Subgroups — #14285004

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66062	C₂	1	1	yes	maximal, characteristic
2	#11747007	C₂	1	1	yes	maximal, characteristic
3	D₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#11746368	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4547	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745861	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745861	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599216	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃×#4092	—	2	1	no	—
11	C₃×Q₈×Dic₅	—	2	1	no	—
12	C₃×#4043	C₄	1	2	yes	—
13	C₃×#4038	—	2	1	no	—
14	C₃×#4036	—	2	1	no	—
15	C₃×#4033	C₄	1	2	yes	—
16	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
17	C₃×#4015	C₂²	1	1	yes	characteristic
18	C₂×#6649	—	2	1	no	—
19	C₂×D₅×C₃⋊C₈	—	2	1	no	—
20	C₄⋊(C₃⋊#168)	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	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	#90982	—	2	1	no	—
25	#90982	—	2	1	no	—
26	#71830	—	5	1	no	maximal, Hall
27	#66062	D₃	1	1	yes	characteristic
28	#65126	—	3	1	no	—
29	D₅×C₄⋊C₈	—	3	1	no	—
30	C₅×#332	—	3	1	no	—
31	#64767	—	3	1	no	—
32	#64701	—	3	1	no	—
33	#64701	—	3	1	no	—
34	C₂×C₃×C₅×Q₈	—	2	1	no	—
35	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
37	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
38	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₃⋊#168	—	2	1	no	—
43	C₂×C₃⋊#168	—	2	1	no	—
44	C₂×C₅×C₃⋊C₈	—	2	1	no	—
45	C₂×C₅×C₃⋊C₈	—	2	1	no	—
46	C₃×#685	—	2	2	no	—
47	C₃×C₄⋊Dic₅	—	2	1	no	—
48	C₃×C₄⋊Dic₅	—	2	1	no	—
49	C₃×C₄⋊Dic₅	—	2	2	no	—
50	C₃×#683	—	2	2	no	—
51	C₃×C₄×Dic₅	—	2	2	no	—
52	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
53	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
54	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
55	#6649	—	4	2	no	—
56	D₅×C₃⋊C₈	—	4	2	no	—
57	C₃×#533	—	5	1	no	—
58	#4924	—	5	1	no	—
59	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
60	#4547	—	5	1	no	—
61	#4547	D₅	1	1	yes	characteristic, complemented
62	#4546	—	5	1	no	—
63	#4546	—	5	1	no	—
64	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
65	#4092	—	2	1	no	—
66	Q₈×Dic₅	—	2	1	no	—
67	C₂×#676	—	6	1	no	—
68	C₂×C₈×D₅	—	6	1	no	—
69	#4043	Dic₃	1	2	yes	—
70	#4038	—	2	1	no	—
71	#4036	—	2	1	no	—
72	#4033	Dic₃	1	2	yes	—
73	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
74	#4015	C₂×D₃	1	1	yes	characteristic
75	C₅×C₄⋊C₈	—	3	1	no	—
76	C₅×C₄⋊C₈	—	3	1	no	—
77	#3946	—	6	1	no	—
78	#3946	—	6	1	no	—
79	C₄⋊#168	—	3	1	no	—
80	C₄⋊#168	—	3	1	no	—
81	#1550	—	15	1	no	Sylow
82	C₂²×C₃×D₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₃×C₅×Q₈	—	4	1	no	—
84	C₃×C₅×Q₈	—	4	1	no	—
85	C₂×C₃×C₄×C₅	—	2	2	no	—
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	C₂×C₃×Dic₅	—	2	2	no	—
90	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
93	C₃×C₄×D₅	—	2	1	no	—
94	C₃×C₄×D₅	—	2	1	no	—
95	C₃×C₄×D₅	—	4	1	no	—
96	C₃×C₄×D₅	SD₁₆	1	2	yes	—
97	C₃×C₄×D₅	Q₁₆	1	2	yes	—
98	C₃×Dic₁₀	—	4	2	no	—
99	C₃⋊#168	—	4	1	no	—
100	C₃⋊#168	—	4	1	no	—
101	C₅×C₃⋊C₈	—	4	1	no	—
102	C₅×C₃⋊C₈	—	4	1	no	—
103	C₃×C₄⋊Q₈	—	5	1	no	—
104	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
105	C₃×#125	—	10	1	no	—
106	C₃×C₂²⋊Q₈	—	5	2	no	—
107	C₃×C₂²⋊Q₈	—	10	1	no	—
108	C₃×C₄×Q₈	—	5	2	no	—
109	C₃×C₄×Q₈	—	10	1	no	—
110	C₃×#117	—	10	1	no	—
111	C₂×C₃×C₄²	—	5	1	no	—
112	C₂×#208	—	10	1	no	—
113	C₂²×C₃⋊C₈	—	10	1	no	—
114	C₄⋊(C₃⋊C₈)	—	5	1	no	—
115	C₄⋊(C₃⋊C₈)	—	5	1	no	—
116	C₄⋊(C₃⋊C₈)	—	10	1	no	—
117	C₄⋊(C₃⋊C₈)	—	10	1	no	—
118	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
119	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
120	C₂×C₅×Q₈	—	2	1	no	—
121	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
122	C₂×C₄×D₅	#67	1	1	yes	characteristic
123	C₂×C₄×D₅	#67	1	1	yes	characteristic
124	C₂×Dic₁₀	C₂×Dic₃	1	2	yes	—
125	C₂×C₅×C₈	—	6	1	no	—
126	C₂×C₅×C₈	—	6	1	no	—
127	C₅×C₄⋊C₄	—	2	1	no	—
128	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
129	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
130	#685	—	2	2	no	—
131	C₄⋊Dic₅	—	2	1	no	—
132	C₄⋊Dic₅	—	2	1	no	—
133	C₄⋊Dic₅	—	2	2	no	—
134	#683	—	2	2	no	—
135	C₄×Dic₅	—	2	2	no	—
136	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
137	C₄×Dic₅	#67	1	1	yes	characteristic
138	C₄×Dic₅	#67	1	1	yes	characteristic
139	C₂×#168	—	6	1	no	—
140	C₂×#168	—	6	1	no	—
141	#676	—	12	2	no	—
142	C₈×D₅	—	12	2	no	—
143	#533	—	5	1	no	—
144	#423	—	15	1	no	—
145	C₂×C₄⋊C₈	—	15	1	no	—
146	#333	—	15	1	no	—
147	#333	—	15	1	no	—
148	#332	—	3	1	no	—
149	#332	—	15	1	no	—
150	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₃×D₅	—	2	1	no	—
152	C₂×C₃×D₅	—	2	1	no	—
153	C₂×C₃×D₅	#103	1	2	yes	—
154	C₃×C₄×C₅	—	2	1	no	—
155	C₃×C₄×C₅	—	4	2	no	—
156	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
157	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
158	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₃×C₄×C₅	#137	1	1	yes	characteristic
160	C₃×Dic₅	—	2	1	no	—
161	C₃×Dic₅	—	2	1	no	—
162	C₃×Dic₅	—	4	2	no	—
163	C₃×Dic₅	#103	1	2	yes	—
164	C₂×C₃×Q₈	—	2	1	no	—
165	C₂×C₃×Q₈	—	5	2	no	—
166	C₂²×C₃×C₄	—	5	1	no	—
167	C₂²×C₃×C₄	—	5	1	no	—
168	C₂²×C₃×C₄	—	5	1	no	—
169	C₃×C₄⋊C₄	—	2	1	no	—
170	C₃×C₄⋊C₄	—	10	2	no	—
171	C₃×C₄⋊C₄	—	10	2	no	—
172	C₃×C₄⋊C₄	—	10	1	no	—
173	C₃×C₄⋊C₄	—	10	1	no	—
174	C₃×C₄⋊C₄	C₄×D₅	1	2	yes	—
175	C₃×C₂²⋊C₄	—	10	2	no	—
176	C₃×C₄²	—	5	1	no	—
177	C₃×C₄²	—	5	1	no	—
178	C₃×C₄²	—	5	1	no	—
179	C₃×C₄²	—	10	2	no	—
180	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
181	#208	—	20	2	no	—
182	C₂×C₃⋊C₈	—	2	1	no	—
183	C₂×C₃⋊C₈	—	2	1	no	—
184	C₂×C₃⋊C₈	—	10	1	no	—
185	C₂×C₃⋊C₈	—	10	1	no	—
186	C₂×C₃⋊C₈	—	20	2	no	—
187	C₂²×D₅	#217	1	1	yes	characteristic
188	C₅×Q₈	—	4	1	no	—
189	C₅×Q₈	—	4	1	no	—
190	C₂×C₄×C₅	—	2	2	no	—
191	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
194	C₂×Dic₅	—	2	2	no	—
195	C₂×Dic₅	#217	1	1	yes	characteristic
196	C₂×Dic₅	#217	1	1	yes	characteristic
197	C₂×Dic₅	#217	1	1	yes	characteristic
198	C₄×D₅	—	2	1	no	—
199	C₄×D₅	—	2	1	no	—
200	C₄×D₅	—	4	1	no	—
201	C₄×D₅	#215	1	2	yes	—
202	C₄×D₅	#216	1	2	yes	—
203	Dic₁₀	—	4	2	no	—
204	C₅×C₈	—	12	1	no	—
205	C₅×C₈	—	12	1	no	—
206	#168	—	12	1	no	—
207	#168	—	12	1	no	—
208	C₂×M₁₆	—	30	1	no	—
209	C₂²×C₈	—	30	1	no	—
210	C₄⋊Q₈	—	5	1	no	—
211	C₄⋊Q₈	D₃×D₅	1	1	yes	characteristic
212	#125	—	10	1	no	—
213	C₂²⋊Q₈	—	5	2	no	—
214	C₂²⋊Q₈	—	10	1	no	—
215	C₄×Q₈	—	5	2	no	—
216	C₄×Q₈	—	10	1	no	—
217	#117	—	10	1	no	—
218	C₂×C₄²	—	5	1	no	—
219	C₄⋊C₈	—	3	1	no	—
220	C₄⋊C₈	—	3	1	no	—
221	C₄⋊C₈	—	15	1	no	—
222	C₄⋊C₈	—	15	1	no	—
223	C₄⋊C₈	—	30	1	no	—
224	C₄⋊C₈	—	30	1	no	—
225	C₂×C₃×C₅	#413	1	1	yes	characteristic
226	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
227	C₂×C₃×C₅	#417	1	1	yes	characteristic
228	C₃×D₅	—	2	2	no	—
229	C₂³×C₃	—	5	1	no	—
230	C₃×Q₈	—	4	1	no	—
231	C₃×Q₈	—	4	1	no	—
232	C₃×Q₈	—	20	2	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	5	1	no	—
235	C₂×C₃×C₄	—	5	2	no	—
236	C₂×C₃×C₄	—	5	2	no	—
237	C₂×C₃×C₄	—	5	1	no	—
238	C₂×C₃×C₄	—	5	1	no	—
239	C₂×C₃×C₄	—	10	1	no	—
240	C₂×C₃×C₄	—	10	1	no	—
241	C₂×C₃×C₄	—	10	2	no	—
242	C₂×C₃×C₄	—	20	1	no	—
243	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
244	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
245	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
246	C₃⋊C₈	—	4	1	no	—
247	C₃⋊C₈	—	4	1	no	—
248	C₃⋊C₈	—	20	1	no	—
249	C₃⋊C₈	—	20	1	no	—
250	C₂²×C₅	C₂×#217	1	1	yes	characteristic
251	C₂×D₅	—	2	1	no	—
252	C₂×D₅	—	2	1	no	—
253	C₂×D₅	#835	1	2	yes	—
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	4	2	no	—
256	C₄×C₅	C₂×#215	1	1	yes	characteristic
257	C₄×C₅	C₂×#216	1	1	yes	characteristic
258	C₄×C₅	#949	1	1	yes	characteristic
259	C₄×C₅	#951	1	1	yes	characteristic
260	Dic₅	—	2	1	no	—
261	Dic₅	—	2	1	no	—
262	Dic₅	—	4	2	no	—
263	Dic₅	#835	1	2	yes	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	5	2	no	—
266	C₂²×C₄	—	5	1	no	—
267	C₂²×C₄	—	5	1	no	—
268	C₂²×C₄	—	5	1	no	—
269	M₁₆	—	60	2	no	—
270	C₂×C₈	—	6	1	no	—
271	C₂×C₈	—	6	1	no	—
272	C₂×C₈	—	30	1	no	—
273	C₂×C₈	—	30	1	no	—
274	C₂×C₈	—	60	2	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	10	2	no	—
277	C₄⋊C₄	—	10	2	no	—
278	C₄⋊C₄	—	10	1	no	—
279	C₄⋊C₄	—	10	1	no	—
280	C₄⋊C₄	D₅×Dic₃	1	2	yes	—
281	C₂²⋊C₄	—	10	2	no	—
282	C₄²	—	5	1	no	—
283	C₄²	—	5	1	no	—
284	C₄²	—	5	1	no	—
285	C₄²	—	10	2	no	—
286	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
287	C₃×C₅	#1550	1	1	yes	characteristic, complemented, Hall
288	C₂²×C₃	—	5	2	no	—
289	C₂²×C₃	—	10	1	no	—
290	C₂²×C₃	—	10	1	no	—
291	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	4	2	no	—
294	C₃×C₄	—	5	2	no	—
295	C₃×C₄	—	10	1	no	—
296	C₃×C₄	—	10	1	no	—
297	C₃×C₄	—	20	2	no	—
298	C₃×C₄	#4057	1	1	yes	characteristic
299	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
300	C₃×C₄	#4061	1	1	yes	characteristic
301	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
302	C₂×C₅	C₂×#835	1	1	yes	characteristic
303	C₂×C₅	#5232	1	1	yes	characteristic
304	C₂×C₅	#5237	1	1	yes	characteristic
305	D₅	—	2	2	no	—
306	C₂³	—	5	1	no	—
307	Q₈	—	4	1	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	20	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	5	1	no	—
312	C₂×C₄	—	5	2	no	—
313	C₂×C₄	—	5	2	no	—
314	C₂×C₄	—	5	1	no	—
315	C₂×C₄	—	5	1	no	—
316	C₂×C₄	—	10	1	no	—
317	C₂×C₄	—	10	1	no	—
318	C₂×C₄	—	10	2	no	—
319	C₂×C₄	—	20	1	no	—
320	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
321	C₂×C₄	D₅×#67	1	1	yes	characteristic
322	C₂×C₄	D₅×#67	1	1	yes	characteristic
323	C₈	—	12	1	no	—
324	C₈	—	12	1	no	—
325	C₈	—	60	1	no	—
326	C₈	—	60	1	no	—
327	C₂×C₃	—	10	2	no	—
328	C₂×C₃	#65040	1	1	yes	characteristic
329	C₂×C₃	#65060	1	1	yes	characteristic
330	C₂×C₃	D₅×#103	1	1	yes	characteristic
331	C₅	#71830	1	1	yes	characteristic, complemented, Sylow
332	C₂²	—	5	2	no	—
333	C₂²	—	10	1	no	—
334	C₂²	—	10	1	no	—
335	C₂²	D₅×#217	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	2	no	—
338	C₄	—	5	2	no	—
339	C₄	—	10	1	no	—
340	C₄	—	10	1	no	—
341	C₄	—	20	2	no	—
342	C₄	#91478	1	1	yes	characteristic
343	C₄	#91484	1	1	yes	characteristic
344	C₄	D₅×#215	1	1	yes	characteristic
345	C₄	D₅×#216	1	1	yes	characteristic
346	C₃	#10599216	1	1	yes	characteristic, complemented, Sylow
347	C₂	—	10	2	no	—
348	C₂	#11748951	1	1	yes	characteristic
349	C₂	D₅×#835	1	1	yes	characteristic
350	C₂	#11749369	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅×#217
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊Q₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₅	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊Q₈	D₃×D₅
hypocenter	C₃×C₅	#1550
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
4. #11746368	C₂	semidirect
5. C₅×#4547	C₂	semidirect
6. #11745861	C₂	semidirect
7. #11745861	C₂	semidirect
61. #4547	D₅	semidirect
287. C₃×C₅	#1550	semidirect
331. C₅	#71830	semidirect
346. C₃	#10599216	semidirect

Maximal subgroups

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

subgroup	index
C₃×#66062	2
#11747007	2
D₅×C₄⋊(C₃⋊C₈)	2
#11746368	2
C₅×#4547	2
#11745861	2
#10599216	3
#71830	5