Subgroups — #14417796

Order 1920.

968 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	C₂×C₅×#904	C₂	1	1	yes	maximal, characteristic
2	#11754513	C₂	1	2	yes	maximal, complemented
3	C₂×#91101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65390	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748390	C₂	1	2	yes	maximal, complemented
6	#11747856	C₂	1	2	yes	maximal, complemented
7	#11747775	C₂	1	2	yes	maximal, complemented
8	Dic₅×#204	C₂	1	4	yes	maximal, complemented
9	#10615630	—	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	2	yes	—
13	C₂×C₅×Dic₁₂	C₂²	1	1	yes	characteristic
14	C₅×#904	C₄	1	8	yes	complemented
15	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
16	C₂×C₅×#204	C₂²	1	2	yes	—
17	C₃×#4072	C₂²	1	1	yes	characteristic
18	C₂×C₃×#681	C₂²	1	1	yes	characteristic
19	#67⋊Dic₅	—	2	4	no	—
20	D₃×C₄⋊Dic₅	—	2	2	no	—
21	Dic₅×D₁₂	C₂²	1	4	yes	—
22	#91338	—	2	2	no	—
23	Dic₅×Dic₆	C₂²	1	4	yes	—
24	#91101	C₂²	1	2	yes	—
25	#91101	C₂²	1	2	yes	—
26	C₃×#3947	C₂²	1	2	yes	—
27	C₃×C₈×Dic₅	C₂²	1	2	yes	—
28	#90973	C₂²	1	4	yes	—
29	#90967	C₂²	1	4	yes	—
30	#78574	—	5	1	no	maximal, Hall
31	C₂×C₅×#135	—	3	1	no	—
32	#66161	—	3	2	no	—
33	#65521	—	3	2	no	—
34	#65473	—	3	2	no	—
35	SD₁₆×Dic₅	—	3	4	no	—
36	#65441	—	3	2	no	—
37	C₂×#3949	—	3	1	no	—
38	#65390	D₃	1	1	yes	characteristic, complemented
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₅	—	2	2	no	—
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	—	2	2	no	—
52	C₄⋊Dic₁₅	C₂³	1	4	yes	—
53	C₅×Dic₁₂	C₂×C₄	1	4	yes	complemented
54	C₅×D₂₄	C₂×C₄	1	4	yes	complemented
55	C₅×#204	C₂×C₄	1	8	yes	complemented
56	C₃×C₂²⋊Dic₅	D₄	1	2	yes	—
57	C₃×C₄⋊Dic₅	D₄	1	2	yes	—
58	C₃×C₄×Dic₅	C₂³	1	2	yes	—
59	C₃×#681	—	2	2	no	—
60	C₂×C₃×#168	C₂³	1	2	yes	—
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	#5523	—	5	2	no	—
66	C₃×#405	—	5	1	no	—
67	#5133	—	5	2	no	—
68	C₂×#817	—	5	1	no	—
69	#4716	—	5	2	no	—
70	#4711	—	5	2	no	—
71	C₄×#204	—	5	4	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	1	yes	characteristic
83	C₂×C₄⋊Dic₅	—	3	2	no	—
84	C₂×C₄⋊Dic₅	—	6	2	no	—
85	C₂×#681	C₂×D₃	1	1	yes	characteristic
86	#3967	—	3	4	no	—
87	#3964	—	3	4	no	—
88	#3949	—	3	2	no	—
89	#3949	—	3	2	no	—
90	#3947	C₂×D₃	1	2	yes	—
91	C₈×Dic₅	C₂×D₃	1	2	yes	—
92	#2952	—	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	—	2	4	no	—
100	C₂×Dic₁₅	—	8	2	no	—
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	C₂×D₄	1	4	yes	—
109	D₃×Dic₅	—	8	4	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	C₂×C₃×M₁₆	—	5	1	no	—
115	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
116	C₃×#117	—	5	1	no	—
117	C₂×C₄⋊Dic₃	—	5	2	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	#892	—	10	2	no	—
123	D₃×C₄⋊C₄	—	10	2	no	—
124	#67⋊C₄	—	10	4	no	—
125	C₄×D₁₂	—	5	4	no	—
126	C₄×Dic₆	—	5	4	no	—
127	C₃×#97	—	5	2	no	—
128	C₃×C₄×C₈	—	5	2	no	—
129	#821	—	5	4	no	—
130	#817	—	5	2	no	—
131	#817	—	5	2	no	—
132	#816	—	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	—	6	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	—	6	4	no	—
149	C₂²⋊Dic₅	D₁₂	1	2	yes	—
150	C₄⋊Dic₅	—	3	4	no	—
151	C₄⋊Dic₅	—	6	2	no	—
152	C₄⋊Dic₅	—	6	4	no	—
153	C₄⋊Dic₅	D₁₂	1	2	yes	—
154	C₄×Dic₅	—	6	4	no	—
155	C₄×Dic₅	C₂²×D₃	1	2	yes	—
156	#681	—	2	2	no	—
157	C₂×#168	C₂²×D₃	1	2	yes	—
158	C₂×#135	—	3	1	no	—
159	#520	—	15	2	no	—
160	#442	—	15	2	no	—
161	#441	—	15	2	no	—
162	C₄×SD₁₆	—	15	4	no	—
163	C₂×#106	—	15	1	no	—
164	#417	—	15	2	no	—
165	#405	—	5	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	2	no	—
171	C₂×C₅×D₃	—	4	2	no	—
172	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
173	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
174	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
175	Dic₁₅	—	4	4	no	—
176	C₃×Dic₅	—	2	4	no	—
177	C₅×Dic₃	—	2	4	no	—
178	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
179	C₂×#67	—	2	2	no	—
180	C₂²×Dic₃	—	10	2	no	—
181	#235	—	2	4	no	—
182	#235	C₂×Dic₅	1	8	yes	complemented
183	C₂×D₁₂	C₂²×D₅	1	2	yes	—
184	C₂×C₄×D₃	—	2	2	no	—
185	C₂×C₄×D₃	—	10	4	no	—
186	C₂×Dic₆	C₂²×D₅	1	2	yes	—
187	C₃×M₁₆	—	10	2	no	—
188	C₂×C₃×C₈	—	5	2	no	—
189	C₂×C₃×C₈	C₂×Dic₅	1	4	yes	complemented
190	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
191	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
192	C₃×C₄⋊C₄	—	5	2	no	—
193	C₃×C₂²⋊C₄	—	5	2	no	—
194	C₃×C₄²	—	5	2	no	—
195	(C₂×D₃)⋊C₄	—	10	4	no	—
196	C₄⋊Dic₃	—	5	4	no	—
197	C₄⋊Dic₃	—	10	2	no	—
198	Dic₃⋊C₄	—	10	4	no	—
199	C₄×Dic₃	—	10	4	no	—
200	Dic₁₂	C₂×Dic₅	1	4	yes	complemented
201	D₂₄	C₂×Dic₅	1	4	yes	complemented
202	#204	C₂×Dic₅	1	8	yes	complemented
203	C₂³×C₅	—	6	2	no	—
204	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
205	C₅×Q₈	—	3	4	no	—
206	C₅×Q₈	—	6	2	no	—
207	C₅×D₄	—	3	4	no	—
208	C₅×D₄	—	6	8	no	—
209	C₅×D₄	—	6	2	no	—
210	C₂×C₄×C₅	—	6	2	no	—
211	C₂×C₄×C₅	—	6	8	no	—
212	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
213	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
214	C₂×C₄×C₅	#235	1	2	yes	—
215	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
216	C₂×Dic₅	—	6	4	no	—
217	C₂×Dic₅	—	24	4	no	—
218	C₂×Dic₅	—	24	2	no	—
219	C₂×Dic₅	C₂×D₁₂	1	4	yes	—
220	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
221	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
222	#168	—	2	2	no	—
223	C₂×#41	—	3	2	no	—
224	#135	—	3	8	no	—
225	C₂×Q₁₆	—	3	1	no	—
226	C₂×SD₁₆	—	3	2	no	—
227	C₂×D₈	—	3	1	no	—
228	C₂×M₁₆	—	5	1	no	—
229	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
230	C₄×Q₈	—	15	4	no	—
231	C₄×D₄	—	15	4	no	—
232	C₄×D₄	—	30	4	no	—
233	#117	—	5	1	no	—
234	#117	—	30	2	no	—
235	C₂×C₄⋊C₄	—	15	2	no	—
236	C₂×C₄⋊C₄	—	30	2	no	—
237	#106	—	15	2	no	—
238	#106	—	15	2	no	—
239	#103	—	15	4	no	—
240	#102	—	15	4	no	—
241	#97	—	5	2	no	—
242	C₄×C₈	—	5	2	no	—
243	C₂×C₃×C₅	—	2	2	no	—
244	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
245	C₂×C₃×C₅	#577	1	2	yes	—
246	C₅×D₃	—	4	4	no	—
247	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
248	C₂²×D₃	—	2	2	no	—
249	C₂×C₃×C₄	—	5	4	no	—
250	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
251	C₂×C₃×C₄	#711	1	2	yes	—
252	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
253	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
254	#67	—	2	8	no	—
255	C₂×Dic₃	—	2	2	no	—
256	C₂×Dic₃	—	10	4	no	—
257	C₂×Dic₃	—	40	2	no	—
258	D₁₂	—	2	2	no	—
259	D₁₂	C₂²×Dic₅	1	4	yes	—
260	C₄×D₃	—	2	8	no	—
261	C₄×D₃	—	40	4	no	—
262	Dic₆	—	2	2	no	—
263	Dic₆	C₂²×Dic₅	1	4	yes	—
264	C₃×C₈	—	10	2	no	—
265	C₃×C₈	C₂²×Dic₅	1	2	yes	—
266	C₃×C₈	C₂²×Dic₅	1	2	yes	—
267	C₂²×C₅	—	2	2	no	—
268	C₂²×C₅	—	6	4	no	—
269	C₂²×C₅	—	12	2	no	—
270	C₂²×C₅	—	12	2	no	—
271	C₂²×C₅	C₄×D₁₂	1	2	yes	—
272	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
273	C₄×C₅	—	6	4	no	—
274	C₄×C₅	C₄×D₁₂	1	2	yes	—
275	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
276	C₄×C₅	C₂×#235	1	1	yes	characteristic
277	Dic₅	—	2	4	no	—
278	Dic₅	—	12	4	no	—
279	#41	—	3	8	no	—
280	#41	—	6	4	no	—
281	C₂×Q₈	—	3	2	no	—
282	C₂×D₄	—	3	2	no	—
283	C₂×D₄	—	6	2	no	—
284	C₂²×C₄	—	6	2	no	—
285	C₂²×C₄	—	30	4	no	—
286	C₂²×C₄	—	30	2	no	—
287	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
288	Q₁₆	—	3	4	no	—
289	SD₁₆	—	3	8	no	—
290	D₈	—	3	4	no	—
291	M₁₆	—	10	2	no	—
292	C₂×C₈	—	5	2	no	—
293	C₂×C₈	D₃×Dic₅	1	4	yes	complemented
294	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
295	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
296	C₄⋊C₄	—	5	2	no	—
297	C₄⋊C₄	—	15	4	no	—
298	C₄⋊C₄	—	30	4	no	—
299	C₄⋊C₄	—	30	2	no	—
300	C₂²⋊C₄	—	5	2	no	—
301	C₂²⋊C₄	—	30	4	no	—
302	C₄²	—	5	2	no	—
303	C₄²	—	30	4	no	—
304	C₃×C₅	#2952	1	1	yes	characteristic, complemented, Hall
305	C₂²×C₃	—	2	2	no	—
306	C₂²×C₃	D₄×Dic₅	1	2	yes	—
307	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
308	C₂×D₃	—	2	4	no	—
309	C₂×D₃	—	4	2	no	—
310	C₂×D₃	—	4	2	no	—
311	C₃×C₄	—	10	4	no	—
312	C₃×C₄	D₄×Dic₅	1	2	yes	—
313	C₃×C₄	C₂×#711	1	1	yes	characteristic
314	C₃×C₄	C₂³×Dic₅	1	1	yes	characteristic
315	Dic₃	—	2	4	no	—
316	Dic₃	—	20	4	no	—
317	C₂×C₅	—	2	2	no	—
318	C₂×C₅	—	12	4	no	—
319	C₂×C₅	C₂×C₄×D₁₂	1	1	yes	characteristic
320	C₂×C₅	#5751	1	2	yes	—
321	C₂³	—	6	2	no	—
322	C₂³	D₅×D₁₂	1	1	yes	characteristic
323	Q₈	—	3	4	no	—
324	Q₈	—	6	2	no	—
325	D₄	—	3	4	no	—
326	D₄	—	6	8	no	—
327	D₄	—	6	2	no	—
328	C₂×C₄	—	5	4	no	—
329	C₂×C₄	—	6	2	no	—
330	C₂×C₄	—	6	8	no	—
331	C₂×C₄	—	30	4	no	—
332	C₂×C₄	—	120	4	no	—
333	C₂×C₄	—	120	2	no	—
334	C₂×C₄	#6772	1	2	yes	—
335	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
336	C₂×C₄	C₂×D₃×Dic₅	1	2	yes	—
337	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
338	C₈	—	10	2	no	—
339	C₈	C₂×D₃×Dic₅	1	2	yes	—
340	C₈	C₂×D₃×Dic₅	1	2	yes	—
341	C₂×C₃	—	2	2	no	—
342	C₂×C₃	#66104	1	2	yes	—
343	C₂×C₃	C₂×D₄×Dic₅	1	1	yes	characteristic
344	D₃	—	4	4	no	—
345	C₅	#78574	1	1	yes	characteristic, complemented, Sylow
346	C₂²	—	2	2	no	—
347	C₂²	—	6	4	no	—
348	C₂²	—	12	2	no	—
349	C₂²	—	12	2	no	—
350	C₂²	Dic₅×D₁₂	1	2	yes	—
351	C₂²	C₂×D₅×D₁₂	1	1	yes	characteristic
352	C₄	—	6	4	no	—
353	C₄	—	10	4	no	—
354	C₄	—	60	4	no	—
355	C₄	Dic₅×D₁₂	1	2	yes	—
356	C₄	C₂×#6772	1	1	yes	characteristic
357	C₄	C₂²×D₃×Dic₅	1	1	yes	characteristic
358	C₃	#10615630	1	1	yes	characteristic, complemented, Sylow
359	C₂	—	2	2	no	—
360	C₂	—	12	4	no	—
361	C₂	#11753962	1	2	yes	—
362	C₂	C₂×Dic₅×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₅×D₁₂
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	#2952
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #11754513	C₂	semidirect
3. C₂×#91101	C₂	semidirect
4. C₃×#65390	C₂	semidirect
5. #11748390	C₂	semidirect
6. #11747856	C₂	semidirect
7. #11747775	C₂	semidirect
8. Dic₅×#204	C₂	semidirect
14. C₅×#904	C₄	semidirect
38. #65390	D₃	semidirect
43. C₅×#235	C₂×C₄	semidirect
48. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
53. C₅×Dic₁₂	C₂×C₄	semidirect
54. C₅×D₂₄	C₂×C₄	semidirect
55. C₅×#204	C₂×C₄	semidirect
119. #904	Dic₅	semidirect
145. C₂×C₅×C₈	C₄×D₃	semidirect
182. #235	C₂×Dic₅	semidirect
189. C₂×C₃×C₈	C₂×Dic₅	semidirect
200. Dic₁₂	C₂×Dic₅	semidirect
201. D₂₄	C₂×Dic₅	semidirect
202. #204	C₂×Dic₅	semidirect
293. C₂×C₈	D₃×Dic₅	semidirect
304. C₃×C₅	#2952	semidirect
345. C₅	#78574	semidirect
358. C₃	#10615630	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
#11754513	2
C₂×#91101	2
C₃×#65390	2
#11748390	2
#11747856	2
#11747775	2
Dic₅×#204	2
#10615630	3
#78574	5