Subgroups — #14285429

Order 1920.

508 conjugacy classes of subgroups.

The subgroup lattice has 388 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	#11747011	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746306	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64701	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745861	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745854	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601298	—	3	1	no	maximal, Hall
9	C₂×#6769	C₄	1	2	yes	—
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	—	2	1	no	—
16	#91428	—	2	1	no	—
17	#91428	—	2	1	no	—
18	#91374	C₄	1	2	yes	—
19	C₂×#6650	—	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	#68489	—	5	1	no	maximal, Hall
27	#65811	—	3	1	no	—
28	C₅×#423	—	3	1	no	—
29	#65213	—	3	1	no	—
30	#64705	—	3	1	no	—
31	#64705	—	3	1	no	—
32	#64701	—	3	1	no	—
33	#64701	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	C₂×C₄	1	2	yes	—
41	C₂×#1203	—	2	2	no	—
42	C₂×#1200	—	2	2	no	—
43	#6769	—	4	4	no	—
44	C₂×C₃×C₅×C₈	—	2	1	no	—
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
49	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
50	C₂×C₅×C₃⋊C₈	—	2	1	no	—
51	C₅×ΓL(1,𝔽₂₅)	—	4	2	no	—
52	C₃×C₄⋊Dic₅	—	2	1	no	—
53	C₃×C₄⋊Dic₅	—	2	1	no	—
54	C₂×C₃×#168	—	2	1	no	—
55	#6667	—	2	2	no	—
56	#6667	—	2	2	no	—
57	#6650	—	4	2	no	—
58	#5725	—	5	1	no	—
59	#4837	—	5	1	no	—
60	#4837	D₅	1	1	yes	characteristic, complemented
61	C₃×#333	—	5	1	no	—
62	#4546	—	5	1	no	—
63	#4485	—	5	1	no	—
64	#4485	—	5	1	no	—
65	C₂×#709	—	3	2	no	—
66	C₂×C₅×M₁₆	—	6	1	no	—
67	C₂×C₄²×C₅	—	3	1	no	—
68	#4076	—	6	1	no	—
69	#4076	—	6	1	no	—
70	C₂×#681	—	6	1	no	—
71	#4021	—	3	2	no	—
72	#4020	—	3	1	no	—
73	C₄×D₂₀	—	6	1	no	—
74	C₄×D₂₀	—	6	1	no	—
75	#4015	C₂×D₃	1	1	yes	characteristic
76	C₅×C₄⋊C₈	—	3	1	no	—
77	C₅×C₄⋊C₈	—	6	1	no	—
78	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
79	C₄⋊#168	—	3	1	no	—
80	C₄⋊#168	—	6	1	no	—
81	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
82	#1560	—	15	1	no	Sylow
83	C₂²×D₁₅	—	2	2	no	—
84	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	D₆₀	—	4	2	no	—
89	D₆₀	—	4	1	no	—
90	D₆₀	—	4	1	no	—
91	D₆₀	—	4	1	no	—
92	D₆₀	—	4	1	no	—
93	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₄×C₅×D₃	—	2	1	no	—
97	C₄×C₅×D₃	—	2	1	no	—
98	C₄×C₅×D₃	—	2	1	no	—
99	C₄×C₅×D₃	—	2	1	no	—
100	C₄×C₅×D₃	—	2	2	no	—
101	C₂×C₃×Dic₅	—	2	2	no	—
102	C₃×Dic₁₀	—	4	2	no	—
103	#1203	—	4	4	no	—
104	#1200	—	4	4	no	—
105	C₃×C₅×C₈	—	4	1	no	—
106	C₃⋊#168	—	4	1	no	—
107	C₃×#168	—	4	1	no	—
108	C₅×C₃⋊C₈	—	4	1	no	—
109	C₂×#239	—	5	2	no	—
110	C₃×C₄⋊Q₈	—	5	1	no	—
111	#947	—	5	2	no	—
112	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
113	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
114	#895	—	10	1	no	—
115	#895	—	10	1	no	—
116	#893	—	10	1	no	—
117	#893	—	10	1	no	—
118	#874	—	5	1	no	—
119	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
120	C₃×C₄⋊C₈	—	5	1	no	—
121	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
122	Dic₃⋊C₈	—	2	1	no	—
123	Dic₃⋊C₈	—	10	1	no	—
124	C₄⋊(C₃⋊C₈)	—	5	1	no	—
125	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
126	C₂²×C₄×C₅	—	3	1	no	—
127	C₂²×C₄×C₅	—	3	1	no	—
128	C₂²×C₄×C₅	—	3	1	no	—
129	C₂×C₂²⋊D₅	—	6	2	no	—
130	#709	—	12	4	no	—
131	C₂×D₂₀	—	3	2	no	—
132	C₂×D₂₀	—	6	1	no	—
133	C₂×D₂₀	—	6	1	no	—
134	C₂×C₄×D₅	—	6	2	no	—
135	C₂×Dic₁₀	C₄×D₃	1	2	yes	—
136	C₅×M₁₆	—	12	2	no	—
137	C₂×C₅×C₈	—	2	1	no	—
138	C₂×C₅×C₈	—	6	1	no	—
139	C₄²×C₅	—	3	1	no	—
140	C₄²×C₅	—	3	1	no	—
141	C₄²×C₅	—	3	1	no	—
142	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
143	#685	—	6	2	no	—
144	#685	—	6	2	no	—
145	C₄⋊Dic₅	—	2	1	no	—
146	C₄⋊Dic₅	—	2	1	no	—
147	#681	—	12	2	no	—
148	C₂×#168	—	2	1	no	—
149	C₂×#168	—	6	1	no	—
150	#532	—	15	1	no	—
151	#423	—	3	1	no	—
152	#423	—	15	1	no	—
153	#333	—	5	1	no	—
154	#333	—	15	1	no	—
155	#331	—	15	1	no	—
156	#331	—	15	1	no	—
157	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×D₁₅	—	4	2	no	—
159	C₂×D₁₅	—	4	2	no	—
160	C₂×D₁₅	—	4	2	no	—
161	C₂×C₅×D₃	—	2	1	no	—
162	C₂×C₅×D₃	—	2	1	no	—
163	C₂×C₅×D₃	—	2	1	no	—
164	C₃×C₄×C₅	—	2	1	no	—
165	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₃×C₄×C₅	#137	1	1	yes	characteristic
168	C₃×C₄×C₅	#137	1	1	yes	characteristic
169	C₃×Dic₅	—	4	2	no	—
170	C₅×Dic₃	—	2	1	no	—
171	C₅×Dic₃	—	2	1	no	—
172	C₅×Dic₃	#100	1	2	yes	—
173	C₂×C₃×Q₈	—	5	2	no	—
174	#239	—	20	4	no	—
175	C₂×D₁₂	—	5	2	no	—
176	C₂×D₁₂	—	10	2	no	—
177	C₂×D₁₂	—	10	1	no	—
178	C₂×D₁₂	—	10	1	no	—
179	C₂×C₄×D₃	—	10	2	no	—
180	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
181	C₂×C₄×D₃	D₂₀	1	1	yes	characteristic
182	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
183	C₂×C₃×C₈	—	2	1	no	—
184	C₂×C₃×C₈	—	10	1	no	—
185	C₃×C₄⋊C₄	—	10	1	no	—
186	C₃×C₄⋊C₄	—	10	1	no	—
187	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
188	(C₂×D₃)⋊C₄	—	10	2	no	—
189	(C₂×D₃)⋊C₄	—	10	2	no	—
190	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
191	C₄×Dic₃	D₂₀	1	1	yes	characteristic
192	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
193	C₂×C₃⋊C₈	—	2	1	no	—
194	C₂×C₃⋊C₈	—	10	1	no	—
195	ΓL(1,𝔽₂₅)	—	4	2	no	—
196	ΓL(1,𝔽₂₅)	—	20	2	no	—
197	C₂³×C₅	—	3	1	no	—
198	C₂²×D₅	—	6	2	no	—
199	C₂×C₄×C₅	—	3	1	no	—
200	C₂×C₄×C₅	—	3	1	no	—
201	C₂×C₄×C₅	—	3	1	no	—
202	C₂×C₄×C₅	—	6	2	no	—
203	C₂×C₄×C₅	—	6	1	no	—
204	C₂×C₄×C₅	—	6	1	no	—
205	C₂×C₄×C₅	—	6	1	no	—
206	C₂×C₄×C₅	—	6	1	no	—
207	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
208	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
209	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
210	C₂²⋊D₅	—	12	4	no	—
211	C₂×Dic₅	—	2	2	no	—
212	D₂₀	—	12	2	no	—
213	D₂₀	—	12	1	no	—
214	D₂₀	—	12	1	no	—
215	D₂₀	—	12	1	no	—
216	D₂₀	—	12	1	no	—
217	C₄×D₅	—	12	4	no	—
218	Dic₁₀	—	4	2	no	—
219	C₅×C₈	—	4	1	no	—
220	C₅×C₈	—	12	1	no	—
221	#168	—	4	1	no	—
222	#168	—	12	1	no	—
223	C₂×#41	—	15	2	no	—
224	C₂×M₁₆	—	6	1	no	—
225	C₂×M₁₆	—	30	1	no	—
226	C₄⋊Q₈	—	5	1	no	—
227	#127	—	15	1	no	—
228	#124	—	15	2	no	—
229	#121	—	30	1	no	—
230	#121	—	30	1	no	—
231	C₄×D₄	—	30	1	no	—
232	C₄×D₄	—	30	1	no	—
233	C₂×C₄²	—	3	1	no	—
234	C₄⋊C₈	—	3	1	no	—
235	C₄⋊C₈	—	5	1	no	—
236	C₄⋊C₈	—	6	1	no	—
237	C₄⋊C₈	—	15	1	no	—
238	C₄⋊C₈	—	30	1	no	—
239	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
240	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
241	C₂×C₃×C₅	#417	1	1	yes	characteristic
242	C₂×C₃×C₅	#417	1	1	yes	characteristic
243	D₁₅	—	8	2	no	—
244	C₅×D₃	—	4	1	no	—
245	C₂²×D₃	—	10	2	no	—
246	C₂²×D₃	#685	1	1	yes	characteristic
247	C₃×Q₈	—	20	2	no	—
248	C₂×C₃×C₄	—	10	2	no	—
249	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
250	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
251	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
252	C₂×Dic₃	#685	1	1	yes	characteristic
253	C₂×Dic₃	#685	1	1	yes	characteristic
254	C₂×Dic₃	#685	1	1	yes	characteristic
255	D₁₂	—	20	2	no	—
256	D₁₂	—	20	4	no	—
257	D₁₂	—	20	1	no	—
258	D₁₂	—	20	1	no	—
259	D₁₂	—	20	1	no	—
260	D₁₂	—	20	1	no	—
261	C₄×D₃	—	2	1	no	—
262	C₄×D₃	—	2	1	no	—
263	C₄×D₃	—	2	1	no	—
264	C₄×D₃	—	2	1	no	—
265	C₄×D₃	—	2	2	no	—
266	C₄×D₃	—	20	4	no	—
267	C₃×C₈	—	4	1	no	—
268	C₃×C₈	—	20	1	no	—
269	C₃⋊C₈	—	4	1	no	—
270	C₃⋊C₈	—	20	1	no	—
271	C₂²×C₅	—	6	1	no	—
272	C₂²×C₅	—	6	1	no	—
273	C₂²×C₅	—	6	1	no	—
274	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×D₅	—	12	2	no	—
276	C₂×D₅	—	12	2	no	—
277	C₂×D₅	—	12	2	no	—
278	C₄×C₅	—	2	1	no	—
279	C₄×C₅	—	3	2	no	—
280	C₄×C₅	—	6	1	no	—
281	C₄×C₅	—	6	1	no	—
282	C₄×C₅	#914	1	1	yes	characteristic
283	C₄×C₅	#914	1	1	yes	characteristic
284	C₄×C₅	#918	1	1	yes	characteristic
285	C₄×C₅	#918	1	1	yes	characteristic
286	Dic₅	—	4	2	no	—
287	#41	—	60	4	no	—
288	C₂×Q₈	—	5	2	no	—
289	C₂×D₄	—	15	2	no	—
290	C₂×D₄	—	30	2	no	—
291	C₂×D₄	—	30	1	no	—
292	C₂×D₄	—	30	1	no	—
293	C₂²×C₄	—	3	1	no	—
294	C₂²×C₄	—	3	1	no	—
295	C₂²×C₄	—	3	1	no	—
296	C₂²×C₄	—	30	2	no	—
297	M₁₆	—	12	2	no	—
298	M₁₆	—	60	2	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	6	1	no	—
301	C₂×C₈	—	10	1	no	—
302	C₂×C₈	—	30	1	no	—
303	C₄⋊C₄	—	10	1	no	—
304	C₄⋊C₄	—	10	1	no	—
305	C₂²⋊C₄	—	30	2	no	—
306	C₂²⋊C₄	—	30	2	no	—
307	C₄²	—	3	1	no	—
308	C₄²	—	3	1	no	—
309	C₄²	—	3	1	no	—
310	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
311	C₃×C₅	#1560	1	1	yes	characteristic, complemented, Hall
312	C₂²×C₃	C₂×#685	1	1	yes	characteristic
313	C₂×D₃	—	2	1	no	—
314	C₂×D₃	—	2	1	no	—
315	C₂×D₃	—	2	1	no	—
316	C₂×D₃	—	20	2	no	—
317	C₂×D₃	—	20	2	no	—
318	C₂×D₃	—	20	2	no	—
319	C₃×C₄	—	2	1	no	—
320	C₃×C₄	—	20	2	no	—
321	C₃×C₄	#4054	1	1	yes	characteristic
322	C₃×C₄	#4055	1	1	yes	characteristic
323	C₃×C₄	#4078	1	1	yes	characteristic
324	C₃×C₄	#4088	1	1	yes	characteristic
325	Dic₃	—	2	1	no	—
326	Dic₃	—	2	1	no	—
327	Dic₃	#3955	1	2	yes	—
328	C₂×C₅	—	12	1	no	—
329	C₂×C₅	D₃×#100	1	1	yes	characteristic
330	C₂×C₅	#4803	1	1	yes	characteristic
331	C₂×C₅	#4803	1	1	yes	characteristic
332	D₅	—	24	2	no	—
333	C₂³	—	3	1	no	—
334	C₂³	—	30	2	no	—
335	Q₈	—	20	2	no	—
336	D₄	—	60	2	no	—
337	D₄	—	60	4	no	—
338	D₄	—	60	1	no	—
339	D₄	—	60	1	no	—
340	D₄	—	60	1	no	—
341	D₄	—	60	1	no	—
342	C₂×C₄	—	3	1	no	—
343	C₂×C₄	—	3	1	no	—
344	C₂×C₄	—	3	1	no	—
345	C₂×C₄	—	6	2	no	—
346	C₂×C₄	—	6	1	no	—
347	C₂×C₄	—	6	1	no	—
348	C₂×C₄	—	6	1	no	—
349	C₂×C₄	—	6	1	no	—
350	C₂×C₄	—	10	2	no	—
351	C₂×C₄	—	60	4	no	—
352	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
353	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
354	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
355	C₈	—	4	1	no	—
356	C₈	—	12	1	no	—
357	C₈	—	20	1	no	—
358	C₈	—	60	1	no	—
359	C₂×C₃	#65255	1	1	yes	characteristic
360	C₂×C₃	C₂×#3955	1	1	yes	characteristic
361	C₂×C₃	#65422	1	1	yes	characteristic
362	D₃	—	4	1	no	—
363	D₃	—	40	2	no	—
364	C₅	#68489	1	1	yes	characteristic, complemented, Sylow
365	C₂²	—	6	1	no	—
366	C₂²	—	6	1	no	—
367	C₂²	—	6	1	no	—
368	C₂²	—	60	2	no	—
369	C₂²	—	60	2	no	—
370	C₂²	—	60	2	no	—
371	C₂²	D₃×#685	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	3	2	no	—
374	C₄	—	6	1	no	—
375	C₄	—	6	1	no	—
376	C₄	—	20	2	no	—
377	C₄	#91253	1	1	yes	characteristic
378	C₄	#91263	1	1	yes	characteristic
379	C₄	#91496	1	1	yes	characteristic
380	C₄	#91520	1	1	yes	characteristic
381	C₃	#10601298	1	1	yes	characteristic, complemented, Sylow
382	C₂	—	12	1	no	—
383	C₂	—	120	2	no	—
384	C₂	#11747505	1	1	yes	characteristic
385	C₂	#11747608	1	1	yes	characteristic
386	C₂	D₃×#3955	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	#1560
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. #11747011	C₂	semidirect
4. #11746306	C₂	semidirect
5. C₃×#64701	C₂	semidirect
6. #11745861	C₂	semidirect
7. #11745854	C₂	semidirect
33. #64701	D₃	semidirect
60. #4837	D₅	semidirect
311. C₃×C₅	#1560	semidirect
364. C₅	#68489	semidirect
381. C₃	#10601298	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
#11747011	2
#11746306	2
C₃×#64701	2
#11745861	2
#11745854	2
#10601298	3
#68489	5