Subgroups — #14285006

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	C₅×#5728	C₂	1	1	yes	maximal, characteristic
2	#11747011	C₂	1	1	yes	maximal, characteristic
3	D₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
4	#11746368	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64767	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745862	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745862	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602623	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₅×#946	—	2	1	no	—
11	C₅×Q₈×Dic₃	—	2	1	no	—
12	C₅×#897	C₄	1	2	yes	—
13	C₅×#892	—	2	1	no	—
14	C₅×#890	—	2	1	no	—
15	C₅×#887	C₄	1	2	yes	—
16	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₅×#869	C₂²	1	1	yes	characteristic
18	C₂×#6650	—	2	1	no	—
19	C₂×D₃×#168	—	2	1	no	—
20	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
24	Dic₃⋊#168	—	2	1	no	—
25	Dic₃⋊#168	—	2	1	no	—
26	#68423	—	5	1	no	maximal, Hall
27	C₅×#533	—	3	1	no	—
28	#65213	—	3	1	no	—
29	C₂×C₄⋊#168	—	3	1	no	—
30	#64767	—	3	1	no	—
31	#64767	D₃	1	1	yes	characteristic, complemented
32	#64766	—	3	1	no	—
33	#64766	—	3	1	no	—
34	C₂×C₃×C₅×Q₈	—	2	1	no	—
35	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
38	C₂×C₅×Dic₆	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	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₅×C₄⋊Dic₃	—	2	1	no	—
47	C₅×C₄⋊Dic₃	—	2	2	no	—
48	C₅×Dic₃⋊C₄	—	2	2	no	—
49	C₄×C₅×Dic₃	—	2	2	no	—
50	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
52	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
53	C₂×C₃×#168	—	2	1	no	—
54	C₂×C₃×#168	—	2	1	no	—
55	#6650	—	4	2	no	—
56	D₃×#168	—	4	2	no	—
57	#5728	D₅	1	1	yes	characteristic
58	#4837	—	5	1	no	—
59	D₃×C₄⋊C₈	—	5	1	no	—
60	C₃×#332	—	5	1	no	—
61	#4547	—	5	1	no	—
62	#4481	—	5	1	no	—
63	#4481	—	5	1	no	—
64	C₅×C₄⋊Q₈	—	3	1	no	—
65	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
66	C₅×#125	—	6	1	no	—
67	C₅×C₂²⋊Q₈	—	3	2	no	—
68	C₅×C₂²⋊Q₈	—	6	1	no	—
69	C₄×C₅×Q₈	—	3	2	no	—
70	C₄×C₅×Q₈	—	6	1	no	—
71	C₅×#117	—	6	1	no	—
72	C₂×C₄²×C₅	—	3	1	no	—
73	C₂×#681	—	6	1	no	—
74	C₂²×#168	—	6	1	no	—
75	C₄⋊#168	—	3	1	no	—
76	C₄⋊#168	—	3	1	no	—
77	C₄⋊#168	—	6	1	no	—
78	C₄⋊#168	—	6	1	no	—
79	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
80	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
81	#1550	—	15	1	no	Sylow
82	C₂²×C₅×D₃	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×C₅×Dic₃	—	2	2	no	—
90	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₅×D₃	—	2	1	no	—
94	C₄×C₅×D₃	—	2	1	no	—
95	C₄×C₅×D₃	—	4	1	no	—
96	C₄×C₅×D₃	SD₁₆	1	2	yes	—
97	C₄×C₅×D₃	Q₁₆	1	2	yes	—
98	C₅×Dic₆	—	4	2	no	—
99	C₃⋊#168	—	4	1	no	—
100	C₃⋊#168	—	4	1	no	—
101	C₃×#168	—	4	1	no	—
102	C₃×#168	—	4	1	no	—
103	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
104	#946	—	2	1	no	—
105	Q₈×Dic₃	—	2	1	no	—
106	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
107	C₂×D₃×C₈	—	10	1	no	—
108	#897	Dic₅	1	2	yes	—
109	#892	—	2	1	no	—
110	#890	—	2	1	no	—
111	#887	Dic₅	1	2	yes	—
112	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
113	#869	C₂×D₅	1	1	yes	characteristic
114	C₃×C₄⋊C₈	—	5	1	no	—
115	C₃×C₄⋊C₈	—	5	1	no	—
116	Dic₃⋊C₈	—	10	1	no	—
117	Dic₃⋊C₈	—	10	1	no	—
118	C₄⋊(C₃⋊C₈)	—	5	1	no	—
119	C₄⋊(C₃⋊C₈)	—	5	1	no	—
120	C₂×C₅×Q₈	—	2	1	no	—
121	C₂×C₅×Q₈	—	3	2	no	—
122	C₂²×C₄×C₅	—	3	1	no	—
123	C₂²×C₄×C₅	—	3	1	no	—
124	C₂²×C₄×C₅	—	3	1	no	—
125	C₅×C₄⋊C₄	—	2	1	no	—
126	C₅×C₄⋊C₄	—	6	2	no	—
127	C₅×C₄⋊C₄	—	6	2	no	—
128	C₅×C₄⋊C₄	—	6	1	no	—
129	C₅×C₄⋊C₄	—	6	1	no	—
130	C₅×C₄⋊C₄	C₄×D₃	1	2	yes	—
131	C₅×C₂²⋊C₄	—	6	2	no	—
132	C₄²×C₅	—	3	1	no	—
133	C₄²×C₅	—	3	1	no	—
134	C₄²×C₅	—	3	1	no	—
135	C₄²×C₅	—	6	2	no	—
136	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
137	#681	—	12	2	no	—
138	C₂×#168	—	2	1	no	—
139	C₂×#168	—	2	1	no	—
140	C₂×#168	—	6	1	no	—
141	C₂×#168	—	6	1	no	—
142	C₂×#168	—	12	2	no	—
143	#533	—	3	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	—	5	1	no	—
149	#332	—	15	1	no	—
150	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₅×D₃	—	2	1	no	—
152	C₂×C₅×D₃	—	2	1	no	—
153	C₂×C₅×D₃	#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	C₅×Dic₃	—	2	1	no	—
161	C₅×Dic₃	—	2	1	no	—
162	C₅×Dic₃	—	4	2	no	—
163	C₅×Dic₃	#103	1	2	yes	—
164	C₂×C₃×Q₈	—	2	1	no	—
165	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
166	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
167	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
168	C₂×Dic₆	C₂×Dic₅	1	2	yes	—
169	C₂×C₃×C₈	—	10	1	no	—
170	C₂×C₃×C₈	—	10	1	no	—
171	C₃×C₄⋊C₄	—	2	1	no	—
172	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
173	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
174	(C₂×D₃)⋊C₄	—	2	2	no	—
175	C₄⋊Dic₃	—	2	1	no	—
176	C₄⋊Dic₃	—	2	1	no	—
177	C₄⋊Dic₃	—	2	2	no	—
178	Dic₃⋊C₄	—	2	2	no	—
179	C₄×Dic₃	—	2	2	no	—
180	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
181	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
182	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
183	C₂×C₃⋊C₈	—	10	1	no	—
184	C₂×C₃⋊C₈	—	10	1	no	—
185	ΓL(1,𝔽₂₅)	—	20	2	no	—
186	D₃×C₈	—	20	2	no	—
187	C₂³×C₅	—	3	1	no	—
188	C₅×Q₈	—	4	1	no	—
189	C₅×Q₈	—	4	1	no	—
190	C₅×Q₈	—	12	2	no	—
191	C₂×C₄×C₅	—	2	2	no	—
192	C₂×C₄×C₅	—	3	1	no	—
193	C₂×C₄×C₅	—	3	2	no	—
194	C₂×C₄×C₅	—	3	2	no	—
195	C₂×C₄×C₅	—	3	1	no	—
196	C₂×C₄×C₅	—	3	1	no	—
197	C₂×C₄×C₅	—	6	1	no	—
198	C₂×C₄×C₅	—	6	1	no	—
199	C₂×C₄×C₅	—	6	2	no	—
200	C₂×C₄×C₅	—	12	1	no	—
201	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
202	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
203	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
204	#168	—	4	1	no	—
205	#168	—	4	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	—	3	1	no	—
211	C₄⋊Q₈	D₃×D₅	1	1	yes	characteristic
212	#125	—	6	1	no	—
213	C₂²⋊Q₈	—	3	2	no	—
214	C₂²⋊Q₈	—	6	1	no	—
215	C₄×Q₈	—	3	2	no	—
216	C₄×Q₈	—	6	1	no	—
217	#117	—	6	1	no	—
218	C₂×C₄²	—	3	1	no	—
219	C₄⋊C₈	—	5	1	no	—
220	C₄⋊C₈	—	5	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	C₅×D₃	—	2	2	no	—
229	C₂²×D₃	C₂²⋊Dic₅	1	1	yes	characteristic
230	C₃×Q₈	—	4	1	no	—
231	C₃×Q₈	—	4	1	no	—
232	C₂×C₃×C₄	—	2	2	no	—
233	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
236	C₂×Dic₃	—	2	2	no	—
237	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
238	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
239	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
240	C₄×D₃	—	2	1	no	—
241	C₄×D₃	—	2	1	no	—
242	C₄×D₃	—	4	1	no	—
243	C₄×D₃	#688	1	2	yes	—
244	C₄×D₃	#689	1	2	yes	—
245	Dic₆	—	4	2	no	—
246	C₃×C₈	—	20	1	no	—
247	C₃×C₈	—	20	1	no	—
248	C₃⋊C₈	—	20	1	no	—
249	C₃⋊C₈	—	20	1	no	—
250	C₂²×C₅	—	3	2	no	—
251	C₂²×C₅	—	6	1	no	—
252	C₂²×C₅	—	6	1	no	—
253	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	3	2	no	—
256	C₄×C₅	—	4	2	no	—
257	C₄×C₅	—	6	1	no	—
258	C₄×C₅	—	6	1	no	—
259	C₄×C₅	—	12	2	no	—
260	C₄×C₅	#911	1	1	yes	characteristic
261	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
262	C₄×C₅	#915	1	1	yes	characteristic
263	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	3	2	no	—
266	C₂²×C₄	—	3	1	no	—
267	C₂²×C₄	—	3	1	no	—
268	C₂²×C₄	—	3	1	no	—
269	M₁₆	—	60	2	no	—
270	C₂×C₈	—	10	1	no	—
271	C₂×C₈	—	10	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	—	6	2	no	—
277	C₄⋊C₄	—	6	2	no	—
278	C₄⋊C₄	—	6	1	no	—
279	C₄⋊C₄	—	6	1	no	—
280	C₄⋊C₄	D₃×Dic₅	1	2	yes	—
281	C₂²⋊C₄	—	6	2	no	—
282	C₄²	—	3	1	no	—
283	C₄²	—	3	1	no	—
284	C₄²	—	3	1	no	—
285	C₄²	—	6	2	no	—
286	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
287	C₃×C₅	#1550	1	1	yes	characteristic, complemented, Hall
288	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
289	C₂×D₃	—	2	1	no	—
290	C₂×D₃	—	2	1	no	—
291	C₂×D₃	#3967	1	2	yes	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	4	2	no	—
294	C₃×C₄	C₂×#688	1	1	yes	characteristic
295	C₃×C₄	C₂×#689	1	1	yes	characteristic
296	C₃×C₄	#4095	1	1	yes	characteristic
297	C₃×C₄	#4097	1	1	yes	characteristic
298	Dic₃	—	2	1	no	—
299	Dic₃	—	2	1	no	—
300	Dic₃	—	4	2	no	—
301	Dic₃	#3967	1	2	yes	—
302	C₂×C₅	—	6	2	no	—
303	C₂×C₅	#4751	1	1	yes	characteristic
304	C₂×C₅	#4771	1	1	yes	characteristic
305	C₂×C₅	D₃×#103	1	1	yes	characteristic
306	C₂³	—	3	1	no	—
307	Q₈	—	4	1	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	12	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	3	1	no	—
312	C₂×C₄	—	3	2	no	—
313	C₂×C₄	—	3	2	no	—
314	C₂×C₄	—	3	1	no	—
315	C₂×C₄	—	3	1	no	—
316	C₂×C₄	—	6	1	no	—
317	C₂×C₄	—	6	1	no	—
318	C₂×C₄	—	6	2	no	—
319	C₂×C₄	—	12	1	no	—
320	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
321	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
322	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
323	C₈	—	20	1	no	—
324	C₈	—	20	1	no	—
325	C₈	—	60	1	no	—
326	C₈	—	60	1	no	—
327	C₂×C₃	C₂×#3967	1	1	yes	characteristic
328	C₂×C₃	#65521	1	1	yes	characteristic
329	C₂×C₃	#65526	1	1	yes	characteristic
330	D₃	—	2	2	no	—
331	C₅	#68423	1	1	yes	characteristic, complemented, Sylow
332	C₂²	—	3	2	no	—
333	C₂²	—	6	1	no	—
334	C₂²	—	6	1	no	—
335	C₂²	D₃×C₂²⋊Dic₅	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	3	2	no	—
338	C₄	—	4	2	no	—
339	C₄	—	6	1	no	—
340	C₄	—	6	1	no	—
341	C₄	—	12	2	no	—
342	C₄	#91480	1	1	yes	characteristic
343	C₄	#91486	1	1	yes	characteristic
344	C₄	D₃×#688	1	1	yes	characteristic
345	C₄	D₃×#689	1	1	yes	characteristic
346	C₃	#10602623	1	1	yes	characteristic, complemented, Sylow
347	C₂	—	6	2	no	—
348	C₂	#11748955	1	1	yes	characteristic
349	C₂	D₃×#3967	1	1	yes	characteristic
350	C₂	#11749372	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×C₂²⋊Dic₅
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×C₅×D₃	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. C₃×#64767	C₂	semidirect
6. #11745862	C₂	semidirect
7. #11745862	C₂	semidirect
31. #64767	D₃	semidirect
287. C₃×C₅	#1550	semidirect
331. C₅	#68423	semidirect
346. C₃	#10602623	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5728	2
#11747011	2
D₃×C₄⋊#168	2
#11746368	2
C₃×#64767	2
#11745862	2
#10602623	3
#68423	5