Subgroups — #11753569

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×#888	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4044	C₂	1	1	yes	maximal, characteristic
4	#91421	C₂	1	1	yes	maximal, characteristic
5	#91421	C₂	1	1	yes	maximal, characteristic
6	#91393	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	#91366	C₂	1	1	yes	maximal, characteristic
10	#91362	C₂	1	1	yes	maximal, characteristic
11	#91361	C₂	1	1	yes	maximal, characteristic
12	#91344	C₂	1	1	yes	maximal, characteristic, complemented
13	#91339	C₂	1	1	yes	maximal, characteristic, complemented
14	#91335	C₂	1	1	yes	maximal, characteristic, complemented
15	Dic₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
16	#66065	—	3	1	no	maximal, Hall
17	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
18	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
19	C₂×D₅×Dic₃	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×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
29	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
30	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
31	C₃×#685	C₂²	1	1	yes	characteristic
32	C₃×#685	C₂²	1	1	yes	characteristic
33	C₃×#685	C₂²	1	1	yes	characteristic
34	C₃×C₄⋊Dic₅	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	Dic₃⋊Dic₅	C₂²	1	1	yes	characteristic
41	#6669	C₂²	1	1	yes	characteristic
42	#6669	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	#6665	C₂²	1	1	yes	characteristic
47	#6665	C₂²	1	1	yes	characteristic
48	#6665	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	#5701	—	5	1	no	maximal, Hall
53	C₅×C₄⋊Q₈	—	3	1	no	—
54	#4092	—	3	1	no	—
55	#4092	—	3	1	no	—
56	Q₈×Dic₅	—	3	1	no	—
57	Q₈×Dic₅	—	3	1	no	—
58	#4044	—	3	1	no	—
59	#4044	—	3	1	no	—
60	#4044	—	3	1	no	—
61	#4044	D₃	1	1	yes	characteristic
62	#4035	—	3	1	no	—
63	#4035	—	3	1	no	—
64	#4035	—	3	1	no	—
65	#4035	—	3	1	no	—
66	#4018	—	3	1	no	—
67	#4016	—	3	1	no	—
68	C₂²×C₃×D₅	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×C₅×Dic₃	C₂³	1	1	yes	characteristic
80	C₅×Dic₆	—	2	2	no	—
81	C₅×Dic₆	—	2	2	no	—
82	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
83	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
84	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
85	D₅×Dic₃	—	2	2	no	—
86	C₃×#126	—	5	1	no	—
87	#890	—	5	1	no	—
88	#889	—	5	1	no	—
89	#889	—	5	1	no	—
90	#889	—	5	1	no	—
91	#888	D₅	1	1	yes	characteristic, complemented
92	#887	—	5	1	no	—
93	#879	—	5	1	no	—
94	#879	—	5	1	no	—
95	#879	—	5	1	no	—
96	#878	—	5	1	no	—
97	#878	—	5	1	no	—
98	#877	—	5	1	no	—
99	#870	—	5	1	no	—
100	C₄×Dic₆	—	5	1	no	—
101	C₂×C₅×Q₈	—	3	1	no	—
102	C₂×C₅×Q₈	—	3	1	no	—
103	C₂×C₄×D₅	—	3	1	no	—
104	C₅×C₄⋊C₄	—	3	1	no	—
105	C₅×C₄⋊C₄	—	3	1	no	—
106	C₅×C₄⋊C₄	—	3	1	no	—
107	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
108	C₄²×C₅	—	3	1	no	—
109	#685	—	3	1	no	—
110	#685	—	3	1	no	—
111	#685	—	3	1	no	—
112	#685	C₂×D₃	1	1	yes	characteristic
113	#685	C₂×D₃	1	1	yes	characteristic
114	#685	C₂×D₃	1	1	yes	characteristic
115	C₄⋊Dic₅	—	3	1	no	—
116	C₄⋊Dic₅	—	3	1	no	—
117	C₄⋊Dic₅	—	3	1	no	—
118	C₄⋊Dic₅	—	3	1	no	—
119	C₄⋊Dic₅	—	3	1	no	—
120	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
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	—	3	1	no	—
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×C₃×D₅	—	2	1	no	—
139	C₂×C₃×D₅	—	2	1	no	—
140	C₂×C₃×D₅	—	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	C₅×Dic₃	—	2	1	no	—
152	C₅×Dic₃	—	2	1	no	—
153	C₅×Dic₃	—	2	1	no	—
154	C₅×Dic₃	#41	1	2	yes	—
155	C₂²×Dic₃	—	5	1	no	—
156	C₂×Dic₆	C₂×D₅	1	1	yes	characteristic
157	C₂×Dic₆	C₂×D₅	1	1	yes	characteristic
158	C₃×C₄⋊C₄	—	5	1	no	—
159	C₃×C₄⋊C₄	—	5	1	no	—
160	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
161	C₃×C₂²⋊C₄	—	5	1	no	—
162	C₃×C₂²⋊C₄	—	5	1	no	—
163	C₃×C₂²⋊C₄	—	5	1	no	—
164	C₃×C₄²	—	5	1	no	—
165	#217	—	5	1	no	—
166	#217	—	5	1	no	—
167	#217	—	5	1	no	—
168	C₄⋊Dic₃	—	5	1	no	—
169	C₄⋊Dic₃	—	5	1	no	—
170	C₄⋊Dic₃	—	5	1	no	—
171	C₄⋊Dic₃	—	5	1	no	—
172	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
173	Dic₃⋊C₄	—	5	1	no	—
174	Dic₃⋊C₄	—	5	1	no	—
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	C₂×D₅	1	1	yes	characteristic
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	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
191	C₅×Q₈	—	6	2	no	—
192	C₅×Q₈	—	6	2	no	—
193	C₂×C₄×C₅	—	3	1	no	—
194	C₂×C₄×C₅	—	3	1	no	—
195	C₂×C₄×C₅	—	3	1	no	—
196	C₂×C₄×C₅	—	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×Dic₅	—	3	1	no	—
201	C₂×Dic₅	—	3	1	no	—
202	C₂×Dic₅	—	3	1	no	—
203	C₂×Dic₅	—	3	1	no	—
204	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
205	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
206	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
207	C₄×D₅	—	6	2	no	—
208	C₄⋊Q₈	—	3	1	no	—
209	#126	—	5	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	C₃×D₅	—	4	1	no	—
227	C₂³×C₃	—	5	1	no	—
228	C₂×C₃×C₄	—	5	1	no	—
229	C₂×C₃×C₄	—	5	1	no	—
230	C₂×C₃×C₄	—	5	1	no	—
231	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
234	C₂×Dic₃	—	5	1	no	—
235	C₂×Dic₃	—	5	1	no	—
236	C₂×Dic₃	—	5	1	no	—
237	C₂×Dic₃	—	5	1	no	—
238	C₂×Dic₃	—	10	2	no	—
239	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
240	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
241	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
242	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
243	Dic₆	—	2	2	no	—
244	Dic₆	—	2	2	no	—
245	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
246	C₂×D₅	—	2	1	no	—
247	C₂×D₅	—	2	1	no	—
248	C₂×D₅	—	2	1	no	—
249	C₄×C₅	—	2	1	no	—
250	C₄×C₅	—	2	1	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	3	2	no	—
253	C₄×C₅	—	6	1	no	—
254	C₄×C₅	—	6	1	no	—
255	C₄×C₅	—	6	1	no	—
256	Dic₅	—	2	1	no	—
257	Dic₅	—	2	1	no	—
258	Dic₅	—	2	1	no	—
259	Dic₅	—	6	1	no	—
260	Dic₅	—	6	1	no	—
261	Dic₅	—	6	1	no	—
262	Dic₅	—	6	1	no	—
263	C₂×Q₈	—	3	1	no	—
264	C₂×Q₈	—	3	1	no	—
265	C₂²×C₄	—	15	1	no	—
266	C₄⋊C₄	—	3	1	no	—
267	C₄⋊C₄	—	3	1	no	—
268	C₄⋊C₄	—	3	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	—	5	1	no	—
287	C₂²⋊C₄	—	5	1	no	—
288	C₂²⋊C₄	—	5	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	—	10	1	no	—
300	C₂²×C₃	—	10	1	no	—
301	C₂²×C₃	—	10	1	no	—
302	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	2	1	no	—
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	—	10	1	no	—
307	C₃×C₄	—	10	1	no	—
308	C₃×C₄	—	10	1	no	—
309	Dic₃	—	2	1	no	—
310	Dic₃	—	2	1	no	—
311	Dic₃	—	2	1	no	—
312	Dic₃	—	10	1	no	—
313	Dic₃	—	10	1	no	—
314	Dic₃	—	10	1	no	—
315	Dic₃	—	10	1	no	—
316	Dic₃	#711	1	2	yes	—
317	C₂×C₅	D₃×#41	1	1	yes	characteristic
318	C₂×C₅	#1010	1	1	yes	characteristic
319	C₂×C₅	#1010	1	1	yes	characteristic
320	D₅	—	4	1	no	—
321	C₂³	—	5	1	no	—
322	Q₈	—	6	2	no	—
323	Q₈	—	6	2	no	—
324	C₂×C₄	—	3	1	no	—
325	C₂×C₄	—	3	1	no	—
326	C₂×C₄	—	3	1	no	—
327	C₂×C₄	—	3	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	—	20	1	no	—
340	C₂×C₃	C₂×#711	1	1	yes	characteristic
341	C₂×C₃	#4147	1	1	yes	characteristic
342	C₂×C₃	#4147	1	1	yes	characteristic
343	C₅	#5701	1	1	yes	characteristic, complemented, Sylow
344	C₂²	—	10	1	no	—
345	C₂²	—	10	1	no	—
346	C₂²	—	10	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	—	3	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	#66065	1	1	yes	characteristic, complemented, Sylow
363	C₂	—	20	1	no	—
364	C₂	#92000	1	1	yes	characteristic
365	C₂	D₃×#711	1	1	yes	characteristic
366	C₂	#92028	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×C₃×D₅	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
2. C₅×#888	C₂	semidirect
7. #91379	C₂	semidirect
8. #91378	C₂	semidirect
12. #91344	C₂	semidirect
13. #91339	C₂	semidirect
14. #91335	C₂	semidirect
15. Dic₅×Dic₆	C₂	semidirect
91. #888	D₅	semidirect
298. C₃×C₅	#541	semidirect
343. C₅	#5701	semidirect
362. C₃	#66065	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×#888	2
C₃×#4044	2
#91421	2
#91393	2
#91379	2
#91378	2
#91366	2
#91362	2
#91361	2
#91344	2
#91339	2
#91335	2
Dic₅×Dic₆	2
#66065	3
#5701	5