Subgroups — C₃×C₅×#2381 #14275415

Order 1920.

848 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#391	C₂	1	4	yes	maximal
3	C₃×C₅×#387	C₂	1	8	yes	maximal
4	C₃×C₅×#383	C₂	1	2	yes	maximal
5	C₅×#2381	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
7	C₃×C₅×C₄⋊Q₈	C₄	1	8	yes	complemented
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	4	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×#95	C₂²	1	8	yes	—
13	C₃×C₅×#95	C₂²	1	4	yes	—
14	C₃×C₅×#95	C₂²	1	8	yes	—
15	C₃×#2381	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
17	C₅×#391	C₂×C₃	1	4	yes	—
18	C₅×#387	C₂×C₃	1	8	yes	—
19	C₅×#383	C₂×C₃	1	2	yes	—
20	C₂×C₃×C₅×Q₈	—	4	2	no	—
21	C₂×C₃×C₅×Q₈	—	4	2	no	—
22	C₂×C₃×C₅×Q₈	C₂×C₄	1	8	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	8	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
26	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₃×C₅×C₄⋊C₄	—	4	4	no	—
28	C₃×C₅×C₄⋊C₄	C₂×C₄	1	16	yes	—
29	C₃×C₄²×C₅	—	4	4	no	—
30	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
31	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
32	C₃×#391	C₂×C₅	1	4	yes	—
33	C₃×#387	C₂×C₅	1	8	yes	—
34	C₃×#383	C₂×C₅	1	2	yes	—
35	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
36	C₅×C₄⋊Q₈	C₃×C₄	1	8	yes	complemented
37	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
38	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
39	C₂×C₄²×C₅	C₂²×C₃	1	4	yes	—
40	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
41	C₅×#95	C₂²×C₃	1	8	yes	—
42	C₅×#95	C₂²×C₃	1	4	yes	—
43	C₅×#95	C₂²×C₃	1	8	yes	—
44	#2381	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
45	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
46	C₃×C₅×Q₈	—	4	8	no	—
47	C₂×C₃×C₄×C₅	—	2	4	no	—
48	C₂×C₃×C₄×C₅	—	2	4	no	—
49	C₂×C₃×C₄×C₅	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	2	4	no	—
51	C₂×C₃×C₄×C₅	—	2	8	no	—
52	C₂×C₃×C₄×C₅	—	2	16	no	—
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
54	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂²×C₄	1	8	yes	—
56	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
57	C₃×C₄⋊Q₈	C₄×C₅	1	8	yes	complemented
58	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
59	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
60	C₂×C₃×C₄²	C₂²×C₅	1	4	yes	—
61	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
62	C₃×#95	C₂²×C₅	1	8	yes	—
63	C₃×#95	C₂²×C₅	1	4	yes	—
64	C₃×#95	C₂²×C₅	1	8	yes	—
65	C₂×C₅×Q₈	—	4	2	no	—
66	C₂×C₅×Q₈	—	4	2	no	—
67	C₂×C₅×Q₈	C₂×C₃×C₄	1	8	yes	—
68	C₂²×C₄×C₅	C₂³×C₃	1	8	yes	—
69	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
70	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
71	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
72	C₅×C₄⋊C₄	—	4	4	no	—
73	C₅×C₄⋊C₄	C₂×C₃×C₄	1	16	yes	—
74	C₄²×C₅	—	4	4	no	—
75	C₄²×C₅	C₂×C₃×C₄	1	4	yes	—
76	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
77	#391	C₂×C₃×C₅	1	4	yes	—
78	#387	C₂×C₃×C₅	1	8	yes	—
79	#383	C₂×C₃×C₅	1	2	yes	—
80	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
81	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
82	C₂²×C₃×C₅	2₋¹⁺⁴	1	4	yes	—
83	C₃×C₄×C₅	—	2	8	no	—
84	C₃×C₄×C₅	—	2	2	no	—
85	C₃×C₄×C₅	—	2	4	no	—
86	C₃×C₄×C₅	—	4	8	no	—
87	C₂×C₃×Q₈	—	4	2	no	—
88	C₂×C₃×Q₈	—	4	2	no	—
89	C₂×C₃×Q₈	C₂×C₄×C₅	1	8	yes	—
90	C₂²×C₃×C₄	C₂³×C₅	1	8	yes	—
91	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
92	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
94	C₃×C₄⋊C₄	—	4	4	no	—
95	C₃×C₄⋊C₄	C₂×C₄×C₅	1	16	yes	—
96	C₃×C₄²	—	4	4	no	—
97	C₃×C₄²	C₂×C₄×C₅	1	4	yes	—
98	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
99	C₅×Q₈	—	4	8	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	2	4	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	8	no	—
105	C₂×C₄×C₅	—	2	16	no	—
106	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
107	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
108	C₂×C₄×C₅	C₂²×C₃×C₄	1	8	yes	—
109	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
110	C₄⋊Q₈	C₃×C₄×C₅	1	8	yes	complemented
111	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
112	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
113	C₂×C₄²	C₂²×C₃×C₅	1	4	yes	—
114	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
115	#95	C₂²×C₃×C₅	1	8	yes	—
116	#95	C₂²×C₃×C₅	1	4	yes	—
117	#95	C₂²×C₃×C₅	1	8	yes	—
118	C₂×C₃×C₅	#518	1	1	yes	characteristic
119	C₂×C₃×C₅	#519	1	2	yes	—
120	C₂×C₃×C₅	#563	1	4	yes	—
121	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
122	C₃×Q₈	—	4	8	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	4	no	—
127	C₂×C₃×C₄	—	2	8	no	—
128	C₂×C₃×C₄	—	2	16	no	—
129	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
130	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
131	C₂×C₃×C₄	C₂²×C₄×C₅	1	8	yes	—
132	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
133	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
134	C₂²×C₅	C₃×2₋¹⁺⁴	1	4	yes	—
135	C₄×C₅	—	2	8	no	—
136	C₄×C₅	—	2	2	no	—
137	C₄×C₅	—	2	4	no	—
138	C₄×C₅	—	4	8	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂×Q₈	—	4	2	no	—
141	C₂×Q₈	C₂×C₃×C₄×C₅	1	8	yes	—
142	C₂²×C₄	C₂³×C₃×C₅	1	8	yes	—
143	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
144	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
145	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	C₂×C₃×C₄×C₅	1	16	yes	—
148	C₄²	—	4	4	no	—
149	C₄²	C₂×C₃×C₄×C₅	1	4	yes	—
150	C₃×C₅	#2381	1	1	yes	characteristic, direct product factor, Hall
151	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
152	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
153	C₂²×C₃	C₅×2₋¹⁺⁴	1	4	yes	—
154	C₃×C₄	—	2	8	no	—
155	C₃×C₄	—	2	2	no	—
156	C₃×C₄	—	2	4	no	—
157	C₃×C₄	—	4	8	no	—
158	C₂×C₅	C₃×#518	1	1	yes	characteristic
159	C₂×C₅	C₃×#519	1	2	yes	—
160	C₂×C₅	C₃×#563	1	4	yes	—
161	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
162	Q₈	—	4	8	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	16	no	—
169	C₂×C₄	C₂²×C₃×C₄×C₅	1	4	yes	—
170	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
171	C₂×C₄	C₂²×C₃×C₄×C₅	1	8	yes	—
172	C₂×C₃	C₅×#518	1	1	yes	characteristic
173	C₂×C₃	C₅×#519	1	2	yes	—
174	C₂×C₃	C₅×#563	1	4	yes	—
175	C₅	C₃×#2381	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
177	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
178	C₂²	C₃×C₅×2₋¹⁺⁴	1	4	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	2	8	no	—
181	C₄	—	2	4	no	—
182	C₄	—	4	8	no	—
183	C₃	C₅×#2381	1	1	yes	characteristic, direct product factor, Sylow
184	C₂	C₃×C₅×#518	1	1	yes	characteristic
185	C₂	C₃×C₅×#519	1	2	yes	—
186	C₂	C₃×C₅×#563	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂⁴
commutator	C₂²	C₂³×C₃×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×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
5. C₅×#2381	C₃	direct (N × H)
7. C₃×C₅×C₄⋊Q₈	C₄	semidirect
15. C₃×#2381	C₅	direct (N × H)
36. C₅×C₄⋊Q₈	C₃×C₄	semidirect
44. #2381	C₃×C₅	direct (N × H)
57. C₃×C₄⋊Q₈	C₄×C₅	semidirect
110. C₄⋊Q₈	C₃×C₄×C₅	semidirect
150. C₃×C₅	#2381	direct (N × H)
175. C₅	C₃×#2381	direct (N × H)
183. C₃	C₅×#2381	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×C₅×C₄⋊Q₈	2
C₃×C₅×#391	2
C₃×C₅×#387	2
C₃×C₅×#383	2
C₅×#2381	3
C₃×#2381	5

Subgroups — C3×C5×#2381 #14275415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2381 #14275415