Subgroups — C₅×#8821 #11931541

Order 1280.

406 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1845	C₂	1	1	yes	maximal, characteristic
2	C₅×#1844	C₂	1	1	yes	maximal, characteristic
3	C₅×#1834	C₂	1	1	yes	maximal, characteristic
4	C₅×#1834	C₂	1	1	yes	maximal, characteristic
5	C₅×#1834	C₂	1	1	yes	maximal, characteristic
6	C₅×#1834	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×#107	—	2	2	no	—
9	C₂×C₅×#106	—	2	2	no	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#384	C₂²	1	1	yes	characteristic
14	C₅×#384	C₂²	1	1	yes	characteristic
15	C₅×#384	C₂²	1	1	yes	characteristic
16	C₅×#384	C₂²	1	1	yes	characteristic
17	C₅×#340	—	2	2	no	—
18	C₅×#340	—	2	2	no	—
19	C₅×#340	—	2	2	no	—
20	C₅×#340	—	2	2	no	—
21	C₅×#322	C₄	1	4	yes	complemented
22	#8821	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×C₈	D₄	1	2	yes	—
24	C₂²×C₅×C₈	D₄	1	2	yes	—
25	C₂²×C₅×C₈	D₄	1	2	yes	—
26	C₂×C₅×C₄⋊C₄	—	2	1	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	—	2	1	no	—
31	C₂×C₅×C₄⋊C₄	—	2	1	no	—
32	C₂×C₅×C₄⋊C₄	—	2	1	no	—
33	C₂×C₅×C₄⋊C₄	—	2	1	no	—
34	C₂×C₅×C₄⋊C₄	—	2	1	no	—
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₄⋊C₄	—	2	1	no	—
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₅×#107	—	4	4	no	—
40	C₅×#106	—	4	4	no	—
41	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
43	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
44	C₄×C₅×C₈	Q₈	1	2	yes	—
45	C₄×C₅×C₈	Q₈	1	2	yes	—
46	C₄×C₅×C₈	Q₈	1	2	yes	—
47	#1845	C₂×C₅	1	1	yes	characteristic
48	#1844	C₂×C₅	1	1	yes	characteristic
49	#1834	C₂×C₅	1	1	yes	characteristic
50	#1834	C₂×C₅	1	1	yes	characteristic
51	#1834	C₂×C₅	1	1	yes	characteristic
52	#1834	C₂×C₅	1	1	yes	characteristic
53	C₂×#322	C₂×C₅	1	1	yes	characteristic
54	C₂²×C₄×C₅	—	4	1	no	—
55	C₂²×C₄×C₅	—	4	1	no	—
56	C₂²×C₄×C₅	—	4	1	no	—
57	C₂²×C₄×C₅	—	4	1	no	—
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₅×C₈	—	2	2	no	—
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	—	2	2	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
68	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
69	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
70	C₅×C₄⋊C₄	—	4	2	no	—
71	C₅×C₄⋊C₄	—	4	2	no	—
72	C₅×C₄⋊C₄	—	4	2	no	—
73	C₅×C₄⋊C₄	—	4	2	no	—
74	C₅×C₄⋊C₄	—	4	2	no	—
75	C₅×C₄⋊C₄	—	4	2	no	—
76	C₅×C₄⋊C₄	—	4	2	no	—
77	C₅×C₄⋊C₄	—	4	2	no	—
78	C₅×C₄⋊C₄	—	4	2	no	—
79	C₅×C₄⋊C₄	—	4	2	no	—
80	C₅×C₄⋊C₄	—	4	2	no	—
81	C₅×C₄⋊C₄	—	4	2	no	—
82	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
86	C₂×#107	—	2	2	no	—
87	C₂×#106	—	2	2	no	—
88	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
91	#384	C₂²×C₅	1	1	yes	characteristic
92	#384	C₂²×C₅	1	1	yes	characteristic
93	#384	C₂²×C₅	1	1	yes	characteristic
94	#384	C₂²×C₅	1	1	yes	characteristic
95	#340	—	2	2	no	—
96	#340	—	2	2	no	—
97	#340	—	2	2	no	—
98	#340	—	2	2	no	—
99	#322	C₄×C₅	1	4	yes	complemented
100	C₂³×C₅	#127	1	1	yes	characteristic
101	C₂×C₄×C₅	—	8	1	no	—
102	C₂×C₄×C₅	—	8	1	no	—
103	C₂×C₄×C₅	—	8	1	no	—
104	C₂×C₄×C₅	—	8	1	no	—
105	C₂×C₄×C₅	—	8	1	no	—
106	C₂×C₄×C₅	—	8	1	no	—
107	C₂×C₄×C₅	—	8	1	no	—
108	C₂×C₄×C₅	—	8	1	no	—
109	C₂×C₄×C₅	—	8	1	no	—
110	C₂×C₄×C₅	—	8	1	no	—
111	C₂×C₄×C₅	—	8	1	no	—
112	C₂×C₄×C₅	—	8	1	no	—
113	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅	#135	1	2	yes	—
120	C₂×C₄×C₅	#135	1	2	yes	—
121	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₅	#137	1	2	yes	—
124	C₂×C₄×C₅	#137	1	2	yes	—
125	C₅×C₈	—	2	4	no	—
126	C₅×C₈	—	2	4	no	—
127	C₅×C₈	—	2	4	no	—
128	C₂²×C₈	C₅×D₄	1	2	yes	—
129	C₂²×C₈	C₅×D₄	1	2	yes	—
130	C₂²×C₈	C₅×D₄	1	2	yes	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₄⋊C₄	—	2	1	no	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄⋊C₄	—	2	1	no	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄⋊C₄	—	2	1	no	—
143	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
144	#107	—	4	4	no	—
145	#106	—	4	4	no	—
146	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
147	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
148	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
149	C₄×C₈	C₅×Q₈	1	2	yes	—
150	C₄×C₈	C₅×Q₈	1	2	yes	—
151	C₄×C₈	C₅×Q₈	1	2	yes	—
152	C₂²×C₅	#384	1	1	yes	characteristic
153	C₂²×C₅	#495	1	1	yes	characteristic
154	C₂²×C₅	#495	1	1	yes	characteristic
155	C₂²×C₅	#496	1	1	yes	characteristic
156	C₂²×C₅	#496	1	1	yes	characteristic
157	C₂²×C₅	#497	1	1	yes	characteristic
158	C₂²×C₅	#497	1	1	yes	characteristic
159	C₄×C₅	—	16	1	no	—
160	C₄×C₅	—	16	1	no	—
161	C₄×C₅	—	16	1	no	—
162	C₄×C₅	—	16	1	no	—
163	C₄×C₅	#427	1	2	yes	—
164	C₄×C₅	#428	1	2	yes	—
165	C₄×C₅	#428	1	2	yes	—
166	C₄×C₅	#499	1	2	yes	—
167	C₄×C₅	#501	1	2	yes	—
168	C₄×C₅	#501	1	2	yes	—
169	C₂²×C₄	—	4	1	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	—	4	1	no	—
173	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
183	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
184	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
198	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
199	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
200	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
201	C₂×C₅	#1717	1	1	yes	characteristic
202	C₂×C₅	#1718	1	1	yes	characteristic
203	C₂×C₅	#1719	1	1	yes	characteristic
204	C₂×C₅	#1720	1	1	yes	characteristic
205	C₂×C₅	#1846	1	1	yes	characteristic
206	C₂×C₅	#1847	1	1	yes	characteristic
207	C₂×C₅	#1847	1	1	yes	characteristic
208	C₂³	C₅×#127	1	1	yes	characteristic
209	C₂×C₄	—	8	1	no	—
210	C₂×C₄	—	8	1	no	—
211	C₂×C₄	—	8	1	no	—
212	C₂×C₄	—	8	1	no	—
213	C₂×C₄	—	8	1	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	8	1	no	—
218	C₂×C₄	—	8	1	no	—
219	C₂×C₄	—	8	1	no	—
220	C₂×C₄	—	8	1	no	—
221	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
222	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
224	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
225	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
226	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
227	C₂×C₄	C₅×#135	1	2	yes	—
228	C₂×C₄	C₅×#135	1	2	yes	—
229	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
230	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
231	C₂×C₄	C₅×#137	1	2	yes	—
232	C₂×C₄	C₅×#137	1	2	yes	—
233	C₈	—	2	4	no	—
234	C₈	—	2	4	no	—
235	C₈	—	2	4	no	—
236	C₅	#8821	1	1	yes	characteristic, direct product factor, Sylow
237	C₂²	C₅×#384	1	1	yes	characteristic
238	C₂²	C₅×#495	1	1	yes	characteristic
239	C₂²	C₅×#495	1	1	yes	characteristic
240	C₂²	C₅×#496	1	1	yes	characteristic
241	C₂²	C₅×#496	1	1	yes	characteristic
242	C₂²	C₅×#497	1	1	yes	characteristic
243	C₂²	C₅×#497	1	1	yes	characteristic
244	C₄	—	16	1	no	—
245	C₄	—	16	1	no	—
246	C₄	—	16	1	no	—
247	C₄	—	16	1	no	—
248	C₄	C₅×#427	1	2	yes	—
249	C₄	C₅×#428	1	2	yes	—
250	C₄	C₅×#428	1	2	yes	—
251	C₄	C₅×#499	1	2	yes	—
252	C₄	C₅×#501	1	2	yes	—
253	C₄	C₅×#501	1	2	yes	—
254	C₂	C₅×#1717	1	1	yes	characteristic
255	C₂	C₅×#1718	1	1	yes	characteristic
256	C₂	C₅×#1719	1	1	yes	characteristic
257	C₂	C₅×#1720	1	1	yes	characteristic
258	C₂	C₅×#1846	1	1	yes	characteristic
259	C₂	C₅×#1847	1	1	yes	characteristic
260	C₂	C₅×#1847	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#127
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
21. C₅×#322	C₄	semidirect
22. #8821	C₅	direct (N × H)
99. #322	C₄×C₅	semidirect
236. C₅	#8821	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#1845	2
C₅×#1844	2
C₅×#1834	2
C₂×C₅×#322	2
#8821	5

Subgroups — C5×#8821 #11931541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8821 #11931541