Subgroups — C₃×D₅×#384 #14320122

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×C₃×D₅×C₄⋊C₄	C₂	1	6	yes	maximal
2	C₂×C₃×C₄²×D₅	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65265	C₂	1	6	yes	maximal, complemented
5	C₃×#65224	C₂	1	1	yes	maximal, characteristic, complemented
6	D₅×#384	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₄×D₅	C₂²	1	4	yes	—
8	C₂²×C₃×C₄×D₅	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×C₃×C₄⋊Dic₅	C₂²	1	6	yes	—
12	C₂×C₃×#683	C₂²	1	12	yes	—
13	C₂×C₃×C₄×Dic₅	C₂²	1	3	yes	—
14	C₃×D₅×C₄⋊C₄	—	2	24	no	—
15	C₃×C₄²×D₅	C₄	1	8	yes	complemented
16	C₂×C₃×#384	—	5	1	no	maximal, Hall
17	C₂×D₅×C₄⋊C₄	C₂×C₃	1	6	yes	—
18	C₂×C₄²×D₅	C₂×C₃	1	1	yes	characteristic
19	C₅×#384	C₂×C₃	1	1	yes	characteristic, complemented
20	#65265	C₂×C₃	1	6	yes	complemented
21	#65224	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂³×C₃×D₅	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×C₃×Dic₅	C₂³	1	4	yes	—
26	C₂²×C₃×Dic₅	C₂³	1	3	yes	—
27	C₂×C₃×C₄×D₅	—	2	24	no	—
28	C₂×C₃×C₄×D₅	C₂×C₄	1	12	yes	—
29	C₂×C₃×C₄×D₅	D₄	1	12	yes	—
30	C₂×C₃×C₄×D₅	Q₈	1	12	yes	—
31	C₃×C₅×C₄⋊C₄	—	2	12	no	—
32	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
33	C₃×C₄⋊Dic₅	—	2	12	no	—
34	C₃×#683	—	2	24	no	—
35	C₃×C₄×Dic₅	C₂×C₄	1	12	yes	complemented
36	C₂²×C₃×C₄⋊C₄	—	5	6	no	—
37	C₂²×C₃×C₄²	—	5	1	no	—
38	C₃×#384	—	5	6	no	—
39	C₃×#384	—	5	1	no	—
40	C₃×#384	D₅	1	1	yes	characteristic, direct product factor
41	C₂²×C₄×D₅	C₂²×C₃	1	4	yes	—
42	C₂²×C₄×D₅	C₂²×C₃	1	3	yes	—
43	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	6	yes	—
44	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₄⋊Dic₅	C₂²×C₃	1	6	yes	—
46	C₂×#683	C₂²×C₃	1	12	yes	—
47	C₂×C₄×Dic₅	C₂²×C₃	1	3	yes	—
48	D₅×C₄⋊C₄	—	2	24	no	—
49	C₄²×D₅	C₃×C₄	1	8	yes	complemented
50	C₂×#384	—	5	1	no	Sylow
51	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
52	C₂²×C₃×D₅	C₂²×C₄	1	2	yes	—
53	C₂²×C₃×D₅	C₂×D₄	1	6	yes	—
54	C₂²×C₃×D₅	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×C₃×Dic₅	—	2	12	no	—
60	C₂×C₃×Dic₅	C₂²×C₄	1	6	yes	—
61	C₂×C₃×Dic₅	C₂×D₄	1	6	yes	—
62	C₂×C₃×Dic₅	C₂×Q₈	1	6	yes	—
63	C₃×C₄×D₅	—	4	16	no	—
64	C₃×C₄×D₅	C₄⋊C₄	1	48	yes	complemented
65	C₂³×C₃×C₄	—	5	3	no	—
66	C₂³×C₃×C₄	—	5	4	no	—
67	C₂×C₃×C₄⋊C₄	—	5	12	no	—
68	C₂×C₃×C₄⋊C₄	—	5	6	no	—
69	C₂×C₃×C₄⋊C₄	—	10	24	no	—
70	C₂×C₃×C₄⋊C₄	C₂×D₅	1	6	yes	—
71	C₂×C₃×C₄²	—	5	3	no	—
72	C₂×C₃×C₄²	—	5	8	no	—
73	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
74	C₂³×D₅	C₂³×C₃	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
76	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
77	C₂²×Dic₅	C₂³×C₃	1	4	yes	—
78	C₂²×Dic₅	C₂³×C₃	1	3	yes	—
79	C₂×C₄×D₅	—	2	24	no	—
80	C₂×C₄×D₅	C₂×C₃×C₄	1	12	yes	—
81	C₂×C₄×D₅	C₃×D₄	1	12	yes	—
82	C₂×C₄×D₅	C₃×Q₈	1	12	yes	—
83	C₅×C₄⋊C₄	—	2	12	no	—
84	C₄²×C₅	C₂×C₃×C₄	1	4	yes	complemented
85	C₄⋊Dic₅	—	2	12	no	—
86	#683	—	2	24	no	—
87	C₄×Dic₅	C₂×C₃×C₄	1	12	yes	complemented
88	C₂²×C₄⋊C₄	—	5	6	no	—
89	C₂²×C₄²	—	5	1	no	—
90	#384	—	5	6	no	—
91	#384	—	5	1	no	—
92	#384	C₃×D₅	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×C₃×D₅	C₂×C₄⋊C₄	1	12	yes	—
97	C₂×C₃×D₅	#127	1	4	yes	—
98	C₂×C₃×D₅	C₄⋊Q₈	1	12	yes	—
99	C₃×C₄×C₅	—	4	4	no	—
100	C₃×C₄×C₅	C₂×C₄⋊C₄	1	12	yes	complemented
101	C₃×Dic₅	—	4	4	no	—
102	C₃×Dic₅	C₂×C₄⋊C₄	1	12	yes	complemented
103	C₂⁴×C₃	—	5	1	no	—
104	C₂²×C₃×C₄	—	5	3	no	—
105	C₂²×C₃×C₄	—	5	12	no	—
106	C₂²×C₃×C₄	—	5	12	no	—
107	C₂²×C₃×C₄	—	5	12	no	—
108	C₂²×C₃×C₄	—	5	4	no	—
109	C₂²×C₃×C₄	—	10	24	no	—
110	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
111	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
112	C₃×C₄⋊C₄	—	2	12	no	—
113	C₃×C₄⋊C₄	—	10	24	no	—
114	C₃×C₄⋊C₄	—	10	12	no	—
115	C₃×C₄²	—	5	12	no	—
116	C₃×C₄²	C₄×D₅	1	4	yes	complemented
117	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
118	C₂²×D₅	C₂²×C₃×C₄	1	2	yes	—
119	C₂²×D₅	C₂×C₃×D₄	1	6	yes	—
120	C₂²×D₅	C₂×C₃×Q₈	1	6	yes	—
121	C₂×C₄×C₅	—	2	12	no	—
122	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
123	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
124	C₂×C₄×C₅	C₂×C₃×Q₈	1	6	yes	—
125	C₂×Dic₅	—	2	12	no	—
126	C₂×Dic₅	C₂²×C₃×C₄	1	6	yes	—
127	C₂×Dic₅	C₂×C₃×D₄	1	6	yes	—
128	C₂×Dic₅	C₂×C₃×Q₈	1	6	yes	—
129	C₄×D₅	—	4	16	no	—
130	C₄×D₅	C₃×C₄⋊C₄	1	48	yes	complemented
131	C₂³×C₄	—	5	3	no	—
132	C₂³×C₄	—	5	4	no	—
133	C₂×C₄⋊C₄	—	5	12	no	—
134	C₂×C₄⋊C₄	—	5	6	no	—
135	C₂×C₄⋊C₄	—	10	24	no	—
136	C₂×C₄⋊C₄	C₂×C₃×D₅	1	6	yes	—
137	C₂×C₄²	—	5	3	no	—
138	C₂×C₄²	—	5	8	no	—
139	C₂×C₄²	C₂×C₃×D₅	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	C₃×D₅	#384	1	8	yes	direct product factor
144	C₂³×C₃	—	5	2	no	—
145	C₂³×C₃	—	5	6	no	—
146	C₂³×C₃	—	5	6	no	—
147	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
148	C₂×C₃×C₄	—	2	12	no	—
149	C₂×C₃×C₄	—	5	6	no	—
150	C₂×C₃×C₄	—	5	6	no	—
151	C₂×C₃×C₄	—	5	6	no	—
152	C₂×C₃×C₄	—	5	48	no	—
153	C₂×C₃×C₄	—	10	12	no	—
154	C₂×C₃×C₄	—	20	16	no	—
155	C₂×C₃×C₄	C₂×C₄×D₅	1	6	yes	—
156	C₂×C₃×C₄	D₄×D₅	1	6	yes	—
157	C₂×C₃×C₄	Q₈×D₅	1	6	yes	—
158	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
159	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
160	C₂²×C₅	C₂²×C₃×Q₈	1	3	yes	—
161	C₂×D₅	C₂×C₃×C₄⋊C₄	1	12	yes	—
162	C₂×D₅	C₃×#127	1	4	yes	—
163	C₂×D₅	C₃×C₄⋊Q₈	1	12	yes	—
164	C₄×C₅	—	4	4	no	—
165	C₄×C₅	C₂×C₃×C₄⋊C₄	1	12	yes	complemented
166	Dic₅	—	4	4	no	—
167	Dic₅	C₂×C₃×C₄⋊C₄	1	12	yes	complemented
168	C₂⁴	—	5	1	no	—
169	C₂²×C₄	—	5	3	no	—
170	C₂²×C₄	—	5	12	no	—
171	C₂²×C₄	—	5	12	no	—
172	C₂²×C₄	—	5	12	no	—
173	C₂²×C₄	—	5	4	no	—
174	C₂²×C₄	—	10	24	no	—
175	C₂²×C₄	C₂²×C₃×D₅	1	4	yes	—
176	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
177	C₄⋊C₄	—	2	12	no	—
178	C₄⋊C₄	—	10	24	no	—
179	C₄⋊C₄	—	10	12	no	—
180	C₄²	—	5	12	no	—
181	C₄²	C₃×C₄×D₅	1	4	yes	complemented
182	C₃×C₅	C₂×#384	1	1	yes	characteristic, complemented, Hall
183	C₂²×C₃	—	5	4	no	—
184	C₂²×C₃	—	5	12	no	—
185	C₂²×C₃	—	5	12	no	—
186	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
187	C₂²×C₃	C₂×D₄×D₅	1	3	yes	—
188	C₂²×C₃	C₂×Q₈×D₅	1	3	yes	—
189	C₃×C₄	—	4	4	no	—
190	C₃×C₄	—	5	12	no	—
191	C₃×C₄	—	20	4	no	—
192	C₃×C₄	D₅×C₄⋊C₄	1	12	yes	complemented
193	C₂×C₅	C₂²×C₃×C₄⋊C₄	1	3	yes	—
194	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
195	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	3	yes	—
196	D₅	C₃×#384	1	8	yes	direct product factor
197	C₂³	—	5	2	no	—
198	C₂³	—	5	6	no	—
199	C₂³	—	5	6	no	—
200	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
201	C₂×C₄	—	2	12	no	—
202	C₂×C₄	—	5	6	no	—
203	C₂×C₄	—	5	6	no	—
204	C₂×C₄	—	5	6	no	—
205	C₂×C₄	—	5	48	no	—
206	C₂×C₄	—	10	12	no	—
207	C₂×C₄	—	20	16	no	—
208	C₂×C₄	C₂×C₃×C₄×D₅	1	6	yes	—
209	C₂×C₄	C₃×D₄×D₅	1	6	yes	—
210	C₂×C₄	C₃×Q₈×D₅	1	6	yes	—
211	C₂×C₃	—	5	8	no	—
212	C₂×C₃	C₂×D₅×C₄⋊C₄	1	3	yes	—
213	C₂×C₃	D₅×#127	1	1	yes	characteristic
214	C₂×C₃	D₅×C₄⋊Q₈	1	3	yes	—
215	C₅	C₂×C₃×#384	1	1	yes	characteristic, complemented, Sylow
216	C₂²	—	5	4	no	—
217	C₂²	—	5	12	no	—
218	C₂²	—	5	12	no	—
219	C₂²	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
220	C₂²	C₂×C₃×D₄×D₅	1	3	yes	—
221	C₂²	C₂×C₃×Q₈×D₅	1	3	yes	—
222	C₄	—	4	4	no	—
223	C₄	—	5	12	no	—
224	C₄	—	20	4	no	—
225	C₄	C₃×D₅×C₄⋊C₄	1	12	yes	complemented
226	C₃	D₅×#384	1	1	yes	characteristic, direct product factor, Sylow
227	C₂	—	5	8	no	—
228	C₂	C₂×C₃×D₅×C₄⋊C₄	1	3	yes	—
229	C₂	C₃×D₅×#127	1	1	yes	characteristic
230	C₂	C₃×D₅×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂³	C₂³×C₃×D₅
Fitting	C₃×C₅×#384	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂³×C₃×D₅	C₂³
cosocle	C₂³×C₅	C₂⁴×C₃
hypercenter	C₃×#384	D₅
hypocenter	C₅	C₂×C₃×#384
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#384

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#384	C₂	semidirect
4. C₃×#65265	C₂	semidirect
5. C₃×#65224	C₂	semidirect
6. D₅×#384	C₃	direct (N × H)
15. C₃×C₄²×D₅	C₄	semidirect
19. C₅×#384	C₂×C₃	semidirect
20. #65265	C₂×C₃	semidirect
21. #65224	C₂×C₃	semidirect
32. C₃×C₄²×C₅	C₂×C₄	semidirect
35. C₃×C₄×Dic₅	C₂×C₄	semidirect
40. C₃×#384	D₅	direct (N × H)
49. C₄²×D₅	C₃×C₄	semidirect
64. C₃×C₄×D₅	C₄⋊C₄	semidirect
84. C₄²×C₅	C₂×C₃×C₄	semidirect
87. C₄×Dic₅	C₂×C₃×C₄	semidirect
92. #384	C₃×D₅	direct (N × H)
100. C₃×C₄×C₅	C₂×C₄⋊C₄	semidirect
102. C₃×Dic₅	C₂×C₄⋊C₄	semidirect
116. C₃×C₄²	C₄×D₅	semidirect
130. C₄×D₅	C₃×C₄⋊C₄	semidirect
143. C₃×D₅	#384	direct (N × H)
165. C₄×C₅	C₂×C₃×C₄⋊C₄	semidirect
167. Dic₅	C₂×C₃×C₄⋊C₄	semidirect
181. C₄²	C₃×C₄×D₅	semidirect
182. C₃×C₅	C₂×#384	semidirect
192. C₃×C₄	D₅×C₄⋊C₄	semidirect
196. D₅	C₃×#384	direct (N × H)
215. C₅	C₂×C₃×#384	semidirect
225. C₄	C₃×D₅×C₄⋊C₄	semidirect
226. C₃	D₅×#384	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×D₅×C₄⋊C₄	2
C₂×C₃×C₄²×D₅	2
C₃×C₅×#384	2
C₃×#65265	2
C₃×#65224	2
D₅×#384	3
C₂×C₃×#384	5

Subgroups — C3×D5×#384 #14320122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₅×#384 #14320122