Subgroups — C₅×#20542 #11943262

Order 1280.

866 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#433	C₂	1	2	yes	maximal
2	C₅×#2296	C₂	1	1	yes	maximal, characteristic
3	C₅×#1771	C₂	1	4	yes	maximal, complemented
4	C₅×#1770	C₂	1	4	yes	maximal, complemented
5	C₅×#1750	C₂	1	2	yes	maximal
6	C₅×#1749	C₂	1	2	yes	maximal
7	C₂×C₅×#117	C₂²	1	1	yes	characteristic
8	C₂×C₅×#117	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₅×#433	C₄	1	16	yes	complemented
11	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₅×#97	C₂²	1	2	yes	—
14	C₂×C₅×#97	C₂²	1	4	yes	—
15	C₂×C₅×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₅×#389	C₄	1	4	yes	—
19	C₅×#385	C₄	1	4	yes	—
20	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
21	C₅×#336	C₂²	1	2	yes	—
22	C₅×#336	C₂²	1	4	yes	—
23	C₅×#336	C₂²	1	2	yes	—
24	#20542	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	4	yes	—
28	C₂²×C₅×C₈	C₂³	1	4	yes	—
29	C₅×#117	—	4	2	no	—
30	C₅×#117	C₂×C₄	1	8	yes	—
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅	—	4	2	no	—
37	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅	C₂³	1	2	yes	—
39	C₂×C₄²×C₅	C₂³	1	2	yes	—
40	C₂×C₄²×C₅	C₂³	1	2	yes	—
41	C₅×C₄⋊C₈	C₂×C₄	1	16	yes	complemented
42	C₅×C₂²⋊C₈	C₂×C₄	1	16	yes	—
43	C₅×#97	—	4	4	no	—
44	C₅×#97	—	4	2	no	—
45	C₅×#97	C₂×C₄	1	8	yes	complemented
46	C₄×C₅×C₈	—	4	2	no	—
47	C₄×C₅×C₈	C₂×C₄	1	8	yes	complemented
48	C₅×#95	C₂×C₄	1	8	yes	—
49	C₂×#433	C₂×C₅	1	2	yes	—
50	#2296	C₂×C₅	1	1	yes	characteristic
51	#1771	C₂×C₅	1	4	yes	complemented
52	#1770	C₂×C₅	1	4	yes	complemented
53	#1750	C₂×C₅	1	2	yes	—
54	#1749	C₂×C₅	1	2	yes	—
55	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	—	4	2	no	—
59	C₂²×C₄×C₅	—	4	4	no	—
60	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
66	C₂×C₅×C₈	—	2	4	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	—	2	4	no	—
69	C₂×C₅×C₈	—	2	4	no	—
70	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
71	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
72	C₅×C₄⋊C₄	—	4	8	no	—
73	C₅×C₄⋊C₄	—	4	2	no	—
74	C₅×C₄⋊C₄	C₄²	1	8	yes	—
75	C₅×C₂²⋊C₄	—	4	2	no	—
76	C₅×C₂²⋊C₄	C₄²	1	8	yes	—
77	C₄²×C₅	—	2	4	no	—
78	C₄²×C₅	—	2	2	no	—
79	C₄²×C₅	—	2	2	no	—
80	C₄²×C₅	—	4	2	no	—
81	C₄²×C₅	C₂²×C₄	1	4	yes	—
82	C₄²×C₅	C₂²×C₄	1	4	yes	—
83	C₂×#117	C₂²×C₅	1	1	yes	characteristic
84	C₂×#117	C₂²×C₅	1	1	yes	characteristic
85	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
86	#433	C₄×C₅	1	16	yes	complemented
87	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
88	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
89	C₂×#97	C₂²×C₅	1	4	yes	—
90	C₂×#97	C₂²×C₅	1	2	yes	—
91	C₂×#97	C₂²×C₅	1	2	yes	—
92	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
93	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
94	#389	C₄×C₅	1	4	yes	—
95	#385	C₄×C₅	1	4	yes	—
96	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
97	#336	C₂²×C₅	1	2	yes	—
98	#336	C₂²×C₅	1	2	yes	—
99	#336	C₂²×C₅	1	4	yes	—
100	C₂³×C₅	—	2	1	no	—
101	C₂³×C₅	—	2	1	no	—
102	C₂³×C₅	—	2	4	no	—
103	C₂³×C₅	C₂×C₄²	1	2	yes	—
104	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	8	no	—
108	C₂×C₄×C₅	—	2	8	no	—
109	C₂×C₄×C₅	—	2	4	no	—
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	2	8	no	—
113	C₂×C₄×C₅	—	2	4	no	—
114	C₂×C₄×C₅	—	4	8	no	—
115	C₂×C₄×C₅	C₂×C₄²	1	8	yes	—
116	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
118	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
119	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
120	C₅×C₈	—	2	8	no	—
121	C₅×C₈	—	2	8	no	—
122	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
123	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
124	C₂²×C₈	C₂³×C₅	1	4	yes	—
125	C₂²×C₈	C₂³×C₅	1	4	yes	—
126	#117	—	4	2	no	—
127	#117	C₂×C₄×C₅	1	8	yes	—
128	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
129	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
130	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
131	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
133	C₂×C₄²	—	4	2	no	—
134	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
135	C₂×C₄²	C₂³×C₅	1	2	yes	—
136	C₂×C₄²	C₂³×C₅	1	2	yes	—
137	C₂×C₄²	C₂³×C₅	1	2	yes	—
138	C₄⋊C₈	C₂×C₄×C₅	1	16	yes	complemented
139	C₂²⋊C₈	C₂×C₄×C₅	1	16	yes	—
140	#97	—	4	4	no	—
141	#97	—	4	2	no	—
142	#97	C₂×C₄×C₅	1	8	yes	complemented
143	C₄×C₈	—	4	2	no	—
144	C₄×C₈	C₂×C₄×C₅	1	8	yes	complemented
145	#95	C₂×C₄×C₅	1	8	yes	—
146	C₂²×C₅	—	2	4	no	—
147	C₂²×C₅	—	2	2	no	—
148	C₂²×C₅	—	4	2	no	—
149	C₂²×C₅	—	4	2	no	—
150	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
151	C₂²×C₅	#518	1	1	yes	characteristic
152	C₂²×C₅	#519	1	1	yes	characteristic
153	C₂²×C₅	#568	1	4	yes	—
154	C₄×C₅	—	2	8	no	—
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	—	4	4	no	—
157	C₄×C₅	—	4	4	no	—
158	C₄×C₅	#520	1	4	yes	—
159	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
165	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
166	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
167	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
168	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
169	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
175	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	8	no	—
178	C₄⋊C₄	C₄²×C₅	1	8	yes	—
179	C₂²⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	C₄²×C₅	1	8	yes	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	C₂²×C₄×C₅	1	4	yes	—
186	C₄²	C₂²×C₄×C₅	1	4	yes	—
187	C₂×C₅	—	4	2	no	—
188	C₂×C₅	#2274	1	1	yes	characteristic
189	C₂×C₅	#2875	1	2	yes	—
190	C₂×C₅	#2978	1	2	yes	—
191	C₂×C₅	#2987	1	2	yes	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	4	no	—
195	C₂³	C₂×C₄²×C₅	1	2	yes	—
196	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	8	no	—
207	C₂×C₄	C₂×C₄²×C₅	1	8	yes	—
208	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
209	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
210	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
211	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
212	C₈	—	2	8	no	—
213	C₈	—	2	8	no	—
214	C₅	#20542	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²	—	2	2	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
220	C₂²	C₅×#518	1	1	yes	characteristic
221	C₂²	C₅×#519	1	1	yes	characteristic
222	C₂²	C₅×#568	1	4	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	2	8	no	—
225	C₄	—	4	4	no	—
226	C₄	—	4	4	no	—
227	C₄	C₅×#520	1	4	yes	—
228	C₂	—	4	2	no	—
229	C₂	C₅×#2274	1	1	yes	characteristic
230	C₂	C₅×#2875	1	2	yes	—
231	C₂	C₅×#2978	1	2	yes	—
232	C₂	C₅×#2987	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅×#1771	C₂	semidirect
4. C₅×#1770	C₂	semidirect
10. C₅×#433	C₄	semidirect
24. #20542	C₅	direct (N × H)
41. C₅×C₄⋊C₈	C₂×C₄	semidirect
45. C₅×#97	C₂×C₄	semidirect
47. C₄×C₅×C₈	C₂×C₄	semidirect
51. #1771	C₂×C₅	semidirect
52. #1770	C₂×C₅	semidirect
86. #433	C₄×C₅	semidirect
138. C₄⋊C₈	C₂×C₄×C₅	semidirect
142. #97	C₂×C₄×C₅	semidirect
144. C₄×C₈	C₂×C₄×C₅	semidirect
214. C₅	#20542	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×#433	2
C₅×#2296	2
C₅×#1771	2
C₅×#1770	2
C₅×#1750	2
C₅×#1749	2
#20542	5

Subgroups — C5×#20542 #11943262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20542 #11943262