Subgroups — #14281886

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5479	C₂	1	1	yes	maximal, characteristic
2	#11751026	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65214	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745842	C₂	1	2	yes	maximal, complemented
5	#11745836	C₂	1	2	yes	maximal, complemented
6	#10602543	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₅×#930	C₄	1	1	yes	characteristic
9	C₅×#929	—	2	1	no	—
10	C₄×C₅×#67	—	2	1	no	—
11	C₅×#924	C₄	1	1	yes	characteristic
12	C₅×#876	—	2	1	no	—
13	C₅×#875	C₄	1	1	yes	characteristic
14	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
15	C₅×#872	—	2	1	no	—
16	C₅×#870	C₄	1	1	yes	characteristic
17	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
18	#91114	—	2	1	no	—
19	C₄⋊(C₃⋊#168)	—	2	1	no	—
20	C₄×C₃⋊#168	C₂²	1	2	yes	—
21	C₃×C₂²⋊#168	—	2	1	no	—
22	C₃×C₄⋊#168	C₂²	1	2	yes	—
23	C₃×C₄×#168	—	2	1	no	—
24	#70759	—	5	1	no	maximal, Hall
25	C₅×#529	—	3	1	no	—
26	#65214	—	3	1	no	—
27	#65214	D₃	1	1	yes	characteristic, complemented
28	Q₈⋊#168	—	3	2	no	—
29	D₄⋊#168	—	3	2	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂×C₅×#67	—	2	1	no	—
34	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅×D₃	—	2	1	no	—
36	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₃⋊#168	—	2	2	no	—
42	C₅×#217	—	2	1	no	—
43	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
44	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
45	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
46	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
47	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₅×Dic₃⋊C₄	—	2	1	no	—
50	C₅×Dic₃⋊C₄	—	2	1	no	—
51	C₅×Dic₃⋊C₄	—	2	1	no	—
52	C₅×Dic₃⋊C₄	—	2	1	no	—
53	C₄×C₅×Dic₃	—	2	1	no	—
54	C₂×C₃×#168	—	2	2	no	—
55	#5479	D₅	1	1	yes	characteristic
56	C₃×#431	—	5	1	no	—
57	#4925	—	5	1	no	—
58	#4484	—	5	2	no	—
59	#4483	—	5	2	no	—
60	C₅×#126	—	6	1	no	—
61	C₅×#125	—	3	1	no	—
62	C₅×#124	—	3	1	no	—
63	C₅×#123	—	6	1	no	—
64	C₅×C₂²⋊Q₈	—	3	1	no	—
65	C₅×#121	—	3	1	no	—
66	C₄×C₅×Q₈	—	3	1	no	—
67	C₄×C₅×D₄	—	3	1	no	—
68	C₄×C₅×D₄	—	6	1	no	—
69	C₅×#117	—	6	1	no	—
70	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
71	C₂²⋊#168	—	2	1	no	—
72	C₂²⋊#168	—	6	1	no	—
73	C₄⋊#168	—	6	1	no	—
74	C₄⋊#168	C₂×D₃	1	2	yes	—
75	C₄×#168	—	2	1	no	—
76	C₄×#168	—	3	2	no	—
77	#1482	—	15	1	no	Sylow
78	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₅×D₃	—	2	1	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	4	1	no	—
83	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₅×#67	—	4	2	no	—
90	C₂×C₅×Dic₃	—	2	1	no	—
91	C₂×C₅×Dic₃	—	2	1	no	—
92	C₂×C₅×Dic₃	—	2	1	no	—
93	C₅×D₁₂	—	2	2	no	—
94	C₄×C₅×D₃	—	4	2	no	—
95	C₅×Dic₆	—	2	2	no	—
96	C₃⋊#168	—	4	2	no	—
97	C₃×#168	—	4	2	no	—
98	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
99	#930	Dic₅	1	1	yes	characteristic
100	#929	—	2	1	no	—
101	C₄×#67	—	2	1	no	—
102	#924	Dic₅	1	1	yes	characteristic
103	#876	—	2	1	no	—
104	#875	Dic₅	1	1	yes	characteristic
105	C₄×D₁₂	C₂×D₅	1	1	yes	characteristic
106	#872	—	2	1	no	—
107	#870	Dic₅	1	1	yes	characteristic
108	C₄×Dic₆	C₂×D₅	1	1	yes	characteristic
109	C₃×C₄⋊C₈	—	5	2	no	—
110	C₃×C₂²⋊C₈	—	10	1	no	—
111	C₃×C₄×C₈	—	10	1	no	—
112	#830	—	10	1	no	—
113	C₄⋊(C₃⋊C₈)	—	10	1	no	—
114	C₄×C₃⋊C₈	—	5	2	no	—
115	C₂×C₅×Q₈	—	3	1	no	—
116	C₂×C₅×D₄	—	3	1	no	—
117	C₂×C₅×D₄	—	6	1	no	—
118	C₂²×C₄×C₅	—	6	1	no	—
119	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
120	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
121	C₂²×C₄×C₅	#67	1	1	yes	characteristic
122	C₅×C₄⋊C₄	—	3	1	no	—
123	C₅×C₄⋊C₄	—	3	1	no	—
124	C₅×C₄⋊C₄	—	6	1	no	—
125	C₅×C₄⋊C₄	—	6	1	no	—
126	C₅×C₄⋊C₄	—	6	1	no	—
127	C₅×C₄⋊C₄	—	6	1	no	—
128	C₅×C₂²⋊C₄	—	6	1	no	—
129	C₅×C₂²⋊C₄	—	6	1	no	—
130	C₅×C₂²⋊C₄	—	6	1	no	—
131	C₅×C₂²⋊C₄	—	6	1	no	—
132	C₅×C₂²⋊C₄	—	6	1	no	—
133	C₄²×C₅	—	6	1	no	—
134	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
135	C₄²×C₅	D₁₂	1	1	yes	characteristic
136	C₄²×C₅	#67	1	1	yes	characteristic
137	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
138	C₂×#168	—	2	2	no	—
139	C₂×#168	—	6	2	no	—
140	#529	—	3	1	no	—
141	#431	—	5	1	no	—
142	#431	—	15	1	no	—
143	Q₈⋊C₈	—	15	2	no	—
144	D₄⋊C₈	—	15	2	no	—
145	C₂²×C₃×C₅	—	2	1	no	—
146	C₂²×C₃×C₅	—	2	1	no	—
147	C₂²×C₃×C₅	—	2	1	no	—
148	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₅×D₃	—	4	1	no	—
150	C₂×C₅×D₃	—	4	1	no	—
151	C₂×C₅×D₃	—	4	1	no	—
152	C₃×C₄×C₅	—	2	1	no	—
153	C₃×C₄×C₅	—	2	1	no	—
154	C₃×C₄×C₅	—	2	1	no	—
155	C₃×C₄×C₅	—	2	1	no	—
156	C₃×C₄×C₅	#131	1	2	yes	—
157	C₃×C₄×C₅	#135	1	2	yes	—
158	C₅×Dic₃	—	4	1	no	—
159	C₅×Dic₃	—	4	1	no	—
160	C₅×Dic₃	—	4	1	no	—
161	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
162	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
163	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
164	C₂×#67	—	2	1	no	—
165	C₂×D₁₂	C₂×Dic₅	1	1	yes	characteristic
166	C₂×C₄×D₃	—	2	1	no	—
167	C₂×Dic₆	C₂×Dic₅	1	1	yes	characteristic
168	C₂×C₃×C₈	—	10	2	no	—
169	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
170	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
171	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
172	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
173	#217	—	2	1	no	—
174	(C₂×D₃)⋊C₄	—	2	1	no	—
175	(C₂×D₃)⋊C₄	—	2	1	no	—
176	(C₂×D₃)⋊C₄	—	2	1	no	—
177	(C₂×D₃)⋊C₄	—	2	1	no	—
178	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
179	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
180	Dic₃⋊C₄	—	2	1	no	—
181	Dic₃⋊C₄	—	2	1	no	—
182	Dic₃⋊C₄	—	2	1	no	—
183	Dic₃⋊C₄	—	2	1	no	—
184	C₄×Dic₃	—	2	1	no	—
185	C₂×C₃⋊C₈	—	10	2	no	—
186	C₂³×C₅	—	6	1	no	—
187	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
188	C₅×Q₈	—	6	2	no	—
189	C₅×D₄	—	6	2	no	—
190	C₅×D₄	—	12	2	no	—
191	C₂×C₄×C₅	—	2	2	no	—
192	C₂×C₄×C₅	—	2	2	no	—
193	C₂×C₄×C₅	—	4	1	no	—
194	C₂×C₄×C₅	—	6	1	no	—
195	C₂×C₄×C₅	—	6	1	no	—
196	C₂×C₄×C₅	—	6	1	no	—
197	C₂×C₄×C₅	—	12	2	no	—
198	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
199	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
200	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
202	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
203	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
204	#168	—	4	2	no	—
205	#168	—	12	2	no	—
206	#126	—	6	1	no	—
207	#125	—	3	1	no	—
208	#124	—	3	1	no	—
209	#123	—	6	1	no	—
210	C₂²⋊Q₈	—	3	1	no	—
211	#121	—	3	1	no	—
212	C₄×Q₈	—	3	1	no	—
213	C₄×D₄	—	3	1	no	—
214	C₄×D₄	—	6	1	no	—
215	#117	—	6	1	no	—
216	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
217	C₄⋊C₈	—	5	2	no	—
218	C₄⋊C₈	—	30	1	no	—
219	C₂²⋊C₈	—	10	1	no	—
220	C₂²⋊C₈	—	30	1	no	—
221	C₄×C₈	—	10	1	no	—
222	C₄×C₈	—	15	2	no	—
223	C₂×C₃×C₅	—	4	1	no	—
224	C₂×C₃×C₅	#408	1	1	yes	characteristic
225	C₂×C₃×C₅	#416	1	1	yes	characteristic
226	C₂×C₃×C₅	#421	1	1	yes	characteristic
227	C₅×D₃	—	8	1	no	—
228	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
229	C₂²×D₃	—	2	1	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	2	2	no	—
232	C₂×C₃×C₄	—	4	1	no	—
233	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
239	#67	—	4	2	no	—
240	C₂×Dic₃	—	2	1	no	—
241	C₂×Dic₃	—	2	1	no	—
242	C₂×Dic₃	—	2	1	no	—
243	D₁₂	—	2	2	no	—
244	C₄×D₃	—	4	2	no	—
245	Dic₆	—	2	2	no	—
246	C₃×C₈	—	20	2	no	—
247	C₃⋊C₈	—	20	2	no	—
248	C₂²×C₅	—	2	1	no	—
249	C₂²×C₅	—	2	1	no	—
250	C₂²×C₅	—	2	1	no	—
251	C₂²×C₅	—	12	1	no	—
252	C₂²×C₅	—	12	1	no	—
253	C₂²×C₅	—	12	1	no	—
254	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
255	C₄×C₅	—	2	1	no	—
256	C₄×C₅	—	2	1	no	—
257	C₄×C₅	—	2	1	no	—
258	C₄×C₅	—	2	1	no	—
259	C₄×C₅	—	12	1	no	—
260	C₄×C₅	—	12	1	no	—
261	C₄×C₅	—	12	1	no	—
262	C₄×C₅	#901	1	2	yes	—
263	C₄×C₅	#950	1	2	yes	—
264	C₂×Q₈	—	3	1	no	—
265	C₂×D₄	—	3	1	no	—
266	C₂×D₄	—	6	1	no	—
267	C₂²×C₄	—	6	1	no	—
268	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
269	C₂²×C₄	#1201	1	1	yes	characteristic
270	C₂²×C₄	#1203	1	1	yes	characteristic
271	C₂×C₈	—	10	2	no	—
272	C₂×C₈	—	30	2	no	—
273	C₄⋊C₄	—	3	1	no	—
274	C₄⋊C₄	—	3	1	no	—
275	C₄⋊C₄	—	6	1	no	—
276	C₄⋊C₄	—	6	1	no	—
277	C₄⋊C₄	—	6	1	no	—
278	C₄⋊C₄	—	6	1	no	—
279	C₂²⋊C₄	—	6	1	no	—
280	C₂²⋊C₄	—	6	1	no	—
281	C₂²⋊C₄	—	6	1	no	—
282	C₂²⋊C₄	—	6	1	no	—
283	C₂²⋊C₄	—	6	1	no	—
284	C₄²	—	6	1	no	—
285	C₄²	D₃×Dic₅	1	1	yes	characteristic
286	C₄²	#1201	1	1	yes	characteristic
287	C₄²	#1203	1	1	yes	characteristic
288	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
289	C₃×C₅	#1482	1	1	yes	characteristic, complemented, Hall
290	C₂²×C₃	—	2	1	no	—
291	C₂²×C₃	—	2	1	no	—
292	C₂²×C₃	—	2	1	no	—
293	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
294	C₂×D₃	—	4	1	no	—
295	C₂×D₃	—	4	1	no	—
296	C₂×D₃	—	4	1	no	—
297	C₃×C₄	—	2	1	no	—
298	C₃×C₄	—	2	1	no	—
299	C₃×C₄	—	2	1	no	—
300	C₃×C₄	—	2	1	no	—
301	C₃×C₄	#4094	1	2	yes	—
302	C₃×C₄	#4096	1	2	yes	—
303	Dic₃	—	4	1	no	—
304	Dic₃	—	4	1	no	—
305	Dic₃	—	4	1	no	—
306	C₂×C₅	—	4	1	no	—
307	C₂×C₅	—	24	1	no	—
308	C₂×C₅	#5002	1	1	yes	characteristic
309	C₂×C₅	#5110	1	1	yes	characteristic
310	C₂×C₅	#5139	1	1	yes	characteristic
311	C₂³	—	6	1	no	—
312	C₂³	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
313	Q₈	—	6	2	no	—
314	D₄	—	6	2	no	—
315	D₄	—	12	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	6	1	no	—
320	C₂×C₄	—	6	1	no	—
321	C₂×C₄	—	6	1	no	—
322	C₂×C₄	—	12	2	no	—
323	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
324	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
325	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
326	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
327	C₂×C₄	C₂×#1201	1	1	yes	characteristic
328	C₂×C₄	C₂×#1203	1	1	yes	characteristic
329	C₈	—	20	2	no	—
330	C₈	—	60	2	no	—
331	C₂×C₃	—	4	1	no	—
332	C₂×C₃	#65522	1	1	yes	characteristic
333	C₂×C₃	#65523	1	1	yes	characteristic
334	C₂×C₃	#65525	1	1	yes	characteristic
335	D₃	—	8	1	no	—
336	C₅	#70759	1	1	yes	characteristic, complemented, Sylow
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	12	1	no	—
341	C₂²	—	12	1	no	—
342	C₂²	—	12	1	no	—
343	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	12	1	no	—
349	C₄	—	12	1	no	—
350	C₄	—	12	1	no	—
351	C₄	#91286	1	2	yes	—
352	C₄	#91297	1	2	yes	—
353	C₃	#10602543	1	1	yes	characteristic, complemented, Sylow
354	C₂	—	4	1	no	—
355	C₂	—	24	1	no	—
356	C₂	#11748345	1	1	yes	characteristic
357	C₂	#11748382	1	1	yes	characteristic
358	C₂	#11748424	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1201
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#930	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1482
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #11751026	C₂	semidirect
3. C₃×#65214	C₂	semidirect
4. #11745842	C₂	semidirect
5. #11745836	C₂	semidirect
27. #65214	D₃	semidirect
289. C₃×C₅	#1482	semidirect
336. C₅	#70759	semidirect
353. C₃	#10602543	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5479	2
#11751026	2
C₃×#65214	2
#11745842	2
#11745836	2
#10602543	3
#70759	5