Subgroups — C₂×C₅×#4096 #13106978

Order 1600.

696 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4096	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#689	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#688	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#687	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#686	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×#168	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅²×#41	—	2	2	no	—
10	C₅²×#41	C₂²	1	4	yes	complemented
11	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₅²×D₄	—	2	1	no	—
13	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅²	—	2	1	no	—
15	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₂²⋊D₅	—	2	1	no	—
17	C₅×#709	—	2	2	no	—
18	C₅×#709	C₂²	1	4	yes	complemented
19	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×D₅	—	2	1	no	—
21	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
22	C₅×#689	C₂²	1	4	yes	complemented
23	C₅×#688	C₂²	1	4	yes	complemented
24	C₅×#687	C₂²	1	4	yes	complemented
25	C₅×#686	C₂²	1	4	yes	complemented
26	C₂×C₅×#168	C₂²	1	1	yes	characteristic
27	C₂×C₅×#168	C₂²	1	1	yes	characteristic
28	C₂×C₅×#168	C₂²	1	4	yes	complemented
29	C₂×C₅×#135	—	5	1	no	maximal
30	C₂×#4096	C₅	1	1	yes	maximal, characteristic, direct product factor
31	C₂³×C₅²	—	2	1	no	—
32	C₂³×C₅²	D₄	1	1	yes	characteristic
33	C₂²×C₅×D₅	—	2	1	no	—
34	C₅²×Q₈	—	2	1	no	—
35	C₅²×Q₈	C₂³	1	2	yes	—
36	C₅²×D₄	—	2	4	no	—
37	C₅²×D₄	—	2	1	no	—
38	C₅²×D₄	C₂³	1	2	yes	—
39	C₂×C₄×C₅²	—	2	1	no	—
40	C₂×C₄×C₅²	—	2	4	no	—
41	C₂×C₄×C₅²	D₄	1	2	yes	—
42	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
43	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₅²	C₂³	1	2	yes	—
45	C₅×C₂²⋊D₅	—	2	4	no	—
46	C₂×C₅×Dic₅	—	2	1	no	—
47	C₅×D₂₀	—	2	1	no	—
48	C₅×D₂₀	C₂³	1	2	yes	—
49	C₄×C₅×D₅	—	2	4	no	—
50	C₅×Dic₁₀	—	2	1	no	—
51	C₅×Dic₁₀	C₂³	1	2	yes	—
52	C₅×#168	C₂³	1	2	yes	—
53	C₅×#168	C₂³	1	2	yes	—
54	C₂×C₅×#41	—	2	2	no	—
55	C₂×C₅×#41	—	5	1	no	—
56	C₂×C₅×#41	D₅	1	1	yes	characteristic, complemented
57	C₂×C₅×#41	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×#709	C₂×C₅	1	1	yes	characteristic, complemented
59	C₅×#135	—	5	8	no	—
60	C₂×C₅×Q₁₆	—	5	1	no	—
61	C₂×C₅×SD₁₆	—	5	1	no	—
62	C₂×C₅×SD₁₆	—	5	1	no	—
63	C₂×C₅×D₈	—	5	1	no	—
64	C₂²×C₅×C₈	—	5	1	no	—
65	#4096	C₂×C₅	1	8	yes	direct product factor
66	C₂×#689	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂×#688	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂×#687	C₂×C₅	1	1	yes	characteristic, complemented
69	C₂×#686	C₂×C₅	1	1	yes	characteristic, complemented
70	C₂²×#168	C₂×C₅	1	1	yes	characteristic, complemented
71	C₂²×C₅²	—	2	2	no	—
72	C₂²×C₅²	—	2	2	no	—
73	C₂²×C₅²	—	4	2	no	—
74	C₂²×C₅²	C₂×D₄	1	2	yes	—
75	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
76	C₂×C₅×D₅	—	2	2	no	—
77	C₂×C₅×D₅	—	4	2	no	—
78	C₄×C₅²	—	2	2	no	—
79	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
80	C₄×C₅²	C₂×D₄	1	2	yes	—
81	C₄×C₅²	C₂⁴	1	1	yes	characteristic
82	C₅×Dic₅	—	2	2	no	—
83	C₅×#41	—	2	8	no	—
84	C₅×#41	—	2	2	no	—
85	C₅×#41	—	2	2	no	—
86	C₅×#41	—	2	8	no	—
87	C₅×#41	—	5	4	no	—
88	C₅×#41	—	10	2	no	—
89	C₅×#41	C₂×D₅	1	4	yes	complemented
90	C₅×#41	C₂²×C₅	1	4	yes	complemented
91	C₂×C₅×Q₈	—	2	2	no	—
92	C₂×C₅×Q₈	—	5	1	no	—
93	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
94	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
95	C₂×C₅×D₄	—	2	4	no	—
96	C₂×C₅×D₄	—	2	1	no	—
97	C₂×C₅×D₄	—	2	1	no	—
98	C₂×C₅×D₄	—	2	2	no	—
99	C₂×C₅×D₄	—	5	1	no	—
100	C₂×C₅×D₄	—	10	1	no	—
101	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
102	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic
103	C₂²×C₄×C₅	—	2	4	no	—
104	C₂²×C₄×C₅	—	2	1	no	—
105	C₂²×C₄×C₅	—	2	1	no	—
106	C₂²×C₄×C₅	—	2	2	no	—
107	C₂²×C₄×C₅	—	10	1	no	—
108	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
109	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₂²⋊D₅	—	2	1	no	—
111	#709	—	2	2	no	—
112	#709	C₂²×C₅	1	4	yes	complemented
113	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
114	C₂×C₄×D₅	—	2	1	no	—
115	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
116	C₅×Q₁₆	—	5	4	no	—
117	C₅×SD₁₆	—	5	4	no	—
118	C₅×SD₁₆	—	5	4	no	—
119	C₅×D₈	—	5	4	no	—
120	C₂×C₅×C₈	—	5	1	no	—
121	C₂×C₅×C₈	—	5	1	no	—
122	C₂×C₅×C₈	—	5	4	no	—
123	#689	C₂²×C₅	1	4	yes	complemented
124	#688	C₂²×C₅	1	4	yes	complemented
125	#687	C₂²×C₅	1	4	yes	complemented
126	#686	C₂²×C₅	1	4	yes	complemented
127	C₂×#168	C₂²×C₅	1	1	yes	characteristic
128	C₂×#168	C₂²×C₅	1	1	yes	characteristic
129	C₂×#168	C₂²×C₅	1	4	yes	complemented
130	C₂×#135	—	5	1	no	Sylow
131	C₂×C₅²	—	2	2	no	—
132	C₂×C₅²	—	4	2	no	—
133	C₂×C₅²	#135	1	2	yes	complemented
134	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
135	C₅×D₅	—	4	2	no	—
136	C₂³×C₅	—	2	4	no	—
137	C₂³×C₅	—	2	1	no	—
138	C₂³×C₅	—	2	1	no	—
139	C₂³×C₅	—	2	2	no	—
140	C₂³×C₅	—	10	1	no	—
141	C₂³×C₅	C₂²⋊D₅	1	1	yes	characteristic
142	C₂³×C₅	C₅×D₄	1	1	yes	characteristic
143	C₂²×D₅	—	2	1	no	—
144	C₅×Q₈	—	2	4	no	—
145	C₅×Q₈	—	2	1	no	—
146	C₅×Q₈	—	2	1	no	—
147	C₅×Q₈	—	2	4	no	—
148	C₅×Q₈	—	5	2	no	—
149	C₅×Q₈	—	10	1	no	—
150	C₅×Q₈	C₂²×D₅	1	2	yes	—
151	C₅×Q₈	C₂³×C₅	1	2	yes	—
152	C₅×D₄	—	2	16	no	—
153	C₅×D₄	—	2	4	no	—
154	C₅×D₄	—	2	4	no	—
155	C₅×D₄	—	2	4	no	—
156	C₅×D₄	—	2	1	no	—
157	C₅×D₄	—	2	1	no	—
158	C₅×D₄	—	2	4	no	—
159	C₅×D₄	—	5	2	no	—
160	C₅×D₄	—	10	4	no	—
161	C₅×D₄	—	10	1	no	—
162	C₅×D₄	C₂²×D₅	1	2	yes	—
163	C₅×D₄	C₂³×C₅	1	2	yes	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	2	16	no	—
166	C₂×C₄×C₅	—	2	1	no	—
167	C₂×C₄×C₅	—	2	1	no	—
168	C₂×C₄×C₅	—	2	4	no	—
169	C₂×C₄×C₅	—	2	4	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	4	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	4	no	—
174	C₂×C₄×C₅	—	10	1	no	—
175	C₂×C₄×C₅	—	10	4	no	—
176	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
177	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
178	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
180	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
182	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
184	C₂²⋊D₅	—	2	4	no	—
185	C₂×Dic₅	—	2	1	no	—
186	D₂₀	—	2	1	no	—
187	D₂₀	C₂³×C₅	1	2	yes	—
188	C₄×D₅	—	2	4	no	—
189	Dic₁₀	—	2	1	no	—
190	Dic₁₀	C₂³×C₅	1	2	yes	—
191	C₅×C₈	—	5	2	no	—
192	C₅×C₈	—	5	2	no	—
193	#168	C₂³×C₅	1	2	yes	—
194	#168	C₂³×C₅	1	2	yes	—
195	C₂×#41	—	5	1	no	—
196	C₂×#41	C₅×D₅	1	1	yes	characteristic, complemented
197	#135	—	5	8	no	—
198	C₂×Q₁₆	—	5	1	no	—
199	C₂×SD₁₆	—	5	1	no	—
200	C₂×SD₁₆	—	5	1	no	—
201	C₂×D₈	—	5	1	no	—
202	C₂²×C₈	—	5	1	no	—
203	C₅²	C₂×#135	1	1	yes	characteristic, complemented, Sylow
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	2	2	no	—
206	C₂²×C₅	—	2	8	no	—
207	C₂²×C₅	—	2	2	no	—
208	C₂²×C₅	—	2	2	no	—
209	C₂²×C₅	—	2	2	no	—
210	C₂²×C₅	—	2	4	no	—
211	C₂²×C₅	—	4	4	no	—
212	C₂²×C₅	—	4	8	no	—
213	C₂²×C₅	—	4	2	no	—
214	C₂²×C₅	—	4	2	no	—
215	C₂²×C₅	—	10	2	no	—
216	C₂²×C₅	—	20	2	no	—
217	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
218	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
219	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
220	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂×D₅	—	2	2	no	—
222	C₂×D₅	—	4	2	no	—
223	C₄×C₅	—	2	8	no	—
224	C₄×C₅	—	2	2	no	—
225	C₄×C₅	—	2	2	no	—
226	C₄×C₅	—	2	4	no	—
227	C₄×C₅	—	2	2	no	—
228	C₄×C₅	—	2	2	no	—
229	C₄×C₅	—	10	2	no	—
230	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
231	C₄×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
232	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
233	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
234	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
235	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
236	Dic₅	—	2	2	no	—
237	#41	—	2	2	no	—
238	#41	—	5	4	no	—
239	#41	—	10	2	no	—
240	#41	C₂×C₅×D₅	1	4	yes	complemented
241	C₂×Q₈	—	5	1	no	—
242	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	5	1	no	—
245	C₂×D₄	—	10	1	no	—
246	C₂×D₄	C₂×C₅×D₅	1	1	yes	characteristic
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	10	1	no	—
249	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
250	Q₁₆	—	5	4	no	—
251	SD₁₆	—	5	4	no	—
252	SD₁₆	—	5	4	no	—
253	D₈	—	5	4	no	—
254	C₂×C₈	—	5	1	no	—
255	C₂×C₈	—	5	1	no	—
256	C₂×C₈	—	5	4	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	—	2	4	no	—
260	C₂×C₅	—	2	2	no	—
261	C₂×C₅	—	4	4	no	—
262	C₂×C₅	—	4	8	no	—
263	C₂×C₅	—	4	2	no	—
264	C₂×C₅	—	4	2	no	—
265	C₂×C₅	—	20	2	no	—
266	C₂×C₅	#4096	1	2	yes	direct product factor
267	C₂×C₅	C₅×#135	1	2	yes	complemented
268	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
269	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
270	D₅	—	4	2	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	10	1	no	—
273	C₂³	C₅×C₂²⋊D₅	1	1	yes	characteristic
274	Q₈	—	2	1	no	—
275	Q₈	—	5	2	no	—
276	Q₈	—	10	1	no	—
277	Q₈	C₂²×C₅×D₅	1	2	yes	—
278	D₄	—	2	4	no	—
279	D₄	—	2	1	no	—
280	D₄	—	5	2	no	—
281	D₄	—	10	4	no	—
282	D₄	—	10	1	no	—
283	D₄	C₂²×C₅×D₅	1	2	yes	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	10	1	no	—
287	C₂×C₄	—	10	4	no	—
288	C₂×C₄	C₅×C₂²⋊D₅	1	1	yes	characteristic
289	C₂×C₄	C₅×C₂²⋊D₅	1	2	yes	—
290	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
292	C₈	—	5	2	no	—
293	C₈	—	5	2	no	—
294	C₅	—	2	2	no	—
295	C₅	C₂×#4096	1	1	yes	characteristic, direct product factor
296	C₅	C₂×C₅×#135	1	1	yes	characteristic, complemented
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	10	2	no	—
301	C₂²	—	20	2	no	—
302	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
303	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	10	2	no	—
306	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
307	C₄	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
308	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
309	C₂	—	2	2	no	—
310	C₂	—	4	2	no	—
311	C₂	—	20	2	no	—
312	C₂	C₅×#4096	1	2	yes	direct product factor
313	C₂	C₂²×C₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₂×C₅²×#41	C₂
socle	C₂²×C₅²	C₂×D₄
hypercenter	C₂×C₅×#41	D₅
hypocenter	C₅	C₂×C₅×#135
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×#41	C₂	semidirect
2. C₂×C₅×#709	C₂	semidirect
3. C₅×#4096	C₂	direct (N × H)
4. C₂×C₅×#689	C₂	semidirect
5. C₂×C₅×#688	C₂	semidirect
6. C₂×C₅×#687	C₂	semidirect
7. C₂×C₅×#686	C₂	semidirect
8. C₂²×C₅×#168	C₂	semidirect
10. C₅²×#41	C₂²	semidirect
18. C₅×#709	C₂²	semidirect
22. C₅×#689	C₂²	semidirect
23. C₅×#688	C₂²	semidirect
24. C₅×#687	C₂²	semidirect
25. C₅×#686	C₂²	semidirect
28. C₂×C₅×#168	C₂²	semidirect
30. C₂×#4096	C₅	direct (N × H)
56. C₂×C₅×#41	D₅	semidirect
57. C₂×C₅×#41	C₂×C₅	semidirect
58. C₂×#709	C₂×C₅	semidirect
65. #4096	C₂×C₅	direct (N × H)
66. C₂×#689	C₂×C₅	semidirect
67. C₂×#688	C₂×C₅	semidirect
68. C₂×#687	C₂×C₅	semidirect
69. C₂×#686	C₂×C₅	semidirect
70. C₂²×#168	C₂×C₅	semidirect
89. C₅×#41	C₂×D₅	semidirect
90. C₅×#41	C₂²×C₅	semidirect
112. #709	C₂²×C₅	semidirect
123. #689	C₂²×C₅	semidirect
124. #688	C₂²×C₅	semidirect
125. #687	C₂²×C₅	semidirect
126. #686	C₂²×C₅	semidirect
129. C₂×#168	C₂²×C₅	semidirect
133. C₂×C₅²	#135	semidirect
196. C₂×#41	C₅×D₅	semidirect
203. C₅²	C₂×#135	semidirect
240. #41	C₂×C₅×D₅	semidirect
266. C₂×C₅	#4096	direct (N × H)
267. C₂×C₅	C₅×#135	semidirect
295. C₅	C₂×#4096	direct (N × H)
296. C₅	C₂×C₅×#135	semidirect
312. C₂	C₅×#4096	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅²×#41	2
C₂×C₅×#709	2
C₅×#4096	2
C₂×C₅×#689	2
C₂×C₅×#688	2
C₂×C₅×#687	2
C₂×C₅×#686	2
C₂²×C₅×#168	2
C₂×C₅×#135	5
C₂×#4096	5

Subgroups — C2×C5×#4096 #13106978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4096 #13106978