Subgroups — #14439143

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	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#91838	C₂	1	4	yes	maximal, complemented
3	C₂×#91763	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751186	C₂	1	2	yes	maximal, complemented
5	#11751174	C₂	1	2	yes	maximal, complemented
6	#11751159	C₂	1	4	yes	maximal, complemented
7	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
8	#10615018	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
10	C₂²×D₆₀	—	2	2	no	—
11	C₂²×D₆₀	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₃×C₅×#127	C₂²	1	1	yes	characteristic, complemented
14	C₃×C₅×#124	C₂²	1	2	yes	complemented
15	C₃×C₅×#121	—	2	2	no	—
16	C₃×C₄×C₅×D₄	—	2	2	no	—
17	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
18	#91838	—	2	16	no	—
19	C₂×#6717	C₂²	1	8	yes	complemented
20	C₂×#6715	C₂²	1	8	yes	complemented
21	C₂×#6710	C₂²	1	2	yes	—
22	#91763	—	2	2	no	—
23	#91763	C₂²	1	2	yes	—
24	#91763	C₂²	1	2	yes	—
25	#91087	C₂²	1	2	yes	complemented
26	C₄×C₃⋊#168	C₂²	1	2	yes	complemented
27	#83287	—	5	1	no	maximal, Hall
28	C₅×#532	D₃	1	1	yes	characteristic, complemented
29	C₂×#4095	—	3	4	no	—
30	C₂×#4020	—	3	1	no	—
31	#65374	—	3	2	no	—
32	#65362	—	3	2	no	—
33	#65347	—	3	4	no	—
34	C₄×#681	—	3	1	no	—
35	C₂³×D₁₅	—	2	2	no	—
36	C₃×C₅×#41	—	2	8	no	—
37	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
38	C₂×C₃×C₅×D₄	—	2	2	no	—
39	C₂×C₃×C₅×D₄	—	2	2	no	—
40	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
41	C₂²×C₃×C₄×C₅	—	2	2	no	—
42	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
43	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
44	C₂×D₆₀	—	2	8	no	—
45	C₂×D₆₀	—	2	2	no	—
46	C₂×D₆₀	—	2	4	no	—
47	C₂×D₆₀	—	2	4	no	—
48	C₂×D₆₀	—	4	8	no	—
49	C₂×D₆₀	C₂³	1	4	yes	—
50	C₃×C₅×C₄⋊C₄	—	2	2	no	—
51	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
52	C₃×C₄²×C₅	D₄	1	2	yes	—
53	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
55	#6717	—	2	16	no	—
56	#6715	—	2	16	no	—
57	#6710	D₄	1	8	yes	complemented
58	C₂×C₃⋊#168	C₂³	1	4	yes	—
59	C₃×#532	D₅	1	1	yes	characteristic, complemented
60	C₂×#949	—	5	4	no	—
61	C₂×#874	—	5	1	no	—
62	#5085	—	5	2	no	—
63	#5073	—	5	2	no	—
64	#5058	—	5	4	no	—
65	C₄×#208	—	5	1	no	—
66	C₂×C₅×#41	C₂×D₃	1	2	yes	—
67	C₂²×D₂₀	—	3	2	no	—
68	C₂²×D₂₀	—	6	2	no	—
69	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic, complemented
70	C₅×#127	C₂×D₃	1	1	yes	characteristic, complemented
71	C₅×#124	C₂×D₃	1	2	yes	complemented
72	C₅×#121	—	2	2	no	—
73	C₄×C₅×D₄	—	2	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	—	3	2	no	—
80	#4020	—	3	2	no	—
81	#4020	—	6	2	no	—
82	#3935	—	3	2	no	—
83	C₄×#168	—	3	2	no	—
84	#3151	—	15	1	no	Sylow
85	C₂³×C₃×C₅	—	2	2	no	—
86	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂²×D₁₅	—	2	4	no	—
88	C₂²×D₁₅	—	4	8	no	—
89	C₂²×D₁₅	—	4	4	no	—
90	C₃×C₅×Q₈	—	2	4	no	—
91	C₃×C₅×D₄	—	2	4	no	—
92	C₃×C₅×D₄	—	4	4	no	—
93	C₃×C₅×D₄	—	4	4	no	—
94	C₂×C₃×C₄×C₅	—	2	4	no	—
95	C₂×C₃×C₄×C₅	—	2	2	no	—
96	C₂×C₃×C₄×C₅	—	4	4	no	—
97	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
98	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
100	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
101	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
102	D₆₀	—	2	8	no	—
103	D₆₀	—	4	8	no	—
104	D₆₀	—	4	4	no	—
105	D₆₀	—	4	8	no	—
106	C₃⋊#168	C₂×D₄	1	8	yes	complemented
107	C₂×C₃×#41	C₂×D₅	1	2	yes	—
108	C₂²×D₁₂	—	5	2	no	—
109	C₂²×D₁₂	—	10	2	no	—
110	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic, complemented
111	C₃×#127	C₂×D₅	1	1	yes	characteristic, complemented
112	C₃×#124	C₂×D₅	1	2	yes	complemented
113	C₃×#121	—	2	2	no	—
114	C₃×C₄×D₄	—	2	2	no	—
115	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
116	#949	—	10	16	no	—
117	C₂×#215	—	5	8	no	—
118	C₂×#213	—	5	8	no	—
119	C₂×#208	—	5	2	no	—
120	#874	—	5	2	no	—
121	#874	—	5	2	no	—
122	#874	—	10	2	no	—
123	#803	—	5	2	no	—
124	C₄×C₃⋊C₈	—	5	2	no	—
125	C₂³×D₅	—	6	2	no	—
126	C₅×#41	—	2	8	no	—
127	C₂×C₅×Q₈	C₂²×D₃	1	2	yes	—
128	C₂×C₅×D₄	—	2	2	no	—
129	C₂×C₅×D₄	—	2	2	no	—
130	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
131	C₂²×C₄×C₅	—	2	2	no	—
132	C₂²×C₄×C₅	#67	1	2	yes	—
133	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
134	C₂×D₂₀	—	3	4	no	—
135	C₂×D₂₀	—	6	8	no	—
136	C₂×D₂₀	—	6	2	no	—
137	C₂×D₂₀	—	6	4	no	—
138	C₂×D₂₀	—	6	4	no	—
139	C₂×D₂₀	—	12	8	no	—
140	C₅×C₄⋊C₄	—	2	2	no	—
141	C₅×C₂²⋊C₄	—	2	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	D₁₅	1	1	yes	characteristic, complemented
151	C₂×#127	—	15	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	—	4	2	no	—
158	C₂²×C₃×C₅	—	4	4	no	—
159	C₂²×C₃×C₅	#127	1	2	yes	—
160	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₁₅	—	4	4	no	—
162	C₂×D₁₅	—	4	8	no	—
163	C₂×D₁₅	—	8	8	no	—
164	C₃×C₄×C₅	—	2	2	no	—
165	C₃×C₄×C₅	—	4	2	no	—
166	C₃×C₄×C₅	#127	1	2	yes	—
167	C₃×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	complemented
168	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
169	C₂³×D₃	—	10	2	no	—
170	C₃×#41	—	2	8	no	—
171	C₂×C₃×Q₈	C₂²×D₅	1	2	yes	—
172	C₂×C₃×D₄	—	2	2	no	—
173	C₂×C₃×D₄	—	2	2	no	—
174	C₂×C₃×D₄	C₂²×D₅	1	2	yes	—
175	C₂²×C₃×C₄	—	2	2	no	—
176	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
177	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
178	C₂×D₁₂	—	5	4	no	—
179	C₂×D₁₂	—	10	8	no	—
180	C₂×D₁₂	—	10	4	no	—
181	C₂×D₁₂	—	10	4	no	—
182	C₂×D₁₂	—	10	2	no	—
183	C₂×D₁₂	—	20	8	no	—
184	C₃×C₄⋊C₄	—	2	2	no	—
185	C₃×C₂²⋊C₄	—	2	4	no	—
186	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
187	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
188	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
189	#215	—	10	16	no	—
190	#213	—	10	16	no	—
191	#208	—	5	8	no	—
192	C₂×C₃⋊C₈	—	5	4	no	—
193	C₂³×C₅	—	2	2	no	—
194	C₂³×C₅	C₂×#67	1	1	yes	characteristic
195	C₂²×D₅	—	6	4	no	—
196	C₂²×D₅	—	12	8	no	—
197	C₂²×D₅	—	12	4	no	—
198	C₅×Q₈	—	2	4	no	—
199	C₅×D₄	—	2	4	no	—
200	C₅×D₄	—	4	4	no	—
201	C₅×D₄	—	4	4	no	—
202	C₂×C₄×C₅	—	2	4	no	—
203	C₂×C₄×C₅	—	2	2	no	—
204	C₂×C₄×C₅	—	4	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	1	yes	characteristic
208	C₂×C₄×C₅	C₂×#67	1	2	yes	—
209	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
210	D₂₀	—	6	8	no	—
211	D₂₀	—	12	8	no	—
212	D₂₀	—	12	4	no	—
213	D₂₀	—	12	8	no	—
214	#168	—	3	8	no	—
215	C₂×#41	C₂×D₁₅	1	2	yes	—
216	C₂²×D₄	—	15	2	no	—
217	C₂²×D₄	—	30	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	C₂×D₁₅	1	1	yes	characteristic, complemented
223	#127	—	15	2	no	—
224	#127	—	15	2	no	—
225	#127	—	30	2	no	—
226	#127	C₂×D₁₅	1	1	yes	characteristic, complemented
227	#124	C₂×D₁₅	1	2	yes	complemented
228	#121	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	C₂×C₄²	C₂×D₁₅	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	—	8	2	no	—
235	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
236	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
237	D₁₅	—	8	4	no	—
238	C₂³×C₃	—	2	2	no	—
239	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
240	C₂²×D₃	—	10	4	no	—
241	C₂²×D₃	—	20	8	no	—
242	C₂²×D₃	—	20	4	no	—
243	C₃×Q₈	—	2	4	no	—
244	C₃×D₄	—	2	4	no	—
245	C₃×D₄	—	4	4	no	—
246	C₃×D₄	—	4	4	no	—
247	C₂×C₃×C₄	—	2	4	no	—
248	C₂×C₃×C₄	—	2	2	no	—
249	C₂×C₃×C₄	—	4	4	no	—
250	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
251	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
252	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
253	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
254	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
255	D₁₂	—	10	8	no	—
256	D₁₂	—	20	8	no	—
257	D₁₂	—	20	4	no	—
258	D₁₂	—	20	8	no	—
259	C₃⋊C₈	—	5	8	no	—
260	C₂²×C₅	—	2	2	no	—
261	C₂²×C₅	—	4	2	no	—
262	C₂²×C₅	—	4	4	no	—
263	C₂²×C₅	#940	1	2	yes	—
264	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
265	C₂×D₅	—	12	4	no	—
266	C₂×D₅	—	12	8	no	—
267	C₂×D₅	—	24	8	no	—
268	C₄×C₅	—	2	2	no	—
269	C₄×C₅	—	4	2	no	—
270	C₄×C₅	#940	1	2	yes	—
271	C₄×C₅	#949	1	4	yes	complemented
272	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
273	C₂⁴	—	30	2	no	—
274	#41	—	2	8	no	—
275	C₂×Q₈	C₂²×D₁₅	1	2	yes	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	15	4	no	—
279	C₂×D₄	—	30	8	no	—
280	C₂×D₄	—	30	4	no	—
281	C₂×D₄	—	30	4	no	—
282	C₂×D₄	—	30	2	no	—
283	C₂×D₄	—	60	8	no	—
284	C₂×D₄	C₂²×D₁₅	1	2	yes	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	#1220	1	2	yes	—
287	C₂²×C₄	C₂²×D₁₅	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	—	2	2	no	—
293	C₂²⋊C₄	—	2	4	no	—
294	C₄²	#1220	1	2	yes	—
295	C₄²	C₂²×D₁₅	1	1	yes	characteristic
296	C₄²	C₂²×D₁₅	1	1	yes	characteristic
297	C₃×C₅	#3151	1	1	yes	characteristic, complemented, Hall
298	C₂²×C₃	—	2	2	no	—
299	C₂²×C₃	—	4	2	no	—
300	C₂²×C₃	—	4	4	no	—
301	C₂²×C₃	#4086	1	2	yes	—
302	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
303	C₂×D₃	—	20	4	no	—
304	C₂×D₃	—	20	8	no	—
305	C₂×D₃	—	40	8	no	—
306	C₃×C₄	—	2	2	no	—
307	C₃×C₄	—	4	2	no	—
308	C₃×C₄	#4086	1	2	yes	—
309	C₃×C₄	#4095	1	4	yes	complemented
310	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
311	C₂×C₅	—	2	2	no	—
312	C₂×C₅	—	8	2	no	—
313	C₂×C₅	C₂×#940	1	1	yes	characteristic
314	C₂×C₅	C₂×#949	1	2	yes	—
315	D₅	—	24	4	no	—
316	C₂³	—	2	2	no	—
317	C₂³	—	30	4	no	—
318	C₂³	—	60	8	no	—
319	C₂³	—	60	4	no	—
320	C₂³	C₂×#1220	1	1	yes	characteristic
321	Q₈	—	2	4	no	—
322	D₄	—	2	4	no	—
323	D₄	—	4	4	no	—
324	D₄	—	4	4	no	—
325	D₄	—	30	8	no	—
326	D₄	—	60	8	no	—
327	D₄	—	60	4	no	—
328	D₄	—	60	8	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	D₄×D₁₅	1	4	yes	—
333	C₂×C₄	C₂×#1220	1	2	yes	—
334	C₂×C₄	C₂×#1220	1	1	yes	characteristic
335	C₂×C₄	C₂×#1220	1	2	yes	—
336	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
337	C₈	—	15	8	no	—
338	C₂×C₃	—	2	2	no	—
339	C₂×C₃	—	8	2	no	—
340	C₂×C₃	C₂×#4086	1	1	yes	characteristic
341	C₂×C₃	C₂×#4095	1	2	yes	—
342	D₃	—	40	4	no	—
343	C₅	#83287	1	1	yes	characteristic, complemented, Sylow
344	C₂²	—	2	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	4	no	—
347	C₂²	—	60	4	no	—
348	C₂²	—	60	8	no	—
349	C₂²	—	120	8	no	—
350	C₂²	#91829	1	2	yes	—
351	C₂²	C₂²×#1220	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	4	2	no	—
354	C₄	#91829	1	2	yes	—
355	C₄	#91838	1	4	yes	complemented
356	C₄	C₂×D₄×D₁₅	1	2	yes	—
357	C₃	#10615018	1	1	yes	characteristic, complemented, Sylow
358	C₂	—	2	2	no	—
359	C₂	—	8	2	no	—
360	C₂	—	120	4	no	—
361	C₂	C₂×#91829	1	1	yes	characteristic
362	C₂	C₂×#91838	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#1220
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₃×C₅×#532	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	#532	D₁₅
hypocenter	C₃×C₅	#3151
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#532	C₂	semidirect
2. C₂×#91838	C₂	semidirect
3. C₂×#91763	C₂	semidirect
4. #11751186	C₂	semidirect
5. #11751174	C₂	semidirect
6. #11751159	C₂	semidirect
7. C₄×#6710	C₂	semidirect
12. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
13. C₃×C₅×#127	C₂²	semidirect
14. C₃×C₅×#124	C₂²	semidirect
19. C₂×#6717	C₂²	semidirect
20. C₂×#6715	C₂²	semidirect
25. #91087	C₂²	semidirect
26. C₄×C₃⋊#168	C₂²	semidirect
28. C₅×#532	D₃	semidirect
57. #6710	D₄	semidirect
59. C₃×#532	D₅	semidirect
69. C₅×C₄⋊Q₈	C₂×D₃	semidirect
70. C₅×#127	C₂×D₃	semidirect
71. C₅×#124	C₂×D₃	semidirect
106. C₃⋊#168	C₂×D₄	semidirect
110. C₃×C₄⋊Q₈	C₂×D₅	semidirect
111. C₃×#127	C₂×D₅	semidirect
112. C₃×#124	C₂×D₅	semidirect
150. #532	D₁₅	semidirect
167. C₃×C₄×C₅	AGL(1,ℤ/8)	semidirect
222. C₄⋊Q₈	C₂×D₁₅	semidirect
226. #127	C₂×D₁₅	semidirect
227. #124	C₂×D₁₅	semidirect
271. C₄×C₅	#949	semidirect
297. C₃×C₅	#3151	semidirect
309. C₃×C₄	#4095	semidirect
343. C₅	#83287	semidirect
355. C₄	#91838	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
C₃×C₅×#532	2
C₂×#91838	2
C₂×#91763	2
#11751186	2
#11751174	2
#11751159	2
C₄×#6710	2
#10615018	3
#83287	5