Subgroups — C₅×#81774 #14458098

Order 1920.

920 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	C₅×#5762	C₂	1	4	yes	maximal, complemented
2	C₃×C₅×#492	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5177	C₂	1	4	yes	maximal, complemented
4	C₅×#5095	C₂	1	2	yes	maximal, complemented
5	C₅×#5077	C₂	1	2	yes	maximal, complemented
6	C₅×#4700	C₂	1	2	yes	maximal, complemented
7	C₅×#3291	—	3	1	no	maximal, Hall
8	C₅×#1007	C₂²	1	2	yes	—
9	C₅×#1004	C₂²	1	2	yes	complemented
10	C₂×C₃×C₅×SD₁₆	C₂²	1	4	yes	complemented
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₃×C₅×#127	C₂²	1	1	yes	characteristic
13	C₅×#942	C₂²	1	4	yes	complemented
14	C₅×#932	C₂²	1	4	yes	—
15	C₅×#916	—	2	4	no	—
16	C₅×#915	—	2	4	no	—
17	C₅×#914	—	2	4	no	—
18	C₅×D₃×SD₁₆	—	2	4	no	—
19	C₅×#904	C₂²	1	2	yes	—
20	C₅×#901	C₂²	1	4	yes	complemented
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
22	C₅×#836	C₂²	1	2	yes	complemented
23	C₅×#833	C₂²	1	2	yes	complemented
24	C₅×#819	C₂²	1	2	yes	complemented
25	C₅×#805	C₂²	1	4	yes	complemented
26	#81774	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#577	—	3	4	no	—
28	C₅×#492	D₃	1	1	yes	characteristic, complemented
29	C₅×#471	—	3	4	no	—
30	C₅×#456	—	3	2	no	—
31	C₅×#453	—	3	2	no	—
32	C₅×#443	—	3	2	no	—
33	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
34	C₂×C₃×C₅×D₄	—	4	1	no	—
35	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
36	C₂×C₅×#67	—	4	2	no	—
37	C₅×#239	—	2	4	no	—
38	C₅×D₃×Q₈	—	2	4	no	—
39	C₅×#237	—	2	4	no	—
40	C₅×D₃×D₄	—	2	4	no	—
41	C₅×#235	—	4	2	no	—
42	C₅×#235	C₂³	1	4	yes	—
43	C₃×C₅×SD₁₆	—	2	4	no	—
44	C₃×C₅×SD₁₆	—	2	4	no	—
45	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
46	C₃×C₅×C₄⋊C₄	—	4	1	no	—
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₅×#216	—	2	4	no	—
49	C₅×#215	—	2	4	no	—
50	C₅×#214	—	2	4	no	—
51	C₅×#213	—	2	4	no	—
52	C₅×#208	C₂³	1	4	yes	—
53	C₅×Dic₁₂	D₄	1	2	yes	complemented
54	C₅×D₂₄	D₄	1	2	yes	complemented
55	C₅×#204	D₄	1	4	yes	complemented
56	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
57	C₅×D₃×C₈	—	2	4	no	—
58	#5762	C₂×C₅	1	4	yes	complemented
59	C₃×#492	C₂×C₅	1	1	yes	characteristic, complemented
60	#5177	C₂×C₅	1	4	yes	complemented
61	#5095	C₂×C₅	1	2	yes	complemented
62	#5077	C₂×C₅	1	2	yes	complemented
63	#4700	C₂×C₅	1	2	yes	complemented
64	C₅×2₋¹⁺⁴	—	3	2	no	—
65	C₅×H(5,𝔽₂)	—	3	2	no	—
66	C₅×#137	—	3	4	no	—
67	C₅×#137	—	6	4	no	—
68	C₅×AGL(1,ℤ/8)	—	3	4	no	—
69	C₅×AGL(1,ℤ/8)	—	6	4	no	—
70	C₅×#135	—	3	2	no	—
71	C₅×#135	—	6	4	no	—
72	C₂×C₅×SD₁₆	—	6	4	no	—
73	C₂×C₅×SD₁₆	C₂×D₃	1	4	yes	complemented
74	C₅×#131	—	3	4	no	—
75	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic, complemented
76	C₅×#127	C₂×D₃	1	1	yes	characteristic
77	C₅×#108	—	3	2	no	—
78	C₅×#104	—	3	4	no	—
79	C₅×#101	—	3	2	no	—
80	C₅×#100	—	3	2	no	—
81	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic, complemented
82	#3291	—	3	1	no	Sylow
83	C₂³×C₃×C₅	—	2	2	no	—
84	C₂²×C₅×D₃	—	4	2	no	—
85	C₃×C₅×Q₈	—	2	4	no	—
86	C₃×C₅×D₄	—	2	4	no	—
87	C₃×C₅×D₄	—	8	2	no	—
88	C₂×C₃×C₄×C₅	—	2	1	no	—
89	C₂×C₃×C₄×C₅	—	2	2	no	—
90	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₅×#67	—	2	4	no	—
92	C₅×#67	—	4	4	no	—
93	C₂×C₅×Dic₃	—	4	2	no	—
94	C₅×D₁₂	—	4	2	no	—
95	C₅×D₁₂	C₂×D₄	1	4	yes	—
96	C₄×C₅×D₃	—	2	4	no	—
97	C₄×C₅×D₃	—	4	4	no	—
98	C₅×Dic₆	—	4	2	no	—
99	C₅×Dic₆	C₂×D₄	1	4	yes	—
100	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
101	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
102	C₅×C₃⋊C₈	—	2	4	no	—
103	#1007	C₂²×C₅	1	2	yes	—
104	#1004	C₂²×C₅	1	2	yes	complemented
105	C₂×C₃×SD₁₆	C₂²×C₅	1	4	yes	complemented
106	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
107	C₃×#127	C₂²×C₅	1	1	yes	characteristic
108	#942	C₂²×C₅	1	4	yes	complemented
109	#932	C₂²×C₅	1	4	yes	—
110	#916	—	2	4	no	—
111	#915	—	2	4	no	—
112	#914	—	2	4	no	—
113	D₃×SD₁₆	—	2	4	no	—
114	#904	C₂²×C₅	1	2	yes	—
115	#901	C₂²×C₅	1	4	yes	complemented
116	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
117	#836	C₂²×C₅	1	2	yes	complemented
118	#833	C₂²×C₅	1	2	yes	complemented
119	#819	C₂²×C₅	1	2	yes	complemented
120	#805	C₂²×C₅	1	4	yes	complemented
121	C₅×#41	—	3	4	no	—
122	C₅×#41	—	6	4	no	—
123	C₅×#41	—	6	4	no	—
124	C₅×#41	—	12	2	no	—
125	C₂×C₅×Q₈	—	6	4	no	—
126	C₂×C₅×Q₈	C₂²×D₃	1	2	yes	—
127	C₂×C₅×D₄	—	4	1	no	—
128	C₂×C₅×D₄	—	6	4	no	—
129	C₂×C₅×D₄	—	12	2	no	—
130	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
131	C₅×Q₁₆	—	3	2	no	—
132	C₅×Q₁₆	—	6	4	no	—
133	C₅×SD₁₆	—	2	4	no	—
134	C₅×SD₁₆	—	2	4	no	—
135	C₅×SD₁₆	—	3	4	no	—
136	C₅×SD₁₆	—	6	4	no	—
137	C₅×SD₁₆	—	6	4	no	—
138	C₅×D₈	—	3	2	no	—
139	C₅×D₈	—	6	4	no	—
140	C₅×M₁₆	—	3	4	no	—
141	C₅×M₁₆	—	6	4	no	—
142	C₂×C₅×C₈	—	6	4	no	—
143	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
144	C₅×C₄⋊C₄	—	4	1	no	—
145	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
146	#577	—	3	4	no	—
147	#492	C₅×D₃	1	1	yes	characteristic, complemented
148	#471	—	3	4	no	—
149	#456	—	3	2	no	—
150	#453	—	3	2	no	—
151	#443	—	3	2	no	—
152	C₂²×C₃×C₅	—	4	2	no	—
153	C₂²×C₃×C₅	—	8	2	no	—
154	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₅×D₃	—	2	4	no	—
156	C₂×C₅×D₃	—	16	2	no	—
157	C₃×C₄×C₅	—	2	2	no	—
158	C₃×C₄×C₅	—	4	2	no	—
159	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
160	C₅×Dic₃	—	2	4	no	—
161	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
162	C₂×C₃×D₄	—	4	1	no	—
163	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
164	C₂×#67	—	4	2	no	—
165	#239	—	2	4	no	—
166	D₃×Q₈	—	2	4	no	—
167	#237	—	2	4	no	—
168	D₃×D₄	—	2	4	no	—
169	#235	—	4	2	no	—
170	#235	C₂³×C₅	1	4	yes	—
171	C₃×SD₁₆	—	2	4	no	—
172	C₃×SD₁₆	—	2	4	no	—
173	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
174	C₃×C₄⋊C₄	—	4	1	no	—
175	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
176	#216	—	2	4	no	—
177	#215	—	2	4	no	—
178	#214	—	2	4	no	—
179	#213	—	2	4	no	—
180	#208	C₂³×C₅	1	4	yes	—
181	Dic₁₂	C₅×D₄	1	2	yes	complemented
182	D₂₄	C₅×D₄	1	2	yes	complemented
183	#204	C₅×D₄	1	4	yes	complemented
184	ΓL(1,𝔽₂₅)	—	2	4	no	—
185	D₃×C₈	—	2	4	no	—
186	C₂³×C₅	—	2	2	no	—
187	C₂³×C₅	—	12	2	no	—
188	C₅×Q₈	—	2	4	no	—
189	C₅×Q₈	—	3	4	no	—
190	C₅×Q₈	—	12	2	no	—
191	C₅×D₄	—	2	4	no	—
192	C₅×D₄	—	3	4	no	—
193	C₅×D₄	—	6	4	no	—
194	C₅×D₄	—	8	2	no	—
195	C₅×D₄	—	12	2	no	—
196	C₅×D₄	—	12	4	no	—
197	C₂×C₄×C₅	—	2	1	no	—
198	C₂×C₄×C₅	—	2	2	no	—
199	C₂×C₄×C₅	—	6	4	no	—
200	C₂×C₄×C₅	—	12	4	no	—
201	C₂×C₄×C₅	—	12	2	no	—
202	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
203	C₅×C₈	—	6	4	no	—
204	C₅×C₈	D₃×D₄	1	2	yes	—
205	C₅×C₈	D₃×D₄	1	2	yes	—
206	2₋¹⁺⁴	—	3	2	no	—
207	H(5,𝔽₂)	—	3	2	no	—
208	#137	—	3	4	no	—
209	#137	—	6	4	no	—
210	AGL(1,ℤ/8)	—	3	4	no	—
211	AGL(1,ℤ/8)	—	6	4	no	—
212	#135	—	3	2	no	—
213	#135	—	6	4	no	—
214	C₂×SD₁₆	—	6	4	no	—
215	C₂×SD₁₆	C₂×C₅×D₃	1	4	yes	complemented
216	#131	—	3	4	no	—
217	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
218	#127	C₂×C₅×D₃	1	1	yes	characteristic
219	#108	—	3	2	no	—
220	#104	—	3	4	no	—
221	#101	—	3	2	no	—
222	#100	—	3	2	no	—
223	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	—	8	2	no	—
226	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
227	C₅×D₃	—	4	4	no	—
228	C₂³×C₃	—	2	2	no	—
229	C₂²×D₃	—	4	2	no	—
230	C₃×Q₈	—	2	4	no	—
231	C₃×D₄	—	2	4	no	—
232	C₃×D₄	—	8	2	no	—
233	C₂×C₃×C₄	—	2	1	no	—
234	C₂×C₃×C₄	—	2	2	no	—
235	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
236	#67	—	2	4	no	—
237	#67	—	4	4	no	—
238	C₂×Dic₃	—	4	2	no	—
239	D₁₂	—	4	2	no	—
240	D₁₂	C₂×C₅×D₄	1	4	yes	—
241	C₄×D₃	—	2	4	no	—
242	C₄×D₃	—	4	4	no	—
243	Dic₆	—	4	2	no	—
244	Dic₆	C₂×C₅×D₄	1	4	yes	—
245	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
246	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
247	C₃⋊C₈	—	2	4	no	—
248	C₂²×C₅	—	4	2	no	—
249	C₂²×C₅	—	6	4	no	—
250	C₂²×C₅	—	8	2	no	—
251	C₂²×C₅	—	48	2	no	—
252	C₂²×C₅	#1004	1	1	yes	characteristic
253	C₄×C₅	—	2	2	no	—
254	C₄×C₅	—	4	2	no	—
255	C₄×C₅	—	6	4	no	—
256	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
257	#41	—	3	4	no	—
258	#41	—	6	4	no	—
259	#41	—	6	4	no	—
260	#41	—	12	2	no	—
261	C₂×Q₈	—	6	4	no	—
262	C₂×Q₈	C₂²×C₅×D₃	1	2	yes	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	6	4	no	—
265	C₂×D₄	—	12	2	no	—
266	C₂×D₄	C₂²×C₅×D₃	1	2	yes	—
267	Q₁₆	—	3	2	no	—
268	Q₁₆	—	6	4	no	—
269	SD₁₆	—	2	4	no	—
270	SD₁₆	—	2	4	no	—
271	SD₁₆	—	3	4	no	—
272	SD₁₆	—	6	4	no	—
273	SD₁₆	—	6	4	no	—
274	D₈	—	3	2	no	—
275	D₈	—	6	4	no	—
276	M₁₆	—	3	4	no	—
277	M₁₆	—	6	4	no	—
278	C₂×C₈	—	6	4	no	—
279	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
282	C₃×C₅	#3291	1	1	yes	characteristic, complemented, Hall
283	C₂²×C₃	—	4	2	no	—
284	C₂²×C₃	—	8	2	no	—
285	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
286	C₂×D₃	—	2	4	no	—
287	C₂×D₃	—	16	2	no	—
288	C₃×C₄	—	2	2	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
291	Dic₃	—	2	4	no	—
292	C₂×C₅	—	2	1	no	—
293	C₂×C₅	—	8	2	no	—
294	C₂×C₅	—	12	4	no	—
295	C₂×C₅	#5717	1	1	yes	characteristic
296	C₂³	—	2	2	no	—
297	C₂³	—	12	2	no	—
298	Q₈	—	2	4	no	—
299	Q₈	—	3	4	no	—
300	Q₈	—	12	2	no	—
301	D₄	—	2	4	no	—
302	D₄	—	3	4	no	—
303	D₄	—	6	4	no	—
304	D₄	—	8	2	no	—
305	D₄	—	12	2	no	—
306	D₄	—	12	4	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	6	4	no	—
310	C₂×C₄	—	12	4	no	—
311	C₂×C₄	—	12	2	no	—
312	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
313	C₈	—	6	4	no	—
314	C₈	C₅×D₃×D₄	1	2	yes	—
315	C₈	C₅×D₃×D₄	1	2	yes	—
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	8	2	no	—
318	C₂×C₃	C₅×D₄²	1	1	yes	characteristic
319	D₃	—	4	4	no	—
320	C₅	#81774	1	1	yes	characteristic, direct product factor, Sylow
321	C₂²	—	4	2	no	—
322	C₂²	—	6	4	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	48	2	no	—
325	C₂²	C₅×#1004	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	6	4	no	—
329	C₄	C₂×C₅×D₃×D₄	1	2	yes	—
330	C₃	C₅×#3291	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	1	no	—
332	C₂	—	8	2	no	—
333	C₂	—	12	4	no	—
334	C₂	C₅×#5717	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	#5717
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#492	C₂
socle	C₂×C₃×C₅	D₄²
hypercenter	C₅×#492	D₃
hypocenter	C₃	C₅×#3291
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#492

Complements

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

N	H ≅ G/N	type
1. C₅×#5762	C₂	semidirect
2. C₃×C₅×#492	C₂	semidirect
3. C₅×#5177	C₂	semidirect
4. C₅×#5095	C₂	semidirect
5. C₅×#5077	C₂	semidirect
6. C₅×#4700	C₂	semidirect
9. C₅×#1004	C₂²	semidirect
10. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
11. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
13. C₅×#942	C₂²	semidirect
20. C₅×#901	C₂²	semidirect
21. C₃×C₄×C₅×C₈	C₂²	semidirect
22. C₅×#836	C₂²	semidirect
23. C₅×#833	C₂²	semidirect
24. C₅×#819	C₂²	semidirect
25. C₅×#805	C₂²	semidirect
26. #81774	C₅	direct (N × H)
28. C₅×#492	D₃	semidirect
53. C₅×Dic₁₂	D₄	semidirect
54. C₅×D₂₄	D₄	semidirect
55. C₅×#204	D₄	semidirect
58. #5762	C₂×C₅	semidirect
59. C₃×#492	C₂×C₅	semidirect
60. #5177	C₂×C₅	semidirect
61. #5095	C₂×C₅	semidirect
62. #5077	C₂×C₅	semidirect
63. #4700	C₂×C₅	semidirect
73. C₂×C₅×SD₁₆	C₂×D₃	semidirect
75. C₅×C₄⋊Q₈	C₂×D₃	semidirect
81. C₄×C₅×C₈	C₂×D₃	semidirect
104. #1004	C₂²×C₅	semidirect
105. C₂×C₃×SD₁₆	C₂²×C₅	semidirect
106. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
108. #942	C₂²×C₅	semidirect
115. #901	C₂²×C₅	semidirect
116. C₃×C₄×C₈	C₂²×C₅	semidirect
117. #836	C₂²×C₅	semidirect
118. #833	C₂²×C₅	semidirect
119. #819	C₂²×C₅	semidirect
120. #805	C₂²×C₅	semidirect
147. #492	C₅×D₃	semidirect
181. Dic₁₂	C₅×D₄	semidirect
182. D₂₄	C₅×D₄	semidirect
183. #204	C₅×D₄	semidirect
215. C₂×SD₁₆	C₂×C₅×D₃	semidirect
217. C₄⋊Q₈	C₂×C₅×D₃	semidirect
223. C₄×C₈	C₂×C₅×D₃	semidirect
282. C₃×C₅	#3291	semidirect
320. C₅	#81774	direct (N × H)
330. C₃	C₅×#3291	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5762	2
C₃×C₅×#492	2
C₅×#5177	2
C₅×#5095	2
C₅×#5077	2
C₅×#4700	2
C₅×#3291	3
#81774	5

Subgroups — C5×#81774 #14458098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#81774 #14458098