Subgroups — C₂×C₃×C₅×H(3,𝔽₄) #14276560

Order 1920.

2816 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×H(3,𝔽₄)	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×C₅×#124	C₂	1	9	yes	maximal, complemented
3	C₂×C₃×C₅×#120	C₂	1	6	yes	maximal, complemented
4	C₂×C₅×H(3,𝔽₄)	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂⁵×C₃×C₅	C₂²	1	2	yes	complemented
6	C₂²×C₃×C₅×Q₈	C₂²	1	3	yes	complemented
7	C₂²×C₃×C₅×D₄	C₂²	1	9	yes	—
8	C₃×C₅×#124	C₂²	1	72	yes	complemented
9	C₃×C₅×#120	C₂²	1	48	yes	complemented
10	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	18	yes	complemented
11	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	complemented
12	C₂×C₃×H(3,𝔽₄)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₅×H(3,𝔽₄)	C₂×C₃	1	16	yes	direct product factor
14	C₂×C₅×#124	C₂×C₃	1	9	yes	complemented
15	C₂×C₅×#120	C₂×C₃	1	6	yes	complemented
16	C₂⁴×C₃×C₅	—	4	12	no	—
17	C₂⁴×C₃×C₅	C₂³	1	6	yes	—
18	C₂⁴×C₃×C₅	C₂³	1	8	yes	complemented
19	C₂×C₃×C₅×Q₈	—	4	6	no	—
20	C₂×C₃×C₅×Q₈	C₂³	1	12	yes	complemented
21	C₂×C₃×C₅×D₄	—	4	18	no	—
22	C₂×C₃×C₅×D₄	C₂³	1	36	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	9	yes	—
24	C₃×C₅×C₂²⋊C₄	C₂³	1	72	yes	complemented
25	C₃×C₄²×C₅	C₂³	1	12	yes	complemented
26	C₃×H(3,𝔽₄)	C₂×C₅	1	16	yes	direct product factor
27	C₂×C₃×#124	C₂×C₅	1	9	yes	complemented
28	C₂×C₃×#120	C₂×C₅	1	6	yes	complemented
29	C₂⁵×C₅	C₂²×C₃	1	2	yes	complemented
30	C₂²×C₅×Q₈	C₂²×C₃	1	3	yes	complemented
31	C₂²×C₅×D₄	C₂²×C₃	1	9	yes	—
32	C₅×#124	C₂²×C₃	1	72	yes	complemented
33	C₅×#120	C₂²×C₃	1	48	yes	complemented
34	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	18	yes	complemented
35	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	complemented
36	C₂×H(3,𝔽₄)	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
37	C₂³×C₃×C₅	—	2	36	no	—
38	C₂³×C₃×C₅	—	4	24	no	—
39	C₂³×C₃×C₅	—	4	32	no	—
40	C₂³×C₃×C₅	C₂⁴	1	12	yes	—
41	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
42	C₃×C₅×Q₈	—	4	12	no	—
43	C₃×C₅×D₄	—	4	36	no	—
44	C₂×C₃×C₄×C₅	—	2	18	no	—
45	C₂×C₃×C₄×C₅	C₂⁴	1	18	yes	—
46	C₂⁵×C₃	C₂²×C₅	1	2	yes	complemented
47	C₂²×C₃×Q₈	C₂²×C₅	1	3	yes	complemented
48	C₂²×C₃×D₄	C₂²×C₅	1	9	yes	—
49	C₃×#124	C₂²×C₅	1	72	yes	complemented
50	C₃×#120	C₂²×C₅	1	48	yes	complemented
51	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	18	yes	complemented
52	C₂×C₃×C₄²	C₂²×C₅	1	3	yes	complemented
53	C₂⁴×C₅	—	4	12	no	—
54	C₂⁴×C₅	C₂³×C₃	1	6	yes	—
55	C₂⁴×C₅	C₂³×C₃	1	8	yes	complemented
56	C₂×C₅×Q₈	—	4	6	no	—
57	C₂×C₅×Q₈	C₂³×C₃	1	12	yes	complemented
58	C₂×C₅×D₄	—	4	18	no	—
59	C₂×C₅×D₄	C₂³×C₃	1	36	yes	—
60	C₂²×C₄×C₅	C₂³×C₃	1	9	yes	—
61	C₅×C₂²⋊C₄	C₂³×C₃	1	72	yes	complemented
62	C₄²×C₅	C₂³×C₃	1	12	yes	complemented
63	H(3,𝔽₄)	C₂×C₃×C₅	1	16	yes	direct product factor
64	C₂×#124	C₂×C₃×C₅	1	9	yes	complemented
65	C₂×#120	C₂×C₃×C₅	1	6	yes	complemented
66	C₂²×C₃×C₅	—	2	36	no	—
67	C₂²×C₃×C₅	—	4	32	no	—
68	C₂²×C₃×C₅	—	4	24	no	—
69	C₂²×C₃×C₅	H(5,𝔽₂)	1	6	yes	—
70	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
71	C₃×C₄×C₅	—	2	18	no	—
72	C₂⁴×C₃	—	4	12	no	—
73	C₂⁴×C₃	C₂³×C₅	1	6	yes	—
74	C₂⁴×C₃	C₂³×C₅	1	8	yes	complemented
75	C₂×C₃×Q₈	—	4	6	no	—
76	C₂×C₃×Q₈	C₂³×C₅	1	12	yes	complemented
77	C₂×C₃×D₄	—	4	18	no	—
78	C₂×C₃×D₄	C₂³×C₅	1	36	yes	—
79	C₂²×C₃×C₄	C₂³×C₅	1	9	yes	—
80	C₃×C₂²⋊C₄	C₂³×C₅	1	72	yes	complemented
81	C₃×C₄²	C₂³×C₅	1	12	yes	complemented
82	C₂³×C₅	—	2	36	no	—
83	C₂³×C₅	—	4	24	no	—
84	C₂³×C₅	—	4	32	no	—
85	C₂³×C₅	C₂⁴×C₃	1	12	yes	—
86	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
87	C₅×Q₈	—	4	12	no	—
88	C₅×D₄	—	4	36	no	—
89	C₂×C₄×C₅	—	2	18	no	—
90	C₂×C₄×C₅	C₂⁴×C₃	1	18	yes	—
91	C₂⁵	C₂²×C₃×C₅	1	2	yes	complemented
92	C₂²×Q₈	C₂²×C₃×C₅	1	3	yes	complemented
93	C₂²×D₄	C₂²×C₃×C₅	1	9	yes	—
94	#124	C₂²×C₃×C₅	1	72	yes	complemented
95	#120	C₂²×C₃×C₅	1	48	yes	complemented
96	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	18	yes	complemented
97	C₂×C₄²	C₂²×C₃×C₅	1	3	yes	complemented
98	C₂×C₃×C₅	—	4	12	no	—
99	C₂×C₃×C₅	H(3,𝔽₄)	1	4	yes	direct product factor
100	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	3	yes	—
101	C₂³×C₃	—	2	36	no	—
102	C₂³×C₃	—	4	24	no	—
103	C₂³×C₃	—	4	32	no	—
104	C₂³×C₃	C₂⁴×C₅	1	12	yes	—
105	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
106	C₃×Q₈	—	4	12	no	—
107	C₃×D₄	—	4	36	no	—
108	C₂×C₃×C₄	—	2	18	no	—
109	C₂×C₃×C₄	C₂⁴×C₅	1	18	yes	—
110	C₂²×C₅	—	2	36	no	—
111	C₂²×C₅	—	4	32	no	—
112	C₂²×C₅	—	4	24	no	—
113	C₂²×C₅	C₃×H(5,𝔽₂)	1	6	yes	—
114	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
115	C₄×C₅	—	2	18	no	—
116	C₂⁴	—	4	12	no	—
117	C₂⁴	C₂³×C₃×C₅	1	6	yes	—
118	C₂⁴	C₂³×C₃×C₅	1	8	yes	complemented
119	C₂×Q₈	—	4	6	no	—
120	C₂×Q₈	C₂³×C₃×C₅	1	12	yes	complemented
121	C₂×D₄	—	4	18	no	—
122	C₂×D₄	C₂³×C₃×C₅	1	36	yes	—
123	C₂²×C₄	C₂³×C₃×C₅	1	9	yes	—
124	C₂²⋊C₄	C₂³×C₃×C₅	1	72	yes	complemented
125	C₄²	C₂³×C₃×C₅	1	12	yes	complemented
126	C₃×C₅	C₂×H(3,𝔽₄)	1	1	yes	characteristic, direct product factor, Hall
127	C₂²×C₃	—	2	36	no	—
128	C₂²×C₃	—	4	32	no	—
129	C₂²×C₃	—	4	24	no	—
130	C₂²×C₃	C₅×H(5,𝔽₂)	1	6	yes	—
131	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
132	C₃×C₄	—	2	18	no	—
133	C₂×C₅	—	4	12	no	—
134	C₂×C₅	C₃×H(3,𝔽₄)	1	4	yes	direct product factor
135	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	3	yes	—
136	C₂³	—	2	36	no	—
137	C₂³	—	4	24	no	—
138	C₂³	—	4	32	no	—
139	C₂³	C₂⁴×C₃×C₅	1	12	yes	—
140	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
141	Q₈	—	4	12	no	—
142	D₄	—	4	36	no	—
143	C₂×C₄	—	2	18	no	—
144	C₂×C₄	C₂⁴×C₃×C₅	1	18	yes	—
145	C₂×C₃	—	4	12	no	—
146	C₂×C₃	C₅×H(3,𝔽₄)	1	4	yes	direct product factor
147	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	3	yes	—
148	C₅	C₂×C₃×H(3,𝔽₄)	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²	—	2	36	no	—
150	C₂²	—	4	32	no	—
151	C₂²	—	4	24	no	—
152	C₂²	C₃×C₅×H(5,𝔽₂)	1	6	yes	—
153	C₂²	C₂⁵×C₃×C₅	1	1	yes	characteristic
154	C₄	—	2	18	no	—
155	C₃	C₂×C₅×H(3,𝔽₄)	1	1	yes	characteristic, direct product factor, Sylow
156	C₂	—	4	12	no	—
157	C₂	C₃×C₅×H(3,𝔽₄)	1	4	yes	direct product factor
158	C₂	C₂×C₃×C₅×H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×H(3,𝔽₄)	C₂	direct (N × H)
2. C₂×C₃×C₅×#124	C₂	semidirect
3. C₂×C₃×C₅×#120	C₂	semidirect
4. C₂×C₅×H(3,𝔽₄)	C₃	direct (N × H)
5. C₂⁵×C₃×C₅	C₂²	semidirect
6. C₂²×C₃×C₅×Q₈	C₂²	semidirect
8. C₃×C₅×#124	C₂²	semidirect
9. C₃×C₅×#120	C₂²	semidirect
10. C₂×C₃×C₅×C₂²⋊C₄	C₂²	semidirect
11. C₂×C₃×C₄²×C₅	C₂²	semidirect
12. C₂×C₃×H(3,𝔽₄)	C₅	direct (N × H)
13. C₅×H(3,𝔽₄)	C₂×C₃	direct (N × H)
14. C₂×C₅×#124	C₂×C₃	semidirect
15. C₂×C₅×#120	C₂×C₃	semidirect
18. C₂⁴×C₃×C₅	C₂³	semidirect
20. C₂×C₃×C₅×Q₈	C₂³	semidirect
24. C₃×C₅×C₂²⋊C₄	C₂³	semidirect
25. C₃×C₄²×C₅	C₂³	semidirect
26. C₃×H(3,𝔽₄)	C₂×C₅	direct (N × H)
27. C₂×C₃×#124	C₂×C₅	semidirect
28. C₂×C₃×#120	C₂×C₅	semidirect
29. C₂⁵×C₅	C₂²×C₃	semidirect
30. C₂²×C₅×Q₈	C₂²×C₃	semidirect
32. C₅×#124	C₂²×C₃	semidirect
33. C₅×#120	C₂²×C₃	semidirect
34. C₂×C₅×C₂²⋊C₄	C₂²×C₃	semidirect
35. C₂×C₄²×C₅	C₂²×C₃	semidirect
36. C₂×H(3,𝔽₄)	C₃×C₅	direct (N × H)
46. C₂⁵×C₃	C₂²×C₅	semidirect
47. C₂²×C₃×Q₈	C₂²×C₅	semidirect
49. C₃×#124	C₂²×C₅	semidirect
50. C₃×#120	C₂²×C₅	semidirect
51. C₂×C₃×C₂²⋊C₄	C₂²×C₅	semidirect
52. C₂×C₃×C₄²	C₂²×C₅	semidirect
55. C₂⁴×C₅	C₂³×C₃	semidirect
57. C₂×C₅×Q₈	C₂³×C₃	semidirect
61. C₅×C₂²⋊C₄	C₂³×C₃	semidirect
62. C₄²×C₅	C₂³×C₃	semidirect
63. H(3,𝔽₄)	C₂×C₃×C₅	direct (N × H)
64. C₂×#124	C₂×C₃×C₅	semidirect
65. C₂×#120	C₂×C₃×C₅	semidirect
74. C₂⁴×C₃	C₂³×C₅	semidirect
76. C₂×C₃×Q₈	C₂³×C₅	semidirect
80. C₃×C₂²⋊C₄	C₂³×C₅	semidirect
81. C₃×C₄²	C₂³×C₅	semidirect
91. C₂⁵	C₂²×C₃×C₅	semidirect
92. C₂²×Q₈	C₂²×C₃×C₅	semidirect
94. #124	C₂²×C₃×C₅	semidirect
95. #120	C₂²×C₃×C₅	semidirect
96. C₂×C₂²⋊C₄	C₂²×C₃×C₅	semidirect
97. C₂×C₄²	C₂²×C₃×C₅	semidirect
99. C₂×C₃×C₅	H(3,𝔽₄)	direct (N × H)
118. C₂⁴	C₂³×C₃×C₅	semidirect
120. C₂×Q₈	C₂³×C₃×C₅	semidirect
124. C₂²⋊C₄	C₂³×C₃×C₅	semidirect
125. C₄²	C₂³×C₃×C₅	semidirect
126. C₃×C₅	C₂×H(3,𝔽₄)	direct (N × H)
134. C₂×C₅	C₃×H(3,𝔽₄)	direct (N × H)
146. C₂×C₃	C₅×H(3,𝔽₄)	direct (N × H)
148. C₅	C₂×C₃×H(3,𝔽₄)	direct (N × H)
155. C₃	C₂×C₅×H(3,𝔽₄)	direct (N × H)
157. C₂	C₃×C₅×H(3,𝔽₄)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×H(3,𝔽₄)	2
C₂×C₃×C₅×#124	2
C₂×C₃×C₅×#120	2
C₂×C₅×H(3,𝔽₄)	3
C₂×C₃×H(3,𝔽₄)	5

Subgroups — C2×C3×C5×H(3,𝔽4) #14276560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×H(3,𝔽₄) #14276560