Subgroups — #14284148

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5680	C₂	1	1	yes	maximal, characteristic
2	#11747005	C₂	1	1	yes	maximal, characteristic, complemented
3	#11746878	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746361	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64760	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745850	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745850	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602540	—	3	1	no	maximal, Hall
9	C₃×C₅×#124	C₂²	1	1	yes	characteristic
10	C₅×#946	C₄	1	1	yes	characteristic
11	C₅×Q₈×Dic₃	C₄	1	1	yes	characteristic
12	C₅×#938	C₄	1	1	yes	characteristic
13	C₅×D₄×Dic₃	C₄	1	1	yes	characteristic
14	C₅×#883	—	2	1	no	—
15	C₅×#67⋊C₄	—	2	1	no	—
16	C₅×#879	—	2	1	no	—
17	C₅×#877	—	2	1	no	—
18	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
19	C₅×#870	C₂²	1	1	yes	characteristic
20	C₂×#6650	—	2	1	no	—
21	#91087	C₂²	1	1	yes	characteristic
22	#91087	C₂²	1	1	yes	characteristic
23	C₃×#3935	C₂²	1	1	yes	characteristic
24	C₃×#3935	C₂²	1	1	yes	characteristic
25	Dic₃⋊#168	—	2	1	no	—
26	(C₂×D₃)⋊#168	—	2	1	no	—
27	Dic₃×#168	—	2	1	no	—
28	#68344	—	5	1	no	maximal, Hall
29	C₅×#529	—	3	1	no	—
30	#65215	—	3	1	no	—
31	C₄×#681	—	3	1	no	—
32	#64763	—	3	1	no	—
33	#64763	—	3	1	no	—
34	#64760	—	3	1	no	—
35	#64760	D₃	1	1	yes	characteristic, complemented
36	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×#67	—	2	1	no	—
39	C₂²×C₅×Dic₃	—	2	1	no	—
40	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
42	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
43	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
44	C₃×C₅×C₂²⋊C₄	—	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	C₅×#217	—	2	1	no	—
49	C₅×#217	—	2	1	no	—
50	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
51	C₅×C₄⋊Dic₃	—	2	1	no	—
52	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
53	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
54	C₅×Dic₃⋊C₄	—	2	1	no	—
55	C₅×Dic₃⋊C₄	—	2	1	no	—
56	C₅×Dic₃⋊C₄	—	2	1	no	—
57	C₄×C₅×Dic₃	—	2	1	no	—
58	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
60	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
61	C₂×C₃×#168	—	2	1	no	—
62	C₂×C₃×#168	—	2	1	no	—
63	#6650	—	2	4	no	—
64	#5680	D₅	1	1	yes	characteristic
65	#4707	—	5	1	no	—
66	#4705	—	5	1	no	—
67	C₃×#329	—	5	1	no	—
68	#4540	—	5	1	no	—
69	#4465	—	5	1	no	—
70	#4465	—	5	1	no	—
71	C₅×#126	—	6	1	no	—
72	C₅×#125	—	3	1	no	—
73	C₅×#124	C₂×D₃	1	1	yes	characteristic
74	C₅×#123	—	6	1	no	—
75	C₅×C₂²⋊Q₈	—	3	1	no	—
76	C₅×#121	—	3	1	no	—
77	C₄×C₅×Q₈	—	3	1	no	—
78	C₄×C₅×D₄	—	3	1	no	—
79	C₄×C₅×D₄	—	6	1	no	—
80	C₅×#117	—	6	1	no	—
81	C₂×C₄²×C₅	—	3	1	no	—
82	C₂×#681	—	6	1	no	—
83	C₂²⋊#168	—	6	1	no	—
84	C₄⋊#168	—	6	1	no	—
85	#3935	—	3	1	no	—
86	#3935	—	3	1	no	—
87	#3935	C₂×D₃	1	1	yes	characteristic
88	#3935	C₂×D₃	1	1	yes	characteristic
89	C₄×#168	—	6	1	no	—
90	#1529	—	15	1	no	Sylow
91	C₂³×C₃×C₅	—	2	1	no	—
92	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₃×C₅×Q₈	—	4	1	no	—
94	C₃×C₅×D₄	—	4	1	no	—
95	C₂×C₃×C₄×C₅	—	2	1	no	—
96	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₅×#67	—	4	2	no	—
100	C₂×C₅×Dic₃	—	2	1	no	—
101	C₂×C₅×Dic₃	—	2	1	no	—
102	C₂×C₅×Dic₃	—	4	2	no	—
103	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×C₅×D₃	—	2	2	no	—
107	C₄×C₅×D₃	—	4	1	no	—
108	C₄×C₅×D₃	—	4	1	no	—
109	C₃⋊#168	—	2	2	no	—
110	C₃⋊#168	—	4	1	no	—
111	C₃×#168	—	2	2	no	—
112	C₃×#168	—	4	1	no	—
113	C₃×#124	C₂×D₅	1	1	yes	characteristic
114	#946	Dic₅	1	1	yes	characteristic
115	Q₈×Dic₃	Dic₅	1	1	yes	characteristic
116	#938	Dic₅	1	1	yes	characteristic
117	D₄×Dic₃	Dic₅	1	1	yes	characteristic
118	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
119	#883	—	2	1	no	—
120	#67⋊C₄	—	2	1	no	—
121	#879	—	2	1	no	—
122	#877	—	2	1	no	—
123	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
124	#870	C₂×D₅	1	1	yes	characteristic
125	C₃×#97	—	5	1	no	—
126	C₃×#97	—	5	1	no	—
127	(C₂×D₃)⋊C₈	—	10	1	no	—
128	Dic₃⋊C₈	—	10	1	no	—
129	C₈×Dic₃	—	10	1	no	—
130	#803	—	5	1	no	—
131	#803	—	5	1	no	—
132	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
133	C₂×C₅×D₄	—	6	1	no	—
134	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
135	C₂²×C₄×C₅	—	3	1	no	—
136	C₂²×C₄×C₅	—	3	1	no	—
137	C₂²×C₄×C₅	—	3	1	no	—
138	C₂²×C₄×C₅	—	6	1	no	—
139	C₅×C₄⋊C₄	—	3	1	no	—
140	C₅×C₄⋊C₄	—	3	1	no	—
141	C₅×C₄⋊C₄	—	6	1	no	—
142	C₅×C₄⋊C₄	—	6	1	no	—
143	C₅×C₄⋊C₄	—	6	1	no	—
144	C₅×C₄⋊C₄	—	6	1	no	—
145	C₅×C₂²⋊C₄	—	2	1	no	—
146	C₅×C₂²⋊C₄	—	2	1	no	—
147	C₅×C₂²⋊C₄	—	6	1	no	—
148	C₅×C₂²⋊C₄	—	6	1	no	—
149	C₅×C₂²⋊C₄	—	6	1	no	—
150	C₄²×C₅	—	3	1	no	—
151	C₄²×C₅	—	3	1	no	—
152	C₄²×C₅	—	3	1	no	—
153	C₄²×C₅	—	6	1	no	—
154	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
155	#681	—	6	4	no	—
156	C₂×#168	—	2	1	no	—
157	C₂×#168	—	2	1	no	—
158	C₂×#168	—	6	1	no	—
159	C₂×#168	—	6	1	no	—
160	#529	—	3	1	no	—
161	#432	—	15	1	no	—
162	C₄×M₁₆	—	15	1	no	—
163	#330	—	15	1	no	—
164	#330	—	15	1	no	—
165	#329	—	5	1	no	—
166	#329	—	15	1	no	—
167	C₂²×C₃×C₅	—	4	1	no	—
168	C₂²×C₃×C₅	—	4	1	no	—
169	C₂²×C₃×C₅	—	4	1	no	—
170	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₅×D₃	—	2	1	no	—
172	C₂×C₅×D₃	—	2	1	no	—
173	C₂×C₅×D₃	—	2	1	no	—
174	C₃×C₄×C₅	—	2	1	no	—
175	C₃×C₄×C₅	—	2	1	no	—
176	C₃×C₄×C₅	—	2	1	no	—
177	C₃×C₄×C₅	—	4	1	no	—
178	C₅×Dic₃	—	2	1	no	—
179	C₅×Dic₃	—	2	1	no	—
180	C₅×Dic₃	—	4	1	no	—
181	C₅×Dic₃	—	4	1	no	—
182	C₅×Dic₃	#104	1	2	yes	—
183	C₂×C₃×Q₈	C₂×Dic₅	1	1	yes	characteristic
184	C₂×C₃×D₄	C₂×Dic₅	1	1	yes	characteristic
185	C₂×#67	—	2	1	no	—
186	C₂²×Dic₃	—	2	1	no	—
187	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
188	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
189	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
190	C₂×C₃×C₈	—	10	1	no	—
191	C₂×C₃×C₈	—	10	1	no	—
192	C₃×C₂²⋊C₄	—	2	1	no	—
193	C₃×C₂²⋊C₄	—	2	1	no	—
194	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
195	#217	—	2	1	no	—
196	#217	—	2	1	no	—
197	(C₂×D₃)⋊C₄	—	2	1	no	—
198	C₄⋊Dic₃	—	2	1	no	—
199	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
200	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
201	Dic₃⋊C₄	—	2	1	no	—
202	Dic₃⋊C₄	—	2	1	no	—
203	Dic₃⋊C₄	—	2	1	no	—
204	C₄×Dic₃	—	2	1	no	—
205	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
206	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
207	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
208	C₂×C₃⋊C₈	—	10	1	no	—
209	C₂×C₃⋊C₈	—	10	1	no	—
210	ΓL(1,𝔽₂₅)	—	10	4	no	—
211	C₂³×C₅	—	2	1	no	—
212	C₂³×C₅	—	3	1	no	—
213	C₅×Q₈	—	4	1	no	—
214	C₅×D₄	—	4	1	no	—
215	C₅×D₄	—	12	2	no	—
216	C₂×C₄×C₅	—	2	1	no	—
217	C₂×C₄×C₅	—	3	1	no	—
218	C₂×C₄×C₅	—	3	1	no	—
219	C₂×C₄×C₅	—	3	1	no	—
220	C₂×C₄×C₅	—	6	2	no	—
221	C₂×C₄×C₅	—	6	1	no	—
222	C₂×C₄×C₅	—	6	1	no	—
223	C₂×C₄×C₅	—	12	1	no	—
224	C₂×C₄×C₅	—	12	1	no	—
225	C₂×C₄×C₅	—	12	2	no	—
226	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
227	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
228	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
229	#168	—	2	2	no	—
230	#168	—	4	1	no	—
231	#168	—	6	2	no	—
232	#168	—	12	1	no	—
233	C₂×M₁₆	—	30	1	no	—
234	#126	—	6	1	no	—
235	#125	—	3	1	no	—
236	#124	D₃×D₅	1	1	yes	characteristic
237	#123	—	6	1	no	—
238	C₂²⋊Q₈	—	3	1	no	—
239	#121	—	3	1	no	—
240	C₄×Q₈	—	3	1	no	—
241	C₄×D₄	—	3	1	no	—
242	C₄×D₄	—	6	1	no	—
243	#117	—	6	1	no	—
244	C₂×C₄²	—	3	1	no	—
245	C₄⋊C₈	—	30	1	no	—
246	C₂²⋊C₈	—	30	1	no	—
247	#97	—	5	1	no	—
248	#97	—	5	1	no	—
249	#97	—	15	1	no	—
250	#97	—	15	1	no	—
251	C₄×C₈	—	30	1	no	—
252	C₂×C₃×C₅	—	8	1	no	—
253	C₂×C₃×C₅	#413	1	1	yes	characteristic
254	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
255	C₂×C₃×C₅	#421	1	1	yes	characteristic
256	C₅×D₃	—	4	1	no	—
257	C₂³×C₃	—	2	1	no	—
258	C₂²×D₃	C₂²⋊Dic₅	1	1	yes	characteristic
259	C₃×Q₈	—	4	1	no	—
260	C₃×D₄	—	4	1	no	—
261	C₂×C₃×C₄	—	2	1	no	—
262	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
263	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
264	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
265	#67	—	4	2	no	—
266	C₂×Dic₃	—	2	1	no	—
267	C₂×Dic₃	—	2	1	no	—
268	C₂×Dic₃	—	4	2	no	—
269	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
270	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
271	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
272	C₄×D₃	—	2	2	no	—
273	C₄×D₃	—	4	1	no	—
274	C₄×D₃	—	4	1	no	—
275	C₃×C₈	—	10	2	no	—
276	C₃×C₈	—	20	1	no	—
277	C₃⋊C₈	—	10	2	no	—
278	C₃⋊C₈	—	20	1	no	—
279	C₂²×C₅	—	4	1	no	—
280	C₂²×C₅	—	4	1	no	—
281	C₂²×C₅	—	4	1	no	—
282	C₂²×C₅	—	6	1	no	—
283	C₂²×C₅	—	6	1	no	—
284	C₂²×C₅	—	6	1	no	—
285	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
286	C₄×C₅	—	2	1	no	—
287	C₄×C₅	—	2	1	no	—
288	C₄×C₅	—	2	1	no	—
289	C₄×C₅	—	3	2	no	—
290	C₄×C₅	—	4	1	no	—
291	C₄×C₅	—	6	1	no	—
292	C₄×C₅	—	6	1	no	—
293	C₄×C₅	—	12	1	no	—
294	C₄×C₅	—	12	1	no	—
295	C₂×Q₈	D₃×Dic₅	1	1	yes	characteristic
296	C₂×D₄	—	6	1	no	—
297	C₂×D₄	D₃×Dic₅	1	1	yes	characteristic
298	C₂²×C₄	—	3	1	no	—
299	C₂²×C₄	—	3	1	no	—
300	C₂²×C₄	—	3	1	no	—
301	C₂²×C₄	—	6	1	no	—
302	M₁₆	—	30	4	no	—
303	C₂×C₈	—	10	1	no	—
304	C₂×C₈	—	10	1	no	—
305	C₂×C₈	—	30	1	no	—
306	C₂×C₈	—	30	1	no	—
307	C₄⋊C₄	—	3	1	no	—
308	C₄⋊C₄	—	3	1	no	—
309	C₄⋊C₄	—	6	1	no	—
310	C₄⋊C₄	—	6	1	no	—
311	C₄⋊C₄	—	6	1	no	—
312	C₄⋊C₄	—	6	1	no	—
313	C₂²⋊C₄	—	2	1	no	—
314	C₂²⋊C₄	—	2	1	no	—
315	C₂²⋊C₄	—	6	1	no	—
316	C₂²⋊C₄	—	6	1	no	—
317	C₂²⋊C₄	—	6	1	no	—
318	C₄²	—	3	1	no	—
319	C₄²	—	3	1	no	—
320	C₄²	—	3	1	no	—
321	C₄²	—	6	1	no	—
322	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
323	C₃×C₅	#1529	1	1	yes	characteristic, complemented, Hall
324	C₂²×C₃	—	4	1	no	—
325	C₂²×C₃	—	4	1	no	—
326	C₂²×C₃	—	4	1	no	—
327	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
328	C₂×D₃	—	2	1	no	—
329	C₂×D₃	—	2	1	no	—
330	C₂×D₃	—	2	1	no	—
331	C₃×C₄	—	2	1	no	—
332	C₃×C₄	—	2	1	no	—
333	C₃×C₄	—	2	1	no	—
334	C₃×C₄	—	4	1	no	—
335	Dic₃	—	2	1	no	—
336	Dic₃	—	2	1	no	—
337	Dic₃	—	4	1	no	—
338	Dic₃	—	4	1	no	—
339	Dic₃	#3969	1	2	yes	—
340	C₂×C₅	—	8	1	no	—
341	C₂×C₅	—	12	1	no	—
342	C₂×C₅	#4745	1	1	yes	characteristic
343	C₂×C₅	D₃×#104	1	1	yes	characteristic
344	C₂×C₅	#4821	1	1	yes	characteristic
345	C₂³	—	2	1	no	—
346	C₂³	—	3	1	no	—
347	Q₈	—	4	1	no	—
348	D₄	—	4	1	no	—
349	D₄	—	12	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	3	1	no	—
352	C₂×C₄	—	3	1	no	—
353	C₂×C₄	—	3	1	no	—
354	C₂×C₄	—	6	2	no	—
355	C₂×C₄	—	6	1	no	—
356	C₂×C₄	—	6	1	no	—
357	C₂×C₄	—	12	1	no	—
358	C₂×C₄	—	12	1	no	—
359	C₂×C₄	—	12	2	no	—
360	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
361	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
362	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
363	C₈	—	10	2	no	—
364	C₈	—	20	1	no	—
365	C₈	—	30	2	no	—
366	C₈	—	60	1	no	—
367	C₂×C₃	—	8	1	no	—
368	C₂×C₃	C₂×#3969	1	1	yes	characteristic
369	C₂×C₃	#65525	1	1	yes	characteristic
370	C₂×C₃	#65526	1	1	yes	characteristic
371	D₃	—	4	1	no	—
372	C₅	#68344	1	1	yes	characteristic, complemented, Sylow
373	C₂²	—	4	1	no	—
374	C₂²	—	4	1	no	—
375	C₂²	—	4	1	no	—
376	C₂²	—	6	1	no	—
377	C₂²	—	6	1	no	—
378	C₂²	—	6	1	no	—
379	C₂²	D₃×C₂²⋊Dic₅	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	3	2	no	—
384	C₄	—	4	1	no	—
385	C₄	—	6	1	no	—
386	C₄	—	6	1	no	—
387	C₄	—	12	1	no	—
388	C₄	—	12	1	no	—
389	C₃	#10602540	1	1	yes	characteristic, complemented, Sylow
390	C₂	—	8	1	no	—
391	C₂	—	12	1	no	—
392	C₂	#11749126	1	1	yes	characteristic
393	C₂	D₃×#3969	1	1	yes	characteristic
394	C₂	#11749429	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×C₂²⋊Dic₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×#124	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#938	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#124	D₃×D₅
hypocenter	C₃×C₅	#1529
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
2. #11747005	C₂	semidirect
3. #11746878	C₂	semidirect
4. #11746361	C₂	semidirect
5. C₃×#64760	C₂	semidirect
6. #11745850	C₂	semidirect
7. #11745850	C₂	semidirect
35. #64760	D₃	semidirect
323. C₃×C₅	#1529	semidirect
372. C₅	#68344	semidirect
389. C₃	#10602540	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5680	2
#11747005	2
#11746878	2
#11746361	2
C₃×#64760	2
#11745850	2
#10602540	3
#68344	5