Group #9859 — subgroups

Subgroups — #9859

Order 256.

195 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 195 conjugacy classes of subgroups — too many to draw. Only its 71 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	#3160	C₂	1	1	yes	maximal, characteristic
2	#1837	C₂	1	1	yes	maximal, characteristic
3	#1493	C₂	1	2	yes	maximal
4	#1493	C₂	1	2	yes	maximal
5	C₂×#322	C₂	1	1	yes	maximal, characteristic
6	#533	C₂²	1	2	yes	—
7	#500	C₄	1	1	yes	characteristic
8	#500	C₄	1	1	yes	characteristic
9	#499	—	2	2	no	—
10	#498	C₄	1	2	yes	—
11	#427	—	2	2	no	—
12	#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#341	—	2	2	no	—
17	Q₈⋊C₈	—	2	4	no	—
18	Q₈⋊C₈	—	2	4	no	—
19	#322	—	2	2	no	—
20	C₂×M₁₆	—	2	2	no	—
21	C₂²×C₈	—	2	1	no	—
22	C₂²×C₈	—	2	1	no	—
23	C₂²×C₈	D₄	1	2	yes	—
24	C₄⋊Q₈	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈	C₂×C₄	1	2	yes	—
26	#125	—	2	2	no	—
27	C₂²⋊Q₈	—	2	2	no	—
28	C₂²⋊Q₈	—	2	2	no	—
29	C₄×Q₈	—	2	2	no	—
30	C₄×Q₈	—	2	2	no	—
31	#117	—	2	2	no	—
32	C₂×C₄²	C₂³	1	1	yes	characteristic
33	#107	—	2	2	no	—
34	#107	—	2	2	no	—
35	#106	—	2	4	no	—
36	C₄⋊C₈	—	2	2	no	—
37	C₄⋊C₈	—	2	2	no	—
38	C₄×C₈	—	2	1	no	—
39	C₄×C₈	—	2	1	no	—
40	C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈	D₄	1	2	yes	—
43	C₄×C₈	D₄	1	2	yes	—
44	C₄×C₈	D₄	1	2	yes	—
45	C₂×Q₈	—	2	2	no	—
46	C₂×Q₈	—	2	2	no	—
47	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄	#41	1	1	yes	characteristic
49	C₂²×C₄	#41	1	1	yes	characteristic
50	M₁₆	—	4	4	no	—
51	C₂×C₈	—	2	4	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	2	4	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	—	4	2	no	—
57	C₂×C₈	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈	C₂²⋊C₄	1	2	yes	—
59	C₄⋊C₄	—	2	2	no	—
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄⋊C₄	—	4	2	no	—
65	C₂²⋊C₄	—	4	2	no	—
66	C₄²	—	4	2	no	—
67	C₄²	C₂²×C₄	1	1	yes	characteristic
68	C₄²	C₂×D₄	1	1	yes	characteristic
69	C₄²	C₂×D₄	1	1	yes	characteristic
70	C₄²	C₂×D₄	1	1	yes	characteristic
71	C₂³	#124	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	4	4	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄	#121	1	1	yes	characteristic
82	C₂×C₄	#121	1	1	yes	characteristic
83	C₂×C₄	#127	1	1	yes	characteristic
84	C₂×C₄	#137	1	4	yes	—
85	C₈	—	2	4	no	—
86	C₈	—	2	2	no	—
87	C₈	—	2	2	no	—
88	C₈	—	4	2	no	—
89	C₈	—	8	2	no	—
90	C₂²	—	2	1	no	—
91	C₂²	—	2	1	no	—
92	C₂²	#390	1	1	yes	characteristic
93	C₂²	#491	1	2	yes	—
94	C₄	—	2	1	no	—
95	C₄	—	2	1	no	—
96	C₄	—	8	2	no	—
97	C₄	—	8	2	no	—
98	C₄	#419	1	2	yes	—
99	C₄	#443	1	2	yes	—
100	C₄	#444	1	2	yes	—
101	C₄	#497	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	#1960	1	1	yes	characteristic
104	C₂	#1974	1	1	yes	characteristic
105	C₂	#1975	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3160

C₂

yes

maximal, characteristic

#1837

C₂

yes

maximal, characteristic

#1493

C₂

yes

maximal

#1493

C₂

yes

maximal

C₂×#322

C₂

yes

maximal, characteristic

#533

C₂²

yes

—

#500

C₄

yes

characteristic

#500

C₄

yes

characteristic

#499

—

#498

C₄

yes

—

#427

—

#423

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#341

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂³

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×D₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₂³

#124

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

#390

yes

characteristic

C₂²

#491

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#419

yes

—

C₄

#443

yes

—

100

C₄

#444

yes

—

101

C₄

#497

yes

—

102

C₂

—

103

C₂

#1960

yes

characteristic

104

C₂

#1974

yes

characteristic

105

C₂

#1975

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#390
prime radical	C₂³	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#127

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ 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
#3160	2
#1837	2
#1493	2
C₂×#322	2

subgroup

index

Subgroups — #9859

Conjugacy classes of subgroups

Maximal subgroups