Subgroups — #14285042

Order 1920.

504 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5716	C₂	1	1	yes	maximal, characteristic
2	D₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	#11746875	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746365	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64764	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745862	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745862	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602600	—	3	1	no	maximal, Hall
9	C₂×C₅×#237	C₄	1	2	yes	—
10	C₃×C₅×#127	C₂²	1	1	yes	characteristic
11	C₅×#938	—	2	1	no	—
12	C₅×#938	—	2	1	no	—
13	C₅×#936	C₄	1	2	yes	—
14	C₅×D₄×Dic₃	—	2	1	no	—
15	C₅×D₄×Dic₃	—	2	1	no	—
16	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₅×#869	C₂²	1	1	yes	characteristic
18	C₂×D₃×#168	—	2	1	no	—
19	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
22	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
23	Dic₃⋊#168	—	2	1	no	—
24	(C₂×D₃)⋊#168	—	2	1	no	—
25	#90953	—	2	1	no	—
26	#68385	—	5	1	no	maximal, Hall
27	C₅×#532	—	3	1	no	—
28	#65215	—	3	1	no	—
29	C₂×C₄⋊#168	—	3	1	no	—
30	#64766	—	3	1	no	—
31	#64766	—	3	1	no	—
32	#64764	—	3	1	no	—
33	#64764	D₃	1	1	yes	characteristic, complemented
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×C₅×#67	—	2	2	no	—
38	C₂²×C₅×Dic₃	—	2	2	no	—
39	C₅×#237	—	4	4	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	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₃⋊#168	—	2	1	no	—
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₅×#217	—	2	2	no	—
48	C₅×#217	—	2	2	no	—
49	C₅×C₄⋊Dic₃	—	2	1	no	—
50	C₅×C₄⋊Dic₃	—	2	1	no	—
51	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
53	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
54	C₂×C₃×#168	—	2	1	no	—
55	C₂×C₃×#168	—	2	1	no	—
56	D₃×#168	—	4	2	no	—
57	#5716	D₅	1	1	yes	characteristic
58	#4835	—	5	1	no	—
59	D₃×C₄⋊C₈	—	5	1	no	—
60	C₃×#331	—	5	1	no	—
61	#4544	—	5	1	no	—
62	#4481	—	5	1	no	—
63	#4481	—	5	1	no	—
64	C₂×C₅×#41	—	3	2	no	—
65	C₅×C₄⋊Q₈	—	3	1	no	—
66	C₅×#127	C₂×D₃	1	1	yes	characteristic
67	C₅×#124	—	3	2	no	—
68	C₅×#121	—	6	1	no	—
69	C₅×#121	—	6	1	no	—
70	C₄×C₅×D₄	—	6	1	no	—
71	C₄×C₅×D₄	—	6	1	no	—
72	C₂×C₄²×C₅	—	3	1	no	—
73	C₂²×#168	—	6	1	no	—
74	C₂²⋊#168	—	6	1	no	—
75	C₄⋊#168	—	3	1	no	—
76	C₄⋊#168	—	3	1	no	—
77	C₄⋊#168	—	6	1	no	—
78	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
79	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
80	#3935	—	6	1	no	—
81	#1551	—	15	1	no	Sylow
82	C₂³×C₃×C₅	—	2	2	no	—
83	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₃×C₅×D₄	—	4	2	no	—
85	C₃×C₅×D₄	—	4	1	no	—
86	C₃×C₅×D₄	—	4	1	no	—
87	C₃×C₅×D₄	—	4	1	no	—
88	C₃×C₅×D₄	—	4	1	no	—
89	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₅×#67	—	4	4	no	—
93	C₂×C₅×Dic₃	—	2	2	no	—
94	C₂×C₅×Dic₃	—	4	4	no	—
95	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×C₅×D₃	—	2	2	no	—
99	C₄×C₅×D₃	—	2	1	no	—
100	C₄×C₅×D₃	—	2	1	no	—
101	C₄×C₅×D₃	—	2	1	no	—
102	C₄×C₅×D₃	—	2	1	no	—
103	C₅×Dic₆	—	4	2	no	—
104	C₃⋊#168	—	4	1	no	—
105	C₃⋊#168	—	4	1	no	—
106	C₃×#168	—	4	1	no	—
107	C₃×#168	—	4	1	no	—
108	C₂×#237	Dic₅	1	2	yes	—
109	C₃×#127	C₂×D₅	1	1	yes	characteristic
110	#938	—	2	1	no	—
111	#938	—	2	1	no	—
112	#936	Dic₅	1	2	yes	—
113	D₄×Dic₃	—	2	1	no	—
114	D₄×Dic₃	—	2	1	no	—
115	C₂×D₃×C₈	—	10	1	no	—
116	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
117	#869	C₂×D₅	1	1	yes	characteristic
118	C₃×C₄⋊C₈	—	5	1	no	—
119	C₃×C₄⋊C₈	—	5	1	no	—
120	(C₂×D₃)⋊C₈	—	10	1	no	—
121	#815	—	10	1	no	—
122	Dic₃⋊C₈	—	10	1	no	—
123	C₄⋊(C₃⋊C₈)	—	5	1	no	—
124	C₄⋊(C₃⋊C₈)	—	5	1	no	—
125	C₅×#41	—	12	4	no	—
126	C₂×C₅×Q₈	—	3	2	no	—
127	C₂×C₅×D₄	—	2	1	no	—
128	C₂×C₅×D₄	—	2	1	no	—
129	C₂×C₅×D₄	—	6	2	no	—
130	C₂×C₅×D₄	C₄×D₃	1	2	yes	—
131	C₂²×C₄×C₅	—	3	1	no	—
132	C₂²×C₄×C₅	—	3	1	no	—
133	C₂²×C₄×C₅	—	3	1	no	—
134	C₂²×C₄×C₅	—	6	2	no	—
135	C₅×C₄⋊C₄	—	6	1	no	—
136	C₅×C₄⋊C₄	—	6	1	no	—
137	C₅×C₂²⋊C₄	—	6	2	no	—
138	C₅×C₂²⋊C₄	—	6	2	no	—
139	C₄²×C₅	—	3	1	no	—
140	C₄²×C₅	—	3	1	no	—
141	C₄²×C₅	—	3	1	no	—
142	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
143	C₂×#168	—	2	1	no	—
144	C₂×#168	—	2	1	no	—
145	C₂×#168	—	6	1	no	—
146	C₂×#168	—	6	1	no	—
147	C₂×#168	—	12	2	no	—
148	#532	—	3	1	no	—
149	#432	—	15	1	no	—
150	C₂×C₄⋊C₈	—	15	1	no	—
151	#333	—	15	1	no	—
152	#333	—	15	1	no	—
153	#331	—	5	1	no	—
154	#331	—	15	1	no	—
155	C₂²×C₃×C₅	—	4	2	no	—
156	C₂²×C₃×C₅	—	4	2	no	—
157	C₂²×C₃×C₅	—	4	2	no	—
158	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₅×D₃	—	2	1	no	—
160	C₂×C₅×D₃	—	2	1	no	—
161	C₂×C₅×D₃	#100	1	2	yes	—
162	C₃×C₄×C₅	—	2	1	no	—
163	C₃×C₄×C₅	#135	1	1	yes	characteristic
164	C₃×C₄×C₅	#135	1	1	yes	characteristic
165	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₃×C₄×C₅	#137	1	1	yes	characteristic
167	C₅×Dic₃	—	2	1	no	—
168	C₅×Dic₃	—	2	1	no	—
169	C₅×Dic₃	—	2	1	no	—
170	C₅×Dic₃	—	4	2	no	—
171	C₂×C₃×D₄	—	2	1	no	—
172	C₂×C₃×D₄	—	2	1	no	—
173	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
174	C₂×#67	—	2	2	no	—
175	C₂²×Dic₃	—	2	2	no	—
176	#237	—	4	4	no	—
177	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
178	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
179	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
180	C₂×Dic₆	C₂×Dic₅	1	2	yes	—
181	C₂×C₃×C₈	—	10	1	no	—
182	C₂×C₃×C₈	—	10	1	no	—
183	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
184	#217	—	2	2	no	—
185	#217	—	2	2	no	—
186	C₄⋊Dic₃	—	2	1	no	—
187	C₄⋊Dic₃	—	2	1	no	—
188	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
189	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
190	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
191	C₂×C₃⋊C₈	—	10	1	no	—
192	C₂×C₃⋊C₈	—	10	1	no	—
193	D₃×C₈	—	20	2	no	—
194	C₂³×C₅	—	2	2	no	—
195	C₂³×C₅	—	3	1	no	—
196	C₅×Q₈	—	12	2	no	—
197	C₅×D₄	—	4	2	no	—
198	C₅×D₄	—	4	1	no	—
199	C₅×D₄	—	4	1	no	—
200	C₅×D₄	—	4	1	no	—
201	C₅×D₄	—	4	1	no	—
202	C₅×D₄	—	12	4	no	—
203	C₂×C₄×C₅	—	3	1	no	—
204	C₂×C₄×C₅	—	3	1	no	—
205	C₂×C₄×C₅	—	3	1	no	—
206	C₂×C₄×C₅	—	6	2	no	—
207	C₂×C₄×C₅	—	6	1	no	—
208	C₂×C₄×C₅	—	6	1	no	—
209	C₂×C₄×C₅	—	6	1	no	—
210	C₂×C₄×C₅	—	6	1	no	—
211	C₂×C₄×C₅	—	6	2	no	—
212	C₂×C₄×C₅	—	12	4	no	—
213	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
214	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
215	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
216	#168	—	4	1	no	—
217	#168	—	4	1	no	—
218	#168	—	12	1	no	—
219	#168	—	12	1	no	—
220	C₂×#41	—	3	2	no	—
221	C₂²×C₈	—	30	1	no	—
222	C₄⋊Q₈	—	3	1	no	—
223	#127	D₃×D₅	1	1	yes	characteristic
224	#124	—	3	2	no	—
225	#121	—	6	1	no	—
226	#121	—	6	1	no	—
227	C₄×D₄	—	6	1	no	—
228	C₄×D₄	—	6	1	no	—
229	C₂×C₄²	—	3	1	no	—
230	C₄⋊C₈	—	5	1	no	—
231	C₄⋊C₈	—	5	1	no	—
232	C₄⋊C₈	—	15	1	no	—
233	C₄⋊C₈	—	15	1	no	—
234	C₄⋊C₈	—	30	1	no	—
235	C₂²⋊C₈	—	30	1	no	—
236	#97	—	30	1	no	—
237	C₂×C₃×C₅	—	8	2	no	—
238	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
239	C₂×C₃×C₅	#416	1	1	yes	characteristic
240	C₂×C₃×C₅	#417	1	1	yes	characteristic
241	C₅×D₃	—	2	2	no	—
242	C₂³×C₃	—	2	2	no	—
243	C₂²×D₃	C₂²⋊Dic₅	1	1	yes	characteristic
244	C₃×D₄	—	4	2	no	—
245	C₃×D₄	—	4	1	no	—
246	C₃×D₄	—	4	1	no	—
247	C₃×D₄	—	4	1	no	—
248	C₃×D₄	—	4	1	no	—
249	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
250	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
251	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
252	#67	—	4	4	no	—
253	C₂×Dic₃	—	2	2	no	—
254	C₂×Dic₃	—	4	4	no	—
255	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
256	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
257	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
258	C₄×D₃	—	2	2	no	—
259	C₄×D₃	—	2	1	no	—
260	C₄×D₃	—	2	1	no	—
261	C₄×D₃	—	2	1	no	—
262	C₄×D₃	—	2	1	no	—
263	Dic₆	—	4	2	no	—
264	C₃×C₈	—	20	1	no	—
265	C₃×C₈	—	20	1	no	—
266	C₃⋊C₈	—	20	1	no	—
267	C₃⋊C₈	—	20	1	no	—
268	C₂²×C₅	—	3	2	no	—
269	C₂²×C₅	—	4	2	no	—
270	C₂²×C₅	—	4	2	no	—
271	C₂²×C₅	—	4	2	no	—
272	C₂²×C₅	—	6	1	no	—
273	C₂²×C₅	—	6	1	no	—
274	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
275	C₄×C₅	—	2	1	no	—
276	C₄×C₅	—	6	1	no	—
277	C₄×C₅	—	6	1	no	—
278	C₄×C₅	—	6	1	no	—
279	C₄×C₅	—	12	2	no	—
280	C₄×C₅	#911	1	1	yes	characteristic
281	C₄×C₅	#912	1	1	yes	characteristic
282	C₄×C₅	#915	1	1	yes	characteristic
283	C₄×C₅	#916	1	1	yes	characteristic
284	#41	—	12	4	no	—
285	C₂×Q₈	—	3	2	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	6	2	no	—
289	C₂×D₄	D₃×Dic₅	1	2	yes	—
290	C₂²×C₄	—	3	1	no	—
291	C₂²×C₄	—	3	1	no	—
292	C₂²×C₄	—	3	1	no	—
293	C₂²×C₄	—	6	2	no	—
294	C₂×C₈	—	10	1	no	—
295	C₂×C₈	—	10	1	no	—
296	C₂×C₈	—	30	1	no	—
297	C₂×C₈	—	30	1	no	—
298	C₂×C₈	—	60	2	no	—
299	C₄⋊C₄	—	6	1	no	—
300	C₄⋊C₄	—	6	1	no	—
301	C₂²⋊C₄	—	6	2	no	—
302	C₂²⋊C₄	—	6	2	no	—
303	C₄²	—	3	1	no	—
304	C₄²	—	3	1	no	—
305	C₄²	—	3	1	no	—
306	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
307	C₃×C₅	#1551	1	1	yes	characteristic, complemented, Hall
308	C₂²×C₃	—	4	2	no	—
309	C₂²×C₃	—	4	2	no	—
310	C₂²×C₃	—	4	2	no	—
311	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
312	C₂×D₃	—	2	1	no	—
313	C₂×D₃	—	2	1	no	—
314	C₂×D₃	#3965	1	2	yes	—
315	C₃×C₄	—	2	1	no	—
316	C₃×C₄	#4095	1	1	yes	characteristic
317	C₃×C₄	#4096	1	1	yes	characteristic
318	C₃×C₄	#4096	1	1	yes	characteristic
319	C₃×C₄	#4097	1	1	yes	characteristic
320	Dic₃	—	2	1	no	—
321	Dic₃	—	2	1	no	—
322	Dic₃	—	2	1	no	—
323	Dic₃	—	4	2	no	—
324	C₂×C₅	—	6	2	no	—
325	C₂×C₅	—	8	2	no	—
326	C₂×C₅	D₃×#100	1	1	yes	characteristic
327	C₂×C₅	#4771	1	1	yes	characteristic
328	C₂×C₅	#4772	1	1	yes	characteristic
329	C₂³	—	2	2	no	—
330	C₂³	—	3	1	no	—
331	Q₈	—	12	2	no	—
332	D₄	—	4	2	no	—
333	D₄	—	4	1	no	—
334	D₄	—	4	1	no	—
335	D₄	—	4	1	no	—
336	D₄	—	4	1	no	—
337	D₄	—	12	4	no	—
338	C₂×C₄	—	3	1	no	—
339	C₂×C₄	—	3	1	no	—
340	C₂×C₄	—	3	1	no	—
341	C₂×C₄	—	6	2	no	—
342	C₂×C₄	—	6	1	no	—
343	C₂×C₄	—	6	1	no	—
344	C₂×C₄	—	6	1	no	—
345	C₂×C₄	—	6	1	no	—
346	C₂×C₄	—	6	2	no	—
347	C₂×C₄	—	12	4	no	—
348	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
349	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
350	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
351	C₈	—	20	1	no	—
352	C₈	—	20	1	no	—
353	C₈	—	60	1	no	—
354	C₈	—	60	1	no	—
355	C₂×C₃	—	8	2	no	—
356	C₂×C₃	C₂×#3965	1	1	yes	characteristic
357	C₂×C₃	#65521	1	1	yes	characteristic
358	C₂×C₃	#65522	1	1	yes	characteristic
359	D₃	—	2	2	no	—
360	C₅	#68385	1	1	yes	characteristic, complemented, Sylow
361	C₂²	—	3	2	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	6	1	no	—
366	C₂²	—	6	1	no	—
367	C₂²	D₃×C₂²⋊Dic₅	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	6	1	no	—
370	C₄	—	6	1	no	—
371	C₄	—	6	1	no	—
372	C₄	—	12	2	no	—
373	C₄	#91480	1	1	yes	characteristic
374	C₄	#91486	1	1	yes	characteristic
375	C₄	#91493	1	1	yes	characteristic
376	C₄	#91495	1	1	yes	characteristic
377	C₃	#10602600	1	1	yes	characteristic, complemented, Sylow
378	C₂	—	6	2	no	—
379	C₂	—	8	2	no	—
380	C₂	#11748955	1	1	yes	characteristic
381	C₂	#11748965	1	1	yes	characteristic
382	C₂	D₃×#3965	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×#127	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#5716	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#127	D₃×D₅
hypocenter	C₃×C₅	#1551
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. D₃×C₄⋊#168	C₂	semidirect
3. #11746875	C₂	semidirect
4. #11746365	C₂	semidirect
5. C₃×#64764	C₂	semidirect
6. #11745862	C₂	semidirect
7. #11745862	C₂	semidirect
33. #64764	D₃	semidirect
307. C₃×C₅	#1551	semidirect
360. C₅	#68385	semidirect
377. C₃	#10602600	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5716	2
D₃×C₄⋊#168	2
#11746875	2
#11746365	2
C₃×#64764	2
#11745862	2
#10602600	3
#68385	5