Subgroups — #14418172

Order 1920.

916 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5748	C₂	1	1	yes	maximal, characteristic
2	#11754125	C₂	1	2	yes	maximal, complemented
3	M₁₆×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65414	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#90978	C₂	1	2	yes	maximal, complemented
6	#11748073	C₂	1	4	yes	maximal, complemented
7	#11748072	C₂	1	4	yes	maximal, complemented
8	#10615731	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
12	C₂×#6710	C₂²	1	1	yes	characteristic
13	C₅×#909	C₄	1	4	yes	complemented
14	C₅×#908	C₄	1	4	yes	complemented
15	C₅×#904	C₂²	1	4	yes	—
16	C₂×C₃×#681	C₂²	1	2	yes	—
17	#91288	C₂²	1	8	yes	complemented
18	D₃×#681	—	2	4	no	—
19	#91286	—	2	4	no	—
20	#91099	C₂²	1	2	yes	complemented
21	C₈×Dic₁₅	C₂²	1	2	yes	complemented
22	C₃×#3951	C₂²	1	4	yes	complemented
23	#90978	C₂²	1	4	yes	complemented
24	#90978	C₂²	1	4	yes	complemented
25	#79258	—	5	1	no	maximal, Hall
26	C₅×#575	—	3	1	no	—
27	#66153	—	3	2	no	—
28	C₂×#3969	—	3	2	no	—
29	#65486	—	3	4	no	—
30	#65485	—	3	4	no	—
31	#65414	D₃	1	1	yes	characteristic, complemented
32	M₁₆×Dic₅	—	3	1	no	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₅	—	2	1	no	—
35	C₂×C₅×#67	—	2	2	no	—
36	C₅×#235	—	2	2	no	—
37	C₅×#235	—	2	2	no	—
38	C₅×#235	C₂×C₄	1	4	yes	—
39	C₅×#235	C₂³	1	4	yes	—
40	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
41	C₂×C₄×C₅×D₃	—	2	2	no	—
42	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
43	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
44	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
45	C₄×Dic₁₅	—	2	1	no	—
46	C₄×Dic₁₅	C₂³	1	1	yes	characteristic
47	C₄×Dic₁₅	C₂³	1	1	yes	characteristic
48	#6710	D₄	1	4	yes	—
49	C₂×C₃⋊#168	C₂³	1	2	yes	—
50	C₅×Dic₁₂	C₂×C₄	1	4	yes	complemented
51	C₅×D₂₄	C₂×C₄	1	4	yes	complemented
52	C₅×#204	C₂×C₄	1	8	yes	complemented
53	C₃×#681	—	2	2	no	—
54	C₃×#681	C₂³	1	4	yes	—
55	C₂×C₃×#168	—	2	2	no	—
56	#6650	—	2	8	no	—
57	D₃×#168	—	2	8	no	—
58	#5748	D₅	1	1	yes	characteristic
59	#5745	—	5	2	no	—
60	C₃×#430	—	5	1	no	—
61	C₂×#825	—	5	2	no	—
62	Dic₃×M₁₆	—	5	1	no	—
63	#4895	—	5	4	no	—
64	#4894	—	5	4	no	—
65	C₂×C₅×#41	—	3	2	no	—
66	C₅×#137	—	3	4	no	—
67	C₅×AGL(1,ℤ/8)	—	3	4	no	—
68	C₅×#135	—	3	4	no	—
69	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
70	#4094	—	3	8	no	—
71	#4094	—	6	4	no	—
72	C₂×C₄×Dic₅	—	3	1	no	—
73	C₂×#681	—	3	1	no	—
74	C₂×#681	—	6	4	no	—
75	C₂×#681	C₂×D₃	1	2	yes	—
76	#3969	—	3	4	no	—
77	#3969	—	3	4	no	—
78	#3951	C₂×D₃	1	4	yes	complemented
79	#3947	—	3	2	no	—
80	C₈×Dic₅	—	3	2	no	—
81	#2956	—	15	1	no	Sylow
82	C₂³×C₃×C₅	#41	1	1	yes	characteristic
83	C₂²×C₅×D₃	—	2	2	no	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₁₅	—	2	1	no	—
89	C₂×Dic₁₅	—	2	1	no	—
90	C₂×Dic₁₅	—	4	2	no	—
91	C₅×#67	—	2	4	no	—
92	C₅×#67	—	2	4	no	—
93	C₂×C₅×Dic₃	—	2	2	no	—
94	C₅×D₁₂	—	2	2	no	—
95	C₅×D₁₂	C₂²×C₄	1	4	yes	—
96	C₄×C₅×D₃	—	2	4	no	—
97	C₄×C₅×D₃	—	2	4	no	—
98	C₅×Dic₆	—	2	2	no	—
99	C₅×Dic₆	C₂²×C₄	1	4	yes	—
100	C₃×C₅×C₈	C₂²×C₄	1	4	yes	—
101	C₃⋊#168	C₂×D₄	1	4	yes	—
102	C₃×#168	—	2	4	no	—
103	C₂×#235	C₂×D₅	1	2	yes	—
104	C₂×C₃×M₁₆	—	5	2	no	—
105	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
106	C₂×C₄×Dic₃	—	5	1	no	—
107	C₂×#208	—	5	1	no	—
108	#909	Dic₅	1	4	yes	complemented
109	#908	Dic₅	1	4	yes	complemented
110	#907	—	5	8	no	—
111	D₃×M₁₆	—	10	4	no	—
112	#904	C₂×D₅	1	4	yes	—
113	#901	—	10	4	no	—
114	C₃×#108	—	5	4	no	—
115	#825	—	5	4	no	—
116	#825	—	5	4	no	—
117	#815	—	5	2	no	—
118	C₈×Dic₃	—	5	2	no	—
119	C₅×#41	—	3	4	no	—
120	C₅×#41	—	3	4	no	—
121	C₅×#41	—	6	2	no	—
122	C₅×#41	—	6	2	no	—
123	C₂×C₅×Q₈	—	3	2	no	—
124	C₂×C₅×D₄	—	3	2	no	—
125	C₂×C₅×D₄	—	6	2	no	—
126	C₂²×C₄×C₅	—	6	2	no	—
127	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
128	C₂²×Dic₅	—	6	1	no	—
129	C₅×Q₁₆	—	3	4	no	—
130	C₅×SD₁₆	—	3	8	no	—
131	C₅×D₈	—	3	4	no	—
132	C₅×M₁₆	C₄×D₃	1	4	yes	—
133	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
134	C₄×Dic₅	—	3	1	no	—
135	C₄×Dic₅	—	3	1	no	—
136	C₄×Dic₅	—	6	1	no	—
137	#681	—	2	2	no	—
138	#681	—	3	4	no	—
139	#681	—	6	8	no	—
140	#681	C₂²×D₃	1	4	yes	—
141	C₂×#168	—	2	2	no	—
142	C₂×#168	—	3	2	no	—
143	C₂×#168	—	6	8	no	—
144	#575	—	3	1	no	—
145	#568	—	15	2	no	—
146	#444	—	15	4	no	—
147	#443	—	15	4	no	—
148	#430	—	5	1	no	—
149	C₂×#104	—	15	2	no	—
150	C₄×M₁₆	—	15	1	no	—
151	C₂²×C₃×C₅	—	4	1	no	—
152	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
153	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
154	C₂×C₅×D₃	—	2	4	no	—
155	C₂×C₅×D₃	—	8	2	no	—
156	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
157	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
158	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
159	Dic₁₅	—	2	2	no	—
160	Dic₁₅	—	4	1	no	—
161	C₅×Dic₃	—	2	4	no	—
162	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
163	C₂×#67	—	2	2	no	—
164	C₂²×Dic₃	—	10	1	no	—
165	#235	—	2	2	no	—
166	#235	—	2	2	no	—
167	#235	C₂×Dic₅	1	4	yes	—
168	#235	C₂²×D₅	1	4	yes	—
169	C₂×D₁₂	C₂×Dic₅	1	2	yes	—
170	C₂×C₄×D₃	—	2	2	no	—
171	C₂×Dic₆	C₂×Dic₅	1	2	yes	—
172	C₃×M₁₆	—	5	4	no	—
173	C₃×M₁₆	—	10	2	no	—
174	C₃×M₁₆	C₂×Dic₅	1	4	yes	—
175	C₂×C₃×C₈	—	10	2	no	—
176	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
177	C₄×Dic₃	—	5	1	no	—
178	C₄×Dic₃	—	5	1	no	—
179	C₄×Dic₃	—	10	1	no	—
180	#208	—	5	4	no	—
181	C₂×C₃⋊C₈	—	5	2	no	—
182	Dic₁₂	C₂×Dic₅	1	4	yes	complemented
183	D₂₄	C₂×Dic₅	1	4	yes	complemented
184	#204	C₂×Dic₅	1	8	yes	complemented
185	ΓL(1,𝔽₂₅)	—	10	8	no	—
186	D₃×C₈	—	10	8	no	—
187	C₂³×C₅	—	6	2	no	—
188	C₂³×C₅	#239	1	1	yes	characteristic
189	C₅×Q₈	—	3	4	no	—
190	C₅×Q₈	—	6	2	no	—
191	C₅×D₄	—	3	4	no	—
192	C₅×D₄	—	6	4	no	—
193	C₅×D₄	—	6	4	no	—
194	C₅×D₄	—	6	2	no	—
195	C₂×C₄×C₅	—	6	2	no	—
196	C₂×C₄×C₅	—	6	4	no	—
197	C₂×C₄×C₅	—	6	4	no	—
198	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
199	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
200	C₂×C₄×C₅	#239	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
202	C₂×Dic₅	—	6	1	no	—
203	C₂×Dic₅	—	6	1	no	—
204	C₂×Dic₅	—	12	2	no	—
205	C₅×C₈	C₂×C₄×D₃	1	4	yes	—
206	#168	—	2	4	no	—
207	#168	—	3	4	no	—
208	C₂×#41	—	3	2	no	—
209	#137	—	3	4	no	—
210	AGL(1,ℤ/8)	—	3	4	no	—
211	#135	—	3	4	no	—
212	#131	—	15	8	no	—
213	#131	—	30	4	no	—
214	C₂×M₁₆	—	5	2	no	—
215	C₂×M₁₆	—	15	1	no	—
216	C₂×M₁₆	—	30	4	no	—
217	C₂×M₁₆	D₃×D₅	1	1	yes	characteristic
218	C₂×C₄²	—	15	1	no	—
219	#108	—	5	4	no	—
220	#104	—	15	4	no	—
221	#104	—	15	4	no	—
222	#97	—	15	2	no	—
223	C₄×C₈	—	15	2	no	—
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	—	2	2	no	—
226	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
227	C₅×D₃	—	4	4	no	—
228	C₂³×C₃	#711	1	1	yes	characteristic
229	C₂²×D₃	—	2	2	no	—
230	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
231	C₂×C₃×C₄	#711	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
233	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
234	#67	—	2	4	no	—
235	#67	—	2	4	no	—
236	C₂×Dic₃	—	2	2	no	—
237	C₂×Dic₃	—	10	1	no	—
238	C₂×Dic₃	—	10	1	no	—
239	C₂×Dic₃	—	20	2	no	—
240	D₁₂	—	2	2	no	—
241	D₁₂	C₂²×Dic₅	1	4	yes	—
242	C₄×D₃	—	2	4	no	—
243	C₄×D₃	—	2	4	no	—
244	Dic₆	—	2	2	no	—
245	Dic₆	C₂²×Dic₅	1	4	yes	—
246	C₃×C₈	—	10	4	no	—
247	C₃×C₈	C₂²×Dic₅	1	4	yes	—
248	C₃⋊C₈	—	5	4	no	—
249	C₂²×C₅	—	4	1	no	—
250	C₂²×C₅	—	6	4	no	—
251	C₂²×C₅	—	24	2	no	—
252	C₂²×C₅	#893	1	2	yes	—
253	C₂²×C₅	C₂×#239	1	1	yes	characteristic
254	C₄×C₅	—	6	4	no	—
255	C₄×C₅	#893	1	2	yes	—
256	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
257	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
258	Dic₅	—	6	2	no	—
259	Dic₅	—	12	1	no	—
260	#41	—	3	4	no	—
261	#41	—	3	4	no	—
262	#41	—	6	2	no	—
263	#41	—	6	2	no	—
264	C₂×Q₈	—	3	2	no	—
265	C₂×D₄	—	3	2	no	—
266	C₂×D₄	—	6	2	no	—
267	C₂²×C₄	—	6	2	no	—
268	C₂²×C₄	—	30	1	no	—
269	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
270	Q₁₆	—	3	4	no	—
271	SD₁₆	—	3	8	no	—
272	D₈	—	3	4	no	—
273	M₁₆	—	5	4	no	—
274	M₁₆	—	10	2	no	—
275	M₁₆	—	15	4	no	—
276	M₁₆	—	30	8	no	—
277	M₁₆	D₃×Dic₅	1	4	yes	—
278	C₂×C₈	—	10	2	no	—
279	C₂×C₈	—	15	2	no	—
280	C₂×C₈	—	30	8	no	—
281	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
282	C₄²	—	15	1	no	—
283	C₄²	—	15	1	no	—
284	C₄²	—	30	1	no	—
285	C₃×C₅	#2956	1	1	yes	characteristic, complemented, Hall
286	C₂²×C₃	—	4	1	no	—
287	C₂²×C₃	D₄×Dic₅	1	2	yes	—
288	C₂²×C₃	C₂×#711	1	1	yes	characteristic
289	C₂×D₃	—	2	4	no	—
290	C₂×D₃	—	8	2	no	—
291	C₃×C₄	D₄×Dic₅	1	2	yes	—
292	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
293	C₃×C₄	C₂³×Dic₅	1	1	yes	characteristic
294	Dic₃	—	2	4	no	—
295	Dic₃	—	10	2	no	—
296	Dic₃	—	20	1	no	—
297	C₂×C₅	—	2	1	no	—
298	C₂×C₅	—	2	2	no	—
299	C₂×C₅	—	12	4	no	—
300	C₂×C₅	C₂×#893	1	1	yes	characteristic
301	C₂³	—	6	2	no	—
302	C₂³	#6768	1	1	yes	characteristic
303	Q₈	—	3	4	no	—
304	Q₈	—	6	2	no	—
305	D₄	—	3	4	no	—
306	D₄	—	6	4	no	—
307	D₄	—	6	4	no	—
308	D₄	—	6	2	no	—
309	C₂×C₄	—	6	2	no	—
310	C₂×C₄	—	6	4	no	—
311	C₂×C₄	—	6	4	no	—
312	C₂×C₄	—	30	1	no	—
313	C₂×C₄	—	30	1	no	—
314	C₂×C₄	—	60	2	no	—
315	C₂×C₄	#6768	1	1	yes	characteristic
316	C₂×C₄	#6777	1	2	yes	—
317	C₂×C₄	C₂×D₃×Dic₅	1	2	yes	—
318	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
319	C₈	—	10	4	no	—
320	C₈	—	15	4	no	—
321	C₈	C₂×D₃×Dic₅	1	4	yes	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	2	2	no	—
324	C₂×C₃	C₂×D₄×Dic₅	1	1	yes	characteristic
325	D₃	—	4	4	no	—
326	C₅	#79258	1	1	yes	characteristic, complemented, Sylow
327	C₂²	—	4	1	no	—
328	C₂²	—	6	4	no	—
329	C₂²	—	24	2	no	—
330	C₂²	#91435	1	2	yes	—
331	C₂²	C₂×#6768	1	1	yes	characteristic
332	C₄	—	6	4	no	—
333	C₄	—	30	2	no	—
334	C₄	—	60	1	no	—
335	C₄	#91435	1	2	yes	—
336	C₄	C₂×#6777	1	1	yes	characteristic
337	C₄	C₂²×D₃×Dic₅	1	1	yes	characteristic
338	C₃	#10615731	1	1	yes	characteristic, complemented, Sylow
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	—	12	4	no	—
342	C₂	C₂×#91435	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#6777
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₅×M₁₆	C₂²
socle	C₂×C₃×C₅	C₂×C₄×D₄
prime radical	C₅×#5748	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×M₁₆	D₃×D₅
hypocenter	C₃×C₅	#2956
LCS	G ▷ C₃×C₄×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. #11754125	C₂	semidirect
3. M₁₆×Dic₁₅	C₂	semidirect
4. C₃×#65414	C₂	semidirect
5. C₂×#90978	C₂	semidirect
6. #11748073	C₂	semidirect
7. #11748072	C₂	semidirect
13. C₅×#909	C₄	semidirect
14. C₅×#908	C₄	semidirect
17. #91288	C₂²	semidirect
20. #91099	C₂²	semidirect
21. C₈×Dic₁₅	C₂²	semidirect
22. C₃×#3951	C₂²	semidirect
23. #90978	C₂²	semidirect
24. #90978	C₂²	semidirect
31. #65414	D₃	semidirect
50. C₅×Dic₁₂	C₂×C₄	semidirect
51. C₅×D₂₄	C₂×C₄	semidirect
52. C₅×#204	C₂×C₄	semidirect
78. #3951	C₂×D₃	semidirect
108. #909	Dic₅	semidirect
109. #908	Dic₅	semidirect
182. Dic₁₂	C₂×Dic₅	semidirect
183. D₂₄	C₂×Dic₅	semidirect
184. #204	C₂×Dic₅	semidirect
285. C₃×C₅	#2956	semidirect
326. C₅	#79258	semidirect
338. C₃	#10615731	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5748	2
#11754125	2
M₁₆×Dic₁₅	2
C₃×#65414	2
C₂×#90978	2
#11748073	2
#11748072	2
#10615731	3
#79258	5