Subgroups — #14412916

Order 1920.

1476 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11757146	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
3	#11756578	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751106	C₂	1	6	yes	maximal, complemented
5	C₂×#91089	C₂	1	6	yes	maximal, complemented
6	#10614267	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	#92101	C₄	1	4	yes	complemented
9	#92100	C₄	1	4	yes	complemented
10	#41×D₁₅	—	2	4	no	—
11	#92098	—	2	1	no	—
12	#92095	—	2	3	no	—
13	C₂×#6817	C₂²	1	6	yes	—
14	C₃×C₄×C₅×Q₈	—	2	1	no	—
15	C₃×C₄×C₅×D₄	—	2	3	no	—
16	C₃×C₅×#117	C₂²	1	3	yes	complemented
17	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	complemented
18	#91837	—	2	4	no	—
19	C₂×#6710	C₂²	1	6	yes	complemented
20	#91089	C₂²	1	4	yes	complemented
21	#91089	C₂²	1	12	yes	complemented
22	#82536	—	5	1	no	maximal, Hall
23	#66283	—	3	1	no	—
24	C₄×C₅×#41	D₃	1	1	yes	characteristic, complemented
25	#66153	—	3	1	no	—
26	#65294	—	3	6	no	—
27	C₂×#3937	—	3	6	no	—
28	C₃×C₅×#41	C₂×C₄	1	4	yes	—
29	C₃×C₅×#41	D₄	1	4	yes	—
30	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×D₄	D₄	1	3	yes	—
32	C₂²×C₃×C₄×C₅	—	2	3	no	—
33	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
34	C₂×#1220	—	2	6	no	—
35	#6821	—	2	4	no	—
36	Q₈×D₁₅	—	2	4	no	—
37	#6819	—	2	12	no	—
38	D₄×D₁₅	—	2	12	no	—
39	#6817	—	2	6	no	—
40	#6817	—	2	6	no	—
41	#6817	C₂×C₄	1	12	yes	complemented
42	#6817	C₂³	1	4	yes	—
43	C₂×D₆₀	C₂×C₄	1	6	yes	complemented
44	C₂×C₄×D₁₅	—	2	6	no	—
45	C₂×Dic₃₀	C₂×C₄	1	6	yes	complemented
46	C₃×C₅×C₄⋊C₄	—	2	3	no	—
47	C₃×C₅×C₂²⋊C₄	—	2	3	no	—
48	C₃×C₄²×C₅	—	2	3	no	—
49	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₃×C₄²×C₅	C₂³	1	3	yes	—
51	#6710	—	2	6	no	—
52	#6710	C₂³	1	4	yes	—
53	C₂×C₃⋊#168	—	2	6	no	—
54	#5964	—	5	1	no	—
55	C₃×C₄×#41	D₅	1	1	yes	characteristic, complemented
56	#5819	—	5	1	no	—
57	#5005	—	5	6	no	—
58	C₂×#805	—	5	6	no	—
59	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
60	#4150	—	3	4	no	—
61	#4149	—	3	4	no	—
62	D₅×#41	—	6	4	no	—
63	#4147	—	6	1	no	—
64	#4144	—	6	3	no	—
65	C₂×#709	—	3	6	no	—
66	C₄×C₅×Q₈	—	2	1	no	—
67	C₄×C₅×D₄	—	2	3	no	—
68	C₅×#117	C₂×D₃	1	3	yes	complemented
69	C₂×C₄²×C₅	C₂×D₃	1	3	yes	complemented
70	#4094	—	6	4	no	—
71	C₂×#681	—	3	6	no	—
72	#3937	—	3	12	no	—
73	#3937	—	3	4	no	—
74	#2901	—	15	1	no	Sylow
75	C₂³×C₃×C₅	C₂×D₄	1	3	yes	—
76	C₂²×D₁₅	—	2	6	no	—
77	C₃×C₅×Q₈	C₂²⋊C₄	1	4	yes	—
78	C₃×C₅×D₄	C₂²⋊C₄	1	12	yes	—
79	C₂×C₃×C₄×C₅	—	2	1	no	—
80	C₂×C₃×C₄×C₅	—	2	3	no	—
81	C₂×C₃×C₄×C₅	—	4	6	no	—
82	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
85	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	#1220	—	2	12	no	—
87	#1220	—	2	4	no	—
88	C₂×Dic₁₅	—	2	6	no	—
89	D₆₀	—	2	6	no	—
90	D₆₀	C₂²×C₄	1	4	yes	—
91	C₄×D₁₅	—	2	12	no	—
92	C₄×D₁₅	—	2	4	no	—
93	Dic₃₀	—	2	6	no	—
94	Dic₃₀	C₂²×C₄	1	4	yes	—
95	C₃⋊#168	—	2	4	no	—
96	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
97	#1010	—	5	4	no	—
98	#1009	—	5	4	no	—
99	D₃×#41	—	10	4	no	—
100	#1007	—	10	1	no	—
101	#1004	—	10	3	no	—
102	C₂×#235	—	5	6	no	—
103	C₃×C₄×Q₈	—	2	1	no	—
104	C₃×C₄×D₄	—	2	3	no	—
105	C₃×#117	C₂×D₅	1	3	yes	complemented
106	C₂×C₃×C₄²	C₂×D₅	1	3	yes	complemented
107	#948	—	10	4	no	—
108	C₂×#208	—	5	6	no	—
109	#805	—	5	4	no	—
110	#805	—	5	12	no	—
111	C₅×#41	C₄×D₃	1	4	yes	—
112	C₅×#41	D₁₂	1	4	yes	—
113	C₂×C₅×Q₈	#67	1	1	yes	characteristic
114	C₂×C₅×D₄	#67	1	3	yes	—
115	C₂²×C₄×C₅	—	2	3	no	—
116	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
117	C₂×C₂²⋊D₅	—	6	6	no	—
118	#713	—	6	4	no	—
119	Q₈×D₅	—	6	4	no	—
120	#711	—	6	12	no	—
121	D₄×D₅	—	6	12	no	—
122	#709	—	3	12	no	—
123	#709	—	3	4	no	—
124	#709	—	6	6	no	—
125	#709	—	6	6	no	—
126	C₂×D₂₀	—	3	6	no	—
127	C₂×C₄×D₅	—	6	6	no	—
128	C₂×Dic₁₀	—	3	6	no	—
129	C₅×C₄⋊C₄	—	2	3	no	—
130	C₅×C₂²⋊C₄	—	2	3	no	—
131	C₄²×C₅	—	2	3	no	—
132	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
133	C₄²×C₅	C₂²×D₃	1	3	yes	—
134	#681	—	3	4	no	—
135	#681	—	6	6	no	—
136	C₂×#168	—	6	6	no	—
137	#585	—	15	1	no	—
138	#568	—	15	1	no	—
139	C₄×#41	D₁₅	1	1	yes	characteristic, complemented
140	#421	—	15	6	no	—
141	C₂×#104	—	15	6	no	—
142	C₂²×C₃×C₅	—	4	3	no	—
143	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	6	yes	—
144	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₅	—	2	4	no	—
146	C₂×D₁₅	—	8	6	no	—
147	C₃×C₄×C₅	—	2	2	no	—
148	C₃×C₄×C₅	—	4	3	no	—
149	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
150	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
151	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
152	Dic₁₅	—	2	4	no	—
153	C₃×#41	C₄×D₅	1	4	yes	—
154	C₃×#41	D₂₀	1	4	yes	—
155	C₂×C₃×Q₈	C₂²⋊D₅	1	1	yes	characteristic
156	C₂×C₃×D₄	C₂²⋊D₅	1	3	yes	—
157	C₂²×C₃×C₄	—	2	3	no	—
158	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
159	C₂×#67	—	10	6	no	—
160	#239	—	10	4	no	—
161	D₃×Q₈	—	10	4	no	—
162	#237	—	10	12	no	—
163	D₃×D₄	—	10	12	no	—
164	#235	—	5	12	no	—
165	#235	—	5	4	no	—
166	#235	—	10	6	no	—
167	#235	—	10	6	no	—
168	C₂×D₁₂	—	5	6	no	—
169	C₂×C₄×D₃	—	10	6	no	—
170	C₂×Dic₆	—	5	6	no	—
171	C₃×C₄⋊C₄	—	2	3	no	—
172	C₃×C₂²⋊C₄	—	2	3	no	—
173	C₃×C₄²	—	2	3	no	—
174	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
175	C₃×C₄²	C₂²×D₅	1	3	yes	—
176	#208	—	5	4	no	—
177	#208	—	10	6	no	—
178	C₂×C₃⋊C₈	—	10	6	no	—
179	C₂³×C₅	C₂×#67	1	3	yes	—
180	C₂²×D₅	—	6	6	no	—
181	C₅×Q₈	(C₂×D₃)⋊C₄	1	4	yes	—
182	C₅×D₄	(C₂×D₃)⋊C₄	1	12	yes	—
183	C₂×C₄×C₅	—	2	1	no	—
184	C₂×C₄×C₅	—	2	3	no	—
185	C₂×C₄×C₅	—	4	6	no	—
186	C₂×C₄×C₅	C₂×C₄×D₃	1	6	yes	—
187	C₂×C₄×C₅	C₂×D₁₂	1	6	yes	—
188	C₂×C₄×C₅	C₂×#67	1	3	yes	—
189	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
190	C₂²⋊D₅	—	6	12	no	—
191	C₂²⋊D₅	—	6	4	no	—
192	C₂×Dic₅	—	6	6	no	—
193	D₂₀	—	3	4	no	—
194	D₂₀	—	6	6	no	—
195	C₄×D₅	—	6	12	no	—
196	C₄×D₅	—	6	4	no	—
197	Dic₁₀	—	3	4	no	—
198	Dic₁₀	—	6	6	no	—
199	#168	—	6	4	no	—
200	2₋¹⁺⁴	—	15	4	no	—
201	2₋¹⁺⁴	—	30	1	no	—
202	H(5,𝔽₂)	—	15	4	no	—
203	H(5,𝔽₂)	—	30	3	no	—
204	C₂×#41	—	15	6	no	—
205	C₂×#41	—	30	4	no	—
206	C₂×#41	C₂×D₁₅	1	1	yes	characteristic
207	#131	—	30	4	no	—
208	C₂×M₁₆	—	15	6	no	—
209	C₄×Q₈	—	2	1	no	—
210	C₄×D₄	—	2	3	no	—
211	#117	C₂×D₁₅	1	3	yes	complemented
212	C₂×C₄²	C₂×D₁₅	1	3	yes	complemented
213	#104	—	15	4	no	—
214	#104	—	15	12	no	—
215	C₂×C₃×C₅	—	2	1	no	—
216	C₂×C₃×C₅	—	2	6	no	—
217	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
218	D₁₅	—	4	4	no	—
219	C₂³×C₃	C₂×C₂²⋊D₅	1	3	yes	—
220	C₂²×D₃	—	10	6	no	—
221	C₃×Q₈	#685	1	4	yes	—
222	C₃×D₄	#685	1	12	yes	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	—	2	3	no	—
225	C₂×C₃×C₄	—	4	6	no	—
226	C₂×C₃×C₄	C₂×C₄×D₅	1	6	yes	—
227	C₂×C₃×C₄	C₂×D₂₀	1	6	yes	—
228	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	3	yes	—
229	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
230	#67	—	10	4	no	—
231	#67	—	10	12	no	—
232	C₂×Dic₃	—	10	6	no	—
233	D₁₂	—	5	4	no	—
234	D₁₂	—	10	6	no	—
235	C₄×D₃	—	10	4	no	—
236	C₄×D₃	—	10	12	no	—
237	Dic₆	—	5	4	no	—
238	Dic₆	—	10	6	no	—
239	C₃⋊C₈	—	10	4	no	—
240	C₂²×C₅	—	4	3	no	—
241	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	6	yes	—
242	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
243	C₂×D₅	—	6	4	no	—
244	C₂×D₅	—	24	6	no	—
245	C₄×C₅	—	2	2	no	—
246	C₄×C₅	—	4	3	no	—
247	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	6	yes	—
248	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
249	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
250	Dic₅	—	6	4	no	—
251	#41	—	15	12	no	—
252	#41	—	15	4	no	—
253	#41	—	30	4	no	—
254	#41	—	30	12	no	—
255	#41	—	30	6	no	—
256	#41	—	30	6	no	—
257	#41	C₄×D₁₅	1	4	yes	—
258	#41	D₆₀	1	4	yes	—
259	C₂×Q₈	—	15	6	no	—
260	C₂×Q₈	—	30	4	no	—
261	C₂×Q₈	#1220	1	1	yes	characteristic
262	C₂×D₄	—	15	6	no	—
263	C₂×D₄	—	30	6	no	—
264	C₂×D₄	—	30	12	no	—
265	C₂×D₄	#1220	1	3	yes	—
266	C₂²×C₄	—	2	3	no	—
267	C₂²×C₄	—	30	6	no	—
268	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
269	M₁₆	—	15	4	no	—
270	M₁₆	—	30	6	no	—
271	C₂×C₈	—	30	6	no	—
272	C₄⋊C₄	—	2	3	no	—
273	C₂²⋊C₄	—	2	3	no	—
274	C₄²	—	2	3	no	—
275	C₄²	C₂²×D₁₅	1	1	yes	characteristic
276	C₄²	C₂²×D₁₅	1	3	yes	—
277	C₃×C₅	#2901	1	1	yes	characteristic, complemented, Hall
278	C₂²×C₃	—	4	3	no	—
279	C₂²×C₃	C₂×#685	1	6	yes	—
280	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
281	C₂×D₃	—	10	4	no	—
282	C₂×D₃	—	40	6	no	—
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	4	3	no	—
285	C₃×C₄	C₂×#685	1	6	yes	—
286	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
287	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
288	Dic₃	—	10	4	no	—
289	C₂×C₅	—	2	1	no	—
290	C₂×C₅	—	2	6	no	—
291	C₂×C₅	C₂²×(C₂×D₃)⋊C₄	1	1	yes	characteristic
292	D₅	—	12	4	no	—
293	C₂³	—	30	6	no	—
294	C₂³	C₂×#1220	1	3	yes	—
295	Q₈	—	15	4	no	—
296	Q₈	—	30	6	no	—
297	Q₈	#6714	1	4	yes	—
298	D₄	—	15	4	no	—
299	D₄	—	30	12	no	—
300	D₄	—	30	4	no	—
301	D₄	—	30	6	no	—
302	D₄	#6714	1	12	yes	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	3	no	—
305	C₂×C₄	—	4	6	no	—
306	C₂×C₄	—	30	6	no	—
307	C₂×C₄	—	30	12	no	—
308	C₂×C₄	—	30	4	no	—
309	C₂×C₄	C₂×C₄×D₁₅	1	6	yes	—
310	C₂×C₄	C₂×D₆₀	1	6	yes	—
311	C₂×C₄	C₂×#1220	1	3	yes	—
312	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
313	C₈	—	30	4	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	—	2	6	no	—
316	C₂×C₃	C₂²×#685	1	1	yes	characteristic
317	D₃	—	20	4	no	—
318	C₅	#82536	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	4	3	no	—
320	C₂²	—	30	4	no	—
321	C₂²	—	120	6	no	—
322	C₂²	C₂×#6714	1	6	yes	—
323	C₂²	C₂²×#1220	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	4	3	no	—
326	C₄	—	30	4	no	—
327	C₄	C₂×#6714	1	6	yes	—
328	C₄	C₂²×C₄×D₁₅	1	1	yes	characteristic
329	C₄	C₂²×D₆₀	1	1	yes	characteristic
330	C₃	#10614267	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	1	no	—
332	C₂	—	2	6	no	—
333	C₂	—	60	4	no	—
334	C₂	C₂²×#6714	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₆₀
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₃×C₄×C₅×#41	C₂
socle	C₂×C₃×C₅	C₂²×C₂²⋊C₄
prime radical	#11757146	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₄×#41	D₁₅
hypocenter	C₃×C₅	#2901
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×#41	C₂	semidirect
3. #11756578	C₂	semidirect
4. #11751106	C₂	semidirect
5. C₂×#91089	C₂	semidirect
8. #92101	C₄	semidirect
9. #92100	C₄	semidirect
16. C₃×C₅×#117	C₂²	semidirect
17. C₂×C₃×C₄²×C₅	C₂²	semidirect
19. C₂×#6710	C₂²	semidirect
20. #91089	C₂²	semidirect
21. #91089	C₂²	semidirect
24. C₄×C₅×#41	D₃	semidirect
41. #6817	C₂×C₄	semidirect
43. C₂×D₆₀	C₂×C₄	semidirect
45. C₂×Dic₃₀	C₂×C₄	semidirect
55. C₃×C₄×#41	D₅	semidirect
68. C₅×#117	C₂×D₃	semidirect
69. C₂×C₄²×C₅	C₂×D₃	semidirect
105. C₃×#117	C₂×D₅	semidirect
106. C₂×C₃×C₄²	C₂×D₅	semidirect
139. C₄×#41	D₁₅	semidirect
211. #117	C₂×D₁₅	semidirect
212. C₂×C₄²	C₂×D₁₅	semidirect
277. C₃×C₅	#2901	semidirect
318. C₅	#82536	semidirect
330. C₃	#10614267	semidirect

Maximal subgroups

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

subgroup	index
#11757146	2
C₃×C₄×C₅×#41	2
#11756578	2
#11751106	2
C₂×#91089	2
#10614267	3
#82536	5