Group #128203 — subgroups

Subgroups — #128203

Order 512.

426 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 322 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	C₄²×C₁₆	C₂	1	2	yes	maximal
2	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
3	#7136	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆	C₄	1	4	yes	—
5	C₂×C₄×C₁₆	C₄	1	4	yes	—
6	C₂×C₄×C₁₆	C₂²	1	4	yes	—
7	C₄²×C₈	C₄	1	2	yes	—
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈²	C₄	1	2	yes	—
10	C₂×C₈²	C₂²	1	2	yes	—
11	C₂²×C₁₆	C₂×C₄	1	8	yes	—
12	C₂×C₄×C₈	C₈	1	4	yes	—
13	C₂×C₄×C₈	C₈	1	4	yes	—
14	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
15	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
19	C₄³	C₂×C₄	1	1	yes	characteristic
20	C₄×C₁₆	—	2	8	no	—
21	C₄×C₁₆	—	2	8	no	—
22	C₄×C₁₆	C₈	1	16	yes	complemented
23	C₈²	—	2	4	no	—
24	C₈²	—	2	4	no	—
25	C₂²×C₈	C₄²	1	2	yes	—
26	C₂²×C₈	C₂×C₈	1	8	yes	—
27	C₂²×C₈	C₂²×C₄	1	2	yes	—
28	C₂×C₄²	C₄²	1	1	yes	characteristic
29	C₂×C₄²	C₄²	1	1	yes	characteristic
30	C₂×C₄²	C₂×C₈	1	2	yes	—
31	C₂×C₄²	C₂×C₈	1	2	yes	—
32	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₁₆	—	2	8	no	—
34	C₂×C₁₆	—	2	8	no	—
35	C₂×C₁₆	C₂×C₈	1	16	yes	—
36	C₄×C₈	—	2	8	no	—
37	C₄×C₈	—	2	4	no	—
38	C₄×C₈	—	2	4	no	—
39	C₄×C₈	C₁₆	1	16	yes	complemented
40	C₄×C₈	C₂×C₈	1	8	yes	—
41	C₄×C₈	M₁₆	1	8	yes	—
42	C₄×C₈	M₁₆	1	4	yes	—
43	C₄×C₈	#41	1	4	yes	—
44	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
45	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
46	C₂²×C₄	C₄×C₈	1	2	yes	—
47	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
48	C₂²×C₄	C₂²×C₈	1	2	yes	—
49	C₂×C₈	—	2	8	no	—
50	C₂×C₈	—	2	8	no	—
51	C₂×C₈	C₄×C₈	1	4	yes	—
52	C₂×C₈	#97	1	4	yes	—
53	C₂×C₈	#97	1	4	yes	—
54	C₂×C₈	C₂×C₁₆	1	16	yes	—
55	C₂×C₈	#117	1	4	yes	—
56	C₂×C₈	C₂²×C₈	1	4	yes	—
57	C₂×C₈	C₂×M₁₆	1	4	yes	—
58	C₄²	C₄×C₈	1	4	yes	—
59	C₄²	#97	1	4	yes	—
60	C₄²	C₂×C₁₆	1	8	yes	—
61	C₄²	M₃₂	1	8	yes	—
62	C₄²	#117	1	2	yes	—
63	C₄²	C₂×M₁₆	1	2	yes	—
64	C₁₆	—	2	16	no	—
65	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
66	C₂×C₄	C₈²	1	2	yes	—
67	C₂×C₄	C₈²	1	2	yes	—
68	C₂×C₄	#322	1	4	yes	—
69	C₂×C₄	C₄×C₁₆	1	4	yes	—
70	C₂×C₄	C₄×C₁₆	1	4	yes	—
71	C₂×C₄	#346	1	4	yes	—
72	C₂×C₄	#346	1	4	yes	—
73	C₂×C₄	#376	1	1	yes	characteristic
74	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
75	C₂×C₄	C₂×#97	1	1	yes	characteristic
76	C₂×C₄	C₂×#97	1	2	yes	—
77	C₂×C₄	#431	1	4	yes	—
78	C₂×C₄	C₂²×C₁₆	1	4	yes	—
79	C₂×C₄	C₂×M₃₂	1	4	yes	—
80	C₈	—	2	16	no	—
81	C₈	#322	1	8	yes	—
82	C₈	#431	1	8	yes	—
83	C₂²	C₂×C₈²	1	1	yes	characteristic
84	C₂²	C₂×#322	1	1	yes	characteristic
85	C₂²	#1744	1	1	yes	characteristic
86	C₂²	C₂×C₄×C₁₆	1	2	yes	—
87	C₂²	C₂×#346	1	2	yes	—
88	C₄	C₈×C₁₆	1	4	yes	—
89	C₄	#1311	1	4	yes	—
90	C₄	#1312	1	8	yes	—
91	C₄	C₂×#322	1	2	yes	—
92	C₄	#1744	1	2	yes	—
93	C₄	#2162	1	8	yes	—
94	C₂	#8131	1	1	yes	characteristic
95	C₂	C₂×C₈×C₁₆	1	1	yes	characteristic
96	C₂	C₂×#1311	1	1	yes	characteristic
97	C₂	C₂×#1312	1	2	yes	—
98	C₂	#12590	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×C₁₆

C₂

yes

maximal

C₄×C₈²

C₂

yes

maximal, characteristic

#7136

C₂

yes

maximal

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

C₄²×C₈

C₄

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₂×C₈²

C₄

yes

—

C₂×C₈²

C₂²

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂×C₄×C₈

C₈

yes

—

C₂×C₄×C₈

C₈

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₈

yes

complemented

C₈²

—

C₈²

—

C₂²×C₈

C₄²

yes

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

C₂²×C₄

yes

—

C₂×C₄²

C₄²

yes

characteristic

C₂×C₄²

C₄²

yes

characteristic

C₂×C₄²

C₂×C₈

yes

—

C₂×C₄²

C₂×C₈

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₈

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₁₆

yes

complemented

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

C₄×C₈

yes

characteristic

C₂²×C₄

C₄×C₈

yes

characteristic

C₂²×C₄

C₄×C₈

yes

—

C₂²×C₄

C₂×C₄²

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

#97

yes

—

C₂×C₈

#97

yes

—

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

#117

yes

—

C₂×C₈

C₂²×C₈

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₄²

C₄×C₈

yes

—

C₄²

#97

yes

—

C₄²

C₂×C₁₆

yes

—

C₄²

M₃₂

yes

—

C₄²

#117

yes

—

C₄²

C₂×M₁₆

yes

—

C₁₆

—

C₂³

C₂×C₄×C₈

yes

characteristic

C₂×C₄

C₈²

yes

—

C₂×C₄

C₈²

yes

—

C₂×C₄

#322

yes

—

C₂×C₄

C₄×C₁₆

yes

—

C₂×C₄

C₄×C₁₆

yes

—

C₂×C₄

#346

yes

—

C₂×C₄

#346

yes

—

C₂×C₄

#376

yes

characteristic

C₂×C₄

C₂×C₄×C₈

yes

—

C₂×C₄

C₂×#97

yes

characteristic

C₂×C₄

C₂×#97

yes

—

C₂×C₄

#431

yes

—

C₂×C₄

C₂²×C₁₆

yes

—

C₂×C₄

C₂×M₃₂

yes

—

C₈

—

C₈

#322

yes

—

C₈

#431

yes

—

C₂²

C₂×C₈²

yes

C₂²

yes

C₂²

yes

C₂²

C₂×C₄×C₁₆

yes

—

C₂²

C₂×#346

yes

—

C₄

C₈×C₁₆

yes

—

C₄

#1311

yes

—

C₄

#1312

yes

—

C₄

C₂×#322

yes

—

C₄

#1744

yes

—

C₄

#2162

yes

—

C₂

#8131

yes

characteristic

C₂

C₂×C₈×C₁₆

yes

C₂

yes

C₂

yes

—

C₂

#12590

yes

—

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

subgroup

G/H

center

C₄²×C₈

C₂²

commutator

C₂

C₂×C₈×C₁₆

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₂×C₄×C₈

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
22. C₄×C₁₆	C₈	semidirect
39. C₄×C₈	C₁₆	semidirect

H ≅ G/N

type

22. C₄×C₁₆

C₈

semidirect

39. C₄×C₈

C₁₆

semidirect

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

subgroup	index
C₄²×C₁₆	2
C₄×C₈²	2
#7136	2

subgroup

index

C₄²×C₁₆

C₄×C₈²

#7136

Subgroups — #128203

Conjugacy classes of subgroups

Complements

Maximal subgroups