Subgroups — C₅×#20360 #11943080

Order 1280.

1996 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#568	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#568	C₂	1	1	yes	maximal, characteristic
3	C₅×#3428	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#344	C₂	1	12	yes	maximal, complemented
5	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
6	C₅×#568	—	2	4	no	—
7	C₅×#568	—	2	4	no	—
8	C₅×#568	—	2	4	no	—
9	C₅×#568	C₄	1	8	yes	complemented
10	C₂×C₅×#131	—	2	4	no	—
11	C₂×C₅×#131	C₄	1	8	yes	complemented
12	C₂²×C₅×M₁₆	C₂²	1	6	yes	—
13	C₂²×C₅×M₁₆	C₂²	1	6	yes	—
14	C₅×#520	—	2	4	no	—
15	C₅×#519	C₄	1	2	yes	—
16	C₅×#518	C₄	1	6	yes	—
17	C₄×C₅×#41	—	2	4	no	—
18	C₂×C₅×#117	C₂²	1	6	yes	—
19	C₅×#344	C₂²	1	16	yes	—
20	#20360	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×#41	C₂×C₄	1	4	yes	—
22	C₂×C₅×#41	C₂×C₄	1	4	yes	—
23	C₂×C₅×#41	D₄	1	4	yes	—
24	C₂×C₅×#41	D₄	1	4	yes	—
25	C₂×C₅×#41	D₄	1	4	yes	—
26	C₂×C₅×#41	Q₈	1	4	yes	—
27	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×D₄	C₂×C₄	1	3	yes	—
29	C₂³×C₄×C₅	C₂³	1	3	yes	—
30	C₅×#131	—	2	32	no	—
31	C₅×#131	—	2	32	no	—
32	C₂×C₅×M₁₆	—	2	6	no	—
33	C₂×C₅×M₁₆	—	2	12	no	—
34	C₂×C₅×M₁₆	—	2	12	no	—
35	C₂×C₅×M₁₆	—	2	12	no	—
36	C₂×C₅×M₁₆	C₂×C₄	1	24	yes	complemented
37	C₂×C₅×M₁₆	C₂×C₄	1	12	yes	complemented
38	C₂×C₅×M₁₆	C₂³	1	4	yes	—
39	C₂×C₅×M₁₆	C₂³	1	4	yes	—
40	C₂²×C₅×C₈	—	2	6	no	—
41	C₂²×C₅×C₈	C₂×C₄	1	12	yes	complemented
42	C₄×C₅×Q₈	—	2	8	no	—
43	C₄×C₅×D₄	—	2	24	no	—
44	C₅×#117	—	2	6	no	—
45	C₅×#117	—	2	6	no	—
46	C₅×#117	C₂×C₄	1	12	yes	—
47	C₅×#117	C₂³	1	4	yes	—
48	C₂×C₅×C₄⋊C₄	—	2	6	no	—
49	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	12	yes	—
50	C₂×C₄²×C₅	—	2	6	no	—
51	C₂×#568	C₂×C₅	1	1	yes	characteristic
52	C₂×#568	C₂×C₅	1	1	yes	characteristic
53	#3428	C₂×C₅	1	1	yes	characteristic
54	C₂×#344	C₂×C₅	1	12	yes	complemented
55	C₂⁴×C₅	C₂²×C₄	1	3	yes	—
56	C₅×#41	C₂²⋊C₄	1	16	yes	—
57	C₅×#41	C₂²⋊C₄	1	16	yes	—
58	C₅×#41	C₄⋊C₄	1	16	yes	—
59	C₅×#41	C₄⋊C₄	1	16	yes	—
60	C₂×C₅×Q₈	C₄²	1	4	yes	—
61	C₂×C₅×Q₈	C₂²⋊C₄	1	4	yes	—
62	C₂×C₅×Q₈	C₄⋊C₄	1	4	yes	—
63	C₂×C₅×D₄	C₄²	1	12	yes	—
64	C₂×C₅×D₄	C₂²⋊C₄	1	12	yes	—
65	C₂×C₅×D₄	C₄⋊C₄	1	12	yes	—
66	C₂²×C₄×C₅	—	2	12	no	—
67	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
70	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
71	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
72	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
73	C₂²×C₄×C₅	C₂×Q₈	1	6	yes	—
74	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₅×M₁₆	—	2	24	no	—
76	C₅×M₁₆	—	2	24	no	—
77	C₅×M₁₆	—	2	4	no	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	2	4	no	—
80	C₅×M₁₆	C₂²×C₄	1	8	yes	—
81	C₂×C₅×C₈	—	2	24	no	—
82	C₂×C₅×C₈	—	2	24	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
85	C₅×C₄⋊C₄	—	2	12	no	—
86	C₅×C₄⋊C₄	—	2	4	no	—
87	C₅×C₂²⋊C₄	—	2	12	no	—
88	C₅×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
89	C₄²×C₅	—	2	12	no	—
90	C₄²×C₅	—	2	4	no	—
91	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
92	#568	—	2	4	no	—
93	#568	—	2	4	no	—
94	#568	—	2	4	no	—
95	#568	C₄×C₅	1	8	yes	complemented
96	C₂×#131	—	2	4	no	—
97	C₂×#131	C₄×C₅	1	8	yes	complemented
98	C₂²×M₁₆	C₂²×C₅	1	6	yes	—
99	C₂²×M₁₆	C₂²×C₅	1	6	yes	—
100	#520	—	2	4	no	—
101	#519	C₄×C₅	1	2	yes	—
102	#518	C₄×C₅	1	6	yes	—
103	C₄×#41	—	2	4	no	—
104	C₂×#117	C₂²×C₅	1	6	yes	—
105	#344	C₂²×C₅	1	16	yes	—
106	C₂³×C₅	—	8	3	no	—
107	C₂³×C₅	C₂×C₄²	1	6	yes	—
108	C₂³×C₅	C₂×C₂²⋊C₄	1	6	yes	—
109	C₂³×C₅	C₂×C₄⋊C₄	1	6	yes	—
110	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
111	C₅×Q₈	#95	1	16	yes	—
112	C₅×D₄	#95	1	48	yes	—
113	C₂×C₄×C₅	—	2	8	no	—
114	C₂×C₄×C₅	—	8	12	no	—
115	C₂×C₄×C₅	C₂×C₄²	1	6	yes	—
116	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	12	yes	—
117	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	12	yes	—
118	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
119	C₂×C₄×C₅	C₂×C₄⋊C₄	1	12	yes	—
120	C₂×C₄×C₅	C₂×C₄⋊C₄	1	12	yes	—
121	C₂×C₄×C₅	C₂×C₄⋊C₄	1	6	yes	—
122	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
128	C₅×C₈	—	2	8	no	—
129	C₅×C₈	—	2	8	no	—
130	C₂×#41	C₂×C₄×C₅	1	4	yes	—
131	C₂×#41	C₂×C₄×C₅	1	4	yes	—
132	C₂×#41	C₅×D₄	1	4	yes	—
133	C₂×#41	C₅×D₄	1	4	yes	—
134	C₂×#41	C₅×D₄	1	4	yes	—
135	C₂×#41	C₅×Q₈	1	4	yes	—
136	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
137	C₂²×D₄	C₂×C₄×C₅	1	3	yes	—
138	C₂³×C₄	C₂³×C₅	1	3	yes	—
139	#131	—	2	32	no	—
140	#131	—	2	32	no	—
141	C₂×M₁₆	—	2	6	no	—
142	C₂×M₁₆	—	2	12	no	—
143	C₂×M₁₆	—	2	12	no	—
144	C₂×M₁₆	—	2	12	no	—
145	C₂×M₁₆	C₂×C₄×C₅	1	24	yes	complemented
146	C₂×M₁₆	C₂×C₄×C₅	1	12	yes	complemented
147	C₂×M₁₆	C₂³×C₅	1	4	yes	—
148	C₂×M₁₆	C₂³×C₅	1	4	yes	—
149	C₂²×C₈	—	2	6	no	—
150	C₂²×C₈	C₂×C₄×C₅	1	12	yes	complemented
151	C₄×Q₈	—	2	8	no	—
152	C₄×D₄	—	2	24	no	—
153	#117	—	2	6	no	—
154	#117	—	2	6	no	—
155	#117	C₂×C₄×C₅	1	12	yes	—
156	#117	C₂³×C₅	1	4	yes	—
157	C₂×C₄⋊C₄	—	2	6	no	—
158	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	12	yes	—
159	C₂×C₄²	—	2	6	no	—
160	C₂²×C₅	—	4	6	no	—
161	C₂²×C₅	—	4	1	no	—
162	C₂²×C₅	—	4	6	no	—
163	C₂²×C₅	—	4	6	no	—
164	C₂²×C₅	C₂×#95	1	12	yes	—
165	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
166	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
168	C₄×C₅	—	4	8	no	—
169	C₄×C₅	C₂×#95	1	12	yes	—
170	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
171	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
172	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
173	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
174	C₂⁴	C₂²×C₄×C₅	1	3	yes	—
175	#41	C₅×C₂²⋊C₄	1	16	yes	—
176	#41	C₅×C₂²⋊C₄	1	16	yes	—
177	#41	C₅×C₄⋊C₄	1	16	yes	—
178	#41	C₅×C₄⋊C₄	1	16	yes	—
179	C₂×Q₈	C₄²×C₅	1	4	yes	—
180	C₂×Q₈	C₅×C₂²⋊C₄	1	4	yes	—
181	C₂×Q₈	C₅×C₄⋊C₄	1	4	yes	—
182	C₂×D₄	C₄²×C₅	1	12	yes	—
183	C₂×D₄	C₅×C₂²⋊C₄	1	12	yes	—
184	C₂×D₄	C₅×C₄⋊C₄	1	12	yes	—
185	C₂²×C₄	—	2	12	no	—
186	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
187	C₂²×C₄	C₂²×C₄×C₅	1	6	yes	—
188	C₂²×C₄	C₂²×C₄×C₅	1	6	yes	—
189	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
190	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
191	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
192	C₂²×C₄	C₂×C₅×Q₈	1	6	yes	—
193	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
194	M₁₆	—	2	24	no	—
195	M₁₆	—	2	24	no	—
196	M₁₆	—	2	4	no	—
197	M₁₆	—	2	4	no	—
198	M₁₆	—	2	4	no	—
199	M₁₆	C₂²×C₄×C₅	1	8	yes	—
200	C₂×C₈	—	2	24	no	—
201	C₂×C₈	—	2	24	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
204	C₄⋊C₄	—	2	12	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	—	2	12	no	—
207	C₂²⋊C₄	C₂²×C₄×C₅	1	8	yes	—
208	C₄²	—	2	12	no	—
209	C₄²	—	2	4	no	—
210	C₂×C₅	—	2	1	no	—
211	C₂×C₅	—	2	1	no	—
212	C₂×C₅	—	2	1	no	—
213	C₂×C₅	—	2	12	no	—
214	C₂×C₅	C₂²×#95	1	1	yes	characteristic
215	C₂³	—	8	3	no	—
216	C₂³	C₂×C₄²×C₅	1	6	yes	—
217	C₂³	C₂×C₅×C₂²⋊C₄	1	6	yes	—
218	C₂³	C₂×C₅×C₄⋊C₄	1	6	yes	—
219	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
220	Q₈	C₅×#95	1	16	yes	—
221	D₄	C₅×#95	1	48	yes	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	8	12	no	—
224	C₂×C₄	C₂×C₄²×C₅	1	6	yes	—
225	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
226	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	12	yes	—
227	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	12	yes	—
228	C₂×C₄	C₂×C₅×C₄⋊C₄	1	6	yes	—
229	C₂×C₄	C₂×C₅×C₄⋊C₄	1	12	yes	—
230	C₂×C₄	C₂×C₅×C₄⋊C₄	1	12	yes	—
231	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
232	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
234	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
237	C₈	—	2	8	no	—
238	C₈	—	2	8	no	—
239	C₅	#20360	1	1	yes	characteristic, direct product factor, Sylow
240	C₂²	—	4	6	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	6	no	—
243	C₂²	—	4	6	no	—
244	C₂²	C₂×C₅×#95	1	12	yes	—
245	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
246	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
247	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
248	C₄	—	4	8	no	—
249	C₄	C₂×C₅×#95	1	12	yes	—
250	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
251	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
252	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
253	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	2	12	no	—
258	C₂	C₂²×C₅×#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂×C₅	C₂²×#95
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₅×#41 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#344	C₂	semidirect
9. C₅×#568	C₄	semidirect
11. C₂×C₅×#131	C₄	semidirect
20. #20360	C₅	direct (N × H)
36. C₂×C₅×M₁₆	C₂×C₄	semidirect
37. C₂×C₅×M₁₆	C₂×C₄	semidirect
41. C₂²×C₅×C₈	C₂×C₄	semidirect
54. C₂×#344	C₂×C₅	semidirect
95. #568	C₄×C₅	semidirect
97. C₂×#131	C₄×C₅	semidirect
145. C₂×M₁₆	C₂×C₄×C₅	semidirect
146. C₂×M₁₆	C₂×C₄×C₅	semidirect
150. C₂²×C₈	C₂×C₄×C₅	semidirect
239. C₅	#20360	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×#568	2
C₅×#3428	2
C₂×C₅×#344	2
#20360	5

Subgroups — C5×#20360 #11943080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20360 #11943080