Group #9824 — subgroups

Subgroups — #9824

Order 256.

363 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₂×#493	C₂	1	1	yes	maximal, characteristic
2	#1957	C₂	1	4	yes	maximal, complemented
3	#1845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈	—	2	2	no	—
6	C₂×#127	C₂²	1	2	yes	—
7	#493	—	2	2	no	—
8	#493	C₄	1	4	yes	complemented
9	C₂×#107	—	2	2	no	—
10	C₂×#102	—	2	8	no	—
11	C₂×C₄×C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#384	C₂²	1	2	yes	—
14	C₈²	—	2	2	no	—
15	C₂²×D₄	—	2	2	no	—
16	C₂²×D₄	—	2	4	no	—
17	C₂×D₈	—	2	8	no	—
18	C₂×D₈	—	4	4	no	—
19	C₂×D₈	—	4	4	no	—
20	C₂²×C₈	D₄	1	4	yes	—
21	C₂²×C₈	D₄	1	2	yes	—
22	#127	—	2	2	no	—
23	#127	—	2	4	no	—
24	#127	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄	—	2	4	no	—
26	C₂×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄²	C₂³	1	1	yes	characteristic
28	#107	—	4	4	no	—
29	#102	—	4	16	no	—
30	C₄×C₈	—	2	2	no	—
31	C₄×C₈	—	2	2	no	—
32	C₄×C₈	C₂×C₄	1	2	yes	—
33	C₄×C₈	D₄	1	2	yes	—
34	C₂⁴	—	4	2	no	—
35	C₂×D₄	—	2	4	no	—
36	C₂×D₄	—	2	8	no	—
37	C₂×D₄	—	4	4	no	—
38	C₂×D₄	—	4	2	no	—
39	C₂×D₄	—	4	8	no	—
40	C₂×D₄	—	4	4	no	—
41	C₂×D₄	—	4	4	no	—
42	C₂×D₄	—	4	8	no	—
43	C₂²×C₄	—	4	2	no	—
44	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄	C₂×D₄	1	2	yes	—
46	D₈	—	4	16	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	2	4	no	—
49	C₂×C₈	C₂²⋊C₄	1	4	yes	—
50	C₂×C₈	D₈	1	4	yes	—
51	C₂×C₈	D₈	1	8	yes	complemented
52	C₂×C₈	SD₁₆	1	4	yes	—
53	C₄⋊C₄	—	4	8	no	—
54	C₄⋊C₄	—	4	4	no	—
55	C₄²	C₂²×C₄	1	1	yes	characteristic
56	C₄²	C₂×D₄	1	1	yes	characteristic
57	C₄²	#41	1	2	yes	—
58	C₂³	—	4	4	no	—
59	C₂³	—	8	2	no	—
60	C₂³	—	8	2	no	—
61	C₂³	—	8	4	no	—
62	C₂³	—	8	4	no	—
63	C₂³	#127	1	1	yes	characteristic
64	D₄	—	4	8	no	—
65	D₄	—	4	16	no	—
66	D₄	—	8	8	no	—
67	D₄	—	8	4	no	—
68	C₂×C₄	—	8	2	no	—
69	C₂×C₄	—	8	4	no	—
70	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄	C₄×D₄	1	2	yes	—
72	C₂×C₄	#121	1	2	yes	—
73	C₂×C₄	#124	1	1	yes	characteristic
74	C₂×C₄	C₂×D₈	1	2	yes	—
75	C₂×C₄	C₂×D₈	1	4	yes	—
76	C₂×C₄	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄	#135	1	4	yes	—
78	C₈	—	2	8	no	—
79	C₈	#102	1	8	yes	complemented
80	C₂²	—	8	4	no	—
81	C₂²	—	8	8	no	—
82	C₂²	—	16	4	no	—
83	C₂²	—	16	2	no	—
84	C₂²	—	16	2	no	—
85	C₂²	#390	1	1	yes	characteristic
86	C₂²	#492	1	1	yes	characteristic
87	C₂²	#493	1	1	yes	characteristic
88	C₂²	#493	1	2	yes	—
89	C₂²	#495	1	2	yes	—
90	C₄	—	16	2	no	—
91	C₄	C₂×#102	1	2	yes	—
92	C₄	C₄×D₈	1	4	yes	—
93	C₄	#466	1	4	yes	—
94	C₄	#486	1	2	yes	—
95	C₂	—	16	4	no	—
96	C₂	#1706	1	2	yes	—
97	C₂	#1709	1	1	yes	characteristic
98	C₂	#1712	1	1	yes	characteristic
99	C₂	#1957	1	1	yes	characteristic
100	C₂	#1970	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#493

C₂

yes

maximal, characteristic

#1957

C₂

yes

maximal, complemented

#1845

C₂

yes

maximal, characteristic, complemented

C₂×C₈²

C₂

yes

maximal, characteristic, complemented

C₂²×D₈

—

C₂×#127

C₂²

yes

—

#493

—

#493

C₄

yes

complemented

C₂×#107

—

C₂×#102

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

—

C₈²

—

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

#127

—

#127

—

#127

C₂×C₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#102

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

D₈

yes

—

C₂×C₈

D₈

yes

complemented

C₂×C₈

SD₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#127

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

#121

yes

—

C₂×C₄

#124

yes

characteristic

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×SD₁₆

yes

—

C₂×C₄

#135

yes

—

C₈

—

C₈

#102

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#495

yes

—

C₄

—

C₄

C₂×#102

yes

—

C₄

C₄×D₈

yes

—

C₄

#466

yes

—

C₄

#486

yes

—

C₂

—

C₂

#1706

yes

—

C₂

yes

C₂

yes

C₂

yes

100

C₂

#1970

yes

—

	subgroup	G/H
center, socle	C₂³	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂³

#127

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #1957	C₂	semidirect
3. #1845	C₂	semidirect
4. C₂×C₈²	C₂	semidirect
8. #493	C₄	semidirect
51. C₂×C₈	D₈	semidirect
79. C₈	#102	semidirect

H ≅ G/N

type

2. #1957

C₂

semidirect

3. #1845

C₂

semidirect

4. C₂×C₈²

C₂

semidirect

8. #493

C₄

semidirect

51. C₂×C₈

D₈

semidirect

79. C₈

#102

semidirect

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

subgroup	index
C₂×#493	2
#1957	2
#1845	2
C₂×C₈²	2

subgroup

index

C₂×#493

#1957

#1845

C₂×C₈²

Subgroups — #9824

Conjugacy classes of subgroups

Complements

Maximal subgroups