Group #19733 — subgroups

Subgroups — #19733

Order 256.

356 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 115 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	#3231	C₂	1	3	yes	maximal, complemented
2	#3208	C₂	1	3	yes	maximal
3	#1719	C₂	1	3	yes	maximal, complemented
4	#1714	C₂	1	3	yes	maximal, complemented
5	#1711	C₂	1	1	yes	maximal, characteristic, complemented
6	#1705	C₂	1	1	yes	maximal, characteristic, complemented
7	#1454	C₂	1	1	yes	maximal, characteristic, complemented
8	#533	C₂²	1	3	yes	—
9	C₂×#127	C₂²	1	1	yes	characteristic
10	#500	C₂²	1	3	yes	complemented
11	#498	C₂²	1	3	yes	complemented
12	#492	C₂²	1	3	yes	—
13	#492	C₂²	1	3	yes	—
14	#487	C₂²	1	3	yes	complemented
15	#487	C₂²	1	3	yes	complemented
16	#486	C₂²	1	3	yes	—
17	#486	C₂²	1	3	yes	—
18	#485	—	2	6	no	—
19	#479	—	2	6	no	—
20	#460	—	2	6	no	—
21	#450	—	2	6	no	—
22	#431	C₂²	1	3	yes	—
23	#322	C₂²	1	3	yes	complemented
24	C₈²	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₄	—	2	3	no	—
26	C₂×SD₁₆	—	2	6	no	—
27	C₂×SD₁₆	—	2	6	no	—
28	C₄⋊Q₈	C₂³	1	3	yes	—
29	C₄⋊Q₈	C₂³	1	3	yes	—
30	#127	—	2	3	no	—
31	#127	C₂³	1	1	yes	characteristic
32	#127	C₂³	1	1	yes	characteristic
33	#125	—	2	3	no	—
34	C₂²⋊Q₈	—	2	3	no	—
35	C₂²⋊Q₈	—	2	3	no	—
36	C₄×Q₈	—	2	3	no	—
37	C₄×Q₈	—	2	3	no	—
38	#117	—	2	3	no	—
39	C₂×C₄²	C₂³	1	1	yes	characteristic
40	#107	—	2	6	no	—
41	#106	—	2	6	no	—
42	C₄⋊C₈	D₄	1	6	yes	—
43	#103	—	2	6	no	—
44	#103	—	2	6	no	—
45	#102	—	2	6	no	—
46	#102	—	2	6	no	—
47	C₂²⋊C₈	D₄	1	6	yes	—
48	C₄×C₈	C₂³	1	3	yes	—
49	C₄×C₈	C₂³	1	3	yes	—
50	C₂⁴	—	4	1	no	—
51	C₂×Q₈	—	2	3	no	—
52	C₂×Q₈	—	2	3	no	—
53	C₂×D₄	—	2	3	no	—
54	C₂×D₄	—	2	3	no	—
55	C₂×D₄	—	4	3	no	—
56	C₂×D₄	—	8	6	no	—
57	C₂²×C₄	C₂×D₄	1	3	yes	—
58	SD₁₆	—	8	6	no	—
59	SD₁₆	—	8	6	no	—
60	C₂×C₈	C₂×D₄	1	6	yes	—
61	C₂×C₈	C₂×D₄	1	6	yes	—
62	C₄⋊C₄	—	2	3	no	—
63	C₄⋊C₄	—	2	3	no	—
64	C₄⋊C₄	—	2	3	no	—
65	C₄⋊C₄	—	2	3	no	—
66	C₄⋊C₄	—	4	3	no	—
67	C₄⋊C₄	—	4	3	no	—
68	C₂²⋊C₄	—	4	3	no	—
69	C₄²	—	4	3	no	—
70	C₄²	C₂×D₄	1	3	yes	—
71	C₄²	C₂⁴	1	1	yes	characteristic
72	C₂³	—	4	1	no	—
73	C₂³	—	4	1	no	—
74	C₂³	—	8	6	no	—
75	C₂³	#127	1	1	yes	characteristic
76	Q₈	—	4	6	no	—
77	Q₈	—	4	6	no	—
78	D₄	—	4	6	no	—
79	D₄	—	4	6	no	—
80	D₄	—	8	6	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	4	3	no	—
83	C₂×C₄	—	4	3	no	—
84	C₂×C₄	#127	1	3	yes	—
85	C₂×C₄	C₂²×D₄	1	3	yes	—
86	C₈	—	2	6	no	—
87	C₈	—	2	6	no	—
88	C₂²	—	2	3	no	—
89	C₂²	—	8	3	no	—
90	C₂²	—	8	3	no	—
91	C₂²	—	16	4	no	—
92	C₂²	C₂×#127	1	1	yes	characteristic
93	C₄	—	2	3	no	—
94	C₄	—	8	3	no	—
95	C₄	—	8	3	no	—
96	C₄	#577	1	6	yes	—
97	C₂	—	4	1	no	—
98	C₂	—	16	1	no	—
99	C₂	—	16	1	no	—
100	C₂	#3154	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3231

C₂

yes

maximal, complemented

#3208

C₂

yes

maximal

#1719

C₂

yes

maximal, complemented

#1714

C₂

yes

maximal, complemented

#1711

C₂

yes

maximal, characteristic, complemented

#1705

C₂

yes

maximal, characteristic, complemented

#1454

C₂

yes

maximal, characteristic, complemented

#533

C₂²

yes

—

C₂×#127

C₂²

yes

characteristic

#500

C₂²

yes

complemented

#498

C₂²

yes

complemented

#492

C₂²

yes

—

#492

C₂²

yes

—

#487

C₂²

yes

complemented

#487

C₂²

yes

complemented

#486

C₂²

yes

—

#486

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#322

C₂²

yes

complemented

C₈²

C₂²

yes

characteristic, complemented

C₂²×D₄

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#127

—

#127

C₂³

yes

characteristic

#127

C₂³

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#106

—

C₄⋊C₈

D₄

yes

—

—

—

—

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂×D₄

yes

—

SD₁₆

—

SD₁₆

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

C₂⁴

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

#127

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#127

yes

—

C₂×C₄

C₂²×D₄

yes

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×#127

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#577

yes

—

C₂

—

C₂

—

C₂

—

100

C₂

#3154

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#127	C₂²
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

C₂²

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3231	C₂	semidirect
3. #1719	C₂	semidirect
4. #1714	C₂	semidirect
5. #1711	C₂	semidirect
6. #1705	C₂	semidirect
7. #1454	C₂	semidirect
10. #500	C₂²	semidirect
11. #498	C₂²	semidirect
14. #487	C₂²	semidirect
15. #487	C₂²	semidirect
23. #322	C₂²	semidirect
24. C₈²	C₂²	semidirect

H ≅ G/N

type

1. #3231

C₂

semidirect

3. #1719

C₂

semidirect

4. #1714

C₂

semidirect

5. #1711

C₂

semidirect

6. #1705

C₂

semidirect

7. #1454

C₂

semidirect

10. #500

C₂²

semidirect

11. #498

C₂²

semidirect

14. #487

C₂²

semidirect

15. #487

C₂²

semidirect

23. #322

C₂²

semidirect

24. C₈²

C₂²

semidirect

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

subgroup	index
#3231	2
#3208	2
#1719	2
#1714	2
#1711	2
#1705	2
#1454	2

subgroup

index

Subgroups — #19733

Conjugacy classes of subgroups

Complements

Maximal subgroups