Group #13488 — subgroups

Subgroups — #13488

Order 256.

151 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 151 conjugacy classes of subgroups — too many to draw. Only its 53 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×#362	C₂	1	2	yes	maximal
2	#1971	C₂	1	1	yes	maximal, characteristic
3	#1849	C₂	1	1	yes	maximal, characteristic
4	#1383	C₂	1	2	yes	maximal
5	#1377	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂×M₃₂	C₂²	1	2	yes	—
8	C₂×#108	—	2	1	no	—
9	C₂×#108	C₂²	1	2	yes	—
10	C₂×#106	C₂²	1	1	yes	characteristic
11	#423	—	2	1	no	—
12	C₂×#103	—	2	1	no	—
13	C₂×#97	C₂²	1	1	yes	characteristic
14	#391	—	2	1	no	—
15	#362	—	2	4	no	—
16	C₂²×Q₈	—	2	1	no	—
17	C₂×Q₁₆	—	2	2	no	—
18	C₂×Q₁₆	—	2	2	no	—
19	C₂×Q₁₆	—	2	1	no	—
20	C₂×Q₁₆	D₄	1	2	yes	—
21	C₂×M₁₆	—	2	2	no	—
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄	—	2	1	no	—
25	C₂×C₄²	D₄	1	1	yes	characteristic
26	M₃₂	—	2	2	no	—
27	M₃₂	—	2	2	no	—
28	C₂×C₁₆	—	2	2	no	—
29	#108	—	2	2	no	—
30	#108	—	2	2	no	—
31	#108	—	4	2	no	—
32	#106	—	2	2	no	—
33	C₄⋊C₈	—	4	2	no	—
34	#103	—	4	2	no	—
35	#97	—	2	2	no	—
36	#95	—	4	2	no	—
37	C₂×Q₈	—	2	2	no	—
38	C₂×Q₈	—	4	1	no	—
39	C₂×Q₈	—	4	1	no	—
40	C₂×Q₈	—	4	1	no	—
41	C₂×Q₈	—	4	1	no	—
42	C₂×Q₈	—	4	1	no	—
43	C₂²×C₄	—	4	1	no	—
44	C₂²×C₄	—	4	1	no	—
45	C₂²×C₄	SD₁₆	1	2	yes	—
46	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
47	Q₁₆	—	2	4	no	—
48	Q₁₆	—	4	2	no	—
49	M₁₆	—	4	2	no	—
50	M₁₆	—	4	2	no	—
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂×C₈	—	4	2	no	—
54	C₂×C₈	D₈	1	2	yes	—
55	C₂×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈	#41	1	2	yes	—
57	C₂×C₈	#41	1	2	yes	—
58	C₂×C₈	#41	1	1	yes	characteristic
59	C₄⋊C₄	—	4	1	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄²	—	2	1	no	—
62	C₄²	—	2	1	no	—
63	C₁₆	—	4	2	no	—
64	C₂³	C₂×SD₁₆	1	1	yes	characteristic
65	Q₈	—	4	2	no	—
66	Q₈	—	4	2	no	—
67	Q₈	—	8	1	no	—
68	Q₈	—	8	1	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	—	8	1	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	#121	1	1	yes	characteristic
77	C₂×C₄	#123	1	1	yes	characteristic
78	C₂×C₄	#124	1	1	yes	characteristic
79	C₂×C₄	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄	#137	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	—	8	2	no	—
84	C₈	#123	1	2	yes	—
85	C₈	#126	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	#461	1	1	yes	characteristic
88	C₂²	#481	1	1	yes	characteristic
89	C₂²	#486	1	1	yes	characteristic
90	C₄	—	4	2	no	—
91	C₄	—	8	2	no	—
92	C₄	—	16	1	no	—
93	C₄	#397	1	1	yes	characteristic
94	C₄	#459	1	1	yes	characteristic
95	C₄	#480	1	1	yes	characteristic
96	C₄	#489	1	1	yes	characteristic
97	C₂	—	2	1	no	—
98	C₂	—	2	1	no	—
99	C₂	#2071	1	1	yes	characteristic
100	C₂	#2239	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#362

C₂

yes

maximal

#1971

C₂

yes

maximal, characteristic

#1849

C₂

yes

maximal, characteristic

#1383

C₂

yes

maximal

#1377

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

—

C₂×#108

—

C₂×#108

C₂²

yes

—

C₂×#106

C₂²

yes

characteristic

#423

—

C₂×#103

—

C₂×#97

C₂²

yes

characteristic

#391

—

#362

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

D₄

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

C₄⋊C₈

—

#103

—

#97

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

SD₁₆

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₈

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₂³

C₂×SD₁₆

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

#123

yes

—

C₈

#126

yes

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

C₄

yes

C₂

—

C₂

—

C₂

#2071

yes

characteristic

100

C₂

#2239

yes

—

	subgroup	G/H
center, socle	C₂²	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#481

commutator, Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

C₂×SD₁₆

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#362	2
#1971	2
#1849	2
#1383	2
#1377	2

subgroup

index

Subgroups — #13488

Conjugacy classes of subgroups

Maximal subgroups