Subgroups — #53169

Order 256.

1648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2846	C₂	1	3	yes	maximal, complemented
2	#2685	C₂	1	21	yes	maximal, complemented
3	#2679	C₂	1	7	yes	maximal, complemented
4	C₂⁶	C₂²	1	1	yes	characteristic, complemented
5	C₂×#126	C₂²	1	7	yes	complemented
6	C₂×#120	C₂²	1	21	yes	—
7	C₂×#120	C₂²	1	7	yes	complemented
8	#399	C₂²	1	7	yes	complemented
9	#396	C₂²	1	7	yes	complemented
10	#394	C₂²	1	21	yes	complemented
11	#394	C₂²	1	21	yes	complemented
12	#390	C₂²	1	21	yes	complemented
13	#388	C₂²	1	21	yes	complemented
14	#379	C₂²	1	21	yes	complemented
15	C₂⁵	—	4	14	no	—
16	C₂⁵	C₂³	1	7	yes	—
17	C₂⁵	C₂³	1	1	yes	characteristic, complemented
18	C₂²×D₄	C₂³	1	21	yes	—
19	#126	—	4	14	no	—
20	#120	—	4	42	no	—
21	#120	—	4	14	no	—
22	C₂×C₄⋊C₄	C₂³	1	21	yes	complemented
23	C₂×C₂²⋊C₄	C₂³	1	21	yes	—
24	C₂×C₂²⋊C₄	C₂³	1	21	yes	—
25	C₂×C₂²⋊C₄	C₂³	1	21	yes	—
26	C₂×C₂²⋊C₄	C₂³	1	21	yes	complemented
27	C₂×C₄²	C₂³	1	7	yes	complemented
28	#95	C₂³	1	7	yes	complemented
29	#95	C₂³	1	7	yes	complemented
30	C₂⁴	—	2	42	no	—
31	C₂⁴	—	4	21	no	—
32	C₂⁴	—	4	7	no	—
33	C₂⁴	—	4	7	no	—
34	C₂⁴	—	4	112	no	—
35	C₂⁴	C₂⁴	1	3	yes	—
36	C₂⁴	C₂⁴	1	7	yes	—
37	C₂×D₄	—	2	42	no	—
38	C₂×D₄	—	4	21	no	—
39	C₂×D₄	—	4	21	no	—
40	C₂²×C₄	C₂⁴	1	21	yes	—
41	C₄⋊C₄	—	4	21	no	—
42	C₂²⋊C₄	—	2	42	no	—
43	C₂²⋊C₄	—	4	21	no	—
44	C₂²⋊C₄	—	4	21	no	—
45	C₂²⋊C₄	—	4	21	no	—
46	C₄²	—	4	7	no	—
47	C₂³	—	2	21	no	—
48	C₂³	—	2	21	no	—
49	C₂³	—	2	21	no	—
50	C₂³	—	4	84	no	—
51	C₂³	—	4	84	no	—
52	C₂³	—	4	128	no	—
53	C₂³	—	4	28	no	—
54	C₂³	—	8	14	no	—
55	C₂³	H(5,𝔽₂)	1	14	yes	—
56	C₂³	C₂⁵	1	1	yes	characteristic
57	D₄	—	8	42	no	—
58	C₂×C₄	—	2	21	no	—
59	C₂×C₄	—	2	21	no	—
60	C₂×C₄	—	2	21	no	—
61	C₂²	—	2	42	no	—
62	C₂²	—	4	112	no	—
63	C₂²	—	4	21	no	—
64	C₂²	—	4	7	no	—
65	C₂²	—	4	21	no	—
66	C₂²	—	8	8	no	—
67	C₂²	C₂×H(5,𝔽₂)	1	7	yes	—
68	C₄	—	4	21	no	—
69	C₂	—	4	14	no	—
70	C₂	—	8	3	no	—
71	C₂	#3532	1	7	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #2846	C₂	semidirect
2. #2685	C₂	semidirect
3. #2679	C₂	semidirect
4. C₂⁶	C₂²	semidirect
5. C₂×#126	C₂²	semidirect
7. C₂×#120	C₂²	semidirect
8. #399	C₂²	semidirect
9. #396	C₂²	semidirect
10. #394	C₂²	semidirect
11. #394	C₂²	semidirect
12. #390	C₂²	semidirect
13. #388	C₂²	semidirect
14. #379	C₂²	semidirect
17. C₂⁵	C₂³	semidirect
22. C₂×C₄⋊C₄	C₂³	semidirect
26. C₂×C₂²⋊C₄	C₂³	semidirect
27. C₂×C₄²	C₂³	semidirect
28. #95	C₂³	semidirect
29. #95	C₂³	semidirect

Maximal subgroups

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

subgroup	index
#2846	2
#2685	2
#2679	2