Subgroups — #11754526

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×C₃×C₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#6772	C₂	1	1	yes	maximal, characteristic, complemented
3	#91813	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#930	C₂	1	1	yes	maximal, characteristic, complemented
5	#91563	C₂	1	2	yes	maximal, complemented
6	Dic₅×#67	C₂	1	2	yes	maximal, complemented
7	#91449	C₂	1	1	yes	maximal, characteristic, complemented
8	#91447	C₂	1	2	yes	maximal, complemented
9	#91440	C₂	1	2	yes	maximal, complemented
10	#91360	C₂	1	1	yes	maximal, characteristic, complemented
11	#91358	C₂	1	1	yes	maximal, characteristic, complemented
12	#65940	—	3	1	no	maximal, Hall
13	C₂³×C₃×D₅	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	C₂×C₅×#67	C₂²	1	2	yes	—
17	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×D₅	—	2	4	no	—
20	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
24	C₂×#1201	C₂²	1	2	yes	—
25	C₂×#1201	C₂²	1	2	yes	—
26	C₂×D₃×Dic₅	C₂²	1	2	yes	complemented
27	#6772	—	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	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
32	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
33	Dic₃⋊Dic₅	C₂²	1	2	yes	complemented
34	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	complemented
35	#6665	C₂²	1	2	yes	—
36	#6665	C₂²	1	2	yes	—
37	Dic₃×Dic₅	C₂²	1	2	yes	complemented
38	#5791	—	5	1	no	maximal, Hall
39	C₂×#711	—	3	1	no	—
40	C₂²×C₄×D₅	D₃	1	1	yes	characteristic, complemented
41	C₅×#121	—	3	1	no	—
42	#4084	—	3	1	no	—
43	#4083	—	3	2	no	—
44	D₄×Dic₅	—	3	2	no	—
45	#4070	—	3	1	no	—
46	#4043	—	3	1	no	—
47	#4038	—	3	1	no	—
48	#4029	—	3	2	no	—
49	#4027	—	3	2	no	—
50	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂²×C₅×D₃	C₂³	1	2	yes	—
52	C₂²×C₃×D₅	—	2	2	no	—
53	C₂²×C₃×D₅	—	2	2	no	—
54	C₂²×C₃×D₅	—	2	2	no	—
55	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
56	C₂²×C₃×D₅	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	C₂×Dic₁₅	C₂³	1	2	yes	—
61	C₂×Dic₁₅	C₂³	1	2	yes	—
62	Dic₃₀	—	2	2	no	—
63	C₅×#67	—	2	4	no	—
64	C₂×C₅×Dic₃	C₂³	1	2	yes	—
65	C₅×D₁₂	—	2	2	no	—
66	C₂×C₃×Dic₅	—	2	2	no	—
67	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
68	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
69	C₃×C₄×D₅	—	2	2	no	—
70	C₃×C₄×D₅	—	2	2	no	—
71	C₃×C₄×D₅	—	2	2	no	—
72	C₃×C₄×D₅	D₄	1	4	yes	complemented
73	#1201	—	2	4	no	—
74	#1201	—	2	4	no	—
75	D₃×Dic₅	—	2	4	no	—
76	C₂³×C₃×C₄	—	5	1	no	—
77	C₂×#235	—	5	1	no	—
78	#953	—	5	2	no	—
79	#930	—	5	1	no	—
80	#930	D₅	1	1	yes	characteristic, complemented
81	#929	—	5	2	no	—
82	C₄×#67	—	5	2	no	—
83	C₄×#67	—	5	2	no	—
84	#926	—	5	1	no	—
85	#924	—	5	1	no	—
86	#924	—	5	1	no	—
87	C₂³×D₅	C₂×D₃	1	1	yes	characteristic
88	C₂×C₅×D₄	—	3	2	no	—
89	C₂×C₅×D₄	—	3	1	no	—
90	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
91	C₂×C₂²⋊D₅	—	3	2	no	—
92	C₂×C₂²⋊D₅	—	3	2	no	—
93	C₂²×Dic₅	—	3	2	no	—
94	C₂²×Dic₅	C₂×D₃	1	1	yes	characteristic
95	#711	—	6	4	no	—
96	C₂×C₄×D₅	—	2	4	no	—
97	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
98	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
99	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
100	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
101	C₂×Dic₁₀	—	3	1	no	—
102	C₅×C₄⋊C₄	—	3	1	no	—
103	C₅×C₂²⋊C₄	—	3	2	no	—
104	C₂²⋊Dic₅	—	3	2	no	—
105	C₂²⋊Dic₅	—	3	2	no	—
106	#685	—	3	2	no	—
107	#685	—	3	2	no	—
108	C₄⋊Dic₅	—	3	1	no	—
109	C₄⋊Dic₅	—	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	2	no	—
118	C₂×C₅×D₃	—	2	2	no	—
119	C₂×C₅×D₃	—	2	2	no	—
120	C₂×C₃×D₅	—	2	8	no	—
121	C₂×C₃×D₅	—	2	1	no	—
122	C₂×C₃×D₅	—	2	1	no	—
123	C₂×C₃×D₅	—	2	1	no	—
124	C₂×C₃×D₅	—	2	1	no	—
125	C₂×C₃×D₅	C₂×D₄	1	2	yes	—
126	C₂×C₃×D₅	#41	1	2	yes	complemented
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	1	no	—
132	C₃×Dic₅	C₂×D₄	1	2	yes	—
133	C₅×Dic₃	—	2	2	no	—
134	C₂⁴×C₃	—	5	1	no	—
135	C₂²×C₃×C₄	—	5	1	no	—
136	C₂²×C₃×C₄	—	5	1	no	—
137	C₂²×C₃×C₄	—	5	1	no	—
138	C₂²×C₃×C₄	—	5	1	no	—
139	C₂²×C₃×C₄	—	5	1	no	—
140	C₂²×C₃×C₄	—	10	4	no	—
141	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
142	C₂×#67	—	5	2	no	—
143	C₂×#67	—	5	2	no	—
144	C₂×#67	C₂×D₅	1	2	yes	—
145	#235	—	10	4	no	—
146	C₂×D₁₂	C₂×D₅	1	1	yes	characteristic, complemented
147	C₂×C₄×D₃	—	5	2	no	—
148	C₂×Dic₆	—	5	1	no	—
149	#217	—	5	2	no	—
150	#217	—	5	2	no	—
151	#217	—	5	2	no	—
152	(C₂×D₃)⋊C₄	—	5	2	no	—
153	(C₂×D₃)⋊C₄	C₂×D₅	1	2	yes	complemented
154	C₄⋊Dic₃	—	5	1	no	—
155	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic, complemented
156	Dic₃⋊C₄	—	5	2	no	—
157	Dic₃⋊C₄	—	5	2	no	—
158	C₄×Dic₃	—	5	2	no	—
159	C₂³×C₅	—	3	2	no	—
160	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
161	C₂²×D₅	—	2	2	no	—
162	C₂²×D₅	—	2	2	no	—
163	C₂²×D₅	—	2	2	no	—
164	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
165	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
166	C₅×D₄	—	6	4	no	—
167	C₅×D₄	—	6	2	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	3	2	no	—
170	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
172	C₂²⋊D₅	—	6	4	no	—
173	C₂²⋊D₅	—	6	4	no	—
174	C₂×Dic₅	—	2	2	no	—
175	C₂×Dic₅	—	3	2	no	—
176	C₂×Dic₅	—	3	2	no	—
177	C₂×Dic₅	—	6	4	no	—
178	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
179	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
180	C₄×D₅	—	2	2	no	—
181	C₄×D₅	—	2	2	no	—
182	C₄×D₅	—	2	2	no	—
183	C₄×D₅	#67	1	4	yes	complemented
184	Dic₁₀	—	6	2	no	—
185	C₂×#41	—	15	1	no	—
186	C₂³×C₄	—	5	1	no	—
187	#123	—	15	2	no	—
188	C₂²⋊Q₈	—	15	1	no	—
189	C₂²⋊Q₈	—	15	1	no	—
190	#121	—	3	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	—	2	2	no	—
201	C₃×D₅	—	2	2	no	—
202	C₅×D₃	—	4	2	no	—
203	C₂³×C₃	—	5	1	no	—
204	C₂³×C₃	—	5	1	no	—
205	C₂³×C₃	—	10	2	no	—
206	C₂³×C₃	—	10	2	no	—
207	C₂³×C₃	—	10	2	no	—
208	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
209	C₂²×D₃	C₂²×D₅	1	2	yes	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	5	1	no	—
212	C₂×C₃×C₄	—	5	4	no	—
213	C₂×C₃×C₄	—	5	1	no	—
214	C₂×C₃×C₄	—	10	2	no	—
215	C₂×C₃×C₄	—	10	2	no	—
216	C₂×C₃×C₄	—	10	2	no	—
217	C₂×C₃×C₄	—	10	2	no	—
218	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
220	#67	—	2	4	no	—
221	#67	—	10	4	no	—
222	#67	—	10	4	no	—
223	C₂×Dic₃	—	5	2	no	—
224	C₂×Dic₃	—	5	2	no	—
225	C₂×Dic₃	C₂²×D₅	1	2	yes	—
226	D₁₂	—	2	2	no	—
227	C₄×D₃	—	10	4	no	—
228	Dic₆	—	10	2	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	—	6	2	no	—
232	C₂²×C₅	—	6	2	no	—
233	C₂²×C₅	—	6	2	no	—
234	C₂²×C₅	#235	1	2	yes	complemented
235	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
236	C₂×D₅	—	2	8	no	—
237	C₂×D₅	—	2	1	no	—
238	C₂×D₅	—	2	1	no	—
239	C₂×D₅	—	2	1	no	—
240	C₂×D₅	—	2	1	no	—
241	C₂×D₅	#235	1	2	yes	complemented
242	C₂×D₅	C₂×#67	1	2	yes	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	6	2	no	—
245	C₄×C₅	C₂×#67	1	2	yes	—
246	Dic₅	—	2	1	no	—
247	Dic₅	—	6	2	no	—
248	Dic₅	—	6	2	no	—
249	Dic₅	C₂×#67	1	2	yes	—
250	C₂⁴	—	5	1	no	—
251	#41	—	30	4	no	—
252	C₂×Q₈	—	15	1	no	—
253	C₂×D₄	—	3	2	no	—
254	C₂×D₄	—	3	1	no	—
255	C₂×D₄	—	15	2	no	—
256	C₂×D₄	—	15	2	no	—
257	C₂²×C₄	—	5	1	no	—
258	C₂²×C₄	—	5	1	no	—
259	C₂²×C₄	—	5	1	no	—
260	C₂²×C₄	—	5	1	no	—
261	C₂²×C₄	—	5	1	no	—
262	C₂²×C₄	—	10	4	no	—
263	C₂²×C₄	—	15	2	no	—
264	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
265	C₄⋊C₄	—	3	1	no	—
266	C₄⋊C₄	—	15	2	no	—
267	C₄⋊C₄	—	15	1	no	—
268	C₄⋊C₄	—	15	2	no	—
269	C₂²⋊C₄	—	3	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	—	5	2	no	—
279	C₂²×C₃	—	5	2	no	—
280	C₂²×C₃	—	10	8	no	—
281	C₂²×C₃	—	10	1	no	—
282	C₂²×C₃	—	10	1	no	—
283	C₂²×C₃	—	10	1	no	—
284	C₂²×C₃	—	10	1	no	—
285	C₂²×C₃	#711	1	2	yes	complemented
286	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
287	C₂×D₃	—	2	2	no	—
288	C₂×D₃	—	2	2	no	—
289	C₂×D₃	—	2	2	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	5	2	no	—
292	C₃×C₄	—	10	1	no	—
293	C₃×C₄	D₄×D₅	1	2	yes	—
294	Dic₃	—	2	2	no	—
295	Dic₃	—	10	2	no	—
296	Dic₃	—	10	2	no	—
297	C₂×C₅	—	2	2	no	—
298	C₂×C₅	—	12	2	no	—
299	C₂×C₅	C₂×#235	1	1	yes	characteristic
300	C₂×C₅	C₂×#235	1	1	yes	characteristic
301	C₂×C₅	C₂²×#67	1	1	yes	characteristic
302	D₅	—	2	2	no	—
303	D₅	—	2	2	no	—
304	C₂³	—	3	2	no	—
305	C₂³	—	5	1	no	—
306	C₂³	—	5	1	no	—
307	C₂³	—	10	2	no	—
308	C₂³	—	10	2	no	—
309	C₂³	—	10	2	no	—
310	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
311	Q₈	—	30	2	no	—
312	D₄	—	6	4	no	—
313	D₄	—	6	2	no	—
314	D₄	—	30	4	no	—
315	D₄	—	30	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	3	2	no	—
318	C₂×C₄	—	5	1	no	—
319	C₂×C₄	—	5	4	no	—
320	C₂×C₄	—	5	1	no	—
321	C₂×C₄	—	10	2	no	—
322	C₂×C₄	—	10	2	no	—
323	C₂×C₄	—	10	2	no	—
324	C₂×C₄	—	10	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	—	10	2	no	—
332	C₂×C₃	—	10	2	no	—
333	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
334	C₂×C₃	C₂×#711	1	1	yes	characteristic
335	C₂×C₃	D₅×#41	1	1	yes	characteristic
336	D₃	—	4	2	no	—
337	C₅	#5791	1	1	yes	characteristic, complemented, Sylow
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	5	2	no	—
341	C₂²	—	5	2	no	—
342	C₂²	—	6	2	no	—
343	C₂²	—	6	2	no	—
344	C₂²	—	6	2	no	—
345	C₂²	—	10	8	no	—
346	C₂²	—	10	1	no	—
347	C₂²	—	10	1	no	—
348	C₂²	—	10	1	no	—
349	C₂²	—	10	1	no	—
350	C₂²	#6772	1	2	yes	complemented
351	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	5	2	no	—
354	C₄	—	6	2	no	—
355	C₄	—	10	1	no	—
356	C₄	—	30	2	no	—
357	C₄	—	30	2	no	—
358	C₄	D₅×#67	1	2	yes	—
359	C₃	#65940	1	1	yes	characteristic, complemented, Sylow
360	C₂	—	2	2	no	—
361	C₂	—	10	2	no	—
362	C₂	—	10	2	no	—
363	C₂	—	12	2	no	—
364	C₂	C₂×#6772	1	1	yes	characteristic
365	C₂	D₅×#235	1	1	yes	characteristic
366	C₂	C₂×D₅×#67	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×C₃×C₄×D₅	C₂	semidirect
2. C₂×#6772	C₂	semidirect
3. #91813	C₂	semidirect
4. C₅×#930	C₂	semidirect
5. #91563	C₂	semidirect
6. Dic₅×#67	C₂	semidirect
7. #91449	C₂	semidirect
8. #91447	C₂	semidirect
9. #91440	C₂	semidirect
10. #91360	C₂	semidirect
11. #91358	C₂	semidirect
15. C₂×Dic₃₀	C₂²	semidirect
17. C₂×C₅×D₁₂	C₂²	semidirect
26. C₂×D₃×Dic₅	C₂²	semidirect
29. C₄⋊Dic₁₅	C₂²	semidirect
30. #6712	C₂²	semidirect
31. C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
32. C₅×C₄⋊Dic₃	C₂²	semidirect
33. Dic₃⋊Dic₅	C₂²	semidirect
34. (C₂×D₃)⋊Dic₅	C₂²	semidirect
37. Dic₃×Dic₅	C₂²	semidirect
40. C₂²×C₄×D₅	D₃	semidirect
72. C₃×C₄×D₅	D₄	semidirect
80. #930	D₅	semidirect
115. C₂²×C₃×C₅	#41	semidirect
126. C₂×C₃×D₅	#41	semidirect
146. C₂×D₁₂	C₂×D₅	semidirect
153. (C₂×D₃)⋊C₄	C₂×D₅	semidirect
155. C₄⋊Dic₃	C₂×D₅	semidirect
183. C₄×D₅	#67	semidirect
234. C₂²×C₅	#235	semidirect
241. C₂×D₅	#235	semidirect
275. C₃×C₅	#525	semidirect
285. C₂²×C₃	#711	semidirect
337. C₅	#5791	semidirect
350. C₂²	#6772	semidirect
359. C₃	#65940	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×D₅	2
C₂×#6772	2
#91813	2
C₅×#930	2
#91563	2
Dic₅×#67	2
#91449	2
#91447	2
#91440	2
#91360	2
#91358	2
#65940	3
#5791	5