Subgroups — C₅×#13160 #11935880

Order 1280.

338 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₅×#1374	C₂	1	2	yes	maximal
6	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
7	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₅×C₄⋊C₁₆	—	2	4	no	—
14	C₅×C₄⋊C₁₆	—	2	2	no	—
15	C₅×#346	—	2	2	no	—
16	C₅×#336	C₄	1	2	yes	—
17	#13160	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
22	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₅×C₁₆	—	2	1	no	—
25	C₂×C₅×C₁₆	—	2	1	no	—
26	C₂×C₅×C₁₆	—	2	1	no	—
27	C₂×C₅×C₁₆	—	2	1	no	—
28	C₂×C₅×C₁₆	—	2	2	no	—
29	C₂×C₅×C₁₆	—	2	2	no	—
30	C₂×C₅×C₁₆	—	2	2	no	—
31	C₂×C₅×C₁₆	D₄	1	2	yes	—
32	C₂×C₅×C₁₆	Q₈	1	2	yes	—
33	C₅×C₄⋊C₈	—	2	2	no	—
34	C₄×C₅×C₈	—	2	2	no	—
35	C₄×C₅×C₈	—	2	2	no	—
36	C₄×C₅×C₈	D₄	1	2	yes	—
37	C₄×C₅×C₈	Q₈	1	2	yes	—
38	C₅×#95	C₈	1	1	yes	characteristic
39	C₅×#95	C₈	1	1	yes	characteristic
40	C₂×C₄⋊C₁₆	C₂×C₅	1	2	yes	—
41	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
42	C₂×#346	C₂×C₅	1	1	yes	characteristic
43	C₈×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
44	#1374	C₂×C₅	1	2	yes	—
45	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	—	2	1	no	—
51	C₂×C₅×C₈	—	2	1	no	—
52	C₂×C₅×C₈	—	2	2	no	—
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
56	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₅×C₈	#41	1	1	yes	characteristic
61	C₂×C₅×C₈	#41	1	1	yes	characteristic
62	C₅×C₄⋊C₄	—	2	4	no	—
63	C₄²×C₅	—	2	2	no	—
64	C₄²×C₅	—	2	2	no	—
65	C₄²×C₅	C₄⋊C₄	1	2	yes	—
66	C₄²×C₅	M₁₆	1	2	yes	—
67	C₅×C₁₆	—	2	2	no	—
68	C₅×C₁₆	—	2	2	no	—
69	C₅×C₁₆	—	4	2	no	—
70	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
71	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
72	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₄⋊C₈	C₄×C₅	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
75	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
76	C₄×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
77	C₄⋊C₁₆	—	2	4	no	—
78	C₄⋊C₁₆	—	2	2	no	—
79	#346	—	2	2	no	—
80	#336	C₄×C₅	1	2	yes	—
81	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
90	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₅	M₃₂	1	4	yes	—
92	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅	#131	1	1	yes	characteristic
97	C₂×C₄×C₅	#131	1	1	yes	characteristic
98	C₅×C₈	—	2	2	no	—
99	C₅×C₈	—	4	2	no	—
100	C₅×C₈	#121	1	1	yes	characteristic
101	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
102	C₅×C₈	#125	1	1	yes	characteristic
103	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
105	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
107	C₂×C₄⋊C₄	C₅×C₈	1	2	yes	—
108	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
109	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
110	C₂×C₁₆	—	2	1	no	—
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	—	2	2	no	—
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	C₅×D₄	1	2	yes	—
118	C₂×C₁₆	C₅×Q₈	1	2	yes	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₅×D₄	1	2	yes	—
123	C₄×C₈	C₅×Q₈	1	2	yes	—
124	#95	C₅×C₈	1	1	yes	characteristic
125	#95	C₅×C₈	1	1	yes	characteristic
126	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
127	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
128	C₂²×C₅	C₈×Q₈	1	1	yes	characteristic
129	C₂²×C₅	C₂×M₃₂	1	2	yes	—
130	C₂²×C₅	#504	1	2	yes	—
131	C₄×C₅	—	2	2	no	—
132	C₄×C₅	—	2	2	no	—
133	C₄×C₅	—	2	4	no	—
134	C₄×C₅	—	4	2	no	—
135	C₄×C₅	#389	1	1	yes	characteristic
136	C₄×C₅	#424	1	1	yes	characteristic
137	C₄×C₅	#436	1	1	yes	characteristic
138	C₄×C₅	#446	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
140	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
143	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
150	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
151	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
152	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
153	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
154	C₂×C₈	C₅×#41	1	1	yes	characteristic
155	C₂×C₈	C₅×#41	1	1	yes	characteristic
156	C₄⋊C₄	—	2	4	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	C₅×C₄⋊C₄	1	2	yes	—
160	C₄²	C₅×M₁₆	1	2	yes	—
161	C₁₆	—	2	2	no	—
162	C₁₆	—	2	2	no	—
163	C₁₆	—	4	2	no	—
164	C₂×C₅	#1939	1	1	yes	characteristic
165	C₂×C₅	#2150	1	1	yes	characteristic
166	C₂×C₅	#2151	1	1	yes	characteristic
167	C₂×C₅	#2168	1	2	yes	—
168	C₂×C₅	#2183	1	2	yes	—
169	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	C₅×C₄⋊C₈	1	2	yes	—
178	C₂×C₄	C₅×C₄⋊C₈	1	2	yes	—
179	C₂×C₄	C₅×M₃₂	1	4	yes	—
180	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
182	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
183	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
184	C₂×C₄	C₅×#131	1	1	yes	characteristic
185	C₂×C₄	C₅×#131	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	4	2	no	—
188	C₈	C₅×#121	1	1	yes	characteristic
189	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
190	C₈	C₅×#125	1	1	yes	characteristic
191	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
192	C₅	#13160	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
194	C₂²	C₅×C₈×D₄	1	1	yes	characteristic
195	C₂²	C₅×C₈×Q₈	1	1	yes	characteristic
196	C₂²	C₂×C₅×M₃₂	1	2	yes	—
197	C₂²	C₅×#504	1	2	yes	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	4	no	—
201	C₄	—	4	2	no	—
202	C₄	C₅×#389	1	1	yes	characteristic
203	C₄	C₅×#424	1	1	yes	characteristic
204	C₄	C₅×#436	1	1	yes	characteristic
205	C₄	C₅×#446	1	1	yes	characteristic
206	C₂	C₅×#1939	1	1	yes	characteristic
207	C₂	C₅×#2150	1	1	yes	characteristic
208	C₂	C₅×#2151	1	1	yes	characteristic
209	C₂	C₅×#2168	1	2	yes	—
210	C₂	C₅×#2183	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
17. #13160	C₅	direct (N × H)
192. C₅	#13160	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×C₄⋊C₁₆	2
C₂×C₅×#346	2
C₅×C₈×C₄⋊C₄	2
C₅×#1374	2
#13160	5

Subgroups — C5×#13160 #11935880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13160 #11935880