Group #13415 — subgroups

Subgroups — #13415

Order 256.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂×#362	C₂	1	2	yes	maximal
3	#2141	C₂	1	2	yes	maximal
4	#1945	C₂	1	1	yes	maximal, characteristic
5	#1376	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆	—	2	1	no	—
7	C₂²×Q₁₆	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈	—	2	1	no	—
9	C₂×M₃₂	C₂²	1	2	yes	—
10	#494	—	2	4	no	—
11	C₂×#108	C₂²	1	1	yes	characteristic
12	#427	C₂²	1	1	yes	characteristic
13	#424	—	2	1	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	#362	—	2	4	no	—
16	#358	—	2	4	no	—
17	#339	—	2	1	no	—
18	C₂²×Q₈	—	2	2	no	—
19	C₂×Q₁₆	—	2	4	no	—
20	C₂×Q₁₆	—	2	2	no	—
21	C₂×Q₁₆	—	2	2	no	—
22	C₂×Q₁₆	—	2	2	no	—
23	C₂×Q₁₆	—	2	2	no	—
24	C₂×Q₁₆	—	2	2	no	—
25	C₂×Q₁₆	—	4	2	no	—
26	C₂×Q₁₆	—	4	2	no	—
27	C₂×Q₁₆	D₄	1	4	yes	—
28	C₂×M₁₆	—	2	1	no	—
29	C₂²×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈	—	2	4	no	—
32	C₄⋊Q₈	—	4	2	no	—
33	#117	—	2	1	no	—
34	C₂×C₄⋊C₄	—	4	2	no	—
35	C₂×C₄²	D₄	1	1	yes	characteristic
36	M₃₂	—	2	2	no	—
37	M₃₂	—	2	2	no	—
38	C₂×C₁₆	—	2	2	no	—
39	#108	—	2	2	no	—
40	#107	D₄	1	2	yes	—
41	#106	—	2	1	no	—
42	C₄⋊C₈	—	4	2	no	—
43	C₄×C₈	—	2	2	no	—
44	C₂×Q₈	—	2	4	no	—
45	C₂×Q₈	—	4	2	no	—
46	C₂×Q₈	—	4	2	no	—
47	C₂×Q₈	—	4	2	no	—
48	C₂×Q₈	—	4	2	no	—
49	C₂×Q₈	—	4	2	no	—
50	C₂²×C₄	—	2	1	no	—
51	C₂²×C₄	—	4	2	no	—
52	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
53	Q₁₆	—	2	4	no	—
54	Q₁₆	—	2	4	no	—
55	Q₁₆	—	4	8	no	—
56	Q₁₆	—	4	2	no	—
57	Q₁₆	—	4	2	no	—
58	M₁₆	—	4	1	no	—
59	M₁₆	—	4	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	4	1	no	—
63	C₂×C₈	D₈	1	2	yes	—
64	C₂×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈	C₂×D₄	1	2	yes	—
66	C₂×C₈	#41	1	2	yes	—
67	C₂×C₈	#41	1	1	yes	characteristic
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	4	8	no	—
70	C₂²⋊C₄	—	4	1	no	—
71	C₄²	—	2	1	no	—
72	C₄²	—	4	1	no	—
73	C₄²	SD₁₆	1	2	yes	—
74	C₁₆	—	4	2	no	—
75	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Q₈	—	4	4	no	—
77	Q₈	—	4	4	no	—
78	Q₈	—	8	2	no	—
79	Q₈	—	8	2	no	—
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	8	2	no	—
85	C₂×C₄	—	8	2	no	—
86	C₂×C₄	#120	1	1	yes	characteristic
87	C₂×C₄	#121	1	1	yes	characteristic
88	C₂×C₄	#124	1	1	yes	characteristic
89	C₂×C₄	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₈	—	2	2	no	—
93	C₈	—	8	1	no	—
94	C₈	#121	1	2	yes	—
95	C₈	#123	1	2	yes	—
96	C₂²	—	2	2	no	—
97	C₂²	#450	1	1	yes	characteristic
98	C₂²	#459	1	1	yes	characteristic
99	C₂²	#490	1	1	yes	characteristic
100	C₄	—	2	4	no	—
101	C₄	—	8	2	no	—
102	C₄	—	8	4	no	—
103	C₄	#394	1	1	yes	characteristic
104	C₄	#447	1	1	yes	characteristic
105	C₄	#460	1	1	yes	characteristic
106	C₄	#486	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	#2041	1	1	yes	characteristic
110	C₂	#2214	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#494

C₂

yes

maximal, characteristic

C₂×#362

C₂

yes

maximal

#2141

C₂

yes

maximal

#1945

C₂

yes

maximal, characteristic

#1376

C₂

yes

maximal, characteristic

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂²

yes

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

C₂²

yes

—

#494

—

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

#424

—

C₂×C₄×C₈

C₂²

yes

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

D₄

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#107

D₄

yes

—

#106

—

C₄⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₈

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

SD₁₆

yes

—

C₁₆

—

C₂³

AGL(1,ℤ/8)

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

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

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

C₈

—

C₈

—

C₈

#121

yes

—

C₈

#123

yes

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

#394

yes

characteristic

104

C₄

#447

yes

characteristic

105

C₄

#460

yes

characteristic

106

C₄

#486

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

#2041

yes

characteristic

110

C₂

#2214

yes

—

	subgroup	G/H
center, socle	C₂²	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#459

commutator, Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

AGL(1,ℤ/8)

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ 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×#494	2
C₂×#362	2
#2141	2
#1945	2
#1376	2

subgroup

index

Subgroups — #13415

Conjugacy classes of subgroups

Maximal subgroups