Subgroups — C₃×C₅×#2341 #14275375

Order 1920.

1744 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₅×Q₈	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#390	C₂	1	6	yes	maximal, complemented
4	C₃×C₅×#380	C₂	1	3	yes	maximal, complemented
5	C₃×C₄×C₅×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
6	C₅×#2341	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	—	2	8	no	—
8	C₂×C₃×C₅×#41	C₄	1	8	yes	complemented
9	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×D₄	C₂²	1	3	yes	—
11	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
12	C₃×C₄×C₅×Q₈	—	2	8	no	—
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
14	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
15	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
16	C₂×C₃×C₄²×C₅	C₂²	1	6	yes	—
17	C₃×C₅×#95	C₂²	1	1	yes	characteristic
18	C₃×C₅×#95	C₂²	1	3	yes	—
19	C₃×#2341	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂²×C₅×#41	C₂×C₃	1	1	yes	characteristic
21	C₂×C₄×C₅×Q₈	C₂×C₃	1	2	yes	complemented
22	C₅×#390	C₂×C₃	1	6	yes	complemented
23	C₅×#380	C₂×C₃	1	3	yes	complemented
24	C₄×C₅×C₂²⋊C₄	C₂×C₃	1	3	yes	complemented
25	C₂⁴×C₃×C₅	C₂³	1	3	yes	—
26	C₃×C₅×#41	—	2	32	no	—
27	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₅×Q₈	D₄	1	8	yes	—
29	C₂×C₃×C₅×D₄	—	2	12	no	—
30	C₂×C₃×C₅×D₄	C₂×C₄	1	12	yes	—
31	C₂²×C₃×C₄×C₅	—	2	12	no	—
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	12	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
37	C₃×C₅×C₄⋊C₄	—	2	12	no	—
38	C₃×C₅×C₂²⋊C₄	—	2	12	no	—
39	C₃×C₅×C₂²⋊C₄	—	4	6	no	—
40	C₃×C₄²×C₅	—	2	12	no	—
41	C₂²×C₃×#41	C₂×C₅	1	1	yes	characteristic
42	C₂×C₃×C₄×Q₈	C₂×C₅	1	2	yes	complemented
43	C₃×#390	C₂×C₅	1	6	yes	complemented
44	C₃×#380	C₂×C₅	1	3	yes	complemented
45	C₃×C₄×C₂²⋊C₄	C₂×C₅	1	3	yes	complemented
46	C₂×C₅×#41	—	2	8	no	—
47	C₂×C₅×#41	C₃×C₄	1	8	yes	complemented
48	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
49	C₂²×C₅×D₄	C₂²×C₃	1	3	yes	—
50	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
51	C₄×C₅×Q₈	—	2	8	no	—
52	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	6	yes	—
53	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
54	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
55	C₂×C₄²×C₅	C₂²×C₃	1	6	yes	—
56	C₅×#95	C₂²×C₃	1	1	yes	characteristic
57	C₅×#95	C₂²×C₃	1	3	yes	—
58	#2341	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
59	C₂³×C₃×C₅	—	2	6	no	—
60	C₂³×C₃×C₅	—	2	6	no	—
61	C₂³×C₃×C₅	—	2	3	no	—
62	C₂³×C₃×C₅	—	2	3	no	—
63	C₂³×C₃×C₅	C₂²×C₄	1	6	yes	—
64	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
65	C₃×C₅×Q₈	C₂²⋊C₄	1	16	yes	complemented
66	C₃×C₅×D₄	—	2	24	no	—
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	24	no	—
71	C₂×C₃×C₄×C₅	—	2	6	no	—
72	C₂×C₃×C₄×C₅	—	2	6	no	—
73	C₂×C₃×C₄×C₅	—	2	6	no	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	—
77	C₂×C₃×C₄×C₅	#41	1	4	yes	complemented
78	C₂×C₃×#41	—	2	8	no	—
79	C₂×C₃×#41	C₄×C₅	1	8	yes	complemented
80	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
81	C₂²×C₃×D₄	C₂²×C₅	1	3	yes	—
82	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
83	C₃×C₄×Q₈	—	2	8	no	—
84	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	6	yes	—
85	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
86	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
87	C₂×C₃×C₄²	C₂²×C₅	1	6	yes	—
88	C₃×#95	C₂²×C₅	1	1	yes	characteristic
89	C₃×#95	C₂²×C₅	1	3	yes	—
90	C₂⁴×C₅	C₂³×C₃	1	3	yes	—
91	C₅×#41	—	2	32	no	—
92	C₂×C₅×Q₈	C₂×C₃×C₄	1	4	yes	—
93	C₂×C₅×Q₈	C₃×D₄	1	8	yes	—
94	C₂×C₅×D₄	—	2	12	no	—
95	C₂×C₅×D₄	C₂×C₃×C₄	1	12	yes	—
96	C₂²×C₄×C₅	—	2	12	no	—
97	C₂²×C₄×C₅	C₂×C₃×C₄	1	12	yes	—
98	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
99	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
101	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
102	C₅×C₄⋊C₄	—	2	12	no	—
103	C₅×C₂²⋊C₄	—	2	12	no	—
104	C₅×C₂²⋊C₄	—	4	6	no	—
105	C₄²×C₅	—	2	12	no	—
106	C₂²×#41	C₂×C₃×C₅	1	1	yes	characteristic
107	C₂×C₄×Q₈	C₂×C₃×C₅	1	2	yes	complemented
108	#390	C₂×C₃×C₅	1	6	yes	complemented
109	#380	C₂×C₃×C₅	1	3	yes	complemented
110	C₄×C₂²⋊C₄	C₂×C₃×C₅	1	3	yes	complemented
111	C₂²×C₃×C₅	—	2	6	no	—
112	C₂²×C₃×C₅	—	2	6	no	—
113	C₂²×C₃×C₅	—	2	6	no	—
114	C₂²×C₃×C₅	—	4	6	no	—
115	C₂²×C₃×C₅	—	4	6	no	—
116	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
117	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
118	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
119	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
121	C₃×C₄×C₅	—	2	2	no	—
122	C₃×C₄×C₅	—	2	4	no	—
123	C₃×C₄×C₅	—	4	6	no	—
124	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	12	yes	—
125	C₂⁴×C₃	C₂³×C₅	1	3	yes	—
126	C₃×#41	—	2	32	no	—
127	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	—
128	C₂×C₃×Q₈	C₅×D₄	1	8	yes	—
129	C₂×C₃×D₄	—	2	12	no	—
130	C₂×C₃×D₄	C₂×C₄×C₅	1	12	yes	—
131	C₂²×C₃×C₄	—	2	12	no	—
132	C₂²×C₃×C₄	C₂×C₄×C₅	1	12	yes	—
133	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
134	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
136	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
137	C₃×C₄⋊C₄	—	2	12	no	—
138	C₃×C₂²⋊C₄	—	2	12	no	—
139	C₃×C₂²⋊C₄	—	4	6	no	—
140	C₃×C₄²	—	2	12	no	—
141	C₂³×C₅	—	2	6	no	—
142	C₂³×C₅	—	2	6	no	—
143	C₂³×C₅	—	2	3	no	—
144	C₂³×C₅	—	2	3	no	—
145	C₂³×C₅	C₂²×C₃×C₄	1	6	yes	—
146	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
147	C₅×Q₈	C₃×C₂²⋊C₄	1	16	yes	complemented
148	C₅×D₄	—	2	24	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	24	no	—
153	C₂×C₄×C₅	—	2	6	no	—
154	C₂×C₄×C₅	—	2	6	no	—
155	C₂×C₄×C₅	—	2	6	no	—
156	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
157	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
158	C₂×C₄×C₅	C₂×C₃×D₄	1	12	yes	—
159	C₂×C₄×C₅	C₃×#41	1	4	yes	complemented
160	C₂×#41	—	2	8	no	—
161	C₂×#41	C₃×C₄×C₅	1	8	yes	complemented
162	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
163	C₂²×D₄	C₂²×C₃×C₅	1	3	yes	—
164	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
165	C₄×Q₈	—	2	8	no	—
166	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	6	yes	—
167	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	6	yes	—
168	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	6	yes	—
169	C₂×C₄²	C₂²×C₃×C₅	1	6	yes	—
170	#95	C₂²×C₃×C₅	1	1	yes	characteristic
171	#95	C₂²×C₃×C₅	1	3	yes	—
172	C₂×C₃×C₅	—	4	6	no	—
173	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
175	C₂×C₃×C₅	#520	1	1	yes	characteristic
176	C₂×C₃×C₅	#548	1	2	yes	—
177	C₂×C₃×C₅	#550	1	2	yes	—
178	C₂³×C₃	—	2	6	no	—
179	C₂³×C₃	—	2	6	no	—
180	C₂³×C₃	—	2	3	no	—
181	C₂³×C₃	—	2	3	no	—
182	C₂³×C₃	C₂²×C₄×C₅	1	6	yes	—
183	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
184	C₃×Q₈	C₅×C₂²⋊C₄	1	16	yes	complemented
185	C₃×D₄	—	2	24	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	24	no	—
190	C₂×C₃×C₄	—	2	6	no	—
191	C₂×C₃×C₄	—	2	6	no	—
192	C₂×C₃×C₄	—	2	6	no	—
193	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
194	C₂×C₃×C₄	C₂²×C₄×C₅	1	6	yes	—
195	C₂×C₃×C₄	C₂×C₅×D₄	1	12	yes	—
196	C₂×C₃×C₄	C₅×#41	1	4	yes	complemented
197	C₂²×C₅	—	2	6	no	—
198	C₂²×C₅	—	2	6	no	—
199	C₂²×C₅	—	2	6	no	—
200	C₂²×C₅	—	4	6	no	—
201	C₂²×C₅	—	4	6	no	—
202	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
203	C₂²×C₅	C₂²×C₃×D₄	1	2	yes	—
204	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
205	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
206	C₂²×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	2	4	no	—
209	C₄×C₅	—	4	6	no	—
210	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	12	yes	—
211	C₂⁴	C₂³×C₃×C₅	1	3	yes	—
212	#41	—	2	32	no	—
213	C₂×Q₈	C₂×C₃×C₄×C₅	1	4	yes	—
214	C₂×Q₈	C₃×C₅×D₄	1	8	yes	—
215	C₂×D₄	—	2	12	no	—
216	C₂×D₄	C₂×C₃×C₄×C₅	1	12	yes	—
217	C₂²×C₄	—	2	12	no	—
218	C₂²×C₄	C₂×C₃×C₄×C₅	1	12	yes	—
219	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
220	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
221	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
222	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
223	C₄⋊C₄	—	2	12	no	—
224	C₂²⋊C₄	—	2	12	no	—
225	C₂²⋊C₄	—	4	6	no	—
226	C₄²	—	2	12	no	—
227	C₃×C₅	#2341	1	1	yes	characteristic, direct product factor, Hall
228	C₂²×C₃	—	2	6	no	—
229	C₂²×C₃	—	2	6	no	—
230	C₂²×C₃	—	2	6	no	—
231	C₂²×C₃	—	4	6	no	—
232	C₂²×C₃	—	4	6	no	—
233	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
234	C₂²×C₃	C₂²×C₅×D₄	1	2	yes	—
235	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
236	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
237	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	2	4	no	—
240	C₃×C₄	—	4	6	no	—
241	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	12	yes	—
242	C₂×C₅	—	4	6	no	—
243	C₂×C₅	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₅	C₃×C₄×#41	1	1	yes	characteristic
245	C₂×C₅	C₃×#520	1	1	yes	characteristic
246	C₂×C₅	C₃×#548	1	2	yes	—
247	C₂×C₅	C₃×#550	1	2	yes	—
248	C₂³	—	2	6	no	—
249	C₂³	—	2	6	no	—
250	C₂³	—	2	3	no	—
251	C₂³	—	2	3	no	—
252	C₂³	C₂²×C₃×C₄×C₅	1	6	yes	—
253	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
254	Q₈	C₃×C₅×C₂²⋊C₄	1	16	yes	complemented
255	D₄	—	2	24	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	24	no	—
260	C₂×C₄	—	2	6	no	—
261	C₂×C₄	—	2	6	no	—
262	C₂×C₄	—	2	6	no	—
263	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
264	C₂×C₄	C₂²×C₃×C₄×C₅	1	6	yes	—
265	C₂×C₄	C₂×C₃×C₅×D₄	1	12	yes	—
266	C₂×C₄	C₃×C₅×#41	1	4	yes	complemented
267	C₂×C₃	—	4	6	no	—
268	C₂×C₃	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₃	C₄×C₅×#41	1	1	yes	characteristic
270	C₂×C₃	C₅×#520	1	1	yes	characteristic
271	C₂×C₃	C₅×#548	1	2	yes	—
272	C₂×C₃	C₅×#550	1	2	yes	—
273	C₅	C₃×#2341	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	—	2	6	no	—
275	C₂²	—	2	6	no	—
276	C₂²	—	2	6	no	—
277	C₂²	—	4	6	no	—
278	C₂²	—	4	6	no	—
279	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
280	C₂²	C₂²×C₃×C₅×D₄	1	2	yes	—
281	C₂²	C₂×C₃×C₅×#41	1	2	yes	—
282	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
283	C₂²	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	4	no	—
286	C₄	—	4	6	no	—
287	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	12	yes	—
288	C₃	C₅×#2341	1	1	yes	characteristic, direct product factor, Sylow
289	C₂	—	4	6	no	—
290	C₂	C₂²×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
291	C₂	C₃×C₄×C₅×#41	1	1	yes	characteristic
292	C₂	C₃×C₅×#520	1	1	yes	characteristic
293	C₂	C₃×C₅×#548	1	2	yes	—
294	C₂	C₃×C₅×#550	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator	C₂²	C₂³×C₃×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂²×C₃×C₅×#41	C₂
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
2. C₂×C₃×C₄×C₅×Q₈	C₂	semidirect
3. C₃×C₅×#390	C₂	semidirect
4. C₃×C₅×#380	C₂	semidirect
5. C₃×C₄×C₅×C₂²⋊C₄	C₂	semidirect
6. C₅×#2341	C₃	direct (N × H)
8. C₂×C₃×C₅×#41	C₄	semidirect
19. C₃×#2341	C₅	direct (N × H)
21. C₂×C₄×C₅×Q₈	C₂×C₃	semidirect
22. C₅×#390	C₂×C₃	semidirect
23. C₅×#380	C₂×C₃	semidirect
24. C₄×C₅×C₂²⋊C₄	C₂×C₃	semidirect
42. C₂×C₃×C₄×Q₈	C₂×C₅	semidirect
43. C₃×#390	C₂×C₅	semidirect
44. C₃×#380	C₂×C₅	semidirect
45. C₃×C₄×C₂²⋊C₄	C₂×C₅	semidirect
47. C₂×C₅×#41	C₃×C₄	semidirect
58. #2341	C₃×C₅	direct (N × H)
65. C₃×C₅×Q₈	C₂²⋊C₄	semidirect
77. C₂×C₃×C₄×C₅	#41	semidirect
79. C₂×C₃×#41	C₄×C₅	semidirect
107. C₂×C₄×Q₈	C₂×C₃×C₅	semidirect
108. #390	C₂×C₃×C₅	semidirect
109. #380	C₂×C₃×C₅	semidirect
110. C₄×C₂²⋊C₄	C₂×C₃×C₅	semidirect
147. C₅×Q₈	C₃×C₂²⋊C₄	semidirect
159. C₂×C₄×C₅	C₃×#41	semidirect
161. C₂×#41	C₃×C₄×C₅	semidirect
184. C₃×Q₈	C₅×C₂²⋊C₄	semidirect
196. C₂×C₃×C₄	C₅×#41	semidirect
227. C₃×C₅	#2341	direct (N × H)
254. Q₈	C₃×C₅×C₂²⋊C₄	semidirect
266. C₂×C₄	C₃×C₅×#41	semidirect
273. C₅	C₃×#2341	direct (N × H)
288. C₃	C₅×#2341	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×#41	2
C₂×C₃×C₄×C₅×Q₈	2
C₃×C₅×#390	2
C₃×C₅×#380	2
C₃×C₄×C₅×C₂²⋊C₄	2
C₅×#2341	3
C₃×#2341	5

Subgroups — C3×C5×#2341 #14275375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2341 #14275375