Subgroups — C₃×D₅⋊#361 #14314180

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₃×D₅⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₅⋊M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64906	C₂	1	4	yes	maximal, complemented
5	D₅⋊#361	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×D₄×D₅	—	2	1	no	—
7	C₃×D₅⋊M₁₆	—	2	1	no	—
8	C₂×C₃×C₅×D₈	C₄	1	1	yes	characteristic
9	C₂×C₃×#686	—	2	1	no	—
10	C₃×D₅×D₈	—	2	1	no	—
11	C₃×D₅×D₈	—	2	1	no	—
12	C₃×D₅×D₈	C₄	1	2	yes	—
13	C₃×D₅×D₈	C₂²	1	2	yes	—
14	C₂×C₃×D₄₀	C₄	1	1	yes	characteristic
15	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
16	C₃×#3998	—	2	1	no	—
17	C₃×#3996	—	2	1	no	—
18	C₃×#3995	C₂²	1	2	yes	complemented
19	C₃×#3990	C₂²	1	2	yes	complemented
20	C₃×D₅⋊C₁₆	—	2	1	no	—
21	C₂×C₃×#361	—	5	1	no	maximal, Hall
22	C₂×D₅×D₈	C₂×C₃	1	1	yes	characteristic
23	D₅⋊#108	C₂×C₃	1	1	yes	characteristic, complemented
24	D₅⋊M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
25	#64906	C₂×C₃	1	4	yes	complemented
26	C₂³×C₃×D₅	—	4	1	no	—
27	C₂×C₃×C₅×D₄	—	2	1	no	—
28	C₂×C₃×C₂²⋊D₅	—	4	1	no	—
29	C₃×D₄×D₅	—	2	2	no	—
30	C₃×D₄×D₅	—	2	2	no	—
31	C₃×D₄×D₅	—	4	1	no	—
32	C₃×D₄×D₅	—	4	1	no	—
33	C₂×C₃×D₂₀	—	2	1	no	—
34	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
35	C₃×#704	—	4	1	no	—
36	C₃×#700	—	2	2	no	—
37	C₃×D₅⋊C₈	—	4	1	no	—
38	C₃×C₅×D₈	—	2	1	no	—
39	C₃×C₅×D₈	C₂×C₄	1	2	yes	—
40	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₃×#686	—	2	2	no	—
42	C₃×#686	—	2	2	no	—
43	C₂×C₃×#168	D₄	1	1	yes	characteristic
44	C₃×D₄₀	—	2	1	no	—
45	C₃×D₄₀	C₂×C₄	1	2	yes	—
46	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
47	C₃×C₈×D₅	D₄	1	1	yes	characteristic
48	C₃×C₈×D₅	D₄	1	1	yes	characteristic
49	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
50	C₃×#674	—	2	2	no	—
51	C₂²×C₃×D₈	—	5	1	no	—
52	C₂×C₃×M₃₂	—	5	1	no	—
53	C₂×C₃×#108	—	5	1	no	—
54	C₃×#361	—	5	4	no	—
55	C₂×D₄×D₅	—	2	1	no	—
56	D₅⋊M₁₆	—	2	1	no	—
57	C₂×C₅×D₈	C₃×C₄	1	1	yes	characteristic
58	C₂×#686	—	2	1	no	—
59	D₅×D₈	—	2	1	no	—
60	D₅×D₈	—	2	1	no	—
61	D₅×D₈	C₃×C₄	1	2	yes	—
62	D₅×D₈	C₂²×C₃	1	2	yes	—
63	C₂×D₄₀	C₃×C₄	1	1	yes	characteristic
64	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
65	#3998	—	2	1	no	—
66	#3996	—	2	1	no	—
67	#3995	C₂²×C₃	1	2	yes	complemented
68	#3990	C₂²×C₃	1	2	yes	complemented
69	D₅⋊C₁₆	—	2	1	no	—
70	C₂×#361	—	5	1	no	Sylow
71	C₂³×C₃×C₅	—	4	1	no	—
72	C₂²×C₃×D₅	—	4	1	no	—
73	C₂²×C₃×D₅	—	4	2	no	—
74	C₂²×C₃×D₅	—	4	2	no	—
75	C₂²×C₃×D₅	—	8	4	no	—
76	C₂²×C₃×D₅	SD₁₆	1	1	yes	characteristic
77	C₃×C₅×D₄	—	2	2	no	—
78	C₃×C₅×D₄	—	4	1	no	—
79	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₃×C₂²⋊D₅	—	4	2	no	—
81	C₃×C₂²⋊D₅	—	4	2	no	—
82	C₂×C₃×Dic₅	D₈	1	1	yes	characteristic
83	C₃×D₂₀	—	2	2	no	—
84	C₃×D₂₀	—	4	1	no	—
85	C₃×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
86	C₃×C₄×D₅	D₈	1	1	yes	characteristic
87	C₃×C₄×D₅	SD₁₆	1	1	yes	characteristic
88	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
89	C₃×#170	—	4	2	no	—
90	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₃×#168	C₂²⋊C₄	1	1	yes	characteristic
93	C₃×#168	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₃×D₄	—	10	1	no	—
95	C₂×C₃×D₈	—	5	2	no	—
96	C₂×C₃×D₈	—	5	2	no	—
97	C₂×C₃×D₈	—	5	1	no	—
98	C₂×C₃×D₈	—	10	1	no	—
99	C₂×C₃×D₈	—	10	1	no	—
100	C₂×C₃×D₈	—	10	1	no	—
101	C₂×C₃×D₈	AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₂×C₃×M₁₆	—	10	1	no	—
103	C₂²×C₃×C₈	—	5	1	no	—
104	C₃×M₃₂	—	5	2	no	—
105	C₃×M₃₂	—	10	1	no	—
106	C₂×C₃×C₁₆	—	10	1	no	—
107	C₃×#108	—	5	2	no	—
108	C₃×#108	—	10	1	no	—
109	C₂³×D₅	—	4	1	no	—
110	C₂×C₅×D₄	—	2	1	no	—
111	C₂×C₂²⋊D₅	—	4	1	no	—
112	D₄×D₅	—	2	2	no	—
113	D₄×D₅	—	2	2	no	—
114	D₄×D₅	—	4	1	no	—
115	D₄×D₅	—	4	1	no	—
116	C₂×D₂₀	—	2	1	no	—
117	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
118	#704	—	4	1	no	—
119	#700	—	2	2	no	—
120	D₅⋊C₈	—	4	1	no	—
121	C₅×D₈	—	2	1	no	—
122	C₅×D₈	C₂×C₃×C₄	1	2	yes	—
123	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
124	#686	—	2	2	no	—
125	#686	—	2	2	no	—
126	C₂×#168	C₃×D₄	1	1	yes	characteristic
127	D₄₀	—	2	1	no	—
128	D₄₀	C₂×C₃×C₄	1	2	yes	—
129	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
130	C₈×D₅	C₃×D₄	1	1	yes	characteristic
131	C₈×D₅	C₃×D₄	1	1	yes	characteristic
132	C₈×D₅	C₂³×C₃	1	1	yes	characteristic
133	#674	—	2	2	no	—
134	C₂²×D₈	—	5	1	no	—
135	C₂×M₃₂	—	5	1	no	—
136	C₂×#108	—	5	1	no	—
137	#361	—	5	4	no	—
138	C₂²×C₃×C₅	—	4	2	no	—
139	C₂²×C₃×C₅	—	8	2	no	—
140	C₂²×C₃×C₅	#102	1	1	yes	characteristic
141	C₂×C₃×D₅	—	2	2	no	—
142	C₂×C₃×D₅	—	4	2	no	—
143	C₂×C₃×D₅	—	8	4	no	—
144	C₂×C₃×D₅	—	8	2	no	—
145	C₂×C₃×D₅	—	8	4	no	—
146	C₂×C₃×D₅	#102	1	1	yes	characteristic
147	C₂×C₃×D₅	C₂×SD₁₆	1	1	yes	characteristic
148	C₃×C₄×C₅	#102	1	1	yes	characteristic
149	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₃×Dic₅	#102	1	1	yes	characteristic
151	C₃×Dic₅	C₂×D₈	1	1	yes	characteristic
152	C₂⁴×C₃	—	20	1	no	—
153	C₂×C₃×D₄	—	2	1	no	—
154	C₂×C₃×D₄	—	10	2	no	—
155	C₂×C₃×D₄	—	10	2	no	—
156	C₂×C₃×D₄	—	10	1	no	—
157	C₂×C₃×D₄	—	20	1	no	—
158	C₂×C₃×D₄	—	20	1	no	—
159	C₂×C₃×D₄	—	20	1	no	—
160	C₂²×C₃×C₄	—	5	1	no	—
161	C₃×D₈	—	2	1	no	—
162	C₃×D₈	—	5	2	no	—
163	C₃×D₈	—	10	2	no	—
164	C₃×D₈	—	10	2	no	—
165	C₃×D₈	—	10	1	no	—
166	C₃×D₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
167	C₃×M₁₆	—	10	2	no	—
168	C₃×M₁₆	—	20	1	no	—
169	C₂×C₃×C₈	—	5	1	no	—
170	C₂×C₃×C₈	—	5	1	no	—
171	C₂×C₃×C₈	—	5	1	no	—
172	C₂×C₃×C₈	—	5	1	no	—
173	C₂×C₃×C₈	—	5	1	no	—
174	C₂×C₃×C₈	—	20	1	no	—
175	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
176	C₃×C₁₆	—	10	2	no	—
177	C₂³×C₅	—	4	1	no	—
178	C₂²×D₅	—	4	1	no	—
179	C₂²×D₅	—	4	2	no	—
180	C₂²×D₅	—	4	2	no	—
181	C₂²×D₅	—	8	4	no	—
182	C₂²×D₅	C₃×SD₁₆	1	1	yes	characteristic
183	C₅×D₄	—	2	2	no	—
184	C₅×D₄	—	4	1	no	—
185	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
186	C₂²⋊D₅	—	4	2	no	—
187	C₂²⋊D₅	—	4	2	no	—
188	C₂×Dic₅	C₃×D₈	1	1	yes	characteristic
189	D₂₀	—	2	2	no	—
190	D₂₀	—	4	1	no	—
191	C₄×D₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
192	C₄×D₅	C₃×D₈	1	1	yes	characteristic
193	C₄×D₅	C₃×SD₁₆	1	1	yes	characteristic
194	C₄×D₅	C₂×C₃×D₄	1	1	yes	characteristic
195	#170	—	4	2	no	—
196	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
197	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
198	#168	C₃×C₂²⋊C₄	1	1	yes	characteristic
199	#168	C₂×C₃×D₄	1	1	yes	characteristic
200	C₂²×D₄	—	10	1	no	—
201	C₂×D₈	—	5	2	no	—
202	C₂×D₈	—	5	2	no	—
203	C₂×D₈	—	5	1	no	—
204	C₂×D₈	—	10	1	no	—
205	C₂×D₈	—	10	1	no	—
206	C₂×D₈	—	10	1	no	—
207	C₂×D₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
208	C₂×M₁₆	—	10	1	no	—
209	C₂²×C₈	—	5	1	no	—
210	M₃₂	—	5	2	no	—
211	M₃₂	—	10	1	no	—
212	C₂×C₁₆	—	10	1	no	—
213	#108	—	5	2	no	—
214	#108	—	10	1	no	—
215	C₂×C₃×C₅	—	2	1	no	—
216	C₂×C₃×C₅	—	8	2	no	—
217	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
218	C₃×D₅	—	2	1	no	—
219	C₃×D₅	—	8	2	no	—
220	C₃×D₅	#361	1	2	yes	complemented
221	C₂³×C₃	—	4	1	no	—
222	C₂³×C₃	—	5	1	no	—
223	C₂³×C₃	—	20	2	no	—
224	C₂³×C₃	—	20	2	no	—
225	C₂³×C₃	—	20	1	no	—
226	C₂³×C₃	—	40	4	no	—
227	C₃×D₄	—	2	2	no	—
228	C₃×D₄	—	4	1	no	—
229	C₃×D₄	—	10	2	no	—
230	C₃×D₄	—	20	2	no	—
231	C₃×D₄	—	20	2	no	—
232	C₃×D₄	—	20	1	no	—
233	C₂×C₃×C₄	—	5	1	no	—
234	C₂×C₃×C₄	—	5	1	no	—
235	C₂×C₃×C₄	—	5	1	no	—
236	C₂×C₃×C₄	—	5	1	no	—
237	C₂×C₃×C₄	—	5	1	no	—
238	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
239	C₃×C₈	—	5	1	no	—
240	C₃×C₈	—	5	1	no	—
241	C₃×C₈	—	20	2	no	—
242	C₃×C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
244	C₂²×C₅	—	4	2	no	—
245	C₂²×C₅	—	8	2	no	—
246	C₂²×C₅	C₃×#102	1	1	yes	characteristic
247	C₂×D₅	—	2	2	no	—
248	C₂×D₅	—	4	2	no	—
249	C₂×D₅	—	8	4	no	—
250	C₂×D₅	—	8	2	no	—
251	C₂×D₅	—	8	4	no	—
252	C₂×D₅	C₃×#102	1	1	yes	characteristic
253	C₂×D₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
254	C₄×C₅	C₃×#102	1	1	yes	characteristic
255	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
256	Dic₅	C₃×#102	1	1	yes	characteristic
257	Dic₅	C₂×C₃×D₈	1	1	yes	characteristic
258	C₂⁴	—	20	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	10	2	no	—
261	C₂×D₄	—	10	2	no	—
262	C₂×D₄	—	10	1	no	—
263	C₂×D₄	—	20	1	no	—
264	C₂×D₄	—	20	1	no	—
265	C₂×D₄	—	20	1	no	—
266	C₂²×C₄	—	5	1	no	—
267	D₈	—	2	1	no	—
268	D₈	—	5	2	no	—
269	D₈	—	10	2	no	—
270	D₈	—	10	2	no	—
271	D₈	—	10	1	no	—
272	D₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
273	M₁₆	—	10	2	no	—
274	M₁₆	—	20	1	no	—
275	C₂×C₈	—	5	1	no	—
276	C₂×C₈	—	5	1	no	—
277	C₂×C₈	—	5	1	no	—
278	C₂×C₈	—	5	1	no	—
279	C₂×C₈	—	5	1	no	—
280	C₂×C₈	—	20	1	no	—
281	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
282	C₁₆	—	10	2	no	—
283	C₃×C₅	C₂×#361	1	1	yes	characteristic, complemented, Hall
284	C₂²×C₃	—	4	2	no	—
285	C₂²×C₃	—	5	1	no	—
286	C₂²×C₃	—	5	1	no	—
287	C₂²×C₃	—	8	2	no	—
288	C₂²×C₃	—	10	2	no	—
289	C₂²×C₃	—	20	2	no	—
290	C₂²×C₃	—	40	2	no	—
291	C₂²×C₃	—	40	4	no	—
292	C₂²×C₃	—	40	4	no	—
293	C₂²×C₃	#4007	1	1	yes	characteristic
294	C₃×C₄	—	5	1	no	—
295	C₃×C₄	—	5	1	no	—
296	C₃×C₄	#4007	1	1	yes	characteristic
297	C₃×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
298	C₂×C₅	—	2	1	no	—
299	C₂×C₅	—	8	2	no	—
300	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
301	D₅	—	2	1	no	—
302	D₅	—	8	2	no	—
303	D₅	C₃×#361	1	2	yes	complemented
304	C₂³	—	4	1	no	—
305	C₂³	—	5	1	no	—
306	C₂³	—	20	2	no	—
307	C₂³	—	20	2	no	—
308	C₂³	—	20	1	no	—
309	C₂³	—	40	4	no	—
310	D₄	—	2	2	no	—
311	D₄	—	4	1	no	—
312	D₄	—	10	2	no	—
313	D₄	—	20	2	no	—
314	D₄	—	20	2	no	—
315	D₄	—	20	1	no	—
316	C₂×C₄	—	5	1	no	—
317	C₂×C₄	—	5	1	no	—
318	C₂×C₄	—	5	1	no	—
319	C₂×C₄	—	5	1	no	—
320	C₂×C₄	—	5	1	no	—
321	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
322	C₈	—	5	1	no	—
323	C₈	—	5	1	no	—
324	C₈	—	20	2	no	—
325	C₈	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
326	C₈	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	5	2	no	—
329	C₂×C₃	—	8	2	no	—
330	C₂×C₃	—	10	1	no	—
331	C₂×C₃	—	40	2	no	—
332	C₂×C₃	C₂×#4007	1	1	yes	characteristic
333	C₅	C₂×C₃×#361	1	1	yes	characteristic, complemented, Sylow
334	C₂²	—	4	2	no	—
335	C₂²	—	5	1	no	—
336	C₂²	—	5	1	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	10	2	no	—
339	C₂²	—	20	2	no	—
340	C₂²	—	40	2	no	—
341	C₂²	—	40	4	no	—
342	C₂²	—	40	4	no	—
343	C₂²	C₃×#4007	1	1	yes	characteristic
344	C₄	—	5	1	no	—
345	C₄	—	5	1	no	—
346	C₄	C₃×#4007	1	1	yes	characteristic
347	C₄	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
348	C₃	D₅⋊#361	1	1	yes	characteristic, direct product factor, Sylow
349	C₂	—	2	1	no	—
350	C₂	—	5	2	no	—
351	C₂	—	8	2	no	—
352	C₂	—	10	1	no	—
353	C₂	—	40	2	no	—
354	C₂	C₂×C₃×#4007	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×#4007
commutator	C₅×C₈	C₂²×C₃×C₄
Frattini	C₈	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×D₈	C₄
socle	C₂×C₃×C₅	C₂×#102
prime radical	C₂×C₃×D₅×D₈	C₂
cosocle	C₈×D₅	C₂³×C₃
hypercenter	C₂×C₃×D₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₃×#361
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×D₈

Complements

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

N	H ≅ G/N	type
2. C₃×D₅⋊#108	C₂	semidirect
3. C₃×D₅⋊M₃₂	C₂	semidirect
4. C₃×#64906	C₂	semidirect
5. D₅⋊#361	C₃	direct (N × H)
18. C₃×#3995	C₂²	semidirect
19. C₃×#3990	C₂²	semidirect
23. D₅⋊#108	C₂×C₃	semidirect
24. D₅⋊M₃₂	C₂×C₃	semidirect
25. #64906	C₂×C₃	semidirect
67. #3995	C₂²×C₃	semidirect
68. #3990	C₂²×C₃	semidirect
220. C₃×D₅	#361	semidirect
283. C₃×C₅	C₂×#361	semidirect
303. D₅	C₃×#361	semidirect
333. C₅	C₂×C₃×#361	semidirect
348. C₃	D₅⋊#361	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×D₈	2
C₃×D₅⋊#108	2
C₃×D₅⋊M₃₂	2
C₃×#64906	2
D₅⋊#361	3
C₂×C₃×#361	5

Subgroups — C3×D5⋊#361 #14314180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₅⋊#361 #14314180