Subgroups — C₂×C₄×H(5,𝔽₂) #62649

Order 256.

6189 conjugacy classes of subgroups.

The subgroup lattice has 124 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₄×H(5,𝔽₂)	C₂	1	32	yes	maximal, direct product factor
3	C₂×#520	C₂	1	6	yes	maximal, complemented
4	C₂×#518	C₂	1	9	yes	maximal, complemented
5	C₂×C₄×#41	C₂	1	6	yes	maximal, complemented
6	C₂²×C₄×D₄	C₂	1	9	yes	maximal, complemented
7	C₂×H(5,𝔽₂)	C₄	1	32	yes	direct product factor
8	C₂×H(5,𝔽₂)	C₂²	1	16	yes	—
9	C₂²×#41	C₂²	1	6	yes	—
10	C₂³×D₄	C₂²	1	9	yes	—
11	C₂⁴×C₄	C₂²	1	6	yes	complemented
12	#520	C₂²	1	96	yes	complemented
13	#518	C₂²	1	144	yes	complemented
14	C₄×#41	C₂²	1	96	yes	complemented
15	C₂×C₄×Q₈	C₂²	1	2	yes	complemented
16	C₂×C₄×Q₈	C₂²	1	6	yes	complemented
17	C₂×C₄×D₄	C₂²	1	18	yes	complemented
18	C₂×C₄×D₄	C₂²	1	36	yes	complemented
19	C₂×C₄×D₄	C₂²	1	18	yes	complemented
20	C₂×C₄×D₄	C₂²	1	144	yes	complemented
21	C₂×#117	C₂²	1	18	yes	complemented
22	C₂²×C₄⋊C₄	C₂²	1	9	yes	complemented
23	C₂²×C₂²⋊C₄	C₂²	1	18	yes	complemented
24	C₂²×C₄²	C₂²	1	9	yes	complemented
25	C₂⁵	C₂³	1	6	yes	—
26	H(5,𝔽₂)	C₂×C₄	1	256	yes	direct product factor
27	C₂×#41	C₂×C₄	1	96	yes	complemented
28	C₂×#41	C₂³	1	48	yes	—
29	C₂²×Q₈	C₂³	1	2	yes	—
30	C₂²×D₄	C₂×C₄	1	144	yes	complemented
31	C₂²×D₄	C₂³	1	18	yes	—
32	C₂²×D₄	C₂³	1	72	yes	—
33	C₂³×C₄	—	4	24	no	—
34	C₂³×C₄	C₂³	1	9	yes	—
35	C₂³×C₄	C₂³	1	9	yes	—
36	C₂³×C₄	C₂³	1	18	yes	—
37	C₂³×C₄	C₂³	1	9	yes	—
38	C₂³×C₄	C₂³	1	48	yes	complemented
39	C₄×Q₈	C₂³	1	16	yes	complemented
40	C₄×Q₈	C₂³	1	48	yes	complemented
41	C₄×D₄	C₂³	1	144	yes	complemented
42	C₄×D₄	C₂³	1	288	yes	complemented
43	C₄×D₄	C₂³	1	144	yes	complemented
44	#117	C₂³	1	144	yes	complemented
45	C₂×C₄⋊C₄	C₂³	1	18	yes	—
46	C₂×C₄⋊C₄	C₂³	1	6	yes	—
47	C₂×C₄⋊C₄	C₂³	1	72	yes	complemented
48	C₂×C₂²⋊C₄	C₂³	1	36	yes	—
49	C₂×C₂²⋊C₄	C₂³	1	144	yes	complemented
50	C₂×C₄²	C₂³	1	18	yes	—
51	C₂×C₄²	C₂³	1	6	yes	—
52	C₂×C₄²	C₂³	1	72	yes	complemented
53	C₂⁴	—	4	12	no	—
54	C₂⁴	—	4	12	no	—
55	C₂⁴	C₂²×C₄	1	48	yes	complemented
56	C₂⁴	C₂⁴	1	9	yes	—
57	C₂⁴	C₂⁴	1	24	yes	—
58	#41	C₂²×C₄	1	384	yes	complemented
59	C₂×Q₈	C₂²×C₄	1	16	yes	complemented
60	C₂×Q₈	C₂⁴	1	8	yes	—
61	C₂×D₄	C₂²×C₄	1	144	yes	complemented
62	C₂×D₄	C₂²×C₄	1	576	yes	complemented
63	C₂×D₄	C₂⁴	1	72	yes	—
64	C₂²×C₄	—	2	36	no	—
65	C₂²×C₄	—	4	96	no	—
66	C₂²×C₄	—	4	36	no	—
67	C₂²×C₄	—	4	18	no	—
68	C₂²×C₄	C₂²×C₄	1	72	yes	complemented
69	C₂²×C₄	C₂⁴	1	6	yes	—
70	C₂²×C₄	C₂⁴	1	9	yes	—
71	C₂²×C₄	C₂⁴	1	36	yes	—
72	C₂²×C₄	C₂⁴	1	36	yes	—
73	C₂²×C₄	C₂⁴	1	72	yes	—
74	C₂²×C₄	C₂⁴	1	6	yes	—
75	C₂²×C₄	C₂⁴	1	36	yes	—
76	C₂²×C₄	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄	C₂⁴	1	72	yes	—
78	C₄⋊C₄	C₂⁴	1	24	yes	—
79	C₂²⋊C₄	C₂⁴	1	144	yes	—
80	C₄²	C₂⁴	1	24	yes	—
81	C₄²	C₂⁴	1	72	yes	—
82	C₂³	—	2	18	no	—
83	C₂³	—	2	18	no	—
84	C₂³	—	4	24	no	—
85	C₂³	—	4	96	no	—
86	C₂³	—	4	24	no	—
87	C₂³	C₂³×C₄	1	36	yes	—
88	C₂³	C₂³×C₄	1	96	yes	complemented
89	C₂³	C₂⁵	1	18	yes	—
90	C₂³	C₂⁵	1	1	yes	characteristic
91	Q₈	C₂³×C₄	1	32	yes	complemented
92	D₄	C₂³×C₄	1	288	yes	complemented
93	C₂×C₄	—	2	18	no	—
94	C₂×C₄	—	2	12	no	—
95	C₂×C₄	—	2	72	no	—
96	C₂×C₄	—	4	72	no	—
97	C₂×C₄	—	4	36	no	—
98	C₂×C₄	C₂³×C₄	1	24	yes	—
99	C₂×C₄	C₂³×C₄	1	144	yes	complemented
100	C₂×C₄	H(5,𝔽₂)	1	4	yes	direct product factor
101	C₂×C₄	C₂⁵	1	12	yes	—
102	C₂×C₄	C₂⁵	1	18	yes	—
103	C₂×C₄	C₂⁵	1	12	yes	—
104	C₂×C₄	C₂⁵	1	2	yes	—
105	C₂²	—	2	18	no	—
106	C₂²	—	2	18	no	—
107	C₂²	—	2	72	no	—
108	C₂²	—	4	96	no	—
109	C₂²	C₂⁴×C₄	1	36	yes	—
110	C₂²	C₂⁴×C₄	1	2	yes	—
111	C₂²	C₂×H(5,𝔽₂)	1	2	yes	—
112	C₂²	#585	1	2	yes	—
113	C₂²	C₂⁶	1	1	yes	characteristic
114	C₄	—	2	18	no	—
115	C₄	—	2	12	no	—
116	C₄	C₂⁴×C₄	1	24	yes	—
117	C₄	C₂×H(5,𝔽₂)	1	4	yes	direct product factor
118	C₂	—	2	36	no	—
119	C₂	C₄×H(5,𝔽₂)	1	4	yes	direct product factor
120	C₂	C₂⁵×C₄	1	1	yes	characteristic
121	C₂	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
122	C₂	C₂×#585	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator	C₂	C₂⁵×C₄
Frattini, cosocle	C₂²	C₂⁶
socle	C₂³	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₄×H(5,𝔽₂)	C₂	direct (N × H)
3. C₂×#520	C₂	semidirect
4. C₂×#518	C₂	semidirect
5. C₂×C₄×#41	C₂	semidirect
6. C₂²×C₄×D₄	C₂	semidirect
7. C₂×H(5,𝔽₂)	C₄	direct (N × H)
11. C₂⁴×C₄	C₂²	semidirect
12. #520	C₂²	semidirect
13. #518	C₂²	semidirect
14. C₄×#41	C₂²	semidirect
15. C₂×C₄×Q₈	C₂²	semidirect
16. C₂×C₄×Q₈	C₂²	semidirect
17. C₂×C₄×D₄	C₂²	semidirect
18. C₂×C₄×D₄	C₂²	semidirect
19. C₂×C₄×D₄	C₂²	semidirect
20. C₂×C₄×D₄	C₂²	semidirect
21. C₂×#117	C₂²	semidirect
22. C₂²×C₄⋊C₄	C₂²	semidirect
23. C₂²×C₂²⋊C₄	C₂²	semidirect
24. C₂²×C₄²	C₂²	semidirect
26. H(5,𝔽₂)	C₂×C₄	direct (N × H)
27. C₂×#41	C₂×C₄	semidirect
30. C₂²×D₄	C₂×C₄	semidirect
38. C₂³×C₄	C₂³	semidirect
39. C₄×Q₈	C₂³	semidirect
40. C₄×Q₈	C₂³	semidirect
41. C₄×D₄	C₂³	semidirect
42. C₄×D₄	C₂³	semidirect
43. C₄×D₄	C₂³	semidirect
44. #117	C₂³	semidirect
47. C₂×C₄⋊C₄	C₂³	semidirect
49. C₂×C₂²⋊C₄	C₂³	semidirect
52. C₂×C₄²	C₂³	semidirect
55. C₂⁴	C₂²×C₄	semidirect
58. #41	C₂²×C₄	semidirect
59. C₂×Q₈	C₂²×C₄	semidirect
61. C₂×D₄	C₂²×C₄	semidirect
62. C₂×D₄	C₂²×C₄	semidirect
68. C₂²×C₄	C₂²×C₄	semidirect
88. C₂³	C₂³×C₄	semidirect
91. Q₈	C₂³×C₄	semidirect
92. D₄	C₂³×C₄	semidirect
99. C₂×C₄	C₂³×C₄	semidirect
100. C₂×C₄	H(5,𝔽₂)	direct (N × H)
117. C₄	C₂×H(5,𝔽₂)	direct (N × H)
119. C₂	C₄×H(5,𝔽₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×H(5,𝔽₂)	2
C₄×H(5,𝔽₂)	2
C₂×#520	2
C₂×#518	2
C₂×C₄×#41	2
C₂²×C₄×D₄	2

Subgroups — C2×C4×H(5,𝔽2) #62649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×H(5,𝔽₂) #62649