Subgroups — C₅×#10453 #11933173

Order 1280.

408 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#476	C₂	1	1	yes	maximal, characteristic
2	C₅×#1800	C₂	1	1	yes	maximal, characteristic
3	C₅×#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1549	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1478	C₂	1	1	yes	maximal, characteristic
7	C₅×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₅×#476	—	2	2	no	—
11	C₅×#476	—	2	2	no	—
12	C₅×#431	C₂²	1	1	yes	characteristic
13	C₅×#431	C₂²	1	1	yes	characteristic
14	C₂×C₅×#106	—	2	1	no	—
15	C₅×#423	C₂²	1	1	yes	characteristic
16	C₅×#423	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₅×#102	—	2	1	no	—
19	C₂×C₅×#100	—	2	1	no	—
20	C₅×#343	—	2	1	no	—
21	C₅×#335	—	2	2	no	—
22	C₅×#333	—	2	2	no	—
23	C₅×#332	—	2	2	no	—
24	C₅×D₄⋊C₈	—	2	2	no	—
25	#10453	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅×Q₈	—	2	1	no	—
27	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅	—	2	1	no	—
29	C₂×C₅×M₁₆	—	2	1	no	—
30	C₂×C₅×M₁₆	—	2	1	no	—
31	C₂²×C₅×C₈	—	2	1	no	—
32	C₅×C₄⋊Q₈	—	2	1	no	—
33	C₅×C₄⋊Q₈	—	2	1	no	—
34	C₅×C₄⋊Q₈	—	2	1	no	—
35	C₅×C₄⋊Q₈	D₄	1	2	yes	—
36	C₄×C₅×D₄	—	2	1	no	—
37	C₄×C₅×D₄	—	2	1	no	—
38	C₄×C₅×D₄	—	2	1	no	—
39	C₄×C₅×D₄	—	2	1	no	—
40	C₂×C₅×C₄⋊C₄	—	2	1	no	—
41	C₂×C₅×C₄⋊C₄	—	2	1	no	—
42	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
43	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×#106	—	2	4	no	—
46	C₅×C₄⋊C₈	—	2	1	no	—
47	C₅×C₄⋊C₈	—	2	1	no	—
48	C₅×C₄⋊C₈	—	2	1	no	—
49	C₅×C₄⋊C₈	—	2	1	no	—
50	C₅×C₄⋊C₈	—	2	1	no	—
51	C₅×C₄⋊C₈	—	2	1	no	—
52	C₅×C₄⋊C₈	D₄	1	2	yes	—
53	C₅×C₄⋊C₈	D₄	1	2	yes	—
54	C₅×#102	—	2	4	no	—
55	C₅×#100	—	4	2	no	—
56	C₅×C₂²⋊C₈	—	2	1	no	—
57	C₅×C₂²⋊C₈	—	2	1	no	—
58	C₄×C₅×C₈	—	2	1	no	—
59	C₄×C₅×C₈	—	2	1	no	—
60	C₂×#476	C₂×C₅	1	1	yes	characteristic
61	#1800	C₂×C₅	1	1	yes	characteristic
62	#1571	C₂×C₅	1	1	yes	characteristic, complemented
63	#1549	C₂×C₅	1	1	yes	characteristic, complemented
64	#1548	C₂×C₅	1	1	yes	characteristic, complemented
65	#1478	C₂×C₅	1	1	yes	characteristic
66	#1301	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂⁴×C₅	—	2	1	no	—
68	C₂×C₅×Q₈	—	2	2	no	—
69	C₂×C₅×Q₈	—	4	1	no	—
70	C₂×C₅×Q₈	—	8	2	no	—
71	C₂×C₅×D₄	—	2	1	no	—
72	C₂×C₅×D₄	—	2	1	no	—
73	C₂×C₅×D₄	—	2	1	no	—
74	C₂×C₅×D₄	—	2	1	no	—
75	C₂×C₅×D₄	—	2	1	no	—
76	C₂×C₅×D₄	Q₁₆	1	2	yes	—
77	C₂²×C₄×C₅	—	2	1	no	—
78	C₂²×C₄×C₅	—	4	1	no	—
79	C₂²×C₄×C₅	—	4	1	no	—
80	C₂²×C₄×C₅	—	4	1	no	—
81	C₂²×C₄×C₅	—	8	2	no	—
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	#41	1	1	yes	characteristic
84	C₂²×C₄×C₅	#41	1	1	yes	characteristic
85	C₅×M₁₆	—	4	1	no	—
86	C₅×M₁₆	—	4	1	no	—
87	C₅×M₁₆	—	4	1	no	—
88	C₅×M₁₆	—	4	1	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	4	1	no	—
92	C₂×C₅×C₈	—	4	1	no	—
93	C₂×C₅×C₈	—	4	1	no	—
94	C₂×C₅×C₈	—	4	1	no	—
95	C₂×C₅×C₈	—	4	1	no	—
96	C₅×C₄⋊C₄	—	2	2	no	—
97	C₅×C₄⋊C₄	—	2	1	no	—
98	C₅×C₄⋊C₄	—	2	1	no	—
99	C₅×C₄⋊C₄	—	2	2	no	—
100	C₅×C₄⋊C₄	—	4	1	no	—
101	C₅×C₄⋊C₄	—	4	1	no	—
102	C₅×C₂²⋊C₄	—	4	1	no	—
103	C₅×C₂²⋊C₄	—	4	1	no	—
104	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₅	#41	1	1	yes	characteristic
108	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
110	#476	—	2	2	no	—
111	#476	—	2	2	no	—
112	#431	C₂²×C₅	1	1	yes	characteristic
113	#431	C₂²×C₅	1	1	yes	characteristic
114	C₂×#106	—	2	1	no	—
115	#423	C₂²×C₅	1	1	yes	characteristic
116	#423	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
118	C₂×#102	—	2	1	no	—
119	C₂×#100	—	2	1	no	—
120	#343	—	2	1	no	—
121	#335	—	2	2	no	—
122	#333	—	2	2	no	—
123	#332	—	2	2	no	—
124	D₄⋊C₈	—	2	2	no	—
125	C₂³×C₅	—	2	2	no	—
126	C₂³×C₅	—	4	1	no	—
127	C₂³×C₅	—	4	1	no	—
128	C₂³×C₅	—	8	1	no	—
129	C₂³×C₅	—	8	1	no	—
130	C₂³×C₅	#123	1	1	yes	characteristic
131	C₅×Q₈	—	8	2	no	—
132	C₅×Q₈	—	8	2	no	—
133	C₅×D₄	—	2	2	no	—
134	C₅×D₄	—	2	2	no	—
135	C₅×D₄	—	4	1	no	—
136	C₅×D₄	—	4	1	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	4	1	no	—
140	C₂×C₄×C₅	—	4	2	no	—
141	C₂×C₄×C₅	—	8	2	no	—
142	C₂×C₄×C₅	—	8	2	no	—
143	C₂×C₄×C₅	—	8	2	no	—
144	C₂×C₄×C₅	—	8	1	no	—
145	C₂×C₄×C₅	#120	1	1	yes	characteristic
146	C₂×C₄×C₅	#121	1	1	yes	characteristic
147	C₂×C₄×C₅	#121	1	1	yes	characteristic
148	C₂×C₄×C₅	#121	1	1	yes	characteristic
149	C₂×C₄×C₅	#123	1	1	yes	characteristic
150	C₂×C₄×C₅	#124	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
152	C₂×C₄×C₅	#135	1	1	yes	characteristic
153	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×C₄×C₅	#137	1	1	yes	characteristic
155	C₅×C₈	—	4	2	no	—
156	C₅×C₈	—	4	2	no	—
157	C₅×C₈	—	8	1	no	—
158	C₅×C₈	—	8	1	no	—
159	C₅×C₈	—	8	1	no	—
160	C₂²×Q₈	—	2	1	no	—
161	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
162	C₂³×C₄	—	2	1	no	—
163	C₂×M₁₆	—	2	1	no	—
164	C₂×M₁₆	—	2	1	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₄⋊Q₈	—	2	1	no	—
167	C₄⋊Q₈	—	2	1	no	—
168	C₄⋊Q₈	—	2	1	no	—
169	C₄⋊Q₈	C₅×D₄	1	2	yes	—
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
177	C₂×C₂²⋊C₄	—	2	1	no	—
178	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
179	#106	—	2	4	no	—
180	C₄⋊C₈	—	2	1	no	—
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	C₅×D₄	1	2	yes	—
187	C₄⋊C₈	C₅×D₄	1	2	yes	—
188	#102	—	2	4	no	—
189	#100	—	4	2	no	—
190	C₂²⋊C₈	—	2	1	no	—
191	C₂²⋊C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	C₂²×C₅	—	2	1	no	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	—	4	2	no	—
197	C₂²×C₅	—	8	2	no	—
198	C₂²×C₅	—	8	2	no	—
199	C₂²×C₅	—	8	1	no	—
200	C₂²×C₅	—	8	1	no	—
201	C₂²×C₅	#394	1	1	yes	characteristic
202	C₂²×C₅	#484	1	1	yes	characteristic
203	C₂²×C₅	#485	1	1	yes	characteristic
204	C₄×C₅	—	2	1	no	—
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	8	2	no	—
208	C₄×C₅	—	8	1	no	—
209	C₄×C₅	#449	1	1	yes	characteristic
210	C₄×C₅	#451	1	1	yes	characteristic
211	C₄×C₅	#467	1	1	yes	characteristic
212	C₄×C₅	#468	1	1	yes	characteristic
213	C₄×C₅	#471	1	2	yes	—
214	C₂⁴	—	2	1	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	—	8	2	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	—	2	1	no	—
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	C₅×Q₁₆	1	2	yes	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	—	8	2	no	—
229	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
230	C₂²×C₄	C₅×#41	1	1	yes	characteristic
231	C₂²×C₄	C₅×#41	1	1	yes	characteristic
232	M₁₆	—	4	1	no	—
233	M₁₆	—	4	1	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	4	1	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₂²⋊C₄	—	4	1	no	—
250	C₂²⋊C₄	—	4	1	no	—
251	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
252	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
253	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
254	C₄²	C₅×#41	1	1	yes	characteristic
255	C₂×C₅	—	2	1	no	—
256	C₂×C₅	—	2	1	no	—
257	C₂×C₅	—	8	2	no	—
258	C₂×C₅	#2036	1	1	yes	characteristic
259	C₂×C₅	#2046	1	1	yes	characteristic
260	C₂×C₅	#2048	1	1	yes	characteristic
261	C₂³	—	2	2	no	—
262	C₂³	—	4	1	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	8	1	no	—
265	C₂³	—	8	1	no	—
266	C₂³	C₅×#123	1	1	yes	characteristic
267	Q₈	—	8	2	no	—
268	Q₈	—	8	2	no	—
269	D₄	—	2	2	no	—
270	D₄	—	2	2	no	—
271	D₄	—	4	1	no	—
272	D₄	—	4	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	8	1	no	—
281	C₂×C₄	C₅×#120	1	1	yes	characteristic
282	C₂×C₄	C₅×#121	1	1	yes	characteristic
283	C₂×C₄	C₅×#121	1	1	yes	characteristic
284	C₂×C₄	C₅×#121	1	1	yes	characteristic
285	C₂×C₄	C₅×#123	1	1	yes	characteristic
286	C₂×C₄	C₅×#124	1	1	yes	characteristic
287	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
288	C₂×C₄	C₅×#135	1	1	yes	characteristic
289	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₂×C₄	C₅×#137	1	1	yes	characteristic
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	8	1	no	—
294	C₈	—	8	1	no	—
295	C₈	—	8	1	no	—
296	C₅	#10453	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	1	no	—
303	C₂²	—	8	1	no	—
304	C₂²	C₅×#394	1	1	yes	characteristic
305	C₂²	C₅×#484	1	1	yes	characteristic
306	C₂²	C₅×#485	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	8	1	no	—
311	C₄	—	8	2	no	—
312	C₄	C₅×#449	1	1	yes	characteristic
313	C₄	C₅×#451	1	1	yes	characteristic
314	C₄	C₅×#467	1	1	yes	characteristic
315	C₄	C₅×#468	1	1	yes	characteristic
316	C₄	C₅×#471	1	2	yes	—
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	—	8	2	no	—
320	C₂	C₅×#2036	1	1	yes	characteristic
321	C₂	C₅×#2046	1	1	yes	characteristic
322	C₂	C₅×#2048	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1571	C₂	semidirect
4. C₅×#1549	C₂	semidirect
5. C₅×#1548	C₂	semidirect
7. C₅×#1301	C₂	semidirect
25. #10453	C₅	direct (N × H)
62. #1571	C₂×C₅	semidirect
63. #1549	C₂×C₅	semidirect
64. #1548	C₂×C₅	semidirect
66. #1301	C₂×C₅	semidirect
296. C₅	#10453	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×#476	2
C₅×#1800	2
C₅×#1571	2
C₅×#1549	2
C₅×#1548	2
C₅×#1478	2
C₅×#1301	2
#10453	5

Subgroups — C5×#10453 #11933173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10453 #11933173