Subgroups — #14478618

Order 1920.

700 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756530	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#508	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751296	C₂	1	2	yes	maximal, complemented
4	D₁₅×Q₃₂	C₂	1	2	yes	maximal, complemented
5	#11751016	C₂	1	4	yes	maximal, complemented
6	#11751013	C₂	1	2	yes	maximal, complemented
7	C₂×Dic₁₂₀	C₂	1	1	yes	maximal, characteristic, complemented
8	#11751002	C₂	1	1	yes	maximal, characteristic, complemented
9	#10613764	—	3	1	no	maximal, Hall
10	#92101	—	2	2	no	—
11	C₃×C₅×#135	C₂²	1	2	yes	—
12	#91840	—	2	2	no	—
13	Q₁₆×D₁₅	C₂²	1	4	yes	—
14	#91804	—	2	4	no	—
15	#91801	C₂²	1	4	yes	—
16	C₂×Dic₆₀	C₂²	1	2	yes	—
17	#91790	C₂²	1	1	yes	characteristic
18	C₃×C₅×Q₃₂	C₂²	1	1	yes	characteristic
19	C₃×C₅×SD₃₂	C₂²	1	2	yes	—
20	C₃×C₅×D₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
22	#91113	C₂²	1	4	yes	—
23	#91111	C₂²	1	4	yes	—
24	#91096	C₂²	1	1	yes	characteristic
25	Dic₁₂₀	C₂²	1	2	yes	—
26	Dic₁₂₀	C₂²	1	2	yes	—
27	#91082	C₂²	1	2	yes	—
28	C₁₆×D₁₅	C₂²	1	2	yes	—
29	#82033	—	5	1	no	maximal, Hall
30	#66105	—	3	2	no	—
31	C₅×#508	D₃	1	1	yes	characteristic, complemented
32	#65484	—	3	2	no	—
33	D₅×Q₃₂	—	3	2	no	—
34	#65204	—	3	4	no	—
35	#65201	—	3	2	no	—
36	C₂×Dic₄₀	—	3	1	no	—
37	#65190	—	3	1	no	—
38	C₃×C₅×#41	—	2	2	no	—
39	Q₈×D₁₅	—	2	4	no	—
40	#6819	—	2	4	no	—
41	#6819	—	4	4	no	—
42	#6817	—	4	2	no	—
43	#6817	D₄	1	1	yes	characteristic
44	C₂×Dic₃₀	—	2	2	no	—
45	C₂×Dic₃₀	—	4	2	no	—
46	C₃×C₅×Q₁₆	C₂³	1	2	yes	—
47	C₃×C₅×SD₁₆	—	2	2	no	—
48	C₃×C₅×D₈	C₂³	1	2	yes	—
49	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
50	#6718	—	2	4	no	—
51	#6716	—	2	4	no	—
52	#6710	D₄	1	1	yes	characteristic
53	Dic₆₀	—	2	2	no	—
54	Dic₆₀	C₂³	1	4	yes	—
55	#6705	D₄	1	2	yes	—
56	C₈×D₁₅	C₂³	1	2	yes	—
57	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
58	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
59	C₃⋊#672	C₂³	1	2	yes	—
60	#5771	—	5	2	no	—
61	C₃×#508	D₅	1	1	yes	characteristic, complemented
62	#5195	—	5	2	no	—
63	D₃×Q₃₂	—	5	2	no	—
64	#4915	—	5	4	no	—
65	#4912	—	5	2	no	—
66	C₂×Dic₂₄	—	5	1	no	—
67	#4901	—	5	1	no	—
68	#4150	—	6	2	no	—
69	C₅×#135	C₂×D₃	1	2	yes	—
70	#4097	—	6	2	no	—
71	D₅×Q₁₆	—	3	4	no	—
72	#4061	—	6	4	no	—
73	#4058	—	3	4	no	—
74	C₂×Dic₂₀	—	3	2	no	—
75	#4047	—	3	1	no	—
76	C₅×Q₃₂	C₂×D₃	1	1	yes	characteristic
77	C₅×SD₃₂	C₂×D₃	1	2	yes	—
78	C₅×D₁₆	C₂×D₃	1	1	yes	characteristic
79	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
80	#3961	—	3	4	no	—
81	#3959	—	3	4	no	—
82	#3944	—	3	1	no	—
83	Dic₄₀	—	3	2	no	—
84	Dic₄₀	—	3	2	no	—
85	#3930	—	3	2	no	—
86	D₅×C₁₆	—	3	2	no	—
87	#3417	—	15	1	no	Sylow
88	C₃×C₅×Q₈	—	2	2	no	—
89	C₃×C₅×D₄	—	2	2	no	—
90	C₃×C₅×D₄	—	4	2	no	—
91	C₂×C₃×C₄×C₅	—	4	2	no	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	#1220	—	4	4	no	—
94	#1220	D₈	1	2	yes	—
95	C₂×Dic₁₅	—	4	2	no	—
96	C₂×Dic₁₅	—	4	4	no	—
97	D₆₀	D₈	1	1	yes	characteristic
98	C₄×D₁₅	—	4	4	no	—
99	C₄×D₁₅	C₂×D₄	1	2	yes	—
100	Dic₃₀	—	2	4	no	—
101	Dic₃₀	—	4	2	no	—
102	Dic₃₀	—	4	4	no	—
103	Dic₃₀	D₈	1	1	yes	characteristic
104	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
106	C₃⋊#168	C₂×D₄	1	2	yes	—
107	#1010	—	10	2	no	—
108	C₃×#135	C₂×D₅	1	2	yes	—
109	#951	—	10	2	no	—
110	D₃×Q₁₆	—	5	4	no	—
111	#915	—	10	4	no	—
112	#912	—	5	4	no	—
113	C₂×Dic₁₂	—	5	2	no	—
114	#901	—	5	1	no	—
115	C₃×Q₃₂	C₂×D₅	1	1	yes	characteristic
116	C₃×SD₃₂	C₂×D₅	1	2	yes	—
117	C₃×D₁₆	C₂×D₅	1	1	yes	characteristic
118	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
119	#829	—	5	4	no	—
120	#827	—	5	4	no	—
121	#812	—	5	1	no	—
122	Dic₂₄	—	5	2	no	—
123	Dic₂₄	—	5	2	no	—
124	#798	—	5	2	no	—
125	D₃×C₁₆	—	5	2	no	—
126	C₅×#41	—	2	2	no	—
127	Q₈×D₅	—	6	4	no	—
128	#711	—	6	4	no	—
129	#711	—	12	4	no	—
130	#709	—	3	1	no	—
131	#709	—	12	2	no	—
132	C₂×Dic₁₀	—	6	2	no	—
133	C₂×Dic₁₀	—	12	2	no	—
134	C₅×Q₁₆	C₂²×D₃	1	2	yes	—
135	C₅×SD₁₆	—	2	2	no	—
136	C₅×D₈	C₂²×D₃	1	2	yes	—
137	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
138	#689	—	6	4	no	—
139	#687	—	6	4	no	—
140	#681	—	3	1	no	—
141	Dic₂₀	—	3	4	no	—
142	Dic₂₀	—	6	2	no	—
143	#676	—	3	2	no	—
144	C₈×D₅	—	3	2	no	—
145	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
146	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
147	#672	—	3	2	no	—
148	#578	—	15	2	no	—
149	#510	—	15	2	no	—
150	#510	—	15	4	no	—
151	#508	—	15	2	no	—
152	#508	D₁₅	1	1	yes	characteristic, complemented
153	C₂×Q₃₂	—	15	2	no	—
154	C₂×Q₃₂	—	15	1	no	—
155	#504	—	15	1	no	—
156	C₂²×C₃×C₅	—	4	2	no	—
157	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
158	C₂×D₁₅	C₂×D₈	1	2	yes	—
159	C₃×C₄×C₅	—	4	2	no	—
160	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
161	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
162	Dic₁₅	—	4	4	no	—
163	Dic₁₅	C₂×D₈	1	2	yes	—
164	C₃×#41	—	2	2	no	—
165	D₃×Q₈	—	10	4	no	—
166	#237	—	10	4	no	—
167	#237	—	20	4	no	—
168	#235	—	5	1	no	—
169	#235	—	20	2	no	—
170	C₂×Dic₆	—	10	2	no	—
171	C₂×Dic₆	—	20	2	no	—
172	C₃×Q₁₆	C₂²×D₅	1	2	yes	—
173	C₃×SD₁₆	—	2	2	no	—
174	C₃×D₈	C₂²×D₅	1	2	yes	—
175	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
176	#216	—	10	4	no	—
177	#214	—	10	4	no	—
178	#208	—	5	1	no	—
179	Dic₁₂	—	5	4	no	—
180	Dic₁₂	—	10	2	no	—
181	ΓL(1,𝔽₂₅)	—	5	2	no	—
182	D₃×C₈	—	5	2	no	—
183	C₃×C₁₆	C₂²×D₅	1	1	yes	characteristic
184	C₃×C₁₆	C₂²×D₅	1	1	yes	characteristic
185	C₃⋊C₁₆	—	5	2	no	—
186	C₅×Q₈	—	2	2	no	—
187	C₅×D₄	—	2	2	no	—
188	C₅×D₄	—	4	2	no	—
189	C₂×C₄×C₅	—	4	2	no	—
190	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
191	C₂²⋊D₅	—	3	2	no	—
192	C₂²⋊D₅	—	12	4	no	—
193	C₂×Dic₅	—	12	4	no	—
194	C₂×Dic₅	—	12	2	no	—
195	D₂₀	—	3	1	no	—
196	C₄×D₅	—	3	2	no	—
197	C₄×D₅	—	12	4	no	—
198	Dic₁₀	—	3	1	no	—
199	Dic₁₀	—	6	4	no	—
200	Dic₁₀	—	12	4	no	—
201	Dic₁₀	—	12	2	no	—
202	C₅×C₈	D₃×D₄	1	1	yes	characteristic
203	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
204	#168	—	3	2	no	—
205	2₋¹⁺⁴	—	30	2	no	—
206	#137	—	30	4	no	—
207	#137	—	30	2	no	—
208	#135	—	15	4	no	—
209	#135	C₂×D₁₅	1	2	yes	—
210	C₂×Q₁₆	—	15	4	no	—
211	C₂×Q₁₆	—	15	2	no	—
212	#131	—	15	1	no	—
213	Q₃₂	—	15	4	no	—
214	Q₃₂	—	15	2	no	—
215	Q₃₂	—	15	2	no	—
216	Q₃₂	C₂×D₁₅	1	1	yes	characteristic
217	SD₃₂	—	15	4	no	—
218	SD₃₂	C₂×D₁₅	1	2	yes	—
219	D₁₆	C₂×D₁₅	1	1	yes	characteristic
220	M₃₂	—	15	1	no	—
221	M₃₂	—	15	2	no	—
222	C₂×C₁₆	—	15	2	no	—
223	C₂×C₁₆	C₂×D₁₅	1	1	yes	characteristic
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	—	8	2	no	—
226	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
227	D₁₅	—	2	2	no	—
228	C₃×Q₈	—	2	2	no	—
229	C₃×D₄	—	2	2	no	—
230	C₃×D₄	—	4	2	no	—
231	C₂×C₃×C₄	—	4	2	no	—
232	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
233	#67	—	5	2	no	—
234	#67	—	20	4	no	—
235	C₂×Dic₃	—	20	4	no	—
236	C₂×Dic₃	—	20	2	no	—
237	D₁₂	—	5	1	no	—
238	C₄×D₃	—	5	2	no	—
239	C₄×D₃	—	20	4	no	—
240	Dic₆	—	5	1	no	—
241	Dic₆	—	10	4	no	—
242	Dic₆	—	20	4	no	—
243	Dic₆	—	20	2	no	—
244	C₃×C₈	D₄×D₅	1	1	yes	characteristic
245	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
246	C₃⋊C₈	—	5	2	no	—
247	C₂²×C₅	—	4	2	no	—
248	C₂²×C₅	D₃×D₈	1	1	yes	characteristic
249	C₂×D₅	—	3	2	no	—
250	C₄×C₅	—	4	2	no	—
251	C₄×C₅	D₃×D₈	1	1	yes	characteristic
252	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
253	Dic₅	—	3	2	no	—
254	Dic₅	—	12	4	no	—
255	#41	—	2	2	no	—
256	#41	—	15	1	no	—
257	#41	—	30	4	no	—
258	#41	—	60	2	no	—
259	#41	—	60	4	no	—
260	C₂×Q₈	—	30	4	no	—
261	C₂×Q₈	—	30	2	no	—
262	C₂×Q₈	—	60	2	no	—
263	Q₁₆	—	15	4	no	—
264	Q₁₆	—	30	4	no	—
265	Q₁₆	—	30	2	no	—
266	Q₁₆	C₂²×D₁₅	1	2	yes	—
267	SD₁₆	—	2	2	no	—
268	SD₁₆	—	30	4	no	—
269	D₈	C₂²×D₁₅	1	2	yes	—
270	M₁₆	—	15	1	no	—
271	M₁₆	—	15	2	no	—
272	C₂×C₈	—	15	2	no	—
273	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
274	C₁₆	—	15	2	no	—
275	C₁₆	C₂²×D₁₅	1	1	yes	characteristic
276	C₁₆	C₂²×D₁₅	1	1	yes	characteristic
277	C₃×C₅	#3417	1	1	yes	characteristic, complemented, Hall
278	C₂²×C₃	—	4	2	no	—
279	C₂²×C₃	D₅×D₈	1	1	yes	characteristic
280	C₂×D₃	—	5	2	no	—
281	C₃×C₄	—	4	2	no	—
282	C₃×C₄	D₅×D₈	1	1	yes	characteristic
283	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
284	Dic₃	—	5	2	no	—
285	Dic₃	—	20	4	no	—
286	C₂×C₅	—	2	1	no	—
287	C₂×C₅	—	8	2	no	—
288	C₂×C₅	C₂×D₃×D₈	1	1	yes	characteristic
289	D₅	—	6	2	no	—
290	Q₈	—	2	2	no	—
291	Q₈	—	15	1	no	—
292	Q₈	—	30	4	no	—
293	Q₈	—	60	4	no	—
294	Q₈	—	60	2	no	—
295	D₄	—	2	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	15	2	no	—
298	D₄	—	15	1	no	—
299	D₄	—	60	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	15	2	no	—
302	C₂×C₄	—	60	4	no	—
303	C₂×C₄	—	60	4	no	—
304	C₂×C₄	—	60	2	no	—
305	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
306	C₈	—	15	2	no	—
307	C₈	D₄×D₁₅	1	1	yes	characteristic
308	C₈	C₂³×D₁₅	1	1	yes	characteristic
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	8	2	no	—
311	C₂×C₃	C₂×D₅×D₈	1	1	yes	characteristic
312	D₃	—	10	2	no	—
313	C₅	#82033	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	4	2	no	—
315	C₂²	—	15	2	no	—
316	C₂²	D₈×D₁₅	1	1	yes	characteristic
317	C₄	—	4	2	no	—
318	C₄	—	15	2	no	—
319	C₄	—	60	4	no	—
320	C₄	D₈×D₁₅	1	1	yes	characteristic
321	C₄	C₂×D₄×D₁₅	1	1	yes	characteristic
322	C₃	#10613764	1	1	yes	characteristic, complemented, Sylow
323	C₂	—	2	1	no	—
324	C₂	—	8	2	no	—
325	C₂	—	30	2	no	—
326	C₂	C₂×D₈×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×D₁₅
commutator, cosocle	C₃×C₅×C₈	C₂⁴
Frattini	C₈	C₂³×D₁₅
Fitting	C₃×C₅×#508	C₂
socle	C₂×C₃×C₅	C₂²×D₈
hypercenter	#508	D₁₅
hypocenter	C₃×C₅	#3417
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #11756530	C₂	semidirect
2. C₃×C₅×#508	C₂	semidirect
3. #11751296	C₂	semidirect
4. D₁₅×Q₃₂	C₂	semidirect
5. #11751016	C₂	semidirect
6. #11751013	C₂	semidirect
7. C₂×Dic₁₂₀	C₂	semidirect
8. #11751002	C₂	semidirect
31. C₅×#508	D₃	semidirect
61. C₃×#508	D₅	semidirect
152. #508	D₁₅	semidirect
277. C₃×C₅	#3417	semidirect
313. C₅	#82033	semidirect
322. C₃	#10613764	semidirect

Maximal subgroups

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

subgroup	index
#11756530	2
C₃×C₅×#508	2
#11751296	2
D₁₅×Q₃₂	2
#11751016	2
#11751013	2
C₂×Dic₁₂₀	2
#11751002	2
#10613764	3
#82033	5