Group #19652 — subgroups

Subgroups — #19652

Order 256.

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

normal subgroups only

This group has 441 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	#3208	C₂	1	4	yes	maximal, complemented
2	#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#493	C₂	1	1	yes	maximal, characteristic
4	#1715	C₂	1	4	yes	maximal, complemented
5	#1706	C₂	1	4	yes	maximal, complemented
6	#1450	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈	—	2	2	no	—
8	#533	C₂²	1	2	yes	—
9	C₂×#127	C₂²	1	2	yes	—
10	#500	C₂²	1	2	yes	complemented
11	#499	—	2	2	no	—
12	#498	C₂²	1	2	yes	complemented
13	#493	—	2	2	no	—
14	#493	C₂²	1	2	yes	—
15	#493	C₂²	1	2	yes	—
16	#492	C₂²	1	8	yes	complemented
17	#486	C₂²	1	8	yes	complemented
18	#460	—	2	8	no	—
19	#450	—	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	complemented
24	C₈²	C₂²	1	2	yes	complemented
25	C₂²×D₄	—	2	2	no	—
26	C₂²×D₄	—	2	4	no	—
27	C₂×SD₁₆	—	2	16	no	—
28	C₂×D₈	—	2	4	no	—
29	C₂×D₈	—	2	4	no	—
30	C₂×D₈	—	4	8	no	—
31	C₂²×C₈	D₄	1	2	yes	—
32	C₄⋊Q₈	C₂³	1	4	yes	—
33	#127	—	2	4	no	—
34	#127	—	2	2	no	—
35	#127	C₂³	1	4	yes	—
36	#125	—	2	2	no	—
37	C₂²⋊Q₈	—	2	4	no	—
38	C₄×Q₈	—	2	4	no	—
39	#117	—	2	2	no	—
40	C₂×C₄²	C₂³	1	1	yes	characteristic
41	#107	—	2	4	no	—
42	#106	—	2	4	no	—
43	C₄⋊C₈	D₄	1	4	yes	—
44	#102	—	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	C₂⁴	—	4	2	no	—
51	C₂×Q₈	—	2	4	no	—
52	C₂×D₄	—	2	8	no	—
53	C₂×D₄	—	2	4	no	—
54	C₂×D₄	—	4	4	no	—
55	C₂×D₄	—	4	2	no	—
56	C₂×D₄	—	4	8	no	—
57	C₂×D₄	—	8	8	no	—
58	C₂²×C₄	C₂×D₄	1	2	yes	—
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	SD₁₆	—	8	16	no	—
61	D₈	—	4	8	no	—
62	D₈	—	4	8	no	—
63	C₂×C₈	SD₁₆	1	8	yes	complemented
64	C₂×C₈	C₂×D₄	1	8	yes	—
65	C₂×C₈	C₂×D₄	1	2	yes	—
66	C₂×C₈	C₂×D₄	1	2	yes	—
67	C₄⋊C₄	—	2	4	no	—
68	C₄⋊C₄	—	2	4	no	—
69	C₄⋊C₄	—	4	4	no	—
70	C₂²⋊C₄	—	4	2	no	—
71	C₄²	—	4	2	no	—
72	C₄²	C₂×D₄	1	1	yes	characteristic
73	C₄²	C₂×D₄	1	2	yes	—
74	C₄²	C₂⁴	1	1	yes	characteristic
75	C₂³	—	4	4	no	—
76	C₂³	—	8	8	no	—
77	C₂³	—	8	4	no	—
78	C₂³	#127	1	1	yes	characteristic
79	Q₈	—	4	8	no	—
80	D₄	—	4	8	no	—
81	D₄	—	4	16	no	—
82	D₄	—	8	8	no	—
83	D₄	—	8	4	no	—
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	—	4	4	no	—
86	C₂×C₄	#127	1	2	yes	—
87	C₂×C₄	#127	1	1	yes	characteristic
88	C₂×C₄	C₂×SD₁₆	1	4	yes	—
89	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₄	1	2	yes	—
91	C₈	—	2	8	no	—
92	C₈	C₂×SD₁₆	1	4	yes	—
93	C₈	C₂×SD₁₆	1	4	yes	—
94	C₂²	—	2	2	no	—
95	C₂²	—	8	4	no	—
96	C₂²	—	8	8	no	—
97	C₂²	—	16	8	no	—
98	C₂²	#492	1	2	yes	—
99	C₂²	C₂×#127	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	8	4	no	—
102	C₄	#492	1	2	yes	—
103	C₄	C₂²×SD₁₆	1	2	yes	—
104	C₄	#576	1	4	yes	—
105	C₂	—	2	2	no	—
106	C₂	—	16	4	no	—
107	C₂	C₂×#492	1	1	yes	characteristic
108	C₂	#3155	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3208

C₂

yes

maximal, complemented

#3160

C₂

yes

maximal, characteristic, complemented

C₂×#493

C₂

yes

maximal, characteristic

#1715

C₂

yes

maximal, complemented

#1706

C₂

yes

maximal, complemented

#1450

C₂

yes

maximal, characteristic, complemented

C₂²×D₈

—

#533

C₂²

yes

—

C₂×#127

C₂²

yes

—

#500

C₂²

yes

complemented

#499

—

#498

C₂²

yes

complemented

#493

—

#493

C₂²

yes

—

#493

C₂²

yes

—

#492

C₂²

yes

complemented

#486

C₂²

yes

—

—

C₂²

yes

—

#427

—

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

complemented

C₈²

C₂²

yes

complemented

C₂²×D₄

—

C₂²×D₄

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂³

yes

—

#127

—

#127

—

#127

C₂³

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#106

—

C₄⋊C₈

D₄

yes

—

#102

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

SD₁₆

yes

complemented

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₄²

C₂⁴

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

#127

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

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

—

C₂²

—

C₂²

—

C₂²

#492

yes

—

C₂²

C₂×#127

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

#492

yes

—

103

C₄

C₂²×SD₁₆

yes

—

104

C₄

#576

yes

—

105

C₂

—

106

C₂

—

107

C₂

C₂×#492

yes

characteristic

108

C₂

#3155

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#493	C₂
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

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3208	C₂	semidirect
2. #3160	C₂	semidirect
4. #1715	C₂	semidirect
5. #1706	C₂	semidirect
6. #1450	C₂	semidirect
10. #500	C₂²	semidirect
12. #498	C₂²	semidirect
16. #492	C₂²	semidirect
17. #486	C₂²	semidirect
23. #322	C₂²	semidirect
24. C₈²	C₂²	semidirect
63. C₂×C₈	SD₁₆	semidirect

H ≅ G/N

type

1. #3208

C₂

semidirect

2. #3160

C₂

semidirect

4. #1715

C₂

semidirect

5. #1706

C₂

semidirect

6. #1450

C₂

semidirect

10. #500

C₂²

semidirect

12. #498

C₂²

semidirect

16. #492

C₂²

semidirect

17. #486

C₂²

semidirect

23. #322

C₂²

semidirect

24. C₈²

C₂²

semidirect

63. C₂×C₈

SD₁₆

semidirect

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

subgroup	index
#3208	2
#3160	2
C₂×#493	2
#1715	2
#1706	2
#1450	2

subgroup

index

Subgroups — #19652

Conjugacy classes of subgroups

Complements

Maximal subgroups