Subgroups — #14281920

Order 1920.

468 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756238	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746305	C₂	1	2	yes	maximal, complemented
5	#11746304	C₂	1	2	yes	maximal, complemented
6	#10601220	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	—	2	1	no	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	#91819	C₄	1	1	yes	characteristic
10	#91818	—	2	1	no	—
11	C₄×#1220	—	2	1	no	—
12	#91813	C₄	1	1	yes	characteristic
13	C₂×#6710	—	2	1	no	—
14	#91765	—	2	1	no	—
15	#91764	C₄	1	1	yes	characteristic
16	C₄×D₆₀	C₂²	1	1	yes	characteristic
17	#91761	—	2	1	no	—
18	#91759	C₄	1	1	yes	characteristic
19	C₄×Dic₃₀	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₈	—	2	1	no	—
21	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
22	#91087	—	2	1	no	—
23	C₄×C₃⋊#168	C₂²	1	2	yes	—
24	#70427	—	5	1	no	maximal, Hall
25	#65813	—	3	1	no	—
26	C₂×C₅×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
27	C₄×#681	—	3	1	no	—
28	#64704	—	3	2	no	—
29	#64703	—	3	2	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂×#1220	—	2	1	no	—
34	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₅	—	2	1	no	—
36	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	—	2	2	no	—
38	C₂×C₃×C₅×C₈	—	4	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	#6719	—	2	1	no	—
44	#6714	—	2	1	no	—
45	#6714	—	2	1	no	—
46	#6714	—	2	1	no	—
47	#6714	—	2	1	no	—
48	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
49	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
50	#6712	—	2	1	no	—
51	#6712	—	2	1	no	—
52	#6712	—	2	1	no	—
53	#6712	—	2	1	no	—
54	C₄×Dic₁₅	—	2	1	no	—
55	#6710	—	2	4	no	—
56	C₂×C₃⋊#168	—	2	2	no	—
57	#5479	—	5	1	no	—
58	C₂×C₃×C₄⋊C₈	D₅	1	1	yes	characteristic, complemented
59	C₄×#208	—	5	1	no	—
60	#4484	—	5	2	no	—
61	#4483	—	5	2	no	—
62	C₂²×C₅×C₈	—	2	1	no	—
63	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
64	#4076	—	3	1	no	—
65	#4075	—	6	1	no	—
66	C₄×C₂²⋊D₅	—	6	1	no	—
67	#4070	—	3	1	no	—
68	C₂×#681	—	6	1	no	—
69	#4022	—	6	1	no	—
70	#4021	—	3	1	no	—
71	C₄×D₂₀	—	3	1	no	—
72	#4018	—	6	1	no	—
73	#4016	—	3	1	no	—
74	C₄×Dic₁₀	—	3	1	no	—
75	C₅×C₄⋊C₈	—	2	1	no	—
76	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	—
77	#3935	—	6	1	no	—
78	C₄×#168	—	3	2	no	—
79	#1483	—	15	1	no	Sylow
80	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×D₁₅	—	2	1	no	—
82	C₂×C₃×C₄×C₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	—	2	2	no	—
84	C₂×C₃×C₄×C₅	—	2	1	no	—
85	C₂×C₃×C₄×C₅	—	2	1	no	—
86	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	#1220	—	4	2	no	—
93	C₂×Dic₁₅	—	2	1	no	—
94	C₂×Dic₁₅	—	2	1	no	—
95	C₂×Dic₁₅	—	2	1	no	—
96	D₆₀	—	2	2	no	—
97	C₄×D₁₅	—	4	1	no	—
98	C₄×D₁₅	—	4	1	no	—
99	Dic₃₀	—	2	2	no	—
100	C₃×C₅×C₈	—	4	2	no	—
101	C₃⋊#168	—	2	4	no	—
102	C₂²×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
104	#930	—	5	1	no	—
105	#929	—	10	1	no	—
106	C₄×#67	—	10	1	no	—
107	#924	—	5	1	no	—
108	C₂×#208	—	10	1	no	—
109	#876	—	10	1	no	—
110	#875	—	5	1	no	—
111	C₄×D₁₂	—	5	1	no	—
112	#872	—	10	1	no	—
113	#870	—	5	1	no	—
114	C₄×Dic₆	—	5	1	no	—
115	C₃×C₄⋊C₈	—	2	1	no	—
116	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	—
117	#803	—	10	1	no	—
118	C₄×C₃⋊C₈	—	5	2	no	—
119	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
120	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
121	C₂²×C₄×C₅	#67	1	1	yes	characteristic
122	C₂×C₂²⋊D₅	—	6	1	no	—
123	C₂×D₂₀	—	3	1	no	—
124	C₂×C₄×D₅	—	6	1	no	—
125	C₂×Dic₁₀	—	3	1	no	—
126	C₂×C₅×C₈	—	2	2	no	—
127	C₂×C₅×C₈	—	4	2	no	—
128	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
129	C₄²×C₅	D₁₂	1	1	yes	characteristic
130	C₄²×C₅	#67	1	1	yes	characteristic
131	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
132	C₂²⋊Dic₅	—	6	1	no	—
133	#685	—	6	1	no	—
134	#685	—	6	1	no	—
135	#685	—	6	1	no	—
136	#685	—	6	1	no	—
137	C₄⋊Dic₅	—	3	1	no	—
138	C₄⋊Dic₅	—	3	1	no	—
139	#683	—	6	1	no	—
140	#683	—	6	1	no	—
141	#683	—	6	1	no	—
142	#683	—	6	1	no	—
143	C₄×Dic₅	—	6	1	no	—
144	#681	—	6	4	no	—
145	C₂×#168	—	6	2	no	—
146	#529	—	15	1	no	—
147	C₂×C₄⋊C₈	D₁₅	1	1	yes	characteristic, complemented
148	C₄×M₁₆	—	15	1	no	—
149	Q₈⋊C₈	—	15	2	no	—
150	D₄⋊C₈	—	15	2	no	—
151	C₂²×C₃×C₅	—	2	1	no	—
152	C₂²×C₃×C₅	—	2	1	no	—
153	C₂²×C₃×C₅	#104	1	2	yes	—
154	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×D₁₅	—	4	1	no	—
156	C₂×D₁₅	—	4	1	no	—
157	C₂×D₁₅	—	4	1	no	—
158	C₃×C₄×C₅	—	2	1	no	—
159	C₃×C₄×C₅	—	2	1	no	—
160	C₃×C₄×C₅	—	2	1	no	—
161	C₃×C₄×C₅	—	2	1	no	—
162	C₃×C₄×C₅	#131	1	2	yes	—
163	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₃×C₄×C₅	#137	1	1	yes	characteristic
165	Dic₁₅	—	4	1	no	—
166	Dic₁₅	—	4	1	no	—
167	Dic₁₅	—	4	1	no	—
168	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
169	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
170	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
171	C₂×#67	—	10	1	no	—
172	C₂×D₁₂	—	5	1	no	—
173	C₂×C₄×D₃	—	10	1	no	—
174	C₂×Dic₆	—	5	1	no	—
175	C₂×C₃×C₈	—	2	2	no	—
176	C₂×C₃×C₈	—	4	2	no	—
177	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
178	C₃×C₄²	D₂₀	1	1	yes	characteristic
179	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
180	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
181	#217	—	10	1	no	—
182	(C₂×D₃)⋊C₄	—	10	1	no	—
183	(C₂×D₃)⋊C₄	—	10	1	no	—
184	(C₂×D₃)⋊C₄	—	10	1	no	—
185	(C₂×D₃)⋊C₄	—	10	1	no	—
186	C₄⋊Dic₃	—	5	1	no	—
187	C₄⋊Dic₃	—	5	1	no	—
188	Dic₃⋊C₄	—	10	1	no	—
189	Dic₃⋊C₄	—	10	1	no	—
190	Dic₃⋊C₄	—	10	1	no	—
191	Dic₃⋊C₄	—	10	1	no	—
192	C₄×Dic₃	—	10	1	no	—
193	#208	—	10	4	no	—
194	C₂×C₃⋊C₈	—	10	2	no	—
195	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
196	C₂²×D₅	—	6	1	no	—
197	C₂×C₄×C₅	—	2	2	no	—
198	C₂×C₄×C₅	—	2	2	no	—
199	C₂×C₄×C₅	—	2	1	no	—
200	C₂×C₄×C₅	—	2	1	no	—
201	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
202	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
204	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
206	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
207	C₂²⋊D₅	—	12	2	no	—
208	C₂×Dic₅	—	6	1	no	—
209	C₂×Dic₅	—	6	1	no	—
210	C₂×Dic₅	—	6	1	no	—
211	D₂₀	—	6	2	no	—
212	C₄×D₅	—	12	1	no	—
213	C₄×D₅	—	12	1	no	—
214	Dic₁₀	—	6	2	no	—
215	C₅×C₈	—	4	2	no	—
216	#168	—	6	4	no	—
217	C₂×M₁₆	—	30	1	no	—
218	C₂²×C₈	—	2	1	no	—
219	#126	—	30	1	no	—
220	#125	—	15	1	no	—
221	#124	—	15	1	no	—
222	#123	—	30	1	no	—
223	C₂²⋊Q₈	—	15	1	no	—
224	#121	—	15	1	no	—
225	C₄×Q₈	—	15	1	no	—
226	C₄×D₄	—	15	1	no	—
227	C₄×D₄	—	30	1	no	—
228	#117	—	30	1	no	—
229	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	C₂×D₁₅	1	2	yes	—
232	#97	—	30	1	no	—
233	C₄×C₈	—	15	2	no	—
234	C₂×C₃×C₅	—	2	2	no	—
235	C₂×C₃×C₅	#408	1	1	yes	characteristic
236	C₂×C₃×C₅	#417	1	1	yes	characteristic
237	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
238	D₁₅	—	8	1	no	—
239	C₂³×C₃	#685	1	1	yes	characteristic
240	C₂²×D₃	—	10	1	no	—
241	C₂×C₃×C₄	—	2	2	no	—
242	C₂×C₃×C₄	—	2	2	no	—
243	C₂×C₃×C₄	—	2	1	no	—
244	C₂×C₃×C₄	—	2	1	no	—
245	C₂×C₃×C₄	#685	1	1	yes	characteristic
246	C₂×C₃×C₄	#685	1	1	yes	characteristic
247	C₂×C₃×C₄	#685	1	1	yes	characteristic
248	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
249	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
250	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
251	#67	—	20	2	no	—
252	C₂×Dic₃	—	10	1	no	—
253	C₂×Dic₃	—	10	1	no	—
254	C₂×Dic₃	—	10	1	no	—
255	D₁₂	—	10	2	no	—
256	C₄×D₃	—	20	1	no	—
257	C₄×D₃	—	20	1	no	—
258	Dic₆	—	10	2	no	—
259	C₃×C₈	—	4	2	no	—
260	C₃⋊C₈	—	10	4	no	—
261	C₂²×C₅	—	2	1	no	—
262	C₂²×C₅	—	2	1	no	—
263	C₂²×C₅	#825	1	2	yes	—
264	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
265	C₂×D₅	—	12	1	no	—
266	C₂×D₅	—	12	1	no	—
267	C₂×D₅	—	12	1	no	—
268	C₄×C₅	—	2	1	no	—
269	C₄×C₅	—	2	1	no	—
270	C₄×C₅	—	2	1	no	—
271	C₄×C₅	—	2	1	no	—
272	C₄×C₅	#901	1	2	yes	—
273	C₄×C₅	#932	1	1	yes	characteristic
274	C₄×C₅	#942	1	1	yes	characteristic
275	Dic₅	—	12	1	no	—
276	Dic₅	—	12	1	no	—
277	Dic₅	—	12	1	no	—
278	C₂×Q₈	—	15	1	no	—
279	C₂×D₄	—	15	1	no	—
280	C₂×D₄	—	30	1	no	—
281	C₂²×C₄	—	30	1	no	—
282	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
283	C₂²×C₄	D₆₀	1	1	yes	characteristic
284	C₂²×C₄	#1220	1	1	yes	characteristic
285	M₁₆	—	30	4	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	2	no	—
288	C₂×C₈	—	30	2	no	—
289	C₄⋊C₄	—	15	1	no	—
290	C₄⋊C₄	—	15	1	no	—
291	C₄⋊C₄	—	30	1	no	—
292	C₄⋊C₄	—	30	1	no	—
293	C₄⋊C₄	—	30	1	no	—
294	C₄⋊C₄	—	30	1	no	—
295	C₂²⋊C₄	—	30	1	no	—
296	C₂²⋊C₄	—	30	1	no	—
297	C₂²⋊C₄	—	30	1	no	—
298	C₂²⋊C₄	—	30	1	no	—
299	C₂²⋊C₄	—	30	1	no	—
300	C₄²	—	30	1	no	—
301	C₄²	C₄×D₁₅	1	1	yes	characteristic
302	C₄²	D₆₀	1	1	yes	characteristic
303	C₄²	#1220	1	1	yes	characteristic
304	C₄²	C₂²×D₁₅	1	1	yes	characteristic
305	C₃×C₅	#1483	1	1	yes	characteristic, complemented, Hall
306	C₂²×C₃	—	2	1	no	—
307	C₂²×C₃	—	2	1	no	—
308	C₂²×C₃	#3957	1	2	yes	—
309	C₂²×C₃	C₂×#685	1	1	yes	characteristic
310	C₂×D₃	—	20	1	no	—
311	C₂×D₃	—	20	1	no	—
312	C₂×D₃	—	20	1	no	—
313	C₃×C₄	—	2	1	no	—
314	C₃×C₄	—	2	1	no	—
315	C₃×C₄	—	2	1	no	—
316	C₃×C₄	—	2	1	no	—
317	C₃×C₄	#4047	1	2	yes	—
318	C₃×C₄	#4078	1	1	yes	characteristic
319	C₃×C₄	#4088	1	1	yes	characteristic
320	Dic₃	—	20	1	no	—
321	Dic₃	—	20	1	no	—
322	Dic₃	—	20	1	no	—
323	C₂×C₅	—	2	2	no	—
324	C₂×C₅	#4966	1	1	yes	characteristic
325	C₂×C₅	#5110	1	1	yes	characteristic
326	C₂×C₅	C₂×#825	1	1	yes	characteristic
327	D₅	—	24	1	no	—
328	C₂³	—	30	1	no	—
329	C₂³	#6714	1	1	yes	characteristic
330	Q₈	—	30	2	no	—
331	D₄	—	30	2	no	—
332	D₄	—	60	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	30	1	no	—
338	C₂×C₄	—	30	1	no	—
339	C₂×C₄	—	30	1	no	—
340	C₂×C₄	—	60	1	no	—
341	C₂×C₄	—	60	1	no	—
342	C₂×C₄	#6714	1	1	yes	characteristic
343	C₂×C₄	#6714	1	1	yes	characteristic
344	C₂×C₄	#6714	1	1	yes	characteristic
345	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
346	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
347	C₂×C₄	C₂×#1220	1	1	yes	characteristic
348	C₈	—	4	2	no	—
349	C₈	—	30	4	no	—
350	C₂×C₃	—	2	2	no	—
351	C₂×C₃	#65255	1	1	yes	characteristic
352	C₂×C₃	#65399	1	1	yes	characteristic
353	C₂×C₃	C₂×#3957	1	1	yes	characteristic
354	D₃	—	40	1	no	—
355	C₅	#70427	1	1	yes	characteristic, complemented, Sylow
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	60	1	no	—
359	C₂²	—	60	1	no	—
360	C₂²	—	60	1	no	—
361	C₂²	#91109	1	2	yes	—
362	C₂²	C₂×#6714	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	60	1	no	—
368	C₄	—	60	1	no	—
369	C₄	—	60	1	no	—
370	C₄	#91790	1	2	yes	—
371	C₄	#91821	1	1	yes	characteristic
372	C₄	#91831	1	1	yes	characteristic
373	C₃	#10601220	1	1	yes	characteristic, complemented, Sylow
374	C₂	—	2	2	no	—
375	C₂	—	120	1	no	—
376	C₂	#11751067	1	1	yes	characteristic
377	C₂	#11751211	1	1	yes	characteristic
378	C₂	C₂×#91109	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1220
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₂×C₃×C₅×C₄⋊C₈	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	#91819	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₅
hypocenter	C₃×C₅	#1483
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×C₄⋊C₈	C₂	semidirect
3. C₄×#6710	C₂	semidirect
4. #11746305	C₂	semidirect
5. #11746304	C₂	semidirect
26. C₂×C₅×C₄⋊C₈	D₃	semidirect
58. C₂×C₃×C₄⋊C₈	D₅	semidirect
147. C₂×C₄⋊C₈	D₁₅	semidirect
305. C₃×C₅	#1483	semidirect
355. C₅	#70427	semidirect
373. C₃	#10601220	semidirect

Maximal subgroups

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

subgroup	index
#11756238	2
C₂×C₃×C₅×C₄⋊C₈	2
C₄×#6710	2
#11746305	2
#11746304	2
#10601220	3
#70427	5