Group #19653 — subgroups

Subgroups — #19653

Order 256.

365 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 139 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	#3229	C₂	1	4	yes	maximal
2	#3160	C₂	1	1	yes	maximal, characteristic
3	C₂×#494	C₂	1	1	yes	maximal, characteristic
4	#1716	C₂	1	4	yes	maximal, complemented
5	#1707	C₂	1	4	yes	maximal, complemented
6	#1450	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆	—	2	2	no	—
8	#533	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
10	#500	C₂²	1	2	yes	—
11	#499	—	2	2	no	—
12	#498	C₂²	1	2	yes	—
13	#498	C₂²	1	8	yes	—
14	#494	—	2	2	no	—
15	#494	C₂²	1	2	yes	—
16	#494	C₂²	1	2	yes	—
17	#487	C₂²	1	8	yes	—
18	#483	—	2	8	no	—
19	#475	—	2	8	no	—
20	#431	C₂²	1	2	yes	—
21	#427	—	2	2	no	—
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	#322	C₂²	1	2	yes	—
24	C₈²	C₂²	1	2	yes	—
25	C₂²×Q₈	—	2	2	no	—
26	C₂×Q₁₆	—	2	4	no	—
27	C₂×Q₁₆	—	2	4	no	—
28	C₂×Q₁₆	—	4	8	no	—
29	C₂²×C₈	D₄	1	2	yes	—
30	C₄⋊Q₈	—	2	4	no	—
31	C₄⋊Q₈	—	2	2	no	—
32	C₄⋊Q₈	C₂³	1	4	yes	—
33	C₄⋊Q₈	C₂³	1	4	yes	—
34	#125	—	2	2	no	—
35	C₂²⋊Q₈	—	2	4	no	—
36	C₄×Q₈	—	2	4	no	—
37	#117	—	2	2	no	—
38	C₂×C₄⋊C₄	—	2	4	no	—
39	C₂×C₄²	C₂³	1	1	yes	characteristic
40	#107	—	2	4	no	—
41	#106	—	2	4	no	—
42	#106	—	2	16	no	—
43	C₄⋊C₈	D₄	1	4	yes	—
44	#103	—	2	16	no	—
45	C₂²⋊C₈	D₄	1	4	yes	—
46	C₄×C₈	D₄	1	2	yes	—
47	C₄×C₈	C₂³	1	1	yes	characteristic
48	C₄×C₈	C₂³	1	4	yes	—
49	C₄×C₈	C₂³	1	1	yes	characteristic
50	C₂×Q₈	—	2	4	no	—
51	C₂×Q₈	—	2	4	no	—
52	C₂×Q₈	—	4	2	no	—
53	C₂×Q₈	—	4	8	no	—
54	C₂²×C₄	—	4	2	no	—
55	C₂²×C₄	C₂×D₄	1	2	yes	—
56	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
57	Q₁₆	—	4	8	no	—
58	Q₁₆	—	4	8	no	—
59	C₂×C₈	SD₁₆	1	8	yes	—
60	C₂×C₈	C₂×D₄	1	8	yes	—
61	C₂×C₈	C₂×D₄	1	2	yes	—
62	C₂×C₈	C₂×D₄	1	2	yes	—
63	C₄⋊C₄	—	2	8	no	—
64	C₄⋊C₄	—	2	4	no	—
65	C₄⋊C₄	—	2	4	no	—
66	C₄⋊C₄	—	4	4	no	—
67	C₄⋊C₄	—	4	4	no	—
68	C₂²⋊C₄	—	4	2	no	—
69	C₄²	—	4	2	no	—
70	C₄²	C₂×D₄	1	1	yes	characteristic
71	C₄²	C₂×D₄	1	2	yes	—
72	C₄²	C₂⁴	1	1	yes	characteristic
73	C₂³	#127	1	1	yes	characteristic
74	Q₈	—	4	8	no	—
75	Q₈	—	4	8	no	—
76	Q₈	—	8	4	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	—	4	4	no	—
80	C₂×C₄	—	8	4	no	—
81	C₂×C₄	#127	1	2	yes	—
82	C₂×C₄	#127	1	1	yes	characteristic
83	C₂×C₄	C₂×SD₁₆	1	4	yes	—
84	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄	C₂²×D₄	1	2	yes	—
86	C₈	—	2	8	no	—
87	C₈	C₂×SD₁₆	1	4	yes	—
88	C₈	C₂×SD₁₆	1	4	yes	—
89	C₂²	—	2	2	no	—
90	C₂²	#492	1	2	yes	—
91	C₂²	C₂×#127	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	8	4	no	—
94	C₄	—	8	4	no	—
95	C₄	#492	1	2	yes	—
96	C₄	C₂²×SD₁₆	1	2	yes	—
97	C₄	#578	1	4	yes	—
98	C₂	—	2	2	no	—
99	C₂	C₂×#492	1	1	yes	characteristic
100	C₂	#3156	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3229

C₂

yes

maximal

#3160

C₂

yes

maximal, characteristic

C₂×#494

C₂

yes

maximal, characteristic

#1716

C₂

yes

maximal, complemented

#1707

C₂

yes

maximal, complemented

#1450

C₂

yes

maximal, characteristic

C₂²×Q₁₆

—

#533

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

#500

C₂²

yes

—

#499

—

#498

C₂²

yes

—

#498

C₂²

yes

—

#494

—

#494

C₂²

yes

—

#494

C₂²

yes

—

#487

C₂²

yes

—

#483

—

#475

—

#431

C₂²

yes

—

#427

—

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

—

C₈²

C₂²

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

—

—

—

C₄⋊C₈

D₄

yes

—

#103

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₄²

C₂⁴

yes

characteristic

C₂³

#127

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#127

yes

—

C₂×C₄

#127

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

—

C₂×C₄

C₂²×D₄

yes

characteristic

C₂×C₄

C₂²×D₄

yes

—

C₈

—

C₈

C₂×SD₁₆

yes

—

C₈

C₂×SD₁₆

yes

—

C₂²

—

C₂²

#492

yes

—

C₂²

C₂×#127

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#492

yes

—

C₄

C₂²×SD₁₆

yes

—

C₄

#578

yes

—

C₂

—

C₂

C₂×#492

yes

characteristic

100

C₂

#3156

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂³	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#127

commutator, Frattini, cosocle

C₄²

C₂⁴

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
4. #1716	C₂	semidirect
5. #1707	C₂	semidirect

H ≅ G/N

type

4. #1716

C₂

semidirect

5. #1707

C₂

semidirect

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

subgroup	index
#3229	2
#3160	2
C₂×#494	2
#1716	2
#1707	2
#1450	2

subgroup

index

Subgroups — #19653

Conjugacy classes of subgroups

Complements

Maximal subgroups