Subgroups — C₅×#13165 #11935885

Order 1280.

326 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅×#1942	C₂	1	2	yes	maximal
5	C₅×#1380	C₂	1	2	yes	maximal
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₅×#107	C₂²	1	2	yes	—
9	C₂×C₅×#107	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₅×#389	—	2	2	no	—
12	C₅×#366	—	2	2	no	—
13	C₅×#366	C₄	1	4	yes	complemented
14	C₄×C₅×C₁₆	—	2	2	no	—
15	C₅×#340	—	2	2	no	—
16	#13165	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₅×C₄⋊C₄	—	2	2	no	—
20	C₂×C₅×C₄⋊C₄	—	2	2	no	—
21	C₂×C₅×C₄⋊C₄	—	4	2	no	—
22	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
23	C₂×C₅×C₁₆	—	2	1	no	—
24	C₂×C₅×C₁₆	—	2	1	no	—
25	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₅×C₁₆	D₄	1	2	yes	—
27	C₂×C₅×C₁₆	Q₈	1	2	yes	—
28	C₂×C₅×C₁₆	Q₈	1	2	yes	—
29	C₅×#107	—	2	2	no	—
30	C₅×#107	—	2	2	no	—
31	C₅×#107	—	2	2	no	—
32	C₅×#107	C₂×C₄	1	4	yes	—
33	C₄×C₅×C₈	—	2	1	no	—
34	C₄×C₅×C₈	—	2	1	no	—
35	C₅×#95	—	4	2	no	—
36	C₂×#366	C₂×C₅	1	1	yes	characteristic
37	C₂×#366	C₂×C₅	1	1	yes	characteristic
38	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
39	#1942	C₂×C₅	1	2	yes	—
40	#1380	C₂×C₅	1	2	yes	—
41	C₂²×C₄×C₅	—	4	2	no	—
42	C₂²×C₄×C₅	—	4	2	no	—
43	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂×C₅×C₈	—	2	1	no	—
47	C₂×C₅×C₈	—	2	1	no	—
48	C₂×C₅×C₈	Q₁₆	1	2	yes	—
49	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₅×C₈	#41	1	1	yes	characteristic
54	C₂×C₅×C₈	#41	1	1	yes	characteristic
55	C₅×C₄⋊C₄	—	2	4	no	—
56	C₅×C₄⋊C₄	—	4	2	no	—
57	C₅×C₄⋊C₄	—	4	2	no	—
58	C₅×C₄⋊C₄	—	4	2	no	—
59	C₅×C₄⋊C₄	—	8	4	no	—
60	C₄²×C₅	—	2	1	no	—
61	C₄²×C₅	—	2	1	no	—
62	C₅×C₁₆	—	2	2	no	—
63	C₅×C₁₆	C₄⋊C₄	1	4	yes	complemented
64	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
65	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
66	C₂×#107	C₂²×C₅	1	2	yes	—
67	C₂×#107	C₂²×C₅	1	2	yes	—
68	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
69	#389	—	2	2	no	—
70	#366	—	2	2	no	—
71	#366	C₄×C₅	1	4	yes	complemented
72	C₄×C₁₆	—	2	2	no	—
73	#340	—	2	2	no	—
74	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₅	—	2	1	no	—
76	C₂×C₄×C₅	—	2	1	no	—
77	C₂×C₄×C₅	—	2	1	no	—
78	C₂×C₄×C₅	—	2	1	no	—
79	C₂×C₄×C₅	—	4	4	no	—
80	C₂×C₄×C₅	—	8	2	no	—
81	C₂×C₄×C₅	—	8	2	no	—
82	C₂×C₄×C₅	—	8	2	no	—
83	C₂×C₄×C₅	—	8	2	no	—
84	C₂×C₄×C₅	—	8	2	no	—
85	C₂×C₄×C₅	D₁₆	1	2	yes	—
86	C₂×C₄×C₅	Q₃₂	1	2	yes	—
87	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅	#121	1	1	yes	characteristic
89	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅	#135	1	1	yes	characteristic
92	C₂×C₄×C₅	#135	1	1	yes	characteristic
93	C₅×C₈	—	2	2	no	—
94	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
96	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
97	C₅×C₈	#125	1	1	yes	characteristic
98	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
99	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	4	2	no	—
103	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	—	2	1	no	—
106	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
107	C₂×C₁₆	C₅×D₄	1	2	yes	—
108	C₂×C₁₆	C₅×Q₈	1	2	yes	—
109	C₂×C₁₆	C₅×Q₈	1	2	yes	—
110	#107	—	2	2	no	—
111	#107	—	2	2	no	—
112	#107	—	2	2	no	—
113	#107	C₂×C₄×C₅	1	4	yes	—
114	C₄×C₈	—	2	1	no	—
115	C₄×C₈	—	2	1	no	—
116	#95	—	4	2	no	—
117	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
118	C₂²×C₅	#466	1	1	yes	characteristic
119	C₂²×C₅	#500	1	1	yes	characteristic
120	C₂²×C₅	C₂×D₁₆	1	1	yes	characteristic
121	C₂²×C₅	C₂×Q₃₂	1	1	yes	characteristic
122	C₂²×C₅	#508	1	1	yes	characteristic
123	C₂²×C₅	#508	1	1	yes	characteristic
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	—	8	4	no	—
127	C₄×C₅	—	16	2	no	—
128	C₄×C₅	#389	1	1	yes	characteristic
129	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
130	C₄×C₅	#467	1	1	yes	characteristic
131	C₄×C₅	#499	1	1	yes	characteristic
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
135	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	C₅×Q₁₆	1	2	yes	—
140	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
141	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
142	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
143	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
144	C₂×C₈	C₅×#41	1	1	yes	characteristic
145	C₂×C₈	C₅×#41	1	1	yes	characteristic
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	8	4	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	2	1	no	—
153	C₁₆	—	2	2	no	—
154	C₁₆	C₅×C₄⋊C₄	1	4	yes	complemented
155	C₂×C₅	#1942	1	1	yes	characteristic
156	C₂×C₅	C₄×D₁₆	1	1	yes	characteristic
157	C₂×C₅	C₄×Q₃₂	1	1	yes	characteristic
158	C₂×C₅	#2215	1	1	yes	characteristic
159	C₂×C₅	#2216	1	1	yes	characteristic
160	C₂×C₅	#2252	1	1	yes	characteristic
161	C₂×C₅	#2253	1	1	yes	characteristic
162	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	C₅×D₁₆	1	2	yes	—
174	C₂×C₄	C₅×Q₃₂	1	2	yes	—
175	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
176	C₂×C₄	C₅×#121	1	1	yes	characteristic
177	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
178	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
179	C₂×C₄	C₅×#135	1	1	yes	characteristic
180	C₂×C₄	C₅×#135	1	1	yes	characteristic
181	C₈	—	2	2	no	—
182	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
183	C₈	C₄×C₅×Q₈	1	1	yes	characteristic
184	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
185	C₈	C₅×#125	1	1	yes	characteristic
186	C₅	#13165	1	1	yes	characteristic, direct product factor, Sylow
187	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
188	C₂²	C₅×#466	1	1	yes	characteristic
189	C₂²	C₅×#500	1	1	yes	characteristic
190	C₂²	C₂×C₅×D₁₆	1	1	yes	characteristic
191	C₂²	C₂×C₅×Q₃₂	1	1	yes	characteristic
192	C₂²	C₅×#508	1	1	yes	characteristic
193	C₂²	C₅×#508	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	8	4	no	—
197	C₄	—	16	2	no	—
198	C₄	C₅×#389	1	1	yes	characteristic
199	C₄	C₄×C₅×Q₁₆	1	1	yes	characteristic
200	C₄	C₅×#467	1	1	yes	characteristic
201	C₄	C₅×#499	1	1	yes	characteristic
202	C₂	C₅×#1942	1	1	yes	characteristic
203	C₂	C₄×C₅×D₁₆	1	1	yes	characteristic
204	C₂	C₄×C₅×Q₃₂	1	1	yes	characteristic
205	C₂	C₅×#2215	1	1	yes	characteristic
206	C₂	C₅×#2216	1	1	yes	characteristic
207	C₂	C₅×#2252	1	1	yes	characteristic
208	C₂	C₅×#2253	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅×#366	C₄	semidirect
16. #13165	C₅	direct (N × H)
63. C₅×C₁₆	C₄⋊C₄	semidirect
71. #366	C₄×C₅	semidirect
154. C₁₆	C₅×C₄⋊C₄	semidirect
186. C₅	#13165	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×#366	2
C₂×C₄×C₅×C₁₆	2
C₅×#1942	2
C₅×#1380	2
#13165	5

Subgroups — C5×#13165 #11935885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13165 #11935885