Subgroups — C₅×#13265 #11935985

Order 1280.

338 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2960	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₅×#1379	C₂	1	4	yes	maximal
5	C₅×#533	C₄	1	1	yes	characteristic
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
8	C₅×#446	—	2	1	no	—
9	C₅×#446	—	2	1	no	—
10	C₅×C₈×Q₈	—	2	1	no	—
11	C₅×C₈×Q₈	—	2	1	no	—
12	C₅×#424	C₄	1	2	yes	—
13	C₅×#423	C₄	1	1	yes	characteristic
14	C₅×#405	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×C₄⋊C₁₆	—	2	1	no	—
17	C₅×C₄⋊C₁₆	—	2	1	no	—
18	C₅×#346	—	2	1	no	—
19	C₄×C₅×C₁₆	—	2	1	no	—
20	#13265	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
24	C₅×C₄⋊Q₈	C₈	1	1	yes	characteristic
25	C₅×C₄⋊Q₈	C₈	1	1	yes	characteristic
26	C₅×#125	C₈	1	2	yes	—
27	C₅×C₂²⋊Q₈	—	2	1	no	—
28	C₅×C₂²⋊Q₈	—	2	1	no	—
29	C₄×C₅×Q₈	—	2	1	no	—
30	C₄×C₅×Q₈	—	2	1	no	—
31	C₅×#117	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₅×M₃₂	—	2	2	no	—
34	C₅×M₃₂	Q₈	1	4	yes	complemented
35	C₂×C₅×C₁₆	—	2	2	no	—
36	C₂×C₅×C₁₆	—	2	2	no	—
37	C₂×C₅×C₁₆	—	2	4	no	—
38	C₅×C₄⋊C₈	—	2	2	no	—
39	C₅×C₄⋊C₈	—	2	2	no	—
40	C₅×C₄⋊C₈	—	2	1	no	—
41	C₅×C₄⋊C₈	—	2	1	no	—
42	C₅×#97	—	2	2	no	—
43	C₄×C₅×C₈	—	2	2	no	—
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	C₄×C₅×C₈	D₄	1	1	yes	characteristic
47	C₄×C₅×C₈	D₄	1	1	yes	characteristic
48	#2960	C₂×C₅	1	1	yes	characteristic
49	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
50	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic
51	#1379	C₂×C₅	1	4	yes	—
52	C₂×C₅×Q₈	—	2	1	no	—
53	C₂×C₅×Q₈	—	2	1	no	—
54	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₅×M₁₆	—	2	4	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
65	C₂×C₅×C₈	#41	1	2	yes	—
66	C₅×C₄⋊C₄	—	2	2	no	—
67	C₅×C₄⋊C₄	—	2	2	no	—
68	C₅×C₄⋊C₄	—	2	1	no	—
69	C₅×C₄⋊C₄	—	2	1	no	—
70	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
71	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
72	C₅×C₂²⋊C₄	—	2	2	no	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
77	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
78	C₅×C₁₆	—	2	4	no	—
79	C₅×C₁₆	—	4	2	no	—
80	#533	C₄×C₅	1	1	yes	characteristic
81	C₂×M₃₂	C₂²×C₅	1	2	yes	—
82	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
83	#446	—	2	1	no	—
84	#446	—	2	1	no	—
85	C₈×Q₈	—	2	1	no	—
86	C₈×Q₈	—	2	1	no	—
87	#424	C₄×C₅	1	2	yes	—
88	#423	C₄×C₅	1	1	yes	characteristic
89	#405	C₂²×C₅	1	2	yes	—
90	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
91	C₄⋊C₁₆	—	2	1	no	—
92	C₄⋊C₁₆	—	2	1	no	—
93	#346	—	2	1	no	—
94	C₄×C₁₆	—	2	1	no	—
95	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
96	C₅×Q₈	—	4	2	no	—
97	C₅×Q₈	—	4	2	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	4	1	no	—
102	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅	#131	1	2	yes	—
110	C₅×C₈	—	2	1	no	—
111	C₅×C₈	—	2	1	no	—
112	C₅×C₈	—	4	2	no	—
113	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
114	C₅×C₈	#124	1	1	yes	characteristic
115	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
117	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
119	C₄⋊Q₈	C₅×C₈	1	1	yes	characteristic
120	C₄⋊Q₈	C₅×C₈	1	1	yes	characteristic
121	#125	C₅×C₈	1	2	yes	—
122	C₂²⋊Q₈	—	2	1	no	—
123	C₂²⋊Q₈	—	2	1	no	—
124	C₄×Q₈	—	2	1	no	—
125	C₄×Q₈	—	2	1	no	—
126	#117	C₂×C₄×C₅	1	2	yes	—
127	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	C₅×Q₈	1	4	yes	complemented
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	2	4	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	1	no	—
136	C₄⋊C₈	—	2	1	no	—
137	#97	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	C₅×D₄	1	1	yes	characteristic
140	C₄×C₈	C₅×D₄	1	1	yes	characteristic
141	C₄×C₈	C₅×D₄	1	1	yes	characteristic
142	C₄×C₈	C₅×D₄	1	1	yes	characteristic
143	C₂²×C₅	—	2	1	no	—
144	C₂²×C₅	—	2	1	no	—
145	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₂²×C₅	#446	1	2	yes	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	4	2	no	—
151	C₄×C₅	—	4	2	no	—
152	C₄×C₅	#350	1	2	yes	—
153	C₄×C₅	#391	1	1	yes	characteristic
154	C₄×C₅	#408	1	1	yes	characteristic
155	C₄×C₅	C₈×Q₈	1	2	yes	—
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	2	1	no	—
158	C₂²×C₄	C₅×M₁₆	1	1	yes	characteristic
159	C₂²×C₄	C₅×M₁₆	1	1	yes	characteristic
160	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
161	M₁₆	—	2	4	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
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	2	yes	—
169	C₂×C₈	C₅×#41	1	2	yes	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	C₂×C₅×C₈	1	2	yes	—
175	C₄⋊C₄	C₂×C₅×C₈	1	2	yes	—
176	C₂²⋊C₄	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
179	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
180	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
181	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
182	C₁₆	—	2	4	no	—
183	C₁₆	—	4	2	no	—
184	C₂×C₅	—	2	2	no	—
185	C₂×C₅	#1984	1	1	yes	characteristic
186	C₂×C₅	C₂×#350	1	1	yes	characteristic
187	C₂×C₅	#2117	1	1	yes	characteristic
188	C₂³	C₂×C₅×M₁₆	1	1	yes	characteristic
189	Q₈	—	4	2	no	—
190	Q₈	—	4	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
196	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
197	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
199	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
201	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
202	C₂×C₄	C₅×#131	1	2	yes	—
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	4	2	no	—
206	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
207	C₈	C₅×#124	1	1	yes	characteristic
208	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
209	C₅	#13265	1	1	yes	characteristic, direct product factor, Sylow
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
213	C₂²	C₅×#446	1	2	yes	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	4	2	no	—
218	C₄	—	4	2	no	—
219	C₄	C₅×#350	1	2	yes	—
220	C₄	C₅×#391	1	1	yes	characteristic
221	C₄	C₅×#408	1	1	yes	characteristic
222	C₄	C₅×C₈×Q₈	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	C₅×#1984	1	1	yes	characteristic
225	C₂	C₂×C₅×#350	1	1	yes	characteristic
226	C₂	C₅×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂³×C₅	C₂×M₁₆
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
20. #13265	C₅	direct (N × H)
34. C₅×M₃₂	Q₈	semidirect
129. M₃₂	C₅×Q₈	semidirect
209. C₅	#13265	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×#2960	2
C₂×C₅×C₄⋊C₁₆	2
C₄×C₅×M₃₂	2
C₅×#1379	2
#13265	5

Subgroups — C5×#13265 #11935985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13265 #11935985