Subgroups — #14300375

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5745	C₂	1	1	yes	maximal, characteristic
2	#11751220	C₂	1	1	yes	maximal, characteristic, complemented
3	#11748625	C₂	1	1	yes	maximal, characteristic
4	#11748433	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748347	C₂	1	1	yes	maximal, characteristic
6	C₃×#64776	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745990	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602819	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	#91815	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
13	C₂×C₅×#208	C₂²	1	1	yes	characteristic
14	C₅×#907	—	2	2	no	—
15	C₅×D₃×M₁₆	—	2	2	no	—
16	C₅×#901	—	2	2	no	—
17	C₃×#4072	C₂²	1	1	yes	characteristic
18	C₂×C₃×#681	C₂²	1	1	yes	characteristic
19	#91098	—	2	2	no	—
20	#90996	C₄	1	4	yes	—
21	#90982	—	2	2	no	—
22	(C₂×D₃)⋊#168	—	2	2	no	—
23	#69034	—	5	1	no	maximal, Hall
24	C₅×#568	—	3	1	no	—
25	#65529	—	3	1	no	—
26	#65523	—	3	1	no	—
27	#65408	—	3	1	no	—
28	#65408	—	3	1	no	—
29	#64776	—	3	1	no	—
30	#64776	D₃	1	1	yes	characteristic, complemented
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₅×#67	C₂×C₄	1	2	yes	—
33	C₅×#235	—	2	1	no	—
34	C₅×#235	—	2	1	no	—
35	C₅×#235	—	2	1	no	—
36	C₅×#235	—	2	1	no	—
37	C₂×C₅×D₁₂	Q₈	1	1	yes	characteristic
38	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
39	C₂×C₅×Dic₆	Q₈	1	1	yes	characteristic
40	C₃×C₅×M₁₆	—	2	1	no	—
41	C₃×C₅×M₁₆	—	2	1	no	—
42	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
43	#6719	C₂×C₄	1	2	yes	—
44	C₄⋊Dic₁₅	—	2	1	no	—
45	C₄×Dic₁₅	—	2	1	no	—
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₂×C₃⋊#168	—	2	1	no	—
48	C₂×C₃⋊#168	—	2	1	no	—
49	C₅×#208	—	2	1	no	—
50	C₅×#208	—	2	1	no	—
51	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
52	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
53	C₅×D₃×C₈	—	2	4	no	—
54	C₃×C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
55	C₃×C₄⋊Dic₅	—	2	1	no	—
56	C₃×C₄×Dic₅	—	2	1	no	—
57	C₃×#681	—	2	1	no	—
58	C₃×#681	—	2	1	no	—
59	C₂×C₃×#168	—	2	1	no	—
60	#5745	D₅	1	1	yes	characteristic
61	#5129	—	5	1	no	—
62	#5119	—	5	1	no	—
63	#5006	—	5	1	no	—
64	#4999	—	5	1	no	—
65	C₃×#344	—	5	1	no	—
66	#4559	—	5	1	no	—
67	C₂×C₅×#41	—	3	1	no	—
68	C₅×#131	—	6	2	no	—
69	C₅×#131	—	6	2	no	—
70	C₂×C₅×M₁₆	—	3	1	no	—
71	C₂×C₅×M₁₆	—	6	2	no	—
72	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
73	#4072	—	3	1	no	—
74	#4072	C₂×D₃	1	1	yes	characteristic
75	C₂×#681	C₂×D₃	1	1	yes	characteristic
76	C₂²×#168	—	3	1	no	—
77	#3966	—	3	4	no	—
78	C₂²⋊#168	—	6	2	no	—
79	#3946	—	6	2	no	—
80	#3946	—	6	2	no	—
81	#1868	—	15	1	no	Sylow
82	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₅×D₃	C₄⋊C₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
90	C₂×Dic₁₅	—	2	2	no	—
91	C₅×#67	—	2	2	no	—
92	C₅×#67	—	2	2	no	—
93	C₂×C₅×Dic₃	C₄⋊C₄	1	2	yes	—
94	C₅×D₁₂	—	2	1	no	—
95	C₅×D₁₂	—	2	1	no	—
96	C₄×C₅×D₃	—	2	2	no	—
97	C₄×C₅×D₃	—	2	2	no	—
98	C₅×Dic₆	—	2	1	no	—
99	C₅×Dic₆	—	2	1	no	—
100	C₂×C₃×Dic₅	—	2	2	no	—
101	C₃×C₅×C₈	—	2	2	no	—
102	C₃⋊#168	—	4	1	no	—
103	C₃×#168	—	4	1	no	—
104	C₅×C₃⋊C₈	—	2	2	no	—
105	C₂×#235	C₂×D₅	1	1	yes	characteristic
106	C₂×C₃×M₁₆	—	5	1	no	—
107	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
108	C₃×#117	—	5	1	no	—
109	#926	—	5	1	no	—
110	C₂×#208	C₂×D₅	1	1	yes	characteristic
111	C₂²×C₃⋊C₈	—	5	1	no	—
112	#907	—	2	2	no	—
113	D₃×M₁₆	—	2	2	no	—
114	#901	—	2	2	no	—
115	(C₂×D₃)⋊C₈	—	10	2	no	—
116	Dic₃⋊C₈	—	10	2	no	—
117	#806	—	5	4	no	—
118	C₄⋊(C₃⋊C₈)	—	10	2	no	—
119	C₅×#41	—	6	1	no	—
120	C₅×#41	—	6	1	no	—
121	C₅×#41	—	6	1	no	—
122	C₅×#41	—	6	1	no	—
123	C₂×C₅×Q₈	—	3	1	no	—
124	C₂×C₅×D₄	—	3	2	no	—
125	C₂×C₅×D₄	—	3	1	no	—
126	C₂²×C₄×C₅	—	3	2	no	—
127	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
128	C₅×M₁₆	—	2	1	no	—
129	C₅×M₁₆	—	2	1	no	—
130	C₅×M₁₆	—	6	4	no	—
131	C₅×M₁₆	—	6	1	no	—
132	C₅×M₁₆	—	6	1	no	—
133	C₂×C₅×C₈	—	3	2	no	—
134	C₂×C₅×C₈	—	6	4	no	—
135	C₂×C₅×C₈	D₁₂	1	2	yes	—
136	C₂²⋊Dic₅	—	3	2	no	—
137	C₂²⋊Dic₅	C₄×D₃	1	2	yes	—
138	C₄⋊Dic₅	—	2	1	no	—
139	C₄⋊Dic₅	—	6	1	no	—
140	C₄×Dic₅	—	2	1	no	—
141	C₄×Dic₅	—	6	1	no	—
142	#681	—	2	1	no	—
143	#681	—	2	1	no	—
144	C₂×#168	—	2	1	no	—
145	C₂×#168	—	6	1	no	—
146	C₂×#168	—	6	1	no	—
147	C₂×#168	—	6	1	no	—
148	#568	—	3	1	no	—
149	#424	—	15	1	no	—
150	#424	—	15	1	no	—
151	#410	—	15	1	no	—
152	#408	—	15	1	no	—
153	#344	—	5	1	no	—
154	#344	—	15	1	no	—
155	C₂²×C₃×C₅	—	4	1	no	—
156	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
158	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
159	C₂×C₅×D₃	—	2	2	no	—
160	C₂×C₅×D₃	—	4	2	no	—
161	C₃×C₄×C₅	#120	1	1	yes	characteristic
162	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
163	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
164	C₃×C₄×C₅	#123	1	1	yes	characteristic
165	Dic₁₅	—	4	2	no	—
166	C₃×Dic₅	—	4	2	no	—
167	C₅×Dic₃	—	2	2	no	—
168	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
169	C₂×#67	C₄×D₅	1	2	yes	—
170	#235	—	2	1	no	—
171	#235	—	2	1	no	—
172	#235	—	2	1	no	—
173	#235	—	2	1	no	—
174	C₂×D₁₂	Dic₁₀	1	1	yes	characteristic
175	C₂×C₄×D₃	C₂²⋊D₅	1	2	yes	—
176	C₂×Dic₆	Dic₁₀	1	1	yes	characteristic
177	C₃×M₁₆	—	2	1	no	—
178	C₃×M₁₆	—	2	1	no	—
179	C₃×M₁₆	—	10	1	no	—
180	C₃×M₁₆	—	10	1	no	—
181	C₂×C₃×C₈	—	10	1	no	—
182	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
183	C₃×C₄⋊C₄	—	10	1	no	—
184	C₃×C₂²⋊C₄	—	5	2	no	—
185	C₃×C₄²	—	10	1	no	—
186	#217	—	5	2	no	—
187	C₄⋊Dic₃	—	10	1	no	—
188	C₄×Dic₃	—	10	1	no	—
189	#208	—	2	1	no	—
190	#208	—	2	1	no	—
191	C₂×C₃⋊C₈	—	10	1	no	—
192	C₂×C₃⋊C₈	—	10	1	no	—
193	C₂×C₃⋊C₈	—	10	1	no	—
194	C₂×C₃⋊C₈	C₂²⋊D₅	1	2	yes	—
195	ΓL(1,𝔽₂₅)	—	2	4	no	—
196	D₃×C₈	—	2	4	no	—
197	C₂³×C₅	—	3	2	no	—
198	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
199	C₅×Q₈	—	6	1	no	—
200	C₅×Q₈	—	6	1	no	—
201	C₅×D₄	—	6	2	no	—
202	C₅×D₄	—	6	2	no	—
203	C₅×D₄	—	6	1	no	—
204	C₅×D₄	—	6	1	no	—
205	C₂×C₄×C₅	—	3	2	no	—
206	C₂×C₄×C₅	—	6	2	no	—
207	C₂×C₄×C₅	—	6	2	no	—
208	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
209	C₂×C₄×C₅	#235	1	1	yes	characteristic
210	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
211	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
212	C₂×C₄×C₅	#237	1	1	yes	characteristic
213	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
214	C₂×Dic₅	—	2	2	no	—
215	C₂×Dic₅	—	6	2	no	—
216	C₅×C₈	—	2	2	no	—
217	C₅×C₈	—	6	2	no	—
218	#168	—	4	1	no	—
219	#168	—	12	1	no	—
220	C₂×#41	—	3	1	no	—
221	#131	—	6	2	no	—
222	#131	—	6	2	no	—
223	C₂×M₁₆	—	3	1	no	—
224	C₂×M₁₆	—	5	1	no	—
225	C₂×M₁₆	—	6	2	no	—
226	C₂×M₁₆	D₃×D₅	1	1	yes	characteristic
227	C₂²×C₈	—	15	1	no	—
228	#117	—	5	1	no	—
229	#117	—	15	1	no	—
230	C₄⋊C₈	—	30	2	no	—
231	C₄⋊C₈	—	30	2	no	—
232	#99	—	15	4	no	—
233	C₂²⋊C₈	—	30	2	no	—
234	C₂×C₃×C₅	—	2	1	no	—
235	C₂×C₃×C₅	—	2	1	no	—
236	C₂×C₃×C₅	—	2	1	no	—
237	C₂×C₃×C₅	#385	1	1	yes	characteristic
238	C₅×D₃	—	4	2	no	—
239	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
240	C₂²×D₃	#683	1	2	yes	—
241	C₂×C₃×C₄	—	10	2	no	—
242	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
243	C₂×C₃×C₄	#709	1	1	yes	characteristic
244	C₂×C₃×C₄	#709	1	1	yes	characteristic
245	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
246	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
247	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
248	#67	—	2	2	no	—
249	#67	—	2	2	no	—
250	C₂×Dic₃	—	10	2	no	—
251	C₂×Dic₃	#683	1	2	yes	—
252	D₁₂	—	2	1	no	—
253	D₁₂	—	2	1	no	—
254	C₄×D₃	—	2	2	no	—
255	C₄×D₃	—	2	2	no	—
256	Dic₆	—	2	1	no	—
257	Dic₆	—	2	1	no	—
258	C₃×C₈	—	2	2	no	—
259	C₃×C₈	—	20	1	no	—
260	C₃⋊C₈	—	2	2	no	—
261	C₃⋊C₈	—	20	1	no	—
262	C₂²×C₅	—	4	1	no	—
263	C₂²×C₅	—	6	2	no	—
264	C₂²×C₅	—	12	2	no	—
265	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
266	C₂²×C₅	#67⋊C₄	1	1	yes	characteristic
267	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
268	C₄×C₅	—	6	2	no	—
269	C₄×C₅	#882	1	1	yes	characteristic
270	C₄×C₅	#883	1	1	yes	characteristic
271	C₄×C₅	#896	1	1	yes	characteristic
272	C₄×C₅	#897	1	1	yes	characteristic
273	Dic₅	—	4	2	no	—
274	Dic₅	—	12	2	no	—
275	#41	—	6	1	no	—
276	#41	—	6	1	no	—
277	#41	—	6	1	no	—
278	#41	—	6	1	no	—
279	C₂×Q₈	—	3	1	no	—
280	C₂×D₄	—	3	2	no	—
281	C₂×D₄	—	3	1	no	—
282	C₂²×C₄	—	3	2	no	—
283	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
284	M₁₆	—	2	1	no	—
285	M₁₆	—	2	1	no	—
286	M₁₆	—	6	4	no	—
287	M₁₆	—	6	1	no	—
288	M₁₆	—	6	1	no	—
289	M₁₆	—	10	1	no	—
290	M₁₆	—	10	1	no	—
291	C₂×C₈	—	3	2	no	—
292	C₂×C₈	—	6	4	no	—
293	C₂×C₈	—	10	1	no	—
294	C₂×C₈	—	30	1	no	—
295	C₂×C₈	—	30	1	no	—
296	C₂×C₈	—	30	1	no	—
297	C₂×C₈	#1203	1	2	yes	—
298	C₄⋊C₄	—	10	1	no	—
299	C₄⋊C₄	—	30	1	no	—
300	C₂²⋊C₄	—	5	2	no	—
301	C₂²⋊C₄	—	15	2	no	—
302	C₄²	—	10	1	no	—
303	C₄²	—	30	1	no	—
304	C₃×C₅	#1868	1	1	yes	characteristic, complemented, Hall
305	C₂²×C₃	—	4	1	no	—
306	C₂²×C₃	C₂×#683	1	1	yes	characteristic
307	C₂²×C₃	C₄×C₂²⋊D₅	1	1	yes	characteristic
308	C₂²×C₃	C₄×C₂²⋊D₅	1	1	yes	characteristic
309	C₂×D₃	—	2	2	no	—
310	C₂×D₃	—	4	2	no	—
311	C₃×C₄	—	20	2	no	—
312	C₃×C₄	#4070	1	1	yes	characteristic
313	C₃×C₄	#4070	1	1	yes	characteristic
314	C₃×C₄	#4075	1	1	yes	characteristic
315	C₃×C₄	#4099	1	1	yes	characteristic
316	Dic₃	—	2	2	no	—
317	Dic₃	—	20	2	no	—
318	C₂×C₅	—	2	1	no	—
319	C₂×C₅	—	2	1	no	—
320	C₂×C₅	—	2	1	no	—
321	C₂×C₅	—	12	2	no	—
322	C₂×C₅	#4668	1	1	yes	characteristic
323	C₂³	—	3	2	no	—
324	C₂³	C₄×D₃×D₅	1	1	yes	characteristic
325	Q₈	—	6	1	no	—
326	Q₈	—	6	1	no	—
327	D₄	—	6	2	no	—
328	D₄	—	6	2	no	—
329	D₄	—	6	1	no	—
330	D₄	—	6	1	no	—
331	C₂×C₄	—	3	2	no	—
332	C₂×C₄	—	6	2	no	—
333	C₂×C₄	—	6	2	no	—
334	C₂×C₄	—	10	2	no	—
335	C₂×C₄	—	30	2	no	—
336	C₂×C₄	#6765	1	1	yes	characteristic
337	C₂×C₄	D₃×Dic₁₀	1	1	yes	characteristic
338	C₂×C₄	#6771	1	1	yes	characteristic
339	C₂×C₄	C₂×#1203	1	1	yes	characteristic
340	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
341	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
342	C₈	—	2	2	no	—
343	C₈	—	6	2	no	—
344	C₈	—	20	1	no	—
345	C₈	—	60	1	no	—
346	C₂×C₃	—	2	1	no	—
347	C₂×C₃	—	2	1	no	—
348	C₂×C₃	—	2	1	no	—
349	C₂×C₃	#65499	1	1	yes	characteristic
350	D₃	—	4	2	no	—
351	C₅	#69034	1	1	yes	characteristic, complemented, Sylow
352	C₂²	—	4	1	no	—
353	C₂²	—	6	2	no	—
354	C₂²	—	12	2	no	—
355	C₂²	D₃×#683	1	1	yes	characteristic
356	C₂²	#91441	1	1	yes	characteristic
357	C₂²	C₄×#1203	1	1	yes	characteristic
358	C₄	—	6	2	no	—
359	C₄	—	20	2	no	—
360	C₄	—	60	2	no	—
361	C₄	#91352	1	1	yes	characteristic
362	C₄	#91365	1	1	yes	characteristic
363	C₄	#91452	1	1	yes	characteristic
364	C₄	#91570	1	1	yes	characteristic
365	C₃	#10602819	1	1	yes	characteristic, complemented, Sylow
366	C₂	—	2	1	no	—
367	C₂	—	2	1	no	—
368	C₂	—	2	1	no	—
369	C₂	—	12	2	no	—
370	C₂	#11748722	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#91570
commutator	C₂³×C₃×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₅×M₁₆	C₂²
socle	C₂×C₃×C₅	#385
prime radical	C₂×C₅×#235	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×M₁₆	D₃×D₅
hypocenter	C₃×C₅	#1868
LCS	G ▷ C₂³×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. #11751220	C₂	semidirect
4. #11748433	C₂	semidirect
6. C₃×#64776	C₂	semidirect
7. #11745990	C₂	semidirect
30. #64776	D₃	semidirect
304. C₃×C₅	#1868	semidirect
351. C₅	#69034	semidirect
365. C₃	#10602819	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5745	2
#11751220	2
#11748625	2
#11748433	2
#11748347	2
C₃×#64776	2
#11745990	2
#10602819	3
#69034	5