Subgroups — C₅×#20831 #11943551

Order 1280.

758 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2920	C₂	1	2	yes	maximal
2	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#1848	C₂	1	2	yes	maximal, complemented
4	C₅×#1837	C₂	1	2	yes	maximal, complemented
5	C₅×#1811	C₂	1	4	yes	maximal
6	C₅×#1778	C₂	1	4	yes	maximal, complemented
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₅×#432	—	2	4	no	—
10	C₅×#431	—	2	4	no	—
11	C₂×C₅×#108	C₂²	1	8	yes	—
12	C₅×#423	C₂²	1	4	yes	—
13	C₅×#407	C₂²	1	4	yes	—
14	C₄×C₅×M₁₆	C₄	1	4	yes	—
15	C₄×C₅×M₁₆	C₄	1	4	yes	—
16	C₄×C₅×M₁₆	C₂²	1	4	yes	—
17	C₂×C₅×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#341	C₂²	1	8	yes	—
21	#20831	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂³×C₄×C₅	Q₈	1	2	yes	—
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₅×M₁₆	—	2	8	no	—
25	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
26	C₂×C₅×M₁₆	C₂³	1	8	yes	—
27	C₂²×C₅×C₈	C₂³	1	2	yes	—
28	C₂²×C₅×C₈	C₂³	1	2	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅	D₄	1	4	yes	—
32	C₂×C₄²×C₅	Q₈	1	2	yes	—
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₅×#108	—	2	16	no	—
36	C₅×C₄⋊C₈	—	2	8	no	—
37	C₅×C₂²⋊C₈	—	2	8	no	—
38	C₅×#97	—	2	4	no	—
39	C₅×#97	C₂×C₄	1	8	yes	—
40	C₄×C₅×C₈	—	2	4	no	—
41	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
43	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
44	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
45	#2920	C₂×C₅	1	2	yes	—
46	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
47	#1848	C₂×C₅	1	2	yes	complemented
48	#1837	C₂×C₅	1	2	yes	complemented
49	#1811	C₂×C₅	1	4	yes	—
50	#1778	C₂×C₅	1	4	yes	complemented
51	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₅	—	2	4	no	—
53	C₂²×C₄×C₅	—	4	4	no	—
54	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
62	C₅×M₁₆	—	2	16	no	—
63	C₅×M₁₆	—	2	16	no	—
64	C₂×C₅×C₈	—	2	4	no	—
65	C₂×C₅×C₈	—	2	8	no	—
66	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
67	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
68	C₂×C₅×C₈	#41	1	8	yes	—
69	C₄²×C₅	C₄⋊C₄	1	4	yes	—
70	C₄²×C₅	C₄⋊C₄	1	4	yes	—
71	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅	C₂²×C₄	1	2	yes	—
74	C₄²×C₅	C₂×D₄	1	2	yes	—
75	C₄²×C₅	C₂×D₄	1	2	yes	—
76	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
77	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
78	#432	—	2	4	no	—
79	#431	—	2	4	no	—
80	C₂×#108	C₂²×C₅	1	8	yes	—
81	#423	C₂²×C₅	1	4	yes	—
82	#407	C₂²×C₅	1	4	yes	—
83	C₄×M₁₆	C₄×C₅	1	4	yes	—
84	C₄×M₁₆	C₄×C₅	1	4	yes	—
85	C₄×M₁₆	C₂²×C₅	1	4	yes	—
86	C₂×#97	C₂²×C₅	1	2	yes	—
87	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
89	#341	C₂²×C₅	1	8	yes	—
90	C₂³×C₅	—	2	2	no	—
91	C₂³×C₅	—	4	2	no	—
92	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅	—	2	2	no	—
95	C₂×C₄×C₅	—	2	4	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	8	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	4	4	no	—
100	C₂×C₄×C₅	—	4	2	no	—
101	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×#41	1	4	yes	—
108	C₅×C₈	—	2	4	no	—
109	C₅×C₈	—	4	8	no	—
110	C₅×C₈	#117	1	8	yes	—
111	C₂³×C₄	C₅×Q₈	1	2	yes	—
112	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
113	C₂×M₁₆	—	2	8	no	—
114	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
115	C₂×M₁₆	C₂³×C₅	1	8	yes	—
116	C₂²×C₈	C₂³×C₅	1	2	yes	—
117	C₂²×C₈	C₂³×C₅	1	2	yes	—
118	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
119	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
120	C₂×C₄²	C₅×D₄	1	4	yes	—
121	C₂×C₄²	C₅×Q₈	1	2	yes	—
122	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
123	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
124	#108	—	2	16	no	—
125	C₄⋊C₈	—	2	8	no	—
126	C₂²⋊C₈	—	2	8	no	—
127	#97	—	2	4	no	—
128	#97	C₂×C₄×C₅	1	8	yes	—
129	C₄×C₈	—	2	4	no	—
130	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
131	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
132	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
133	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
134	C₂²×C₅	—	2	2	no	—
135	C₂²×C₅	—	2	4	no	—
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	—	4	4	no	—
138	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₅	#533	1	2	yes	—
140	C₄×C₅	—	2	4	no	—
141	C₄×C₅	—	2	2	no	—
142	C₄×C₅	—	2	2	no	—
143	C₄×C₅	#430	1	4	yes	—
144	C₄×C₅	C₂×#117	1	2	yes	—
145	C₄×C₅	#532	1	2	yes	—
146	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	4	4	no	—
149	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
150	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
151	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
152	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
153	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
155	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
156	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
157	M₁₆	—	2	16	no	—
158	M₁₆	—	2	16	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
162	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
163	C₂×C₈	C₅×#41	1	8	yes	—
164	C₄²	C₅×C₄⋊C₄	1	4	yes	—
165	C₄²	C₅×C₄⋊C₄	1	4	yes	—
166	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
167	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
168	C₄²	C₂²×C₄×C₅	1	2	yes	—
169	C₄²	C₂×C₅×D₄	1	2	yes	—
170	C₄²	C₂×C₅×D₄	1	2	yes	—
171	C₂×C₅	—	2	2	no	—
172	C₂×C₅	—	4	2	no	—
173	C₂×C₅	#2285	1	1	yes	characteristic
174	C₂×C₅	C₂×#430	1	2	yes	—
175	C₂³	—	2	2	no	—
176	C₂³	—	4	2	no	—
177	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
178	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
187	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
188	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
189	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
190	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
192	C₂×C₄	C₂×C₅×#41	1	4	yes	—
193	C₈	—	2	4	no	—
194	C₈	—	4	8	no	—
195	C₈	C₅×#117	1	8	yes	—
196	C₅	#20831	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	4	no	—
200	C₂²	—	4	4	no	—
201	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
202	C₂²	C₅×#533	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	4	no	—
205	C₄	—	2	2	no	—
206	C₄	C₅×#430	1	4	yes	—
207	C₄	C₂×C₅×#117	1	2	yes	—
208	C₄	C₅×#532	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	4	2	no	—
211	C₂	C₅×#2285	1	1	yes	characteristic
212	C₂	C₂×C₅×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1848	C₂	semidirect
4. C₅×#1837	C₂	semidirect
6. C₅×#1778	C₂	semidirect
21. #20831	C₅	direct (N × H)
47. #1848	C₂×C₅	semidirect
48. #1837	C₂×C₅	semidirect
50. #1778	C₂×C₅	semidirect
196. C₅	#20831	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×#2920	2
C₂×C₄×C₅×M₁₆	2
C₅×#1848	2
C₅×#1837	2
C₅×#1811	2
C₅×#1778	2
#20831	5

Subgroups — C5×#20831 #11943551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20831 #11943551