Subgroups — #14416610

Order 1920.

984 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#904	C₂	1	1	yes	maximal, characteristic
2	#11754455	C₂	1	2	yes	maximal, complemented
3	C₂×#91099	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65395	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748382	C₂	1	2	yes	maximal, complemented
6	#11747888	C₂	1	2	yes	maximal, complemented
7	#11747816	C₂	1	4	yes	maximal, complemented
8	#11747813	C₂	1	2	yes	maximal, complemented
9	#10615628	—	3	1	no	maximal, Hall
10	C₂×C₅×#235	C₂²	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
14	C₂×C₅×Dic₁₂	C₂²	1	1	yes	characteristic
15	C₅×#904	C₄	1	8	yes	complemented
16	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
17	C₂×C₅×#204	C₂²	1	2	yes	—
18	C₃×#4072	C₂²	1	2	yes	—
19	#67⋊Dic₅	—	2	4	no	—
20	#91435	C₂²	1	4	yes	—
21	C₄×D₃×Dic₅	—	2	2	no	—
22	#91371	—	2	2	no	—
23	#91344	C₂²	1	4	yes	—
24	#91099	C₂²	1	2	yes	—
25	#91099	C₂²	1	2	yes	—
26	C₃×#3950	C₂²	1	2	yes	—
27	C₃×#3949	C₂²	1	2	yes	—
28	#90972	C₂²	1	4	yes	—
29	#90966	C₂²	1	4	yes	—
30	#79166	—	5	1	no	maximal, Hall
31	C₂×C₅×#135	—	3	1	no	—
32	#41×Dic₅	—	3	2	no	—
33	#65522	—	3	2	no	—
34	#65473	—	3	2	no	—
35	#65453	—	3	4	no	—
36	#65441	—	3	2	no	—
37	#65395	D₃	1	1	yes	characteristic, complemented
38	C₂×#3947	—	3	1	no	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₅	D₄	1	2	yes	—
41	C₂×C₅×#67	—	2	2	no	—
42	C₅×#235	—	2	4	no	—
43	C₅×#235	C₂×C₄	1	8	yes	complemented
44	C₂×C₅×D₁₂	C₂³	1	2	yes	—
45	C₂×C₄×C₅×D₃	—	2	2	no	—
46	C₂×C₅×Dic₆	C₂³	1	2	yes	—
47	C₂×D₃×Dic₅	—	2	4	no	—
48	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
49	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₄×Dic₁₅	D₄	1	2	yes	—
52	C₄×Dic₁₅	C₂³	1	2	yes	—
53	C₂×C₃⋊#168	—	2	2	no	—
54	C₂×C₃⋊#168	C₂³	1	2	yes	—
55	C₅×Dic₁₂	C₂×C₄	1	4	yes	complemented
56	C₅×D₂₄	C₂×C₄	1	4	yes	complemented
57	C₅×#204	C₂×C₄	1	8	yes	complemented
58	C₃×C₂²⋊Dic₅	—	2	2	no	—
59	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
60	C₃×C₄×Dic₅	—	2	2	no	—
61	Dic₃⋊Dic₅	—	2	4	no	—
62	(C₂×D₃)⋊Dic₅	—	2	4	no	—
63	Dic₃×Dic₅	—	2	4	no	—
64	C₂×#904	D₅	1	1	yes	characteristic
65	#5522	—	5	2	no	—
66	C₃×#427	—	5	1	no	—
67	C₂×#815	—	5	1	no	—
68	#5002	—	5	2	no	—
69	#4889	—	5	4	no	—
70	#4869	—	5	2	no	—
71	#4853	—	5	2	no	—
72	C₂×C₅×#41	—	3	2	no	—
73	C₅×#135	—	3	8	no	—
74	C₂×C₅×Q₁₆	—	3	1	no	—
75	C₂×C₅×SD₁₆	—	3	2	no	—
76	C₂×C₅×D₈	—	3	1	no	—
77	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
78	Q₈×Dic₅	—	3	4	no	—
79	D₄×Dic₅	—	3	4	no	—
80	D₄×Dic₅	—	6	4	no	—
81	#4072	—	6	2	no	—
82	#4072	C₂×D₃	1	2	yes	—
83	C₂×C₄×Dic₅	—	3	1	no	—
84	C₂×C₄×Dic₅	—	6	2	no	—
85	C₂²×#168	—	3	1	no	—
86	#3967	—	3	4	no	—
87	#3964	—	3	4	no	—
88	#3950	C₂×D₃	1	2	yes	—
89	#3949	C₂×D₃	1	2	yes	—
90	#3947	—	3	2	no	—
91	#3947	—	3	2	no	—
92	#2943	—	15	1	no	Sylow
93	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₅×D₃	—	2	2	no	—
95	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×C₄×C₅	#41	1	2	yes	—
98	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₁₅	—	4	2	no	—
100	C₂×Dic₁₅	C₂×D₄	1	4	yes	—
101	C₅×#67	—	2	8	no	—
102	C₂×C₅×Dic₃	—	2	2	no	—
103	C₅×D₁₂	—	2	2	no	—
104	C₅×D₁₂	C₂²×C₄	1	4	yes	—
105	C₄×C₅×D₃	—	2	8	no	—
106	C₅×Dic₆	—	2	2	no	—
107	C₅×Dic₆	C₂²×C₄	1	4	yes	—
108	C₂×C₃×Dic₅	—	2	4	no	—
109	D₃×Dic₅	—	4	8	no	—
110	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
111	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
112	C₃⋊#168	—	2	2	no	—
113	C₂×#235	C₂×D₅	1	2	yes	—
114	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
115	C₃×#117	—	5	2	no	—
116	C₂×C₄×Dic₃	—	5	1	no	—
117	C₂²×C₃⋊C₈	—	5	1	no	—
118	C₂×Dic₁₂	C₂×D₅	1	1	yes	characteristic
119	#904	Dic₅	1	8	yes	complemented
120	C₂×D₂₄	C₂×D₅	1	1	yes	characteristic
121	C₂×#204	C₂×D₅	1	2	yes	—
122	#893	—	5	4	no	—
123	#887	—	5	4	no	—
124	#67⋊C₄	—	10	4	no	—
125	#872	—	10	2	no	—
126	C₄²×D₃	—	10	2	no	—
127	C₃×#107	—	5	2	no	—
128	C₃×#106	—	5	2	no	—
129	#815	—	5	2	no	—
130	#815	—	5	2	no	—
131	#810	—	5	4	no	—
132	#809	—	5	4	no	—
133	C₅×#41	—	3	8	no	—
134	C₅×#41	—	6	4	no	—
135	C₂×C₅×Q₈	—	3	2	no	—
136	C₂×C₅×D₄	—	3	2	no	—
137	C₂×C₅×D₄	—	6	2	no	—
138	C₂²×C₄×C₅	—	6	2	no	—
139	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
140	C₂²×Dic₅	—	3	2	no	—
141	C₂²×Dic₅	—	6	4	no	—
142	C₅×Q₁₆	—	3	4	no	—
143	C₅×SD₁₆	—	3	8	no	—
144	C₅×D₈	—	3	4	no	—
145	C₂×C₅×C₈	C₄×D₃	1	4	yes	complemented
146	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
147	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
148	C₂²⋊Dic₅	—	2	2	no	—
149	C₂²⋊Dic₅	—	6	4	no	—
150	C₄⋊Dic₅	—	6	4	no	—
151	C₄⋊Dic₅	C₂²×D₃	1	4	yes	—
152	C₄×Dic₅	—	2	2	no	—
153	C₄×Dic₅	—	3	2	no	—
154	C₄×Dic₅	—	3	2	no	—
155	C₄×Dic₅	—	6	4	no	—
156	C₂×#168	—	3	2	no	—
157	C₂×#168	—	6	2	no	—
158	C₂×#135	—	3	1	no	—
159	C₄×#41	—	15	2	no	—
160	#442	—	15	2	no	—
161	#441	—	15	2	no	—
162	#440	—	15	4	no	—
163	#427	—	5	1	no	—
164	#416	—	15	2	no	—
165	C₂×#97	—	15	1	no	—
166	C₂²×C₃×C₅	—	2	2	no	—
167	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
168	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₅×D₃	—	2	4	no	—
170	C₂×C₅×D₃	—	4	4	no	—
171	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
172	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
173	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
174	Dic₁₅	—	2	4	no	—
175	C₃×Dic₅	—	4	4	no	—
176	C₅×Dic₃	—	2	4	no	—
177	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
178	C₂×#67	—	2	2	no	—
179	C₂²×Dic₃	—	5	2	no	—
180	#235	—	2	4	no	—
181	#235	C₂×Dic₅	1	8	yes	complemented
182	C₂×D₁₂	C₂²×D₅	1	2	yes	—
183	C₂×C₄×D₃	—	2	2	no	—
184	C₂×C₄×D₃	—	10	4	no	—
185	C₂×Dic₆	C₂²×D₅	1	2	yes	—
186	C₂×C₃×C₈	C₂×Dic₅	1	4	yes	complemented
187	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
188	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
189	C₃×C₄⋊C₄	—	5	4	no	—
190	C₃×C₂²⋊C₄	—	10	2	no	—
191	C₃×C₄²	—	10	2	no	—
192	(C₂×D₃)⋊C₄	—	10	4	no	—
193	Dic₃⋊C₄	—	10	4	no	—
194	C₄×Dic₃	—	5	2	no	—
195	C₄×Dic₃	—	5	2	no	—
196	C₄×Dic₃	—	10	4	no	—
197	C₂×C₃⋊C₈	—	5	2	no	—
198	C₂×C₃⋊C₈	—	10	2	no	—
199	Dic₁₂	C₂×Dic₅	1	4	yes	complemented
200	D₂₄	C₂×Dic₅	1	4	yes	complemented
201	#204	C₂×Dic₅	1	8	yes	complemented
202	C₂³×C₅	—	6	2	no	—
203	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
204	C₅×Q₈	—	3	4	no	—
205	C₅×Q₈	—	6	2	no	—
206	C₅×D₄	—	3	4	no	—
207	C₅×D₄	—	6	8	no	—
208	C₅×D₄	—	6	2	no	—
209	C₂×C₄×C₅	—	6	2	no	—
210	C₂×C₄×C₅	—	6	8	no	—
211	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
212	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
213	C₂×C₄×C₅	#239	1	2	yes	—
214	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
215	C₂×Dic₅	—	2	4	no	—
216	C₂×Dic₅	—	3	4	no	—
217	C₂×Dic₅	—	12	8	no	—
218	C₂×Dic₅	—	12	2	no	—
219	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
220	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
221	#168	—	6	2	no	—
222	C₂×#41	—	3	2	no	—
223	#135	—	3	8	no	—
224	C₂×Q₁₆	—	3	1	no	—
225	C₂×SD₁₆	—	3	2	no	—
226	C₂×D₈	—	3	1	no	—
227	C₂²×C₈	—	15	1	no	—
228	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
229	C₄×Q₈	—	15	4	no	—
230	C₄×D₄	—	15	4	no	—
231	C₄×D₄	—	30	4	no	—
232	#117	—	5	2	no	—
233	#117	—	30	2	no	—
234	C₂×C₄²	—	15	1	no	—
235	C₂×C₄²	—	30	2	no	—
236	#107	—	5	2	no	—
237	#106	—	5	2	no	—
238	#103	—	15	4	no	—
239	#102	—	15	4	no	—
240	#97	—	15	2	no	—
241	#97	—	15	2	no	—
242	C₂×C₃×C₅	—	2	2	no	—
243	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
244	C₂×C₃×C₅	#575	1	2	yes	—
245	C₅×D₃	—	4	4	no	—
246	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
247	C₂²×D₃	—	2	2	no	—
248	C₂×C₃×C₄	—	10	4	no	—
249	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
250	C₂×C₃×C₄	#711	1	2	yes	—
251	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
252	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
253	#67	—	2	8	no	—
254	C₂×Dic₃	—	2	2	no	—
255	C₂×Dic₃	—	5	4	no	—
256	C₂×Dic₃	—	20	2	no	—
257	D₁₂	—	2	2	no	—
258	D₁₂	C₂²×Dic₅	1	4	yes	—
259	C₄×D₃	—	2	8	no	—
260	C₄×D₃	—	20	8	no	—
261	Dic₆	—	2	2	no	—
262	Dic₆	C₂²×Dic₅	1	4	yes	—
263	C₃×C₈	C₂²×Dic₅	1	2	yes	—
264	C₃×C₈	C₂²×Dic₅	1	2	yes	—
265	C₃⋊C₈	—	10	2	no	—
266	C₂²×C₅	—	2	2	no	—
267	C₂²×C₅	—	6	4	no	—
268	C₂²×C₅	—	12	4	no	—
269	C₂²×C₅	#893	1	2	yes	—
270	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
271	C₄×C₅	—	6	4	no	—
272	C₄×C₅	#893	1	2	yes	—
273	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
274	C₄×C₅	C₂×#239	1	1	yes	characteristic
275	Dic₅	—	4	4	no	—
276	Dic₅	—	6	4	no	—
277	#41	—	3	8	no	—
278	#41	—	6	4	no	—
279	C₂×Q₈	—	3	2	no	—
280	C₂×D₄	—	3	2	no	—
281	C₂×D₄	—	6	2	no	—
282	C₂²×C₄	—	6	2	no	—
283	C₂²×C₄	—	15	2	no	—
284	C₂²×C₄	—	30	4	no	—
285	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
286	Q₁₆	—	3	4	no	—
287	SD₁₆	—	3	8	no	—
288	D₈	—	3	4	no	—
289	C₂×C₈	—	15	2	no	—
290	C₂×C₈	—	30	2	no	—
291	C₂×C₈	D₃×Dic₅	1	4	yes	complemented
292	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
293	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
294	C₄⋊C₄	—	5	4	no	—
295	C₄⋊C₄	—	30	4	no	—
296	C₂²⋊C₄	—	10	2	no	—
297	C₂²⋊C₄	—	30	4	no	—
298	C₄²	—	10	2	no	—
299	C₄²	—	15	2	no	—
300	C₄²	—	15	2	no	—
301	C₄²	—	30	4	no	—
302	C₃×C₅	#2943	1	1	yes	characteristic, complemented, Hall
303	C₂²×C₃	—	2	2	no	—
304	C₂²×C₃	D₄×Dic₅	1	2	yes	—
305	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
306	C₂×D₃	—	2	4	no	—
307	C₂×D₃	—	4	4	no	—
308	C₃×C₄	—	20	4	no	—
309	C₃×C₄	D₄×Dic₅	1	2	yes	—
310	C₃×C₄	C₂×#711	1	1	yes	characteristic
311	C₃×C₄	C₂³×Dic₅	1	1	yes	characteristic
312	Dic₃	—	2	4	no	—
313	Dic₃	—	10	4	no	—
314	C₂×C₅	—	2	2	no	—
315	C₂×C₅	—	12	4	no	—
316	C₂×C₅	C₂×#893	1	1	yes	characteristic
317	C₂×C₅	#5768	1	2	yes	—
318	C₂³	—	6	2	no	—
319	C₂³	#6777	1	1	yes	characteristic
320	Q₈	—	3	4	no	—
321	Q₈	—	6	2	no	—
322	D₄	—	3	4	no	—
323	D₄	—	6	8	no	—
324	D₄	—	6	2	no	—
325	C₂×C₄	—	6	2	no	—
326	C₂×C₄	—	6	8	no	—
327	C₂×C₄	—	10	4	no	—
328	C₂×C₄	—	15	4	no	—
329	C₂×C₄	—	60	8	no	—
330	C₂×C₄	—	60	2	no	—
331	C₂×C₄	#6768	1	2	yes	—
332	C₂×C₄	#6777	1	1	yes	characteristic
333	C₂×C₄	C₂×D₃×Dic₅	1	2	yes	—
334	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
335	C₈	—	30	2	no	—
336	C₈	C₂×D₃×Dic₅	1	2	yes	—
337	C₈	C₂×D₃×Dic₅	1	2	yes	—
338	C₂×C₃	—	2	2	no	—
339	C₂×C₃	#66102	1	2	yes	—
340	C₂×C₃	C₂×D₄×Dic₅	1	1	yes	characteristic
341	D₃	—	4	4	no	—
342	C₅	#79166	1	1	yes	characteristic, complemented, Sylow
343	C₂²	—	2	2	no	—
344	C₂²	—	6	4	no	—
345	C₂²	—	12	4	no	—
346	C₂²	#91435	1	2	yes	—
347	C₂²	C₂×#6777	1	1	yes	characteristic
348	C₄	—	6	4	no	—
349	C₄	—	20	4	no	—
350	C₄	—	30	4	no	—
351	C₄	#91435	1	2	yes	—
352	C₄	C₂×#6768	1	1	yes	characteristic
353	C₄	C₂²×D₃×Dic₅	1	1	yes	characteristic
354	C₃	#10615628	1	1	yes	characteristic, complemented, Sylow
355	C₂	—	2	2	no	—
356	C₂	—	12	4	no	—
357	C₂	#11753979	1	2	yes	—
358	C₂	C₂×#91435	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6777
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×C₅×#904	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2943
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #11754455	C₂	semidirect
3. C₂×#91099	C₂	semidirect
4. C₃×#65395	C₂	semidirect
5. #11748382	C₂	semidirect
6. #11747888	C₂	semidirect
7. #11747816	C₂	semidirect
8. #11747813	C₂	semidirect
15. C₅×#904	C₄	semidirect
37. #65395	D₃	semidirect
43. C₅×#235	C₂×C₄	semidirect
48. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
55. C₅×Dic₁₂	C₂×C₄	semidirect
56. C₅×D₂₄	C₂×C₄	semidirect
57. C₅×#204	C₂×C₄	semidirect
119. #904	Dic₅	semidirect
145. C₂×C₅×C₈	C₄×D₃	semidirect
181. #235	C₂×Dic₅	semidirect
186. C₂×C₃×C₈	C₂×Dic₅	semidirect
199. Dic₁₂	C₂×Dic₅	semidirect
200. D₂₄	C₂×Dic₅	semidirect
201. #204	C₂×Dic₅	semidirect
291. C₂×C₈	D₃×Dic₅	semidirect
302. C₃×C₅	#2943	semidirect
342. C₅	#79166	semidirect
354. C₃	#10615628	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#904	2
#11754455	2
C₂×#91099	2
C₃×#65395	2
#11748382	2
#11747888	2
#11747816	2
#11747813	2
#10615628	3
#79166	5