Group #13315 — subgroups

Subgroups — #13315

Order 256.

258 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 59 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	C₂²×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#473	C₂	1	2	yes	maximal
3	C₂×#362	C₂	1	2	yes	maximal, complemented
4	#2113	C₂	1	1	yes	maximal, characteristic
5	#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#137	—	2	2	no	—
7	C₂²×Q₁₆	C₂²	1	2	yes	—
8	C₂×#131	C₂²	1	1	yes	characteristic
9	C₂×Q₃₂	—	2	4	no	—
10	C₂×Q₃₂	—	2	2	no	—
11	C₂×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆	C₂²	1	1	yes	characteristic
13	#473	—	2	4	no	—
14	#473	—	2	4	no	—
15	C₂×#108	C₂²	1	2	yes	—
16	C₂×#101	—	2	2	no	—
17	#362	—	2	4	no	—
18	C₂²⋊C₁₆	—	2	2	no	—
19	C₂×#41	D₄	1	1	yes	characteristic
20	C₂²×Q₈	—	2	2	no	—
21	#137	—	2	8	no	—
22	#137	—	4	4	no	—
23	C₂×Q₁₆	—	2	2	no	—
24	C₂×Q₁₆	—	2	2	no	—
25	C₂×Q₁₆	—	2	2	no	—
26	C₂×Q₁₆	—	2	2	no	—
27	C₂×Q₁₆	—	2	2	no	—
28	C₂×Q₁₆	—	4	2	no	—
29	C₂×Q₁₆	D₄	1	4	yes	—
30	C₂×SD₁₆	—	4	2	no	—
31	#131	—	2	2	no	—
32	#131	—	2	2	no	—
33	C₂×M₁₆	—	2	1	no	—
34	C₂×M₁₆	—	2	2	no	—
35	C₂×M₁₆	D₄	1	1	yes	characteristic
36	C₂²×C₈	—	2	1	no	—
37	C₂²×C₈	C₂³	1	1	yes	characteristic
38	Q₃₂	—	2	8	no	—
39	M₃₂	—	2	2	no	—
40	C₂×C₁₆	—	2	2	no	—
41	C₂×C₁₆	—	2	1	no	—
42	C₂×C₁₆	D₄	1	2	yes	—
43	#108	—	2	2	no	—
44	#108	D₄	1	4	yes	—
45	#101	—	2	8	no	—
46	#41	—	2	2	no	—
47	#41	—	2	2	no	—
48	C₂×Q₈	—	2	4	no	—
49	C₂×Q₈	—	4	2	no	—
50	C₂×Q₈	—	4	2	no	—
51	C₂×Q₈	—	4	2	no	—
52	C₂×Q₈	—	4	2	no	—
53	C₂×Q₈	—	4	2	no	—
54	C₂×Q₈	D₈	1	1	yes	characteristic
55	C₂×D₄	—	2	1	no	—
56	C₂×D₄	D₈	1	1	yes	characteristic
57	C₂²×C₄	—	2	1	no	—
58	C₂²×C₄	—	4	2	no	—
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	Q₁₆	—	2	4	no	—
61	Q₁₆	—	2	4	no	—
62	Q₁₆	—	4	8	no	—
63	Q₁₆	—	4	2	no	—
64	Q₁₆	—	4	2	no	—
65	SD₁₆	—	4	8	no	—
66	M₁₆	—	2	2	no	—
67	M₁₆	—	2	4	no	—
68	M₁₆	—	4	2	no	—
69	M₁₆	—	4	2	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	4	2	no	—
72	C₂×C₈	—	4	1	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	C₂×D₄	1	2	yes	—
75	C₂×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₈	#41	1	1	yes	characteristic
78	C₁₆	—	2	2	no	—
79	C₁₆	—	4	1	no	—
80	C₂³	—	2	1	no	—
81	C₂³	#135	1	1	yes	characteristic
82	Q₈	—	2	2	no	—
83	Q₈	—	4	4	no	—
84	Q₈	—	4	4	no	—
85	Q₈	—	8	2	no	—
86	Q₈	—	8	2	no	—
87	D₄	—	2	2	no	—
88	D₄	—	4	2	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	4	4	no	—
93	C₂×C₄	—	8	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	#120	1	2	yes	—
96	C₂×C₄	#121	1	1	yes	characteristic
97	C₂×C₄	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
99	C₈	—	4	1	no	—
100	C₈	—	4	4	no	—
101	C₈	#120	1	1	yes	characteristic
102	C₈	#121	1	2	yes	—
103	C₈	#127	1	1	yes	characteristic
104	C₂²	—	2	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	—	4	1	no	—
107	C₂²	#449	1	2	yes	—
108	C₂²	#466	1	1	yes	characteristic
109	C₄	—	4	1	no	—
110	C₄	—	8	4	no	—
111	C₄	#392	1	1	yes	characteristic
112	C₄	#447	1	2	yes	—
113	C₄	#469	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	—	8	1	no	—
116	C₂	#1999	1	1	yes	characteristic
117	C₂	#2222	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂

C₂

yes

maximal, characteristic, complemented

C₂×#473

C₂

yes

maximal

C₂×#362

C₂

yes

maximal, complemented

#2113

C₂

yes

maximal, characteristic

#1381

C₂

yes

maximal, characteristic, complemented

C₂×#137

—

C₂²×Q₁₆

C₂²

yes

—

C₂×#131

C₂²

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

—

C₂²

yes

—

C₂×#101

—

#362

—

C₂²⋊C₁₆

—

C₂×#41

D₄

yes

characteristic

C₂²×Q₈

—

#137

—

#137

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

D₄

yes

—

C₂×SD₁₆

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂³

yes

characteristic

Q₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

—

#108

—

#108

D₄

yes

—

#101

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₈

yes

characteristic

C₂×D₄

—

C₂×D₄

D₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

#41

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

—

C₂³

#135

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#120

yes

—

C₂×C₄

#121

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

AGL(1,ℤ/8)

yes

—

C₈

—

100

C₈

—

101

C₈

#120

yes

characteristic

102

C₈

#121

yes

—

103

C₈

#127

yes

characteristic

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

#449

yes

—

108

C₂²

#466

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

#392

yes

characteristic

112

C₄

#447

yes

—

113

C₄

#469

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#1999

yes

characteristic

117

C₂

#2222

yes

—

	subgroup	G/H
center, socle	C₂²	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂×#41

D₄

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂²×Q₃₂	C₂	semidirect
3. C₂×#362	C₂	semidirect
5. #1381	C₂	semidirect

H ≅ G/N

type

1. C₂²×Q₃₂

C₂

semidirect

3. C₂×#362

C₂

semidirect

5. #1381

C₂

semidirect

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

subgroup	index
C₂²×Q₃₂	2
C₂×#473	2
C₂×#362	2
#2113	2
#1381	2

subgroup

index

C₂²×Q₃₂

Subgroups — #13315

Conjugacy classes of subgroups

Complements

Maximal subgroups