Group #9860 — subgroups

Subgroups — #9860

Order 256.

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

normal subgroups only

This group has 249 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	#3147	C₂	1	1	yes	maximal, characteristic
2	#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	#1492	C₂	1	2	yes	maximal, complemented
4	#1492	C₂	1	2	yes	maximal, complemented
5	C₂×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#135	—	2	2	no	—
7	#532	C₂²	1	2	yes	—
8	#495	—	2	2	no	—
9	#494	C₄	1	1	yes	characteristic
10	#493	C₄	1	1	yes	characteristic
11	#492	C₄	1	2	yes	—
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	D₄⋊C₈	—	2	4	no	—
18	D₄⋊C₈	—	2	4	no	—
19	#322	—	2	2	no	—
20	C₂×#41	—	2	2	no	—
21	#135	—	4	8	no	—
22	C₂×Q₁₆	—	2	2	no	—
23	C₂×SD₁₆	—	2	4	no	—
24	C₂×D₈	—	2	2	no	—
25	C₂×M₁₆	—	2	2	no	—
26	C₂²×C₈	—	2	1	no	—
27	C₂²×C₈	—	2	1	no	—
28	C₂²×C₈	D₄	1	2	yes	—
29	C₄⋊Q₈	C₂×C₄	1	2	yes	—
30	#127	C₂×C₄	1	2	yes	—
31	#124	—	2	2	no	—
32	#121	—	2	2	no	—
33	#121	—	2	2	no	—
34	C₄×D₄	—	2	2	no	—
35	C₄×D₄	—	2	2	no	—
36	C₂×C₄²	C₂³	1	1	yes	characteristic
37	C₄⋊C₈	—	2	2	no	—
38	C₄⋊C₈	—	2	2	no	—
39	C₄×C₈	—	2	1	no	—
40	C₄×C₈	—	2	1	no	—
41	C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈	D₄	1	2	yes	—
44	C₄×C₈	D₄	1	2	yes	—
45	C₄×C₈	D₄	1	2	yes	—
46	#41	—	4	8	no	—
47	C₂×Q₈	—	2	2	no	—
48	C₂×D₄	—	2	2	no	—
49	C₂×D₄	—	2	2	no	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	—	4	2	no	—
52	C₂²×C₄	—	4	2	no	—
53	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄	#41	1	1	yes	characteristic
55	C₂²×C₄	#41	1	1	yes	characteristic
56	Q₁₆	—	4	4	no	—
57	SD₁₆	—	4	8	no	—
58	D₈	—	4	4	no	—
59	M₁₆	—	4	4	no	—
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	—	4	2	no	—
66	C₂×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈	C₂²⋊C₄	1	2	yes	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₂²⋊C₄	—	4	2	no	—
71	C₂²⋊C₄	—	4	2	no	—
72	C₄²	C₂²×C₄	1	1	yes	characteristic
73	C₄²	C₂×D₄	1	1	yes	characteristic
74	C₄²	C₂×D₄	1	1	yes	characteristic
75	C₄²	C₂×D₄	1	1	yes	characteristic
76	C₂³	—	4	2	no	—
77	C₂³	#124	1	1	yes	characteristic
78	Q₈	—	4	4	no	—
79	D₄	—	4	4	no	—
80	D₄	—	4	4	no	—
81	D₄	—	4	4	no	—
82	D₄	—	8	4	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	8	4	no	—
87	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄	#121	1	1	yes	characteristic
91	C₂×C₄	#121	1	1	yes	characteristic
92	C₂×C₄	#127	1	1	yes	characteristic
93	C₂×C₄	AGL(1,ℤ/8)	1	4	yes	—
94	C₈	—	2	4	no	—
95	C₈	—	2	2	no	—
96	C₈	—	2	2	no	—
97	C₈	—	4	2	no	—
98	C₈	—	8	2	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	8	2	no	—
102	C₂²	—	8	2	no	—
103	C₂²	—	8	2	no	—
104	C₂²	#390	1	1	yes	characteristic
105	C₂²	#490	1	2	yes	—
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	8	2	no	—
109	C₄	#418	1	2	yes	—
110	C₄	#443	1	2	yes	—
111	C₄	#444	1	2	yes	—
112	C₄	#496	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	—	16	2	no	—
115	C₂	#1961	1	1	yes	characteristic
116	C₂	#1974	1	1	yes	characteristic
117	C₂	#1975	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3147

C₂

yes

maximal, characteristic

#1837

C₂

yes

maximal, characteristic, complemented

#1492

C₂

yes

maximal, complemented

#1492

C₂

yes

maximal, complemented

C₂×#322

C₂

yes

maximal, characteristic, complemented

C₂×#135

—

#532

C₂²

yes

—

#495

—

#494

C₄

yes

characteristic

#493

C₄

yes

characteristic

#492

C₄

yes

—

#423

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#341

—

D₄⋊C₈

—

D₄⋊C₈

—

#322

—

C₂×#41

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

#127

C₂×C₄

yes

—

#124

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

C₂³

yes

characteristic

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

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

Q₁₆

—

SD₁₆

—

D₈

—

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

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₂³

—

C₂³

#124

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

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

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

#390

yes

characteristic

105

C₂²

#490

yes

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#418

yes

—

110

C₄

#443

yes

—

111

C₄

#444

yes

—

112

C₄

#496

yes

—

113

C₂

—

114

C₂

—

115

C₂

#1961

yes

characteristic

116

C₂

#1974

yes

characteristic

117

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	#3147	C₂
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

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #1837	C₂	semidirect
3. #1492	C₂	semidirect
4. #1492	C₂	semidirect
5. C₂×#322	C₂	semidirect

H ≅ G/N

type

2. #1837

C₂

semidirect

3. #1492

C₂

semidirect

4. #1492

C₂

semidirect

5. C₂×#322

C₂

semidirect

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

subgroup	index
#3147	2
#1837	2
#1492	2
C₂×#322	2

subgroup

index

Subgroups — #9860

Conjugacy classes of subgroups

Complements

Maximal subgroups