Group #151738 — subgroups

Subgroups — #151738

Order 512.

208 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 62 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	#13727	C₂	1	2	yes	maximal, complemented
2	#13719	C₂	1	2	yes	maximal, complemented
3	#12515	C₂	1	1	yes	maximal, characteristic
4	#12504	C₂	1	1	yes	maximal, characteristic, complemented
5	#7331	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×SD₆₄	—	2	4	no	—
7	#2252	C₂²	1	1	yes	characteristic
8	#2247	C₂²	1	1	yes	characteristic
9	#2246	—	2	1	no	—
10	#2246	C₂²	1	2	yes	—
11	#2241	—	2	1	no	—
12	#1712	—	2	1	no	—
13	#1707	—	2	1	no	—
14	#1415	—	2	4	no	—
15	C₄×C₃₂	C₂²	1	2	yes	—
16	C₈×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×Q₃₂	—	2	2	no	—
18	C₂×D₁₆	—	2	2	no	—
19	C₂×D₁₆	—	2	2	no	—
20	C₂×D₁₆	—	4	2	no	—
21	#500	—	2	1	no	—
22	#498	—	4	1	no	—
23	#494	—	2	1	no	—
24	#493	—	2	2	no	—
25	#493	—	4	1	no	—
26	#493	—	4	1	no	—
27	#487	—	4	1	no	—
28	SD₆₄	—	4	8	no	—
29	C₂×C₃₂	D₄	1	4	yes	—
30	#366	—	2	2	no	—
31	#358	—	4	2	no	—
32	C₄×C₁₆	D₄	1	1	yes	characteristic
33	C₄×C₁₆	C₂³	1	1	yes	characteristic
34	C₈²	D₄	1	1	yes	characteristic
35	C₂×Q₁₆	—	4	1	no	—
36	C₂×Q₁₆	—	4	1	no	—
37	C₂×D₈	—	4	2	no	—
38	C₂×D₈	—	4	2	no	—
39	C₂×D₈	—	8	2	no	—
40	C₂×D₈	—	8	2	no	—
41	C₄⋊Q₈	—	4	1	no	—
42	C₄⋊Q₈	—	4	1	no	—
43	#127	—	4	2	no	—
44	Q₃₂	—	4	4	no	—
45	D₁₆	—	4	4	no	—
46	D₁₆	—	4	4	no	—
47	D₁₆	—	8	4	no	—
48	C₂×C₁₆	D₈	1	2	yes	—
49	C₂×C₁₆	C₂×D₄	1	2	yes	—
50	#107	—	4	1	no	—
51	#107	—	4	1	no	—
52	#106	—	8	2	no	—
53	#103	—	8	2	no	—
54	C₄×C₈	D₈	1	1	yes	characteristic
55	C₄×C₈	D₈	1	1	yes	characteristic
56	C₄×C₈	C₂×D₄	1	1	yes	characteristic
57	C₃₂	—	2	4	no	—
58	C₂×Q₈	—	8	1	no	—
59	C₂×Q₈	—	8	1	no	—
60	C₂×D₄	—	8	2	no	—
61	C₂×D₄	—	8	2	no	—
62	C₂×D₄	—	8	2	no	—
63	Q₁₆	—	8	2	no	—
64	Q₁₆	—	8	2	no	—
65	D₈	—	8	2	no	—
66	D₈	—	8	2	no	—
67	D₈	—	8	4	no	—
68	D₈	—	16	4	no	—
69	D₈	—	16	4	no	—
70	C₂×C₈	D₁₆	1	2	yes	—
71	C₂×C₈	D₁₆	1	2	yes	—
72	C₂×C₈	#127	1	1	yes	characteristic
73	C₂×C₈	C₂×D₈	1	1	yes	characteristic
74	C₄⋊C₄	—	8	1	no	—
75	C₄⋊C₄	—	8	1	no	—
76	C₄⋊C₄	—	8	1	no	—
77	C₄⋊C₄	—	8	1	no	—
78	C₄²	C₂×D₈	1	1	yes	characteristic
79	C₁₆	—	2	2	no	—
80	C₁₆	AGL(1,ℤ/8)	1	4	yes	—
81	C₂³	—	16	2	no	—
82	Q₈	—	16	1	no	—
83	Q₈	—	16	1	no	—
84	Q₈	—	16	2	no	—
85	D₄	—	16	2	no	—
86	D₄	—	16	2	no	—
87	D₄	—	16	4	no	—
88	D₄	—	16	4	no	—
89	C₂×C₄	—	16	1	no	—
90	C₂×C₄	—	16	1	no	—
91	C₂×C₄	#493	1	1	yes	characteristic
92	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
93	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
94	C₈	—	2	2	no	—
95	C₈	SD₆₄	1	4	yes	complemented
96	C₈	#496	1	2	yes	—
97	C₈	#509	1	2	yes	—
98	C₂²	—	32	2	no	—
99	C₂²	—	32	2	no	—
100	C₂²	—	32	2	no	—
101	C₂²	#2246	1	1	yes	characteristic
102	C₄	—	32	1	no	—
103	C₄	—	32	1	no	—
104	C₄	#1721	1	1	yes	characteristic
105	C₄	#2250	1	1	yes	characteristic
106	C₄	C₂×SD₆₄	1	2	yes	—
107	C₄	#2263	1	2	yes	—
108	C₂	—	64	2	no	—
109	C₂	#12525	1	1	yes	characteristic
110	C₂	#13727	1	1	yes	characteristic
111	C₂	#13729	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13727

C₂

yes

maximal, complemented

#13719

C₂

yes

maximal, complemented

#12515

C₂

yes

maximal, characteristic

#12504

C₂

yes

maximal, characteristic, complemented

#7331

C₂

yes

maximal, characteristic, complemented

C₂×SD₆₄

—

#2252

C₂²

yes

characteristic

#2247

C₂²

yes

characteristic

#2246

—

#2246

C₂²

yes

—

—

—

—

—

C₄×C₃₂

C₂²

yes

—

C₈×C₁₆

C₂²

yes

characteristic

C₂×Q₃₂

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

—

—

—

—

—

—

—

SD₆₄

—

C₂×C₃₂

D₄

yes

—

#366

—

#358

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

C₂³

yes

characteristic

C₈²

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

D₁₆

—

D₁₆

—

D₁₆

—

C₂×C₁₆

D₈

yes

—

C₂×C₁₆

C₂×D₄

yes

—

—

—

—

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

#127

yes

characteristic

C₂×C₈

C₂×D₈

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₈

yes

characteristic

C₁₆

—

C₁₆

AGL(1,ℤ/8)

yes

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#493

yes

characteristic

C₂×C₄

C₂×D₁₆

yes

characteristic

C₂×C₄

C₂×D₁₆

yes

characteristic

C₈

—

C₈

SD₆₄

yes

complemented

C₈

#496

yes

—

C₈

#509

yes

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#2246

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

#1721

yes

characteristic

105

C₄

#2250

yes

characteristic

106

C₄

C₂×SD₆₄

yes

—

107

C₄

#2263

yes

—

108

C₂

—

109

C₂

#12525

yes

characteristic

110

C₂

#13727

yes

characteristic

111

C₂

#13729

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2246
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	#12515	C₂
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2246

commutator, Frattini, cosocle

C₄×C₁₆

C₂³

prime radical

#12515

C₂

LCS

G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G

LECS

G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #13727	C₂	semidirect
2. #13719	C₂	semidirect
4. #12504	C₂	semidirect
5. #7331	C₂	semidirect
95. C₈	SD₆₄	semidirect

H ≅ G/N

type

1. #13727

C₂

semidirect

2. #13719

C₂

semidirect

4. #12504

C₂

semidirect

5. #7331

C₂

semidirect

95. C₈

SD₆₄

semidirect

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

subgroup	index
#13727	2
#13719	2
#12515	2
#12504	2
#7331	2

subgroup

index

Subgroups — #151738

Conjugacy classes of subgroups

Complements

Maximal subgroups