Subgroups — #14439184

Order 1920.

1312 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	C₅×#5477	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4020	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91298	C₂	1	4	yes	maximal, complemented
4	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
5	#11747079	C₂	1	2	yes	maximal, complemented
6	#11747041	C₂	1	4	yes	maximal, complemented
7	#11747039	C₂	1	2	yes	maximal, complemented
8	#10615018	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	2	yes	—
10	C₂²×C₃×D₂₀	—	2	2	no	—
11	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×#6710	C₂²	1	2	yes	—
14	C₅×#930	—	2	2	no	—
15	C₅×#875	C₂²	1	2	yes	complemented
16	C₅×#874	C₂²	1	1	yes	characteristic, complemented
17	C₄×C₅×D₁₂	—	2	2	no	—
18	C₅×#869	C₂²	1	1	yes	characteristic, complemented
19	C₃×#4020	—	2	2	no	—
20	C₃×#4020	C₂²	1	2	yes	—
21	C₃×#4020	C₂²	1	2	yes	—
22	C₂×#6655	C₂²	1	8	yes	complemented
23	#91298	—	2	16	no	—
24	C₂×#6652	C₂²	1	8	yes	complemented
25	#91087	C₂²	1	2	yes	complemented
26	C₄×C₃⋊#168	C₂²	1	2	yes	complemented
27	#83163	—	5	1	no	maximal, Hall
28	C₅×#532	—	3	1	no	—
29	C₂×#4095	—	3	4	no	—
30	C₂×#4020	D₃	1	1	yes	characteristic, complemented
31	#65374	—	3	2	no	—
32	#65362	—	3	2	no	—
33	#65347	—	3	4	no	—
34	C₄×#681	—	3	1	no	—
35	C₂³×C₃×D₅	—	2	2	no	—
36	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₅×#67	—	2	2	no	—
39	C₅×#235	—	2	8	no	—
40	C₂×C₅×D₁₂	—	2	2	no	—
41	C₂×C₅×D₁₂	C₂³	1	2	yes	—
42	C₂×C₄×C₅×D₃	—	2	2	no	—
43	C₂×C₅×Dic₆	C₂³	1	2	yes	—
44	C₂×C₃×D₂₀	—	2	8	no	—
45	C₂×C₃×D₂₀	—	2	4	no	—
46	C₂×C₃×D₂₀	—	2	4	no	—
47	C₂×C₃×D₂₀	—	2	2	no	—
48	C₂×C₃×D₂₀	—	4	8	no	—
49	C₂×C₃×D₂₀	C₂³	1	4	yes	—
50	C₃×C₄²×C₅	D₄	1	2	yes	—
51	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
53	#6710	D₄	1	8	yes	complemented
54	C₂×C₃⋊#168	C₂³	1	4	yes	—
55	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
56	C₅×C₄⋊Dic₃	—	2	2	no	—
57	#6655	—	2	16	no	—
58	#6652	—	2	16	no	—
59	C₂×C₃×#127	—	5	1	no	—
60	C₂×#932	—	5	4	no	—
61	#5477	D₅	1	1	yes	characteristic, complemented
62	#5076	—	5	2	no	—
63	#5074	—	5	4	no	—
64	#5073	—	5	2	no	—
65	C₄×#208	—	5	1	no	—
66	C₂×C₅×#41	—	3	2	no	—
67	C₂²×D₂₀	—	2	2	no	—
68	C₂²×D₂₀	C₂×D₃	1	2	yes	—
69	C₅×C₄⋊Q₈	—	3	1	no	—
70	C₅×#127	—	3	1	no	—
71	C₅×#124	—	3	2	no	—
72	C₅×#121	—	6	2	no	—
73	C₄×C₅×D₄	—	6	2	no	—
74	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
75	#4095	—	6	16	no	—
76	C₂×#688	—	3	8	no	—
77	C₂×#686	—	3	8	no	—
78	C₂×#681	—	3	2	no	—
79	#4020	—	2	2	no	—
80	#4020	C₂×D₃	1	2	yes	—
81	#4020	C₂×D₃	1	2	yes	—
82	#3935	—	3	2	no	—
83	C₄×#168	—	3	2	no	—
84	#3151	—	15	1	no	Sylow
85	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₅×D₃	—	2	2	no	—
87	C₂²×C₃×D₅	—	2	4	no	—
88	C₂²×C₃×D₅	—	4	8	no	—
89	C₂²×C₃×D₅	—	4	4	no	—
90	C₂×C₃×C₄×C₅	—	2	4	no	—
91	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
94	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
95	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	C₅×#67	—	4	4	no	—
97	C₂×C₅×Dic₃	—	2	2	no	—
98	C₅×D₁₂	—	2	4	no	—
99	C₅×D₁₂	—	4	4	no	—
100	C₄×C₅×D₃	—	4	4	no	—
101	C₅×Dic₆	—	2	4	no	—
102	C₃×D₂₀	—	2	8	no	—
103	C₃×D₂₀	—	4	8	no	—
104	C₃×D₂₀	—	4	4	no	—
105	C₃×D₂₀	—	4	8	no	—
106	C₃⋊#168	C₂×D₄	1	8	yes	complemented
107	C₂²×C₃×D₄	—	5	2	no	—
108	C₂²×C₃×D₄	—	10	2	no	—
109	C₂×#235	C₂×D₅	1	2	yes	—
110	C₃×#127	—	5	2	no	—
111	C₃×#127	—	5	2	no	—
112	C₃×#127	—	10	2	no	—
113	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
114	C₂×#214	—	5	8	no	—
115	#932	—	10	16	no	—
116	C₂×#213	—	5	8	no	—
117	#930	—	2	2	no	—
118	C₂×#208	—	5	2	no	—
119	#875	C₂×D₅	1	2	yes	complemented
120	#874	C₂×D₅	1	1	yes	characteristic, complemented
121	C₄×D₁₂	—	2	2	no	—
122	#869	C₂×D₅	1	1	yes	characteristic, complemented
123	#803	—	5	2	no	—
124	C₄×C₃⋊C₈	—	5	2	no	—
125	C₂³×D₅	—	2	2	no	—
126	C₅×#41	—	6	8	no	—
127	C₂×C₅×Q₈	—	3	2	no	—
128	C₂×C₅×D₄	—	3	2	no	—
129	C₂×C₅×D₄	—	6	2	no	—
130	C₂×C₅×D₄	—	6	2	no	—
131	C₂²×C₄×C₅	—	6	2	no	—
132	C₂²×C₄×C₅	#67	1	2	yes	—
133	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
134	C₂×D₂₀	—	2	8	no	—
135	C₂×D₂₀	—	2	4	no	—
136	C₂×D₂₀	—	2	4	no	—
137	C₂×D₂₀	—	2	2	no	—
138	C₂×D₂₀	—	4	8	no	—
139	C₂×D₂₀	C₂²×D₃	1	4	yes	—
140	C₅×C₄⋊C₄	—	6	2	no	—
141	C₅×C₂²⋊C₄	—	6	4	no	—
142	C₄²×C₅	#67	1	2	yes	—
143	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
144	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
145	#688	—	6	16	no	—
146	#686	—	6	16	no	—
147	#681	—	3	8	no	—
148	C₂×#168	—	3	4	no	—
149	C₂×AGL(1,ℤ/8)	—	15	4	no	—
150	#532	—	3	1	no	—
151	C₂×#127	—	5	1	no	—
152	#496	—	15	4	no	—
153	#493	—	15	2	no	—
154	#492	—	15	2	no	—
155	C₄×M₁₆	—	15	1	no	—
156	C₂²×C₃×C₅	—	2	2	no	—
157	C₂²×C₃×C₅	#127	1	2	yes	—
158	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₅×D₃	—	4	2	no	—
160	C₂×C₅×D₃	—	4	4	no	—
161	C₂×C₃×D₅	—	4	4	no	—
162	C₂×C₃×D₅	—	4	8	no	—
163	C₂×C₃×D₅	—	8	8	no	—
164	C₃×C₄×C₅	—	2	2	no	—
165	C₃×C₄×C₅	#127	1	2	yes	—
166	C₃×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	complemented
167	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
168	C₅×Dic₃	—	4	2	no	—
169	C₂⁴×C₃	—	10	2	no	—
170	C₂×C₃×D₄	—	5	4	no	—
171	C₂×C₃×D₄	—	10	8	no	—
172	C₂×C₃×D₄	—	10	4	no	—
173	C₂×C₃×D₄	—	10	4	no	—
174	C₂×C₃×D₄	—	10	2	no	—
175	C₂×C₃×D₄	—	20	8	no	—
176	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
177	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
178	C₂×#67	—	2	2	no	—
179	#235	—	2	8	no	—
180	C₂×D₁₂	—	2	2	no	—
181	C₂×D₁₂	C₂²×D₅	1	2	yes	—
182	C₂×C₄×D₃	—	2	2	no	—
183	C₂×Dic₆	C₂²×D₅	1	2	yes	—
184	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
185	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
186	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
187	#214	—	10	16	no	—
188	#213	—	10	16	no	—
189	(C₂×D₃)⋊C₄	—	2	4	no	—
190	C₄⋊Dic₃	—	2	2	no	—
191	#208	—	5	8	no	—
192	C₂×C₃⋊C₈	—	5	4	no	—
193	C₂³×C₅	—	6	2	no	—
194	C₂³×C₅	C₂×#67	1	1	yes	characteristic
195	C₂²×D₅	—	2	4	no	—
196	C₂²×D₅	—	4	8	no	—
197	C₂²×D₅	—	4	4	no	—
198	C₅×Q₈	—	6	4	no	—
199	C₅×D₄	—	6	4	no	—
200	C₅×D₄	—	12	4	no	—
201	C₅×D₄	—	12	4	no	—
202	C₂×C₄×C₅	—	2	4	no	—
203	C₂×C₄×C₅	—	6	2	no	—
204	C₂×C₄×C₅	—	12	4	no	—
205	C₂×C₄×C₅	D₃×D₄	1	4	yes	—
206	C₂×C₄×C₅	C₂×#67	1	2	yes	—
207	C₂×C₄×C₅	C₂×#67	1	2	yes	—
208	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
209	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
210	D₂₀	—	2	8	no	—
211	D₂₀	—	4	8	no	—
212	D₂₀	—	4	8	no	—
213	D₂₀	—	4	4	no	—
214	#168	—	3	8	no	—
215	C₂×#41	—	3	2	no	—
216	C₂²×D₄	—	5	2	no	—
217	C₂²×D₄	—	10	2	no	—
218	AGL(1,ℤ/8)	—	30	16	no	—
219	C₂×SD₁₆	—	15	8	no	—
220	C₂×D₈	—	15	8	no	—
221	C₂×M₁₆	—	15	2	no	—
222	C₄⋊Q₈	—	3	1	no	—
223	#127	—	3	1	no	—
224	#127	—	5	2	no	—
225	#127	—	5	2	no	—
226	#127	—	10	2	no	—
227	#124	—	3	2	no	—
228	#121	—	6	2	no	—
229	C₄×D₄	—	6	2	no	—
230	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
231	#97	—	15	2	no	—
232	C₄×C₈	—	15	2	no	—
233	C₂×C₃×C₅	—	2	2	no	—
234	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
235	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
236	C₃×D₅	—	8	4	no	—
237	C₅×D₃	—	8	2	no	—
238	C₂³×C₃	—	10	4	no	—
239	C₂³×C₃	—	20	8	no	—
240	C₂³×C₃	—	20	4	no	—
241	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
242	C₂²×D₃	—	2	2	no	—
243	C₃×D₄	—	10	8	no	—
244	C₃×D₄	—	20	8	no	—
245	C₃×D₄	—	20	4	no	—
246	C₃×D₄	—	20	8	no	—
247	C₂×C₃×C₄	—	2	4	no	—
248	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
249	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
250	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
251	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
252	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
253	#67	—	4	4	no	—
254	C₂×Dic₃	—	2	2	no	—
255	D₁₂	—	2	4	no	—
256	D₁₂	—	4	4	no	—
257	C₄×D₃	—	4	4	no	—
258	Dic₆	—	2	4	no	—
259	C₃⋊C₈	—	5	8	no	—
260	C₂²×C₅	—	2	2	no	—
261	C₂²×C₅	—	12	4	no	—
262	C₂²×C₅	—	12	2	no	—
263	C₂²×C₅	#940	1	2	yes	—
264	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
265	C₂×D₅	—	4	8	no	—
266	C₂×D₅	—	4	4	no	—
267	C₂×D₅	—	8	8	no	—
268	C₄×C₅	—	2	2	no	—
269	C₄×C₅	—	12	2	no	—
270	C₄×C₅	#932	1	4	yes	complemented
271	C₄×C₅	#940	1	2	yes	—
272	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
273	C₂⁴	—	10	2	no	—
274	#41	—	6	8	no	—
275	C₂×Q₈	—	3	2	no	—
276	C₂×D₄	—	3	2	no	—
277	C₂×D₄	—	5	4	no	—
278	C₂×D₄	—	6	2	no	—
279	C₂×D₄	—	6	2	no	—
280	C₂×D₄	—	10	8	no	—
281	C₂×D₄	—	10	4	no	—
282	C₂×D₄	—	10	4	no	—
283	C₂×D₄	—	10	2	no	—
284	C₂×D₄	—	20	8	no	—
285	C₂²×C₄	—	6	2	no	—
286	C₂²×C₄	#1201	1	2	yes	—
287	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
288	SD₁₆	—	30	16	no	—
289	D₈	—	30	16	no	—
290	M₁₆	—	15	8	no	—
291	C₂×C₈	—	15	4	no	—
292	C₄⋊C₄	—	6	2	no	—
293	C₂²⋊C₄	—	6	4	no	—
294	C₄²	#1201	1	2	yes	—
295	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
296	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
297	C₃×C₅	#3151	1	1	yes	characteristic, complemented, Hall
298	C₂²×C₃	—	2	2	no	—
299	C₂²×C₃	—	20	4	no	—
300	C₂²×C₃	—	20	8	no	—
301	C₂²×C₃	—	40	8	no	—
302	C₂²×C₃	#4086	1	2	yes	—
303	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
304	C₂×D₃	—	4	2	no	—
305	C₂×D₃	—	4	4	no	—
306	C₃×C₄	—	2	2	no	—
307	C₃×C₄	#4086	1	2	yes	—
308	C₃×C₄	#4095	1	4	yes	complemented
309	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
310	Dic₃	—	4	2	no	—
311	C₂×C₅	—	2	2	no	—
312	C₂×C₅	—	24	2	no	—
313	C₂×C₅	C₂×#932	1	2	yes	—
314	C₂×C₅	C₂×#940	1	1	yes	characteristic
315	D₅	—	8	4	no	—
316	C₂³	—	6	2	no	—
317	C₂³	—	10	4	no	—
318	C₂³	—	20	8	no	—
319	C₂³	—	20	4	no	—
320	C₂³	C₂×#1201	1	1	yes	characteristic
321	Q₈	—	6	4	no	—
322	D₄	—	6	4	no	—
323	D₄	—	10	8	no	—
324	D₄	—	12	4	no	—
325	D₄	—	12	4	no	—
326	D₄	—	20	8	no	—
327	D₄	—	20	4	no	—
328	D₄	—	20	8	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	6	2	no	—
331	C₂×C₄	—	12	4	no	—
332	C₂×C₄	#6777	1	4	yes	—
333	C₂×C₄	C₂×#1201	1	2	yes	—
334	C₂×C₄	C₂×#1201	1	1	yes	characteristic
335	C₂×C₄	C₂×#1201	1	2	yes	—
336	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
337	C₈	—	15	8	no	—
338	C₂×C₃	—	2	2	no	—
339	C₂×C₃	—	40	4	no	—
340	C₂×C₃	C₂×#4086	1	1	yes	characteristic
341	C₂×C₃	C₂×#4095	1	2	yes	—
342	D₃	—	8	2	no	—
343	C₅	#83163	1	1	yes	characteristic, complemented, Sylow
344	C₂²	—	2	2	no	—
345	C₂²	—	12	2	no	—
346	C₂²	—	12	4	no	—
347	C₂²	—	20	4	no	—
348	C₂²	—	20	8	no	—
349	C₂²	—	40	8	no	—
350	C₂²	#91463	1	2	yes	—
351	C₂²	C₂²×#1201	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	12	2	no	—
354	C₄	#91298	1	4	yes	complemented
355	C₄	#91463	1	2	yes	—
356	C₄	C₂×#6777	1	2	yes	—
357	C₃	#10615018	1	1	yes	characteristic, complemented, Sylow
358	C₂	—	2	2	no	—
359	C₂	—	24	2	no	—
360	C₂	—	40	4	no	—
361	C₂	C₂×#91298	1	2	yes	—
362	C₂	C₂×#91463	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#1201
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#3151
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₅×#5477	C₂	semidirect
2. C₂×C₃×#4020	C₂	semidirect
3. C₂×#91298	C₂	semidirect
4. C₄×#6710	C₂	semidirect
5. #11747079	C₂	semidirect
6. #11747041	C₂	semidirect
7. #11747039	C₂	semidirect
15. C₅×#875	C₂²	semidirect
16. C₅×#874	C₂²	semidirect
18. C₅×#869	C₂²	semidirect
22. C₂×#6655	C₂²	semidirect
24. C₂×#6652	C₂²	semidirect
25. #91087	C₂²	semidirect
26. C₄×C₃⋊#168	C₂²	semidirect
30. C₂×#4020	D₃	semidirect
53. #6710	D₄	semidirect
61. #5477	D₅	semidirect
106. C₃⋊#168	C₂×D₄	semidirect
119. #875	C₂×D₅	semidirect
120. #874	C₂×D₅	semidirect
122. #869	C₂×D₅	semidirect
166. C₃×C₄×C₅	AGL(1,ℤ/8)	semidirect
270. C₄×C₅	#932	semidirect
297. C₃×C₅	#3151	semidirect
308. C₃×C₄	#4095	semidirect
343. C₅	#83163	semidirect
354. C₄	#91298	semidirect
357. C₃	#10615018	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5477	2
C₂×C₃×#4020	2
C₂×#91298	2
C₄×#6710	2
#11747079	2
#11747041	2
#11747039	2
#10615018	3
#83163	5