Subgroups — C₅×D₃×#384 #14320121

Order 1920.

1776 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₃×C₄⋊C₄	C₂	1	6	yes	maximal
2	C₂×C₄²×C₅×D₃	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4975	C₂	1	6	yes	maximal, complemented
5	C₅×#4935	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#384	—	3	1	no	maximal, Hall
7	C₂²×C₄×C₅×D₃	C₂²	1	4	yes	—
8	C₂²×C₄×C₅×D₃	C₂²	1	3	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊Dic₃	C₂²	1	6	yes	—
12	C₂×C₅×Dic₃⋊C₄	C₂²	1	12	yes	—
13	C₂×C₄×C₅×Dic₃	C₂²	1	3	yes	—
14	C₅×D₃×C₄⋊C₄	—	2	24	no	—
15	C₄²×C₅×D₃	C₄	1	8	yes	complemented
16	D₃×#384	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂²×C₅×C₄⋊C₄	—	3	6	no	—
18	C₂²×C₄²×C₅	—	3	1	no	—
19	C₅×#384	—	3	6	no	—
20	C₅×#384	—	3	1	no	—
21	C₅×#384	D₃	1	1	yes	characteristic, direct product factor
22	C₂³×C₅×D₃	C₂³	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
25	C₂²×C₅×Dic₃	C₂³	1	4	yes	—
26	C₂²×C₅×Dic₃	C₂³	1	3	yes	—
27	C₂×C₄×C₅×D₃	—	2	24	no	—
28	C₂×C₄×C₅×D₃	C₂×C₄	1	12	yes	—
29	C₂×C₄×C₅×D₃	D₄	1	12	yes	—
30	C₂×C₄×C₅×D₃	Q₈	1	12	yes	—
31	C₃×C₅×C₄⋊C₄	—	2	12	no	—
32	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
33	C₅×C₄⋊Dic₃	—	2	12	no	—
34	C₅×Dic₃⋊C₄	—	2	24	no	—
35	C₄×C₅×Dic₃	C₂×C₄	1	12	yes	complemented
36	C₂×D₃×C₄⋊C₄	C₂×C₅	1	6	yes	—
37	C₂×C₄²×D₃	C₂×C₅	1	1	yes	characteristic
38	C₃×#384	C₂×C₅	1	1	yes	characteristic, complemented
39	#4975	C₂×C₅	1	6	yes	complemented
40	#4935	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂³×C₄×C₅	—	3	3	no	—
42	C₂³×C₄×C₅	—	3	4	no	—
43	C₂×C₅×C₄⋊C₄	—	3	12	no	—
44	C₂×C₅×C₄⋊C₄	—	3	6	no	—
45	C₂×C₅×C₄⋊C₄	—	6	24	no	—
46	C₂×C₅×C₄⋊C₄	C₂×D₃	1	6	yes	—
47	C₂×C₄²×C₅	—	3	3	no	—
48	C₂×C₄²×C₅	—	3	8	no	—
49	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
50	C₂×#384	—	3	1	no	Sylow
51	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
52	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
53	C₂²×C₅×D₃	C₂×D₄	1	6	yes	—
54	C₂²×C₅×D₃	C₂×Q₈	1	6	yes	—
55	C₂×C₃×C₄×C₅	—	2	12	no	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
57	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
58	C₂×C₃×C₄×C₅	C₂×Q₈	1	6	yes	—
59	C₂×C₅×Dic₃	—	2	12	no	—
60	C₂×C₅×Dic₃	C₂²×C₄	1	6	yes	—
61	C₂×C₅×Dic₃	C₂×D₄	1	6	yes	—
62	C₂×C₅×Dic₃	C₂×Q₈	1	6	yes	—
63	C₄×C₅×D₃	—	4	16	no	—
64	C₄×C₅×D₃	C₄⋊C₄	1	48	yes	complemented
65	C₂²×C₄×D₃	C₂²×C₅	1	4	yes	—
66	C₂²×C₄×D₃	C₂²×C₅	1	3	yes	—
67	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	6	yes	—
68	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
69	C₂×C₄⋊Dic₃	C₂²×C₅	1	6	yes	—
70	C₂×Dic₃⋊C₄	C₂²×C₅	1	12	yes	—
71	C₂×C₄×Dic₃	C₂²×C₅	1	3	yes	—
72	D₃×C₄⋊C₄	—	2	24	no	—
73	C₄²×D₃	C₄×C₅	1	8	yes	complemented
74	C₂⁴×C₅	—	3	1	no	—
75	C₂²×C₄×C₅	—	3	3	no	—
76	C₂²×C₄×C₅	—	3	12	no	—
77	C₂²×C₄×C₅	—	3	12	no	—
78	C₂²×C₄×C₅	—	3	12	no	—
79	C₂²×C₄×C₅	—	3	4	no	—
80	C₂²×C₄×C₅	—	6	24	no	—
81	C₂²×C₄×C₅	C₂²×D₃	1	4	yes	—
82	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
83	C₅×C₄⋊C₄	—	2	12	no	—
84	C₅×C₄⋊C₄	—	6	24	no	—
85	C₅×C₄⋊C₄	—	6	12	no	—
86	C₄²×C₅	—	3	12	no	—
87	C₄²×C₅	C₄×D₃	1	4	yes	complemented
88	C₂²×C₄⋊C₄	—	3	6	no	—
89	C₂²×C₄²	—	3	1	no	—
90	#384	—	3	6	no	—
91	#384	—	3	1	no	—
92	#384	C₅×D₃	1	1	yes	characteristic, direct product factor
93	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
94	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
95	C₂²×C₃×C₅	C₂²×Q₈	1	3	yes	—
96	C₂×C₅×D₃	C₂×C₄⋊C₄	1	12	yes	—
97	C₂×C₅×D₃	#127	1	4	yes	—
98	C₂×C₅×D₃	C₄⋊Q₈	1	12	yes	—
99	C₃×C₄×C₅	—	4	4	no	—
100	C₃×C₄×C₅	C₂×C₄⋊C₄	1	12	yes	complemented
101	C₅×Dic₃	—	4	4	no	—
102	C₅×Dic₃	C₂×C₄⋊C₄	1	12	yes	complemented
103	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
105	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
106	C₂²×Dic₃	C₂³×C₅	1	4	yes	—
107	C₂²×Dic₃	C₂³×C₅	1	3	yes	—
108	C₂×C₄×D₃	—	2	24	no	—
109	C₂×C₄×D₃	C₂×C₄×C₅	1	12	yes	—
110	C₂×C₄×D₃	C₅×D₄	1	12	yes	—
111	C₂×C₄×D₃	C₅×Q₈	1	12	yes	—
112	C₃×C₄⋊C₄	—	2	12	no	—
113	C₃×C₄²	C₂×C₄×C₅	1	4	yes	complemented
114	C₄⋊Dic₃	—	2	12	no	—
115	Dic₃⋊C₄	—	2	24	no	—
116	C₄×Dic₃	C₂×C₄×C₅	1	12	yes	complemented
117	C₂³×C₅	—	3	2	no	—
118	C₂³×C₅	—	3	6	no	—
119	C₂³×C₅	—	3	6	no	—
120	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
121	C₂×C₄×C₅	—	2	12	no	—
122	C₂×C₄×C₅	—	3	6	no	—
123	C₂×C₄×C₅	—	3	6	no	—
124	C₂×C₄×C₅	—	3	6	no	—
125	C₂×C₄×C₅	—	3	48	no	—
126	C₂×C₄×C₅	—	6	12	no	—
127	C₂×C₄×C₅	—	12	16	no	—
128	C₂×C₄×C₅	C₂×C₄×D₃	1	6	yes	—
129	C₂×C₄×C₅	D₃×D₄	1	6	yes	—
130	C₂×C₄×C₅	D₃×Q₈	1	6	yes	—
131	C₂³×C₄	—	3	3	no	—
132	C₂³×C₄	—	3	4	no	—
133	C₂×C₄⋊C₄	—	3	12	no	—
134	C₂×C₄⋊C₄	—	3	6	no	—
135	C₂×C₄⋊C₄	—	6	24	no	—
136	C₂×C₄⋊C₄	C₂×C₅×D₃	1	6	yes	—
137	C₂×C₄²	—	3	3	no	—
138	C₂×C₄²	—	3	8	no	—
139	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
140	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	3	yes	—
141	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
142	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	3	yes	—
143	C₅×D₃	#384	1	8	yes	direct product factor
144	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
145	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
146	C₂²×D₃	C₂×C₅×D₄	1	6	yes	—
147	C₂²×D₃	C₂×C₅×Q₈	1	6	yes	—
148	C₂×C₃×C₄	—	2	12	no	—
149	C₂×C₃×C₄	C₂²×C₄×C₅	1	6	yes	—
150	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
151	C₂×C₃×C₄	C₂×C₅×Q₈	1	6	yes	—
152	C₂×Dic₃	—	2	12	no	—
153	C₂×Dic₃	C₂²×C₄×C₅	1	6	yes	—
154	C₂×Dic₃	C₂×C₅×D₄	1	6	yes	—
155	C₂×Dic₃	C₂×C₅×Q₈	1	6	yes	—
156	C₄×D₃	—	4	16	no	—
157	C₄×D₃	C₅×C₄⋊C₄	1	48	yes	complemented
158	C₂²×C₅	—	3	4	no	—
159	C₂²×C₅	—	3	12	no	—
160	C₂²×C₅	—	3	12	no	—
161	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
162	C₂²×C₅	C₂×D₃×D₄	1	3	yes	—
163	C₂²×C₅	C₂×D₃×Q₈	1	3	yes	—
164	C₄×C₅	—	3	12	no	—
165	C₄×C₅	—	4	4	no	—
166	C₄×C₅	—	12	4	no	—
167	C₄×C₅	D₃×C₄⋊C₄	1	12	yes	complemented
168	C₂⁴	—	3	1	no	—
169	C₂²×C₄	—	3	3	no	—
170	C₂²×C₄	—	3	12	no	—
171	C₂²×C₄	—	3	12	no	—
172	C₂²×C₄	—	3	12	no	—
173	C₂²×C₄	—	3	4	no	—
174	C₂²×C₄	—	6	24	no	—
175	C₂²×C₄	C₂²×C₅×D₃	1	4	yes	—
176	C₂²×C₄	C₂²×C₅×D₃	1	3	yes	—
177	C₄⋊C₄	—	2	12	no	—
178	C₄⋊C₄	—	6	24	no	—
179	C₄⋊C₄	—	6	12	no	—
180	C₄²	—	3	12	no	—
181	C₄²	C₄×C₅×D₃	1	4	yes	complemented
182	C₃×C₅	C₂×#384	1	1	yes	characteristic, complemented, Hall
183	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
184	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
185	C₂²×C₃	C₂²×C₅×Q₈	1	3	yes	—
186	C₂×D₃	C₂×C₅×C₄⋊C₄	1	12	yes	—
187	C₂×D₃	C₅×#127	1	4	yes	—
188	C₂×D₃	C₅×C₄⋊Q₈	1	12	yes	—
189	C₃×C₄	—	4	4	no	—
190	C₃×C₄	C₂×C₅×C₄⋊C₄	1	12	yes	complemented
191	Dic₃	—	4	4	no	—
192	Dic₃	C₂×C₅×C₄⋊C₄	1	12	yes	complemented
193	C₂×C₅	—	3	8	no	—
194	C₂×C₅	C₂×D₃×C₄⋊C₄	1	3	yes	—
195	C₂×C₅	D₃×#127	1	1	yes	characteristic
196	C₂×C₅	D₃×C₄⋊Q₈	1	3	yes	—
197	C₂³	—	3	2	no	—
198	C₂³	—	3	6	no	—
199	C₂³	—	3	6	no	—
200	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
201	C₂×C₄	—	2	12	no	—
202	C₂×C₄	—	3	6	no	—
203	C₂×C₄	—	3	6	no	—
204	C₂×C₄	—	3	6	no	—
205	C₂×C₄	—	3	48	no	—
206	C₂×C₄	—	6	12	no	—
207	C₂×C₄	—	12	16	no	—
208	C₂×C₄	C₂×C₄×C₅×D₃	1	6	yes	—
209	C₂×C₄	C₅×D₃×D₄	1	6	yes	—
210	C₂×C₄	C₅×D₃×Q₈	1	6	yes	—
211	C₂×C₃	C₂²×C₅×C₄⋊C₄	1	3	yes	—
212	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
213	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	3	yes	—
214	D₃	C₅×#384	1	8	yes	direct product factor
215	C₅	D₃×#384	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²	—	3	4	no	—
217	C₂²	—	3	12	no	—
218	C₂²	—	3	12	no	—
219	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
220	C₂²	C₂×C₅×D₃×D₄	1	3	yes	—
221	C₂²	C₂×C₅×D₃×Q₈	1	3	yes	—
222	C₄	—	3	12	no	—
223	C₄	—	4	4	no	—
224	C₄	—	12	4	no	—
225	C₄	C₅×D₃×C₄⋊C₄	1	12	yes	complemented
226	C₃	C₂×C₅×#384	1	1	yes	characteristic, complemented, Sylow
227	C₂	—	3	8	no	—
228	C₂	C₂×C₅×D₃×C₄⋊C₄	1	3	yes	—
229	C₂	C₅×D₃×#127	1	1	yes	characteristic
230	C₂	C₅×D₃×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#384	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂³×C₅×D₃	C₂³
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#384	D₃
hypocenter	C₃	C₂×C₅×#384
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#384

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#384	C₂	semidirect
4. C₅×#4975	C₂	semidirect
5. C₅×#4935	C₂	semidirect
15. C₄²×C₅×D₃	C₄	semidirect
16. D₃×#384	C₅	direct (N × H)
21. C₅×#384	D₃	direct (N × H)
32. C₃×C₄²×C₅	C₂×C₄	semidirect
35. C₄×C₅×Dic₃	C₂×C₄	semidirect
38. C₃×#384	C₂×C₅	semidirect
39. #4975	C₂×C₅	semidirect
40. #4935	C₂×C₅	semidirect
64. C₄×C₅×D₃	C₄⋊C₄	semidirect
73. C₄²×D₃	C₄×C₅	semidirect
87. C₄²×C₅	C₄×D₃	semidirect
92. #384	C₅×D₃	direct (N × H)
100. C₃×C₄×C₅	C₂×C₄⋊C₄	semidirect
102. C₅×Dic₃	C₂×C₄⋊C₄	semidirect
113. C₃×C₄²	C₂×C₄×C₅	semidirect
116. C₄×Dic₃	C₂×C₄×C₅	semidirect
143. C₅×D₃	#384	direct (N × H)
157. C₄×D₃	C₅×C₄⋊C₄	semidirect
167. C₄×C₅	D₃×C₄⋊C₄	semidirect
181. C₄²	C₄×C₅×D₃	semidirect
182. C₃×C₅	C₂×#384	semidirect
190. C₃×C₄	C₂×C₅×C₄⋊C₄	semidirect
192. Dic₃	C₂×C₅×C₄⋊C₄	semidirect
214. D₃	C₅×#384	direct (N × H)
215. C₅	D₃×#384	direct (N × H)
225. C₄	C₅×D₃×C₄⋊C₄	semidirect
226. C₃	C₂×C₅×#384	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₃×C₄⋊C₄	2
C₂×C₄²×C₅×D₃	2
C₃×C₅×#384	2
C₅×#4975	2
C₅×#4935	2
C₂×C₅×#384	3
D₃×#384	5

Subgroups — C5×D3×#384 #14320121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₃×#384 #14320121