Subgroups — #14295950

Order 1920.

600 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	C₂×#91286	C₂	1	1	yes	maximal, characteristic
2	C₅×#5002	C₂	1	1	yes	maximal, characteristic
3	C₃×#65286	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90978	C₂	1	1	yes	maximal, characteristic
5	C₂×#90951	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746353	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745983	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601082	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	1	yes	characteristic
10	C₃×C₅×#117	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
12	C₂×#6710	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×#681	C₂²	1	1	yes	characteristic
15	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
16	C₂×#6650	—	2	1	no	—
17	#91286	—	2	2	no	—
18	#91286	C₄	1	4	yes	complemented
19	C₂×D₃×#168	—	2	1	no	—
20	C₅×#810	C₄	1	2	yes	complemented
21	C₅×#809	C₄	1	2	yes	complemented
22	C₃×#3935	—	2	1	no	—
23	C₃×C₄×#168	—	2	1	no	—
24	#90978	C₄	1	4	yes	complemented
25	#90951	—	2	2	no	—
26	#69887	—	5	1	no	maximal, Hall
27	C₂×#4094	—	3	1	no	—
28	C₅×#416	—	3	1	no	—
29	C₂×#3969	—	3	1	no	—
30	C₂×C₈×Dic₅	—	3	1	no	—
31	#65286	D₃	1	1	yes	characteristic, complemented
32	#64772	—	3	1	no	—
33	#64752	—	3	1	no	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₅	—	2	1	no	—
36	C₂×C₅×#67	—	2	1	no	—
37	C₅×#235	—	2	1	no	—
38	C₅×#235	—	2	1	no	—
39	C₅×#235	C₂×C₄	1	2	yes	—
40	C₅×#235	C₂×C₄	1	2	yes	—
41	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₅×D₃	—	2	1	no	—
43	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
44	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
46	C₃×C₄²×C₅	—	2	1	no	—
47	C₄×Dic₁₅	—	2	1	no	—
48	C₄×Dic₁₅	C₂×C₄	1	2	yes	—
49	#6710	—	2	1	no	—
50	#6710	C₂×C₄	1	2	yes	—
51	C₂×C₃⋊#168	—	2	1	no	—
52	C₂×C₅×C₃⋊C₈	—	2	1	no	—
53	C₂×C₅×C₃⋊C₈	—	2	1	no	—
54	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
55	C₃×#681	—	2	1	no	—
56	C₃×#681	C₂×C₄	1	2	yes	—
57	C₂×C₃×#168	—	2	1	no	—
58	C₂×C₃×#168	C₂×C₄	1	2	yes	—
59	C₂×C₃×#168	D₄	1	2	yes	—
60	C₂×C₃×#168	D₄	1	1	yes	characteristic
61	C₂×C₃×#168	D₄	1	1	yes	characteristic
62	#6650	—	2	4	no	—
63	D₃×#168	—	2	4	no	—
64	C₂×#901	—	5	1	no	—
65	C₃×#405	—	5	1	no	—
66	C₂×#825	—	5	1	no	—
67	C₂×C₈×Dic₃	—	5	1	no	—
68	#5002	D₅	1	1	yes	characteristic
69	#4558	—	5	1	no	—
70	#4532	—	5	1	no	—
71	C₂×C₅×#41	—	3	1	no	—
72	C₂²×C₅×C₈	—	3	1	no	—
73	C₅×#117	C₂×D₃	1	1	yes	characteristic
74	#4094	—	3	4	no	—
75	#4094	—	6	2	no	—
76	C₂×C₄×Dic₅	—	3	1	no	—
77	C₂×#681	—	3	1	no	—
78	C₂×#681	—	6	1	no	—
79	C₂×#681	C₂×D₃	1	1	yes	characteristic
80	C₂²×#168	—	6	1	no	—
81	C₂²×#168	C₂×D₃	1	1	yes	characteristic
82	C₅×#103	—	3	2	no	—
83	C₅×#102	—	3	2	no	—
84	#3969	—	3	4	no	—
85	C₈×Dic₅	—	6	2	no	—
86	#3935	—	2	1	no	—
87	C₄×#168	—	2	1	no	—
88	#1764	—	15	1	no	Sylow
89	C₂³×C₃×C₅	#41	1	1	yes	characteristic
90	C₂²×C₅×D₃	—	2	1	no	—
91	C₂×C₃×C₄×C₅	—	2	2	no	—
92	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
98	C₂×Dic₁₅	—	2	4	no	—
99	C₂×Dic₁₅	—	2	2	no	—
100	C₅×#67	—	2	2	no	—
101	C₅×#67	—	2	2	no	—
102	C₂×C₅×Dic₃	—	2	1	no	—
103	C₅×D₁₂	—	2	1	no	—
104	C₅×D₁₂	C₄²	1	2	yes	—
105	C₄×C₅×D₃	—	2	2	no	—
106	C₄×C₅×D₃	—	2	2	no	—
107	C₅×Dic₆	—	2	1	no	—
108	C₅×Dic₆	C₄²	1	2	yes	—
109	C₃⋊#168	—	2	2	no	—
110	C₃×#168	—	2	2	no	—
111	C₃×#168	C₂²⋊C₄	1	2	yes	—
112	C₃×#168	C₂²⋊C₄	1	2	yes	—
113	C₅×C₃⋊C₈	—	2	2	no	—
114	C₂×#235	C₂×D₅	1	1	yes	characteristic
115	C₂×C₃×M₁₆	—	5	1	no	—
116	C₂²×C₃×C₈	—	5	1	no	—
117	C₃×#117	C₂×D₅	1	1	yes	characteristic
118	C₂×C₄×Dic₃	—	5	1	no	—
119	C₂×#208	—	5	1	no	—
120	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
121	#901	—	5	4	no	—
122	#901	—	10	2	no	—
123	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
124	C₂×D₃×C₈	—	10	1	no	—
125	C₃×#97	—	10	1	no	—
126	C₃×C₄×C₈	—	10	1	no	—
127	#825	—	5	4	no	—
128	C₈×Dic₃	—	10	2	no	—
129	#810	Dic₅	1	2	yes	complemented
130	#809	Dic₅	1	2	yes	complemented
131	C₅×#41	—	3	2	no	—
132	C₅×#41	—	3	2	no	—
133	C₅×#41	—	6	1	no	—
134	C₅×#41	—	6	1	no	—
135	C₂×C₅×Q₈	—	3	1	no	—
136	C₂×C₅×D₄	—	3	1	no	—
137	C₂×C₅×D₄	—	6	1	no	—
138	C₂²×C₄×C₅	—	6	1	no	—
139	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
140	C₂²×Dic₅	—	6	1	no	—
141	C₂×C₅×C₈	—	3	2	no	—
142	C₂×C₅×C₈	—	6	1	no	—
143	C₂×C₅×C₈	—	6	1	no	—
144	C₅×C₄⋊C₄	C₄×D₃	1	2	yes	—
145	C₅×C₂²⋊C₄	—	2	1	no	—
146	C₄²×C₅	—	2	1	no	—
147	C₄×Dic₅	—	3	2	no	—
148	C₄×Dic₅	—	6	1	no	—
149	#681	—	2	1	no	—
150	#681	—	3	2	no	—
151	#681	—	6	1	no	—
152	#681	—	6	4	no	—
153	#681	C₄×D₃	1	2	yes	—
154	C₂×#168	—	2	1	no	—
155	C₂×#168	—	6	1	no	—
156	C₂×#168	—	6	4	no	—
157	C₂×#168	C₄×D₃	1	2	yes	—
158	C₂×#168	D₁₂	1	2	yes	—
159	C₂×#168	#67	1	1	yes	characteristic
160	C₂×#168	#67	1	1	yes	characteristic
161	C₂×#131	—	15	1	no	—
162	C₂×#104	—	15	1	no	—
163	#416	—	3	1	no	—
164	#405	—	5	1	no	—
165	C₂×C₄×C₈	—	15	1	no	—
166	#341	—	15	1	no	—
167	#339	—	15	1	no	—
168	C₂²×C₃×C₅	—	2	2	no	—
169	C₂²×C₃×C₅	#117	1	1	yes	characteristic
170	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
171	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
172	C₂×C₅×D₃	—	2	2	no	—
173	C₂×C₅×D₃	—	4	2	no	—
174	C₃×C₄×C₅	—	4	2	no	—
175	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
176	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
178	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
179	Dic₁₅	—	2	4	no	—
180	C₅×Dic₃	—	2	2	no	—
181	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
182	C₂×#67	—	2	1	no	—
183	C₂²×Dic₃	—	10	1	no	—
184	#235	—	2	1	no	—
185	#235	—	2	1	no	—
186	#235	C₄×D₅	1	2	yes	—
187	#235	C₄×D₅	1	2	yes	—
188	C₂×D₁₂	C₂×Dic₅	1	1	yes	characteristic
189	C₂×C₄×D₃	—	2	1	no	—
190	C₂×Dic₆	C₂×Dic₅	1	1	yes	characteristic
191	C₃×M₁₆	—	5	2	no	—
192	C₃×M₁₆	—	10	1	no	—
193	C₂×C₃×C₈	—	5	2	no	—
194	C₂×C₃×C₈	—	5	2	no	—
195	C₂×C₃×C₈	—	5	1	no	—
196	C₂×C₃×C₈	—	5	1	no	—
197	C₂×C₃×C₈	—	10	1	no	—
198	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
199	C₃×C₂²⋊C₄	—	2	1	no	—
200	C₃×C₄²	—	2	1	no	—
201	C₄×Dic₃	—	5	2	no	—
202	C₄×Dic₃	—	10	1	no	—
203	#208	—	5	2	no	—
204	#208	—	10	1	no	—
205	C₂×C₃⋊C₈	—	2	1	no	—
206	C₂×C₃⋊C₈	—	2	1	no	—
207	C₂×C₃⋊C₈	—	10	1	no	—
208	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
209	ΓL(1,𝔽₂₅)	—	10	4	no	—
210	D₃×C₈	—	10	4	no	—
211	C₂³×C₅	—	6	1	no	—
212	C₂³×C₅	#235	1	1	yes	characteristic
213	C₅×Q₈	—	3	2	no	—
214	C₅×Q₈	—	6	1	no	—
215	C₅×D₄	—	3	2	no	—
216	C₅×D₄	—	6	2	no	—
217	C₅×D₄	—	6	2	no	—
218	C₅×D₄	—	6	1	no	—
219	C₂×C₄×C₅	—	2	2	no	—
220	C₂×C₄×C₅	—	6	1	no	—
221	C₂×C₄×C₅	—	6	2	no	—
222	C₂×C₄×C₅	—	6	2	no	—
223	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
224	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
225	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
226	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
227	C₂×C₄×C₅	#235	1	1	yes	characteristic
228	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
229	C₂×Dic₅	—	6	4	no	—
230	C₂×Dic₅	—	6	2	no	—
231	C₅×C₈	—	6	2	no	—
232	#168	—	2	2	no	—
233	#168	—	6	2	no	—
234	#168	(C₂×D₃)⋊C₄	1	2	yes	—
235	#168	(C₂×D₃)⋊C₄	1	2	yes	—
236	C₂×#41	—	3	1	no	—
237	#131	—	15	4	no	—
238	#131	—	30	2	no	—
239	C₂×M₁₆	—	5	1	no	—
240	C₂×M₁₆	—	15	1	no	—
241	C₂×M₁₆	—	30	1	no	—
242	C₂²×C₈	—	3	1	no	—
243	C₂²×C₈	—	5	1	no	—
244	C₂²×C₈	—	30	1	no	—
245	#117	D₃×D₅	1	1	yes	characteristic
246	C₂×C₄²	—	15	1	no	—
247	#104	—	15	4	no	—
248	#103	—	3	2	no	—
249	#102	—	3	2	no	—
250	#97	—	10	1	no	—
251	C₄×C₈	—	10	1	no	—
252	C₄×C₈	—	30	2	no	—
253	C₂×C₃×C₅	—	2	1	no	—
254	C₂×C₃×C₅	—	2	1	no	—
255	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₃×C₅	#443	1	2	yes	—
257	C₅×D₃	—	4	2	no	—
258	C₂³×C₃	#711	1	1	yes	characteristic
259	C₂²×D₃	—	2	1	no	—
260	C₂×C₃×C₄	—	2	2	no	—
261	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
262	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
263	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
264	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
265	C₂×C₃×C₄	#711	1	1	yes	characteristic
266	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
267	#67	—	2	2	no	—
268	#67	—	2	2	no	—
269	C₂×Dic₃	—	2	1	no	—
270	C₂×Dic₃	—	10	4	no	—
271	C₂×Dic₃	—	10	2	no	—
272	D₁₂	—	2	1	no	—
273	D₁₂	C₄×Dic₅	1	2	yes	—
274	C₄×D₃	—	2	2	no	—
275	C₄×D₃	—	2	2	no	—
276	Dic₆	—	2	1	no	—
277	Dic₆	C₄×Dic₅	1	2	yes	—
278	C₃×C₈	—	5	2	no	—
279	C₃×C₈	—	5	2	no	—
280	C₃×C₈	—	10	2	no	—
281	C₃⋊C₈	—	2	2	no	—
282	C₃⋊C₈	—	10	2	no	—
283	C₂²×C₅	—	2	2	no	—
284	C₂²×C₅	—	6	2	no	—
285	C₂²×C₅	—	12	2	no	—
286	C₂²×C₅	#872	1	1	yes	characteristic
287	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
288	C₂²×C₅	C₄×#67	1	1	yes	characteristic
289	C₄×C₅	—	4	2	no	—
290	C₄×C₅	—	6	2	no	—
291	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
292	C₄×C₅	C₄×D₁₂	1	1	yes	characteristic
293	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
294	C₄×C₅	C₄×#67	1	1	yes	characteristic
295	Dic₅	—	6	4	no	—
296	#41	—	3	2	no	—
297	#41	—	3	2	no	—
298	#41	—	6	1	no	—
299	#41	—	6	1	no	—
300	C₂×Q₈	—	3	1	no	—
301	C₂×D₄	—	3	1	no	—
302	C₂×D₄	—	6	1	no	—
303	C₂²×C₄	—	6	1	no	—
304	C₂²×C₄	—	30	1	no	—
305	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
306	M₁₆	—	5	2	no	—
307	M₁₆	—	10	1	no	—
308	M₁₆	—	15	2	no	—
309	M₁₆	—	30	4	no	—
310	M₁₆	—	30	1	no	—
311	C₂×C₈	—	3	2	no	—
312	C₂×C₈	—	5	2	no	—
313	C₂×C₈	—	5	2	no	—
314	C₂×C₈	—	5	1	no	—
315	C₂×C₈	—	5	1	no	—
316	C₂×C₈	—	6	1	no	—
317	C₂×C₈	—	6	1	no	—
318	C₂×C₈	—	10	1	no	—
319	C₂×C₈	—	30	4	no	—
320	C₂×C₈	—	30	1	no	—
321	C₄⋊C₄	D₃×Dic₅	1	2	yes	—
322	C₂²⋊C₄	—	2	1	no	—
323	C₄²	—	2	1	no	—
324	C₄²	—	15	2	no	—
325	C₄²	—	30	1	no	—
326	C₃×C₅	#1764	1	1	yes	characteristic, complemented, Hall
327	C₂²×C₃	—	2	2	no	—
328	C₂²×C₃	#4023	1	1	yes	characteristic
329	C₂²×C₃	#4027	1	1	yes	characteristic
330	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
331	C₂×D₃	—	2	2	no	—
332	C₂×D₃	—	4	2	no	—
333	C₃×C₄	—	4	2	no	—
334	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
335	C₃×C₄	#4027	1	1	yes	characteristic
336	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
337	C₃×C₄	D₄×Dic₅	1	1	yes	characteristic
338	Dic₃	—	2	2	no	—
339	Dic₃	—	10	4	no	—
340	C₂×C₅	—	2	1	no	—
341	C₂×C₅	—	2	1	no	—
342	C₂×C₅	—	12	2	no	—
343	C₂×C₅	C₄×(C₂×D₃)⋊C₄	1	1	yes	characteristic
344	C₂×C₅	#5144	1	2	yes	—
345	C₂³	—	6	1	no	—
346	C₂³	#6782	1	1	yes	characteristic
347	Q₈	—	3	2	no	—
348	Q₈	—	6	1	no	—
349	D₄	—	3	2	no	—
350	D₄	—	6	2	no	—
351	D₄	—	6	2	no	—
352	D₄	—	6	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	6	1	no	—
355	C₂×C₄	—	6	2	no	—
356	C₂×C₄	—	6	2	no	—
357	C₂×C₄	—	30	4	no	—
358	C₂×C₄	—	30	2	no	—
359	C₂×C₄	#6772	1	1	yes	characteristic
360	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
361	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
362	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
363	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
364	C₂×C₄	D₅×#67	1	1	yes	characteristic
365	C₈	—	5	2	no	—
366	C₈	—	5	2	no	—
367	C₈	—	6	2	no	—
368	C₈	—	10	2	no	—
369	C₈	—	30	2	no	—
370	C₂×C₃	—	2	1	no	—
371	C₂×C₃	—	2	1	no	—
372	C₂×C₃	C₂²⋊C₄×Dic₅	1	1	yes	characteristic
373	C₂×C₃	#65485	1	2	yes	—
374	D₃	—	4	2	no	—
375	C₅	#69887	1	1	yes	characteristic, complemented, Sylow
376	C₂²	—	2	2	no	—
377	C₂²	—	6	2	no	—
378	C₂²	—	12	2	no	—
379	C₂²	#91398	1	1	yes	characteristic
380	C₂²	#91405	1	1	yes	characteristic
381	C₂²	Dic₅×#67	1	1	yes	characteristic
382	C₄	—	4	2	no	—
383	C₄	—	6	2	no	—
384	C₄	—	30	4	no	—
385	C₄	C₄×D₃×Dic₅	1	1	yes	characteristic
386	C₄	Dic₅×D₁₂	1	1	yes	characteristic
387	C₄	#91440	1	1	yes	characteristic
388	C₄	D₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
389	C₃	#10601082	1	1	yes	characteristic, complemented, Sylow
390	C₂	—	2	1	no	—
391	C₂	—	2	1	no	—
392	C₂	—	12	2	no	—
393	C₂	Dic₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
394	C₂	#11749432	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₅×#67
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₃×C₅×#117	C₂²
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₅×#235	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#117	D₃×D₅
hypocenter	C₃×C₅	#1764
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. C₃×#65286	C₂	semidirect
5. C₂×#90951	C₂	semidirect
6. #11746353	C₂	semidirect
7. #11745983	C₂	semidirect
18. #91286	C₄	semidirect
20. C₅×#810	C₄	semidirect
21. C₅×#809	C₄	semidirect
24. #90978	C₄	semidirect
31. #65286	D₃	semidirect
129. #810	Dic₅	semidirect
130. #809	Dic₅	semidirect
326. C₃×C₅	#1764	semidirect
375. C₅	#69887	semidirect
389. C₃	#10601082	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91286	2
C₅×#5002	2
C₃×#65286	2
C₂×#90978	2
C₂×#90951	2
#11746353	2
#11745983	2
#10601082	3
#69887	5