Subgroups — C₅×#13251 #11935971

Order 1280.

478 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#359	C₂	1	4	yes	maximal, complemented
4	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#135	—	2	1	no	—
6	C₂×C₅×#135	C₂²	1	2	yes	—
7	C₅×#532	—	2	1	no	—
8	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
9	C₅×#495	—	2	1	no	—
10	C₅×#495	C₄	1	2	yes	—
11	C₅×#494	C₄	1	1	yes	characteristic
12	C₅×#493	C₄	1	1	yes	characteristic
13	C₅×#492	—	2	1	no	—
14	C₂×C₅×#108	C₂²	1	2	yes	—
15	C₄×C₅×M₁₆	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×C₄⋊C₁₆	—	2	1	no	—
18	C₅×C₄⋊C₁₆	—	2	1	no	—
19	C₅×#359	—	2	8	no	—
20	C₅×#341	—	2	1	no	—
21	#13251	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×#41	—	2	2	no	—
23	C₅×#135	—	2	4	no	—
24	C₅×#135	—	2	4	no	—
25	C₅×#135	—	4	4	no	—
26	C₂×C₅×Q₁₆	—	2	1	no	—
27	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅×SD₁₆	—	2	2	no	—
29	C₂×C₅×SD₁₆	—	2	2	no	—
30	C₂×C₅×D₈	—	2	1	no	—
31	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
32	C₂×C₅×M₁₆	—	2	2	no	—
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₅×C₄⋊Q₈	—	2	1	no	—
36	C₅×#127	—	2	1	no	—
37	C₅×#124	—	2	2	no	—
38	C₅×#121	—	4	1	no	—
39	C₅×#121	—	4	1	no	—
40	C₄×C₅×D₄	—	4	1	no	—
41	C₄×C₅×D₄	—	4	1	no	—
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₂×C₅×C₁₆	—	2	4	no	—
44	C₂×C₅×C₁₆	—	2	2	no	—
45	C₅×#108	—	2	2	no	—
46	C₅×#108	D₄	1	4	yes	complemented
47	C₅×#97	—	4	1	no	—
48	C₄×C₅×C₈	—	4	1	no	—
49	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₄×C₅×C₈	D₄	1	1	yes	characteristic
53	#3147	C₂×C₅	1	1	yes	characteristic
54	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×#359	C₂×C₅	1	4	yes	complemented
56	C₄×#108	C₂×C₅	1	1	yes	characteristic, complemented
57	C₅×#41	—	4	4	no	—
58	C₅×#41	—	4	4	no	—
59	C₂×C₅×Q₈	—	2	2	no	—
60	C₂×C₅×D₄	—	2	2	no	—
61	C₂×C₅×D₄	—	4	2	no	—
62	C₂×C₅×D₄	—	4	1	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂²×C₄×C₅	—	4	2	no	—
65	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₅×Q₁₆	—	2	4	no	—
69	C₅×Q₁₆	—	4	2	no	—
70	C₅×SD₁₆	—	4	4	no	—
71	C₅×SD₁₆	—	4	2	no	—
72	C₅×SD₁₆	—	4	2	no	—
73	C₅×D₈	—	2	4	no	—
74	C₅×D₈	—	4	2	no	—
75	C₅×M₁₆	—	2	4	no	—
76	C₅×M₁₆	—	4	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	4	2	no	—
79	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×C₈	#41	1	2	yes	—
85	C₅×C₄⋊C₄	—	4	1	no	—
86	C₅×C₄⋊C₄	—	4	1	no	—
87	C₅×C₂²⋊C₄	—	4	2	no	—
88	C₅×C₂²⋊C₄	—	4	2	no	—
89	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
90	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₅	D₈	1	1	yes	characteristic
92	C₄²×C₅	D₈	1	1	yes	characteristic
93	C₅×C₁₆	—	4	2	no	—
94	C₂×#135	—	2	1	no	—
95	C₂×#135	C₂²×C₅	1	2	yes	—
96	#532	—	2	1	no	—
97	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
98	#495	—	2	1	no	—
99	#495	C₄×C₅	1	2	yes	—
100	#494	C₄×C₅	1	1	yes	characteristic
101	#493	C₄×C₅	1	1	yes	characteristic
102	#492	—	2	1	no	—
103	C₂×#108	C₂²×C₅	1	2	yes	—
104	C₄×M₁₆	—	2	1	no	—
105	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
106	C₄⋊C₁₆	—	2	1	no	—
107	C₄⋊C₁₆	—	2	1	no	—
108	#359	—	2	8	no	—
109	#341	—	2	1	no	—
110	C₂³×C₅	—	4	2	no	—
111	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
112	C₅×Q₈	—	4	2	no	—
113	C₅×Q₈	—	4	2	no	—
114	C₅×D₄	—	4	2	no	—
115	C₅×D₄	—	4	2	no	—
116	C₅×D₄	—	8	2	no	—
117	C₅×D₄	—	8	4	no	—
118	C₅×D₄	—	8	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	4	1	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	—	8	4	no	—
123	C₂×C₄×C₅	#102	1	1	yes	characteristic
124	C₂×C₄×C₅	#102	1	1	yes	characteristic
125	C₂×C₄×C₅	#102	1	1	yes	characteristic
126	C₂×C₄×C₅	#102	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	#121	1	2	yes	—
129	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
131	C₅×C₈	—	2	1	no	—
132	C₅×C₈	—	2	1	no	—
133	C₅×C₈	—	4	4	no	—
134	C₅×C₈	C₄×D₄	1	2	yes	—
135	C₅×C₈	#124	1	1	yes	characteristic
136	C₅×C₈	#127	1	1	yes	characteristic
137	C₂×#41	—	2	2	no	—
138	#135	—	2	4	no	—
139	#135	—	2	4	no	—
140	#135	—	4	4	no	—
141	C₂×Q₁₆	—	2	1	no	—
142	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
143	C₂×SD₁₆	—	2	2	no	—
144	C₂×SD₁₆	—	2	2	no	—
145	C₂×D₈	—	2	1	no	—
146	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
150	C₄⋊Q₈	—	2	1	no	—
151	#127	—	2	1	no	—
152	#124	—	2	2	no	—
153	#121	—	4	1	no	—
154	#121	—	4	1	no	—
155	C₄×D₄	—	4	1	no	—
156	C₄×D₄	—	4	1	no	—
157	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
158	C₂×C₁₆	—	2	4	no	—
159	C₂×C₁₆	—	2	2	no	—
160	#108	—	2	2	no	—
161	#108	C₅×D₄	1	4	yes	complemented
162	#97	—	4	1	no	—
163	C₄×C₈	—	4	1	no	—
164	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
165	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
166	C₄×C₈	C₅×D₄	1	1	yes	characteristic
167	C₄×C₈	C₅×D₄	1	1	yes	characteristic
168	C₂²×C₅	—	2	1	no	—
169	C₂²×C₅	—	2	1	no	—
170	C₂²×C₅	—	8	2	no	—
171	C₂²×C₅	—	8	2	no	—
172	C₂²×C₅	—	8	2	no	—
173	C₂²×C₅	C₂×#102	1	1	yes	characteristic
174	C₂²×C₅	#460	1	2	yes	—
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	—	2	1	no	—
178	C₄×C₅	—	8	2	no	—
179	C₄×C₅	#359	1	2	yes	complemented
180	C₄×C₅	#390	1	1	yes	characteristic
181	C₄×C₅	#418	1	1	yes	characteristic
182	C₄×C₅	#459	1	2	yes	—
183	#41	—	4	4	no	—
184	#41	—	4	4	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
192	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	Q₁₆	—	2	4	no	—
195	Q₁₆	—	4	2	no	—
196	SD₁₆	—	4	4	no	—
197	SD₁₆	—	4	2	no	—
198	SD₁₆	—	4	2	no	—
199	D₈	—	2	4	no	—
200	D₈	—	4	2	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	4	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
208	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
209	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂×C₈	C₅×#41	1	2	yes	—
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₂²⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
216	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
217	C₄²	C₅×D₈	1	1	yes	characteristic
218	C₄²	C₅×D₈	1	1	yes	characteristic
219	C₁₆	—	4	2	no	—
220	C₂×C₅	—	2	2	no	—
221	C₂×C₅	—	16	2	no	—
222	C₂×C₅	#1982	1	1	yes	characteristic
223	C₂×C₅	C₂×#359	1	1	yes	characteristic
224	C₂×C₅	#2143	1	1	yes	characteristic
225	C₂³	—	4	2	no	—
226	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
227	Q₈	—	4	2	no	—
228	Q₈	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	8	2	no	—
232	D₄	—	8	4	no	—
233	D₄	—	8	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	C₅×#102	1	1	yes	characteristic
239	C₂×C₄	C₅×#102	1	1	yes	characteristic
240	C₂×C₄	C₅×#102	1	1	yes	characteristic
241	C₂×C₄	C₅×#102	1	1	yes	characteristic
242	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₅×#121	1	2	yes	—
244	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
245	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	4	4	no	—
249	C₈	C₄×C₅×D₄	1	2	yes	—
250	C₈	C₅×#124	1	1	yes	characteristic
251	C₈	C₅×#127	1	1	yes	characteristic
252	C₅	#13251	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	8	2	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	C₂×C₅×#102	1	1	yes	characteristic
259	C₂²	C₅×#460	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	8	2	no	—
264	C₄	C₅×#359	1	2	yes	complemented
265	C₄	C₅×#390	1	1	yes	characteristic
266	C₄	C₅×#418	1	1	yes	characteristic
267	C₄	C₅×#459	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	16	2	no	—
270	C₂	C₅×#1982	1	1	yes	characteristic
271	C₂	C₂×C₅×#359	1	1	yes	characteristic
272	C₂	C₅×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#102
prime radical	C₅×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
2. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₅×#359	C₂	semidirect
4. C₄×C₅×#108	C₂	semidirect
21. #13251	C₅	direct (N × H)
46. C₅×#108	D₄	semidirect
54. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
55. C₂×#359	C₂×C₅	semidirect
56. C₄×#108	C₂×C₅	semidirect
161. #108	C₅×D₄	semidirect
179. C₄×C₅	#359	semidirect
252. C₅	#13251	direct (N × H)
264. C₄	C₅×#359	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#3147	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#359	2
C₄×C₅×#108	2
#13251	5

Subgroups — C5×#13251 #11935971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13251 #11935971