Subgroups — #11756467

Order 960.

428 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#126	C₂	1	1	yes	maximal, characteristic, complemented
2	#91785	C₂	1	1	yes	maximal, characteristic, complemented
3	#91781	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊C₄×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
5	#91779	C₂	1	1	yes	maximal, characteristic, complemented
6	#91778	C₂	1	1	yes	maximal, characteristic, complemented
7	#91778	C₂	1	1	yes	maximal, characteristic, complemented
8	#91772	C₂	1	1	yes	maximal, characteristic
9	#91772	C₂	1	1	yes	maximal, characteristic
10	#91770	C₂	1	1	yes	maximal, characteristic
11	#91768	C₂	1	1	yes	maximal, characteristic, complemented
12	#91767	C₂	1	1	yes	maximal, characteristic, complemented
13	#91766	C₂	1	1	yes	maximal, characteristic, complemented
14	#91761	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
16	#66042	—	3	1	no	maximal, Hall
17	C₂×#1220	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₅	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
22	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
23	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
30	#6719	C₂²	1	1	yes	characteristic
31	#6719	C₂²	1	1	yes	characteristic
32	#6714	C₂²	1	1	yes	characteristic
33	#6714	C₂²	1	1	yes	characteristic
34	#6714	C₂²	1	1	yes	characteristic
35	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
36	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
37	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
38	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
39	#6712	C₂²	1	1	yes	characteristic
40	#6712	C₂²	1	1	yes	characteristic
41	#6712	C₂²	1	1	yes	characteristic
42	#6712	C₂²	1	1	yes	characteristic
43	#6712	C₂²	1	1	yes	characteristic
44	#6712	C₂²	1	1	yes	characteristic
45	#6712	C₂²	1	1	yes	characteristic
46	#6712	C₂²	1	1	yes	characteristic
47	#6712	C₂²	1	1	yes	characteristic
48	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
49	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
50	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
51	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
52	#5708	—	5	1	no	maximal, Hall
53	C₅×#126	D₃	1	1	yes	characteristic, complemented
54	#4042	—	3	1	no	—
55	#4038	—	3	1	no	—
56	D₅×C₄⋊C₄	—	3	1	no	—
57	#4036	—	3	1	no	—
58	#4035	—	3	1	no	—
59	#4035	—	3	1	no	—
60	#4029	—	3	1	no	—
61	#4029	—	3	1	no	—
62	#4027	—	3	1	no	—
63	#4025	—	3	1	no	—
64	#4024	—	3	1	no	—
65	#4023	—	3	1	no	—
66	#4018	—	3	1	no	—
67	C₄×Dic₁₀	—	3	1	no	—
68	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
69	C₂²×D₁₅	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×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
76	#1220	—	2	2	no	—
77	C₂×Dic₁₅	—	2	2	no	—
78	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
80	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
81	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
83	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
84	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
85	C₄×D₁₅	—	2	2	no	—
86	C₄×D₁₅	—	2	2	no	—
87	C₄×D₁₅	—	2	2	no	—
88	Dic₃₀	—	2	2	no	—
89	C₃×#126	D₅	1	1	yes	characteristic, complemented
90	#896	—	5	1	no	—
91	#892	—	5	1	no	—
92	D₃×C₄⋊C₄	—	5	1	no	—
93	#890	—	5	1	no	—
94	#889	—	5	1	no	—
95	#889	—	5	1	no	—
96	#883	—	5	1	no	—
97	#883	—	5	1	no	—
98	#67⋊C₄	—	5	1	no	—
99	#879	—	5	1	no	—
100	#878	—	5	1	no	—
101	#877	—	5	1	no	—
102	#872	—	5	1	no	—
103	C₄×Dic₆	—	5	1	no	—
104	C₂×C₂²⋊D₅	—	3	1	no	—
105	C₂²×Dic₅	—	3	1	no	—
106	C₂×C₄×D₅	—	3	1	no	—
107	C₂×C₄×D₅	—	3	1	no	—
108	C₂×C₄×D₅	—	3	1	no	—
109	C₂×Dic₁₀	—	3	1	no	—
110	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
111	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
112	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
113	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
114	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
115	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
116	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
117	C₂²⋊Dic₅	—	3	1	no	—
118	C₂²⋊Dic₅	—	3	1	no	—
119	#685	—	3	1	no	—
120	#685	—	3	1	no	—
121	#685	—	3	1	no	—
122	C₄⋊Dic₅	—	3	1	no	—
123	C₄⋊Dic₅	—	3	1	no	—
124	C₄⋊Dic₅	—	3	1	no	—
125	C₄⋊Dic₅	—	3	1	no	—
126	#683	—	3	1	no	—
127	#683	—	3	1	no	—
128	#683	—	3	1	no	—
129	#683	—	3	1	no	—
130	#683	—	3	1	no	—
131	#683	—	3	1	no	—
132	#683	—	3	1	no	—
133	#683	—	3	1	no	—
134	#683	—	3	1	no	—
135	C₄×Dic₅	—	3	1	no	—
136	C₄×Dic₅	—	3	1	no	—
137	C₄×Dic₅	—	3	1	no	—
138	C₄×Dic₅	—	3	1	no	—
139	#552	—	15	1	no	Sylow
140	C₂²×C₃×C₅	—	2	1	no	—
141	C₂²×C₃×C₅	—	2	1	no	—
142	C₂²×C₃×C₅	—	2	1	no	—
143	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
144	C₂×D₁₅	—	2	1	no	—
145	C₂×D₁₅	—	2	1	no	—
146	C₂×D₁₅	#41	1	2	yes	—
147	C₃×C₄×C₅	—	2	1	no	—
148	C₃×C₄×C₅	—	2	1	no	—
149	C₃×C₄×C₅	—	2	1	no	—
150	C₃×C₄×C₅	—	2	1	no	—
151	C₃×C₄×C₅	—	2	1	no	—
152	C₃×C₄×C₅	—	2	1	no	—
153	Dic₁₅	—	2	1	no	—
154	Dic₁₅	—	2	1	no	—
155	Dic₁₅	—	2	1	no	—
156	Dic₁₅	—	2	1	no	—
157	Dic₁₅	—	2	1	no	—
158	Dic₁₅	—	2	1	no	—
159	Dic₁₅	#41	1	2	yes	—
160	C₂×#67	—	5	1	no	—
161	C₂²×Dic₃	—	5	1	no	—
162	C₂×C₄×D₃	—	5	1	no	—
163	C₂×C₄×D₃	—	5	1	no	—
164	C₂×C₄×D₃	—	5	1	no	—
165	C₂×Dic₆	—	5	1	no	—
166	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
167	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
168	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
169	C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
170	C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
171	C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
172	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
173	#217	—	5	1	no	—
174	#217	—	5	1	no	—
175	(C₂×D₃)⋊C₄	—	5	1	no	—
176	(C₂×D₃)⋊C₄	—	5	1	no	—
177	(C₂×D₃)⋊C₄	—	5	1	no	—
178	C₄⋊Dic₃	—	5	1	no	—
179	C₄⋊Dic₃	—	5	1	no	—
180	C₄⋊Dic₃	—	5	1	no	—
181	C₄⋊Dic₃	—	5	1	no	—
182	Dic₃⋊C₄	—	5	1	no	—
183	Dic₃⋊C₄	—	5	1	no	—
184	Dic₃⋊C₄	—	5	1	no	—
185	Dic₃⋊C₄	—	5	1	no	—
186	Dic₃⋊C₄	—	5	1	no	—
187	Dic₃⋊C₄	—	5	1	no	—
188	Dic₃⋊C₄	—	5	1	no	—
189	Dic₃⋊C₄	—	5	1	no	—
190	Dic₃⋊C₄	—	5	1	no	—
191	C₄×Dic₃	—	5	1	no	—
192	C₄×Dic₃	—	5	1	no	—
193	C₄×Dic₃	—	5	1	no	—
194	C₄×Dic₃	—	5	1	no	—
195	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
196	C₂²×D₅	—	3	1	no	—
197	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
199	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
200	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
202	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
203	C₂²⋊D₅	—	6	2	no	—
204	C₂×Dic₅	—	3	1	no	—
205	C₂×Dic₅	—	3	1	no	—
206	C₂×Dic₅	—	3	1	no	—
207	C₂×Dic₅	—	3	1	no	—
208	C₂×Dic₅	—	3	1	no	—
209	C₂×Dic₅	—	3	1	no	—
210	C₂×Dic₅	—	3	1	no	—
211	C₂×Dic₅	—	6	2	no	—
212	C₄×D₅	—	6	2	no	—
213	C₄×D₅	—	6	2	no	—
214	C₄×D₅	—	6	2	no	—
215	Dic₁₀	—	6	2	no	—
216	#126	—	15	1	no	—
217	#126	D₁₅	1	1	yes	characteristic, complemented
218	#125	—	15	1	no	—
219	#125	—	15	1	no	—
220	#125	—	15	1	no	—
221	#123	—	15	1	no	—
222	#123	—	15	1	no	—
223	C₂²⋊Q₈	—	15	1	no	—
224	C₂²⋊Q₈	—	15	1	no	—
225	C₄×Q₈	—	15	1	no	—
226	C₄×D₄	—	15	1	no	—
227	#117	—	15	1	no	—
228	#117	—	15	1	no	—
229	#117	—	15	1	no	—
230	C₂×C₄⋊C₄	—	15	1	no	—
231	C₂×C₃×C₅	—	4	1	no	—
232	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
233	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
234	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
235	D₁₅	—	2	2	no	—
236	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
237	C₂²×D₃	—	5	1	no	—
238	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
241	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
244	#67	—	10	2	no	—
245	C₂×Dic₃	—	5	1	no	—
246	C₂×Dic₃	—	5	1	no	—
247	C₂×Dic₃	—	5	1	no	—
248	C₂×Dic₃	—	5	1	no	—
249	C₂×Dic₃	—	5	1	no	—
250	C₂×Dic₃	—	5	1	no	—
251	C₂×Dic₃	—	5	1	no	—
252	C₂×Dic₃	—	10	2	no	—
253	C₄×D₃	—	10	2	no	—
254	C₄×D₃	—	10	2	no	—
255	C₄×D₃	—	10	2	no	—
256	Dic₆	—	10	2	no	—
257	C₂²×C₅	—	2	1	no	—
258	C₂²×C₅	—	2	1	no	—
259	C₂²×C₅	—	2	1	no	—
260	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
261	C₂×D₅	—	3	2	no	—
262	C₂×D₅	—	6	1	no	—
263	C₂×D₅	—	6	1	no	—
264	C₄×C₅	—	2	1	no	—
265	C₄×C₅	—	2	1	no	—
266	C₄×C₅	—	2	1	no	—
267	C₄×C₅	—	2	1	no	—
268	C₄×C₅	—	2	1	no	—
269	C₄×C₅	—	2	1	no	—
270	Dic₅	—	3	2	no	—
271	Dic₅	—	6	1	no	—
272	Dic₅	—	6	1	no	—
273	Dic₅	—	6	1	no	—
274	Dic₅	—	6	1	no	—
275	Dic₅	—	6	1	no	—
276	Dic₅	—	6	1	no	—
277	C₂×Q₈	—	15	1	no	—
278	C₂×D₄	—	15	1	no	—
279	C₂²×C₄	—	15	1	no	—
280	C₂²×C₄	—	15	1	no	—
281	C₂²×C₄	—	15	1	no	—
282	C₂²×C₄	—	15	1	no	—
283	C₄⋊C₄	—	15	1	no	—
284	C₄⋊C₄	—	15	1	no	—
285	C₄⋊C₄	—	15	1	no	—
286	C₄⋊C₄	—	15	1	no	—
287	C₄⋊C₄	—	15	1	no	—
288	C₄⋊C₄	—	15	1	no	—
289	C₄⋊C₄	—	15	1	no	—
290	C₄⋊C₄	—	15	1	no	—
291	C₄⋊C₄	—	15	1	no	—
292	C₄⋊C₄	—	15	1	no	—
293	C₄⋊C₄	—	15	1	no	—
294	C₄⋊C₄	—	15	1	no	—
295	C₄⋊C₄	—	15	1	no	—
296	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
297	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
298	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
299	C₂²⋊C₄	—	15	1	no	—
300	C₂²⋊C₄	—	15	1	no	—
301	C₂²⋊C₄	—	15	1	no	—
302	C₂²⋊C₄	—	15	1	no	—
303	C₂²⋊C₄	—	15	1	no	—
304	C₂²⋊C₄	C₂×D₁₅	1	1	yes	characteristic
305	C₂²⋊C₄	C₂×D₁₅	1	1	yes	characteristic
306	C₂²⋊C₄	C₂×D₁₅	1	1	yes	characteristic
307	C₄²	—	15	1	no	—
308	C₄²	—	15	1	no	—
309	C₄²	—	15	1	no	—
310	C₄²	—	15	1	no	—
311	C₄²	C₂×D₁₅	1	1	yes	characteristic
312	C₃×C₅	#552	1	1	yes	characteristic, complemented, Hall
313	C₂²×C₃	—	2	1	no	—
314	C₂²×C₃	—	2	1	no	—
315	C₂²×C₃	—	2	1	no	—
316	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
317	C₂×D₃	—	5	2	no	—
318	C₂×D₃	—	10	1	no	—
319	C₂×D₃	—	10	1	no	—
320	C₃×C₄	—	2	1	no	—
321	C₃×C₄	—	2	1	no	—
322	C₃×C₄	—	2	1	no	—
323	C₃×C₄	—	2	1	no	—
324	C₃×C₄	—	2	1	no	—
325	C₃×C₄	—	2	1	no	—
326	Dic₃	—	5	2	no	—
327	Dic₃	—	10	1	no	—
328	Dic₃	—	10	1	no	—
329	Dic₃	—	10	1	no	—
330	Dic₃	—	10	1	no	—
331	Dic₃	—	10	1	no	—
332	Dic₃	—	10	1	no	—
333	C₂×C₅	—	4	1	no	—
334	C₂×C₅	D₃×#41	1	1	yes	characteristic
335	C₂×C₅	D₃×#41	1	1	yes	characteristic
336	C₂×C₅	#1010	1	1	yes	characteristic
337	D₅	—	6	2	no	—
338	C₂³	—	15	1	no	—
339	C₂³	C₂²×D₁₅	1	1	yes	characteristic
340	Q₈	—	30	2	no	—
341	D₄	—	30	2	no	—
342	C₂×C₄	—	15	1	no	—
343	C₂×C₄	—	15	1	no	—
344	C₂×C₄	—	15	1	no	—
345	C₂×C₄	—	15	1	no	—
346	C₂×C₄	—	15	1	no	—
347	C₂×C₄	—	15	1	no	—
348	C₂×C₄	—	15	1	no	—
349	C₂×C₄	—	30	2	no	—
350	C₂×C₄	—	30	2	no	—
351	C₂×C₄	—	30	2	no	—
352	C₂×C₄	—	30	2	no	—
353	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
357	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
358	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
359	C₂×C₃	—	4	1	no	—
360	C₂×C₃	D₅×#41	1	1	yes	characteristic
361	C₂×C₃	D₅×#41	1	1	yes	characteristic
362	C₂×C₃	#4150	1	1	yes	characteristic
363	D₃	—	10	2	no	—
364	C₅	#5708	1	1	yes	characteristic, complemented, Sylow
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	15	2	no	—
369	C₂²	—	30	1	no	—
370	C₂²	—	30	1	no	—
371	C₂²	C₂³×D₁₅	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	15	2	no	—
379	C₄	—	30	1	no	—
380	C₄	—	30	1	no	—
381	C₄	—	30	1	no	—
382	C₄	—	30	1	no	—
383	C₄	—	30	1	no	—
384	C₄	—	30	1	no	—
385	C₃	#66042	1	1	yes	characteristic, complemented, Sylow
386	C₂	—	4	1	no	—
387	C₂	—	30	2	no	—
388	C₂	#41×D₁₅	1	1	yes	characteristic
389	C₂	#41×D₁₅	1	1	yes	characteristic
390	C₂	#92101	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×#126	C₂
prime radical	C₂×#1220	C₂²
hypercenter	#126	D₁₅
hypocenter	C₃×C₅	#552
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ #126

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#126	C₂	semidirect
2. #91785	C₂	semidirect
3. #91781	C₂	semidirect
4. C₄⋊C₄×D₁₅	C₂	semidirect
5. #91779	C₂	semidirect
6. #91778	C₂	semidirect
7. #91778	C₂	semidirect
11. #91768	C₂	semidirect
12. #91767	C₂	semidirect
13. #91766	C₂	semidirect
14. #91761	C₂	semidirect
15. C₄×Dic₃₀	C₂	semidirect
53. C₅×#126	D₃	semidirect
89. C₃×#126	D₅	semidirect
217. #126	D₁₅	semidirect
312. C₃×C₅	#552	semidirect
364. C₅	#5708	semidirect
385. C₃	#66042	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#126	2
#91785	2
#91781	2
C₄⋊C₄×D₁₅	2
#91779	2
#91778	2
#91772	2
#91770	2
#91768	2
#91767	2
#91766	2
#91761	2
C₄×Dic₃₀	2
#66042	3
#5708	5