Subgroups — C₂×#2273 #33994

Order 256.

2895 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊C₄	C₂	1	3	yes	maximal
2	#2273	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₂²⋊C₄	C₂	1	12	yes	maximal, complemented
4	C₂⁶	C₂²	1	1	yes	characteristic
5	C₂⁴×C₄	C₂²	1	6	yes	—
6	C₂²×C₂²⋊C₄	C₄	1	48	yes	complemented
7	C₂²×C₂²⋊C₄	C₂²	1	12	yes	—
8	C₂²×C₂²⋊C₄	C₂²	1	24	yes	—
9	C₂²×C₄²	C₂²	1	8	yes	complemented
10	C₄×C₂²⋊C₄	C₂²	1	96	yes	complemented
11	C₂×#95	C₂²	1	8	yes	complemented
12	C₂⁵	—	4	8	no	—
13	C₂⁵	C₂×C₄	1	24	yes	—
14	C₂⁵	C₂³	1	3	yes	—
15	C₂⁵	C₂³	1	4	yes	—
16	C₂³×C₄	—	4	24	no	—
17	C₂³×C₄	C₂×C₄	1	48	yes	complemented
18	C₂³×C₄	C₂³	1	12	yes	—
19	C₂³×C₄	C₂³	1	24	yes	—
20	C₂×C₂²⋊C₄	C₂×C₄	1	96	yes	complemented
21	C₂×C₂²⋊C₄	C₂×C₄	1	192	yes	complemented
22	C₂×C₂²⋊C₄	C₂³	1	48	yes	—
23	C₂×C₄²	—	4	16	no	—
24	C₂×C₄²	C₂³	1	32	yes	complemented
25	#95	C₂³	1	32	yes	complemented
26	C₂⁴	—	2	24	no	—
27	C₂⁴	—	4	96	no	—
28	C₂⁴	—	4	16	no	—
29	C₂⁴	C₄²	1	64	yes	complemented
30	C₂⁴	C₂²×C₄	1	24	yes	—
31	C₂⁴	C₂²×C₄	1	12	yes	—
32	C₂⁴	C₂²×C₄	1	48	yes	—
33	C₂⁴	C₂⁴	1	6	yes	—
34	C₂⁴	C₂⁴	1	1	yes	characteristic
35	C₂²×C₄	—	2	48	no	—
36	C₂²×C₄	—	4	96	no	—
37	C₂²×C₄	—	4	48	no	—
38	C₂²×C₄	C₂²×C₄	1	48	yes	—
39	C₂²×C₄	C₂²×C₄	1	96	yes	complemented
40	C₂²×C₄	C₂⁴	1	24	yes	—
41	C₂²⋊C₄	C₂²×C₄	1	192	yes	complemented
42	C₄²	—	4	32	no	—
43	C₂³	—	2	48	no	—
44	C₂³	—	2	96	no	—
45	C₂³	—	2	24	no	—
46	C₂³	—	4	128	no	—
47	C₂³	—	4	96	no	—
48	C₂³	C₂×C₄²	1	48	yes	—
49	C₂³	C₂×C₄²	1	64	yes	complemented
50	C₂³	C₂³×C₄	1	24	yes	—
51	C₂³	C₂³×C₄	1	12	yes	—
52	C₂³	C₂³×C₄	1	6	yes	—
53	C₂³	H(5,𝔽₂)	1	8	yes	—
54	C₂³	C₂⁵	1	1	yes	characteristic
55	C₂×C₄	—	2	96	no	—
56	C₂×C₄	—	2	48	no	—
57	C₂×C₄	—	4	96	no	—
58	C₂×C₄	C₂³×C₄	1	48	yes	—
59	C₂²	—	2	24	no	—
60	C₂²	—	2	48	no	—
61	C₂²	—	2	96	no	—
62	C₂²	—	4	64	no	—
63	C₂²	C₂²×C₄²	1	24	yes	—
64	C₂²	C₂²×C₄²	1	4	yes	—
65	C₂²	#518	1	24	yes	—
66	C₂²	C₂⁴×C₄	1	3	yes	—
67	C₂²	C₂×H(5,𝔽₂)	1	4	yes	—
68	C₄	—	2	48	no	—
69	C₂	—	2	24	no	—
70	C₂	#2273	1	8	yes	direct product factor
71	C₂	C₂³×C₄²	1	1	yes	characteristic
72	C₂	C₂×#518	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator	C₂	C₂³×C₄²
Frattini, cosocle	C₂³	C₂⁵
prime radical	C₂⁶	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2273	C₂	direct (N × H)
3. C₂×C₄×C₂²⋊C₄	C₂	semidirect
6. C₂²×C₂²⋊C₄	C₄	semidirect
9. C₂²×C₄²	C₂²	semidirect
10. C₄×C₂²⋊C₄	C₂²	semidirect
11. C₂×#95	C₂²	semidirect
17. C₂³×C₄	C₂×C₄	semidirect
20. C₂×C₂²⋊C₄	C₂×C₄	semidirect
21. C₂×C₂²⋊C₄	C₂×C₄	semidirect
24. C₂×C₄²	C₂³	semidirect
25. #95	C₂³	semidirect
29. C₂⁴	C₄²	semidirect
39. C₂²×C₄	C₂²×C₄	semidirect
41. C₂²⋊C₄	C₂²×C₄	semidirect
49. C₂³	C₂×C₄²	semidirect
70. C₂	#2273	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂²⋊C₄	2
#2273	2
C₂×C₄×C₂²⋊C₄	2

Subgroups — C2×#2273 #33994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2273 #33994