no	—
36	#95	—	2	4	no	—
37	#95	—	2	2	no	—
38	#95	D₄	1	4	yes	—
39	C₂×Q₈	—	2	2	no	—
40	C₂×Q₈	—	4	4	no	—
41	C₂×Q₈	—	4	8	no	—
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	—	2	1	no	—
44	C₂²×C₄	—	2	4	no	—
45	C₂²×C₄	C₂×D₄	1	2	yes	—
46	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
47	M₁₆	—	4	1	no	—
48	M₁₆	—	4	1	no	—
49	C₂×C₈	—	4	2	no	—
50	C₂×C₈	—	4	1	no	—
51	C₄⋊C₄	—	2	2	no	—
52	C₄⋊C₄	—	2	2	no	—
53	C₄⋊C₄	—	4	4	no	—
54	C₄⋊C₄	—	4	2	no	—
55	C₄⋊C₄	—	4	2	no	—
56	C₄⋊C₄	—	4	2	no	—
57	C₂²⋊C₄	—	4	1	no	—
58	C₄²	—	2	1	no	—
59	C₄²	—	2	1	no	—
60	C₄²	—	4	4	no	—
61	C₄²	—	4	1	no	—
62	C₄²	SD₁₆	1	2	yes	—
63	C₄²	Q₁₆	1	2	yes	—
64	C₄²	C₂×Q₈	1	1	yes	characteristic
65	C₄²	#41	1	1	yes	characteristic
66	C₄²	#41	1	2	yes	—
67	C₂³	#120	1	1	yes	characteristic
68	Q₈	—	8	2	no	—
69	Q₈	—	8	8	no	—
70	C₂×C₄	—	2	4	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	4	8	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	4	4	no	—
78	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
79	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄	#124	1	2	yes	—
81	C₂×C₄	#124	1	1	yes	characteristic
82	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₂×C₄	#137	1	1	yes	characteristic
86	C₈	—	8	2	no	—
87	C₈	—	8	1	no	—
88	C₂²	—	2	2	no	—
89	C₂²	#393	1	1	yes	characteristic
90	C₂²	#450	1	1	yes	characteristic
91	C₂²	#451	1	1	yes	characteristic
92	C₄	—	2	4	no	—
93	C₄	—	2	2	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	8	4	no	—
97	C₄	—	8	2	no	—
98	C₄	#475	1	1	yes	characteristic
99	C₄	#477	1	1	yes	characteristic
100	C₄	#487	1	1	yes	characteristic
101	C₄	#489	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	2	1	no	—
104	C₂	#2024	1	1	yes	characteristic
105	C₂	#2031	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#2862

C₂

yes

maximal, characteristic

#1994

C₂

yes

maximal, characteristic

#1518

C₂

yes

maximal

#1286

C₂

yes

maximal

#1276

C₂

yes

maximal, characteristic

#533

C₂²

yes

characteristic

C₂×C₄⋊Q₈

—

#431

C₂²

yes

—

#423

C₂²

yes

—

—

C₂²

yes

—

C₄³

C₂²

yes

—

—

—

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Q₈

yes

complemented

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#95

—

#95

—

#95

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

SD₁₆

yes

—

C₄²

Q₁₆

yes

—

C₄²

C₂×Q₈

yes

C₄²

yes

C₄²

yes

—

C₂³

#120

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Q₈

yes

—

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

#124

yes

—

C₂×C₄

#124

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

100

C₄

#487

yes

characteristic

101

C₄

#489

yes

characteristic

102

C₂

—

103

C₂

—

104

C₂

#2024

yes

characteristic

105

C₂

#2031

yes

—

	subgroup	G/H
center, socle	C₂²	#393
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#393

commutator, Frattini, cosocle

C₂×C₄²

C₂³

prime radical

C₂³

#120

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
21. C₄⋊Q₈	Q₈	semidirect

H ≅ G/N

type

21. C₄⋊Q₈

Q₈

semidirect

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

subgroup	index
#2862	2
#1994	2
#1518	2
#1286	2
#1276	2

subgroup

index

Subgroups — #10363

Conjugacy classes of subgroups

Complements

Maximal subgroups