Subgroups — C₅×#20793 #11943513

Order 1280.

1114 conjugacy classes of subgroups.

The subgroup lattice has 280 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	4	yes	maximal
2	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
5	C₅×#1843	C₂	1	4	yes	maximal, complemented
6	C₅×#1842	C₂	1	4	yes	maximal, complemented
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₅×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₅×#424	—	2	16	no	—
12	C₅×#423	—	2	4	no	—
13	C₂×C₅×C₄⋊C₈	—	2	4	no	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊C₈	C₂²	1	16	yes	—
17	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
19	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
20	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
21	C₄×C₅×C₂²⋊C₄	C₄	1	2	yes	—
22	C₄×C₅×C₂²⋊C₄	C₄	1	2	yes	—
23	C₅×#376	C₂²	1	2	yes	—
24	C₄³×C₅	C₂²	1	2	yes	—
25	#20793	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₅×M₁₆	—	2	4	no	—
30	C₂×C₅×M₁₆	—	2	8	no	—
31	C₂²×C₅×C₈	—	2	4	no	—
32	C₂²×C₅×C₈	—	2	8	no	—
33	C₂²×C₅×C₈	C₂³	1	4	yes	—
34	C₂²×C₅×C₈	C₂³	1	4	yes	—
35	C₅×#117	D₄	1	8	yes	—
36	C₅×#117	Q₈	1	8	yes	—
37	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	4	yes	—
44	C₂×C₄²×C₅	Q₈	1	4	yes	—
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	4	yes	—
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₅×C₄⋊C₈	—	2	4	no	—
49	C₅×C₄⋊C₈	—	2	4	no	—
50	C₅×C₄⋊C₈	—	2	4	no	—
51	C₅×C₄⋊C₈	—	2	4	no	—
52	C₅×C₄⋊C₈	—	2	32	no	—
53	C₅×#95	C₂×C₄	1	4	yes	—
54	C₅×#95	C₂×C₄	1	4	yes	—
55	C₂×#424	C₂×C₅	1	4	yes	—
56	C₂×#423	C₂×C₅	1	1	yes	characteristic
57	C₂²×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
58	C₄×#117	C₂×C₅	1	1	yes	characteristic
59	#1843	C₂×C₅	1	4	yes	complemented
60	#1842	C₂×C₅	1	4	yes	complemented
61	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	4	2	no	—
64	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
77	C₅×M₁₆	—	4	8	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	2	8	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	8	no	—
82	C₂×C₅×C₈	—	4	8	no	—
83	C₅×C₄⋊C₄	C₄⋊C₄	1	16	yes	—
84	C₅×C₂²⋊C₄	C₄⋊C₄	1	16	yes	—
85	C₄²×C₅	—	2	8	no	—
86	C₄²×C₅	—	4	4	no	—
87	C₄²×C₅	C₄⋊C₄	1	4	yes	—
88	C₄²×C₅	C₄⋊C₄	1	4	yes	—
89	C₄²×C₅	C₂×D₄	1	2	yes	—
90	C₄²×C₅	C₂×D₄	1	2	yes	—
91	C₄²×C₅	C₂×D₄	1	8	yes	—
92	C₄²×C₅	C₂×Q₈	1	2	yes	—
93	C₄²×C₅	C₂×Q₈	1	2	yes	—
94	C₄²×C₅	C₂×Q₈	1	8	yes	—
95	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
96	C₂³×C₈	C₂²×C₅	1	2	yes	—
97	C₂×#117	C₂²×C₅	1	2	yes	—
98	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
99	#424	—	2	16	no	—
100	#423	—	2	4	no	—
101	C₂×C₄⋊C₈	—	2	4	no	—
102	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
103	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
104	C₂×C₄⋊C₈	C₂²×C₅	1	16	yes	—
105	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
106	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
107	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
108	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
109	C₄×C₂²⋊C₄	C₄×C₅	1	2	yes	—
110	C₄×C₂²⋊C₄	C₄×C₅	1	2	yes	—
111	#376	C₂²×C₅	1	2	yes	—
112	C₄³	C₂²×C₅	1	2	yes	—
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	—	2	2	no	—
115	C₂³×C₅	C₂×C₄⋊C₄	1	4	yes	—
116	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
117	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅	—	2	8	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	8	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	8	no	—
124	C₂×C₄×C₅	—	4	4	no	—
125	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₅	C₂×C₄⋊C₄	1	16	yes	—
129	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
131	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₅	#127	1	4	yes	—
133	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
134	C₂×C₄×C₅	C₄⋊Q₈	1	8	yes	—
135	C₂×C₄×C₅	#131	1	4	yes	—
136	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
140	C₅×C₈	—	4	4	no	—
141	C₅×C₈	—	4	4	no	—
142	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
143	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
144	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
145	C₂×M₁₆	—	2	4	no	—
146	C₂×M₁₆	—	2	8	no	—
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	—	2	8	no	—
149	C₂²×C₈	C₂³×C₅	1	4	yes	—
150	C₂²×C₈	C₂³×C₅	1	4	yes	—
151	#117	C₅×D₄	1	8	yes	—
152	#117	C₅×Q₈	1	8	yes	—
153	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
154	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
155	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
156	C₂×C₄²	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	1	yes	characteristic
163	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
164	C₄⋊C₈	—	2	4	no	—
165	C₄⋊C₈	—	2	4	no	—
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	2	32	no	—
169	#95	C₂×C₄×C₅	1	4	yes	—
170	#95	C₂×C₄×C₅	1	4	yes	—
171	C₂²×C₅	—	2	2	no	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	2	8	no	—
174	C₂²×C₅	#384	1	4	yes	—
175	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×C₅	C₂²×C₄⋊C₄	1	2	yes	—
177	C₂²×C₅	C₂×#131	1	2	yes	—
178	C₂²×C₅	#568	1	2	yes	—
179	C₄×C₅	—	2	2	no	—
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	16	no	—
183	C₄×C₅	#384	1	4	yes	—
184	C₄×C₅	#424	1	4	yes	complemented
185	C₄×C₅	C₂×#127	1	1	yes	characteristic
186	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
187	C₄×C₅	C₂×C₄⋊Q₈	1	2	yes	—
188	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
192	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
193	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
194	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
196	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
200	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
201	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
202	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
203	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
204	M₁₆	—	4	8	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	8	no	—
209	C₂×C₈	—	4	8	no	—
210	C₄⋊C₄	C₅×C₄⋊C₄	1	16	yes	—
211	C₂²⋊C₄	C₅×C₄⋊C₄	1	16	yes	—
212	C₄²	—	2	8	no	—
213	C₄²	—	4	4	no	—
214	C₄²	C₅×C₄⋊C₄	1	4	yes	—
215	C₄²	C₅×C₄⋊C₄	1	4	yes	—
216	C₄²	C₂×C₅×D₄	1	2	yes	—
217	C₄²	C₂×C₅×D₄	1	2	yes	—
218	C₄²	C₂×C₅×D₄	1	8	yes	—
219	C₄²	C₂×C₅×Q₈	1	2	yes	—
220	C₄²	C₂×C₅×Q₈	1	2	yes	—
221	C₄²	C₂×C₅×Q₈	1	8	yes	—
222	C₂×C₅	—	2	4	no	—
223	C₂×C₅	C₂×#384	1	1	yes	characteristic
224	C₂×C₅	C₂×#424	1	1	yes	characteristic
225	C₂×C₅	#2905	1	1	yes	characteristic
226	C₂×C₅	#2906	1	4	yes	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	C₂×C₅×C₄⋊C₄	1	4	yes	—
230	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
231	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
240	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
241	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
242	C₂×C₄	C₂×C₅×C₄⋊C₄	1	16	yes	—
243	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
244	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
245	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
246	C₂×C₄	C₅×#127	1	4	yes	—
247	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
248	C₂×C₄	C₅×C₄⋊Q₈	1	8	yes	—
249	C₂×C₄	C₅×#131	1	4	yes	—
250	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
252	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
254	C₈	—	4	4	no	—
255	C₈	—	4	4	no	—
256	C₅	#20793	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	C₅×#384	1	4	yes	—
261	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
262	C₂²	C₂²×C₅×C₄⋊C₄	1	2	yes	—
263	C₂²	C₂×C₅×#131	1	2	yes	—
264	C₂²	C₅×#568	1	2	yes	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	16	no	—
269	C₄	C₅×#384	1	4	yes	—
270	C₄	C₅×#424	1	4	yes	complemented
271	C₄	C₂×C₅×#127	1	1	yes	characteristic
272	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
273	C₄	C₂×C₅×C₄⋊Q₈	1	2	yes	—
274	C₂	—	2	4	no	—
275	C₂	C₂×C₅×#384	1	1	yes	characteristic
276	C₂	C₂×C₅×#424	1	1	yes	characteristic
277	C₂	C₅×#2905	1	1	yes	characteristic
278	C₂	C₅×#2906	1	4	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×#1843	C₂	semidirect
6. C₅×#1842	C₂	semidirect
25. #20793	C₅	direct (N × H)
59. #1843	C₂×C₅	semidirect
60. #1842	C₂×C₅	semidirect
184. C₄×C₅	#424	semidirect
256. C₅	#20793	direct (N × H)
270. 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×C₅×C₄⋊C₈	2
C₄×C₅×#117	2
C₅×#1843	2
C₅×#1842	2
#20793	5

Subgroups — C5×#20793 #11943513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20793 #11943513