Group #13537 — subgroups

Subgroups — #13537

Order 256.

147 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 147 conjugacy classes of subgroups — too many to draw. Only its 65 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	#1942	C₂	1	1	yes	maximal, characteristic
2	#1845	C₂	1	1	yes	maximal, characteristic
3	#1380	C₂	1	2	yes	maximal
4	#1380	C₂	1	2	yes	maximal
5	#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₆	C₂²	1	2	yes	—
7	C₂×#107	—	2	1	no	—
8	C₂×#107	C₂²	1	2	yes	—
9	C₂×#107	C₂²	1	1	yes	characteristic
10	C₂×#107	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#389	—	2	1	no	—
13	#384	—	2	1	no	—
14	#340	—	2	1	no	—
15	C₂²×C₈	D₄	1	1	yes	characteristic
16	C₂²×C₈	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊C₄	—	2	1	no	—
18	C₂×C₄⋊C₄	—	2	1	no	—
19	C₂×C₄⋊C₄	—	2	2	no	—
20	C₂×C₄⋊C₄	—	4	1	no	—
21	C₂×C₄⋊C₄	—	4	1	no	—
22	C₂×C₄⋊C₄	—	4	1	no	—
23	C₂×C₄²	D₄	1	1	yes	characteristic
24	C₂×C₁₆	—	2	2	no	—
25	C₂×C₁₆	—	2	2	no	—
26	C₂×C₁₆	—	2	2	no	—
27	#107	—	2	1	no	—
28	#107	—	2	2	no	—
29	#107	—	2	4	no	—
30	#107	—	4	2	no	—
31	#107	Q₈	1	2	yes	—
32	C₄×C₈	—	2	2	no	—
33	#95	—	4	1	no	—
34	C₂²×C₄	—	4	1	no	—
35	C₂²×C₄	—	4	1	no	—
36	C₂²×C₄	—	4	2	no	—
37	C₂²×C₄	D₈	1	1	yes	characteristic
38	C₂²×C₄	D₈	1	1	yes	characteristic
39	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	SD₁₆	1	2	yes	—
43	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈	#41	1	2	yes	—
45	C₂×C₈	#41	1	2	yes	—
46	C₂×C₈	#41	1	1	yes	characteristic
47	C₄⋊C₄	—	2	2	no	—
48	C₄⋊C₄	—	4	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₄⋊C₄	—	4	2	no	—
51	C₄⋊C₄	—	4	2	no	—
52	C₄⋊C₄	—	4	1	no	—
53	C₄⋊C₄	—	8	2	no	—
54	C₄⋊C₄	—	8	2	no	—
55	C₄⋊C₄	—	8	2	no	—
56	C₄²	—	2	1	no	—
57	C₄²	—	2	1	no	—
58	C₁₆	—	4	2	no	—
59	C₂³	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	8	1	no	—
66	C₂×C₄	—	8	1	no	—
67	C₂×C₄	—	8	1	no	—
68	C₂×C₄	—	8	2	no	—
69	C₂×C₄	—	8	1	no	—
70	C₂×C₄	—	8	2	no	—
71	C₂×C₄	—	8	2	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	D₁₆	1	2	yes	—
74	C₂×C₄	SD₃₂	1	2	yes	—
75	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄	#123	1	1	yes	characteristic
77	C₂×C₄	#124	1	1	yes	characteristic
78	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄	#137	1	1	yes	characteristic
80	C₂×C₄	#137	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₈	#125	1	2	yes	—
83	C₈	#126	1	2	yes	—
84	C₂²	#474	1	1	yes	characteristic
85	C₂²	#480	1	1	yes	characteristic
86	C₂²	#486	1	1	yes	characteristic
87	C₂²	C₂×D₁₆	1	1	yes	characteristic
88	C₂²	C₂×SD₃₂	1	1	yes	characteristic
89	C₂²	#510	1	1	yes	characteristic
90	C₂²	#510	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	8	2	no	—
94	C₄	—	16	1	no	—
95	C₄	—	16	2	no	—
96	C₄	#400	1	1	yes	characteristic
97	C₄	#475	1	1	yes	characteristic
98	C₄	#483	1	1	yes	characteristic
99	C₄	#491	1	1	yes	characteristic
100	C₂	#2099	1	1	yes	characteristic
101	C₂	#2224	1	1	yes	characteristic
102	C₂	#2225	1	1	yes	characteristic
103	C₂	#2232	1	1	yes	characteristic
104	C₂	#2235	1	1	yes	characteristic
105	C₂	#2240	1	1	yes	characteristic
106	C₂	#2245	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1942

C₂

yes

maximal, characteristic

#1845

C₂

yes

maximal, characteristic

#1380

C₂

yes

maximal

#1380

C₂

yes

maximal

#1374

C₂

yes

maximal, characteristic

C₂²×C₁₆

C₂²

yes

—

C₂×#107

—

C₂×#107

C₂²

yes

—

C₂×#107

C₂²

yes

characteristic

C₂×#107

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

—

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

Q₈

yes

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₈

yes

characteristic

C₂²×C₄

D₈

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₂³

C₂×D₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₁₆

yes

—

C₂×C₄

SD₃₂

yes

—

C₂×C₄

C₂²⋊Q₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×SD₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

#125

yes

—

C₈

#126

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

C₂×D₁₆

yes

characteristic

C₂²

C₂×SD₃₂

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

C₄

yes

100

C₂

#2099

yes

characteristic

101

C₂

#2224

yes

characteristic

102

C₂

#2225

yes

characteristic

103

C₂

#2232

yes

characteristic

104

C₂

#2235

yes

characteristic

105

C₂

#2240

yes

characteristic

106

C₂

#2245

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
#1942	2
#1845	2
#1380	2
#1374	2

subgroup

index

Subgroups — #13537

Conjugacy classes of subgroups

Maximal subgroups