Subgroups — C₂×H(3,𝔽₄)×C₂₂₃ #43252541

Order 28544.

1408 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₄)×C₂₂₃	C₂	1	16	yes	maximal, direct product factor
2	C₂×#124×C₂₂₃	C₂	1	9	yes	maximal, complemented
3	C₂×#120×C₂₂₃	C₂	1	6	yes	maximal, complemented
4	C₂⁵×C₂₂₃	C₂²	1	2	yes	complemented
5	C₂²×Q₈×C₂₂₃	C₂²	1	3	yes	complemented
6	C₂²×D₄×C₂₂₃	C₂²	1	9	yes	—
7	#124×C₂₂₃	C₂²	1	72	yes	complemented
8	#120×C₂₂₃	C₂²	1	48	yes	complemented
9	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	18	yes	complemented
10	C₂×C₄²×C₂₂₃	C₂²	1	3	yes	complemented
11	C₂⁴×C₂₂₃	—	4	12	no	—
12	C₂⁴×C₂₂₃	C₂³	1	6	yes	—
13	C₂⁴×C₂₂₃	C₂³	1	8	yes	complemented
14	C₂×Q₈×C₂₂₃	—	4	6	no	—
15	C₂×Q₈×C₂₂₃	C₂³	1	12	yes	complemented
16	C₂×D₄×C₂₂₃	—	4	18	no	—
17	C₂×D₄×C₂₂₃	C₂³	1	36	yes	—
18	C₂²×C₄×C₂₂₃	C₂³	1	9	yes	—
19	C₂²⋊C₄×C₂₂₃	C₂³	1	72	yes	complemented
20	C₄²×C₂₂₃	C₂³	1	12	yes	complemented
21	C₂³×C₂₂₃	—	2	36	no	—
22	C₂³×C₂₂₃	—	4	24	no	—
23	C₂³×C₂₂₃	—	4	32	no	—
24	C₂³×C₂₂₃	C₂⁴	1	12	yes	—
25	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
26	Q₈×C₂₂₃	—	4	12	no	—
27	D₄×C₂₂₃	—	4	36	no	—
28	C₂×C₄×C₂₂₃	—	2	18	no	—
29	C₂×C₄×C₂₂₃	C₂⁴	1	18	yes	—
30	C₂²×C₂₂₃	—	2	36	no	—
31	C₂²×C₂₂₃	—	4	32	no	—
32	C₂²×C₂₂₃	—	4	24	no	—
33	C₂²×C₂₂₃	H(5,𝔽₂)	1	6	yes	—
34	C₂²×C₂₂₃	C₂⁵	1	1	yes	characteristic
35	C₄×C₂₂₃	—	2	18	no	—
36	C₂×C₂₂₃	—	4	12	no	—
37	C₂×C₂₂₃	H(3,𝔽₄)	1	4	yes	direct product factor
38	C₂×C₂₂₃	C₂×H(5,𝔽₂)	1	3	yes	—
39	C₂₂₃	C₂×H(3,𝔽₄)	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×H(3,𝔽₄)	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	H(3,𝔽₄)	C₂×C₂₂₃	1	16	yes	direct product factor
42	C₂×#124	C₂×C₂₂₃	1	9	yes	complemented
43	C₂×#120	C₂×C₂₂₃	1	6	yes	complemented
44	C₂⁵	C₂²×C₂₂₃	1	2	yes	complemented
45	C₂²×Q₈	C₂²×C₂₂₃	1	3	yes	complemented
46	C₂²×D₄	C₂²×C₂₂₃	1	9	yes	—
47	#124	C₂²×C₂₂₃	1	72	yes	complemented
48	#120	C₂²×C₂₂₃	1	48	yes	complemented
49	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	18	yes	complemented
50	C₂×C₄²	C₂²×C₂₂₃	1	3	yes	complemented
51	C₂⁴	—	4	12	no	—
52	C₂⁴	C₂³×C₂₂₃	1	6	yes	—
53	C₂⁴	C₂³×C₂₂₃	1	8	yes	complemented
54	C₂×Q₈	—	4	6	no	—
55	C₂×Q₈	C₂³×C₂₂₃	1	12	yes	complemented
56	C₂×D₄	—	4	18	no	—
57	C₂×D₄	C₂³×C₂₂₃	1	36	yes	—
58	C₂²×C₄	C₂³×C₂₂₃	1	9	yes	—
59	C₂²⋊C₄	C₂³×C₂₂₃	1	72	yes	complemented
60	C₄²	C₂³×C₂₂₃	1	12	yes	complemented
61	C₂³	—	2	36	no	—
62	C₂³	—	4	24	no	—
63	C₂³	—	4	32	no	—
64	C₂³	C₂⁴×C₂₂₃	1	12	yes	—
65	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
66	Q₈	—	4	12	no	—
67	D₄	—	4	36	no	—
68	C₂×C₄	—	2	18	no	—
69	C₂×C₄	C₂⁴×C₂₂₃	1	18	yes	—
70	C₂²	—	2	36	no	—
71	C₂²	—	4	32	no	—
72	C₂²	—	4	24	no	—
73	C₂²	H(5,𝔽₂)×C₂₂₃	1	6	yes	—
74	C₂²	C₂⁵×C₂₂₃	1	1	yes	characteristic
75	C₄	—	2	18	no	—
76	C₂	—	4	12	no	—
77	C₂	H(3,𝔽₄)×C₂₂₃	1	4	yes	direct product factor
78	C₂	C₂×H(5,𝔽₂)×C₂₂₃	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. H(3,𝔽₄)×C₂₂₃	C₂	direct (N × H)
2. C₂×#124×C₂₂₃	C₂	semidirect
3. C₂×#120×C₂₂₃	C₂	semidirect
4. C₂⁵×C₂₂₃	C₂²	semidirect
5. C₂²×Q₈×C₂₂₃	C₂²	semidirect
7. #124×C₂₂₃	C₂²	semidirect
8. #120×C₂₂₃	C₂²	semidirect
9. C₂×C₂²⋊C₄×C₂₂₃	C₂²	semidirect
10. C₂×C₄²×C₂₂₃	C₂²	semidirect
13. C₂⁴×C₂₂₃	C₂³	semidirect
15. C₂×Q₈×C₂₂₃	C₂³	semidirect
19. C₂²⋊C₄×C₂₂₃	C₂³	semidirect
20. C₄²×C₂₂₃	C₂³	semidirect
37. C₂×C₂₂₃	H(3,𝔽₄)	direct (N × H)
39. C₂₂₃	C₂×H(3,𝔽₄)	direct (N × H)
40. C₂×H(3,𝔽₄)	C₂₂₃	direct (N × H)
41. H(3,𝔽₄)	C₂×C₂₂₃	direct (N × H)
42. C₂×#124	C₂×C₂₂₃	semidirect
43. C₂×#120	C₂×C₂₂₃	semidirect
44. C₂⁵	C₂²×C₂₂₃	semidirect
45. C₂²×Q₈	C₂²×C₂₂₃	semidirect
47. #124	C₂²×C₂₂₃	semidirect
48. #120	C₂²×C₂₂₃	semidirect
49. C₂×C₂²⋊C₄	C₂²×C₂₂₃	semidirect
50. C₂×C₄²	C₂²×C₂₂₃	semidirect
53. C₂⁴	C₂³×C₂₂₃	semidirect
55. C₂×Q₈	C₂³×C₂₂₃	semidirect
59. C₂²⋊C₄	C₂³×C₂₂₃	semidirect
60. C₄²	C₂³×C₂₂₃	semidirect
77. C₂	H(3,𝔽₄)×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
H(3,𝔽₄)×C₂₂₃	2
C₂×#124×C₂₂₃	2
C₂×#120×C₂₂₃	2
C₂×H(3,𝔽₄)	223

Subgroups — C2×H(3,𝔽4)×C223 #43252541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×H(3,𝔽₄)×C₂₂₃ #43252541