Group #19603 — subgroups

Subgroups — #19603

Order 256.

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

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 147 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	#3169	C₂	1	4	yes	maximal, complemented
2	#3147	C₂	1	2	yes	maximal, complemented
3	#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	#1709	C₂	1	2	yes	maximal, complemented
6	#1706	C₂	1	4	yes	maximal, complemented
7	C₂×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135	—	2	4	no	—
9	#532	C₂²	1	4	yes	—
10	#500	C₂²	1	4	yes	—
11	#495	—	2	4	no	—
12	#494	C₂²	1	2	yes	—
13	#493	C₂²	1	2	yes	—
14	#493	C₂²	1	4	yes	—
15	#492	C₂²	1	4	yes	—
16	#486	C₂²	1	8	yes	—
17	#466	—	2	8	no	—
18	C₄×D₈	—	2	8	no	—
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂²	1	2	yes	—
21	C₈²	C₂²	1	2	yes	—
22	C₈²	C₂²	1	1	yes	characteristic
23	C₈²	C₂²	1	1	yes	characteristic
24	C₂×#41	—	2	4	no	—
25	#135	—	4	16	no	—
26	C₂×Q₁₆	—	2	4	no	—
27	C₂×SD₁₆	—	2	8	no	—
28	C₂×D₈	—	2	4	no	—
29	C₂×D₈	—	2	8	no	—
30	C₂²×C₈	D₄	1	4	yes	—
31	C₂²×C₈	D₄	1	2	yes	—
32	C₄⋊Q₈	C₂³	1	4	yes	—
33	#127	C₂³	1	4	yes	—
34	#124	—	2	4	no	—
35	#121	—	2	8	no	—
36	C₄×D₄	—	2	8	no	—
37	C₂×C₄²	C₂³	1	1	yes	characteristic
38	#107	—	2	8	no	—
39	#102	—	2	16	no	—
40	C₄×C₈	D₄	1	4	yes	—
41	C₄×C₈	D₄	1	2	yes	—
42	C₄×C₈	C₂³	1	2	yes	—
43	C₄×C₈	C₂³	1	1	yes	characteristic
44	C₄×C₈	C₂³	1	1	yes	characteristic
45	C₄×C₈	C₂³	1	2	yes	—
46	#41	—	4	16	no	—
47	C₂×Q₈	—	2	4	no	—
48	C₂×D₄	—	2	8	no	—
49	C₂×D₄	—	2	4	no	—
50	C₂×D₄	—	4	4	no	—
51	C₂²×C₄	—	4	4	no	—
52	C₂²×C₄	C₂×D₄	1	2	yes	—
53	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
54	Q₁₆	—	4	8	no	—
55	SD₁₆	—	4	16	no	—
56	D₈	—	4	8	no	—
57	D₈	—	4	16	no	—
58	C₂×C₈	—	2	8	no	—
59	C₂×C₈	D₈	1	8	yes	complemented
60	C₂×C₈	C₂×D₄	1	4	yes	—
61	C₂×C₈	C₂×D₄	1	2	yes	—
62	C₂×C₈	C₂×D₄	1	2	yes	—
63	C₂×C₈	C₂×D₄	1	4	yes	—
64	C₄⋊C₄	—	2	8	no	—
65	C₂²⋊C₄	—	4	8	no	—
66	C₄²	C₂×D₄	1	1	yes	characteristic
67	C₄²	C₂×D₄	1	2	yes	—
68	C₄²	C₂⁴	1	1	yes	characteristic
69	C₂³	—	4	4	no	—
70	C₂³	#127	1	1	yes	characteristic
71	Q₈	—	4	8	no	—
72	D₄	—	4	8	no	—
73	D₄	—	4	16	no	—
74	D₄	—	8	8	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	4	4	no	—
77	C₂×C₄	—	8	8	no	—
78	C₂×C₄	#127	1	2	yes	—
79	C₂×C₄	#127	1	1	yes	characteristic
80	C₂×C₄	C₂×D₈	1	4	yes	—
81	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄	C₂²×D₄	1	2	yes	—
83	C₈	—	2	4	no	—
84	C₈	C₂×D₈	1	4	yes	—
85	C₈	C₂×D₈	1	4	yes	—
86	C₈	#135	1	8	yes	complemented
87	C₂²	—	2	2	no	—
88	C₂²	—	8	4	no	—
89	C₂²	—	8	8	no	—
90	C₂²	#493	1	2	yes	—
91	C₂²	C₂×#127	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	8	4	no	—
94	C₄	#493	1	2	yes	—
95	C₄	C₂²×D₈	1	2	yes	—
96	C₄	C₂×#135	1	4	yes	—
97	C₂	—	2	2	no	—
98	C₂	—	16	4	no	—
99	C₂	C₂×#493	1	1	yes	characteristic
100	C₂	#3147	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3169

C₂

yes

maximal, complemented

#3147

C₂

yes

maximal, complemented

#1713

C₂

yes

maximal, characteristic, complemented

#1712

C₂

yes

maximal, characteristic, complemented

#1709

C₂

yes

maximal, complemented

#1706

C₂

yes

maximal, complemented

C₂×C₈²

C₂

yes

maximal, characteristic, complemented

C₂×#135

—

#532

C₂²

yes

—

#500

C₂²

yes

—

#495

—

#494

C₂²

yes

—

#493

C₂²

yes

—

#493

C₂²

yes

—

#492

C₂²

yes

—

#486

C₂²

yes

—

#466

—

C₄×D₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₈²

C₂²

yes

—

C₈²

C₂²

yes

characteristic

C₈²

C₂²

yes

characteristic

C₂×#41

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂³

yes

—

#127

C₂³

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#102

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

D₈

yes

complemented

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₄²

C₂⁴

yes

characteristic

C₂³

—

C₂³

#127

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#127

yes

—

C₂×C₄

#127

yes

characteristic

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂²×D₄

yes

characteristic

C₂×C₄

C₂²×D₄

yes

—

C₈

—

C₈

C₂×D₈

yes

—

C₈

C₂×D₈

yes

—

C₈

#135

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

#493

yes

—

C₂²

C₂×#127

yes

characteristic

C₄

—

C₄

—

C₄

#493

yes

—

C₄

C₂²×D₈

yes

—

C₄

C₂×#135

yes

—

C₂

—

C₂

—

C₂

C₂×#493

yes

characteristic

100

C₂

#3147

yes

—

	subgroup	G/H
center	C₂×C₄	#127
commutator, Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#127

commutator, Frattini, cosocle

C₄²

C₂⁴

C₂²

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3169	C₂	semidirect
2. #3147	C₂	semidirect
3. #1713	C₂	semidirect
4. #1712	C₂	semidirect
5. #1709	C₂	semidirect
6. #1706	C₂	semidirect
7. C₂×C₈²	C₂	semidirect
59. C₂×C₈	D₈	semidirect
86. C₈	#135	semidirect

H ≅ G/N

type

1. #3169

C₂

semidirect

2. #3147

C₂

semidirect

3. #1713

C₂

semidirect

4. #1712

C₂

semidirect

5. #1709

C₂

semidirect

6. #1706

C₂

semidirect

7. C₂×C₈²

C₂

semidirect

59. C₂×C₈

D₈

semidirect

86. C₈

#135

semidirect

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

subgroup	index
#3169	2
#3147	2
#1713	2
#1712	2
#1709	2
#1706	2
C₂×C₈²	2

subgroup

index

C₂×C₈²

Subgroups — #19603

Conjugacy classes of subgroups

Complements

Maximal subgroups