Subgroups — #11753519

Order 960.

384 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#91778	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#898	C₂	1	1	yes	maximal, characteristic
3	C₃×#4034	C₂	1	1	yes	maximal, characteristic, complemented
4	#91410	C₂	1	1	yes	maximal, characteristic
5	#91410	C₂	1	1	yes	maximal, characteristic
6	#91398	C₂	1	1	yes	maximal, characteristic
7	#91379	C₂	1	1	yes	maximal, characteristic, complemented
8	#91378	C₂	1	1	yes	maximal, characteristic, complemented
9	#91367	C₂	1	1	yes	maximal, characteristic
10	#91346	C₂	1	1	yes	maximal, characteristic, complemented
11	#91342	C₂	1	1	yes	maximal, characteristic
12	#91341	C₂	1	1	yes	maximal, characteristic
13	#91336	C₂	1	1	yes	maximal, characteristic, complemented
14	#91331	C₂	1	1	yes	maximal, characteristic, complemented
15	Dic₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
16	#66035	—	3	1	no	maximal, Hall
17	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
19	C₂×D₃×Dic₅	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
22	#6712	C₂²	1	1	yes	characteristic
23	#6712	C₂²	1	1	yes	characteristic
24	#6712	C₂²	1	1	yes	characteristic
25	#6712	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
27	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
28	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
29	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
30	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
31	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
33	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
34	C₃×#683	C₂²	1	1	yes	characteristic
35	C₃×#683	C₂²	1	1	yes	characteristic
36	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
37	Dic₃⋊Dic₅	C₂²	1	1	yes	characteristic
38	Dic₃⋊Dic₅	C₂²	1	1	yes	characteristic
39	Dic₃⋊Dic₅	C₂²	1	1	yes	characteristic
40	#6669	C₂²	1	1	yes	characteristic
41	#6669	C₂²	1	1	yes	characteristic
42	#6668	C₂²	1	1	yes	characteristic
43	#6668	C₂²	1	1	yes	characteristic
44	#6668	C₂²	1	1	yes	characteristic
45	#6668	C₂²	1	1	yes	characteristic
46	(C₂×D₃)⋊Dic₅	C₂²	1	1	yes	characteristic
47	(C₂×D₃)⋊Dic₅	C₂²	1	1	yes	characteristic
48	(C₂×D₃)⋊Dic₅	C₂²	1	1	yes	characteristic
49	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
50	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
51	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
52	#5731	—	5	1	no	maximal, Hall
53	C₅×#126	—	3	1	no	—
54	#4036	—	3	1	no	—
55	#4035	—	3	1	no	—
56	#4035	—	3	1	no	—
57	#4035	—	3	1	no	—
58	#4034	D₃	1	1	yes	characteristic, complemented
59	#4033	—	3	1	no	—
60	#4025	—	3	1	no	—
61	#4025	—	3	1	no	—
62	#4025	—	3	1	no	—
63	#4024	—	3	1	no	—
64	#4024	—	3	1	no	—
65	#4023	—	3	1	no	—
66	#4016	—	3	1	no	—
67	C₄×Dic₁₀	—	3	1	no	—
68	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
73	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
74	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
76	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
77	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
78	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
80	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
81	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
82	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
83	C₃×Dic₁₀	—	2	2	no	—
84	C₃×Dic₁₀	—	2	2	no	—
85	D₃×Dic₅	—	2	2	no	—
86	C₃×C₄⋊Q₈	—	5	1	no	—
87	#946	—	5	1	no	—
88	#946	—	5	1	no	—
89	Q₈×Dic₃	—	5	1	no	—
90	Q₈×Dic₃	—	5	1	no	—
91	#898	—	5	1	no	—
92	#898	—	5	1	no	—
93	#898	—	5	1	no	—
94	#898	D₅	1	1	yes	characteristic
95	#889	—	5	1	no	—
96	#889	—	5	1	no	—
97	#889	—	5	1	no	—
98	#889	—	5	1	no	—
99	#872	—	5	1	no	—
100	#870	—	5	1	no	—
101	C₂²×Dic₅	—	3	1	no	—
102	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic
103	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic
104	C₅×C₄⋊C₄	—	3	1	no	—
105	C₅×C₄⋊C₄	—	3	1	no	—
106	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
107	C₅×C₂²⋊C₄	—	3	1	no	—
108	C₅×C₂²⋊C₄	—	3	1	no	—
109	C₅×C₂²⋊C₄	—	3	1	no	—
110	C₄²×C₅	—	3	1	no	—
111	C₂²⋊Dic₅	—	3	1	no	—
112	C₂²⋊Dic₅	—	3	1	no	—
113	C₂²⋊Dic₅	—	3	1	no	—
114	C₄⋊Dic₅	—	3	1	no	—
115	C₄⋊Dic₅	—	3	1	no	—
116	C₄⋊Dic₅	—	3	1	no	—
117	C₄⋊Dic₅	—	3	1	no	—
118	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
119	#683	—	3	1	no	—
120	#683	—	3	1	no	—
121	#683	—	3	1	no	—
122	#683	—	3	1	no	—
123	#683	—	3	1	no	—
124	#683	—	3	1	no	—
125	#683	—	3	1	no	—
126	#683	—	3	1	no	—
127	#683	—	3	1	no	—
128	#683	—	3	1	no	—
129	#683	C₂×D₃	1	1	yes	characteristic
130	#683	C₂×D₃	1	1	yes	characteristic
131	C₄×Dic₅	—	3	1	no	—
132	C₄×Dic₅	—	3	1	no	—
133	C₄×Dic₅	—	3	1	no	—
134	C₄×Dic₅	—	3	1	no	—
135	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
136	#541	—	15	1	no	Sylow
137	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
138	C₂×C₅×D₃	—	2	1	no	—
139	C₂×C₅×D₃	—	2	1	no	—
140	C₂×C₅×D₃	—	2	1	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	2	1	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	Dic₁₅	—	2	1	no	—
145	Dic₁₅	—	2	1	no	—
146	Dic₁₅	—	2	1	no	—
147	Dic₁₅	—	2	1	no	—
148	C₃×Dic₅	—	2	1	no	—
149	C₃×Dic₅	—	2	1	no	—
150	C₃×Dic₅	—	2	1	no	—
151	C₃×Dic₅	#41	1	2	yes	—
152	C₅×Dic₃	—	2	1	no	—
153	C₅×Dic₃	—	2	1	no	—
154	C₅×Dic₃	—	2	1	no	—
155	C₂×C₃×Q₈	—	5	1	no	—
156	C₂×C₃×Q₈	—	5	1	no	—
157	C₂×C₄×D₃	—	5	1	no	—
158	C₃×C₄⋊C₄	—	5	1	no	—
159	C₃×C₄⋊C₄	—	5	1	no	—
160	C₃×C₄⋊C₄	—	5	1	no	—
161	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
162	C₃×C₄²	—	5	1	no	—
163	(C₂×D₃)⋊C₄	—	5	1	no	—
164	(C₂×D₃)⋊C₄	—	5	1	no	—
165	(C₂×D₃)⋊C₄	—	5	1	no	—
166	(C₂×D₃)⋊C₄	C₂×D₅	1	1	yes	characteristic
167	(C₂×D₃)⋊C₄	C₂×D₅	1	1	yes	characteristic
168	(C₂×D₃)⋊C₄	C₂×D₅	1	1	yes	characteristic
169	C₄⋊Dic₃	—	5	1	no	—
170	C₄⋊Dic₃	—	5	1	no	—
171	C₄⋊Dic₃	—	5	1	no	—
172	C₄⋊Dic₃	—	5	1	no	—
173	C₄⋊Dic₃	—	5	1	no	—
174	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
175	Dic₃⋊C₄	—	5	1	no	—
176	Dic₃⋊C₄	—	5	1	no	—
177	Dic₃⋊C₄	—	5	1	no	—
178	Dic₃⋊C₄	—	5	1	no	—
179	Dic₃⋊C₄	—	5	1	no	—
180	Dic₃⋊C₄	—	5	1	no	—
181	Dic₃⋊C₄	—	5	1	no	—
182	Dic₃⋊C₄	—	5	1	no	—
183	Dic₃⋊C₄	—	5	1	no	—
184	Dic₃⋊C₄	C₂×D₅	1	1	yes	characteristic
185	C₄×Dic₃	—	5	1	no	—
186	C₄×Dic₃	—	5	1	no	—
187	C₄×Dic₃	—	5	1	no	—
188	C₄×Dic₃	—	5	1	no	—
189	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
190	C₂³×C₅	—	3	1	no	—
191	C₂×C₄×C₅	—	3	1	no	—
192	C₂×C₄×C₅	—	3	1	no	—
193	C₂×C₄×C₅	—	3	1	no	—
194	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
196	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
197	C₂×Dic₅	—	3	1	no	—
198	C₂×Dic₅	—	3	1	no	—
199	C₂×Dic₅	—	3	1	no	—
200	C₂×Dic₅	—	3	1	no	—
201	C₂×Dic₅	—	6	2	no	—
202	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
203	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
204	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
205	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
206	Dic₁₀	—	2	2	no	—
207	Dic₁₀	—	2	2	no	—
208	C₄⋊Q₈	—	5	1	no	—
209	#126	—	3	1	no	—
210	#126	—	15	1	no	—
211	#126	—	15	1	no	—
212	#126	—	15	1	no	—
213	#125	—	15	1	no	—
214	#125	—	15	1	no	—
215	#125	—	15	1	no	—
216	#125	—	15	1	no	—
217	#125	—	15	1	no	—
218	C₂²⋊Q₈	—	15	1	no	—
219	C₂²⋊Q₈	—	15	1	no	—
220	C₄×Q₈	—	15	1	no	—
221	C₄×Q₈	—	15	1	no	—
222	#117	—	15	1	no	—
223	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
224	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
225	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
226	C₅×D₃	—	4	1	no	—
227	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
228	C₃×Q₈	—	10	2	no	—
229	C₃×Q₈	—	10	2	no	—
230	C₂×C₃×C₄	—	5	1	no	—
231	C₂×C₃×C₄	—	5	1	no	—
232	C₂×C₃×C₄	—	5	1	no	—
233	C₂×C₃×C₄	—	5	1	no	—
234	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
237	C₂×Dic₃	—	5	1	no	—
238	C₂×Dic₃	—	5	1	no	—
239	C₂×Dic₃	—	5	1	no	—
240	C₂×Dic₃	—	5	1	no	—
241	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
242	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
243	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
244	C₄×D₃	—	10	2	no	—
245	C₂²×C₅	—	6	1	no	—
246	C₂²×C₅	—	6	1	no	—
247	C₂²×C₅	—	6	1	no	—
248	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
249	C₄×C₅	—	2	1	no	—
250	C₄×C₅	—	2	1	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	6	1	no	—
253	C₄×C₅	—	6	1	no	—
254	C₄×C₅	—	6	1	no	—
255	Dic₅	—	2	1	no	—
256	Dic₅	—	2	1	no	—
257	Dic₅	—	2	1	no	—
258	Dic₅	—	6	1	no	—
259	Dic₅	—	6	1	no	—
260	Dic₅	—	6	1	no	—
261	Dic₅	—	6	1	no	—
262	Dic₅	#237	1	2	yes	—
263	C₂×Q₈	—	5	1	no	—
264	C₂×Q₈	—	5	1	no	—
265	C₂²×C₄	—	15	1	no	—
266	C₄⋊C₄	—	3	1	no	—
267	C₄⋊C₄	—	3	1	no	—
268	C₄⋊C₄	—	5	1	no	—
269	C₄⋊C₄	—	5	1	no	—
270	C₄⋊C₄	—	5	1	no	—
271	C₄⋊C₄	—	15	1	no	—
272	C₄⋊C₄	—	15	1	no	—
273	C₄⋊C₄	—	15	1	no	—
274	C₄⋊C₄	—	15	1	no	—
275	C₄⋊C₄	—	15	1	no	—
276	C₄⋊C₄	—	15	1	no	—
277	C₄⋊C₄	—	15	1	no	—
278	C₄⋊C₄	—	15	1	no	—
279	C₄⋊C₄	—	15	1	no	—
280	C₄⋊C₄	—	15	1	no	—
281	C₄⋊C₄	—	15	1	no	—
282	C₄⋊C₄	—	15	1	no	—
283	C₄⋊C₄	—	15	1	no	—
284	C₄⋊C₄	—	15	1	no	—
285	C₄⋊C₄	D₃×D₅	1	1	yes	characteristic
286	C₂²⋊C₄	—	3	1	no	—
287	C₂²⋊C₄	—	3	1	no	—
288	C₂²⋊C₄	—	3	1	no	—
289	C₂²⋊C₄	—	15	1	no	—
290	C₂²⋊C₄	—	15	1	no	—
291	C₂²⋊C₄	—	15	1	no	—
292	C₄²	—	3	1	no	—
293	C₄²	—	5	1	no	—
294	C₄²	—	15	1	no	—
295	C₄²	—	15	1	no	—
296	C₄²	—	15	1	no	—
297	C₄²	—	15	1	no	—
298	C₃×C₅	#541	1	1	yes	characteristic, complemented, Hall
299	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
300	C₂×D₃	—	2	1	no	—
301	C₂×D₃	—	2	1	no	—
302	C₂×D₃	—	2	1	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	2	1	no	—
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	—	5	2	no	—
307	C₃×C₄	—	10	1	no	—
308	C₃×C₄	—	10	1	no	—
309	C₃×C₄	—	10	1	no	—
310	Dic₃	—	2	1	no	—
311	Dic₃	—	2	1	no	—
312	Dic₃	—	2	1	no	—
313	Dic₃	—	10	1	no	—
314	Dic₃	—	10	1	no	—
315	Dic₃	—	10	1	no	—
316	Dic₃	—	10	1	no	—
317	C₂×C₅	—	12	1	no	—
318	C₂×C₅	C₂×#237	1	1	yes	characteristic
319	C₂×C₅	#1007	1	1	yes	characteristic
320	C₂×C₅	#1007	1	1	yes	characteristic
321	C₂³	—	3	1	no	—
322	Q₈	—	10	2	no	—
323	Q₈	—	10	2	no	—
324	C₂×C₄	—	3	1	no	—
325	C₂×C₄	—	3	1	no	—
326	C₂×C₄	—	3	1	no	—
327	C₂×C₄	—	5	1	no	—
328	C₂×C₄	—	5	1	no	—
329	C₂×C₄	—	5	1	no	—
330	C₂×C₄	—	5	1	no	—
331	C₂×C₄	—	15	1	no	—
332	C₂×C₄	—	15	1	no	—
333	C₂×C₄	—	15	1	no	—
334	C₂×C₄	—	15	1	no	—
335	C₂×C₄	—	30	2	no	—
336	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
337	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
338	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
339	C₂×C₃	D₅×#41	1	1	yes	characteristic
340	C₂×C₃	#4150	1	1	yes	characteristic
341	C₂×C₃	#4150	1	1	yes	characteristic
342	D₃	—	4	1	no	—
343	C₅	#5731	1	1	yes	characteristic, complemented, Sylow
344	C₂²	—	6	1	no	—
345	C₂²	—	6	1	no	—
346	C₂²	—	6	1	no	—
347	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	5	2	no	—
352	C₄	—	6	1	no	—
353	C₄	—	6	1	no	—
354	C₄	—	6	1	no	—
355	C₄	—	10	1	no	—
356	C₄	—	10	1	no	—
357	C₄	—	10	1	no	—
358	C₄	—	30	1	no	—
359	C₄	—	30	1	no	—
360	C₄	—	30	1	no	—
361	C₄	—	30	1	no	—
362	C₃	#66035	1	1	yes	characteristic, complemented, Sylow
363	C₂	—	12	1	no	—
364	C₂	#92004	1	1	yes	characteristic
365	C₂	D₅×#237	1	1	yes	characteristic
366	C₂	#92030	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×C₄⋊C₄	C₂²
prime radical	C₂²×C₅×D₃	C₂³
hypercenter	C₄⋊C₄	D₃×D₅
hypocenter	C₃×C₅	#541
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #91778	C₂	semidirect
3. C₃×#4034	C₂	semidirect
7. #91379	C₂	semidirect
8. #91378	C₂	semidirect
10. #91346	C₂	semidirect
13. #91336	C₂	semidirect
14. #91331	C₂	semidirect
15. Dic₃×Dic₁₀	C₂	semidirect
58. #4034	D₃	semidirect
298. C₃×C₅	#541	semidirect
343. C₅	#5731	semidirect
362. C₃	#66035	semidirect

Maximal subgroups

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

subgroup	index
#91778	2
C₅×#898	2
C₃×#4034	2
#91410	2
#91398	2
#91379	2
#91378	2
#91367	2
#91346	2
#91342	2
#91341	2
#91336	2
#91331	2
Dic₃×Dic₁₀	2
#66035	3
#5731	5