Subgroups — C₅×#13038 #11935758

Order 1280.

474 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#510	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#441	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#360	C₂	1	1	yes	maximal, characteristic
5	C₅×#1975	C₂	1	1	yes	maximal, characteristic
6	C₅×#1383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×Q₈	—	2	1	no	—
11	C₅×#510	—	2	2	no	—
12	C₅×#510	—	2	2	no	—
13	C₂×C₅×Q₃₂	C₄	1	2	yes	complemented
14	C₂×C₅×SD₃₂	C₄	1	2	yes	complemented
15	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
17	C₅×#441	—	2	2	no	—
18	C₅×#441	—	2	2	no	—
19	C₂×C₅×#108	C₂²	1	1	yes	characteristic
20	C₂×C₅×#106	C₂²	1	1	yes	characteristic
21	C₂×C₅×#104	—	2	1	no	—
22	C₂×C₅×#103	—	2	1	no	—
23	C₅×#408	—	2	1	no	—
24	C₂×C₅×#97	C₂²	1	1	yes	characteristic
25	C₅×#362	—	2	2	no	—
26	C₅×#360	—	2	2	no	—
27	#13038	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅×#41	—	2	1	no	—
29	C₂²×C₅×Q₈	—	2	1	no	—
30	C₅×#135	—	2	1	no	—
31	C₅×#135	—	2	1	no	—
32	C₅×#135	—	2	1	no	—
33	C₅×#135	—	2	1	no	—
34	C₂×C₅×Q₁₆	—	2	1	no	—
35	C₂×C₅×Q₁₆	—	2	1	no	—
36	C₂×C₅×Q₁₆	—	2	1	no	—
37	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₅×Q₁₆	D₄	1	2	yes	—
40	C₂×C₅×Q₁₆	D₄	1	2	yes	—
41	C₂×C₅×SD₁₆	—	2	1	no	—
42	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₅×M₁₆	—	2	1	no	—
44	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₄×C₅×Q₈	—	2	4	no	—
47	C₄×C₅×Q₈	—	4	2	no	—
48	C₂×C₅×C₄⋊C₄	—	2	1	no	—
49	C₂×C₅×C₄⋊C₄	—	4	1	no	—
50	C₂×C₄²×C₅	—	4	1	no	—
51	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
52	C₅×Q₃₂	—	2	4	no	—
53	C₅×SD₃₂	—	2	4	no	—
54	C₅×M₃₂	—	2	1	no	—
55	C₅×M₃₂	—	2	1	no	—
56	C₅×M₃₂	—	2	1	no	—
57	C₅×M₃₂	—	2	1	no	—
58	C₂×C₅×C₁₆	—	2	1	no	—
59	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
60	C₅×#108	—	2	1	no	—
61	C₅×#108	—	2	1	no	—
62	C₅×#106	—	2	1	no	—
63	C₅×#106	D₄	1	2	yes	—
64	C₅×#104	—	4	2	no	—
65	C₅×#103	—	2	4	no	—
66	C₅×C₂²⋊C₈	—	4	2	no	—
67	C₅×#97	—	2	1	no	—
68	C₅×#97	—	2	1	no	—
69	C₂×#510	C₂×C₅	1	1	yes	characteristic
70	C₂×#441	C₂×C₅	1	1	yes	characteristic, complemented
71	C₂×#362	C₂×C₅	1	1	yes	characteristic, complemented
72	C₂×#360	C₂×C₅	1	1	yes	characteristic
73	#1975	C₂×C₅	1	1	yes	characteristic
74	#1383	C₂×C₅	1	1	yes	characteristic, complemented
75	#1375	C₂×C₅	1	1	yes	characteristic, complemented
76	C₅×#41	—	4	1	no	—
77	C₅×#41	—	4	1	no	—
78	C₅×#41	—	4	1	no	—
79	C₅×#41	—	4	1	no	—
80	C₂×C₅×Q₈	—	2	2	no	—
81	C₂×C₅×Q₈	—	2	1	no	—
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	2	2	no	—
84	C₂×C₅×Q₈	—	4	1	no	—
85	C₂×C₅×Q₈	—	4	1	no	—
86	C₂×C₅×Q₈	—	4	1	no	—
87	C₂×C₅×D₄	—	2	1	no	—
88	C₂×C₅×D₄	—	4	1	no	—
89	C₂²×C₄×C₅	—	2	1	no	—
90	C₂²×C₄×C₅	—	4	1	no	—
91	C₂²×C₄×C₅	—	4	1	no	—
92	C₂²×C₄×C₅	—	4	1	no	—
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₅×Q₁₆	—	2	1	no	—
95	C₅×Q₁₆	—	2	2	no	—
96	C₅×Q₁₆	—	2	1	no	—
97	C₅×Q₁₆	—	2	2	no	—
98	C₅×Q₁₆	—	2	2	no	—
99	C₅×Q₁₆	C₂²⋊C₄	1	4	yes	—
100	C₅×SD₁₆	—	4	1	no	—
101	C₅×SD₁₆	—	4	1	no	—
102	C₅×D₈	—	2	1	no	—
103	C₅×D₈	—	2	1	no	—
104	C₅×M₁₆	—	4	1	no	—
105	C₅×M₁₆	—	4	1	no	—
106	C₂×C₅×C₈	—	2	1	no	—
107	C₂×C₅×C₈	—	2	1	no	—
108	C₂×C₅×C₈	—	2	1	no	—
109	C₂×C₅×C₈	—	4	1	no	—
110	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	#41	1	1	yes	characteristic
115	C₂×C₅×C₈	#41	1	1	yes	characteristic
116	C₅×C₄⋊C₄	—	2	2	no	—
117	C₅×C₄⋊C₄	—	4	4	no	—
118	C₅×C₄⋊C₄	—	4	1	no	—
119	C₄²×C₅	—	2	1	no	—
120	C₄²×C₅	—	4	4	no	—
121	C₄²×C₅	D₈	1	2	yes	—
122	C₅×C₁₆	—	2	2	no	—
123	C₅×C₁₆	—	4	1	no	—
124	C₂×#135	C₂²×C₅	1	1	yes	characteristic
125	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
126	C₂×C₄×Q₈	—	2	1	no	—
127	#510	—	2	2	no	—
128	#510	—	2	2	no	—
129	C₂×Q₃₂	C₄×C₅	1	2	yes	complemented
130	C₂×SD₃₂	C₄×C₅	1	2	yes	complemented
131	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
132	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
133	#441	—	2	2	no	—
134	#441	—	2	2	no	—
135	C₂×#108	C₂²×C₅	1	1	yes	characteristic
136	C₂×#106	C₂²×C₅	1	1	yes	characteristic
137	C₂×#104	—	2	1	no	—
138	C₂×#103	—	2	1	no	—
139	#408	—	2	1	no	—
140	C₂×#97	C₂²×C₅	1	1	yes	characteristic
141	#362	—	2	2	no	—
142	#360	—	2	2	no	—
143	C₂³×C₅	—	4	1	no	—
144	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₅×Q₈	—	2	4	no	—
146	C₅×Q₈	—	4	2	no	—
147	C₅×Q₈	—	4	2	no	—
148	C₅×Q₈	—	4	2	no	—
149	C₅×Q₈	—	4	1	no	—
150	C₅×Q₈	—	4	1	no	—
151	C₅×D₄	—	4	1	no	—
152	C₅×D₄	—	4	1	no	—
153	C₅×D₄	—	8	1	no	—
154	C₅×D₄	—	8	1	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	—	4	2	no	—
159	C₂×C₄×C₅	—	4	2	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	—	8	2	no	—
163	C₂×C₄×C₅	—	8	1	no	—
164	C₂×C₄×C₅	—	8	1	no	—
165	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅	#120	1	1	yes	characteristic
167	C₂×C₄×C₅	#121	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
169	C₂×C₄×C₅	#135	1	1	yes	characteristic
170	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₅×C₈	—	4	1	no	—
172	C₅×C₈	—	8	1	no	—
173	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₅×C₈	C₄×D₄	1	1	yes	characteristic
175	C₅×C₈	#121	1	1	yes	characteristic
176	C₅×C₈	#123	1	1	yes	characteristic
177	C₂×#41	—	2	1	no	—
178	C₂²×Q₈	—	2	1	no	—
179	#135	—	2	1	no	—
180	#135	—	2	1	no	—
181	#135	—	2	1	no	—
182	#135	—	2	1	no	—
183	C₂×Q₁₆	—	2	1	no	—
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
187	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
188	C₂×Q₁₆	C₅×D₄	1	2	yes	—
189	C₂×Q₁₆	C₅×D₄	1	2	yes	—
190	C₂×SD₁₆	—	2	1	no	—
191	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
192	C₂×M₁₆	—	2	1	no	—
193	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
195	C₄×Q₈	—	2	4	no	—
196	C₄×Q₈	—	4	2	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	4	1	no	—
199	C₂×C₄²	—	4	1	no	—
200	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
201	Q₃₂	—	2	4	no	—
202	SD₃₂	—	2	4	no	—
203	M₃₂	—	2	1	no	—
204	M₃₂	—	2	1	no	—
205	M₃₂	—	2	1	no	—
206	M₃₂	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
209	#108	—	2	1	no	—
210	#108	—	2	1	no	—
211	#106	—	2	1	no	—
212	#106	C₅×D₄	1	2	yes	—
213	#104	—	4	2	no	—
214	#103	—	2	4	no	—
215	C₂²⋊C₈	—	4	2	no	—
216	#97	—	2	1	no	—
217	#97	—	2	1	no	—
218	C₂²×C₅	—	2	2	no	—
219	C₂²×C₅	—	8	1	no	—
220	C₂²×C₅	—	8	2	no	—
221	C₂²×C₅	#442	1	1	yes	characteristic
222	C₂²×C₅	#447	1	1	yes	characteristic
223	C₂²×C₅	#463	1	1	yes	characteristic
224	C₄×C₅	—	4	2	no	—
225	C₄×C₅	—	4	4	no	—
226	C₄×C₅	—	8	2	no	—
227	C₄×C₅	—	8	1	no	—
228	C₄×C₅	#386	1	1	yes	characteristic
229	C₄×C₅	C₄×D₈	1	1	yes	characteristic
230	C₄×C₅	#449	1	1	yes	characteristic
231	C₄×C₅	#459	1	1	yes	characteristic
232	#41	—	4	1	no	—
233	#41	—	4	1	no	—
234	#41	—	4	1	no	—
235	#41	—	4	1	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	2	1	no	—
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	C₅×C₂²⋊C₄	1	4	yes	—
256	SD₁₆	—	4	1	no	—
257	SD₁₆	—	4	1	no	—
258	D₈	—	2	1	no	—
259	D₈	—	2	1	no	—
260	M₁₆	—	4	1	no	—
261	M₁₆	—	4	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
267	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
268	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
269	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
270	C₂×C₈	C₅×#41	1	1	yes	characteristic
271	C₂×C₈	C₅×#41	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	4	4	no	—
277	C₄²	C₅×D₈	1	2	yes	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	4	1	no	—
280	C₂×C₅	—	2	1	no	—
281	C₂×C₅	—	2	1	no	—
282	C₂×C₅	—	16	1	no	—
283	C₂×C₅	#1879	1	1	yes	characteristic
284	C₂×C₅	#2180	1	2	yes	—
285	C₂³	—	4	1	no	—
286	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
287	Q₈	—	2	4	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	4	1	no	—
293	D₄	—	4	1	no	—
294	D₄	—	4	1	no	—
295	D₄	—	8	1	no	—
296	D₄	—	8	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
308	C₂×C₄	C₅×#120	1	1	yes	characteristic
309	C₂×C₄	C₅×#121	1	1	yes	characteristic
310	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
311	C₂×C₄	C₅×#135	1	1	yes	characteristic
312	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₈	—	4	1	no	—
314	C₈	—	8	1	no	—
315	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
316	C₈	C₄×C₅×D₄	1	1	yes	characteristic
317	C₈	C₅×#121	1	1	yes	characteristic
318	C₈	C₅×#123	1	1	yes	characteristic
319	C₅	#13038	1	1	yes	characteristic, direct product factor, Sylow
320	C₂²	—	2	2	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	8	2	no	—
323	C₂²	C₅×#442	1	1	yes	characteristic
324	C₂²	C₅×#447	1	1	yes	characteristic
325	C₂²	C₅×#463	1	1	yes	characteristic
326	C₄	—	4	2	no	—
327	C₄	—	4	4	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	1	no	—
330	C₄	C₅×#386	1	1	yes	characteristic
331	C₄	C₄×C₅×D₈	1	1	yes	characteristic
332	C₄	C₅×#449	1	1	yes	characteristic
333	C₄	C₅×#459	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	2	1	no	—
336	C₂	—	16	1	no	—
337	C₂	C₅×#1879	1	1	yes	characteristic
338	C₂	C₅×#2180	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#442
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#441	C₂	semidirect
3. C₂×C₅×#362	C₂	semidirect
6. C₅×#1383	C₂	semidirect
7. C₅×#1375	C₂	semidirect
13. C₂×C₅×Q₃₂	C₄	semidirect
14. C₂×C₅×SD₃₂	C₄	semidirect
27. #13038	C₅	direct (N × H)
70. C₂×#441	C₂×C₅	semidirect
71. C₂×#362	C₂×C₅	semidirect
74. #1383	C₂×C₅	semidirect
75. #1375	C₂×C₅	semidirect
129. C₂×Q₃₂	C₄×C₅	semidirect
130. C₂×SD₃₂	C₄×C₅	semidirect
319. C₅	#13038	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×#510	2
C₂×C₅×#441	2
C₂×C₅×#362	2
C₂×C₅×#360	2
C₅×#1975	2
C₅×#1383	2
C₅×#1375	2
#13038	5

Subgroups — C5×#13038 #11935758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13038 #11935758