Subgroups — #14284596

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66050	C₂	1	1	yes	maximal, characteristic
2	D₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	#11746869	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746365	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4544	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745856	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745855	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599174	—	3	1	no	maximal, Hall
9	C₂×C₃×#711	—	2	1	no	—
10	C₃×C₅×#127	C₂²	1	1	yes	characteristic
11	C₃×#4084	—	2	1	no	—
12	C₃×#4084	C₄	1	2	yes	—
13	C₃×#4082	—	2	1	no	—
14	C₃×D₄×Dic₅	—	2	1	no	—
15	C₃×D₄×Dic₅	C₄	1	2	yes	—
16	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
17	C₃×#4015	C₂²	1	1	yes	characteristic
18	C₂×D₅×C₃⋊C₈	—	2	1	no	—
19	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
21	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	#90982	—	2	1	no	—
24	#90955	—	2	1	no	—
25	Dic₅×C₃⋊C₈	—	2	1	no	—
26	#71804	—	5	1	no	maximal, Hall
27	#66050	D₃	1	1	yes	characteristic
28	#65122	—	3	1	no	—
29	D₅×C₄⋊C₈	—	3	1	no	—
30	C₅×#331	—	3	1	no	—
31	#64764	—	3	1	no	—
32	#64706	—	3	1	no	—
33	#64702	—	3	1	no	—
34	C₂×C₃×C₅×D₄	—	2	1	no	—
35	C₂×C₃×C₅×D₄	—	2	1	no	—
36	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
38	C₂²×C₃×Dic₅	—	2	2	no	—
39	C₃×#711	—	4	2	no	—
40	C₃×#711	—	4	2	no	—
41	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
43	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
44	C₂×C₃×Dic₁₀	—	2	1	no	—
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₂×C₃⋊#168	—	2	1	no	—
48	C₂×C₅×C₃⋊C₈	—	2	1	no	—
49	C₂×C₅×C₃⋊C₈	—	2	1	no	—
50	C₃×C₂²⋊Dic₅	—	2	2	no	—
51	C₃×C₂²⋊Dic₅	—	2	2	no	—
52	C₃×C₄⋊Dic₅	—	2	1	no	—
53	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
54	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
55	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
56	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
57	D₅×C₃⋊C₈	—	4	2	no	—
58	C₃×#532	—	5	1	no	—
59	#4925	—	5	1	no	—
60	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
61	#4547	—	5	1	no	—
62	#4547	—	5	1	no	—
63	#4544	—	5	1	no	—
64	#4544	D₅	1	1	yes	characteristic, complemented
65	C₂×#711	—	2	1	no	—
66	C₅×#127	C₂×D₃	1	1	yes	characteristic
67	#4084	—	2	1	no	—
68	#4084	Dic₃	1	2	yes	—
69	#4082	—	2	1	no	—
70	D₄×Dic₅	—	2	1	no	—
71	D₄×Dic₅	Dic₃	1	2	yes	—
72	C₂×C₈×D₅	—	6	1	no	—
73	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
74	#4015	C₂×D₃	1	1	yes	characteristic
75	C₅×C₄⋊C₈	—	3	1	no	—
76	C₅×C₄⋊C₈	—	3	1	no	—
77	#3952	—	6	1	no	—
78	#3946	—	6	1	no	—
79	C₈×Dic₅	—	6	1	no	—
80	C₄⋊#168	—	3	1	no	—
81	C₄⋊#168	—	3	1	no	—
82	#1540	—	15	1	no	Sylow
83	C₂³×C₃×C₅	—	2	2	no	—
84	C₂²×C₃×D₅	C₂²⋊C₄	1	1	yes	characteristic
85	C₃×C₅×D₄	—	4	2	no	—
86	C₃×C₅×D₄	—	4	1	no	—
87	C₃×C₅×D₄	—	4	1	no	—
88	C₃×C₅×D₄	—	4	1	no	—
89	C₃×C₅×D₄	—	4	1	no	—
90	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₃×C₂²⋊D₅	—	4	4	no	—
94	C₂×C₃×Dic₅	—	2	2	no	—
95	C₂×C₃×Dic₅	—	4	4	no	—
96	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
99	C₃×C₄×D₅	—	2	1	no	—
100	C₃×C₄×D₅	—	2	1	no	—
101	C₃×C₄×D₅	—	4	1	no	—
102	C₃×C₄×D₅	D₈	1	2	yes	—
103	C₃×C₄×D₅	SD₁₆	1	2	yes	—
104	C₃×Dic₁₀	—	4	1	no	—
105	C₃×Dic₁₀	—	4	1	no	—
106	C₃⋊#168	—	4	1	no	—
107	C₃⋊#168	—	4	1	no	—
108	C₅×C₃⋊C₈	—	4	1	no	—
109	C₅×C₃⋊C₈	—	4	1	no	—
110	C₂×C₃×#41	—	10	1	no	—
111	C₃×C₄⋊Q₈	—	5	1	no	—
112	C₃×#127	C₂×D₅	1	1	yes	characteristic
113	C₃×#124	—	10	1	no	—
114	C₃×#121	—	5	2	no	—
115	C₃×#121	—	10	1	no	—
116	C₃×C₄×D₄	—	5	2	no	—
117	C₃×C₄×D₄	—	10	1	no	—
118	C₂×C₃×C₄²	—	5	1	no	—
119	C₂²×C₃⋊C₈	—	10	1	no	—
120	#830	—	10	1	no	—
121	C₄⋊(C₃⋊C₈)	—	5	1	no	—
122	C₄⋊(C₃⋊C₈)	—	5	1	no	—
123	C₄⋊(C₃⋊C₈)	—	10	1	no	—
124	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
125	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
126	C₄×C₃⋊C₈	—	10	1	no	—
127	C₂×C₅×D₄	—	2	1	no	—
128	C₂×C₅×D₄	—	2	1	no	—
129	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
130	C₂×C₂²⋊D₅	—	2	2	no	—
131	C₂²×Dic₅	—	2	2	no	—
132	#711	—	4	2	no	—
133	#711	—	4	2	no	—
134	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
135	C₂×C₄×D₅	#67	1	1	yes	characteristic
136	C₂×C₄×D₅	#67	1	1	yes	characteristic
137	C₂×Dic₁₀	—	2	1	no	—
138	C₂×C₅×C₈	—	6	1	no	—
139	C₂×C₅×C₈	—	6	1	no	—
140	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
141	C₂²⋊Dic₅	—	2	2	no	—
142	C₂²⋊Dic₅	—	2	2	no	—
143	C₄⋊Dic₅	—	2	1	no	—
144	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
145	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
146	C₄×Dic₅	#67	1	1	yes	characteristic
147	C₄×Dic₅	#67	1	1	yes	characteristic
148	C₂×#168	—	6	1	no	—
149	C₂×#168	—	6	1	no	—
150	C₈×D₅	—	12	2	no	—
151	#532	—	5	1	no	—
152	#431	—	15	1	no	—
153	C₂×C₄⋊C₈	—	15	1	no	—
154	#332	—	15	1	no	—
155	#332	—	15	1	no	—
156	#331	—	3	1	no	—
157	#331	—	15	1	no	—
158	C₂²×C₃×C₅	—	4	2	no	—
159	C₂²×C₃×C₅	—	4	2	no	—
160	C₂²×C₃×C₅	—	4	2	no	—
161	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₃×D₅	—	2	1	no	—
163	C₂×C₃×D₅	—	2	1	no	—
164	C₂×C₃×D₅	#102	1	2	yes	—
165	C₃×C₄×C₅	—	2	1	no	—
166	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
167	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
168	C₃×C₄×C₅	#135	1	1	yes	characteristic
169	C₃×C₄×C₅	#135	1	1	yes	characteristic
170	C₃×Dic₅	—	2	1	no	—
171	C₃×Dic₅	—	2	1	no	—
172	C₃×Dic₅	—	4	2	no	—
173	C₃×Dic₅	#102	1	2	yes	—
174	C₃×#41	—	20	2	no	—
175	C₃×#41	—	20	2	no	—
176	C₂×C₃×Q₈	—	10	1	no	—
177	C₂×C₃×D₄	—	2	1	no	—
178	C₂×C₃×D₄	—	2	1	no	—
179	C₂×C₃×D₄	—	10	2	no	—
180	C₂×C₃×D₄	C₄×D₅	1	2	yes	—
181	C₂²×C₃×C₄	—	5	1	no	—
182	C₂²×C₃×C₄	—	5	1	no	—
183	C₂²×C₃×C₄	—	5	1	no	—
184	C₂²×C₃×C₄	—	10	2	no	—
185	C₃×C₄⋊C₄	—	5	2	no	—
186	C₃×C₄⋊C₄	—	10	1	no	—
187	C₃×C₂²⋊C₄	—	10	2	no	—
188	C₃×C₂²⋊C₄	—	10	2	no	—
189	C₃×C₄²	—	5	1	no	—
190	C₃×C₄²	—	5	1	no	—
191	C₃×C₄²	—	5	1	no	—
192	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
193	C₂×C₃⋊C₈	—	2	1	no	—
194	C₂×C₃⋊C₈	—	2	1	no	—
195	C₂×C₃⋊C₈	—	10	1	no	—
196	C₂×C₃⋊C₈	—	10	1	no	—
197	C₂×C₃⋊C₈	—	20	2	no	—
198	C₂³×C₅	—	2	2	no	—
199	C₂²×D₅	#217	1	1	yes	characteristic
200	C₅×D₄	—	4	2	no	—
201	C₅×D₄	—	4	1	no	—
202	C₅×D₄	—	4	1	no	—
203	C₅×D₄	—	4	1	no	—
204	C₅×D₄	—	4	1	no	—
205	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
206	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
207	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
208	C₂²⋊D₅	—	4	4	no	—
209	C₂×Dic₅	—	2	2	no	—
210	C₂×Dic₅	—	4	4	no	—
211	C₂×Dic₅	#217	1	1	yes	characteristic
212	C₂×Dic₅	#217	1	1	yes	characteristic
213	C₂×Dic₅	#217	1	1	yes	characteristic
214	C₄×D₅	—	2	1	no	—
215	C₄×D₅	—	2	1	no	—
216	C₄×D₅	—	4	1	no	—
217	C₄×D₅	#213	1	2	yes	—
218	C₄×D₅	#214	1	2	yes	—
219	Dic₁₀	—	4	1	no	—
220	Dic₁₀	—	4	1	no	—
221	C₅×C₈	—	12	1	no	—
222	C₅×C₈	—	12	1	no	—
223	#168	—	12	1	no	—
224	#168	—	12	1	no	—
225	C₂×#41	—	10	1	no	—
226	C₂²×C₈	—	30	1	no	—
227	C₄⋊Q₈	—	5	1	no	—
228	#127	D₃×D₅	1	1	yes	characteristic
229	#124	—	10	1	no	—
230	#121	—	5	2	no	—
231	#121	—	10	1	no	—
232	C₄×D₄	—	5	2	no	—
233	C₄×D₄	—	10	1	no	—
234	C₂×C₄²	—	5	1	no	—
235	C₄⋊C₈	—	3	1	no	—
236	C₄⋊C₈	—	3	1	no	—
237	C₄⋊C₈	—	15	1	no	—
238	C₄⋊C₈	—	15	1	no	—
239	C₄⋊C₈	—	30	1	no	—
240	C₂²⋊C₈	—	30	1	no	—
241	C₄×C₈	—	30	1	no	—
242	C₂×C₃×C₅	—	8	2	no	—
243	C₂×C₃×C₅	#413	1	1	yes	characteristic
244	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
245	C₂×C₃×C₅	#416	1	1	yes	characteristic
246	C₃×D₅	—	2	2	no	—
247	C₂³×C₃	—	2	2	no	—
248	C₂³×C₃	—	5	1	no	—
249	C₃×Q₈	—	20	1	no	—
250	C₃×Q₈	—	20	1	no	—
251	C₃×D₄	—	4	2	no	—
252	C₃×D₄	—	4	1	no	—
253	C₃×D₄	—	4	1	no	—
254	C₃×D₄	—	4	1	no	—
255	C₃×D₄	—	4	1	no	—
256	C₃×D₄	—	20	4	no	—
257	C₂×C₃×C₄	—	5	1	no	—
258	C₂×C₃×C₄	—	5	2	no	—
259	C₂×C₃×C₄	—	5	2	no	—
260	C₂×C₃×C₄	—	5	1	no	—
261	C₂×C₃×C₄	—	5	1	no	—
262	C₂×C₃×C₄	—	10	1	no	—
263	C₂×C₃×C₄	—	10	1	no	—
264	C₂×C₃×C₄	—	10	2	no	—
265	C₂×C₃×C₄	—	20	1	no	—
266	C₂×C₃×C₄	—	20	4	no	—
267	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
268	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
269	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
270	C₃⋊C₈	—	4	1	no	—
271	C₃⋊C₈	—	4	1	no	—
272	C₃⋊C₈	—	20	1	no	—
273	C₃⋊C₈	—	20	1	no	—
274	C₂²×C₅	—	4	2	no	—
275	C₂²×C₅	—	4	2	no	—
276	C₂²×C₅	—	4	2	no	—
277	C₂²×C₅	C₂×#217	1	1	yes	characteristic
278	C₂×D₅	—	2	1	no	—
279	C₂×D₅	—	2	1	no	—
280	C₂×D₅	#832	1	2	yes	—
281	C₄×C₅	—	2	1	no	—
282	C₄×C₅	C₂×#213	1	1	yes	characteristic
283	C₄×C₅	C₂×#214	1	1	yes	characteristic
284	C₄×C₅	#950	1	1	yes	characteristic
285	C₄×C₅	#950	1	1	yes	characteristic
286	Dic₅	—	2	1	no	—
287	Dic₅	—	2	1	no	—
288	Dic₅	—	4	2	no	—
289	Dic₅	#832	1	2	yes	—
290	#41	—	20	2	no	—
291	#41	—	20	2	no	—
292	C₂×Q₈	—	10	1	no	—
293	C₂×D₄	—	2	1	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	—	10	2	no	—
296	C₂×D₄	D₅×Dic₃	1	2	yes	—
297	C₂²×C₄	—	5	1	no	—
298	C₂²×C₄	—	5	1	no	—
299	C₂²×C₄	—	5	1	no	—
300	C₂²×C₄	—	10	2	no	—
301	C₂×C₈	—	6	1	no	—
302	C₂×C₈	—	6	1	no	—
303	C₂×C₈	—	30	1	no	—
304	C₂×C₈	—	30	1	no	—
305	C₂×C₈	—	60	2	no	—
306	C₄⋊C₄	—	5	2	no	—
307	C₄⋊C₄	—	10	1	no	—
308	C₂²⋊C₄	—	10	2	no	—
309	C₂²⋊C₄	—	10	2	no	—
310	C₄²	—	5	1	no	—
311	C₄²	—	5	1	no	—
312	C₄²	—	5	1	no	—
313	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
314	C₃×C₅	#1540	1	1	yes	characteristic, complemented, Hall
315	C₂²×C₃	—	4	2	no	—
316	C₂²×C₃	—	4	2	no	—
317	C₂²×C₃	—	4	2	no	—
318	C₂²×C₃	—	5	2	no	—
319	C₂²×C₃	—	10	1	no	—
320	C₂²×C₃	—	10	1	no	—
321	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
322	C₃×C₄	—	2	1	no	—
323	C₃×C₄	—	5	2	no	—
324	C₃×C₄	—	10	1	no	—
325	C₃×C₄	—	10	1	no	—
326	C₃×C₄	—	20	2	no	—
327	C₃×C₄	D₅×D₈	1	1	yes	characteristic
328	C₃×C₄	#4058	1	1	yes	characteristic
329	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
330	C₃×C₄	#4062	1	1	yes	characteristic
331	C₂×C₅	—	8	2	no	—
332	C₂×C₅	C₂×#832	1	1	yes	characteristic
333	C₂×C₅	#5233	1	1	yes	characteristic
334	C₂×C₅	#5237	1	1	yes	characteristic
335	D₅	—	2	2	no	—
336	C₂³	—	2	2	no	—
337	C₂³	—	5	1	no	—
338	Q₈	—	20	1	no	—
339	Q₈	—	20	1	no	—
340	D₄	—	4	2	no	—
341	D₄	—	4	1	no	—
342	D₄	—	4	1	no	—
343	D₄	—	4	1	no	—
344	D₄	—	4	1	no	—
345	D₄	—	20	4	no	—
346	C₂×C₄	—	5	1	no	—
347	C₂×C₄	—	5	2	no	—
348	C₂×C₄	—	5	2	no	—
349	C₂×C₄	—	5	1	no	—
350	C₂×C₄	—	5	1	no	—
351	C₂×C₄	—	10	1	no	—
352	C₂×C₄	—	10	1	no	—
353	C₂×C₄	—	10	2	no	—
354	C₂×C₄	—	20	1	no	—
355	C₂×C₄	—	20	4	no	—
356	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
357	C₂×C₄	D₅×#67	1	1	yes	characteristic
358	C₂×C₄	D₅×#67	1	1	yes	characteristic
359	C₈	—	12	1	no	—
360	C₈	—	12	1	no	—
361	C₈	—	60	1	no	—
362	C₈	—	60	1	no	—
363	C₂×C₃	—	8	2	no	—
364	C₂×C₃	—	10	2	no	—
365	C₂×C₃	#65034	1	1	yes	characteristic
366	C₂×C₃	D₅×#102	1	1	yes	characteristic
367	C₂×C₃	#65061	1	1	yes	characteristic
368	C₅	#71804	1	1	yes	characteristic, complemented, Sylow
369	C₂²	—	4	2	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	5	2	no	—
373	C₂²	—	10	1	no	—
374	C₂²	—	10	1	no	—
375	C₂²	D₅×#217	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	5	2	no	—
378	C₄	—	10	1	no	—
379	C₄	—	10	1	no	—
380	C₄	—	20	2	no	—
381	C₄	D₅×#213	1	1	yes	characteristic
382	C₄	D₅×#214	1	1	yes	characteristic
383	C₄	#91490	1	1	yes	characteristic
384	C₄	#91492	1	1	yes	characteristic
385	C₃	#10599174	1	1	yes	characteristic, complemented, Sylow
386	C₂	—	8	2	no	—
387	C₂	—	10	2	no	—
388	C₂	D₅×#832	1	1	yes	characteristic
389	C₂	#11748961	1	1	yes	characteristic
390	C₂	#11749123	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅×#217
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×#127	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#66050	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#127	D₃×D₅
hypocenter	C₃×C₅	#1540
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
2. D₅×C₄⋊(C₃⋊C₈)	C₂	semidirect
3. #11746869	C₂	semidirect
4. #11746365	C₂	semidirect
5. C₅×#4544	C₂	semidirect
6. #11745856	C₂	semidirect
7. #11745855	C₂	semidirect
64. #4544	D₅	semidirect
314. C₃×C₅	#1540	semidirect
368. C₅	#71804	semidirect
385. C₃	#10599174	semidirect

Maximal subgroups

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

subgroup	index
C₃×#66050	2
D₅×C₄⋊(C₃⋊C₈)	2
#11746869	2
#11746365	2
C₅×#4544	2
#11745856	2
#11745855	2
#10599174	3
#71804	5