Subgroups — C₅×#12825 #11935545

Order 1280.

400 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2148	C₂	1	2	yes	maximal
4	C₅×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1385	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#135	C₂²	1	1	yes	characteristic
7	C₂×C₅×#131	—	2	1	no	—
8	C₂×C₅×M₃₂	C₂²	1	2	yes	—
9	C₅×#443	—	2	1	no	—
10	C₅×#443	—	2	1	no	—
11	C₅×#443	C₄	1	2	yes	—
12	C₅×#443	C₄	1	2	yes	—
13	C₂×C₅×#108	—	2	1	no	—
14	C₂×C₅×#108	C₂²	1	2	yes	—
15	C₂×C₅×#108	C₂²	1	1	yes	characteristic
16	C₅×#423	—	2	1	no	—
17	C₂×C₅×#104	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#364	—	2	1	no	—
20	C₅×#364	—	2	1	no	—
21	C₅×#362	—	2	2	no	—
22	C₅×#361	—	2	2	no	—
23	#12825	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×#41	—	2	1	no	—
25	C₅×#135	—	2	1	no	—
26	C₅×#135	—	2	1	no	—
27	C₅×#135	C₂×C₄	1	2	yes	—
28	C₅×#135	Q₈	1	2	yes	—
29	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
30	C₂×C₅×SD₁₆	—	2	1	no	—
31	C₂×C₅×D₈	D₄	1	1	yes	characteristic
32	C₅×#131	—	2	2	no	—
33	C₅×#131	—	2	2	no	—
34	C₅×#131	—	4	1	no	—
35	C₅×#131	—	4	1	no	—
36	C₂×C₅×M₁₆	—	2	1	no	—
37	C₂×C₅×M₁₆	—	2	2	no	—
38	C₂×C₅×M₁₆	—	4	1	no	—
39	C₂²×C₅×C₈	—	4	1	no	—
40	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₅×M₃₂	—	2	2	no	—
44	C₅×M₃₂	—	2	2	no	—
45	C₂×C₅×C₁₆	—	2	2	no	—
46	C₅×#108	—	2	1	no	—
47	C₅×#108	—	2	2	no	—
48	C₅×#108	—	2	2	no	—
49	C₅×#108	—	4	1	no	—
50	C₅×#108	—	4	1	no	—
51	C₅×#108	C₂×C₄	1	2	yes	—
52	C₅×C₄⋊C₈	—	4	2	no	—
53	C₅×#104	—	2	2	no	—
54	C₅×#104	—	2	2	no	—
55	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
57	C₄×C₅×C₈	D₄	1	1	yes	characteristic
58	C₄×C₅×C₈	D₄	1	1	yes	characteristic
59	C₂×#443	C₂×C₅	1	1	yes	characteristic
60	C₂×#364	C₂×C₅	1	1	yes	characteristic, complemented
61	#2148	C₂×C₅	1	2	yes	—
62	#1848	C₂×C₅	1	1	yes	characteristic, complemented
63	#1385	C₂×C₅	1	2	yes	complemented
64	C₅×#41	—	2	2	no	—
65	C₅×#41	—	2	2	no	—
66	C₅×#41	—	4	1	no	—
67	C₅×#41	—	4	1	no	—
68	C₂×C₅×Q₈	—	2	1	no	—
69	C₂×C₅×D₄	—	2	1	no	—
70	C₂×C₅×D₄	—	4	1	no	—
71	C₂²×C₄×C₅	—	2	1	no	—
72	C₂²×C₄×C₅	—	4	1	no	—
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₅×Q₁₆	—	2	1	no	—
75	C₅×Q₁₆	C₄⋊C₄	1	2	yes	—
76	C₅×SD₁₆	—	2	2	no	—
77	C₅×SD₁₆	—	2	2	no	—
78	C₅×D₈	—	2	1	no	—
79	C₅×D₈	C₄⋊C₄	1	2	yes	—
80	C₅×M₁₆	—	2	2	no	—
81	C₅×M₁₆	—	4	1	no	—
82	C₅×M₁₆	—	4	2	no	—
83	C₅×M₁₆	—	4	2	no	—
84	C₅×M₁₆	—	4	2	no	—
85	C₅×M₁₆	—	4	2	no	—
86	C₂×C₅×C₈	—	4	1	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	—	4	2	no	—
90	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	D₈	1	1	yes	characteristic
93	C₂×C₅×C₈	D₈	1	1	yes	characteristic
94	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
95	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₅×C₈	#41	1	1	yes	characteristic
101	C₂×C₅×C₈	#41	1	1	yes	characteristic
102	C₄²×C₅	—	2	1	no	—
103	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
105	C₅×C₁₆	—	4	2	no	—
106	C₂×#135	C₂²×C₅	1	1	yes	characteristic
107	C₂×#131	—	2	1	no	—
108	C₂×M₃₂	C₂²×C₅	1	2	yes	—
109	#443	—	2	1	no	—
110	#443	—	2	1	no	—
111	#443	C₄×C₅	1	2	yes	—
112	#443	C₄×C₅	1	2	yes	—
113	C₂×#108	—	2	1	no	—
114	C₂×#108	C₂²×C₅	1	2	yes	—
115	C₂×#108	C₂²×C₅	1	1	yes	characteristic
116	#423	—	2	1	no	—
117	C₂×#104	—	2	1	no	—
118	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
119	#364	—	2	1	no	—
120	#364	—	2	1	no	—
121	#362	—	2	2	no	—
122	#361	—	2	2	no	—
123	C₂³×C₅	—	4	1	no	—
124	C₂³×C₅	#137	1	1	yes	characteristic
125	C₅×Q₈	—	2	2	no	—
126	C₅×Q₈	—	4	1	no	—
127	C₅×D₄	—	2	2	no	—
128	C₅×D₄	—	4	2	no	—
129	C₅×D₄	—	4	2	no	—
130	C₅×D₄	—	4	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	#121	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₅×C₈	—	4	2	no	—
144	C₅×C₈	—	8	2	no	—
145	C₅×C₈	#102	1	1	yes	characteristic
146	C₅×C₈	#102	1	1	yes	characteristic
147	C₅×C₈	#102	1	1	yes	characteristic
148	C₅×C₈	#102	1	1	yes	characteristic
149	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₅×C₈	#117	1	1	yes	characteristic
151	C₅×C₈	#121	1	1	yes	characteristic
152	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
153	C₂×#41	—	2	1	no	—
154	#135	—	2	1	no	—
155	#135	—	2	1	no	—
156	#135	C₂×C₄×C₅	1	2	yes	—
157	#135	C₅×Q₈	1	2	yes	—
158	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
159	C₂×SD₁₆	—	2	1	no	—
160	C₂×D₈	C₅×D₄	1	1	yes	characteristic
161	#131	—	2	2	no	—
162	#131	—	2	2	no	—
163	#131	—	4	1	no	—
164	#131	—	4	1	no	—
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	4	1	no	—
168	C₂²×C₈	—	4	1	no	—
169	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
171	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
172	M₃₂	—	2	2	no	—
173	M₃₂	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	#108	—	2	1	no	—
176	#108	—	2	2	no	—
177	#108	—	2	2	no	—
178	#108	—	4	1	no	—
179	#108	—	4	1	no	—
180	#108	C₂×C₄×C₅	1	2	yes	—
181	C₄⋊C₈	—	4	2	no	—
182	#104	—	2	2	no	—
183	#104	—	2	2	no	—
184	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
185	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
186	C₄×C₈	C₅×D₄	1	1	yes	characteristic
187	C₄×C₈	C₅×D₄	1	1	yes	characteristic
188	C₂²×C₅	—	4	1	no	—
189	C₂²×C₅	—	4	2	no	—
190	C₂²×C₅	—	8	2	no	—
191	C₂²×C₅	#417	1	1	yes	characteristic
192	C₂²×C₅	#461	1	1	yes	characteristic
193	C₂²×C₅	#474	1	1	yes	characteristic
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	4	1	no	—
196	C₄×C₅	—	4	2	no	—
197	C₄×C₅	#380	1	1	yes	characteristic
198	C₄×C₅	C₂×#102	1	1	yes	characteristic
199	C₄×C₅	#459	1	1	yes	characteristic
200	C₄×C₅	#476	1	1	yes	characteristic
201	#41	—	2	2	no	—
202	#41	—	2	2	no	—
203	#41	—	4	1	no	—
204	#41	—	4	1	no	—
205	C₂×Q₈	—	2	1	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
211	Q₁₆	—	2	1	no	—
212	Q₁₆	C₅×C₄⋊C₄	1	2	yes	—
213	SD₁₆	—	2	2	no	—
214	SD₁₆	—	2	2	no	—
215	D₈	—	2	1	no	—
216	D₈	C₅×C₄⋊C₄	1	2	yes	—
217	M₁₆	—	2	2	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₈	C₅×D₈	1	1	yes	characteristic
230	C₂×C₈	C₅×D₈	1	1	yes	characteristic
231	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
232	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
233	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
234	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
235	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
236	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
237	C₂×C₈	C₅×#41	1	1	yes	characteristic
238	C₂×C₈	C₅×#41	1	1	yes	characteristic
239	C₄²	—	2	1	no	—
240	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
241	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
242	C₁₆	—	4	2	no	—
243	C₂×C₅	—	2	1	no	—
244	C₂×C₅	—	2	1	no	—
245	C₂×C₅	—	2	1	no	—
246	C₂×C₅	—	8	2	no	—
247	C₂×C₅	#1802	1	1	yes	characteristic
248	C₂³	—	4	1	no	—
249	C₂³	C₅×#137	1	1	yes	characteristic
250	Q₈	—	2	2	no	—
251	Q₈	—	4	1	no	—
252	D₄	—	2	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₅×#121	1	1	yes	characteristic
264	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
265	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
266	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
267	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
268	C₈	—	4	2	no	—
269	C₈	—	8	2	no	—
270	C₈	C₅×#102	1	1	yes	characteristic
271	C₈	C₅×#102	1	1	yes	characteristic
272	C₈	C₅×#102	1	1	yes	characteristic
273	C₈	C₅×#102	1	1	yes	characteristic
274	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
275	C₈	C₅×#117	1	1	yes	characteristic
276	C₈	C₅×#121	1	1	yes	characteristic
277	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
278	C₅	#12825	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	C₅×#417	1	1	yes	characteristic
283	C₂²	C₅×#461	1	1	yes	characteristic
284	C₂²	C₅×#474	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	2	no	—
288	C₄	C₅×#380	1	1	yes	characteristic
289	C₄	C₂×C₅×#102	1	1	yes	characteristic
290	C₄	C₅×#459	1	1	yes	characteristic
291	C₄	C₅×#476	1	1	yes	characteristic
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	2	no	—
296	C₂	C₅×#1802	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#476
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1802
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×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×#364	C₂	semidirect
4. C₅×#1848	C₂	semidirect
5. C₅×#1385	C₂	semidirect
23. #12825	C₅	direct (N × H)
60. C₂×#364	C₂×C₅	semidirect
62. #1848	C₂×C₅	semidirect
63. #1385	C₂×C₅	semidirect
278. C₅	#12825	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×#443	2
C₂×C₅×#364	2
C₅×#2148	2
C₅×#1848	2
C₅×#1385	2
#12825	5

Subgroups — C5×#12825 #11935545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12825 #11935545