Subgroups — #14285426

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	#11752672	C₂	1	1	yes	maximal, characteristic
2	C₃×#65126	C₂	1	1	yes	maximal, characteristic, complemented
3	#11747007	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746302	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4485	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745863	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745853	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599182	—	3	1	no	maximal, Hall
9	C₂×#6772	C₄	1	2	yes	—
10	#91758	C₂²	1	1	yes	characteristic
11	C₅×#874	C₂²	1	1	yes	characteristic
12	C₂×C₃×#676	—	2	1	no	—
13	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
14	#91449	—	2	1	no	—
15	#91449	—	2	1	no	—
16	Dic₅×D₁₂	—	2	1	no	—
17	Dic₅×D₁₂	—	2	1	no	—
18	#91373	C₄	1	2	yes	—
19	C₂×#6649	—	2	1	no	—
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₃×#3946	—	2	1	no	—
24	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
25	#90982	—	2	1	no	—
26	#70505	—	5	1	no	maximal, Hall
27	#66050	—	3	1	no	—
28	#65126	—	3	1	no	—
29	#65126	D₃	1	1	yes	characteristic, complemented
30	C₅×#331	—	3	1	no	—
31	#64764	—	3	1	no	—
32	#64701	—	3	1	no	—
33	#64701	—	3	1	no	—
34	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
35	C₂×C₅×D₁₂	—	2	1	no	—
36	C₂×C₅×D₁₂	—	2	1	no	—
37	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
41	C₂×#1201	—	2	2	no	—
42	C₂×D₃×Dic₅	—	2	2	no	—
43	#6772	—	4	4	no	—
44	C₂×C₃×C₅×C₈	—	2	1	no	—
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₄⋊Dic₁₅	—	2	1	no	—
47	C₄⋊Dic₁₅	—	2	1	no	—
48	C₂×C₃⋊#168	—	2	1	no	—
49	C₂×C₅×C₃⋊C₈	—	2	1	no	—
50	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
51	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
52	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
53	C₂×C₃×#168	—	2	1	no	—
54	C₃×#676	—	4	2	no	—
55	(C₂×D₃)⋊Dic₅	—	2	2	no	—
56	(C₂×D₃)⋊Dic₅	—	2	2	no	—
57	#6649	—	4	2	no	—
58	#5477	—	5	1	no	—
59	C₃×#423	—	5	1	no	—
60	#4924	—	5	1	no	—
61	#4485	—	5	1	no	—
62	#4485	D₅	1	1	yes	characteristic, complemented
63	#4481	—	5	1	no	—
64	#4481	—	5	1	no	—
65	C₂×#711	—	3	2	no	—
66	C₅×#127	—	3	1	no	—
67	#4084	—	6	1	no	—
68	#4084	—	6	1	no	—
69	#4082	—	3	2	no	—
70	D₄×Dic₅	—	6	1	no	—
71	D₄×Dic₅	—	6	1	no	—
72	C₂×#676	—	2	1	no	—
73	C₂×#676	—	6	1	no	—
74	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
75	#4015	—	3	1	no	—
76	C₅×C₄⋊C₈	—	3	1	no	—
77	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
78	#3946	—	2	1	no	—
79	#3946	—	6	1	no	—
80	C₄⋊#168	—	3	1	no	—
81	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
82	#1560	—	15	1	no	Sylow
83	C₂²×C₅×D₃	—	2	2	no	—
84	C₂²×C₃×D₅	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	C₂×Dic₁₅	—	2	2	no	—
89	Dic₃₀	—	4	2	no	—
90	C₅×D₁₂	—	4	2	no	—
91	C₅×D₁₂	—	4	1	no	—
92	C₅×D₁₂	—	4	1	no	—
93	C₅×D₁₂	—	4	1	no	—
94	C₅×D₁₂	—	4	1	no	—
95	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
98	C₃×C₄×D₅	—	2	1	no	—
99	C₃×C₄×D₅	—	2	1	no	—
100	C₃×C₄×D₅	—	2	1	no	—
101	C₃×C₄×D₅	—	2	1	no	—
102	C₃×C₄×D₅	—	2	2	no	—
103	#1201	—	4	4	no	—
104	D₃×Dic₅	—	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×#235	—	5	2	no	—
110	C₂×C₃×M₁₆	—	10	1	no	—
111	C₂×C₃×C₄²	—	5	1	no	—
112	#930	—	10	1	no	—
113	#930	—	10	1	no	—
114	C₂×#208	—	10	1	no	—
115	#875	—	5	2	no	—
116	#874	C₂×D₅	1	1	yes	characteristic
117	C₄×D₁₂	—	10	1	no	—
118	C₄×D₁₂	—	10	1	no	—
119	#869	—	5	1	no	—
120	C₃×C₄⋊C₈	—	5	1	no	—
121	C₃×C₄⋊C₈	—	10	1	no	—
122	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
123	C₄⋊(C₃⋊C₈)	—	5	1	no	—
124	C₄⋊(C₃⋊C₈)	—	10	1	no	—
125	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
126	C₂×C₅×D₄	—	3	2	no	—
127	C₂×C₅×D₄	—	6	1	no	—
128	C₂×C₅×D₄	—	6	1	no	—
129	C₂×C₂²⋊D₅	—	6	2	no	—
130	C₂²×Dic₅	—	6	2	no	—
131	#711	—	12	4	no	—
132	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
133	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
134	C₂×C₄×D₅	#67	1	1	yes	characteristic
135	C₂×Dic₁₀	—	3	2	no	—
136	C₂×C₅×C₈	—	2	1	no	—
137	C₂×C₅×C₈	—	6	1	no	—
138	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
139	C₂²⋊Dic₅	—	6	2	no	—
140	C₂²⋊Dic₅	—	6	2	no	—
141	C₄⋊Dic₅	—	6	1	no	—
142	C₄⋊Dic₅	—	6	1	no	—
143	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
144	C₄×Dic₅	D₁₂	1	1	yes	characteristic
145	C₄×Dic₅	#67	1	1	yes	characteristic
146	C₂×#168	—	2	1	no	—
147	C₂×#168	—	6	1	no	—
148	#676	—	4	2	no	—
149	#676	—	12	2	no	—
150	#532	—	15	1	no	—
151	#423	—	5	1	no	—
152	#423	—	15	1	no	—
153	#333	—	15	1	no	—
154	#333	—	15	1	no	—
155	#331	—	3	1	no	—
156	#331	—	15	1	no	—
157	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₅×D₃	—	4	2	no	—
159	C₂×C₅×D₃	—	4	2	no	—
160	C₂×C₅×D₃	—	4	2	no	—
161	C₂×C₃×D₅	—	2	1	no	—
162	C₂×C₃×D₅	—	2	1	no	—
163	C₂×C₃×D₅	—	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	Dic₁₅	—	4	2	no	—
170	C₃×Dic₅	—	2	1	no	—
171	C₃×Dic₅	—	2	1	no	—
172	C₃×Dic₅	#100	1	2	yes	—
173	C₂²×C₃×C₄	—	5	1	no	—
174	C₂²×C₃×C₄	—	5	1	no	—
175	C₂²×C₃×C₄	—	5	1	no	—
176	C₂×#67	—	10	2	no	—
177	#235	—	20	4	no	—
178	C₂×D₁₂	—	2	1	no	—
179	C₂×D₁₂	—	2	1	no	—
180	C₂×D₁₂	C₄×D₅	1	2	yes	—
181	C₂×C₄×D₃	—	10	2	no	—
182	C₂×Dic₆	—	5	2	no	—
183	C₃×M₁₆	—	20	2	no	—
184	C₂×C₃×C₈	—	2	1	no	—
185	C₂×C₃×C₈	—	10	1	no	—
186	C₃×C₄²	—	5	1	no	—
187	C₃×C₄²	—	5	1	no	—
188	C₃×C₄²	—	5	1	no	—
189	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
190	(C₂×D₃)⋊C₄	—	10	2	no	—
191	(C₂×D₃)⋊C₄	—	10	2	no	—
192	C₄⋊Dic₃	—	10	1	no	—
193	C₄⋊Dic₃	—	10	1	no	—
194	#208	—	20	2	no	—
195	C₂×C₃⋊C₈	—	2	1	no	—
196	C₂×C₃⋊C₈	—	10	1	no	—
197	C₂³×C₅	—	6	2	no	—
198	C₂²×D₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
199	C₅×D₄	—	12	2	no	—
200	C₅×D₄	—	12	1	no	—
201	C₅×D₄	—	12	1	no	—
202	C₅×D₄	—	12	1	no	—
203	C₅×D₄	—	12	1	no	—
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	4	no	—
208	C₂×Dic₅	—	6	2	no	—
209	C₂×Dic₅	—	12	4	no	—
210	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
211	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
212	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
213	C₄×D₅	—	2	1	no	—
214	C₄×D₅	—	2	1	no	—
215	C₄×D₅	—	2	1	no	—
216	C₄×D₅	—	2	1	no	—
217	C₄×D₅	—	2	2	no	—
218	Dic₁₀	—	12	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	—	10	1	no	—
225	C₂×M₁₆	—	30	1	no	—
226	C₄⋊Q₈	—	15	1	no	—
227	#127	—	3	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	—	5	1	no	—
234	C₄⋊C₈	—	3	1	no	—
235	C₄⋊C₈	—	5	1	no	—
236	C₄⋊C₈	—	10	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	C₃×D₅	—	4	1	no	—
244	C₅×D₃	—	8	2	no	—
245	C₂³×C₃	—	5	1	no	—
246	C₂²×D₃	—	2	2	no	—
247	C₂×C₃×C₄	—	5	1	no	—
248	C₂×C₃×C₄	—	5	1	no	—
249	C₂×C₃×C₄	—	5	1	no	—
250	C₂×C₃×C₄	—	10	2	no	—
251	C₂×C₃×C₄	—	10	1	no	—
252	C₂×C₃×C₄	—	10	1	no	—
253	C₂×C₃×C₄	—	10	1	no	—
254	C₂×C₃×C₄	—	10	1	no	—
255	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
256	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
257	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
258	#67	—	20	4	no	—
259	C₂×Dic₃	—	10	2	no	—
260	D₁₂	—	4	2	no	—
261	D₁₂	—	4	1	no	—
262	D₁₂	—	4	1	no	—
263	D₁₂	—	4	1	no	—
264	D₁₂	—	4	1	no	—
265	C₄×D₃	—	20	4	no	—
266	Dic₆	—	20	2	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	—	12	2	no	—
272	C₂²×C₅	—	12	2	no	—
273	C₂²×C₅	—	12	2	no	—
274	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
275	C₂×D₅	—	2	1	no	—
276	C₂×D₅	—	2	1	no	—
277	C₂×D₅	—	2	1	no	—
278	C₄×C₅	—	2	1	no	—
279	C₄×C₅	#908	1	1	yes	characteristic
280	C₄×C₅	#909	1	1	yes	characteristic
281	C₄×C₅	#932	1	1	yes	characteristic
282	C₄×C₅	#942	1	1	yes	characteristic
283	Dic₅	—	2	1	no	—
284	Dic₅	—	2	1	no	—
285	Dic₅	—	12	2	no	—
286	Dic₅	#823	1	2	yes	—
287	#41	—	60	4	no	—
288	C₂×Q₈	—	15	2	no	—
289	C₂×D₄	—	3	2	no	—
290	C₂×D₄	—	6	1	no	—
291	C₂×D₄	—	6	1	no	—
292	C₂×D₄	—	30	2	no	—
293	C₂²×C₄	—	5	1	no	—
294	C₂²×C₄	—	5	1	no	—
295	C₂²×C₄	—	5	1	no	—
296	C₂²×C₄	—	30	2	no	—
297	M₁₆	—	20	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	—	30	1	no	—
304	C₄⋊C₄	—	30	1	no	—
305	C₂²⋊C₄	—	30	2	no	—
306	C₂²⋊C₄	—	30	2	no	—
307	C₄²	—	5	1	no	—
308	C₄²	—	5	1	no	—
309	C₄²	—	5	1	no	—
310	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
311	C₃×C₅	#1560	1	1	yes	characteristic, complemented, Hall
312	C₂²×C₃	—	10	1	no	—
313	C₂²×C₃	—	10	1	no	—
314	C₂²×C₃	—	10	1	no	—
315	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
316	C₂×D₃	—	4	2	no	—
317	C₂×D₃	—	4	2	no	—
318	C₂×D₃	—	4	2	no	—
319	C₃×C₄	—	2	1	no	—
320	C₃×C₄	—	5	2	no	—
321	C₃×C₄	—	10	1	no	—
322	C₃×C₄	—	10	1	no	—
323	C₃×C₄	#4057	1	1	yes	characteristic
324	C₃×C₄	#4057	1	1	yes	characteristic
325	C₃×C₄	#4061	1	1	yes	characteristic
326	C₃×C₄	#4061	1	1	yes	characteristic
327	Dic₃	—	20	2	no	—
328	C₂×C₅	—	24	2	no	—
329	C₂×C₅	#4966	1	1	yes	characteristic
330	C₂×C₅	C₂×#823	1	1	yes	characteristic
331	C₂×C₅	#5133	1	1	yes	characteristic
332	D₅	—	4	1	no	—
333	C₂³	—	5	1	no	—
334	C₂³	—	6	2	no	—
335	Q₈	—	60	2	no	—
336	D₄	—	12	2	no	—
337	D₄	—	12	1	no	—
338	D₄	—	12	1	no	—
339	D₄	—	12	1	no	—
340	D₄	—	12	1	no	—
341	D₄	—	60	4	no	—
342	C₂×C₄	—	5	1	no	—
343	C₂×C₄	—	5	1	no	—
344	C₂×C₄	—	5	1	no	—
345	C₂×C₄	—	10	2	no	—
346	C₂×C₄	—	10	1	no	—
347	C₂×C₄	—	10	1	no	—
348	C₂×C₄	—	10	1	no	—
349	C₂×C₄	—	10	1	no	—
350	C₂×C₄	—	30	2	no	—
351	C₂×C₄	—	60	4	no	—
352	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
353	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
354	C₂×C₄	D₅×#67	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	—	20	1	no	—
360	C₂×C₃	D₅×#100	1	1	yes	characteristic
361	C₂×C₃	#65060	1	1	yes	characteristic
362	C₂×C₃	#65060	1	1	yes	characteristic
363	D₃	—	8	2	no	—
364	C₅	#70505	1	1	yes	characteristic, complemented, Sylow
365	C₂²	—	10	1	no	—
366	C₂²	—	10	1	no	—
367	C₂²	—	10	1	no	—
368	C₂²	—	12	2	no	—
369	C₂²	—	12	2	no	—
370	C₂²	—	12	2	no	—
371	C₂²	D₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	5	2	no	—
374	C₄	—	10	1	no	—
375	C₄	—	10	1	no	—
376	C₄	—	60	2	no	—
377	C₄	#91250	1	1	yes	characteristic
378	C₄	#91260	1	1	yes	characteristic
379	C₄	#91489	1	1	yes	characteristic
380	C₄	#91513	1	1	yes	characteristic
381	C₃	#10599182	1	1	yes	characteristic, complemented, Sylow
382	C₂	—	20	1	no	—
383	C₂	—	24	2	no	—
384	C₂	#11747498	1	1	yes	characteristic
385	C₂	#11747605	1	1	yes	characteristic
386	C₂	D₅×#823	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅×(C₂×D₃)⋊C₄
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	#11752672	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. C₃×#65126	C₂	semidirect
3. #11747007	C₂	semidirect
4. #11746302	C₂	semidirect
5. C₅×#4485	C₂	semidirect
6. #11745863	C₂	semidirect
7. #11745853	C₂	semidirect
29. #65126	D₃	semidirect
62. #4485	D₅	semidirect
311. C₃×C₅	#1560	semidirect
364. C₅	#70505	semidirect
381. C₃	#10599182	semidirect

Maximal subgroups

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

subgroup	index
#11752672	2
C₃×#65126	2
#11747007	2
#11746302	2
C₅×#4485	2
#11745863	2
#11745853	2
#10599182	3
#70505	5