Subgroups — C₅×#21143 #11943863

Order 1280.

1038 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#424	C₂	1	2	yes	maximal
2	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#2285	C₂	1	1	yes	maximal, characteristic
5	C₅×#1939	C₂	1	8	yes	maximal, complemented
6	C₅×#1729	C₂	1	2	yes	maximal
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#117	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₅×#424	—	2	8	no	—
13	C₅×#423	—	2	4	no	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	8	yes	—
16	C₂×C₄×C₅×C₈	—	2	4	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
18	C₅×#384	C₄	1	2	yes	—
19	C₅×#382	C₄	1	2	yes	—
20	C₅×#380	C₄	1	4	yes	—
21	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
22	C₅×#336	C₂²	1	8	yes	—
23	#21143	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₅×M₁₆	—	2	4	no	—
27	C₂×C₅×M₁₆	—	2	4	no	—
28	C₂×C₅×M₁₆	—	2	4	no	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	—	2	2	no	—
33	C₂²×C₅×C₈	D₄	1	4	yes	—
34	C₂²×C₅×C₈	Q₈	1	4	yes	—
35	C₂²×C₅×C₈	C₂³	1	4	yes	—
36	C₂²×C₅×C₈	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₅×#117	—	2	4	no	—
39	C₅×#117	—	2	4	no	—
40	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
41	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
42	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	D₄	1	4	yes	—
46	C₂×C₄²×C₅	Q₈	1	4	yes	—
47	C₂×C₄²×C₅	C₂³	1	4	yes	—
48	C₂×C₄²×C₅	C₂³	1	2	yes	—
49	C₅×C₄⋊C₈	—	2	16	no	—
50	C₅×C₄⋊C₈	—	2	8	no	—
51	C₄×C₅×C₈	—	2	8	no	—
52	C₅×#95	C₂×C₄	1	8	yes	—
53	C₂×#424	C₂×C₅	1	2	yes	—
54	C₂×#423	C₂×C₅	1	1	yes	characteristic
55	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
56	#2285	C₂×C₅	1	1	yes	characteristic
57	#1939	C₂×C₅	1	8	yes	complemented
58	#1729	C₂×C₅	1	2	yes	—
59	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	—	2	8	no	—
61	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₅	#41	1	2	yes	—
71	C₂²×C₄×C₅	#41	1	2	yes	—
72	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
73	C₅×M₁₆	—	4	8	no	—
74	C₂×C₅×C₈	—	2	8	no	—
75	C₂×C₅×C₈	—	2	8	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
84	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
85	C₅×C₄⋊C₄	—	2	8	no	—
86	C₅×C₄⋊C₄	—	2	16	no	—
87	C₅×C₂²⋊C₄	—	2	8	no	—
88	C₄²×C₅	—	2	4	no	—
89	C₄²×C₅	—	2	4	no	—
90	C₄²×C₅	C₄⋊C₄	1	4	yes	—
91	C₄²×C₅	C₄⋊C₄	1	4	yes	—
92	C₄²×C₅	C₂×D₄	1	4	yes	—
93	C₄²×C₅	C₂×Q₈	1	4	yes	—
94	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
95	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
96	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
97	C₂×#117	C₂²×C₅	1	2	yes	—
98	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
99	#424	—	2	8	no	—
100	#423	—	2	4	no	—
101	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
102	C₂×C₄⋊C₈	C₂²×C₅	1	8	yes	—
103	C₂×C₄×C₈	—	2	4	no	—
104	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
105	#384	C₄×C₅	1	2	yes	—
106	#382	C₄×C₅	1	2	yes	—
107	#380	C₄×C₅	1	4	yes	—
108	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
109	#336	C₂²×C₅	1	8	yes	—
110	C₂³×C₅	—	2	4	no	—
111	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
112	C₂³×C₅	C₄×D₄	1	2	yes	—
113	C₂³×C₅	C₄×Q₈	1	2	yes	—
114	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	—	2	4	no	—
116	C₂×C₄×C₅	—	2	16	no	—
117	C₂×C₄×C₅	—	2	8	no	—
118	C₂×C₄×C₅	—	2	16	no	—
119	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₅	#121	1	4	yes	—
125	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
126	C₂×C₄×C₅	#125	1	4	yes	—
127	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
128	C₂×C₄×C₅	#131	1	8	yes	—
129	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
135	C₅×C₈	—	2	4	no	—
136	C₅×C₈	—	2	4	no	—
137	C₅×C₈	—	4	4	no	—
138	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
139	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	—	2	4	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	C₅×D₄	1	4	yes	—
148	C₂²×C₈	C₅×Q₈	1	4	yes	—
149	C₂²×C₈	C₂³×C₅	1	4	yes	—
150	C₂²×C₈	C₂³×C₅	1	2	yes	—
151	C₂²×C₈	C₂³×C₅	1	2	yes	—
152	#117	—	2	4	no	—
153	#117	—	2	4	no	—
154	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
155	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
156	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
157	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
158	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
159	C₂×C₄²	C₅×D₄	1	4	yes	—
160	C₂×C₄²	C₅×Q₈	1	4	yes	—
161	C₂×C₄²	C₂³×C₅	1	4	yes	—
162	C₂×C₄²	C₂³×C₅	1	2	yes	—
163	C₄⋊C₈	—	2	16	no	—
164	C₄⋊C₈	—	2	8	no	—
165	C₄×C₈	—	2	8	no	—
166	#95	C₂×C₄×C₅	1	8	yes	—
167	C₂²×C₅	—	2	4	no	—
168	C₂²×C₅	—	2	4	no	—
169	C₂²×C₅	—	2	4	no	—
170	C₂²×C₅	#389	1	4	yes	—
171	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
174	C₂²×C₅	C₂×#131	1	4	yes	—
175	C₄×C₅	—	2	4	no	—
176	C₄×C₅	—	4	8	no	—
177	C₄×C₅	#389	1	4	yes	—
178	C₄×C₅	#424	1	8	yes	complemented
179	C₄×C₅	C₂×#121	1	1	yes	characteristic
180	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
181	C₄×C₅	C₂×#125	1	1	yes	characteristic
182	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
183	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	C₅×C₄⋊C₄	1	8	yes	—
186	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
187	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
188	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
189	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
190	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
191	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
192	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
193	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
194	C₂²×C₄	C₅×#41	1	2	yes	—
195	C₂²×C₄	C₅×#41	1	2	yes	—
196	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
197	M₁₆	—	4	8	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	—	2	8	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
208	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
209	C₄⋊C₄	—	2	8	no	—
210	C₄⋊C₄	—	2	16	no	—
211	C₂²⋊C₄	—	2	8	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	2	4	no	—
214	C₄²	C₅×C₄⋊C₄	1	4	yes	—
215	C₄²	C₅×C₄⋊C₄	1	4	yes	—
216	C₄²	C₂×C₅×D₄	1	4	yes	—
217	C₄²	C₂×C₅×Q₈	1	4	yes	—
218	C₂×C₅	—	2	4	no	—
219	C₂×C₅	C₂×#389	1	1	yes	characteristic
220	C₂×C₅	C₂×#424	1	2	yes	—
221	C₂×C₅	#2931	1	2	yes	—
222	C₂×C₅	#2960	1	2	yes	—
223	C₂³	—	2	4	no	—
224	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
225	C₂³	C₄×C₅×D₄	1	2	yes	—
226	C₂³	C₄×C₅×Q₈	1	2	yes	—
227	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	16	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
233	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
234	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
235	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
236	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
237	C₂×C₄	C₅×#121	1	4	yes	—
238	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
239	C₂×C₄	C₅×#125	1	4	yes	—
240	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
241	C₂×C₄	C₅×#131	1	8	yes	—
242	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
246	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
247	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
248	C₈	—	2	4	no	—
249	C₈	—	2	4	no	—
250	C₈	—	4	4	no	—
251	C₅	#21143	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	C₅×#389	1	4	yes	—
256	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
257	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
258	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
259	C₂²	C₂×C₅×#131	1	4	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	C₅×#389	1	4	yes	—
263	C₄	C₅×#424	1	8	yes	complemented
264	C₄	C₂×C₅×#121	1	1	yes	characteristic
265	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
266	C₄	C₂×C₅×#125	1	1	yes	characteristic
267	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
268	C₂	—	2	4	no	—
269	C₂	C₂×C₅×#389	1	1	yes	characteristic
270	C₂	C₂×C₅×#424	1	2	yes	—
271	C₂	C₅×#2931	1	2	yes	—
272	C₂	C₅×#2960	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₅×#1939	C₂	semidirect
23. #21143	C₅	direct (N × H)
57. #1939	C₂×C₅	semidirect
178. C₄×C₅	#424	semidirect
251. C₅	#21143	direct (N × H)
263. C₄	C₅×#424	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#424	2
C₂×C₅×#423	2
C₂²×C₄×C₅×C₈	2
C₅×#2285	2
C₅×#1939	2
C₅×#1729	2
#21143	5

Subgroups — C5×#21143 #11943863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21143 #11943863