Group #9798 — subgroups

Subgroups — #9798

Order 256.

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

normal subgroups only

This group has 237 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×#494	C₂	1	1	yes	maximal, characteristic
2	#1997	C₂	1	2	yes	maximal
3	#1567	C₂	1	1	yes	maximal, characteristic
4	#1548	C₂	1	2	yes	maximal
5	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆	C₄	1	4	yes	—
7	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
8	#494	—	2	1	no	—
9	#494	—	2	1	no	—
10	#494	—	2	2	no	—
11	#431	C₂²	1	1	yes	characteristic
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	#333	—	2	4	no	—
18	#322	—	2	1	no	—
19	C₈²	—	2	1	no	—
20	C₂²×Q₈	C₂×C₄	1	4	yes	—
21	C₂×Q₁₆	—	2	4	no	—
22	C₂×Q₁₆	—	2	4	no	—
23	C₂×Q₁₆	—	4	8	no	—
24	C₂×M₁₆	—	2	2	no	—
25	C₂×M₁₆	D₄	1	2	yes	—
26	C₂²×C₈	C₂×C₄	1	2	yes	—
27	C₄⋊Q₈	—	2	2	no	—
28	C₄⋊Q₈	—	2	2	no	—
29	C₄⋊Q₈	—	2	2	no	—
30	C₄⋊Q₈	—	2	2	no	—
31	C₂×C₄⋊C₄	—	2	2	no	—
32	C₂×C₄⋊C₄	—	2	2	no	—
33	C₂×C₄²	C₂³	1	1	yes	characteristic
34	C₄⋊C₈	—	2	2	no	—
35	C₄⋊C₈	—	2	2	no	—
36	C₄⋊C₈	D₄	1	2	yes	—
37	C₂²⋊C₈	—	2	1	no	—
38	#97	—	2	1	no	—
39	C₄×C₈	—	2	1	no	—
40	C₄×C₈	—	2	1	no	—
41	C₄×C₈	D₄	1	1	yes	characteristic
42	C₄×C₈	D₄	1	1	yes	characteristic
43	C₄×C₈	D₄	1	2	yes	—
44	C₂×Q₈	—	2	4	no	—
45	C₂×Q₈	—	2	4	no	—
46	C₂×Q₈	—	4	8	no	—
47	C₂²×C₄	—	2	4	no	—
48	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄	#41	1	1	yes	characteristic
51	Q₁₆	—	4	8	no	—
52	Q₁₆	—	4	8	no	—
53	M₁₆	—	2	2	no	—
54	M₁₆	—	4	4	no	—
55	M₁₆	Q₁₆	1	4	yes	complemented
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	4	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	4	2	no	—
60	C₂×C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈	C₂²⋊C₄	1	2	yes	—
62	C₄⋊C₄	—	4	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄⋊C₄	—	4	2	no	—
65	C₄⋊C₄	—	4	2	no	—
66	C₄²	C₂×D₄	1	1	yes	characteristic
67	C₄²	C₂×D₄	1	1	yes	characteristic
68	C₄²	C₂×D₄	1	1	yes	characteristic
69	C₄²	#41	1	1	yes	characteristic
70	C₂³	C₄×D₄	1	1	yes	characteristic
71	Q₈	—	4	8	no	—
72	Q₈	—	8	4	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	4	4	no	—
76	C₂×C₄	—	8	4	no	—
77	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄	#121	1	1	yes	characteristic
80	C₂×C₄	#121	1	1	yes	characteristic
81	C₂×C₄	#124	1	1	yes	characteristic
82	C₂×C₄	#127	1	1	yes	characteristic
83	C₂×C₄	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄	#135	1	2	yes	—
85	C₈	—	2	2	no	—
86	C₈	—	2	4	no	—
87	C₈	—	4	2	no	—
88	C₈	—	8	2	no	—
89	C₈	#100	1	4	yes	—
90	C₂²	—	2	1	no	—
91	C₂²	—	2	1	no	—
92	C₂²	#390	1	1	yes	characteristic
93	C₂²	C₄×Q₁₆	1	2	yes	—
94	C₄	—	2	1	no	—
95	C₄	—	2	1	no	—
96	C₄	—	8	4	no	—
97	C₄	C₂×#100	1	2	yes	—
98	C₄	#467	1	2	yes	—
99	C₄	#473	1	2	yes	—
100	C₄	#494	1	1	yes	characteristic
101	C₄	#495	1	1	yes	characteristic
102	C₂	—	2	2	no	—
103	C₂	#1965	1	1	yes	characteristic
104	C₂	#1969	1	1	yes	characteristic
105	C₂	#1979	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#494

C₂

yes

maximal, characteristic

#1997

C₂

yes

maximal

#1567

C₂

yes

maximal, characteristic

#1548

C₂

yes

maximal

C₈×M₁₆

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₄

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#423

C₂²

yes

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

—

—

—

C₈²

—

C₂²×Q₈

C₂×C₄

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

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×C₄²

C₂³

yes

characteristic

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

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

Q₁₆

yes

complemented

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

#100

yes

—

C₂²

—

C₂²

—

C₂²

#390

yes

characteristic

C₂²

C₄×Q₁₆

yes

—

C₄

—

C₄

—

C₄

—

C₄

C₂×#100

yes

—

C₄

#467

yes

—

C₄

#473

yes

—

100

C₄

#494

yes

characteristic

101

C₄

#495

yes

characteristic

102

C₂

—

103

C₂

#1965

yes

characteristic

104

C₂

#1969

yes

characteristic

105

C₂

#1979

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

N	H ≅ G/N	type
55. M₁₆	Q₁₆	semidirect

H ≅ G/N

type

55. M₁₆

Q₁₆

semidirect

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

subgroup	index
C₂×#494	2
#1997	2
#1567	2
#1548	2
C₈×M₁₆	2

subgroup

index

C₈×M₁₆

Subgroups — #9798

Conjugacy classes of subgroups

Complements

Maximal subgroups