Subgroups — C₅×#21732 #11944452

Order 1280.

702 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3160	C₂	1	1	yes	maximal, characteristic
2	C₅×#2960	C₂	1	2	yes	maximal
3	C₅×#1994	C₂	1	2	yes	maximal
4	C₅×#1945	C₂	1	4	yes	maximal
5	C₅×#1848	C₂	1	1	yes	maximal, characteristic
6	C₅×#1772	C₂	1	4	yes	maximal
7	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
8	C₅×#533	C₂²	1	2	yes	—
9	C₅×#500	C₄	1	2	yes	complemented
10	C₅×#499	C₄	1	4	yes	complemented
11	C₅×#498	C₄	1	2	yes	complemented
12	C₅×#446	—	2	4	no	—
13	C₅×C₈×Q₈	—	2	4	no	—
14	C₂×C₅×#108	C₂²	1	4	yes	—
15	C₅×#427	C₂²	1	4	yes	—
16	C₅×#424	C₂²	1	4	yes	—
17	C₅×#423	C₂²	1	2	yes	—
18	C₅×#405	C₂²	1	4	yes	—
19	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₄³×C₅	C₂²	1	1	yes	characteristic
24	C₅×#339	C₂²	1	8	yes	—
25	#21732	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×M₁₆	C₂³	1	4	yes	—
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	2	yes	—
29	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
31	C₅×#125	C₂×C₄	1	4	yes	—
32	C₅×C₂²⋊Q₈	—	2	4	no	—
33	C₄×C₅×Q₈	—	2	4	no	—
34	C₅×#117	C₂³	1	4	yes	—
35	C₂×C₄²×C₅	—	2	2	no	—
36	C₂×C₄²×C₅	C₂³	1	2	yes	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×#108	—	2	8	no	—
39	C₅×#107	C₂×C₄	1	8	yes	—
40	C₅×#106	C₂×C₄	1	8	yes	—
41	C₅×C₄⋊C₈	—	2	8	no	—
42	C₅×C₄⋊C₈	—	2	4	no	—
43	C₅×#97	—	2	4	no	—
44	C₄×C₅×C₈	—	2	4	no	—
45	C₄×C₅×C₈	—	2	4	no	—
46	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
48	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₅×C₈	D₄	1	2	yes	—
50	C₄×C₅×C₈	D₄	1	2	yes	—
51	C₄×C₅×C₈	Q₈	1	8	yes	complemented
52	#3160	C₂×C₅	1	1	yes	characteristic
53	#2960	C₂×C₅	1	2	yes	—
54	#1994	C₂×C₅	1	2	yes	—
55	#1945	C₂×C₅	1	4	yes	—
56	#1848	C₂×C₅	1	1	yes	characteristic
57	#1772	C₂×C₅	1	4	yes	—
58	C₄²×C₈	C₂×C₅	1	1	yes	characteristic
59	C₂×C₅×Q₈	—	2	4	no	—
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	#41	1	2	yes	—
62	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
63	C₅×M₁₆	—	2	8	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	4	no	—
66	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
67	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
72	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
73	C₂×C₅×C₈	#41	1	4	yes	—
74	C₅×C₄⋊C₄	—	2	8	no	—
75	C₅×C₄⋊C₄	—	2	4	no	—
76	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
77	C₅×C₂²⋊C₄	—	2	4	no	—
78	C₄²×C₅	—	2	8	no	—
79	C₄²×C₅	—	2	4	no	—
80	C₄²×C₅	—	2	2	no	—
81	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅	C₂×D₄	1	2	yes	—
84	C₄²×C₅	C₂×Q₈	1	4	yes	—
85	#533	C₂²×C₅	1	2	yes	—
86	#500	C₄×C₅	1	2	yes	complemented
87	#499	C₄×C₅	1	4	yes	complemented
88	#498	C₄×C₅	1	2	yes	complemented
89	#446	—	2	4	no	—
90	C₈×Q₈	—	2	4	no	—
91	C₂×#108	C₂²×C₅	1	4	yes	—
92	#427	C₂²×C₅	1	4	yes	—
93	#424	C₂²×C₅	1	4	yes	—
94	#423	C₂²×C₅	1	2	yes	—
95	#405	C₂²×C₅	1	4	yes	—
96	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
97	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
100	C₄³	C₂²×C₅	1	1	yes	characteristic
101	#339	C₂²×C₅	1	8	yes	—
102	C₂³×C₅	C₂×#41	1	1	yes	characteristic
103	C₅×Q₈	—	4	8	no	—
104	C₂×C₄×C₅	—	2	8	no	—
105	C₂×C₄×C₅	—	2	4	no	—
106	C₂×C₄×C₅	—	2	4	no	—
107	C₂×C₄×C₅	—	2	8	no	—
108	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
113	C₂×C₄×C₅	C₂×#41	1	2	yes	—
114	C₅×C₈	—	2	2	no	—
115	C₅×C₈	—	4	4	no	—
116	C₅×C₈	C₄×Q₈	1	4	yes	—
117	C₅×C₈	C₄×Q₈	1	4	yes	—
118	C₅×C₈	C₄⋊Q₈	1	2	yes	—
119	C₅×C₈	C₄⋊Q₈	1	2	yes	—
120	C₂×M₁₆	C₂³×C₅	1	4	yes	—
121	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
122	C₂²×C₈	C₂³×C₅	1	2	yes	—
123	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
124	C₄⋊Q₈	C₂×C₄×C₅	1	4	yes	—
125	#125	C₂×C₄×C₅	1	4	yes	—
126	C₂²⋊Q₈	—	2	4	no	—
127	C₄×Q₈	—	2	4	no	—
128	#117	C₂³×C₅	1	4	yes	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	C₂³×C₅	1	2	yes	—
131	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
132	#108	—	2	8	no	—
133	#107	C₂×C₄×C₅	1	8	yes	—
134	#106	C₂×C₄×C₅	1	8	yes	—
135	C₄⋊C₈	—	2	8	no	—
136	C₄⋊C₈	—	2	4	no	—
137	#97	—	2	4	no	—
138	C₄×C₈	—	2	4	no	—
139	C₄×C₈	—	2	4	no	—
140	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
141	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
142	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
143	C₄×C₈	C₅×D₄	1	2	yes	—
144	C₄×C₈	C₅×D₄	1	2	yes	—
145	C₄×C₈	C₅×Q₈	1	8	yes	complemented
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂²×C₅	#533	1	2	yes	—
149	C₄×C₅	—	2	8	no	—
150	C₄×C₅	—	2	2	no	—
151	C₄×C₅	—	4	8	no	—
152	C₄×C₅	#443	1	4	yes	—
153	C₄×C₅	C₂×C₄×Q₈	1	2	yes	—
154	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
155	C₄×C₅	#532	1	1	yes	characteristic
156	C₂×Q₈	—	2	4	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₅×#41	1	2	yes	—
159	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
160	M₁₆	—	2	8	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
164	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
165	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
166	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
167	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
168	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
169	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
170	C₂×C₈	C₅×#41	1	4	yes	—
171	C₄⋊C₄	—	2	8	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₄²	—	2	8	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	2	no	—
178	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
179	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
180	C₄²	C₂×C₅×D₄	1	2	yes	—
181	C₄²	C₂×C₅×Q₈	1	4	yes	—
182	C₂×C₅	—	2	2	no	—
183	C₂×C₅	C₄×C₄⋊Q₈	1	1	yes	characteristic
184	C₂×C₅	C₂×#443	1	2	yes	—
185	C₂³	C₂×C₅×#41	1	1	yes	characteristic
186	Q₈	—	4	8	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
192	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
193	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
194	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
196	C₂×C₄	C₂×C₅×#41	1	2	yes	—
197	C₈	—	2	2	no	—
198	C₈	—	4	4	no	—
199	C₈	C₄×C₅×Q₈	1	4	yes	—
200	C₈	C₄×C₅×Q₈	1	4	yes	—
201	C₈	C₅×C₄⋊Q₈	1	2	yes	—
202	C₈	C₅×C₄⋊Q₈	1	2	yes	—
203	C₅	#21732	1	1	yes	characteristic, direct product factor, Sylow
204	C₂²	—	2	2	no	—
205	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
206	C₂²	C₅×#533	1	2	yes	—
207	C₄	—	2	8	no	—
208	C₄	—	2	2	no	—
209	C₄	—	4	8	no	—
210	C₄	C₅×#443	1	4	yes	—
211	C₄	C₂×C₄×C₅×Q₈	1	2	yes	—
212	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
213	C₄	C₅×#532	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	C₄×C₅×C₄⋊Q₈	1	1	yes	characteristic
216	C₂	C₂×C₅×#443	1	2	yes	—

	subgroup	G/H
center	C₂×C₅×C₈	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₅×#500	C₄	semidirect
10. C₅×#499	C₄	semidirect
11. C₅×#498	C₄	semidirect
25. #21732	C₅	direct (N × H)
51. C₄×C₅×C₈	Q₈	semidirect
86. #500	C₄×C₅	semidirect
87. #499	C₄×C₅	semidirect
88. #498	C₄×C₅	semidirect
145. C₄×C₈	C₅×Q₈	semidirect
203. C₅	#21732	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×#3160	2
C₅×#2960	2
C₅×#1994	2
C₅×#1945	2
C₅×#1848	2
C₅×#1772	2
C₄²×C₅×C₈	2
#21732	5

Subgroups — C5×#21732 #11944452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21732 #11944452