Subgroups — #11754382

Order 960.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#6769	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#67	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4070	C₂	1	1	yes	maximal, characteristic, complemented
5	#91573	C₂	1	2	yes	maximal, complemented
6	#91561	C₂	1	2	yes	maximal, complemented
7	#91460	C₂	1	1	yes	maximal, characteristic, complemented
8	#91433	C₂	1	2	yes	maximal, complemented
9	#91428	C₂	1	1	yes	maximal, characteristic, complemented
10	#91420	C₂	1	2	yes	maximal, complemented
11	#91374	C₂	1	1	yes	maximal, characteristic, complemented
12	#65884	—	3	1	no	maximal, Hall
13	C₂³×D₁₅	C₂²	1	2	yes	—
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂×D₆₀	—	2	4	no	—
16	C₂×D₆₀	C₂²	1	2	yes	—
17	C₂×D₆₀	C₂²	1	1	yes	characteristic
18	C₂×D₆₀	C₂²	1	1	yes	characteristic
19	C₂×C₅×#67	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
22	C₂×#1203	C₂²	1	2	yes	complemented
23	C₂×#1203	C₂²	1	2	yes	complemented
24	C₂×#1200	C₂²	1	2	yes	complemented
25	C₂×#1200	C₂²	1	2	yes	complemented
26	#6769	—	2	4	no	—
27	C₅×#217	C₂²	1	1	yes	characteristic
28	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic, complemented
29	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic, complemented
30	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
31	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
32	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
33	C₃×#683	C₂²	1	2	yes	complemented
34	#6667	C₂²	1	2	yes	complemented
35	#6667	C₂²	1	2	yes	complemented
36	#6667	C₂²	1	2	yes	complemented
37	#6667	C₂²	1	2	yes	complemented
38	#5630	—	5	1	no	maximal, Hall
39	C₂×#709	—	3	1	no	—
40	C₂²×D₂₀	—	3	1	no	—
41	C₄×C₅×D₄	—	3	1	no	—
42	#4083	—	3	2	no	—
43	#4076	—	3	1	no	—
44	#4070	D₃	1	1	yes	characteristic, complemented
45	#4040	—	3	2	no	—
46	#4030	—	3	2	no	—
47	D₅×C₂²⋊C₄	—	3	2	no	—
48	#4021	—	3	1	no	—
49	C₄×D₂₀	—	3	1	no	—
50	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂²×D₁₅	—	2	4	no	—
52	C₂²×D₁₅	—	2	4	no	—
53	C₂²×D₁₅	—	4	2	no	—
54	C₂²×D₁₅	C₂³	1	2	yes	—
55	C₂²×D₁₅	C₂³	1	2	yes	—
56	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
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	D₆₀	—	2	4	no	—
61	D₆₀	—	2	2	no	—
62	D₆₀	D₄	1	4	yes	complemented
63	C₅×#67	—	2	2	no	—
64	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
65	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
66	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
67	C₄×C₅×D₃	—	2	2	no	—
68	C₂×C₃×Dic₅	C₂³	1	2	yes	—
69	C₂×C₃×Dic₅	C₂³	1	2	yes	—
70	C₃×Dic₁₀	—	2	2	no	—
71	#1203	—	2	4	no	—
72	#1203	—	4	2	no	—
73	#1200	—	2	4	no	—
74	#1200	—	4	2	no	—
75	C₂×#239	—	5	1	no	—
76	C₂²×D₁₂	—	5	1	no	—
77	C₃×C₂²⋊Q₈	—	5	1	no	—
78	#947	—	5	1	no	—
79	C₄×#67	D₅	1	1	yes	characteristic, complemented
80	#895	—	5	1	no	—
81	#895	—	5	2	no	—
82	#894	—	5	2	no	—
83	#893	—	5	1	no	—
84	#882	—	5	2	no	—
85	D₃×C₂²⋊C₄	—	5	2	no	—
86	C₂³×D₅	—	3	2	no	—
87	C₂×C₅×D₄	—	3	1	no	—
88	C₂²×C₄×C₅	—	3	1	no	—
89	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
90	C₂×C₂²⋊D₅	—	3	2	no	—
91	C₂×C₂²⋊D₅	—	3	2	no	—
92	#709	—	6	4	no	—
93	C₂×D₂₀	—	3	1	no	—
94	C₂×D₂₀	—	3	1	no	—
95	C₂×D₂₀	—	3	2	no	—
96	C₂×D₂₀	—	6	4	no	—
97	C₂×C₄×D₅	—	3	2	no	—
98	C₂×C₄×D₅	—	3	2	no	—
99	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic, complemented
100	C₅×C₄⋊C₄	—	3	1	no	—
101	C₅×C₂²⋊C₄	—	3	1	no	—
102	C₅×C₂²⋊C₄	—	3	1	no	—
103	C₄²×C₅	—	3	1	no	—
104	C₂²⋊Dic₅	C₂×D₃	1	2	yes	—
105	#685	—	3	2	no	—
106	#685	—	3	2	no	—
107	#685	—	3	2	no	—
108	#685	—	3	2	no	—
109	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic, complemented
110	#683	C₂×D₃	1	2	yes	complemented
111	#546	—	15	1	no	Sylow
112	C₂²×C₃×C₅	—	2	1	no	—
113	C₂²×C₃×C₅	—	2	1	no	—
114	C₂²×C₃×C₅	#41	1	2	yes	complemented
115	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
116	C₂×D₁₅	—	2	2	no	—
117	C₂×D₁₅	—	2	2	no	—
118	C₂×D₁₅	—	2	2	no	—
119	C₂×D₁₅	—	2	2	no	—
120	C₂×D₁₅	—	2	2	no	—
121	C₂×D₁₅	—	4	8	no	—
122	C₂×D₁₅	C₂×D₄	1	4	yes	—
123	C₂×C₅×D₃	—	2	1	no	—
124	C₂×C₅×D₃	—	2	1	no	—
125	C₂×C₅×D₃	—	2	1	no	—
126	C₃×C₄×C₅	—	2	1	no	—
127	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
128	C₃×Dic₅	—	2	2	no	—
129	C₃×Dic₅	—	2	2	no	—
130	C₅×Dic₃	—	2	1	no	—
131	C₅×Dic₃	—	2	1	no	—
132	C₅×Dic₃	—	2	1	no	—
133	C₂³×D₃	—	5	2	no	—
134	C₂×C₃×Q₈	—	5	1	no	—
135	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
136	C₂×#67	C₂×D₅	1	1	yes	characteristic
137	#239	—	10	4	no	—
138	C₂×D₁₂	—	5	1	no	—
139	C₂×D₁₂	—	5	2	no	—
140	C₂×D₁₂	—	5	2	no	—
141	C₂×D₁₂	—	5	1	no	—
142	C₂×D₁₂	—	5	2	no	—
143	C₂×D₁₂	—	10	4	no	—
144	C₂×C₄×D₃	—	5	2	no	—
145	C₂×C₄×D₃	—	5	2	no	—
146	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic, complemented
147	C₃×C₄⋊C₄	—	5	2	no	—
148	C₃×C₄⋊C₄	—	5	1	no	—
149	C₃×C₂²⋊C₄	—	5	2	no	—
150	#217	C₂×D₅	1	1	yes	characteristic
151	(C₂×D₃)⋊C₄	—	5	2	no	—
152	(C₂×D₃)⋊C₄	—	5	2	no	—
153	(C₂×D₃)⋊C₄	—	5	2	no	—
154	(C₂×D₃)⋊C₄	—	5	2	no	—
155	(C₂×D₃)⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
156	Dic₃⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
157	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic, complemented
158	C₂³×C₅	—	3	1	no	—
159	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
160	C₂²×D₅	—	3	2	no	—
161	C₂²×D₅	—	3	2	no	—
162	C₂²×D₅	—	6	4	no	—
163	C₂²×D₅	—	6	4	no	—
164	C₂²×D₅	—	12	2	no	—
165	C₅×D₄	—	6	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	3	1	no	—
168	C₂×C₄×C₅	—	3	1	no	—
169	C₂×C₄×C₅	—	3	1	no	—
170	C₂×C₄×C₅	—	6	2	no	—
171	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
173	C₂²⋊D₅	—	6	4	no	—
174	C₂²⋊D₅	—	12	2	no	—
175	C₂×Dic₅	C₂²×D₃	1	2	yes	—
176	C₂×Dic₅	C₂²×D₃	1	2	yes	—
177	D₂₀	—	3	4	no	—
178	D₂₀	—	6	4	no	—
179	D₂₀	—	6	2	no	—
180	C₄×D₅	—	6	4	no	—
181	C₄×D₅	—	12	2	no	—
182	Dic₁₀	—	2	2	no	—
183	C₂×#41	—	15	1	no	—
184	C₂²×D₄	—	15	1	no	—
185	#124	—	15	1	no	—
186	#123	—	15	2	no	—
187	C₂²⋊Q₈	—	5	1	no	—
188	#121	—	15	2	no	—
189	#121	—	15	1	no	—
190	#120	—	15	2	no	—
191	C₄×D₄	—	3	1	no	—
192	C₄×D₄	—	15	1	no	—
193	C₂×C₂²⋊C₄	—	15	2	no	—
194	C₂×C₃×C₅	—	2	2	no	—
195	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
197	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
198	D₁₅	—	2	4	no	—
199	D₁₅	—	4	2	no	—
200	C₅×D₃	—	4	1	no	—
201	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
202	C₂²×D₃	—	5	2	no	—
203	C₂²×D₃	—	5	2	no	—
204	C₂²×D₃	—	10	4	no	—
205	C₂²×D₃	—	10	4	no	—
206	C₂²×D₃	—	20	2	no	—
207	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
208	C₃×Q₈	—	10	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	5	2	no	—
211	C₂×C₃×C₄	—	5	2	no	—
212	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
214	#67	—	2	2	no	—
215	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
216	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
217	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
218	D₁₂	—	5	4	no	—
219	D₁₂	—	10	4	no	—
220	D₁₂	—	10	4	no	—
221	D₁₂	—	10	2	no	—
222	D₁₂	—	20	2	no	—
223	C₄×D₃	—	2	2	no	—
224	C₄×D₃	—	10	4	no	—
225	C₄×D₃	—	20	2	no	—
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	—	6	1	no	—
229	C₂²×C₅	—	6	1	no	—
230	C₂²×C₅	—	6	1	no	—
231	C₂²×C₅	#239	1	2	yes	complemented
232	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
233	C₂×D₅	—	3	4	no	—
234	C₂×D₅	—	6	2	no	—
235	C₂×D₅	—	6	2	no	—
236	C₂×D₅	—	6	2	no	—
237	C₂×D₅	—	6	2	no	—
238	C₂×D₅	—	6	2	no	—
239	C₂×D₅	—	12	8	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	6	1	no	—
242	C₄×C₅	—	6	1	no	—
243	C₄×C₅	—	6	1	no	—
244	C₄×C₅	D₃×D₄	1	2	yes	—
245	Dic₅	—	2	2	no	—
246	Dic₅	—	2	2	no	—
247	C₂⁴	—	15	2	no	—
248	#41	—	30	4	no	—
249	C₂×Q₈	—	5	1	no	—
250	C₂×D₄	—	3	1	no	—
251	C₂×D₄	—	15	2	no	—
252	C₂×D₄	—	15	2	no	—
253	C₂×D₄	—	15	1	no	—
254	C₂×D₄	—	15	2	no	—
255	C₂×D₄	—	15	1	no	—
256	C₂×D₄	—	30	4	no	—
257	C₂²×C₄	—	3	1	no	—
258	C₂²×C₄	—	15	2	no	—
259	C₂²×C₄	—	15	2	no	—
260	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
261	C₄⋊C₄	—	3	1	no	—
262	C₄⋊C₄	—	5	1	no	—
263	C₄⋊C₄	—	5	2	no	—
264	C₂²⋊C₄	—	3	1	no	—
265	C₂²⋊C₄	—	3	1	no	—
266	C₂²⋊C₄	—	5	2	no	—
267	C₂²⋊C₄	—	15	2	no	—
268	C₂²⋊C₄	—	15	2	no	—
269	C₂²⋊C₄	—	15	2	no	—
270	C₂²⋊C₄	—	15	2	no	—
271	C₄²	—	3	1	no	—
272	C₃×C₅	#546	1	1	yes	characteristic, complemented, Hall
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	#709	1	2	yes	complemented
276	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
277	C₂×D₃	—	2	1	no	—
278	C₂×D₃	—	2	1	no	—
279	C₂×D₃	—	2	1	no	—
280	C₂×D₃	—	5	4	no	—
281	C₂×D₃	—	10	2	no	—
282	C₂×D₃	—	10	2	no	—
283	C₂×D₃	—	10	2	no	—
284	C₂×D₃	—	10	2	no	—
285	C₂×D₃	—	10	2	no	—
286	C₂×D₃	—	20	8	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	10	2	no	—
289	C₃×C₄	—	10	2	no	—
290	C₃×C₄	D₄×D₅	1	2	yes	—
291	Dic₃	—	2	1	no	—
292	Dic₃	—	2	1	no	—
293	Dic₃	—	2	1	no	—
294	C₂×C₅	—	2	2	no	—
295	C₂×C₅	—	12	1	no	—
296	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
297	C₂×C₅	C₂×#239	1	1	yes	characteristic
298	C₂×C₅	#1009	1	1	yes	characteristic
299	D₅	—	6	4	no	—
300	D₅	—	12	2	no	—
301	C₂³	—	3	1	no	—
302	C₂³	—	15	2	no	—
303	C₂³	—	15	2	no	—
304	C₂³	—	30	4	no	—
305	C₂³	—	30	4	no	—
306	C₂³	—	60	2	no	—
307	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
308	Q₈	—	10	2	no	—
309	D₄	—	6	2	no	—
310	D₄	—	15	4	no	—
311	D₄	—	30	4	no	—
312	D₄	—	30	4	no	—
313	D₄	—	30	2	no	—
314	D₄	—	60	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	3	1	no	—
317	C₂×C₄	—	3	1	no	—
318	C₂×C₄	—	3	1	no	—
319	C₂×C₄	—	5	2	no	—
320	C₂×C₄	—	5	2	no	—
321	C₂×C₄	—	6	2	no	—
322	C₂×C₄	—	30	4	no	—
323	C₂×C₄	—	60	2	no	—
324	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
325	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
326	C₂×C₃	—	2	2	no	—
327	C₂×C₃	C₂×#709	1	1	yes	characteristic
328	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
329	C₂×C₃	#4149	1	1	yes	characteristic
330	D₃	—	4	1	no	—
331	D₃	—	10	4	no	—
332	D₃	—	20	2	no	—
333	C₅	#5630	1	1	yes	characteristic, complemented, Sylow
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	6	1	no	—
337	C₂²	—	6	1	no	—
338	C₂²	—	6	1	no	—
339	C₂²	—	15	4	no	—
340	C₂²	—	30	2	no	—
341	C₂²	—	30	2	no	—
342	C₂²	—	30	2	no	—
343	C₂²	—	30	2	no	—
344	C₂²	—	30	2	no	—
345	C₂²	—	60	8	no	—
346	C₂²	#6769	1	2	yes	complemented
347	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	6	1	no	—
350	C₄	—	6	1	no	—
351	C₄	—	6	1	no	—
352	C₄	—	10	2	no	—
353	C₄	—	10	2	no	—
354	C₄	#6790	1	2	yes	—
355	C₃	#65884	1	1	yes	characteristic, complemented, Sylow
356	C₂	—	2	2	no	—
357	C₂	—	12	1	no	—
358	C₂	—	30	4	no	—
359	C₂	—	60	2	no	—
360	C₂	C₂×#6769	1	1	yes	characteristic
361	C₂	#92019	1	1	yes	characteristic
362	C₂	C₂×#6790	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	#546
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×D₆₀	C₂	semidirect
2. C₂×#6769	C₂	semidirect
3. C₄×C₅×#67	C₂	semidirect
4. C₃×#4070	C₂	semidirect
5. #91573	C₂	semidirect
6. #91561	C₂	semidirect
7. #91460	C₂	semidirect
8. #91433	C₂	semidirect
9. #91428	C₂	semidirect
10. #91420	C₂	semidirect
11. #91374	C₂	semidirect
20. C₂×C₄×C₅×D₃	C₂²	semidirect
21. C₂×C₃×Dic₁₀	C₂²	semidirect
22. C₂×#1203	C₂²	semidirect
23. C₂×#1203	C₂²	semidirect
24. C₂×#1200	C₂²	semidirect
25. C₂×#1200	C₂²	semidirect
28. C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
29. C₅×Dic₃⋊C₄	C₂²	semidirect
30. C₄×C₅×Dic₃	C₂²	semidirect
32. C₃×C₄⋊Dic₅	C₂²	semidirect
33. C₃×#683	C₂²	semidirect
34. #6667	C₂²	semidirect
35. #6667	C₂²	semidirect
36. #6667	C₂²	semidirect
37. #6667	C₂²	semidirect
44. #4070	D₃	semidirect
62. D₆₀	D₄	semidirect
79. C₄×#67	D₅	semidirect
99. C₂×Dic₁₀	C₂×D₃	semidirect
109. C₄⋊Dic₅	C₂×D₃	semidirect
110. #683	C₂×D₃	semidirect
114. C₂²×C₃×C₅	#41	semidirect
146. C₂×C₄×D₃	C₂×D₅	semidirect
155. (C₂×D₃)⋊C₄	C₂×D₅	semidirect
156. Dic₃⋊C₄	C₂×D₅	semidirect
157. C₄×Dic₃	C₂×D₅	semidirect
231. C₂²×C₅	#239	semidirect
272. C₃×C₅	#546	semidirect
275. C₂²×C₃	#709	semidirect
333. C₅	#5630	semidirect
346. C₂²	#6769	semidirect
355. C₃	#65884	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₆₀	2
C₂×#6769	2
C₄×C₅×#67	2
C₃×#4070	2
#91573	2
#91561	2
#91460	2
#91433	2
#91428	2
#91420	2
#91374	2
#65884	3
#5630	5