Group #9801 — subgroups

Subgroups — #9801

Order 256.

205 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 79 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×#500	C₂	1	1	yes	maximal, characteristic
2	#1997	C₂	1	2	yes	maximal
3	#1567	C₂	1	1	yes	maximal, characteristic
4	#1549	C₂	1	2	yes	maximal
5	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
7	#500	—	2	1	no	—
8	#500	—	2	1	no	—
9	#500	—	2	2	no	—
10	#431	C₂²	1	1	yes	characteristic
11	C₂×#107	C₄	1	4	yes	—
12	#423	C₂²	1	2	yes	—
13	C₄×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	#343	—	2	4	no	—
16	#335	—	2	2	no	—
17	#332	—	2	4	no	—
18	#322	—	2	1	no	—
19	C₈²	—	2	1	no	—
20	C₂²×Q₈	—	2	2	no	—
21	C₂×M₁₆	—	2	2	no	—
22	C₂×M₁₆	D₄	1	2	yes	—
23	C₂²×C₈	C₂×C₄	1	2	yes	—
24	C₄⋊Q₈	—	2	2	no	—
25	C₄⋊Q₈	—	2	2	no	—
26	C₄⋊Q₈	—	2	2	no	—
27	C₄⋊Q₈	—	2	2	no	—
28	C₂×C₄⋊C₄	—	2	2	no	—
29	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₄²	C₂³	1	1	yes	characteristic
31	#107	—	2	4	no	—
32	#107	—	2	4	no	—
33	C₄⋊C₈	—	2	2	no	—
34	C₄⋊C₈	—	2	2	no	—
35	C₄⋊C₈	D₄	1	2	yes	—
36	C₂²⋊C₈	—	2	1	no	—
37	#97	—	2	1	no	—
38	C₄×C₈	—	2	1	no	—
39	C₄×C₈	—	2	1	no	—
40	C₄×C₈	D₄	1	1	yes	characteristic
41	C₄×C₈	D₄	1	1	yes	characteristic
42	C₄×C₈	D₄	1	2	yes	—
43	C₂×Q₈	—	4	2	no	—
44	C₂×Q₈	—	4	2	no	—
45	C₂×Q₈	—	8	4	no	—
46	C₂²×C₄	—	2	4	no	—
47	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄	#41	1	1	yes	characteristic
50	M₁₆	—	2	2	no	—
51	M₁₆	—	4	4	no	—
52	M₁₆	Q₁₆	1	4	yes	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	2	4	no	—
55	C₂×C₈	—	2	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	4	no	—
60	C₄⋊C₄	—	2	4	no	—
61	C₄⋊C₄	—	4	2	no	—
62	C₄⋊C₄	—	4	2	no	—
63	C₄²	C₂×D₄	1	1	yes	characteristic
64	C₄²	C₂×D₄	1	1	yes	characteristic
65	C₄²	C₂×D₄	1	1	yes	characteristic
66	C₄²	#41	1	1	yes	characteristic
67	C₂³	C₄×D₄	1	1	yes	characteristic
68	Q₈	—	8	4	no	—
69	Q₈	—	8	4	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	4	no	—
73	C₂×C₄	—	8	4	no	—
74	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄	#121	1	1	yes	characteristic
77	C₂×C₄	#121	1	1	yes	characteristic
78	C₂×C₄	#124	1	1	yes	characteristic
79	C₂×C₄	#127	1	1	yes	characteristic
80	C₂×C₄	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄	#135	1	2	yes	—
82	C₈	—	2	2	no	—
83	C₈	—	2	4	no	—
84	C₈	—	4	2	no	—
85	C₈	—	8	2	no	—
86	C₈	#101	1	4	yes	—
87	C₂²	—	2	1	no	—
88	C₂²	—	2	1	no	—
89	C₂²	#390	1	1	yes	characteristic
90	C₂²	C₄×Q₁₆	1	2	yes	—
91	C₄	—	2	1	no	—
92	C₄	—	2	1	no	—
93	C₄	—	8	4	no	—
94	C₄	C₂×#101	1	2	yes	—
95	C₄	#467	1	2	yes	—
96	C₄	#472	1	2	yes	—
97	C₄	#494	1	1	yes	characteristic
98	C₄	#495	1	1	yes	characteristic
99	C₂	—	2	2	no	—
100	C₂	#1966	1	1	yes	characteristic
101	C₂	#1969	1	1	yes	characteristic
102	C₂	#1978	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, characteristic

#1997

C₂

yes

maximal

#1567

C₂

yes

maximal, characteristic

#1549

C₂

yes

maximal

C₈×M₁₆

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#107

C₄

yes

—

#423

C₂²

yes

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

—

—

—

C₈²

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

Q₁₆

yes

—

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

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

#135

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#101

yes

—

C₂²

—

C₂²

—

C₂²

#390

yes

characteristic

C₂²

C₄×Q₁₆

yes

—

C₄

—

C₄

—

C₄

—

C₄

C₂×#101

yes

—

C₄

#467

yes

—

C₄

#472

yes

—

C₄

yes

C₄

yes

C₂

—

100

C₂

#1966

yes

characteristic

101

C₂

#1969

yes

characteristic

102

C₂

#1978

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#390
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

C₂²

C₂²×C₄

C₂×C₄²

C₂³

prime radical

C₂³

C₄×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ 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
C₂×#500	2
#1997	2
#1567	2
#1549	2
C₈×M₁₆	2

subgroup

index

C₈×M₁₆

Subgroups — #9801

Conjugacy classes of subgroups

Maximal subgroups