Subgroups — #11752799

Order 960.

412 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	#91767	C₂	1	1	yes	maximal, characteristic
2	C₅×#878	C₂	1	1	yes	maximal, characteristic
3	C₃×#4023	C₂	1	1	yes	maximal, characteristic
4	#91576	C₂	1	1	yes	maximal, characteristic
5	#91575	C₂	1	1	yes	maximal, characteristic
6	C₂×#6668	C₂	1	1	yes	maximal, characteristic
7	#91528	C₂	1	1	yes	maximal, characteristic
8	#91468	C₂	1	1	yes	maximal, characteristic, complemented
9	#91378	C₂	1	1	yes	maximal, characteristic, complemented
10	#91378	C₂	1	1	yes	maximal, characteristic, complemented
11	#91376	C₂	1	1	yes	maximal, characteristic, complemented
12	#91335	C₂	1	1	yes	maximal, characteristic, complemented
13	#91334	C₂	1	1	yes	maximal, characteristic, complemented
14	#91329	C₂	1	1	yes	maximal, characteristic, complemented
15	#91328	C₂	1	1	yes	maximal, characteristic, complemented
16	#65957	—	3	1	no	maximal, Hall
17	C₂²×Dic₁₅	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
19	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
21	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
22	C₂×#1204	C₂²	1	1	yes	characteristic
23	C₂×#1204	C₂²	1	1	yes	characteristic
24	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	#6719	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
27	#6712	C₂²	1	1	yes	characteristic
28	#6712	C₂²	1	1	yes	characteristic
29	C₅×#217	C₂²	1	1	yes	characteristic
30	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
31	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
33	C₃×C₂²⋊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	C₃×C₄×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	#6668	C₂²	1	1	yes	characteristic
47	#6668	C₂²	1	1	yes	characteristic
48	#6668	C₂²	1	1	yes	characteristic
49	#6668	C₂²	1	1	yes	characteristic
50	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
51	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
52	#5519	—	5	1	no	maximal, Hall
53	C₅×C₂²⋊Q₈	—	3	1	no	—
54	#4090	—	3	1	no	—
55	#4090	—	3	1	no	—
56	#4070	—	3	1	no	—
57	C₂×#683	—	3	1	no	—
58	#4035	—	3	1	no	—
59	#4035	—	3	1	no	—
60	#4035	—	3	1	no	—
61	#4035	—	3	1	no	—
62	#4034	—	3	1	no	—
63	#4034	—	3	1	no	—
64	#4024	—	3	1	no	—
65	#4024	—	3	1	no	—
66	#4023	—	3	1	no	—
67	#4023	D₃	1	1	yes	characteristic
68	C₂³×C₃×C₅	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×Dic₁₅	—	4	1	no	—
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	Dic₃₀	—	4	1	no	—
77	C₂×C₅×Dic₃	—	4	1	no	—
78	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
79	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
80	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
81	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
82	C₅×Dic₆	—	4	1	no	—
83	C₂×C₃×Dic₅	Q₈	1	4	yes	complemented
84	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
85	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
86	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
87	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
88	#1204	—	4	1	no	—
89	#1204	—	4	1	no	—
90	C₃×#117	—	5	1	no	—
91	#926	—	5	1	no	—
92	C₂×C₄⋊Dic₃	—	5	1	no	—
93	#890	—	5	1	no	—
94	#890	—	5	1	no	—
95	#888	—	5	1	no	—
96	#888	—	5	1	no	—
97	#878	—	5	1	no	—
98	#878	—	5	1	no	—
99	#878	—	5	1	no	—
100	#878	D₅	1	1	yes	characteristic
101	#870	—	5	1	no	—
102	#870	—	5	1	no	—
103	#869	—	5	1	no	—
104	#869	—	5	1	no	—
105	C₂×C₅×Q₈	—	3	1	no	—
106	C₂²×C₄×C₅	—	3	1	no	—
107	C₂²×Dic₅	—	3	1	no	—
108	C₂²×Dic₅	C₂×D₃	1	1	yes	characteristic
109	C₂×Dic₁₀	—	3	1	no	—
110	C₂×Dic₁₀	—	3	1	no	—
111	C₂×Dic₁₀	—	3	1	no	—
112	C₅×C₄⋊C₄	—	3	1	no	—
113	C₅×C₄⋊C₄	—	3	1	no	—
114	C₅×C₄⋊C₄	—	3	1	no	—
115	C₅×C₂²⋊C₄	—	3	1	no	—
116	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
117	C₂²⋊Dic₅	—	3	1	no	—
118	C₂²⋊Dic₅	C₂×D₃	1	1	yes	characteristic
119	C₄⋊Dic₅	—	3	1	no	—
120	C₄⋊Dic₅	—	3	1	no	—
121	C₄⋊Dic₅	—	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	—	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	C₂×D₃	1	1	yes	characteristic
135	#683	C₂×D₃	1	1	yes	characteristic
136	C₄×Dic₅	—	3	1	no	—
137	C₄×Dic₅	—	3	1	no	—
138	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
139	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
140	#544	—	15	1	no	Sylow
141	C₂²×C₃×C₅	—	2	1	no	—
142	C₂²×C₃×C₅	—	2	1	no	—
143	C₂²×C₃×C₅	C₂×Q₈	1	2	yes	—
144	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
145	C₃×C₄×C₅	—	2	1	no	—
146	C₃×C₄×C₅	—	2	1	no	—
147	Dic₁₅	—	2	1	no	—
148	Dic₁₅	—	2	1	no	—
149	Dic₁₅	—	2	1	no	—
150	Dic₁₅	—	2	1	no	—
151	C₃×Dic₅	—	2	1	no	—
152	C₃×Dic₅	—	2	1	no	—
153	C₃×Dic₅	C₂×Q₈	1	2	yes	—
154	C₃×Dic₅	C₂×Q₈	1	2	yes	—
155	C₅×Dic₃	—	2	1	no	—
156	C₅×Dic₃	—	2	1	no	—
157	C₅×Dic₃	—	2	1	no	—
158	C₅×Dic₃	—	2	1	no	—
159	C₂²×C₃×C₄	—	5	1	no	—
160	C₂²×Dic₃	—	5	1	no	—
161	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
162	C₂×Dic₆	—	5	1	no	—
163	C₂×Dic₆	—	5	1	no	—
164	C₂×Dic₆	—	5	1	no	—
165	C₂×Dic₆	C₂×D₅	1	1	yes	characteristic
166	C₃×C₄⋊C₄	—	5	1	no	—
167	C₃×C₄⋊C₄	—	5	1	no	—
168	C₃×C₂²⋊C₄	—	5	1	no	—
169	C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
170	C₃×C₄²	—	5	1	no	—
171	C₃×C₄²	—	5	1	no	—
172	#217	—	5	1	no	—
173	#217	C₂×D₅	1	1	yes	characteristic
174	C₄⋊Dic₃	—	5	1	no	—
175	C₄⋊Dic₃	—	5	1	no	—
176	C₄⋊Dic₃	—	5	1	no	—
177	C₄⋊Dic₃	—	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	C₄⋊Dic₃	—	5	1	no	—
183	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
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	C₂×D₅	1	1	yes	characteristic
191	Dic₃⋊C₄	C₂×D₅	1	1	yes	characteristic
192	C₄×Dic₃	—	5	1	no	—
193	C₄×Dic₃	—	5	1	no	—
194	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
195	C₅×Q₈	—	12	1	no	—
196	C₂×C₄×C₅	—	3	1	no	—
197	C₂×C₄×C₅	—	3	1	no	—
198	C₂×C₄×C₅	—	3	1	no	—
199	C₂×C₄×C₅	—	3	1	no	—
200	C₂×C₄×C₅	—	12	1	no	—
201	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
202	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
203	C₂×Dic₅	—	3	1	no	—
204	C₂×Dic₅	—	3	1	no	—
205	C₂×Dic₅	—	3	1	no	—
206	C₂×Dic₅	—	3	1	no	—
207	C₂×Dic₅	—	12	1	no	—
208	C₂×Dic₅	Dic₆	1	4	yes	complemented
209	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
210	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
211	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
212	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
213	Dic₁₀	—	12	1	no	—
214	Dic₁₀	—	12	1	no	—
215	Dic₁₀	—	12	1	no	—
216	C₄⋊Q₈	—	15	1	no	—
217	C₄⋊Q₈	—	15	1	no	—
218	C₄⋊Q₈	—	15	1	no	—
219	C₄⋊Q₈	—	15	1	no	—
220	#125	—	15	1	no	—
221	#125	—	15	1	no	—
222	#125	—	15	1	no	—
223	#125	—	15	1	no	—
224	C₂²⋊Q₈	—	3	1	no	—
225	C₂²⋊Q₈	—	15	1	no	—
226	C₂²⋊Q₈	—	15	1	no	—
227	C₂²⋊Q₈	—	15	1	no	—
228	#117	—	5	1	no	—
229	#117	—	15	1	no	—
230	C₂×C₄⋊C₄	—	15	1	no	—
231	C₂×C₃×C₅	—	2	2	no	—
232	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
233	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
234	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
235	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
236	C₂×C₃×C₄	—	5	4	no	—
237	C₂×C₃×C₄	—	5	1	no	—
238	C₂×C₃×C₄	—	5	1	no	—
239	C₂×C₃×C₄	—	5	1	no	—
240	C₂×C₃×C₄	—	5	1	no	—
241	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
243	C₂×Dic₃	—	4	1	no	—
244	C₂×Dic₃	—	5	1	no	—
245	C₂×Dic₃	—	5	1	no	—
246	C₂×Dic₃	—	5	1	no	—
247	C₂×Dic₃	—	5	1	no	—
248	C₂×Dic₃	—	20	1	no	—
249	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
250	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
251	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
252	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
253	Dic₆	—	4	1	no	—
254	Dic₆	—	20	1	no	—
255	Dic₆	—	20	1	no	—
256	Dic₆	—	20	1	no	—
257	C₂²×C₅	—	2	1	no	—
258	C₂²×C₅	—	2	1	no	—
259	C₂²×C₅	C₂×Dic₆	1	2	yes	—
260	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
261	C₄×C₅	—	2	1	no	—
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	—	6	1	no	—
264	C₄×C₅	—	6	1	no	—
265	C₄×C₅	—	6	1	no	—
266	C₄×C₅	—	6	1	no	—
267	Dic₅	—	2	1	no	—
268	Dic₅	—	2	1	no	—
269	Dic₅	—	6	1	no	—
270	Dic₅	—	6	1	no	—
271	Dic₅	—	6	1	no	—
272	Dic₅	—	6	1	no	—
273	Dic₅	C₂×Dic₆	1	2	yes	—
274	Dic₅	C₂×Dic₆	1	2	yes	—
275	C₂×Q₈	—	3	1	no	—
276	C₂×Q₈	—	15	1	no	—
277	C₂×Q₈	—	15	1	no	—
278	C₂×Q₈	—	15	1	no	—
279	C₂²×C₄	—	3	1	no	—
280	C₂²×C₄	—	5	1	no	—
281	C₂²×C₄	—	15	1	no	—
282	C₄⋊C₄	—	3	1	no	—
283	C₄⋊C₄	—	3	1	no	—
284	C₄⋊C₄	—	3	1	no	—
285	C₄⋊C₄	—	5	1	no	—
286	C₄⋊C₄	—	5	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	—	15	1	no	—
297	C₄⋊C₄	—	15	1	no	—
298	C₄⋊C₄	—	15	1	no	—
299	C₄⋊C₄	—	15	1	no	—
300	C₄⋊C₄	—	15	1	no	—
301	C₄⋊C₄	—	15	1	no	—
302	C₂²⋊C₄	—	3	1	no	—
303	C₂²⋊C₄	—	5	1	no	—
304	C₂²⋊C₄	—	15	1	no	—
305	C₂²⋊C₄	D₃×D₅	1	1	yes	characteristic
306	C₄²	—	5	1	no	—
307	C₄²	—	5	1	no	—
308	C₄²	—	15	1	no	—
309	C₄²	—	15	1	no	—
310	C₃×C₅	#544	1	1	yes	characteristic, complemented, Hall
311	C₂²×C₃	—	2	1	no	—
312	C₂²×C₃	—	2	1	no	—
313	C₂²×C₃	Q₈×D₅	1	2	yes	—
314	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
315	C₃×C₄	—	2	1	no	—
316	C₃×C₄	—	2	1	no	—
317	C₃×C₄	—	5	2	no	—
318	C₃×C₄	—	5	2	no	—
319	C₃×C₄	—	10	1	no	—
320	C₃×C₄	—	10	1	no	—
321	Dic₃	—	2	1	no	—
322	Dic₃	—	2	1	no	—
323	Dic₃	—	2	1	no	—
324	Dic₃	—	2	1	no	—
325	Dic₃	—	10	1	no	—
326	Dic₃	—	10	1	no	—
327	Dic₃	—	10	1	no	—
328	Dic₃	—	10	1	no	—
329	C₂×C₅	—	2	2	no	—
330	C₂×C₅	C₂²×Dic₆	1	1	yes	characteristic
331	C₂×C₅	#1009	1	1	yes	characteristic
332	C₂×C₅	#1010	1	1	yes	characteristic
333	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
334	Q₈	—	12	1	no	—
335	Q₈	—	60	1	no	—
336	Q₈	—	60	1	no	—
337	Q₈	—	60	1	no	—
338	C₂×C₄	—	3	1	no	—
339	C₂×C₄	—	3	1	no	—
340	C₂×C₄	—	3	1	no	—
341	C₂×C₄	—	3	1	no	—
342	C₂×C₄	—	5	4	no	—
343	C₂×C₄	—	5	1	no	—
344	C₂×C₄	—	5	1	no	—
345	C₂×C₄	—	5	1	no	—
346	C₂×C₄	—	5	1	no	—
347	C₂×C₄	—	12	1	no	—
348	C₂×C₄	—	15	1	no	—
349	C₂×C₄	—	15	1	no	—
350	C₂×C₄	—	15	1	no	—
351	C₂×C₄	—	15	1	no	—
352	C₂×C₄	—	60	1	no	—
353	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
354	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
355	C₂×C₃	—	2	2	no	—
356	C₂×C₃	#4144	1	1	yes	characteristic
357	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
358	C₂×C₃	#4150	1	1	yes	characteristic
359	C₅	#5519	1	1	yes	characteristic, complemented, Sylow
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	D₅×Dic₆	1	2	yes	—
363	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	5	2	no	—
367	C₄	—	5	2	no	—
368	C₄	—	6	1	no	—
369	C₄	—	6	1	no	—
370	C₄	—	6	1	no	—
371	C₄	—	6	1	no	—
372	C₄	—	10	1	no	—
373	C₄	—	10	1	no	—
374	C₄	—	30	1	no	—
375	C₄	—	30	1	no	—
376	C₄	—	30	1	no	—
377	C₄	—	30	1	no	—
378	C₃	#65957	1	1	yes	characteristic, complemented, Sylow
379	C₂	—	2	2	no	—
380	C₂	C₂×D₅×Dic₆	1	1	yes	characteristic
381	C₂	#92020	1	1	yes	characteristic
382	C₂	#92027	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×C₂²⋊C₄	C₂²
prime radical	C₂³×C₃×C₅	C₂³
hypercenter	C₂²⋊C₄	D₃×D₅
hypocenter	C₃×C₅	#544
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
8. #91468	C₂	semidirect
9. #91378	C₂	semidirect
10. #91378	C₂	semidirect
11. #91376	C₂	semidirect
12. #91335	C₂	semidirect
13. #91334	C₂	semidirect
14. #91329	C₂	semidirect
15. #91328	C₂	semidirect
83. C₂×C₃×Dic₅	Q₈	semidirect
208. C₂×Dic₅	Dic₆	semidirect
310. C₃×C₅	#544	semidirect
359. C₅	#5519	semidirect
378. C₃	#65957	semidirect

Maximal subgroups

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

subgroup	index
#91767	2
C₅×#878	2
C₃×#4023	2
#91576	2
#91575	2
C₂×#6668	2
#91528	2
#91468	2
#91378	2
#91376	2
#91335	2
#91334	2
#91329	2
#91328	2
#65957	3
#5519	5