Subgroups — #14284792

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4020	C₂	1	1	yes	maximal, characteristic
2	#11751026	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4924	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745852	C₂	1	4	yes	maximal, complemented
5	#10601295	—	3	1	no	maximal, Hall
6	C₂²×C₃×D₂₀	—	2	1	no	—
7	C₂²×C₃×D₂₀	—	2	1	no	—
8	C₂²×C₃×D₂₀	C₄	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×#208	—	2	1	no	—
11	C₃×#4020	—	2	1	no	—
12	C₃×#4020	—	2	1	no	—
13	C₃×#4020	C₄	1	2	yes	—
14	C₃×#4020	C₂²	1	2	yes	—
15	#91114	—	2	1	no	—
16	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	complemented
17	C₄×C₃⋊#168	—	2	1	no	—
18	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
19	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	complemented
20	#71759	—	5	1	no	maximal, Hall
21	C₂×#4020	D₃	1	1	yes	characteristic
22	C₅×#423	—	3	1	no	—
23	#65214	—	3	1	no	—
24	#64705	—	3	4	no	—
25	C₂³×C₃×D₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×C₃×D₂₀	—	2	2	no	—
30	C₂×C₃×D₂₀	—	2	2	no	—
31	C₂×C₃×D₂₀	—	2	2	no	—
32	C₂×C₃×D₂₀	—	2	2	no	—
33	C₂×C₃×D₂₀	—	2	2	no	—
34	C₂×C₃×D₂₀	—	4	4	no	—
35	C₂×C₃×D₂₀	—	4	2	no	—
36	C₂×C₃×D₂₀	—	4	2	no	—
37	C₂×C₃×D₂₀	—	4	4	no	—
38	C₂×C₃×D₂₀	C₂×C₄	1	4	yes	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₃⋊#168	—	2	2	no	—
44	C₅×#208	—	4	2	no	—
45	C₂×C₅×C₃⋊C₈	—	2	2	no	—
46	C₂×C₃×#127	—	5	1	no	—
47	#4925	—	5	1	no	—
48	#4924	D₅	1	1	yes	characteristic, complemented
49	#4544	—	5	4	no	—
50	C₂²×D₂₀	—	2	1	no	—
51	C₂²×D₂₀	—	2	1	no	—
52	C₂²×D₂₀	Dic₃	1	2	yes	—
53	C₂×C₅×M₁₆	—	6	1	no	—
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	#4020	—	2	1	no	—
56	#4020	—	2	1	no	—
57	#4020	Dic₃	1	2	yes	—
58	#4020	C₂×D₃	1	2	yes	—
59	C₅×C₄⋊C₈	—	3	2	no	—
60	C₅×C₄⋊C₈	—	6	1	no	—
61	C₂²⋊#168	—	6	1	no	—
62	C₄⋊#168	—	3	2	no	—
63	C₄×#168	—	6	1	no	—
64	#1545	—	15	1	no	Sylow
65	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₃×D₅	—	2	4	no	—
67	C₂²×C₃×D₅	—	4	4	no	—
68	C₂²×C₃×D₅	—	4	4	no	—
69	C₂²×C₃×D₅	—	4	4	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
76	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₃×D₂₀	—	4	4	no	—
81	C₃×D₂₀	—	4	4	no	—
82	C₃×D₂₀	—	4	4	no	—
83	C₃×D₂₀	—	4	4	no	—
84	C₃×D₂₀	—	4	2	no	—
85	C₃×D₂₀	—	4	2	no	—
86	C₃×D₂₀	—	4	4	no	—
87	C₃⋊#168	—	4	2	no	—
88	C₅×C₃⋊C₈	—	4	2	no	—
89	C₂²×C₃×D₄	—	5	2	no	—
90	C₂²×C₃×D₄	—	10	1	no	—
91	C₂²×C₃×D₄	—	10	1	no	—
92	C₃×#127	—	5	2	no	—
93	C₃×#127	—	5	2	no	—
94	C₃×#127	—	10	1	no	—
95	C₃×#127	—	10	1	no	—
96	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
97	C₂×#208	—	2	1	no	—
98	#830	—	10	1	no	—
99	C₄⋊(C₃⋊C₈)	—	2	1	no	—
100	C₄⋊(C₃⋊C₈)	—	5	2	no	—
101	C₄⋊(C₃⋊C₈)	C₂×D₅	1	2	yes	complemented
102	C₄×C₃⋊C₈	—	10	1	no	—
103	C₂³×D₅	—	2	2	no	—
104	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
105	C₂²×C₄×C₅	#67	1	1	yes	characteristic
106	C₂²×C₄×C₅	#67	1	1	yes	characteristic
107	C₂×D₂₀	—	2	2	no	—
108	C₂×D₂₀	—	2	2	no	—
109	C₂×D₂₀	—	2	2	no	—
110	C₂×D₂₀	—	2	2	no	—
111	C₂×D₂₀	—	2	2	no	—
112	C₂×D₂₀	—	4	4	no	—
113	C₂×D₂₀	—	4	2	no	—
114	C₂×D₂₀	—	4	2	no	—
115	C₂×D₂₀	—	4	4	no	—
116	C₂×D₂₀	C₂×Dic₃	1	4	yes	—
117	C₅×M₁₆	—	12	2	no	—
118	C₂×C₅×C₈	—	6	2	no	—
119	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
120	C₄²×C₅	#67	1	1	yes	characteristic
121	C₄²×C₅	#67	1	1	yes	characteristic
122	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
123	C₂×#168	—	6	2	no	—
124	C₂×#127	—	5	1	no	—
125	#431	—	15	1	no	—
126	#423	—	3	1	no	—
127	#331	—	15	4	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	#102	1	2	yes	—
131	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₃×D₅	—	4	4	no	—
133	C₂×C₃×D₅	—	4	4	no	—
134	C₂×C₃×D₅	—	4	4	no	—
135	C₂×C₃×D₅	—	8	8	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	#100	1	2	yes	—
139	C₃×C₄×C₅	#102	1	2	yes	—
140	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
141	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
142	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
143	C₂⁴×C₃	—	10	2	no	—
144	C₂×C₃×D₄	—	5	4	no	—
145	C₂×C₃×D₄	—	10	2	no	—
146	C₂×C₃×D₄	—	10	2	no	—
147	C₂×C₃×D₄	—	10	2	no	—
148	C₂×C₃×D₄	—	10	2	no	—
149	C₂×C₃×D₄	—	10	2	no	—
150	C₂×C₃×D₄	—	20	4	no	—
151	C₂×C₃×D₄	—	20	4	no	—
152	C₂×C₃×D₄	—	20	2	no	—
153	C₂×C₃×D₄	—	20	2	no	—
154	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
155	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
156	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
157	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
158	C₃×C₄²	D₂₀	1	1	yes	characteristic
159	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
160	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
161	#208	—	4	2	no	—
162	C₂×C₃⋊C₈	—	2	2	no	—
163	C₂×C₃⋊C₈	—	10	2	no	—
164	C₂³×C₅	#217	1	1	yes	characteristic
165	C₂²×D₅	—	2	4	no	—
166	C₂²×D₅	—	4	4	no	—
167	C₂²×D₅	—	4	4	no	—
168	C₂²×D₅	—	4	4	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	#213	1	2	yes	—
172	C₂×C₄×C₅	#214	1	2	yes	—
173	C₂×C₄×C₅	#217	1	1	yes	characteristic
174	C₂×C₄×C₅	#217	1	1	yes	characteristic
175	C₂×C₄×C₅	#217	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
179	D₂₀	—	4	4	no	—
180	D₂₀	—	4	4	no	—
181	D₂₀	—	4	4	no	—
182	D₂₀	—	4	4	no	—
183	D₂₀	—	4	2	no	—
184	D₂₀	—	4	4	no	—
185	D₂₀	—	4	2	no	—
186	C₅×C₈	—	12	2	no	—
187	#168	—	12	2	no	—
188	C₂²×D₄	—	5	2	no	—
189	C₂²×D₄	—	10	1	no	—
190	C₂²×D₄	—	10	1	no	—
191	C₂×M₁₆	—	6	1	no	—
192	#127	—	5	2	no	—
193	#127	—	5	2	no	—
194	#127	—	10	1	no	—
195	#127	—	10	1	no	—
196	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
197	C₄⋊C₈	—	3	2	no	—
198	C₄⋊C₈	—	6	1	no	—
199	C₄⋊C₈	—	15	2	no	—
200	C₂²⋊C₈	—	30	1	no	—
201	C₄×C₈	—	30	1	no	—
202	C₂×C₃×C₅	—	2	2	no	—
203	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
204	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
205	C₂×C₃×C₅	#418	1	1	yes	characteristic
206	C₃×D₅	—	8	4	no	—
207	C₂³×C₃	—	10	4	no	—
208	C₂³×C₃	—	20	4	no	—
209	C₂³×C₃	—	20	4	no	—
210	C₂³×C₃	—	20	4	no	—
211	C₂³×C₃	#685	1	1	yes	characteristic
212	C₃×D₄	—	20	4	no	—
213	C₃×D₄	—	20	4	no	—
214	C₃×D₄	—	20	4	no	—
215	C₃×D₄	—	20	4	no	—
216	C₃×D₄	—	20	4	no	—
217	C₃×D₄	—	20	2	no	—
218	C₃×D₄	—	20	2	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	#685	1	1	yes	characteristic
222	C₂×C₃×C₄	#685	1	1	yes	characteristic
223	C₂×C₃×C₄	#685	1	1	yes	characteristic
224	C₂×C₃×C₄	#686	1	2	yes	—
225	C₂×C₃×C₄	#688	1	2	yes	—
226	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
229	C₃⋊C₈	—	4	2	no	—
230	C₃⋊C₈	—	20	2	no	—
231	C₂²×C₅	—	2	1	no	—
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	#832	1	2	yes	—
234	C₂²×C₅	C₂×#217	1	1	yes	characteristic
235	C₂×D₅	—	4	4	no	—
236	C₂×D₅	—	4	4	no	—
237	C₂×D₅	—	4	4	no	—
238	C₂×D₅	—	8	8	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	#832	1	2	yes	—
242	C₄×C₅	#833	1	2	yes	—
243	C₄×C₅	C₂×#213	1	1	yes	characteristic
244	C₄×C₅	#932	1	2	yes	—
245	C₄×C₅	C₂×#214	1	1	yes	characteristic
246	C₂⁴	—	10	2	no	—
247	C₂×D₄	—	5	4	no	—
248	C₂×D₄	—	10	2	no	—
249	C₂×D₄	—	10	2	no	—
250	C₂×D₄	—	10	2	no	—
251	C₂×D₄	—	10	2	no	—
252	C₂×D₄	—	10	2	no	—
253	C₂×D₄	—	20	4	no	—
254	C₂×D₄	—	20	4	no	—
255	C₂×D₄	—	20	2	no	—
256	C₂×D₄	—	20	2	no	—
257	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
258	C₂²×C₄	#1201	1	1	yes	characteristic
259	C₂²×C₄	#1202	1	1	yes	characteristic
260	M₁₆	—	12	2	no	—
261	C₂×C₈	—	6	2	no	—
262	C₂×C₈	—	30	2	no	—
263	C₄²	D₅×Dic₃	1	1	yes	characteristic
264	C₄²	#1201	1	1	yes	characteristic
265	C₄²	#1202	1	1	yes	characteristic
266	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
267	C₃×C₅	#1545	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	20	4	no	—
271	C₂²×C₃	—	20	4	no	—
272	C₂²×C₃	—	20	4	no	—
273	C₂²×C₃	—	40	8	no	—
274	C₂²×C₃	#3941	1	2	yes	—
275	C₂²×C₃	C₂×#685	1	1	yes	characteristic
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	#3941	1	2	yes	—
279	C₃×C₄	#3955	1	2	yes	—
280	C₃×C₄	#4054	1	2	yes	—
281	C₃×C₄	C₂×#686	1	1	yes	characteristic
282	C₃×C₄	C₂×#688	1	1	yes	characteristic
283	C₂×C₅	—	2	2	no	—
284	C₂×C₅	C₂×#832	1	1	yes	characteristic
285	C₂×C₅	#5215	1	1	yes	characteristic
286	C₂×C₅	C₂×#833	1	1	yes	characteristic
287	D₅	—	8	4	no	—
288	C₂³	—	10	4	no	—
289	C₂³	—	20	4	no	—
290	C₂³	—	20	4	no	—
291	C₂³	—	20	4	no	—
292	C₂³	#6665	1	1	yes	characteristic
293	D₄	—	20	4	no	—
294	D₄	—	20	4	no	—
295	D₄	—	20	4	no	—
296	D₄	—	20	4	no	—
297	D₄	—	20	2	no	—
298	D₄	—	20	2	no	—
299	D₄	—	20	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	#6652	1	2	yes	—
303	C₂×C₄	#6655	1	2	yes	—
304	C₂×C₄	#6665	1	1	yes	characteristic
305	C₂×C₄	#6665	1	1	yes	characteristic
306	C₂×C₄	#6665	1	1	yes	characteristic
307	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
308	C₂×C₄	C₂×#1201	1	1	yes	characteristic
309	C₂×C₄	C₂×#1202	1	1	yes	characteristic
310	C₈	—	12	2	no	—
311	C₈	—	60	2	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	—	40	4	no	—
314	C₂×C₃	C₂×#3941	1	1	yes	characteristic
315	C₂×C₃	C₂×#3955	1	1	yes	characteristic
316	C₂×C₃	#65420	1	1	yes	characteristic
317	C₅	#71759	1	1	yes	characteristic, complemented, Sylow
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	20	4	no	—
321	C₂²	—	20	4	no	—
322	C₂²	—	20	4	no	—
323	C₂²	—	40	8	no	—
324	C₂²	#90964	1	2	yes	—
325	C₂²	C₂×#6665	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	#90958	1	2	yes	—
329	C₄	#90964	1	2	yes	—
330	C₄	C₂×#6652	1	1	yes	characteristic
331	C₄	C₂×#6655	1	1	yes	characteristic
332	C₄	#91312	1	2	yes	—
333	C₃	#10601295	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	2	2	no	—
335	C₂	—	40	4	no	—
336	C₂	C₂×#90958	1	1	yes	characteristic
337	C₂	C₂×#90964	1	1	yes	characteristic
338	C₂	#11748367	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6665
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₃×#4020	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1545
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #11751026	C₂	semidirect
3. C₅×#4924	C₂	semidirect
4. #11745852	C₂	semidirect
16. C₄⋊(C₃⋊#168)	C₂²	semidirect
19. C₅×C₄⋊(C₃⋊C₈)	C₂²	semidirect
48. #4924	D₅	semidirect
101. C₄⋊(C₃⋊C₈)	C₂×D₅	semidirect
267. C₃×C₅	#1545	semidirect
317. C₅	#71759	semidirect
333. C₃	#10601295	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4020	2
#11751026	2
C₅×#4924	2
#11745852	2
#10601295	3
#71759	5