Subgroups — #14478615

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	#11756525	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751280	C₂	1	2	yes	maximal, complemented
4	#11751020	C₂	1	2	yes	maximal, complemented
5	#11751019	C₂	1	4	yes	maximal, complemented
6	D₁₅×Q₃₂	C₂	1	2	yes	maximal, complemented
7	#11751005	C₂	1	1	yes	maximal, characteristic, complemented
8	#11751002	C₂	1	1	yes	maximal, characteristic, complemented
9	#10613759	—	3	1	no	maximal, Hall
10	#92098	—	2	2	no	—
11	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
12	#91831	—	2	2	no	—
13	#91808	C₂²	1	4	yes	—
14	#91807	—	2	4	no	—
15	Q₁₆×D₁₅	C₂²	1	4	yes	—
16	#91793	C₂²	1	2	yes	—
17	#91790	C₂²	1	1	yes	characteristic
18	C₃×C₅×Q₃₂	C₂²	1	2	yes	—
19	C₃×C₅×Q₃₂	C₂²	1	2	yes	—
20	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
21	#91113	C₂²	1	4	yes	—
22	#91112	C₂²	1	4	yes	—
23	#91096	C₂²	1	1	yes	characteristic
24	Dic₁₂₀	C₂²	1	1	yes	characteristic
25	#91084	C₂²	1	2	yes	—
26	D₂₄₀	C₂²	1	1	yes	characteristic
27	#91082	C₂²	1	2	yes	—
28	C₁₆×D₁₅	C₂²	1	2	yes	—
29	#82028	—	5	1	no	maximal, Hall
30	#66100	—	3	2	no	—
31	C₂×C₅×Q₃₂	D₃	1	1	yes	characteristic, complemented
32	#65468	—	3	2	no	—
33	#65208	—	3	2	no	—
34	#65207	—	3	4	no	—
35	D₅×Q₃₂	—	3	2	no	—
36	#65193	—	3	1	no	—
37	#65190	—	3	1	no	—
38	C₂×C₃×C₅×Q₈	—	2	2	no	—
39	#6821	—	2	4	no	—
40	#6821	—	4	2	no	—
41	Q₈×D₁₅	—	2	4	no	—
42	Q₈×D₁₅	—	4	2	no	—
43	#6817	—	2	2	no	—
44	#6817	—	4	4	no	—
45	#6817	D₄	1	1	yes	characteristic
46	C₃×C₅×Q₁₆	—	2	2	no	—
47	C₃×C₅×Q₁₆	C₂³	1	4	yes	—
48	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
49	#6718	—	2	4	no	—
50	#6717	—	2	4	no	—
51	#6710	D₄	1	1	yes	characteristic
52	Dic₆₀	C₂³	1	2	yes	—
53	D₁₂₀	C₂³	1	2	yes	—
54	#6706	—	2	2	no	—
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	#5766	—	5	2	no	—
61	C₂×C₃×Q₃₂	D₅	1	1	yes	characteristic, complemented
62	#5179	—	5	2	no	—
63	#4919	—	5	2	no	—
64	#4918	—	5	4	no	—
65	D₃×Q₃₂	—	5	2	no	—
66	#4904	—	5	1	no	—
67	#4901	—	5	1	no	—
68	#4147	—	6	2	no	—
69	C₂×C₅×Q₁₆	C₂×D₃	1	2	yes	—
70	#4088	—	6	2	no	—
71	#4065	—	3	4	no	—
72	#4064	—	6	4	no	—
73	D₅×Q₁₆	—	3	4	no	—
74	#4050	—	3	2	no	—
75	#4047	—	3	1	no	—
76	C₅×Q₃₂	C₂×D₃	1	2	yes	—
77	C₅×Q₃₂	C₂×D₃	1	2	yes	—
78	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
79	#3961	—	3	4	no	—
80	#3960	—	3	4	no	—
81	#3944	—	3	1	no	—
82	Dic₄₀	—	3	1	no	—
83	#3932	—	3	2	no	—
84	D₈₀	—	3	1	no	—
85	#3930	—	3	2	no	—
86	D₅×C₁₆	—	3	2	no	—
87	#3417	—	15	1	no	Sylow
88	C₃×C₅×Q₈	—	2	4	no	—
89	C₃×C₅×Q₈	—	4	2	no	—
90	C₂×C₃×C₄×C₅	—	4	2	no	—
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	#1220	—	4	2	no	—
93	#1220	D₈	1	2	yes	—
94	D₆₀	—	2	2	no	—
95	D₆₀	—	4	4	no	—
96	D₆₀	D₈	1	1	yes	characteristic
97	C₄×D₁₅	—	4	4	no	—
98	C₄×D₁₅	—	4	2	no	—
99	C₄×D₁₅	—	4	4	no	—
100	C₄×D₁₅	C₂×D₄	1	2	yes	—
101	Dic₃₀	—	2	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	#1007	—	10	2	no	—
108	C₂×C₃×Q₁₆	C₂×D₅	1	2	yes	—
109	#942	—	10	2	no	—
110	#919	—	5	4	no	—
111	#918	—	10	4	no	—
112	D₃×Q₁₆	—	5	4	no	—
113	#904	—	5	2	no	—
114	#901	—	5	1	no	—
115	C₃×Q₃₂	C₂×D₅	1	2	yes	—
116	C₃×Q₃₂	C₂×D₅	1	2	yes	—
117	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
118	#829	—	5	4	no	—
119	#828	—	5	4	no	—
120	#812	—	5	1	no	—
121	Dic₂₄	—	5	1	no	—
122	#800	—	5	2	no	—
123	D₄₈	—	5	1	no	—
124	#798	—	5	2	no	—
125	D₃×C₁₆	—	5	2	no	—
126	C₂×C₅×Q₈	—	2	2	no	—
127	#713	—	6	4	no	—
128	#713	—	12	2	no	—
129	Q₈×D₅	—	6	4	no	—
130	Q₈×D₅	—	12	2	no	—
131	#709	—	3	1	no	—
132	#709	—	6	2	no	—
133	#709	—	12	4	no	—
134	C₅×Q₁₆	—	2	2	no	—
135	C₅×Q₁₆	C₂²×D₃	1	4	yes	—
136	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
137	#689	—	6	4	no	—
138	#688	—	6	4	no	—
139	#681	—	3	1	no	—
140	Dic₂₀	—	3	2	no	—
141	D₄₀	—	3	2	no	—
142	#677	—	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	—	15	1	no	—
153	C₂×Q₃₂	—	15	2	no	—
154	C₂×Q₃₂	D₁₅	1	1	yes	characteristic, complemented
155	#504	—	15	1	no	—
156	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
157	C₂×D₁₅	—	4	2	no	—
158	C₂×D₁₅	C₂×D₈	1	2	yes	—
159	C₃×C₄×C₅	—	4	4	no	—
160	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
161	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
162	Dic₁₅	—	4	2	no	—
163	Dic₁₅	C₂×D₈	1	2	yes	—
164	C₂×C₃×Q₈	—	2	2	no	—
165	#239	—	10	4	no	—
166	#239	—	20	2	no	—
167	D₃×Q₈	—	10	4	no	—
168	D₃×Q₈	—	20	2	no	—
169	#235	—	5	1	no	—
170	#235	—	10	2	no	—
171	#235	—	20	4	no	—
172	C₃×Q₁₆	—	2	2	no	—
173	C₃×Q₁₆	C₂²×D₅	1	4	yes	—
174	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
175	#216	—	10	4	no	—
176	#215	—	10	4	no	—
177	#208	—	5	1	no	—
178	Dic₁₂	—	5	2	no	—
179	D₂₄	—	5	2	no	—
180	#204	—	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	4	no	—
187	C₅×Q₈	—	4	2	no	—
188	C₂×C₄×C₅	—	4	2	no	—
189	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
190	C₂²⋊D₅	—	3	2	no	—
191	C₂²⋊D₅	—	12	2	no	—
192	D₂₀	—	3	1	no	—
193	D₂₀	—	6	2	no	—
194	D₂₀	—	12	4	no	—
195	C₄×D₅	—	3	2	no	—
196	C₄×D₅	—	12	4	no	—
197	C₄×D₅	—	12	4	no	—
198	C₄×D₅	—	12	2	no	—
199	Dic₁₀	—	3	1	no	—
200	Dic₁₀	—	6	2	no	—
201	Dic₁₀	—	12	4	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	—	15	2	no	—
210	C₂×Q₁₆	—	15	4	no	—
211	C₂×Q₁₆	C₂×D₁₅	1	2	yes	—
212	#131	—	15	1	no	—
213	Q₃₂	—	15	4	no	—
214	Q₃₂	—	15	1	no	—
215	Q₃₂	C₂×D₁₅	1	2	yes	—
216	Q₃₂	C₂×D₁₅	1	2	yes	—
217	SD₃₂	—	15	4	no	—
218	SD₃₂	—	15	2	no	—
219	D₁₆	—	15	1	no	—
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	C₂²×D₈	1	1	yes	characteristic
226	D₁₅	—	2	2	no	—
227	D₁₅	—	8	2	no	—
228	C₃×Q₈	—	2	4	no	—
229	C₃×Q₈	—	4	2	no	—
230	C₂×C₃×C₄	—	4	2	no	—
231	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
232	#67	—	5	2	no	—
233	#67	—	20	2	no	—
234	D₁₂	—	5	1	no	—
235	D₁₂	—	10	2	no	—
236	D₁₂	—	20	4	no	—
237	C₄×D₃	—	5	2	no	—
238	C₄×D₃	—	20	4	no	—
239	C₄×D₃	—	20	4	no	—
240	C₄×D₃	—	20	2	no	—
241	Dic₆	—	5	1	no	—
242	Dic₆	—	10	2	no	—
243	Dic₆	—	20	4	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	D₃×D₈	1	1	yes	characteristic
248	C₂×D₅	—	3	2	no	—
249	C₂×D₅	—	12	2	no	—
250	C₄×C₅	—	4	4	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	2	no	—
255	#41	—	15	1	no	—
256	#41	—	30	4	no	—
257	#41	—	30	2	no	—
258	#41	—	60	4	no	—
259	#41	—	60	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	30	4	no	—
262	C₂×Q₈	—	60	2	no	—
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	15	2	no	—
265	Q₁₆	—	30	4	no	—
266	Q₁₆	C₂²×D₁₅	1	4	yes	—
267	SD₁₆	—	30	4	no	—
268	SD₁₆	—	30	2	no	—
269	D₈	—	15	2	no	—
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	D₅×D₈	1	1	yes	characteristic
279	C₂×D₃	—	5	2	no	—
280	C₂×D₃	—	20	2	no	—
281	C₃×C₄	—	4	4	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	2	no	—
286	C₂×C₅	—	2	1	no	—
287	C₂×C₅	C₂×D₃×D₈	1	1	yes	characteristic
288	D₅	—	6	2	no	—
289	D₅	—	24	2	no	—
290	Q₈	—	2	4	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	15	1	no	—
293	Q₈	—	30	2	no	—
294	Q₈	—	60	4	no	—
295	D₄	—	15	2	no	—
296	D₄	—	15	1	no	—
297	D₄	—	30	2	no	—
298	D₄	—	60	4	no	—
299	D₄	—	60	2	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	C₂×D₅×D₈	1	1	yes	characteristic
311	D₃	—	10	2	no	—
312	D₃	—	40	2	no	—
313	C₅	#82028	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	15	2	no	—
315	C₂²	—	60	2	no	—
316	C₂²	D₈×D₁₅	1	1	yes	characteristic
317	C₄	—	4	4	no	—
318	C₄	—	15	2	no	—
319	C₄	—	60	2	no	—
320	C₄	D₈×D₁₅	1	1	yes	characteristic
321	C₄	C₂×D₄×D₁₅	1	1	yes	characteristic
322	C₃	#10613759	1	1	yes	characteristic, complemented, Sylow
323	C₂	—	2	1	no	—
324	C₂	—	30	2	no	—
325	C₂	—	120	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	C₂×C₃×C₅×Q₃₂	C₂
socle	C₂×C₃×C₅	C₂²×D₈
hypercenter	C₂×Q₃₂	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 ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
1. #11756525	C₂	semidirect
2. C₂×C₃×C₅×Q₃₂	C₂	semidirect
3. #11751280	C₂	semidirect
4. #11751020	C₂	semidirect
5. #11751019	C₂	semidirect
6. D₁₅×Q₃₂	C₂	semidirect
7. #11751005	C₂	semidirect
8. #11751002	C₂	semidirect
31. C₂×C₅×Q₃₂	D₃	semidirect
61. C₂×C₃×Q₃₂	D₅	semidirect
154. C₂×Q₃₂	D₁₅	semidirect
277. C₃×C₅	#3417	semidirect
313. C₅	#82028	semidirect
322. C₃	#10613759	semidirect

Maximal subgroups

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

subgroup	index
#11756525	2
C₂×C₃×C₅×Q₃₂	2
#11751280	2
#11751020	2
#11751019	2
D₁₅×Q₃₂	2
#11751005	2
#11751002	2
#10613759	3
#82028	5