Subgroups — #14413468

Order 1920.

952 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×#901	C₂	1	1	yes	maximal, characteristic
2	#11754492	C₂	1	2	yes	maximal
3	#11751207	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#3949	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748453	C₂	1	2	yes	maximal, complemented
6	#11747679	C₂	1	2	yes	maximal, complemented
7	#11747628	C₂	1	4	yes	maximal, complemented
8	D₃×#3949	C₂	1	2	yes	maximal, complemented
9	#10615515	—	3	1	no	maximal, Hall
10	C₂×C₅×#235	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	#91815	C₂²	1	2	yes	—
13	C₂×C₅×#208	C₂²	1	1	yes	characteristic
14	C₅×#901	C₄	1	8	yes	complemented
15	C₂×C₅×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
16	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
18	Dic₅×#67	—	2	4	no	—
19	#91435	—	2	2	no	—
20	D₃×C₄⋊Dic₅	C₂²	1	4	yes	—
21	#91344	—	2	2	no	—
22	#91338	C₂²	1	4	yes	—
23	#91102	C₂²	1	2	yes	—
24	#91101	C₂²	1	1	yes	characteristic
25	#91101	C₂²	1	1	yes	characteristic
26	C₃×#3949	C₂²	1	2	yes	—
27	C₃×#3949	C₂²	1	2	yes	—
28	#90990	C₂²	1	4	yes	—
29	#90987	C₂²	1	4	yes	—
30	#79116	—	5	1	no	maximal, Hall
31	C₂×C₅×#131	—	3	1	no	—
32	#66161	—	3	2	no	—
33	#65410	—	3	2	no	—
34	#65410	—	3	4	no	—
35	#65395	—	3	2	no	—
36	#65395	—	3	1	no	—
37	C₂×#3949	—	3	2	no	—
38	C₂×#3949	D₃	1	1	yes	characteristic, complemented
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₅×#67	D₄	1	2	yes	—
41	C₅×#235	C₂×C₄	1	4	yes	—
42	C₅×#235	Q₈	1	4	yes	—
43	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
44	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
45	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
46	C₂²×C₃×Dic₅	—	2	2	no	—
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	#6719	—	2	2	no	—
52	C₄⋊Dic₁₅	C₂³	1	2	yes	—
53	C₄⋊Dic₁₅	C₂³	1	2	yes	—
54	C₄×Dic₁₅	—	2	2	no	—
55	C₅×#208	C₂×C₄	1	4	yes	complemented
56	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
57	C₅×ΓL(1,𝔽₂₅)	C₂×C₄	1	8	yes	complemented
58	C₅×D₃×C₈	C₂×C₄	1	8	yes	complemented
59	C₃×C₄⋊Dic₅	—	2	2	no	—
60	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
61	Dic₃⋊Dic₅	—	2	4	no	—
62	(C₂×D₃)⋊Dic₅	—	2	4	no	—
63	Dic₃×Dic₅	—	2	4	no	—
64	C₂×#901	D₅	1	1	yes	characteristic
65	#5504	—	5	2	no	—
66	C₂×C₃×#106	—	5	1	no	—
67	#5106	—	5	1	no	—
68	#4964	—	5	2	no	—
69	#4861	—	5	4	no	—
70	#4860	—	5	2	no	—
71	D₃×#106	—	5	2	no	—
72	C₂×C₅×#41	—	3	1	no	—
73	C₅×#131	—	3	8	no	—
74	C₂×C₅×M₁₆	—	3	1	no	—
75	C₂×C₅×M₁₆	—	3	2	no	—
76	C₂²×C₅×C₈	—	3	2	no	—
77	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
78	Q₈×Dic₅	—	6	2	no	—
79	D₄×Dic₅	—	6	2	no	—
80	D₄×Dic₅	—	6	4	no	—
81	#4072	—	3	2	no	—
82	#4072	—	3	4	no	—
83	C₂×C₄⋊Dic₅	—	3	4	no	—
84	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
85	#3950	—	3	2	no	—
86	#3950	—	3	4	no	—
87	#3949	—	3	1	no	—
88	#3949	—	3	1	no	—
89	#3949	—	3	4	no	—
90	#3949	C₂×D₃	1	2	yes	—
91	#3949	C₂×D₃	1	2	yes	—
92	#2912	—	15	1	no	Sylow
93	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₅×D₃	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	C₂×Dic₁₅	—	2	2	no	—
100	C₂×Dic₁₅	—	2	2	no	—
101	C₅×#67	C₄⋊C₄	1	8	yes	—
102	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
103	C₅×D₁₂	C₄⋊C₄	1	4	yes	—
104	C₄×C₅×D₃	C₂²×C₄	1	4	yes	—
105	C₄×C₅×D₃	C₂×Q₈	1	4	yes	—
106	C₅×Dic₆	C₄⋊C₄	1	4	yes	—
107	C₂×C₃×Dic₅	—	2	4	no	—
108	C₂×C₃×Dic₅	—	8	2	no	—
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	C₅×C₃⋊C₈	C₂²×C₄	1	4	yes	—
113	C₂×#235	C₂×D₅	1	1	yes	characteristic
114	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
115	C₂×C₃×C₄⋊C₄	—	5	2	no	—
116	C₄×#67	—	10	4	no	—
117	#926	—	5	2	no	—
118	C₂×#208	C₂×D₅	1	1	yes	characteristic
119	#901	Dic₅	1	8	yes	complemented
120	C₂×ΓL(1,𝔽₂₅)	C₂×D₅	1	2	yes	—
121	C₂×D₃×C₈	C₂×D₅	1	2	yes	—
122	#893	—	10	2	no	—
123	#892	—	5	4	no	—
124	D₃×C₄⋊C₄	—	5	4	no	—
125	#887	—	10	2	no	—
126	C₃×#106	—	5	2	no	—
127	C₃×#106	—	5	2	no	—
128	#818	—	5	2	no	—
129	#817	—	5	1	no	—
130	#817	—	5	1	no	—
131	#808	—	5	4	no	—
132	#807	—	5	4	no	—
133	C₅×#41	—	3	4	no	—
134	C₅×#41	—	3	4	no	—
135	C₂×C₅×Q₈	—	3	1	no	—
136	C₂×C₅×D₄	—	3	2	no	—
137	C₂×C₅×D₄	—	3	1	no	—
138	C₂²×C₄×C₅	—	3	2	no	—
139	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
140	C₂²×Dic₅	—	2	2	no	—
141	C₂²×Dic₅	—	6	4	no	—
142	C₅×M₁₆	—	3	4	no	—
143	C₅×M₁₆	—	3	8	no	—
144	C₂×C₅×C₈	—	3	2	no	—
145	C₂×C₅×C₈	—	3	8	no	—
146	C₂×C₅×C₈	C₄×D₃	1	4	yes	complemented
147	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
148	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
149	C₂²⋊Dic₅	—	6	4	no	—
150	C₂²⋊Dic₅	—	6	2	no	—
151	C₄⋊Dic₅	—	2	2	no	—
152	C₄⋊Dic₅	—	3	2	no	—
153	C₄⋊Dic₅	—	3	2	no	—
154	C₄⋊Dic₅	—	6	4	no	—
155	C₄⋊Dic₅	C₂²×D₃	1	4	yes	—
156	C₄×Dic₅	—	6	4	no	—
157	C₄×Dic₅	—	6	2	no	—
158	C₂×#131	—	3	1	no	—
159	#520	—	15	2	no	—
160	#428	—	15	4	no	—
161	#428	—	15	2	no	—
162	#427	—	15	2	no	—
163	#427	—	15	1	no	—
164	C₂×#106	—	5	1	no	—
165	C₂×#106	—	15	2	no	—
166	C₂²×C₃×C₅	—	2	2	no	—
167	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
168	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₅×D₃	—	2	4	no	—
170	C₂×C₅×D₃	C₂×C₄⋊C₄	1	4	yes	—
171	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
172	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
173	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
174	Dic₁₅	—	4	2	no	—
175	Dic₁₅	—	4	2	no	—
176	C₃×Dic₅	—	4	4	no	—
177	C₅×Dic₃	C₂×C₄⋊C₄	1	4	yes	—
178	C₂²×C₃×C₄	—	10	2	no	—
179	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
180	C₂×#67	D₂₀	1	2	yes	—
181	#235	Dic₁₀	1	4	yes	—
182	#235	C₂×Dic₅	1	4	yes	—
183	C₂×D₁₂	D₂₀	1	1	yes	characteristic
184	C₂×C₄×D₃	—	10	4	no	—
185	C₂×C₄×D₃	C₂²×D₅	1	2	yes	—
186	C₂×Dic₆	D₂₀	1	1	yes	characteristic
187	C₂×C₃×C₈	C₂×Dic₅	1	4	yes	complemented
188	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
189	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
190	C₃×C₄⋊C₄	—	5	4	no	—
191	C₃×C₄⋊C₄	—	10	2	no	—
192	#217	—	10	2	no	—
193	(C₂×D₃)⋊C₄	—	10	4	no	—
194	C₄⋊Dic₃	—	5	2	no	—
195	C₄⋊Dic₃	—	5	2	no	—
196	Dic₃⋊C₄	—	10	4	no	—
197	C₄×Dic₃	—	10	2	no	—
198	C₄×Dic₃	—	10	4	no	—
199	#208	C₂×Dic₅	1	4	yes	complemented
200	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
201	ΓL(1,𝔽₂₅)	C₂×Dic₅	1	8	yes	complemented
202	D₃×C₈	C₂×Dic₅	1	8	yes	complemented
203	C₂³×C₅	—	3	2	no	—
204	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
205	C₅×Q₈	—	3	4	no	—
206	C₅×D₄	—	3	8	no	—
207	C₅×D₄	—	3	4	no	—
208	C₂×C₄×C₅	—	3	2	no	—
209	C₂×C₄×C₅	—	3	4	no	—
210	C₂×C₄×C₅	—	3	4	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	D₃×Q₈	1	2	yes	—
214	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
215	C₂×Dic₅	—	2	4	no	—
216	C₂×Dic₅	—	6	2	no	—
217	C₂×Dic₅	—	6	2	no	—
218	C₂×Dic₅	—	8	2	no	—
219	C₂×Dic₅	—	24	4	no	—
220	C₅×C₈	—	3	4	no	—
221	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
222	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
223	C₂×#41	—	3	1	no	—
224	#131	—	3	8	no	—
225	C₂×M₁₆	—	3	1	no	—
226	C₂×M₁₆	—	3	2	no	—
227	C₂²×C₈	—	3	2	no	—
228	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
229	C₄×Q₈	—	30	2	no	—
230	C₄×D₄	—	30	2	no	—
231	C₄×D₄	—	30	4	no	—
232	#117	—	15	4	no	—
233	#117	—	15	2	no	—
234	C₂×C₄⋊C₄	—	5	2	no	—
235	C₂×C₄⋊C₄	—	15	4	no	—
236	#107	—	15	4	no	—
237	#107	—	15	2	no	—
238	#106	—	5	2	no	—
239	#106	—	5	2	no	—
240	#106	—	15	4	no	—
241	#106	—	15	1	no	—
242	#106	—	15	1	no	—
243	C₂×C₃×C₅	—	2	2	no	—
244	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
245	C₂×C₃×C₅	#577	1	2	yes	—
246	C₅×D₃	—	2	4	no	—
247	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
248	C₂²×D₃	C₂×D₂₀	1	2	yes	—
249	C₂×C₃×C₄	—	10	4	no	—
250	C₂×C₃×C₄	—	40	2	no	—
251	C₂×C₃×C₄	C₂×Dic₁₀	1	2	yes	—
252	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
253	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
254	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
255	#67	C₄⋊Dic₅	1	8	yes	—
256	C₂×Dic₃	—	10	2	no	—
257	C₂×Dic₃	—	10	2	no	—
258	C₂×Dic₃	C₂×D₂₀	1	2	yes	—
259	D₁₂	C₄⋊Dic₅	1	4	yes	—
260	C₄×D₃	—	40	4	no	—
261	C₄×D₃	C₂×Dic₁₀	1	4	yes	—
262	C₄×D₃	C₂²×Dic₅	1	4	yes	—
263	Dic₆	C₄⋊Dic₅	1	4	yes	—
264	C₃×C₈	C₂²×Dic₅	1	2	yes	—
265	C₃×C₈	C₂²×Dic₅	1	2	yes	—
266	C₃⋊C₈	C₂²×Dic₅	1	4	yes	—
267	C₂²×C₅	—	2	2	no	—
268	C₂²×C₅	—	3	4	no	—
269	C₂²×C₅	—	6	4	no	—
270	C₂²×C₅	D₃×C₄⋊C₄	1	2	yes	—
271	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
272	C₄×C₅	—	3	4	no	—
273	C₄×C₅	D₃×C₄⋊C₄	1	2	yes	—
274	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
275	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
276	Dic₅	—	4	4	no	—
277	Dic₅	—	12	2	no	—
278	Dic₅	—	12	2	no	—
279	#41	—	3	4	no	—
280	#41	—	3	4	no	—
281	C₂×Q₈	—	3	1	no	—
282	C₂×D₄	—	3	2	no	—
283	C₂×D₄	—	3	1	no	—
284	C₂²×C₄	—	3	2	no	—
285	C₂²×C₄	—	10	2	no	—
286	C₂²×C₄	—	30	4	no	—
287	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
288	M₁₆	—	3	4	no	—
289	M₁₆	—	3	8	no	—
290	C₂×C₈	—	3	2	no	—
291	C₂×C₈	—	3	8	no	—
292	C₂×C₈	D₃×Dic₅	1	4	yes	complemented
293	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
294	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
295	C₄⋊C₄	—	5	4	no	—
296	C₄⋊C₄	—	10	2	no	—
297	C₄⋊C₄	—	15	2	no	—
298	C₄⋊C₄	—	15	2	no	—
299	C₄⋊C₄	—	30	4	no	—
300	C₂²⋊C₄	—	30	4	no	—
301	C₂²⋊C₄	—	30	2	no	—
302	C₄²	—	30	4	no	—
303	C₄²	—	30	2	no	—
304	C₃×C₅	#2912	1	1	yes	characteristic, complemented, Hall
305	C₂²×C₃	—	2	2	no	—
306	C₂²×C₃	C₂×C₄⋊Dic₅	1	2	yes	—
307	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
308	C₂×D₃	—	2	4	no	—
309	C₂×D₃	C₂×C₄⋊Dic₅	1	4	yes	—
310	C₃×C₄	—	20	4	no	—
311	C₃×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
312	C₃×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
313	C₃×C₄	C₂³×Dic₅	1	1	yes	characteristic
314	Dic₃	—	20	2	no	—
315	Dic₃	—	20	2	no	—
316	Dic₃	C₂×C₄⋊Dic₅	1	4	yes	—
317	C₂×C₅	—	2	2	no	—
318	C₂×C₅	—	6	4	no	—
319	C₂×C₅	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
320	C₂×C₅	#5762	1	2	yes	—
321	C₂³	—	3	2	no	—
322	C₂³	D₃×D₂₀	1	1	yes	characteristic
323	Q₈	—	3	4	no	—
324	D₄	—	3	8	no	—
325	D₄	—	3	4	no	—
326	C₂×C₄	—	3	2	no	—
327	C₂×C₄	—	3	4	no	—
328	C₂×C₄	—	3	4	no	—
329	C₂×C₄	—	10	4	no	—
330	C₂×C₄	—	30	2	no	—
331	C₂×C₄	—	30	2	no	—
332	C₂×C₄	—	40	2	no	—
333	C₂×C₄	—	120	4	no	—
334	C₂×C₄	D₃×Dic₁₀	1	2	yes	—
335	C₂×C₄	D₃×D₂₀	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	—	3	4	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	#66085	1	2	yes	—
343	C₂×C₃	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
344	D₃	—	2	4	no	—
345	C₅	#79116	1	1	yes	characteristic, complemented, Sylow
346	C₂²	—	2	2	no	—
347	C₂²	—	3	4	no	—
348	C₂²	—	6	4	no	—
349	C₂²	D₃×C₄⋊Dic₅	1	2	yes	—
350	C₂²	C₂×D₃×D₂₀	1	1	yes	characteristic
351	C₄	—	3	4	no	—
352	C₄	—	20	4	no	—
353	C₄	—	60	2	no	—
354	C₄	—	60	2	no	—
355	C₄	D₃×C₄⋊Dic₅	1	2	yes	—
356	C₄	C₂×D₃×Dic₁₀	1	1	yes	characteristic
357	C₄	C₂²×D₃×Dic₅	1	1	yes	characteristic
358	C₃	#10615515	1	1	yes	characteristic, complemented, Sylow
359	C₂	—	2	2	no	—
360	C₂	—	6	4	no	—
361	C₂	#11753970	1	2	yes	—
362	C₂	C₂×D₃×C₄⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₃×D₂₀
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×#235	C₂²
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2912
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. #11751207	C₂	semidirect
4. C₂×C₃×#3949	C₂	semidirect
5. #11748453	C₂	semidirect
6. #11747679	C₂	semidirect
7. #11747628	C₂	semidirect
8. D₃×#3949	C₂	semidirect
14. C₅×#901	C₄	semidirect
38. C₂×#3949	D₃	semidirect
48. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
55. C₅×#208	C₂×C₄	semidirect
57. C₅×ΓL(1,𝔽₂₅)	C₂×C₄	semidirect
58. C₅×D₃×C₈	C₂×C₄	semidirect
119. #901	Dic₅	semidirect
146. C₂×C₅×C₈	C₄×D₃	semidirect
187. C₂×C₃×C₈	C₂×Dic₅	semidirect
199. #208	C₂×Dic₅	semidirect
201. ΓL(1,𝔽₂₅)	C₂×Dic₅	semidirect
202. D₃×C₈	C₂×Dic₅	semidirect
292. C₂×C₈	D₃×Dic₅	semidirect
304. C₃×C₅	#2912	semidirect
345. C₅	#79116	semidirect
358. C₃	#10615515	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#901	2
#11754492	2
#11751207	2
C₂×C₃×#3949	2
#11748453	2
#11747679	2
#11747628	2
D₃×#3949	2
#10615515	3
#79116	5