Subgroups — C₅×#82019 #14478485

Order 1920.

740 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	C₅×#5757	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5147	C₂	1	2	yes	maximal, complemented
4	C₅×#4912	C₂	1	2	yes	maximal, complemented
5	C₅×#4911	C₂	1	4	yes	maximal, complemented
6	C₅×D₃×D₁₆	C₂	1	2	yes	maximal, complemented
7	C₅×#4904	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4901	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#3415	—	3	1	no	maximal, Hall
10	C₅×#1004	—	2	2	no	—
11	C₂×C₃×C₅×D₈	C₂²	1	2	yes	complemented
12	C₅×#932	—	2	2	no	—
13	C₅×#912	C₂²	1	4	yes	complemented
14	C₅×#911	—	2	4	no	—
15	C₅×D₃×D₈	C₂²	1	4	yes	complemented
16	C₅×#904	C₂²	1	2	yes	complemented
17	C₅×#901	C₂²	1	1	yes	characteristic
18	C₃×C₅×D₁₆	C₂²	1	2	yes	complemented
19	C₃×C₅×D₁₆	C₂²	1	2	yes	complemented
20	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
21	C₅×#827	C₂²	1	4	yes	complemented
22	C₅×#826	C₂²	1	4	yes	complemented
23	C₅×#812	C₂²	1	1	yes	characteristic, complemented
24	C₅×Dic₂₄	C₂²	1	1	yes	characteristic, complemented
25	C₅×#800	C₂²	1	2	yes	complemented
26	C₅×D₄₈	C₂²	1	1	yes	characteristic, complemented
27	C₅×#798	C₂²	1	2	yes	complemented
28	C₅×D₃×C₁₆	C₂²	1	2	yes	complemented
29	#82019	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₅×#576	—	3	2	no	—
31	C₅×#509	—	3	4	no	—
32	C₅×#509	—	3	2	no	—
33	C₅×#508	—	3	2	no	—
34	C₅×#508	—	3	1	no	—
35	C₂×C₅×D₁₆	—	3	2	no	—
36	C₂×C₅×D₁₆	D₃	1	1	yes	characteristic, complemented
37	C₅×#504	—	3	1	no	—
38	C₂×C₃×C₅×D₄	—	2	2	no	—
39	C₂×C₅×#67	—	4	4	no	—
40	C₅×#237	—	2	4	no	—
41	C₅×#237	—	4	2	no	—
42	C₅×D₃×D₄	—	2	4	no	—
43	C₅×D₃×D₄	—	4	2	no	—
44	C₅×#235	—	2	2	no	—
45	C₅×#235	D₄	1	1	yes	characteristic
46	C₃×C₅×D₈	—	2	2	no	—
47	C₃×C₅×D₈	C₂³	1	4	yes	—
48	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₅×#214	—	2	4	no	—
50	C₅×#213	—	2	4	no	—
51	C₅×#208	D₄	1	1	yes	characteristic
52	C₅×Dic₁₂	C₂³	1	2	yes	—
53	C₅×D₂₄	C₂³	1	2	yes	—
54	C₅×#204	—	2	2	no	—
55	C₅×ΓL(1,𝔽₂₅)	D₄	1	2	yes	—
56	C₅×D₃×C₈	C₂³	1	2	yes	—
57	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
58	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
59	C₅×C₃⋊C₁₆	C₂³	1	2	yes	—
60	#5757	C₂×C₅	1	2	yes	complemented
61	C₂×C₃×D₁₆	C₂×C₅	1	1	yes	characteristic, complemented
62	#5147	C₂×C₅	1	2	yes	complemented
63	#4912	C₂×C₅	1	2	yes	complemented
64	#4911	C₂×C₅	1	4	yes	complemented
65	D₃×D₁₆	C₂×C₅	1	2	yes	complemented
66	#4904	C₂×C₅	1	1	yes	characteristic, complemented
67	#4901	C₂×C₅	1	1	yes	characteristic, complemented
68	C₅×H(5,𝔽₂)	—	6	2	no	—
69	C₅×AGL(1,ℤ/8)	—	6	4	no	—
70	C₅×AGL(1,ℤ/8)	—	6	2	no	—
71	C₅×#135	—	3	4	no	—
72	C₅×#135	—	3	2	no	—
73	C₂×C₅×D₈	—	3	4	no	—
74	C₂×C₅×D₈	C₂×D₃	1	2	yes	complemented
75	C₅×#131	—	3	1	no	—
76	C₅×Q₃₂	—	3	1	no	—
77	C₅×SD₃₂	—	3	4	no	—
78	C₅×SD₃₂	—	3	2	no	—
79	C₅×D₁₆	—	3	4	no	—
80	C₅×D₁₆	—	3	1	no	—
81	C₅×D₁₆	C₂×D₃	1	2	yes	complemented
82	C₅×D₁₆	C₂×D₃	1	2	yes	complemented
83	C₅×M₃₂	—	3	1	no	—
84	C₅×M₃₂	—	3	2	no	—
85	C₂×C₅×C₁₆	—	3	2	no	—
86	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic, complemented
87	#3415	—	3	1	no	Sylow
88	C₂³×C₃×C₅	—	4	2	no	—
89	C₂²×C₅×D₃	—	4	4	no	—
90	C₃×C₅×D₄	—	2	4	no	—
91	C₃×C₅×D₄	—	4	2	no	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₅×#67	—	4	2	no	—
94	C₅×#67	—	4	4	no	—
95	C₅×#67	—	4	4	no	—
96	C₅×#67	D₈	1	2	yes	—
97	C₂×C₅×Dic₃	—	4	4	no	—
98	C₅×D₁₂	—	2	2	no	—
99	C₅×D₁₂	D₈	1	1	yes	characteristic
100	C₄×C₅×D₃	—	4	2	no	—
101	C₄×C₅×D₃	C₂×D₄	1	2	yes	—
102	C₅×Dic₆	—	2	2	no	—
103	C₅×Dic₆	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	C₅×C₃⋊C₈	C₂×D₄	1	2	yes	—
107	#1004	—	2	2	no	—
108	C₂×C₃×D₈	C₂²×C₅	1	2	yes	complemented
109	#932	—	2	2	no	—
110	#912	C₂²×C₅	1	4	yes	complemented
111	#911	—	2	4	no	—
112	D₃×D₈	C₂²×C₅	1	4	yes	complemented
113	#904	C₂²×C₅	1	2	yes	complemented
114	#901	C₂²×C₅	1	1	yes	characteristic
115	C₃×D₁₆	C₂²×C₅	1	2	yes	complemented
116	C₃×D₁₆	C₂²×C₅	1	2	yes	complemented
117	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
118	#827	C₂²×C₅	1	4	yes	complemented
119	#826	C₂²×C₅	1	4	yes	complemented
120	#812	C₂²×C₅	1	1	yes	characteristic, complemented
121	Dic₂₄	C₂²×C₅	1	1	yes	characteristic, complemented
122	#800	C₂²×C₅	1	2	yes	complemented
123	D₄₈	C₂²×C₅	1	1	yes	characteristic, complemented
124	#798	C₂²×C₅	1	2	yes	complemented
125	D₃×C₁₆	C₂²×C₅	1	2	yes	complemented
126	C₅×#41	—	3	1	no	—
127	C₅×#41	—	6	4	no	—
128	C₅×#41	—	6	2	no	—
129	C₅×#41	—	12	2	no	—
130	C₂×C₅×D₄	—	2	2	no	—
131	C₂×C₅×D₄	—	6	4	no	—
132	C₂×C₅×D₄	—	12	2	no	—
133	C₂×C₅×D₄	—	12	4	no	—
134	C₅×Q₁₆	—	3	2	no	—
135	C₅×SD₁₆	—	6	4	no	—
136	C₅×SD₁₆	—	6	2	no	—
137	C₅×D₈	—	2	2	no	—
138	C₅×D₈	—	3	2	no	—
139	C₅×D₈	—	6	4	no	—
140	C₅×D₈	C₂²×D₃	1	4	yes	—
141	C₅×M₁₆	—	3	1	no	—
142	C₅×M₁₆	—	3	2	no	—
143	C₂×C₅×C₈	—	3	2	no	—
144	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
145	C₅×C₁₆	—	3	2	no	—
146	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
147	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
148	#576	—	3	2	no	—
149	#509	—	3	2	no	—
150	#509	—	3	4	no	—
151	#508	—	3	2	no	—
152	#508	—	3	1	no	—
153	C₂×D₁₆	—	3	2	no	—
154	C₂×D₁₆	C₅×D₃	1	1	yes	characteristic, complemented
155	#504	—	3	1	no	—
156	C₂²×C₃×C₅	—	4	4	no	—
157	C₂²×C₃×C₅	—	16	2	no	—
158	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
159	C₂×C₅×D₃	—	4	2	no	—
160	C₂×C₅×D₃	—	16	4	no	—
161	C₂×C₅×D₃	C₂×D₈	1	2	yes	—
162	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
163	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
164	C₅×Dic₃	—	4	2	no	—
165	C₅×Dic₃	C₂×D₈	1	2	yes	—
166	C₂×C₃×D₄	—	2	2	no	—
167	C₂×#67	—	4	4	no	—
168	#237	—	2	4	no	—
169	#237	—	4	2	no	—
170	D₃×D₄	—	2	4	no	—
171	D₃×D₄	—	4	2	no	—
172	#235	—	2	2	no	—
173	#235	C₅×D₄	1	1	yes	characteristic
174	C₃×D₈	—	2	2	no	—
175	C₃×D₈	C₂³×C₅	1	4	yes	—
176	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
177	#214	—	2	4	no	—
178	#213	—	2	4	no	—
179	#208	C₅×D₄	1	1	yes	characteristic
180	Dic₁₂	C₂³×C₅	1	2	yes	—
181	D₂₄	C₂³×C₅	1	2	yes	—
182	#204	—	2	2	no	—
183	ΓL(1,𝔽₂₅)	C₅×D₄	1	2	yes	—
184	D₃×C₈	C₂³×C₅	1	2	yes	—
185	C₃×C₁₆	C₂³×C₅	1	1	yes	characteristic
186	C₃×C₁₆	C₂³×C₅	1	1	yes	characteristic
187	C₃⋊C₁₆	C₂³×C₅	1	2	yes	—
188	C₂³×C₅	—	4	2	no	—
189	C₂³×C₅	—	12	4	no	—
190	C₅×Q₈	—	3	1	no	—
191	C₅×Q₈	—	6	2	no	—
192	C₅×D₄	—	2	4	no	—
193	C₅×D₄	—	3	2	no	—
194	C₅×D₄	—	3	1	no	—
195	C₅×D₄	—	4	2	no	—
196	C₅×D₄	—	6	2	no	—
197	C₅×D₄	—	12	4	no	—
198	C₅×D₄	—	12	4	no	—
199	C₅×D₄	—	12	2	no	—
200	C₂×C₄×C₅	—	3	2	no	—
201	C₂×C₄×C₅	—	12	4	no	—
202	C₂×C₄×C₅	—	12	2	no	—
203	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
204	C₅×C₈	—	3	2	no	—
205	C₅×C₈	D₃×D₄	1	1	yes	characteristic
206	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
207	H(5,𝔽₂)	—	6	2	no	—
208	AGL(1,ℤ/8)	—	6	4	no	—
209	AGL(1,ℤ/8)	—	6	2	no	—
210	#135	—	3	4	no	—
211	#135	—	3	2	no	—
212	C₂×D₈	—	3	4	no	—
213	C₂×D₈	C₂×C₅×D₃	1	2	yes	complemented
214	#131	—	3	1	no	—
215	Q₃₂	—	3	1	no	—
216	SD₃₂	—	3	4	no	—
217	SD₃₂	—	3	2	no	—
218	D₁₆	—	3	4	no	—
219	D₁₆	—	3	1	no	—
220	D₁₆	C₂×C₅×D₃	1	2	yes	complemented
221	D₁₆	C₂×C₅×D₃	1	2	yes	complemented
222	M₃₂	—	3	1	no	—
223	M₃₂	—	3	2	no	—
224	C₂×C₁₆	—	3	2	no	—
225	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic, complemented
226	C₂×C₃×C₅	—	2	1	no	—
227	C₂×C₃×C₅	—	8	4	no	—
228	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
229	C₅×D₃	—	2	2	no	—
230	C₅×D₃	—	8	2	no	—
231	C₂³×C₃	—	4	2	no	—
232	C₂²×D₃	—	4	4	no	—
233	C₃×D₄	—	2	4	no	—
234	C₃×D₄	—	4	2	no	—
235	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
236	#67	—	4	2	no	—
237	#67	—	4	4	no	—
238	#67	—	4	4	no	—
239	#67	C₅×D₈	1	2	yes	—
240	C₂×Dic₃	—	4	4	no	—
241	D₁₂	—	2	2	no	—
242	D₁₂	C₅×D₈	1	1	yes	characteristic
243	C₄×D₃	—	4	2	no	—
244	C₄×D₃	C₂×C₅×D₄	1	2	yes	—
245	Dic₆	—	2	2	no	—
246	Dic₆	C₅×D₈	1	1	yes	characteristic
247	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
248	C₃×C₈	C₂⁴×C₅	1	1	yes	characteristic
249	C₃⋊C₈	C₂×C₅×D₄	1	2	yes	—
250	C₂²×C₅	—	3	2	no	—
251	C₂²×C₅	—	4	4	no	—
252	C₂²×C₅	—	12	2	no	—
253	C₂²×C₅	—	16	2	no	—
254	C₂²×C₅	—	48	4	no	—
255	C₂²×C₅	D₃×D₈	1	1	yes	characteristic
256	C₄×C₅	—	3	2	no	—
257	C₄×C₅	—	12	2	no	—
258	C₄×C₅	D₃×D₈	1	1	yes	characteristic
259	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
260	#41	—	3	1	no	—
261	#41	—	6	4	no	—
262	#41	—	6	2	no	—
263	#41	—	12	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	6	4	no	—
266	C₂×D₄	—	12	2	no	—
267	C₂×D₄	—	12	4	no	—
268	Q₁₆	—	3	2	no	—
269	SD₁₆	—	6	4	no	—
270	SD₁₆	—	6	2	no	—
271	D₈	—	2	2	no	—
272	D₈	—	3	2	no	—
273	D₈	—	6	4	no	—
274	D₈	C₂²×C₅×D₃	1	4	yes	—
275	M₁₆	—	3	1	no	—
276	M₁₆	—	3	2	no	—
277	C₂×C₈	—	3	2	no	—
278	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
279	C₁₆	—	3	2	no	—
280	C₁₆	C₂²×C₅×D₃	1	1	yes	characteristic
281	C₁₆	C₂²×C₅×D₃	1	1	yes	characteristic
282	C₃×C₅	#3415	1	1	yes	characteristic, complemented, Hall
283	C₂²×C₃	—	4	4	no	—
284	C₂²×C₃	—	16	2	no	—
285	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
286	C₂×D₃	—	4	2	no	—
287	C₂×D₃	—	16	4	no	—
288	C₂×D₃	C₂×C₅×D₈	1	2	yes	—
289	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
290	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
291	Dic₃	—	4	2	no	—
292	Dic₃	C₂×C₅×D₈	1	2	yes	—
293	C₂×C₅	—	2	1	no	—
294	C₂×C₅	—	6	2	no	—
295	C₂×C₅	—	8	4	no	—
296	C₂×C₅	—	24	2	no	—
297	C₂×C₅	C₂×D₃×D₈	1	1	yes	characteristic
298	C₂³	—	4	2	no	—
299	C₂³	—	12	4	no	—
300	Q₈	—	3	1	no	—
301	Q₈	—	6	2	no	—
302	D₄	—	2	4	no	—
303	D₄	—	3	2	no	—
304	D₄	—	3	1	no	—
305	D₄	—	4	2	no	—
306	D₄	—	6	2	no	—
307	D₄	—	12	4	no	—
308	D₄	—	12	4	no	—
309	D₄	—	12	2	no	—
310	C₂×C₄	—	3	2	no	—
311	C₂×C₄	—	12	4	no	—
312	C₂×C₄	—	12	2	no	—
313	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
314	C₈	—	3	2	no	—
315	C₈	C₅×D₃×D₄	1	1	yes	characteristic
316	C₈	C₂³×C₅×D₃	1	1	yes	characteristic
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	8	4	no	—
319	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
320	D₃	—	2	2	no	—
321	D₃	—	8	2	no	—
322	C₅	#82019	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²	—	3	2	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	12	2	no	—
326	C₂²	—	16	2	no	—
327	C₂²	—	48	4	no	—
328	C₂²	C₅×D₃×D₈	1	1	yes	characteristic
329	C₄	—	3	2	no	—
330	C₄	—	12	2	no	—
331	C₄	C₅×D₃×D₈	1	1	yes	characteristic
332	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
333	C₃	C₅×#3415	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	2	1	no	—
335	C₂	—	6	2	no	—
336	C₂	—	8	4	no	—
337	C₂	—	24	2	no	—
338	C₂	C₂×C₅×D₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂×D₃×D₈
commutator, cosocle	C₃×C₈	C₂⁴×C₅
Frattini	C₈	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×D₁₆	C₂
socle	C₂×C₃×C₅	C₂²×D₈
hypercenter	C₂×C₅×D₁₆	D₃
hypocenter	C₃	C₅×#3415
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₅×#5757	C₂	semidirect
2. C₂×C₃×C₅×D₁₆	C₂	semidirect
3. C₅×#5147	C₂	semidirect
4. C₅×#4912	C₂	semidirect
5. C₅×#4911	C₂	semidirect
6. C₅×D₃×D₁₆	C₂	semidirect
7. C₅×#4904	C₂	semidirect
8. C₅×#4901	C₂	semidirect
11. C₂×C₃×C₅×D₈	C₂²	semidirect
13. C₅×#912	C₂²	semidirect
15. C₅×D₃×D₈	C₂²	semidirect
16. C₅×#904	C₂²	semidirect
18. C₃×C₅×D₁₆	C₂²	semidirect
19. C₃×C₅×D₁₆	C₂²	semidirect
20. C₂×C₃×C₅×C₁₆	C₂²	semidirect
21. C₅×#827	C₂²	semidirect
22. C₅×#826	C₂²	semidirect
23. C₅×#812	C₂²	semidirect
24. C₅×Dic₂₄	C₂²	semidirect
25. C₅×#800	C₂²	semidirect
26. C₅×D₄₈	C₂²	semidirect
27. C₅×#798	C₂²	semidirect
28. C₅×D₃×C₁₆	C₂²	semidirect
29. #82019	C₅	direct (N × H)
36. C₂×C₅×D₁₆	D₃	semidirect
60. #5757	C₂×C₅	semidirect
61. C₂×C₃×D₁₆	C₂×C₅	semidirect
62. #5147	C₂×C₅	semidirect
63. #4912	C₂×C₅	semidirect
64. #4911	C₂×C₅	semidirect
65. D₃×D₁₆	C₂×C₅	semidirect
66. #4904	C₂×C₅	semidirect
67. #4901	C₂×C₅	semidirect
74. C₂×C₅×D₈	C₂×D₃	semidirect
81. C₅×D₁₆	C₂×D₃	semidirect
82. C₅×D₁₆	C₂×D₃	semidirect
86. C₂×C₅×C₁₆	C₂×D₃	semidirect
108. C₂×C₃×D₈	C₂²×C₅	semidirect
110. #912	C₂²×C₅	semidirect
112. D₃×D₈	C₂²×C₅	semidirect
113. #904	C₂²×C₅	semidirect
115. C₃×D₁₆	C₂²×C₅	semidirect
116. C₃×D₁₆	C₂²×C₅	semidirect
117. C₂×C₃×C₁₆	C₂²×C₅	semidirect
118. #827	C₂²×C₅	semidirect
119. #826	C₂²×C₅	semidirect
120. #812	C₂²×C₅	semidirect
121. Dic₂₄	C₂²×C₅	semidirect
122. #800	C₂²×C₅	semidirect
123. D₄₈	C₂²×C₅	semidirect
124. #798	C₂²×C₅	semidirect
125. D₃×C₁₆	C₂²×C₅	semidirect
154. C₂×D₁₆	C₅×D₃	semidirect
213. C₂×D₈	C₂×C₅×D₃	semidirect
220. D₁₆	C₂×C₅×D₃	semidirect
221. D₁₆	C₂×C₅×D₃	semidirect
225. C₂×C₁₆	C₂×C₅×D₃	semidirect
282. C₃×C₅	#3415	semidirect
322. C₅	#82019	direct (N × H)
333. C₃	C₅×#3415	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5757	2
C₂×C₃×C₅×D₁₆	2
C₅×#5147	2
C₅×#4912	2
C₅×#4911	2
C₅×D₃×D₁₆	2
C₅×#4904	2
C₅×#4901	2
C₅×#3415	3
#82019	5

Subgroups — C5×#82019 #14478485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#82019 #14478485