Subgroups — #11754341

Order 960.

660 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	C₂²×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#6765	C₂	1	1	yes	maximal, characteristic, complemented
3	#91813	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4076	C₂	1	1	yes	maximal, characteristic, complemented
5	#91568	C₂	1	2	yes	maximal, complemented
6	Dic₃×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
7	#91456	C₂	1	1	yes	maximal, characteristic, complemented
8	#91452	C₂	1	2	yes	maximal, complemented
9	#91441	C₂	1	2	yes	maximal, complemented
10	#91365	C₂	1	1	yes	maximal, characteristic, complemented
11	#91338	C₂	1	1	yes	maximal, characteristic, complemented
12	#66125	—	3	1	no	maximal, Hall
13	C₂³×C₅×D₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₀	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₃	—	2	4	no	—
18	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
23	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic, complemented
24	C₂×#1201	C₂²	1	2	yes	—
25	C₂×#1201	C₂²	1	2	yes	—
26	C₂×D₅×Dic₃	C₂²	1	2	yes	complemented
27	#6765	—	2	4	no	—
28	#6719	C₂²	1	2	yes	—
29	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic, complemented
30	#6712	C₂²	1	2	yes	complemented
31	C₃×#685	C₂²	1	2	yes	complemented
32	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
33	#6668	C₂²	1	2	yes	complemented
34	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
35	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
36	#6665	C₂²	1	2	yes	complemented
37	Dic₃×Dic₅	C₂²	1	2	yes	complemented
38	#5606	—	5	1	no	maximal, Hall
39	C₂³×C₄×C₅	—	3	1	no	—
40	C₂×#709	—	3	1	no	—
41	#4099	—	3	2	no	—
42	#4076	—	3	1	no	—
43	#4076	D₃	1	1	yes	characteristic, complemented
44	#4075	—	3	2	no	—
45	C₄×C₂²⋊D₅	—	3	2	no	—
46	C₄×C₂²⋊D₅	—	3	2	no	—
47	#4072	—	3	1	no	—
48	#4070	—	3	1	no	—
49	#4070	—	3	1	no	—
50	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂²×C₅×D₃	—	2	2	no	—
52	C₂²×C₅×D₃	—	2	2	no	—
53	C₂²×C₅×D₃	—	2	2	no	—
54	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
55	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
56	C₂²×C₃×D₅	C₂³	1	2	yes	—
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₅	C₂³	1	2	yes	—
61	C₂×Dic₁₅	C₂³	1	2	yes	—
62	Dic₃₀	—	2	2	no	—
63	C₂×C₅×Dic₃	—	2	2	no	—
64	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
65	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
66	C₄×C₅×D₃	—	2	2	no	—
67	C₄×C₅×D₃	—	2	2	no	—
68	C₄×C₅×D₃	—	2	2	no	—
69	C₄×C₅×D₃	D₄	1	4	yes	complemented
70	C₃×C₂²⋊D₅	—	2	4	no	—
71	C₂×C₃×Dic₅	C₂³	1	2	yes	—
72	C₃×D₂₀	—	2	2	no	—
73	#1201	—	2	4	no	—
74	#1201	—	2	4	no	—
75	D₅×Dic₃	—	2	4	no	—
76	C₂×#237	—	5	1	no	—
77	C₂²×C₄×D₃	D₅	1	1	yes	characteristic, complemented
78	C₃×#121	—	5	1	no	—
79	#938	—	5	1	no	—
80	#937	—	5	2	no	—
81	D₄×Dic₃	—	5	2	no	—
82	#924	—	5	1	no	—
83	#897	—	5	1	no	—
84	#892	—	5	1	no	—
85	#883	—	5	2	no	—
86	#67⋊C₄	—	5	2	no	—
87	C₂⁴×C₅	—	3	1	no	—
88	C₂²×C₄×C₅	—	3	1	no	—
89	C₂²×C₄×C₅	—	3	1	no	—
90	C₂²×C₄×C₅	—	3	1	no	—
91	C₂²×C₄×C₅	—	3	1	no	—
92	C₂²×C₄×C₅	—	3	1	no	—
93	C₂²×C₄×C₅	—	6	4	no	—
94	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
95	C₂×C₂²⋊D₅	—	3	2	no	—
96	C₂×C₂²⋊D₅	—	3	2	no	—
97	C₂×C₂²⋊D₅	C₂×D₃	1	2	yes	—
98	#709	—	6	4	no	—
99	C₂×D₂₀	C₂×D₃	1	1	yes	characteristic, complemented
100	C₂×C₄×D₅	—	3	2	no	—
101	C₂×Dic₁₀	—	3	1	no	—
102	C₂²⋊Dic₅	—	3	2	no	—
103	C₂²⋊Dic₅	—	3	2	no	—
104	C₂²⋊Dic₅	—	3	2	no	—
105	#685	—	3	2	no	—
106	#685	C₂×D₃	1	2	yes	complemented
107	C₄⋊Dic₅	—	3	1	no	—
108	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic, complemented
109	#683	—	3	2	no	—
110	#683	—	3	2	no	—
111	C₄×Dic₅	—	3	2	no	—
112	#525	—	15	1	no	Sylow
113	C₂²×C₃×C₅	—	2	1	no	—
114	C₂²×C₃×C₅	—	2	1	no	—
115	C₂²×C₃×C₅	#41	1	2	yes	complemented
116	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
117	C₂×C₅×D₃	—	2	8	no	—
118	C₂×C₅×D₃	—	2	1	no	—
119	C₂×C₅×D₃	—	2	1	no	—
120	C₂×C₅×D₃	—	2	1	no	—
121	C₂×C₅×D₃	—	2	1	no	—
122	C₂×C₅×D₃	C₂×D₄	1	2	yes	—
123	C₂×C₅×D₃	#41	1	2	yes	complemented
124	C₂×C₃×D₅	—	2	2	no	—
125	C₂×C₃×D₅	—	2	2	no	—
126	C₂×C₃×D₅	—	2	2	no	—
127	C₃×C₄×C₅	—	2	1	no	—
128	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
129	Dic₁₅	—	2	2	no	—
130	Dic₁₅	—	2	2	no	—
131	C₃×Dic₅	—	2	2	no	—
132	C₅×Dic₃	—	2	1	no	—
133	C₅×Dic₃	C₂×D₄	1	2	yes	—
134	C₂³×D₃	C₂×D₅	1	1	yes	characteristic
135	C₂×C₃×D₄	—	5	2	no	—
136	C₂×C₃×D₄	—	5	1	no	—
137	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
138	C₂×#67	—	5	2	no	—
139	C₂×#67	—	5	2	no	—
140	C₂²×Dic₃	—	5	2	no	—
141	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
142	#237	—	10	4	no	—
143	C₂×C₄×D₃	—	2	4	no	—
144	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
145	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
146	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
147	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
148	C₂×Dic₆	—	5	1	no	—
149	C₃×C₄⋊C₄	—	5	1	no	—
150	C₃×C₂²⋊C₄	—	5	2	no	—
151	#217	—	5	2	no	—
152	#217	—	5	2	no	—
153	(C₂×D₃)⋊C₄	—	5	2	no	—
154	(C₂×D₃)⋊C₄	—	5	2	no	—
155	C₄⋊Dic₃	—	5	1	no	—
156	C₄⋊Dic₃	—	5	2	no	—
157	Dic₃⋊C₄	—	5	2	no	—
158	C₄×Dic₃	—	5	2	no	—
159	C₂³×C₅	—	3	1	no	—
160	C₂³×C₅	—	3	1	no	—
161	C₂³×C₅	—	6	2	no	—
162	C₂³×C₅	—	6	2	no	—
163	C₂³×C₅	—	6	2	no	—
164	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
165	C₂²×D₅	C₂²×D₃	1	2	yes	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	3	1	no	—
168	C₂×C₄×C₅	—	3	4	no	—
169	C₂×C₄×C₅	—	3	1	no	—
170	C₂×C₄×C₅	—	6	2	no	—
171	C₂×C₄×C₅	—	6	2	no	—
172	C₂×C₄×C₅	—	6	2	no	—
173	C₂×C₄×C₅	—	6	2	no	—
174	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
176	C₂²⋊D₅	—	2	4	no	—
177	C₂²⋊D₅	—	6	4	no	—
178	C₂²⋊D₅	—	6	4	no	—
179	C₂×Dic₅	—	3	2	no	—
180	C₂×Dic₅	—	3	2	no	—
181	C₂×Dic₅	C₂²×D₃	1	2	yes	—
182	D₂₀	—	2	2	no	—
183	C₄×D₅	—	6	4	no	—
184	Dic₁₀	—	6	2	no	—
185	C₂×#41	—	15	1	no	—
186	C₂³×C₄	—	3	1	no	—
187	#123	—	15	2	no	—
188	C₂²⋊Q₈	—	15	1	no	—
189	C₂²⋊Q₈	—	15	1	no	—
190	#121	—	5	1	no	—
191	#121	—	15	1	no	—
192	#120	—	15	2	no	—
193	C₄×D₄	—	15	2	no	—
194	C₄×D₄	—	15	2	no	—
195	#117	—	15	1	no	—
196	C₂×C₃×C₅	—	2	2	no	—
197	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
200	C₃×D₅	—	4	2	no	—
201	C₅×D₃	—	2	2	no	—
202	C₅×D₃	—	2	2	no	—
203	C₂³×C₃	—	5	2	no	—
204	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
205	C₂²×D₃	—	2	2	no	—
206	C₂²×D₃	—	2	2	no	—
207	C₂²×D₃	—	2	2	no	—
208	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
209	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
210	C₃×D₄	—	10	4	no	—
211	C₃×D₄	—	10	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	5	2	no	—
214	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
216	#67	—	10	4	no	—
217	#67	—	10	4	no	—
218	C₂×Dic₃	—	2	2	no	—
219	C₂×Dic₃	—	5	2	no	—
220	C₂×Dic₃	—	5	2	no	—
221	C₂×Dic₃	—	10	4	no	—
222	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
223	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
224	C₄×D₃	—	2	2	no	—
225	C₄×D₃	—	2	2	no	—
226	C₄×D₃	—	2	2	no	—
227	C₄×D₃	C₂²⋊D₅	1	4	yes	complemented
228	Dic₆	—	10	2	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	—	3	2	no	—
232	C₂²×C₅	—	3	2	no	—
233	C₂²×C₅	—	6	8	no	—
234	C₂²×C₅	—	6	1	no	—
235	C₂²×C₅	—	6	1	no	—
236	C₂²×C₅	—	6	1	no	—
237	C₂²×C₅	—	6	1	no	—
238	C₂²×C₅	#237	1	2	yes	complemented
239	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
240	C₂×D₅	—	2	2	no	—
241	C₂×D₅	—	2	2	no	—
242	C₂×D₅	—	2	2	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	3	2	no	—
245	C₄×C₅	—	6	1	no	—
246	C₄×C₅	D₃×D₄	1	2	yes	—
247	Dic₅	—	2	2	no	—
248	Dic₅	—	6	2	no	—
249	Dic₅	—	6	2	no	—
250	C₂⁴	—	3	1	no	—
251	#41	—	30	4	no	—
252	C₂×Q₈	—	15	1	no	—
253	C₂×D₄	—	5	2	no	—
254	C₂×D₄	—	5	1	no	—
255	C₂×D₄	—	15	2	no	—
256	C₂×D₄	—	15	2	no	—
257	C₂²×C₄	—	3	1	no	—
258	C₂²×C₄	—	3	1	no	—
259	C₂²×C₄	—	3	1	no	—
260	C₂²×C₄	—	3	1	no	—
261	C₂²×C₄	—	3	1	no	—
262	C₂²×C₄	—	6	4	no	—
263	C₂²×C₄	—	15	2	no	—
264	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
265	C₄⋊C₄	—	5	1	no	—
266	C₄⋊C₄	—	15	2	no	—
267	C₄⋊C₄	—	15	1	no	—
268	C₄⋊C₄	—	15	2	no	—
269	C₂²⋊C₄	—	5	2	no	—
270	C₂²⋊C₄	—	15	2	no	—
271	C₂²⋊C₄	—	15	2	no	—
272	C₂²⋊C₄	—	15	2	no	—
273	C₂²⋊C₄	—	15	2	no	—
274	C₄²	—	15	2	no	—
275	C₃×C₅	#525	1	1	yes	characteristic, complemented, Hall
276	C₂²×C₃	—	2	1	no	—
277	C₂²×C₃	—	2	1	no	—
278	C₂²×C₃	—	10	2	no	—
279	C₂²×C₃	—	10	2	no	—
280	C₂²×C₃	—	10	2	no	—
281	C₂²×C₃	#709	1	2	yes	complemented
282	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
283	C₂×D₃	—	2	8	no	—
284	C₂×D₃	—	2	1	no	—
285	C₂×D₃	—	2	1	no	—
286	C₂×D₃	—	2	1	no	—
287	C₂×D₃	—	2	1	no	—
288	C₂×D₃	#709	1	2	yes	complemented
289	C₂×D₃	C₂×C₂²⋊D₅	1	2	yes	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	10	2	no	—
292	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
293	Dic₃	—	2	1	no	—
294	Dic₃	—	10	2	no	—
295	Dic₃	—	10	2	no	—
296	Dic₃	C₂×C₂²⋊D₅	1	2	yes	—
297	C₂×C₅	—	2	2	no	—
298	C₂×C₅	—	6	2	no	—
299	C₂×C₅	—	6	2	no	—
300	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
301	C₂×C₅	C₂×#237	1	1	yes	characteristic
302	C₂×C₅	D₃×#41	1	1	yes	characteristic
303	D₅	—	4	2	no	—
304	C₂³	—	3	1	no	—
305	C₂³	—	3	1	no	—
306	C₂³	—	5	2	no	—
307	C₂³	—	6	2	no	—
308	C₂³	—	6	2	no	—
309	C₂³	—	6	2	no	—
310	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
311	Q₈	—	30	2	no	—
312	D₄	—	10	4	no	—
313	D₄	—	10	2	no	—
314	D₄	—	30	4	no	—
315	D₄	—	30	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	3	1	no	—
318	C₂×C₄	—	3	4	no	—
319	C₂×C₄	—	3	1	no	—
320	C₂×C₄	—	5	2	no	—
321	C₂×C₄	—	6	2	no	—
322	C₂×C₄	—	6	2	no	—
323	C₂×C₄	—	6	2	no	—
324	C₂×C₄	—	6	2	no	—
325	C₂×C₄	—	15	2	no	—
326	C₂×C₄	—	15	2	no	—
327	C₂×C₄	—	30	4	no	—
328	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
329	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
330	C₂×C₃	—	2	2	no	—
331	C₂×C₃	—	20	2	no	—
332	C₂×C₃	C₂×#709	1	1	yes	characteristic
333	C₂×C₃	C₂×#709	1	1	yes	characteristic
334	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
335	D₃	—	2	2	no	—
336	D₃	—	2	2	no	—
337	C₅	#5606	1	1	yes	characteristic, complemented, Sylow
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	3	2	no	—
341	C₂²	—	3	2	no	—
342	C₂²	—	6	8	no	—
343	C₂²	—	6	1	no	—
344	C₂²	—	6	1	no	—
345	C₂²	—	6	1	no	—
346	C₂²	—	6	1	no	—
347	C₂²	—	10	2	no	—
348	C₂²	—	10	2	no	—
349	C₂²	—	10	2	no	—
350	C₂²	#6765	1	2	yes	complemented
351	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	3	2	no	—
354	C₄	—	6	1	no	—
355	C₄	—	10	2	no	—
356	C₄	—	30	2	no	—
357	C₄	—	30	2	no	—
358	C₄	D₃×C₂²⋊D₅	1	2	yes	—
359	C₃	#66125	1	1	yes	characteristic, complemented, Sylow
360	C₂	—	2	2	no	—
361	C₂	—	6	2	no	—
362	C₂	—	6	2	no	—
363	C₂	—	20	2	no	—
364	C₂	C₂×#6765	1	1	yes	characteristic
365	C₂	D₃×#709	1	1	yes	characteristic
366	C₂	C₂×D₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₂²×C₃×C₄×C₅	C₂²
hypercenter	C₂²×C₄	D₃×D₅
hypocenter	C₃×C₅	#525
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄×C₅×D₃	C₂	semidirect
2. C₂×#6765	C₂	semidirect
3. #91813	C₂	semidirect
4. C₃×#4076	C₂	semidirect
5. #91568	C₂	semidirect
6. Dic₃×C₂²⋊D₅	C₂	semidirect
7. #91456	C₂	semidirect
8. #91452	C₂	semidirect
9. #91441	C₂	semidirect
10. #91365	C₂	semidirect
11. #91338	C₂	semidirect
15. C₂×Dic₃₀	C₂²	semidirect
23. C₂×C₃×D₂₀	C₂²	semidirect
26. C₂×D₅×Dic₃	C₂²	semidirect
29. C₄⋊Dic₁₅	C₂²	semidirect
30. #6712	C₂²	semidirect
31. C₃×#685	C₂²	semidirect
32. C₃×C₄⋊Dic₅	C₂²	semidirect
33. #6668	C₂²	semidirect
36. #6665	C₂²	semidirect
37. Dic₃×Dic₅	C₂²	semidirect
43. #4076	D₃	semidirect
69. C₄×C₅×D₃	D₄	semidirect
77. C₂²×C₄×D₃	D₅	semidirect
99. C₂×D₂₀	C₂×D₃	semidirect
106. #685	C₂×D₃	semidirect
108. C₄⋊Dic₅	C₂×D₃	semidirect
115. C₂²×C₃×C₅	#41	semidirect
123. C₂×C₅×D₃	#41	semidirect
227. C₄×D₃	C₂²⋊D₅	semidirect
238. C₂²×C₅	#237	semidirect
275. C₃×C₅	#525	semidirect
281. C₂²×C₃	#709	semidirect
288. C₂×D₃	#709	semidirect
337. C₅	#5606	semidirect
350. C₂²	#6765	semidirect
359. C₃	#66125	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×D₃	2
C₂×#6765	2
#91813	2
C₃×#4076	2
#91568	2
Dic₃×C₂²⋊D₅	2
#91456	2
#91452	2
#91441	2
#91365	2
#91338	2
#66125	3
#5606	5