Group #13530 — subgroups

Subgroups — #13530

Order 256.

179 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 179 conjugacy classes of subgroups — too many to draw. Only its 69 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×#366	C₂	1	2	yes	maximal
2	C₂×#358	C₂	1	2	yes	maximal
3	#1969	C₂	1	1	yes	maximal, characteristic
4	#1845	C₂	1	1	yes	maximal, characteristic
5	#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆	C₂²	1	2	yes	—
8	C₂×#107	—	2	1	no	—
9	C₂×#107	C₂²	1	2	yes	—
10	C₂×#107	C₂²	1	1	yes	characteristic
11	C₂×#103	—	2	1	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#391	—	2	1	no	—
14	#384	—	2	1	no	—
15	#366	—	2	4	no	—
16	#358	—	2	4	no	—
17	C₂²×Q₈	—	2	1	no	—
18	C₂×Q₁₆	—	2	2	no	—
19	C₂×Q₁₆	—	2	2	no	—
20	C₂×Q₁₆	—	2	1	no	—
21	C₂×Q₁₆	Q₈	1	2	yes	—
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄	—	2	2	no	—
25	C₂×C₄⋊C₄	—	2	1	no	—
26	C₂×C₄⋊C₄	—	4	2	no	—
27	C₂×C₄²	D₄	1	1	yes	characteristic
28	C₂×C₁₆	—	2	2	no	—
29	C₂×C₁₆	—	2	2	no	—
30	C₂×C₁₆	D₄	1	4	yes	—
31	#107	—	2	2	no	—
32	#107	—	2	2	no	—
33	#107	—	2	2	no	—
34	#107	—	4	2	no	—
35	#103	—	4	2	no	—
36	C₄×C₈	—	2	2	no	—
37	#95	—	4	2	no	—
38	C₂×Q₈	—	2	2	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	C₂×Q₈	—	4	1	no	—
44	C₂²×C₄	—	4	1	no	—
45	C₂²×C₄	—	4	1	no	—
46	C₂²×C₄	—	4	2	no	—
47	C₂²×C₄	D₈	1	2	yes	—
48	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
49	Q₁₆	—	2	4	no	—
50	Q₁₆	—	4	2	no	—
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂×C₈	D₈	1	2	yes	—
54	C₂×C₈	C₂×D₄	1	2	yes	—
55	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈	#41	1	2	yes	—
57	C₂×C₈	#41	1	1	yes	characteristic
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	4	1	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄⋊C₄	—	4	2	no	—
62	C₄⋊C₄	—	8	4	no	—
63	C₄²	—	2	1	no	—
64	C₄²	—	2	1	no	—
65	C₁₆	—	2	4	no	—
66	C₂³	C₂×D₈	1	1	yes	characteristic
67	Q₈	—	4	2	no	—
68	Q₈	—	4	2	no	—
69	Q₈	—	8	1	no	—
70	Q₈	—	8	1	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	8	1	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	—	8	1	no	—
77	C₂×C₄	—	8	2	no	—
78	C₂×C₄	—	8	4	no	—
79	C₂×C₄	—	8	2	no	—
80	C₂×C₄	Q₃₂	1	4	yes	—
81	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄	#123	1	1	yes	characteristic
83	C₂×C₄	#127	1	1	yes	characteristic
84	C₂×C₄	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄	#137	1	1	yes	characteristic
86	C₂×C₄	#137	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₈	C₂²⋊Q₈	1	2	yes	—
89	C₈	#123	1	2	yes	—
90	C₂²	#474	1	1	yes	characteristic
91	C₂²	#480	1	1	yes	characteristic
92	C₂²	#493	1	1	yes	characteristic
93	C₂²	C₂×Q₃₂	1	2	yes	—
94	C₂²	#509	1	2	yes	—
95	C₄	—	2	4	no	—
96	C₄	—	8	2	no	—
97	C₄	—	16	2	no	—
98	C₄	—	16	1	no	—
99	C₄	#399	1	1	yes	characteristic
100	C₄	#474	1	1	yes	characteristic
101	C₄	#480	1	1	yes	characteristic
102	C₄	#497	1	1	yes	characteristic
103	C₂	#2093	1	1	yes	characteristic
104	C₂	#2227	1	2	yes	—
105	C₂	#2236	1	2	yes	—
106	C₂	#2247	1	1	yes	characteristic
107	C₂	#2250	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#366

C₂

yes

maximal

C₂×#358

C₂

yes

maximal

#1969

C₂

yes

maximal, characteristic

#1845

C₂

yes

maximal, characteristic

#1374

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

C₂×#107

—

C₂×#107

C₂²

yes

—

C₂×#107

C₂²

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Q₈

yes

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

—

—

—

—

—

—

C₄×C₈

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₈

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₈

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₂³

C₂×D₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Q₃₂

yes

—

C₂×C₄

C₂²⋊Q₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

C₂²⋊Q₈

yes

—

C₈

#123

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

C₂×Q₃₂

yes

—

C₂²

#509

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#399

yes

characteristic

100

C₄

#474

yes

characteristic

101

C₄

#480

yes

characteristic

102

C₄

#497

yes

characteristic

103

C₂

#2093

yes

characteristic

104

C₂

#2227

yes

—

105

C₂

#2236

yes

—

106

C₂

#2247

yes

characteristic

107

C₂

#2250

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₈

commutator, Frattini, cosocle

C₂²×C₈

C₂³

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ 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×#366	2
C₂×#358	2
#1969	2
#1845	2
#1374	2

subgroup

index

Subgroups — #13530

Conjugacy classes of subgroups

Maximal subgroups