Subgroups — #14413522

Order 1920.

952 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91790	C₂	1	1	yes	maximal, characteristic
2	#11754506	C₂	1	2	yes	maximal
3	C₂×C₅×#818	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65395	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748472	C₂	1	2	yes	maximal, complemented
6	#11747690	C₂	1	2	yes	maximal, complemented
7	#11747638	C₂	1	4	yes	maximal, complemented
8	#11747636	C₂	1	2	yes	maximal, complemented
9	#10610877	—	3	1	no	maximal, Hall
10	C₂×#6817	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×#6710	C₂²	1	1	yes	characteristic
13	#91790	C₄	1	8	yes	complemented
14	C₂×#6705	C₂²	1	2	yes	—
15	C₂×C₈×D₁₅	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
17	C₃×#4072	C₂²	1	2	yes	—
18	#91556	—	2	4	no	—
19	#91437	C₂²	1	4	yes	—
20	#91418	—	2	2	no	—
21	#91347	C₂²	1	4	yes	—
22	#91333	—	2	2	no	—
23	C₅×#818	C₂²	1	2	yes	—
24	C₅×#818	C₂²	1	2	yes	—
25	C₃×#3950	C₂²	1	1	yes	characteristic
26	C₃×#3950	C₂²	1	1	yes	characteristic
27	C₃×#3949	C₂²	1	2	yes	—
28	#90989	C₂²	1	4	yes	—
29	#90986	C₂²	1	4	yes	—
30	#79179	—	5	1	no	maximal, Hall
31	C₂×#4047	—	3	1	no	—
32	#65838	—	3	2	no	—
33	C₂×C₅×#107	—	3	1	no	—
34	#65395	D₃	1	1	yes	characteristic, complemented
35	#65253	—	3	2	no	—
36	#65170	—	3	4	no	—
37	#65169	—	3	2	no	—
38	D₅×#107	—	3	2	no	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×#1220	D₄	1	2	yes	—
41	#6817	C₂×C₄	1	4	yes	—
42	#6817	Q₈	1	4	yes	—
43	C₂×D₆₀	D₄	1	1	yes	characteristic
44	C₂×C₄×D₁₅	C₂³	1	2	yes	—
45	C₂×Dic₃₀	D₄	1	1	yes	characteristic
46	C₂²×C₅×Dic₃	—	2	2	no	—
47	C₂×#1200	—	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	#6710	C₂×C₄	1	4	yes	complemented
52	C₂×C₃⋊#168	C₂³	1	2	yes	—
53	#6705	C₂×C₄	1	8	yes	complemented
54	C₈×D₁₅	C₂×C₄	1	8	yes	complemented
55	C₅×C₄⋊Dic₃	—	2	2	no	—
56	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
57	C₃×C₂²⋊Dic₅	—	2	2	no	—
58	C₃×C₄⋊Dic₅	C₂³	1	2	yes	—
59	C₃×C₄⋊Dic₅	C₂³	1	2	yes	—
60	C₃×C₄×Dic₅	—	2	2	no	—
61	#6669	—	2	4	no	—
62	#6667	—	2	4	no	—
63	Dic₃×Dic₅	—	2	4	no	—
64	C₂×#901	—	5	1	no	—
65	#5523	—	5	2	no	—
66	C₃×#427	—	5	1	no	—
67	C₂×#818	D₅	1	1	yes	characteristic, complemented
68	#4995	—	5	2	no	—
69	#4880	—	5	2	no	—
70	D₃×#107	—	5	2	no	—
71	#4861	—	5	4	no	—
72	C₂×#709	—	3	1	no	—
73	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
74	C₂×C₅×C₄⋊C₄	—	3	2	no	—
75	C₄×C₂²⋊D₅	—	6	4	no	—
76	#4072	C₂×D₃	1	2	yes	—
77	C₂×#681	—	3	1	no	—
78	#4047	—	3	8	no	—
79	C₂×#676	—	3	2	no	—
80	C₂×C₈×D₅	—	3	2	no	—
81	#4039	—	6	2	no	—
82	#4038	—	3	4	no	—
83	D₅×C₄⋊C₄	—	3	4	no	—
84	#4033	—	6	2	no	—
85	C₅×#107	—	3	2	no	—
86	C₅×#107	—	3	2	no	—
87	#3950	C₂×D₃	1	1	yes	characteristic
88	#3950	C₂×D₃	1	1	yes	characteristic
89	#3949	C₂×D₃	1	2	yes	—
90	#3940	—	3	4	no	—
91	#3939	—	3	4	no	—
92	#2913	—	15	1	no	Sylow
93	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×D₁₅	C₂×D₄	1	2	yes	—
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	C₂×Q₈	1	2	yes	—
98	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
99	#1220	C₄⋊C₄	1	8	yes	—
100	C₂×Dic₁₅	C₂×D₄	1	2	yes	—
101	D₆₀	C₄⋊C₄	1	4	yes	—
102	C₄×D₁₅	C₂²×C₄	1	4	yes	—
103	C₄×D₁₅	C₂×Q₈	1	4	yes	—
104	Dic₃₀	C₄⋊C₄	1	4	yes	—
105	C₂×C₅×Dic₃	—	2	4	no	—
106	C₂×C₅×Dic₃	—	8	2	no	—
107	C₂×C₃×Dic₅	—	2	2	no	—
108	C₂×C₃×Dic₅	—	2	2	no	—
109	#1200	—	8	4	no	—
110	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
111	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
112	C₃⋊#168	C₂²×C₄	1	4	yes	—
113	C₂×#235	—	5	1	no	—
114	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
115	C₃×#117	—	5	2	no	—
116	C₂×C₄⋊Dic₃	C₂×D₅	1	2	yes	—
117	C₂×#208	—	5	1	no	—
118	#901	—	5	8	no	—
119	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
120	C₂×D₃×C₈	—	5	2	no	—
121	#892	—	5	4	no	—
122	D₃×C₄⋊C₄	—	5	4	no	—
123	#67⋊C₄	—	10	4	no	—
124	C₄×D₁₂	—	10	2	no	—
125	C₄×Dic₆	—	10	2	no	—
126	C₃×#107	—	5	1	no	—
127	C₃×#107	—	5	1	no	—
128	C₃×#106	—	5	2	no	—
129	#818	C₂×D₅	1	2	yes	—
130	#818	C₂×D₅	1	2	yes	—
131	#808	—	5	4	no	—
132	#807	—	5	4	no	—
133	C₂²×C₄×C₅	—	6	2	no	—
134	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₂²⋊D₅	—	3	2	no	—
136	#709	—	3	4	no	—
137	#709	—	3	4	no	—
138	C₂×D₂₀	—	3	1	no	—
139	C₂×C₄×D₅	—	3	2	no	—
140	C₂×C₄×D₅	—	6	4	no	—
141	C₂×Dic₁₀	—	3	1	no	—
142	C₂×C₅×C₈	C₄×D₃	1	4	yes	complemented
143	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
144	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
145	C₅×C₄⋊C₄	—	3	4	no	—
146	C₅×C₄⋊C₄	—	6	2	no	—
147	C₂²⋊Dic₅	—	2	2	no	—
148	#685	—	6	4	no	—
149	C₄⋊Dic₅	C₂²×D₃	1	2	yes	—
150	C₄⋊Dic₅	C₂²×D₃	1	2	yes	—
151	#683	—	6	4	no	—
152	C₄×Dic₅	—	2	2	no	—
153	C₄×Dic₅	—	6	4	no	—
154	#681	—	3	4	no	—
155	C₂×#168	—	3	2	no	—
156	#676	—	3	8	no	—
157	C₈×D₅	—	3	8	no	—
158	C₂×#131	—	15	1	no	—
159	#520	—	15	2	no	—
160	#428	—	15	4	no	—
161	#428	—	15	2	no	—
162	#427	—	5	1	no	—
163	#427	—	15	2	no	—
164	C₂×#107	—	3	1	no	—
165	C₂×#107	—	15	2	no	—
166	C₂²×C₃×C₅	—	2	1	no	—
167	C₂²×C₃×C₅	—	2	1	no	—
168	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
169	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
170	C₂×D₁₅	—	2	2	no	—
171	C₂×D₁₅	—	2	2	no	—
172	C₂×D₁₅	C₂×C₄⋊C₄	1	4	yes	—
173	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
174	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
175	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
176	Dic₁₅	C₂×C₄⋊C₄	1	4	yes	—
177	C₃×Dic₅	—	4	2	no	—
178	C₃×Dic₅	—	4	2	no	—
179	C₅×Dic₃	—	4	4	no	—
180	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
181	C₂×#67	—	5	2	no	—
182	C₂²×Dic₃	—	2	2	no	—
183	#235	—	5	4	no	—
184	#235	—	5	4	no	—
185	C₂×D₁₂	—	5	1	no	—
186	C₂×C₄×D₃	—	5	2	no	—
187	C₂×C₄×D₃	—	10	4	no	—
188	C₂×Dic₆	—	5	1	no	—
189	C₂×C₃×C₈	C₄×D₅	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×C₄⋊C₄	—	5	2	no	—
194	C₃×C₂²⋊C₄	—	10	2	no	—
195	C₃×C₄²	—	10	2	no	—
196	(C₂×D₃)⋊C₄	—	10	4	no	—
197	C₄⋊Dic₃	—	2	2	no	—
198	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
199	Dic₃⋊C₄	—	10	4	no	—
200	C₄×Dic₃	—	10	4	no	—
201	#208	—	5	4	no	—
202	C₂×C₃⋊C₈	—	5	2	no	—
203	ΓL(1,𝔽₂₅)	—	5	8	no	—
204	D₃×C₈	—	5	8	no	—
205	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
206	C₂²×D₅	—	3	2	no	—
207	C₂×C₄×C₅	—	6	4	no	—
208	C₂×C₄×C₅	—	24	2	no	—
209	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
210	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
211	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
212	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
213	C₂²⋊D₅	—	3	8	no	—
214	C₂×Dic₅	—	2	2	no	—
215	C₂×Dic₅	—	2	2	no	—
216	C₂×Dic₅	—	3	2	no	—
217	D₂₀	—	3	4	no	—
218	C₄×D₅	—	3	4	no	—
219	C₄×D₅	—	3	4	no	—
220	C₄×D₅	—	24	4	no	—
221	Dic₁₀	—	3	4	no	—
222	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
223	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
224	#168	—	3	4	no	—
225	C₂×#41	—	15	1	no	—
226	#131	—	15	8	no	—
227	C₂×M₁₆	—	15	1	no	—
228	C₂×M₁₆	—	15	2	no	—
229	C₂²×C₈	—	15	2	no	—
230	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
231	C₄×Q₈	—	30	2	no	—
232	C₄×D₄	—	30	2	no	—
233	C₄×D₄	—	30	4	no	—
234	#117	—	5	2	no	—
235	#117	—	15	4	no	—
236	C₂×C₄⋊C₄	—	3	2	no	—
237	C₂×C₄⋊C₄	—	15	4	no	—
238	#107	—	3	2	no	—
239	#107	—	3	2	no	—
240	#107	—	5	1	no	—
241	#107	—	5	1	no	—
242	#107	—	15	4	no	—
243	#106	—	5	2	no	—
244	#106	—	15	4	no	—
245	C₂×C₃×C₅	—	2	2	no	—
246	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
247	C₂×C₃×C₅	#576	1	1	yes	characteristic
248	C₂×C₃×C₅	#578	1	1	yes	characteristic
249	D₁₅	—	2	4	no	—
250	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
251	C₂²×D₃	—	5	2	no	—
252	C₂×C₃×C₄	—	10	2	no	—
253	C₂×C₃×C₄	—	10	2	no	—
254	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
255	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
256	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
257	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
258	#67	—	5	8	no	—
259	C₂×Dic₃	—	2	4	no	—
260	C₂×Dic₃	—	5	2	no	—
261	C₂×Dic₃	—	8	2	no	—
262	D₁₂	—	5	4	no	—
263	C₄×D₃	—	5	4	no	—
264	C₄×D₃	—	5	4	no	—
265	C₄×D₃	—	40	4	no	—
266	Dic₆	—	5	4	no	—
267	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
268	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
269	C₃⋊C₈	—	5	4	no	—
270	C₂²×C₅	—	2	1	no	—
271	C₂²×C₅	—	2	1	no	—
272	C₂²×C₅	D₃×C₄⋊C₄	1	2	yes	—
273	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
274	C₂×D₅	—	3	4	no	—
275	C₂×D₅	—	6	2	no	—
276	C₂×D₅	—	6	2	no	—
277	C₄×C₅	—	12	4	no	—
278	C₄×C₅	D₃×C₄⋊C₄	1	2	yes	—
279	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
280	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
281	Dic₅	—	3	4	no	—
282	Dic₅	—	4	2	no	—
283	Dic₅	—	4	2	no	—
284	#41	—	15	4	no	—
285	#41	—	15	4	no	—
286	C₂×Q₈	—	15	1	no	—
287	C₂×D₄	—	15	2	no	—
288	C₂×D₄	—	15	1	no	—
289	C₂²×C₄	—	6	2	no	—
290	C₂²×C₄	—	15	2	no	—
291	C₂²×C₄	—	30	4	no	—
292	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
293	M₁₆	—	15	4	no	—
294	M₁₆	—	15	8	no	—
295	C₂×C₈	—	15	2	no	—
296	C₂×C₈	—	15	8	no	—
297	C₂×C₈	#1200	1	4	yes	complemented
298	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
299	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
300	C₄⋊C₄	—	3	4	no	—
301	C₄⋊C₄	—	5	2	no	—
302	C₄⋊C₄	—	5	2	no	—
303	C₄⋊C₄	—	6	2	no	—
304	C₄⋊C₄	—	30	4	no	—
305	C₂²⋊C₄	—	10	2	no	—
306	C₂²⋊C₄	—	30	4	no	—
307	C₄²	—	10	2	no	—
308	C₄²	—	30	4	no	—
309	C₃×C₅	#2913	1	1	yes	characteristic, complemented, Hall
310	C₂²×C₃	—	2	1	no	—
311	C₂²×C₃	—	2	1	no	—
312	C₂²×C₃	D₅×C₄⋊C₄	1	2	yes	—
313	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
314	C₂×D₃	—	5	4	no	—
315	C₂×D₃	—	10	2	no	—
316	C₂×D₃	—	10	2	no	—
317	C₃×C₄	—	20	2	no	—
318	C₃×C₄	—	20	2	no	—
319	C₃×C₄	D₅×C₄⋊C₄	1	2	yes	—
320	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
321	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
322	Dic₃	—	4	4	no	—
323	Dic₃	—	5	4	no	—
324	C₂×C₅	—	2	2	no	—
325	C₂×C₅	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
326	C₂×C₅	#5769	1	1	yes	characteristic
327	C₂×C₅	#5771	1	1	yes	characteristic
328	D₅	—	6	4	no	—
329	C₂³	—	15	2	no	—
330	C₂³	#6777	1	1	yes	characteristic
331	Q₈	—	15	4	no	—
332	D₄	—	15	8	no	—
333	D₄	—	15	4	no	—
334	C₂×C₄	—	6	4	no	—
335	C₂×C₄	—	10	2	no	—
336	C₂×C₄	—	10	2	no	—
337	C₂×C₄	—	15	2	no	—
338	C₂×C₄	—	15	4	no	—
339	C₂×C₄	—	15	4	no	—
340	C₂×C₄	—	24	2	no	—
341	C₂×C₄	—	120	4	no	—
342	C₂×C₄	D₁₅⋊Q₈	1	2	yes	—
343	C₂×C₄	#6777	1	1	yes	characteristic
344	C₂×C₄	C₂×#1200	1	2	yes	—
345	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
346	C₈	—	15	4	no	—
347	C₈	C₂×#1200	1	2	yes	—
348	C₈	C₂×#1200	1	2	yes	—
349	C₂×C₃	—	2	2	no	—
350	C₂×C₃	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
351	C₂×C₃	#66091	1	1	yes	characteristic
352	C₂×C₃	#66100	1	1	yes	characteristic
353	D₃	—	10	4	no	—
354	C₅	#79179	1	1	yes	characteristic, complemented, Sylow
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	15	4	no	—
358	C₂²	—	30	2	no	—
359	C₂²	—	30	2	no	—
360	C₂²	#91437	1	2	yes	—
361	C₂²	C₂×#6777	1	1	yes	characteristic
362	C₄	—	12	4	no	—
363	C₄	—	15	4	no	—
364	C₄	—	20	2	no	—
365	C₄	—	20	2	no	—
366	C₄	#91437	1	2	yes	—
367	C₄	C₂×D₁₅⋊Q₈	1	1	yes	characteristic
368	C₄	C₂²×#1200	1	1	yes	characteristic
369	C₃	#10610877	1	1	yes	characteristic, complemented, Sylow
370	C₂	—	2	2	no	—
371	C₂	—	30	4	no	—
372	C₂	#11753983	1	1	yes	characteristic
373	C₂	#11754008	1	1	yes	characteristic
374	C₂	C₂×#91437	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#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×#6817	C₂²
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2913
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
3. C₂×C₅×#818	C₂	semidirect
4. C₃×#65395	C₂	semidirect
5. #11748472	C₂	semidirect
6. #11747690	C₂	semidirect
7. #11747638	C₂	semidirect
8. #11747636	C₂	semidirect
13. #91790	C₄	semidirect
34. #65395	D₃	semidirect
48. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
51. #6710	C₂×C₄	semidirect
53. #6705	C₂×C₄	semidirect
54. C₈×D₁₅	C₂×C₄	semidirect
67. C₂×#818	D₅	semidirect
142. C₂×C₅×C₈	C₄×D₃	semidirect
189. C₂×C₃×C₈	C₄×D₅	semidirect
297. C₂×C₈	#1200	semidirect
309. C₃×C₅	#2913	semidirect
354. C₅	#79179	semidirect
369. C₃	#10610877	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91790	2
#11754506	2
C₂×C₅×#818	2
C₃×#65395	2
#11748472	2
#11747690	2
#11747638	2
#11747636	2
#10610877	3
#79179	5