Subgroups — C₅×#17989 #11940709

Order 1280.

624 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅×#2981	C₂	1	2	yes	maximal, complemented
3	C₅×#1586	C₂	1	4	yes	maximal, complemented
4	C₅×#1581	C₂	1	4	yes	maximal, complemented
5	C₅×#1568	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1496	C₂	1	2	yes	maximal, complemented
7	C₅×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#532	C₂²	1	2	yes	—
9	C₅×#500	C₄	1	1	yes	characteristic
10	C₅×#493	C₄	1	1	yes	characteristic
11	C₅×#486	C₄	1	2	yes	—
12	C₅×#466	C₄	1	2	yes	—
13	C₅×#466	C₄	1	2	yes	—
14	C₄×C₅×D₈	C₂²	1	2	yes	—
15	C₄×C₅×D₈	C₂²	1	2	yes	—
16	C₅×#436	C₂²	1	4	yes	—
17	C₅×#435	—	2	4	no	—
18	C₅×C₈×D₄	C₂²	1	4	yes	—
19	C₅×#432	C₂²	1	2	yes	—
20	C₅×#431	C₂²	1	1	yes	characteristic
21	C₅×#431	C₂²	1	1	yes	characteristic
22	C₅×#408	—	2	4	no	—
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×#335	C₂²	1	2	yes	—
25	C₅×#334	C₂²	1	2	yes	—
26	C₅×D₄⋊C₈	C₂²	1	8	yes	—
27	C₅×#322	C₂²	1	2	yes	—
28	C₅×#322	C₂²	1	2	yes	—
29	#17989	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×#41	—	2	2	no	—
31	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
32	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
33	C₂×C₅×M₁₆	—	2	4	no	—
34	C₂²×C₅×C₈	—	2	4	no	—
35	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₅×#127	C₂×C₄	1	2	yes	—
39	C₅×#124	—	2	2	no	—
40	C₅×#121	—	2	2	no	—
41	C₅×#121	C₂×C₄	1	4	yes	—
42	C₄×C₅×D₄	—	2	2	no	—
43	C₄×C₅×D₄	C₂³	1	4	yes	—
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×#107	C₂×C₄	1	2	yes	—
46	C₅×#107	C₂×C₄	1	2	yes	—
47	C₅×C₄⋊C₈	—	2	1	no	—
48	C₅×C₄⋊C₈	D₄	1	2	yes	—
49	C₅×C₄⋊C₈	C₂³	1	4	yes	—
50	C₅×#102	C₂×C₄	1	4	yes	—
51	C₅×#102	C₂×C₄	1	4	yes	—
52	C₅×C₂²⋊C₈	—	2	2	no	—
53	C₅×C₂²⋊C₈	—	2	1	no	—
54	C₅×C₂²⋊C₈	—	2	4	no	—
55	C₅×C₂²⋊C₈	—	2	4	no	—
56	C₅×C₂²⋊C₈	D₄	1	2	yes	—
57	C₅×#97	—	2	2	no	—
58	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₅×C₈	C₂³	1	2	yes	—
61	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂³	1	2	yes	—
64	#3169	C₂×C₅	1	1	yes	characteristic
65	#2981	C₂×C₅	1	2	yes	complemented
66	#1586	C₂×C₅	1	4	yes	complemented
67	#1581	C₂×C₅	1	4	yes	complemented
68	#1568	C₂×C₅	1	1	yes	characteristic, complemented
69	#1496	C₂×C₅	1	2	yes	complemented
70	#1452	C₂×C₅	1	1	yes	characteristic, complemented
71	C₅×#41	—	8	4	no	—
72	C₂×C₅×Q₈	—	2	2	no	—
73	C₂×C₅×D₄	—	2	4	no	—
74	C₂×C₅×D₄	—	2	2	no	—
75	C₂×C₅×D₄	—	2	2	no	—
76	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₅	—	2	4	no	—
78	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	#41	1	1	yes	characteristic
81	C₅×D₈	—	4	2	no	—
82	C₅×D₈	—	4	2	no	—
83	C₅×M₁₆	—	8	4	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	4	1	no	—
87	C₂×C₅×C₈	—	4	1	no	—
88	C₂×C₅×C₈	—	8	4	no	—
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
96	C₅×C₂²⋊C₄	—	2	4	no	—
97	C₅×C₂²⋊C₄	—	2	4	no	—
98	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₅	#41	1	1	yes	characteristic
101	C₄²×C₅	C₂⁴	1	1	yes	characteristic
102	#532	C₂²×C₅	1	2	yes	—
103	#500	C₄×C₅	1	1	yes	characteristic
104	#493	C₄×C₅	1	1	yes	characteristic
105	#486	C₄×C₅	1	2	yes	—
106	#466	C₄×C₅	1	2	yes	—
107	#466	C₄×C₅	1	2	yes	—
108	C₄×D₈	C₂²×C₅	1	2	yes	—
109	C₄×D₈	C₂²×C₅	1	2	yes	—
110	#436	C₂²×C₅	1	4	yes	—
111	#435	—	2	4	no	—
112	C₈×D₄	C₂²×C₅	1	4	yes	—
113	#432	C₂²×C₅	1	2	yes	—
114	#431	C₂²×C₅	1	1	yes	characteristic
115	#431	C₂²×C₅	1	1	yes	characteristic
116	#408	—	2	4	no	—
117	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
118	#335	C₂²×C₅	1	2	yes	—
119	#334	C₂²×C₅	1	2	yes	—
120	D₄⋊C₈	C₂²×C₅	1	8	yes	—
121	#322	C₂²×C₅	1	2	yes	—
122	#322	C₂²×C₅	1	2	yes	—
123	C₂³×C₅	—	2	4	no	—
124	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
125	C₅×Q₈	—	8	2	no	—
126	C₅×D₄	—	2	4	no	—
127	C₅×D₄	—	2	4	no	—
128	C₅×D₄	—	8	2	no	—
129	C₅×D₄	—	8	2	no	—
130	C₅×D₄	—	8	4	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	4	4	no	—
137	C₂×C₄×C₅	—	4	4	no	—
138	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
144	C₅×C₈	—	2	1	no	—
145	C₅×C₈	—	2	1	no	—
146	C₅×C₈	—	2	1	no	—
147	C₅×C₈	—	2	1	no	—
148	C₅×C₈	—	2	4	no	—
149	C₅×C₈	—	4	2	no	—
150	C₅×C₈	—	4	4	no	—
151	C₂×#41	—	2	2	no	—
152	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
153	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
154	C₂×M₁₆	—	2	4	no	—
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
158	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
159	#127	C₂×C₄×C₅	1	2	yes	—
160	#124	—	2	2	no	—
161	#121	—	2	2	no	—
162	#121	C₂×C₄×C₅	1	4	yes	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	C₂³×C₅	1	4	yes	—
165	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
166	#107	C₂×C₄×C₅	1	2	yes	—
167	#107	C₂×C₄×C₅	1	2	yes	—
168	C₄⋊C₈	—	2	1	no	—
169	C₄⋊C₈	C₅×D₄	1	2	yes	—
170	C₄⋊C₈	C₂³×C₅	1	4	yes	—
171	#102	C₂×C₄×C₅	1	4	yes	—
172	#102	C₂×C₄×C₅	1	4	yes	—
173	C₂²⋊C₈	—	2	2	no	—
174	C₂²⋊C₈	—	2	1	no	—
175	C₂²⋊C₈	—	2	4	no	—
176	C₂²⋊C₈	—	2	4	no	—
177	C₂²⋊C₈	C₅×D₄	1	2	yes	—
178	#97	—	2	2	no	—
179	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
180	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
181	C₄×C₈	C₂³×C₅	1	2	yes	—
182	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
183	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
184	C₄×C₈	C₂³×C₅	1	2	yes	—
185	C₂²×C₅	—	2	1	no	—
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	—	2	1	no	—
188	C₂²×C₅	—	4	4	no	—
189	C₂²×C₅	—	4	4	no	—
190	C₂²×C₅	—	4	4	no	—
191	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	4	4	no	—
197	C₄×C₅	#568	1	1	yes	characteristic
198	C₄×C₅	#568	1	1	yes	characteristic
199	C₄×C₅	#576	1	2	yes	—
200	#41	—	8	4	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂²×C₄	C₅×#41	1	1	yes	characteristic
210	D₈	—	4	2	no	—
211	D₈	—	4	2	no	—
212	M₁₆	—	8	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	8	4	no	—
218	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
219	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
220	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
221	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
222	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
228	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
229	C₄²	C₅×#41	1	1	yes	characteristic
230	C₄²	C₂⁴×C₅	1	1	yes	characteristic
231	C₂×C₅	—	4	1	no	—
232	C₂×C₅	—	8	4	no	—
233	C₂×C₅	#2932	1	1	yes	characteristic
234	C₂×C₅	#2948	1	1	yes	characteristic
235	C₂×C₅	#2957	1	1	yes	characteristic
236	C₂³	—	2	4	no	—
237	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
238	Q₈	—	8	2	no	—
239	D₄	—	2	4	no	—
240	D₄	—	2	4	no	—
241	D₄	—	8	2	no	—
242	D₄	—	8	2	no	—
243	D₄	—	8	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
252	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
253	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
254	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
256	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	4	no	—
262	C₈	—	4	2	no	—
263	C₈	—	4	4	no	—
264	C₅	#17989	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	C₅×#568	1	1	yes	characteristic
278	C₄	C₅×#568	1	1	yes	characteristic
279	C₄	C₅×#576	1	2	yes	—
280	C₂	—	4	1	no	—
281	C₂	—	8	4	no	—
282	C₂	C₅×#2932	1	1	yes	characteristic
283	C₂	C₅×#2948	1	1	yes	characteristic
284	C₂	C₅×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₅×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2981	C₂	semidirect
3. C₅×#1586	C₂	semidirect
4. C₅×#1581	C₂	semidirect
5. C₅×#1568	C₂	semidirect
6. C₅×#1496	C₂	semidirect
7. C₅×#1452	C₂	semidirect
29. #17989	C₅	direct (N × H)
65. #2981	C₂×C₅	semidirect
66. #1586	C₂×C₅	semidirect
67. #1581	C₂×C₅	semidirect
68. #1568	C₂×C₅	semidirect
69. #1496	C₂×C₅	semidirect
70. #1452	C₂×C₅	semidirect
264. C₅	#17989	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×#3169	2
C₅×#2981	2
C₅×#1586	2
C₅×#1581	2
C₅×#1568	2
C₅×#1496	2
C₅×#1452	2
#17989	5

Subgroups — C5×#17989 #11940709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17989 #11940709