Subgroups — C₃×C₄×H(5,𝔽₂) #87313

Order 384.

1660 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	C₂×C₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₃×#520	C₂	1	6	yes	maximal, complemented
3	C₃×#518	C₂	1	9	yes	maximal, complemented
4	C₃×C₄×#41	C₂	1	6	yes	maximal, complemented
5	C₂×C₃×C₄×D₄	C₂	1	9	yes	maximal, complemented
6	C₄×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃×H(5,𝔽₂)	C₄	1	16	yes	direct product factor
8	C₂×C₃×#41	C₂²	1	6	yes	—
9	C₂²×C₃×D₄	C₂²	1	9	yes	—
10	C₂³×C₃×C₄	C₂²	1	6	yes	complemented
11	C₃×C₄×Q₈	C₂²	1	2	yes	complemented
12	C₃×C₄×Q₈	C₂²	1	6	yes	complemented
13	C₃×C₄×D₄	C₂²	1	18	yes	complemented
14	C₃×C₄×D₄	C₂²	1	36	yes	complemented
15	C₃×C₄×D₄	C₂²	1	18	yes	complemented
16	C₃×#117	C₂²	1	18	yes	complemented
17	C₂×C₃×C₄⋊C₄	C₂²	1	9	yes	complemented
18	C₂×C₃×C₂²⋊C₄	C₂²	1	18	yes	complemented
19	C₂×C₃×C₄²	C₂²	1	9	yes	complemented
20	C₂×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic
21	#520	C₂×C₃	1	6	yes	complemented
22	#518	C₂×C₃	1	9	yes	complemented
23	C₄×#41	C₂×C₃	1	6	yes	complemented
24	C₂×C₄×D₄	C₂×C₃	1	9	yes	complemented
25	C₂⁴×C₃	C₂³	1	6	yes	—
26	C₃×#41	C₂×C₄	1	48	yes	complemented
27	C₂×C₃×Q₈	C₂³	1	2	yes	—
28	C₂×C₃×D₄	C₂×C₄	1	72	yes	complemented
29	C₂×C₃×D₄	C₂³	1	18	yes	—
30	C₂²×C₃×C₄	—	4	12	no	—
31	C₂²×C₃×C₄	C₂³	1	9	yes	—
32	C₂²×C₃×C₄	C₂³	1	9	yes	—
33	C₂²×C₃×C₄	C₂³	1	18	yes	—
34	C₂²×C₃×C₄	C₂³	1	9	yes	—
35	C₃×C₄⋊C₄	C₂³	1	18	yes	—
36	C₃×C₄⋊C₄	C₂³	1	6	yes	—
37	C₃×C₂²⋊C₄	C₂³	1	36	yes	—
38	C₃×C₄²	C₂³	1	18	yes	—
39	C₃×C₄²	C₂³	1	6	yes	—
40	H(5,𝔽₂)	C₃×C₄	1	16	yes	direct product factor
41	C₂×#41	C₂²×C₃	1	6	yes	—
42	C₂²×D₄	C₂²×C₃	1	9	yes	—
43	C₂³×C₄	C₂²×C₃	1	6	yes	complemented
44	C₄×Q₈	C₂²×C₃	1	2	yes	complemented
45	C₄×Q₈	C₂²×C₃	1	6	yes	complemented
46	C₄×D₄	C₂²×C₃	1	18	yes	complemented
47	C₄×D₄	C₂²×C₃	1	36	yes	complemented
48	C₄×D₄	C₂²×C₃	1	18	yes	complemented
49	#117	C₂²×C₃	1	18	yes	complemented
50	C₂×C₄⋊C₄	C₂²×C₃	1	9	yes	complemented
51	C₂×C₂²⋊C₄	C₂²×C₃	1	18	yes	complemented
52	C₂×C₄²	C₂²×C₃	1	9	yes	complemented
53	C₂³×C₃	—	4	6	no	—
54	C₂³×C₃	—	4	6	no	—
55	C₂³×C₃	C₂²×C₄	1	24	yes	complemented
56	C₂³×C₃	C₂⁴	1	9	yes	—
57	C₃×Q₈	C₂²×C₄	1	8	yes	complemented
58	C₃×D₄	C₂²×C₄	1	72	yes	complemented
59	C₂×C₃×C₄	—	2	18	no	—
60	C₂×C₃×C₄	—	4	18	no	—
61	C₂×C₃×C₄	—	4	9	no	—
62	C₂×C₃×C₄	C₂²×C₄	1	36	yes	complemented
63	C₂×C₃×C₄	C₂⁴	1	6	yes	—
64	C₂×C₃×C₄	C₂⁴	1	9	yes	—
65	C₂×C₃×C₄	C₂⁴	1	6	yes	—
66	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
67	C₂⁴	C₂³×C₃	1	6	yes	—
68	#41	C₂×C₃×C₄	1	48	yes	complemented
69	C₂×Q₈	C₂³×C₃	1	2	yes	—
70	C₂×D₄	C₂×C₃×C₄	1	72	yes	complemented
71	C₂×D₄	C₂³×C₃	1	18	yes	—
72	C₂²×C₄	—	4	12	no	—
73	C₂²×C₄	C₂³×C₃	1	9	yes	—
74	C₂²×C₄	C₂³×C₃	1	9	yes	—
75	C₂²×C₄	C₂³×C₃	1	18	yes	—
76	C₂²×C₄	C₂³×C₃	1	9	yes	—
77	C₄⋊C₄	C₂³×C₃	1	18	yes	—
78	C₄⋊C₄	C₂³×C₃	1	6	yes	—
79	C₂²⋊C₄	C₂³×C₃	1	36	yes	—
80	C₄²	C₂³×C₃	1	6	yes	—
81	C₄²	C₂³×C₃	1	18	yes	—
82	C₂²×C₃	—	2	9	no	—
83	C₂²×C₃	—	2	9	no	—
84	C₂²×C₃	—	4	24	no	—
85	C₂²×C₃	C₂³×C₄	1	18	yes	—
86	C₂²×C₃	C₂⁵	1	1	yes	characteristic
87	C₃×C₄	—	2	9	no	—
88	C₃×C₄	—	2	6	no	—
89	C₃×C₄	C₂³×C₄	1	12	yes	—
90	C₃×C₄	H(5,𝔽₂)	1	2	yes	direct product factor
91	C₂³	—	4	6	no	—
92	C₂³	—	4	6	no	—
93	C₂³	C₂²×C₃×C₄	1	24	yes	complemented
94	C₂³	C₂⁴×C₃	1	9	yes	—
95	Q₈	C₂²×C₃×C₄	1	8	yes	complemented
96	D₄	C₂²×C₃×C₄	1	72	yes	complemented
97	C₂×C₄	—	2	18	no	—
98	C₂×C₄	—	4	18	no	—
99	C₂×C₄	—	4	9	no	—
100	C₂×C₄	C₂²×C₃×C₄	1	36	yes	complemented
101	C₂×C₄	C₂⁴×C₃	1	6	yes	—
102	C₂×C₄	C₂⁴×C₃	1	9	yes	—
103	C₂×C₄	C₂⁴×C₃	1	6	yes	—
104	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
105	C₂×C₃	—	2	18	no	—
106	C₂×C₃	C₂⁴×C₄	1	1	yes	characteristic
107	C₂×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
108	C₂×C₃	#585	1	1	yes	characteristic
109	C₂²	—	2	9	no	—
110	C₂²	—	2	9	no	—
111	C₂²	—	4	24	no	—
112	C₂²	C₂³×C₃×C₄	1	18	yes	—
113	C₂²	C₂⁵×C₃	1	1	yes	characteristic
114	C₄	—	2	9	no	—
115	C₄	—	2	6	no	—
116	C₄	C₂³×C₃×C₄	1	12	yes	—
117	C₄	C₃×H(5,𝔽₂)	1	2	yes	direct product factor
118	C₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
119	C₂	—	2	18	no	—
120	C₂	C₂⁴×C₃×C₄	1	1	yes	characteristic
121	C₂	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
122	C₂	C₃×#585	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂⁴
commutator	C₂	C₂⁴×C₃×C₄
Frattini, cosocle	C₂²	C₂⁵×C₃
socle	C₂²×C₃	C₂⁵
prime radical	C₂×C₃×H(5,𝔽₂)	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#520	C₂	semidirect
3. C₃×#518	C₂	semidirect
4. C₃×C₄×#41	C₂	semidirect
5. C₂×C₃×C₄×D₄	C₂	semidirect
6. C₄×H(5,𝔽₂)	C₃	direct (N × H)
7. C₃×H(5,𝔽₂)	C₄	direct (N × H)
10. C₂³×C₃×C₄	C₂²	semidirect
11. C₃×C₄×Q₈	C₂²	semidirect
12. C₃×C₄×Q₈	C₂²	semidirect
13. C₃×C₄×D₄	C₂²	semidirect
14. C₃×C₄×D₄	C₂²	semidirect
15. C₃×C₄×D₄	C₂²	semidirect
16. C₃×#117	C₂²	semidirect
17. C₂×C₃×C₄⋊C₄	C₂²	semidirect
18. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
19. C₂×C₃×C₄²	C₂²	semidirect
21. #520	C₂×C₃	semidirect
22. #518	C₂×C₃	semidirect
23. C₄×#41	C₂×C₃	semidirect
24. C₂×C₄×D₄	C₂×C₃	semidirect
26. C₃×#41	C₂×C₄	semidirect
28. C₂×C₃×D₄	C₂×C₄	semidirect
40. H(5,𝔽₂)	C₃×C₄	direct (N × H)
43. C₂³×C₄	C₂²×C₃	semidirect
44. C₄×Q₈	C₂²×C₃	semidirect
45. C₄×Q₈	C₂²×C₃	semidirect
46. C₄×D₄	C₂²×C₃	semidirect
47. C₄×D₄	C₂²×C₃	semidirect
48. C₄×D₄	C₂²×C₃	semidirect
49. #117	C₂²×C₃	semidirect
50. C₂×C₄⋊C₄	C₂²×C₃	semidirect
51. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
52. C₂×C₄²	C₂²×C₃	semidirect
55. C₂³×C₃	C₂²×C₄	semidirect
57. C₃×Q₈	C₂²×C₄	semidirect
58. C₃×D₄	C₂²×C₄	semidirect
62. C₂×C₃×C₄	C₂²×C₄	semidirect
68. #41	C₂×C₃×C₄	semidirect
70. C₂×D₄	C₂×C₃×C₄	semidirect
90. C₃×C₄	H(5,𝔽₂)	direct (N × H)
93. C₂³	C₂²×C₃×C₄	semidirect
95. Q₈	C₂²×C₃×C₄	semidirect
96. D₄	C₂²×C₃×C₄	semidirect
100. C₂×C₄	C₂²×C₃×C₄	semidirect
117. C₄	C₃×H(5,𝔽₂)	direct (N × H)
118. C₃	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
C₂×C₃×H(5,𝔽₂)	2
C₃×#520	2
C₃×#518	2
C₃×C₄×#41	2
C₂×C₃×C₄×D₄	2
C₄×H(5,𝔽₂)	3

Subgroups — C3×C4×H(5,𝔽2) #87313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×H(5,𝔽₂) #87313