Subgroups — C₅×#33319 #11956039

Order 1280.

806 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	C₂×C₅×#501	C₂	1	2	yes	maximal
2	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#499	C₂	1	1	yes	maximal, characteristic
4	C₅×#2660	C₂	1	2	yes	maximal
5	C₅×#1847	C₂	1	4	yes	maximal
6	C₅×#1846	C₂	1	2	yes	maximal
7	C₅×#1845	C₂	1	2	yes	maximal
8	C₅×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₅×#125	C₂²	1	2	yes	—
11	C₅×#501	—	2	8	no	—
12	C₅×#500	—	2	4	no	—
13	C₅×#499	—	2	4	no	—
14	C₂×C₅×#107	C₂²	1	1	yes	characteristic
15	C₂×C₅×#107	C₂²	1	2	yes	—
16	C₂×C₅×#107	C₂²	1	4	yes	—
17	C₂×C₅×#107	C₂²	1	1	yes	characteristic
18	C₂×C₅×#107	C₂²	1	2	yes	—
19	C₂×C₅×#106	C₂²	1	4	yes	—
20	C₂×C₅×#106	C₂²	1	2	yes	—
21	C₂×C₅×#97	C₂²	1	1	yes	characteristic
22	C₂×C₅×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
25	C₅×#398	—	2	4	no	—
26	C₅×#395	—	2	4	no	—
27	C₅×#384	C₂²	1	4	yes	—
28	C₅×#382	C₂²	1	4	yes	—
29	C₅×#376	C₂²	1	1	yes	characteristic
30	#33319	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₅×Q₈	—	2	2	no	—
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₅×C₄⋊Q₈	—	2	4	no	—
36	C₅×C₄⋊Q₈	—	2	4	no	—
37	C₅×#125	—	2	4	no	—
38	C₅×#125	—	2	4	no	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₄⋊C₄	—	2	2	no	—
41	C₂×C₅×C₄⋊C₄	—	2	2	no	—
42	C₂×C₅×C₄⋊C₄	—	2	8	no	—
43	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
45	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂³	1	2	yes	—
48	C₅×#107	—	2	2	no	—
49	C₅×#107	—	2	2	no	—
50	C₅×#107	—	2	4	no	—
51	C₅×#107	—	2	4	no	—
52	C₅×#107	—	2	8	no	—
53	C₅×#106	—	2	4	no	—
54	C₅×#106	—	2	8	no	—
55	C₅×#97	D₄	1	4	yes	—
56	C₅×#97	Q₈	1	8	yes	complemented
57	C₄×C₅×C₈	D₄	1	4	yes	—
58	C₄×C₅×C₈	Q₈	1	8	yes	complemented
59	C₅×#95	—	2	8	no	—
60	C₅×#95	D₄	1	2	yes	—
61	C₅×#95	D₄	1	2	yes	—
62	C₂×#501	C₂×C₅	1	2	yes	—
63	C₂×#500	C₂×C₅	1	1	yes	characteristic
64	C₂×#499	C₂×C₅	1	1	yes	characteristic
65	#2660	C₂×C₅	1	2	yes	—
66	#1847	C₂×C₅	1	4	yes	—
67	#1846	C₂×C₅	1	2	yes	—
68	#1845	C₂×C₅	1	2	yes	—
69	#1726	C₂×C₅	1	1	yes	characteristic
70	C₂×C₅×Q₈	—	4	4	no	—
71	C₂×C₅×Q₈	—	8	4	no	—
72	C₂²×C₄×C₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	2	4	no	—
74	C₂²×C₄×C₅	—	2	2	no	—
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
80	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
81	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
82	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
83	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
84	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
85	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
86	C₅×C₄⋊C₄	—	2	2	no	—
87	C₅×C₄⋊C₄	—	2	2	no	—
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₄⋊C₄	—	4	4	no	—
93	C₅×C₄⋊C₄	—	4	4	no	—
94	C₅×C₄⋊C₄	—	4	4	no	—
95	C₅×C₄⋊C₄	—	8	8	no	—
96	C₄²×C₅	C₂×D₄	1	2	yes	—
97	C₄²×C₅	C₂×D₄	1	2	yes	—
98	C₄²×C₅	C₂×Q₈	1	4	yes	—
99	C₄²×C₅	C₂×Q₈	1	4	yes	—
100	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
101	C₂×#125	C₂²×C₅	1	2	yes	—
102	#501	—	2	8	no	—
103	#500	—	2	4	no	—
104	#499	—	2	4	no	—
105	C₂×#107	C₂²×C₅	1	1	yes	characteristic
106	C₂×#107	C₂²×C₅	1	2	yes	—
107	C₂×#107	C₂²×C₅	1	4	yes	—
108	C₂×#107	C₂²×C₅	1	1	yes	characteristic
109	C₂×#107	C₂²×C₅	1	2	yes	—
110	C₂×#106	C₂²×C₅	1	4	yes	—
111	C₂×#106	C₂²×C₅	1	2	yes	—
112	C₂×#97	C₂²×C₅	1	1	yes	characteristic
113	C₂×#97	C₂²×C₅	1	2	yes	—
114	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
116	#398	—	2	4	no	—
117	#395	—	2	4	no	—
118	#384	C₂²×C₅	1	4	yes	—
119	#382	C₂²×C₅	1	4	yes	—
120	#376	C₂²×C₅	1	1	yes	characteristic
121	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
122	C₅×Q₈	—	8	8	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	2	no	—
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	4	8	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	#127	1	2	yes	—
134	C₂×C₄×C₅	#127	1	2	yes	—
135	C₂×C₄×C₅	C₄⋊Q₈	1	8	yes	—
136	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
139	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
140	C₅×C₈	C₄⋊Q₈	1	4	yes	—
141	C₅×C₈	C₄⋊Q₈	1	4	yes	—
142	C₅×C₈	C₄⋊Q₈	1	8	yes	—
143	C₂²×Q₈	—	2	2	no	—
144	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
145	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₅	1	2	yes	—
147	C₄⋊Q₈	—	2	4	no	—
148	C₄⋊Q₈	—	2	4	no	—
149	#125	—	2	4	no	—
150	#125	—	2	4	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	8	no	—
155	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
156	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
157	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
158	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
159	C₂×C₄²	C₂³×C₅	1	2	yes	—
160	#107	—	2	2	no	—
161	#107	—	2	2	no	—
162	#107	—	2	4	no	—
163	#107	—	2	4	no	—
164	#107	—	2	8	no	—
165	#106	—	2	4	no	—
166	#106	—	2	8	no	—
167	#97	C₅×D₄	1	4	yes	—
168	#97	C₅×Q₈	1	8	yes	complemented
169	C₄×C₈	C₅×D₄	1	4	yes	—
170	C₄×C₈	C₅×Q₈	1	8	yes	complemented
171	#95	—	2	8	no	—
172	#95	C₅×D₄	1	2	yes	—
173	#95	C₅×D₄	1	2	yes	—
174	C₂²×C₅	C₂×#127	1	1	yes	characteristic
175	C₂²×C₅	C₂×C₄⋊Q₈	1	2	yes	—
176	C₂²×C₅	#576	1	2	yes	—
177	C₂²×C₅	#578	1	2	yes	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	—	2	2	no	—
180	C₄×C₅	—	2	8	no	—
181	C₄×C₅	—	8	2	no	—
182	C₄×C₅	—	8	4	no	—
183	C₄×C₅	—	8	2	no	—
184	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
185	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
186	C₄×C₅	C₂×C₄⋊Q₈	1	2	yes	—
187	C₂×Q₈	—	4	4	no	—
188	C₂×Q₈	—	8	4	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
195	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
196	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
197	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
198	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
199	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
200	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
201	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
202	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	—	8	8	no	—
213	C₄²	C₂×C₅×D₄	1	2	yes	—
214	C₄²	C₂×C₅×D₄	1	2	yes	—
215	C₄²	C₂×C₅×Q₈	1	4	yes	—
216	C₄²	C₂×C₅×Q₈	1	4	yes	—
217	C₂×C₅	#2868	1	1	yes	characteristic
218	C₂×C₅	#3155	1	1	yes	characteristic
219	C₂×C₅	#3156	1	1	yes	characteristic
220	C₂×C₅	#3164	1	4	yes	—
221	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
222	Q₈	—	8	8	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	8	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	C₅×#127	1	2	yes	—
234	C₂×C₄	C₅×#127	1	2	yes	—
235	C₂×C₄	C₅×C₄⋊Q₈	1	8	yes	—
236	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
237	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
239	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
240	C₈	C₅×C₄⋊Q₈	1	4	yes	—
241	C₈	C₅×C₄⋊Q₈	1	4	yes	—
242	C₈	C₅×C₄⋊Q₈	1	8	yes	—
243	C₅	#33319	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²	C₂×C₅×#127	1	1	yes	characteristic
245	C₂²	C₂×C₅×C₄⋊Q₈	1	2	yes	—
246	C₂²	C₅×#576	1	2	yes	—
247	C₂²	C₅×#578	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	8	no	—
251	C₄	—	8	4	no	—
252	C₄	—	8	2	no	—
253	C₄	—	8	2	no	—
254	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
255	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
256	C₄	C₂×C₅×C₄⋊Q₈	1	2	yes	—
257	C₂	C₅×#2868	1	1	yes	characteristic
258	C₂	C₅×#3155	1	1	yes	characteristic
259	C₂	C₅×#3156	1	1	yes	characteristic
260	C₂	C₅×#3164	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#376 ◁ G

Complements

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

N	H ≅ G/N	type
30. #33319	C₅	direct (N × H)
56. C₅×#97	Q₈	semidirect
58. C₄×C₅×C₈	Q₈	semidirect
168. #97	C₅×Q₈	semidirect
170. C₄×C₈	C₅×Q₈	semidirect
243. C₅	#33319	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×C₅×#501	2
C₂×C₅×#500	2
C₂×C₅×#499	2
C₅×#2660	2
C₅×#1847	2
C₅×#1846	2
C₅×#1845	2
C₅×#1726	2
#33319	5

Subgroups — C5×#33319 #11956039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33319 #11956039