Subgroups — #14284876

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752550	C₂	1	1	yes	maximal, characteristic
2	C₅×#4837	C₂	1	1	yes	maximal, characteristic, complemented
3	D₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746306	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64702	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745856	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745854	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601296	—	3	1	no	maximal, Hall
9	C₂×#6769	—	2	1	no	—
10	#91763	C₂²	1	1	yes	characteristic
11	C₂×C₅×ΓL(1,𝔽₂₅)	—	2	1	no	—
12	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
13	C₃×#4015	C₂²	1	1	yes	characteristic
14	#91460	—	2	1	no	—
15	#91460	C₄	1	2	yes	—
16	#91428	—	2	1	no	—
17	#91428	C₄	1	2	yes	—
18	#91374	—	2	1	no	—
19	C₂×D₃×#168	—	2	1	no	—
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₅×Dic₃⋊C₈	—	2	1	no	—
23	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
25	Dic₃⋊#168	—	2	1	no	—
26	#68421	—	5	1	no	maximal, Hall
27	#65811	—	3	1	no	—
28	C₅×#423	—	3	1	no	—
29	C₂×C₄⋊#168	—	3	1	no	—
30	#64705	—	3	1	no	—
31	#64705	—	3	1	no	—
32	#64702	—	3	1	no	—
33	#64702	D₃	1	1	yes	characteristic, complemented
34	C₂×D₆₀	—	2	1	no	—
35	C₂×D₆₀	—	2	1	no	—
36	C₂×D₆₀	C₂×C₄	1	2	yes	—
37	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
39	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
40	C₂×C₃×Dic₁₀	—	2	1	no	—
41	C₂×#1203	—	2	2	no	—
42	C₂×#1200	—	2	2	no	—
43	#6769	—	4	2	no	—
44	#6769	—	4	2	no	—
45	C₂×C₃×C₅×C₈	—	2	1	no	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₃⋊#168	—	2	1	no	—
48	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
50	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
51	C₂×C₅×C₃⋊C₈	—	2	1	no	—
52	C₅×ΓL(1,𝔽₂₅)	—	4	2	no	—
53	C₃×C₄⋊Dic₅	—	2	1	no	—
54	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
55	C₂×C₃×#168	—	2	1	no	—
56	#6667	—	2	2	no	—
57	#6667	—	2	2	no	—
58	D₃×#168	—	4	2	no	—
59	#5725	—	5	1	no	—
60	#4837	D₅	1	1	yes	characteristic, complemented
61	D₃×C₄⋊C₈	—	5	1	no	—
62	C₃×#332	—	5	1	no	—
63	#4547	—	5	1	no	—
64	#4485	—	5	1	no	—
65	#4485	—	5	1	no	—
66	C₂×#709	—	6	1	no	—
67	C₂×C₅×M₁₆	—	6	1	no	—
68	C₂×C₄²×C₅	—	3	1	no	—
69	#4076	—	3	2	no	—
70	#4076	—	6	1	no	—
71	C₂²×#168	—	6	1	no	—
72	#4021	—	6	1	no	—
73	#4020	—	3	1	no	—
74	C₄×D₂₀	—	3	2	no	—
75	C₄×D₂₀	—	6	1	no	—
76	#4015	C₂×D₃	1	1	yes	characteristic
77	C₅×C₄⋊C₈	—	3	1	no	—
78	C₅×C₄⋊C₈	—	6	1	no	—
79	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
80	C₄⋊#168	—	3	1	no	—
81	C₄⋊#168	—	6	1	no	—
82	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
83	#1547	—	15	1	no	Sylow
84	C₂²×D₁₅	—	2	2	no	—
85	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	D₆₀	—	4	2	no	—
90	D₆₀	—	4	1	no	—
91	D₆₀	—	4	1	no	—
92	D₆₀	—	4	1	no	—
93	D₆₀	—	4	1	no	—
94	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₅×D₃	—	2	1	no	—
98	C₄×C₅×D₃	—	2	1	no	—
99	C₄×C₅×D₃	—	4	1	no	—
100	C₄×C₅×D₃	D₈	1	2	yes	—
101	C₄×C₅×D₃	SD₁₆	1	2	yes	—
102	C₂×C₃×Dic₅	—	2	2	no	—
103	C₃×Dic₁₀	—	4	1	no	—
104	C₃×Dic₁₀	—	4	1	no	—
105	#1203	—	4	4	no	—
106	#1200	—	4	4	no	—
107	C₃×C₅×C₈	—	4	1	no	—
108	C₃⋊#168	—	4	1	no	—
109	C₃×#168	—	4	1	no	—
110	C₅×C₃⋊C₈	—	4	1	no	—
111	C₂×#239	—	10	1	no	—
112	C₃×C₄⋊Q₈	—	5	1	no	—
113	#947	—	10	1	no	—
114	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
115	C₂×D₃×C₈	—	10	1	no	—
116	#895	—	5	2	no	—
117	#895	—	10	1	no	—
118	#893	—	5	2	no	—
119	#893	—	10	1	no	—
120	#874	—	5	1	no	—
121	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
122	C₃×C₄⋊C₈	—	5	1	no	—
123	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
124	Dic₃⋊C₈	—	2	1	no	—
125	Dic₃⋊C₈	—	10	1	no	—
126	C₄⋊(C₃⋊C₈)	—	5	1	no	—
127	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
128	C₂²×C₄×C₅	—	3	1	no	—
129	C₂²×C₄×C₅	—	3	1	no	—
130	C₂²×C₄×C₅	—	3	1	no	—
131	C₂×C₂²⋊D₅	—	6	2	no	—
132	#709	—	12	2	no	—
133	#709	—	12	2	no	—
134	C₂×D₂₀	—	3	2	no	—
135	C₂×D₂₀	—	6	1	no	—
136	C₂×D₂₀	—	6	1	no	—
137	C₂×C₄×D₅	—	6	2	no	—
138	C₂×Dic₁₀	—	2	1	no	—
139	C₅×M₁₆	—	12	2	no	—
140	C₂×C₅×C₈	—	2	1	no	—
141	C₂×C₅×C₈	—	6	1	no	—
142	C₄²×C₅	—	3	1	no	—
143	C₄²×C₅	—	3	1	no	—
144	C₄²×C₅	—	3	1	no	—
145	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
146	#685	—	6	2	no	—
147	#685	—	6	2	no	—
148	C₄⋊Dic₅	—	2	1	no	—
149	C₄⋊Dic₅	C₄×D₃	1	2	yes	—
150	C₂×#168	—	2	1	no	—
151	C₂×#168	—	6	1	no	—
152	C₂×#168	—	12	2	no	—
153	#532	—	15	1	no	—
154	#423	—	3	1	no	—
155	C₂×C₄⋊C₈	—	15	1	no	—
156	#332	—	5	1	no	—
157	#332	—	15	1	no	—
158	#331	—	15	1	no	—
159	#331	—	15	1	no	—
160	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×D₁₅	—	4	2	no	—
162	C₂×D₁₅	—	4	2	no	—
163	C₂×D₁₅	—	4	2	no	—
164	C₂×C₅×D₃	—	2	1	no	—
165	C₂×C₅×D₃	—	2	1	no	—
166	C₂×C₅×D₃	#102	1	2	yes	—
167	C₃×C₄×C₅	—	2	1	no	—
168	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
169	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
170	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₃×C₄×C₅	#137	1	1	yes	characteristic
172	C₃×Dic₅	—	4	2	no	—
173	C₅×Dic₃	—	2	1	no	—
174	C₅×Dic₃	—	2	1	no	—
175	C₅×Dic₃	#102	1	2	yes	—
176	C₂×C₃×Q₈	—	10	1	no	—
177	#239	—	20	2	no	—
178	#239	—	20	2	no	—
179	C₂×D₁₂	—	5	2	no	—
180	C₂×D₁₂	—	10	2	no	—
181	C₂×D₁₂	—	10	1	no	—
182	C₂×D₁₂	—	10	1	no	—
183	C₂×C₄×D₃	—	10	2	no	—
184	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
185	C₂×C₄×D₃	D₂₀	1	1	yes	characteristic
186	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
187	C₂×C₃×C₈	—	2	1	no	—
188	C₂×C₃×C₈	—	10	1	no	—
189	C₃×C₄⋊C₄	—	5	2	no	—
190	C₃×C₄⋊C₄	—	10	1	no	—
191	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
192	(C₂×D₃)⋊C₄	—	10	2	no	—
193	(C₂×D₃)⋊C₄	—	10	2	no	—
194	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
195	C₄×Dic₃	D₂₀	1	1	yes	characteristic
196	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
197	C₂×C₃⋊C₈	—	2	1	no	—
198	C₂×C₃⋊C₈	—	10	1	no	—
199	ΓL(1,𝔽₂₅)	—	4	2	no	—
200	D₃×C₈	—	20	2	no	—
201	C₂³×C₅	—	3	1	no	—
202	C₂²×D₅	—	6	2	no	—
203	C₂×C₄×C₅	—	3	1	no	—
204	C₂×C₄×C₅	—	3	2	no	—
205	C₂×C₄×C₅	—	3	2	no	—
206	C₂×C₄×C₅	—	3	1	no	—
207	C₂×C₄×C₅	—	3	1	no	—
208	C₂×C₄×C₅	—	6	1	no	—
209	C₂×C₄×C₅	—	6	1	no	—
210	C₂×C₄×C₅	—	12	1	no	—
211	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
212	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
213	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
214	C₂²⋊D₅	—	12	4	no	—
215	C₂×Dic₅	—	2	2	no	—
216	D₂₀	—	12	2	no	—
217	D₂₀	—	12	1	no	—
218	D₂₀	—	12	1	no	—
219	D₂₀	—	12	1	no	—
220	D₂₀	—	12	1	no	—
221	C₄×D₅	—	12	4	no	—
222	Dic₁₀	—	4	1	no	—
223	Dic₁₀	—	4	1	no	—
224	C₅×C₈	—	4	1	no	—
225	C₅×C₈	—	12	1	no	—
226	#168	—	4	1	no	—
227	#168	—	12	1	no	—
228	C₂×#41	—	30	1	no	—
229	C₂×M₁₆	—	6	1	no	—
230	C₂²×C₈	—	30	1	no	—
231	C₄⋊Q₈	—	5	1	no	—
232	#127	—	15	1	no	—
233	#124	—	30	1	no	—
234	#121	—	15	2	no	—
235	#121	—	30	1	no	—
236	C₄×D₄	—	15	2	no	—
237	C₄×D₄	—	30	1	no	—
238	C₂×C₄²	—	3	1	no	—
239	C₄⋊C₈	—	3	1	no	—
240	C₄⋊C₈	—	5	1	no	—
241	C₄⋊C₈	—	6	1	no	—
242	C₄⋊C₈	—	15	1	no	—
243	C₄⋊C₈	—	30	1	no	—
244	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
245	C₂×C₃×C₅	#413	1	1	yes	characteristic
246	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
247	C₂×C₃×C₅	#417	1	1	yes	characteristic
248	D₁₅	—	8	2	no	—
249	C₅×D₃	—	2	2	no	—
250	C₂²×D₃	—	10	2	no	—
251	C₂²×D₃	#685	1	1	yes	characteristic
252	C₃×Q₈	—	20	1	no	—
253	C₃×Q₈	—	20	1	no	—
254	C₂×C₃×C₄	—	10	2	no	—
255	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
256	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
257	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
258	C₂×Dic₃	#685	1	1	yes	characteristic
259	C₂×Dic₃	#685	1	1	yes	characteristic
260	C₂×Dic₃	#685	1	1	yes	characteristic
261	D₁₂	—	20	2	no	—
262	D₁₂	—	20	4	no	—
263	D₁₂	—	20	1	no	—
264	D₁₂	—	20	1	no	—
265	D₁₂	—	20	1	no	—
266	D₁₂	—	20	1	no	—
267	C₄×D₃	—	2	1	no	—
268	C₄×D₃	—	2	1	no	—
269	C₄×D₃	—	4	1	no	—
270	C₄×D₃	—	20	4	no	—
271	C₄×D₃	#686	1	2	yes	—
272	C₄×D₃	#688	1	2	yes	—
273	C₃×C₈	—	4	1	no	—
274	C₃×C₈	—	20	1	no	—
275	C₃⋊C₈	—	4	1	no	—
276	C₃⋊C₈	—	20	1	no	—
277	C₂²×C₅	—	3	2	no	—
278	C₂²×C₅	—	6	1	no	—
279	C₂²×C₅	—	6	1	no	—
280	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×D₅	—	12	2	no	—
282	C₂×D₅	—	12	2	no	—
283	C₂×D₅	—	12	2	no	—
284	C₄×C₅	—	2	1	no	—
285	C₄×C₅	—	3	2	no	—
286	C₄×C₅	—	6	1	no	—
287	C₄×C₅	—	6	1	no	—
288	C₄×C₅	D₃×D₈	1	1	yes	characteristic
289	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
290	C₄×C₅	#914	1	1	yes	characteristic
291	C₄×C₅	#918	1	1	yes	characteristic
292	Dic₅	—	4	2	no	—
293	#41	—	60	2	no	—
294	#41	—	60	2	no	—
295	C₂×Q₈	—	10	1	no	—
296	C₂×D₄	—	15	2	no	—
297	C₂×D₄	—	30	2	no	—
298	C₂×D₄	—	30	1	no	—
299	C₂×D₄	—	30	1	no	—
300	C₂²×C₄	—	3	1	no	—
301	C₂²×C₄	—	3	1	no	—
302	C₂²×C₄	—	3	1	no	—
303	C₂²×C₄	—	30	2	no	—
304	M₁₆	—	12	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	6	1	no	—
307	C₂×C₈	—	10	1	no	—
308	C₂×C₈	—	30	1	no	—
309	C₂×C₈	—	60	2	no	—
310	C₄⋊C₄	—	5	2	no	—
311	C₄⋊C₄	—	10	1	no	—
312	C₂²⋊C₄	—	30	2	no	—
313	C₂²⋊C₄	—	30	2	no	—
314	C₄²	—	3	1	no	—
315	C₄²	—	3	1	no	—
316	C₄²	—	3	1	no	—
317	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
318	C₃×C₅	#1547	1	1	yes	characteristic, complemented, Hall
319	C₂²×C₃	C₂×#685	1	1	yes	characteristic
320	C₂×D₃	—	2	1	no	—
321	C₂×D₃	—	2	1	no	—
322	C₂×D₃	—	20	2	no	—
323	C₂×D₃	—	20	2	no	—
324	C₂×D₃	—	20	2	no	—
325	C₂×D₃	#3941	1	2	yes	—
326	C₃×C₄	—	2	1	no	—
327	C₃×C₄	—	20	2	no	—
328	C₃×C₄	#4054	1	1	yes	characteristic
329	C₃×C₄	#4055	1	1	yes	characteristic
330	C₃×C₄	C₂×#686	1	1	yes	characteristic
331	C₃×C₄	C₂×#688	1	1	yes	characteristic
332	Dic₃	—	2	1	no	—
333	Dic₃	—	2	1	no	—
334	Dic₃	#3941	1	2	yes	—
335	C₂×C₅	—	6	2	no	—
336	C₂×C₅	#4751	1	1	yes	characteristic
337	C₂×C₅	D₃×#102	1	1	yes	characteristic
338	C₂×C₅	#4803	1	1	yes	characteristic
339	D₅	—	24	2	no	—
340	C₂³	—	3	1	no	—
341	C₂³	—	30	2	no	—
342	Q₈	—	20	1	no	—
343	Q₈	—	20	1	no	—
344	D₄	—	60	2	no	—
345	D₄	—	60	4	no	—
346	D₄	—	60	1	no	—
347	D₄	—	60	1	no	—
348	D₄	—	60	1	no	—
349	D₄	—	60	1	no	—
350	C₂×C₄	—	3	1	no	—
351	C₂×C₄	—	3	2	no	—
352	C₂×C₄	—	3	2	no	—
353	C₂×C₄	—	3	1	no	—
354	C₂×C₄	—	3	1	no	—
355	C₂×C₄	—	6	1	no	—
356	C₂×C₄	—	6	1	no	—
357	C₂×C₄	—	10	2	no	—
358	C₂×C₄	—	12	1	no	—
359	C₂×C₄	—	60	4	no	—
360	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
361	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
362	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
363	C₈	—	4	1	no	—
364	C₈	—	12	1	no	—
365	C₈	—	20	1	no	—
366	C₈	—	60	1	no	—
367	C₂×C₃	C₂×#3941	1	1	yes	characteristic
368	C₂×C₃	#65421	1	1	yes	characteristic
369	C₂×C₃	#65422	1	1	yes	characteristic
370	D₃	—	2	2	no	—
371	D₃	—	40	2	no	—
372	C₅	#68421	1	1	yes	characteristic, complemented, Sylow
373	C₂²	—	3	2	no	—
374	C₂²	—	6	1	no	—
375	C₂²	—	6	1	no	—
376	C₂²	—	60	2	no	—
377	C₂²	—	60	2	no	—
378	C₂²	—	60	2	no	—
379	C₂²	D₃×#685	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	3	2	no	—
382	C₄	—	6	1	no	—
383	C₄	—	6	1	no	—
384	C₄	—	20	2	no	—
385	C₄	#91253	1	1	yes	characteristic
386	C₄	#91263	1	1	yes	characteristic
387	C₄	D₃×#686	1	1	yes	characteristic
388	C₄	D₃×#688	1	1	yes	characteristic
389	C₃	#10601296	1	1	yes	characteristic, complemented, Sylow
390	C₂	—	6	2	no	—
391	C₂	—	120	2	no	—
392	C₂	D₃×#3941	1	1	yes	characteristic
393	C₂	#11747608	1	1	yes	characteristic
394	C₂	#11748235	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×#685
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	#11752550	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊C₈	D₃×D₅
hypocenter	C₃×C₅	#1547
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₅×#4837	C₂	semidirect
3. D₃×C₄⋊#168	C₂	semidirect
4. #11746306	C₂	semidirect
5. C₃×#64702	C₂	semidirect
6. #11745856	C₂	semidirect
7. #11745854	C₂	semidirect
33. #64702	D₃	semidirect
60. #4837	D₅	semidirect
318. C₃×C₅	#1547	semidirect
372. C₅	#68421	semidirect
389. C₃	#10601296	semidirect

Maximal subgroups

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

subgroup	index
#11752550	2
C₅×#4837	2
D₃×C₄⋊#168	2
#11746306	2
C₃×#64702	2
#11745856	2
#11745854	2
#10601296	3
#68421	5