Subgroups — C₅×#21722 #11944442

Order 1280.

734 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×#499	C₂	1	1	yes	maximal, characteristic
2	C₅×#2330	C₂	1	2	yes	maximal
3	C₅×#1992	C₂	1	2	yes	maximal
4	C₅×#1943	C₂	1	2	yes	maximal
5	C₅×#1941	C₂	1	2	yes	maximal
6	C₅×#1844	C₂	1	1	yes	maximal, characteristic
7	C₅×#1776	C₂	1	2	yes	maximal
8	C₄×C₅×#106	C₂	1	2	yes	maximal
9	C₅×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×#125	C₂²	1	2	yes	—
11	C₅×#499	C₄	1	8	yes	complemented
12	C₂×C₅×#107	C₂²	1	2	yes	—
13	C₂×C₅×#106	C₂²	1	2	yes	—
14	C₂×C₅×#106	C₂²	1	2	yes	—
15	C₂×C₅×#106	C₂²	1	2	yes	—
16	C₂×C₅×#97	C₂²	1	1	yes	characteristic
17	C₂×C₅×#97	C₂²	1	1	yes	characteristic
18	C₂×C₅×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×#389	—	2	4	no	—
23	C₅×#389	C₂²	1	4	yes	—
24	C₅×#387	—	2	4	no	—
25	C₅×#384	C₂²	1	2	yes	—
26	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
27	C₅×#376	C₂²	1	1	yes	characteristic
28	C₅×#340	C₂²	1	4	yes	—
29	C₅×#340	C₂²	1	4	yes	—
30	#21722	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₅×#125	—	2	4	no	—
36	C₅×#125	C₂×C₄	1	8	yes	—
37	C₂×C₅×C₄⋊C₄	—	2	4	no	—
38	C₂×C₅×C₄⋊C₄	—	2	4	no	—
39	C₂×C₅×C₄⋊C₄	—	2	4	no	—
40	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
42	C₂×C₄²×C₅	—	2	4	no	—
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₅×#107	C₂×C₄	1	8	yes	—
47	C₅×#106	—	2	4	no	—
48	C₅×#106	—	2	4	no	—
49	C₅×#106	C₂×C₄	1	8	yes	—
50	C₅×#97	—	2	2	no	—
51	C₅×#97	—	2	2	no	—
52	C₅×#97	Q₈	1	4	yes	—
53	C₄×C₅×C₈	—	2	2	no	—
54	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
55	C₄×C₅×C₈	Q₈	1	4	yes	—
56	C₅×#95	—	2	4	no	—
57	C₅×#95	—	2	4	no	—
58	C₅×#95	D₄	1	4	yes	—
59	C₂×#499	C₂×C₅	1	1	yes	characteristic
60	#2330	C₂×C₅	1	2	yes	—
61	#1992	C₂×C₅	1	2	yes	—
62	#1943	C₂×C₅	1	2	yes	—
63	#1941	C₂×C₅	1	2	yes	—
64	#1844	C₂×C₅	1	1	yes	characteristic
65	#1776	C₂×C₅	1	2	yes	—
66	C₄×#106	C₂×C₅	1	2	yes	—
67	#1726	C₂×C₅	1	1	yes	characteristic
68	C₂²×C₄×C₅	—	2	4	no	—
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
74	C₂×C₅×C₈	—	2	1	no	—
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	—	2	1	no	—
78	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
81	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
82	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
83	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
84	C₂×C₅×C₈	#41	1	2	yes	—
85	C₂×C₅×C₈	#41	1	2	yes	—
86	C₅×C₄⋊C₄	—	2	16	no	—
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	—	2	4	no	—
89	C₅×C₄⋊C₄	—	2	4	no	—
90	C₅×C₄⋊C₄	—	8	4	no	—
91	C₅×C₄⋊C₄	—	8	4	no	—
92	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
93	C₄²×C₅	—	2	1	no	—
94	C₄²×C₅	—	2	1	no	—
95	C₄²×C₅	—	8	4	no	—
96	C₄²×C₅	C₂²×C₄	1	2	yes	—
97	C₄²×C₅	C₂×Q₈	1	2	yes	—
98	C₄²×C₅	C₂×Q₈	1	2	yes	—
99	C₄²×C₅	#41	1	2	yes	—
100	C₂×#125	C₂²×C₅	1	2	yes	—
101	#499	C₄×C₅	1	8	yes	complemented
102	C₂×#107	C₂²×C₅	1	2	yes	—
103	C₂×#106	C₂²×C₅	1	2	yes	—
104	C₂×#106	C₂²×C₅	1	2	yes	—
105	C₂×#106	C₂²×C₅	1	2	yes	—
106	C₂×#97	C₂²×C₅	1	1	yes	characteristic
107	C₂×#97	C₂²×C₅	1	1	yes	characteristic
108	C₂×#97	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
112	#389	—	2	4	no	—
113	#389	C₂²×C₅	1	4	yes	—
114	#387	—	2	4	no	—
115	#384	C₂²×C₅	1	2	yes	—
116	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
117	#376	C₂²×C₅	1	1	yes	characteristic
118	#340	C₂²×C₅	1	4	yes	—
119	#340	C₂²×C₅	1	4	yes	—
120	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	8	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	—	4	4	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
132	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
133	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
139	C₅×C₈	—	2	2	no	—
140	C₅×C₈	—	2	2	no	—
141	C₅×C₈	C₄×Q₈	1	4	yes	—
142	C₅×C₈	C₄×Q₈	1	4	yes	—
143	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
145	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
147	#125	—	2	4	no	—
148	#125	C₂×C₄×C₅	1	8	yes	—
149	C₂×C₄⋊C₄	—	2	4	no	—
150	C₂×C₄⋊C₄	—	2	4	no	—
151	C₂×C₄⋊C₄	—	2	4	no	—
152	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
153	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
154	C₂×C₄²	—	2	4	no	—
155	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
156	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
157	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
158	#107	C₂×C₄×C₅	1	8	yes	—
159	#106	—	2	4	no	—
160	#106	—	2	4	no	—
161	#106	C₂×C₄×C₅	1	8	yes	—
162	#97	—	2	2	no	—
163	#97	—	2	2	no	—
164	#97	C₅×Q₈	1	4	yes	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
167	C₄×C₈	C₅×Q₈	1	4	yes	—
168	#95	—	2	4	no	—
169	#95	—	2	4	no	—
170	#95	C₅×D₄	1	4	yes	—
171	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
173	C₂²×C₅	#532	1	1	yes	characteristic
174	C₂²×C₅	#577	1	4	yes	—
175	C₄×C₅	—	2	4	no	—
176	C₄×C₅	—	2	4	no	—
177	C₄×C₅	—	4	2	no	—
178	C₄×C₅	—	4	8	no	—
179	C₄×C₅	—	8	4	no	—
180	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
181	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
182	C₄×C₅	#533	1	1	yes	characteristic
183	C₄×C₅	#533	1	1	yes	characteristic
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
187	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
188	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
189	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
195	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
196	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
197	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
198	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
199	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
200	C₂×C₈	C₅×#41	1	2	yes	—
201	C₂×C₈	C₅×#41	1	2	yes	—
202	C₄⋊C₄	—	2	16	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	8	4	no	—
207	C₄⋊C₄	—	8	4	no	—
208	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	8	4	no	—
212	C₄²	C₂²×C₄×C₅	1	2	yes	—
213	C₄²	C₂×C₅×Q₈	1	2	yes	—
214	C₄²	C₂×C₅×Q₈	1	2	yes	—
215	C₄²	C₅×#41	1	2	yes	—
216	C₂×C₅	C₄×C₄⋊Q₈	1	1	yes	characteristic
217	C₂×C₅	#2952	1	2	yes	—
218	C₂×C₅	#3163	1	2	yes	—
219	C₂×C₅	#3174	1	2	yes	—
220	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
232	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
233	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
234	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
236	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
237	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
238	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	2	2	no	—
241	C₈	C₄×C₅×Q₈	1	4	yes	—
242	C₈	C₄×C₅×Q₈	1	4	yes	—
243	C₅	#21722	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
245	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
246	C₂²	C₅×#532	1	1	yes	characteristic
247	C₂²	C₅×#577	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	4	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	8	no	—
252	C₄	—	8	4	no	—
253	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
254	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
255	C₄	C₅×#533	1	1	yes	characteristic
256	C₄	C₅×#533	1	1	yes	characteristic
257	C₂	C₄×C₅×C₄⋊Q₈	1	1	yes	characteristic
258	C₂	C₅×#2952	1	2	yes	—
259	C₂	C₅×#3163	1	2	yes	—
260	C₂	C₅×#3174	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂×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
11. C₅×#499	C₄	semidirect
30. #21722	C₅	direct (N × H)
101. #499	C₄×C₅	semidirect
243. C₅	#21722	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×#499	2
C₅×#2330	2
C₅×#1992	2
C₅×#1943	2
C₅×#1941	2
C₅×#1844	2
C₅×#1776	2
C₄×C₅×#106	2
C₅×#1726	2
#21722	5

Subgroups — C5×#21722 #11944442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21722 #11944442