Subgroups — C₅×#9294 #11932014

Order 1280.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3208	C₂	1	1	yes	maximal, characteristic
2	C₅×#2961	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1961	C₂	1	1	yes	maximal, characteristic
4	C₅×#1545	C₂	1	1	yes	maximal, characteristic
5	C₅×#1518	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1286	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1275	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#533	C₂²	1	1	yes	characteristic
9	C₂×C₅×#127	C₂²	1	1	yes	characteristic
10	C₅×#492	C₄	1	2	yes	complemented
11	C₅×#486	C₄	1	2	yes	complemented
12	C₅×#460	—	2	2	no	—
13	C₅×#450	—	2	2	no	—
14	C₅×#446	—	2	2	no	—
15	C₅×#446	—	2	2	no	—
16	C₅×#431	C₂²	1	1	yes	characteristic
17	C₅×#431	C₂²	1	1	yes	characteristic
18	C₅×#424	—	2	1	no	—
19	C₅×#423	C₂²	1	1	yes	characteristic
20	C₂×C₅×#102	—	2	1	no	—
21	C₂×C₅×#102	—	2	1	no	—
22	C₅×#405	—	2	1	no	—
23	C₂×C₅×#97	C₂²	1	1	yes	characteristic
24	C₅×#382	C₂²	1	1	yes	characteristic
25	C₅×#331	—	2	2	no	—
26	C₅×#328	—	2	2	no	—
27	#9294	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₅×D₄	—	2	1	no	—
29	C₂²×C₅×D₄	—	2	1	no	—
30	C₂²×C₅×D₄	—	2	1	no	—
31	C₂×C₅×SD₁₆	—	2	4	no	—
32	C₂×C₅×M₁₆	—	2	1	no	—
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
36	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₅×#127	—	2	1	no	—
38	C₅×#127	—	2	1	no	—
39	C₅×#127	—	2	1	no	—
40	C₅×#127	C₂×C₄	1	2	yes	—
41	C₅×#125	—	2	1	no	—
42	C₅×C₂²⋊Q₈	—	2	1	no	—
43	C₅×C₂²⋊Q₈	D₄	1	2	yes	—
44	C₄×C₅×Q₈	—	2	1	no	—
45	C₄×C₅×Q₈	D₄	1	2	yes	—
46	C₅×#117	—	2	1	no	—
47	C₂×C₅×C₄⋊C₄	—	2	1	no	—
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₅×C₄⋊C₈	—	2	4	no	—
50	C₅×C₄⋊C₈	—	2	1	no	—
51	C₅×C₄⋊C₈	—	2	1	no	—
52	C₅×C₄⋊C₈	—	2	1	no	—
53	C₅×C₄⋊C₈	D₄	1	2	yes	—
54	C₅×#102	—	2	4	no	—
55	C₅×#102	—	4	2	no	—
56	C₅×#102	—	4	2	no	—
57	C₅×C₂²⋊C₈	—	2	1	no	—
58	C₅×C₂²⋊C₈	—	2	1	no	—
59	C₅×#97	—	2	2	no	—
60	C₅×#97	—	2	1	no	—
61	C₅×#97	—	2	1	no	—
62	C₄×C₅×C₈	—	2	2	no	—
63	C₄×C₅×C₈	—	2	1	no	—
64	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
65	C₅×#95	—	2	1	no	—
66	C₅×#95	—	2	1	no	—
67	#3208	C₂×C₅	1	1	yes	characteristic
68	#2961	C₂×C₅	1	1	yes	characteristic, complemented
69	#1961	C₂×C₅	1	1	yes	characteristic
70	#1545	C₂×C₅	1	1	yes	characteristic
71	#1518	C₂×C₅	1	1	yes	characteristic, complemented
72	#1286	C₂×C₅	1	1	yes	characteristic, complemented
73	#1275	C₂×C₅	1	1	yes	characteristic, complemented
74	C₂⁴×C₅	—	4	1	no	—
75	C₂×C₅×Q₈	—	2	1	no	—
76	C₂×C₅×Q₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₅×D₄	—	2	2	no	—
78	C₂×C₅×D₄	—	2	2	no	—
79	C₂×C₅×D₄	—	2	2	no	—
80	C₂×C₅×D₄	—	4	1	no	—
81	C₂×C₅×D₄	—	4	1	no	—
82	C₂×C₅×D₄	—	4	1	no	—
83	C₂×C₅×D₄	—	4	1	no	—
84	C₂×C₅×D₄	—	4	1	no	—
85	C₂×C₅×D₄	—	4	1	no	—
86	C₂×C₅×D₄	—	4	1	no	—
87	C₂×C₅×D₄	—	8	2	no	—
88	C₂×C₅×D₄	—	8	2	no	—
89	C₂²×C₄×C₅	—	4	1	no	—
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	#41	1	1	yes	characteristic
93	C₅×SD₁₆	—	8	4	no	—
94	C₅×M₁₆	—	4	1	no	—
95	C₅×M₁₆	—	4	1	no	—
96	C₂×C₅×C₈	—	2	1	no	—
97	C₂×C₅×C₈	—	2	1	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	1	no	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	—	2	1	no	—
102	C₂×C₅×C₈	—	2	1	no	—
103	C₂×C₅×C₈	—	4	1	no	—
104	C₂×C₅×C₈	SD₁₆	1	2	yes	complemented
105	C₅×C₄⋊C₄	—	2	1	no	—
106	C₅×C₄⋊C₄	—	2	1	no	—
107	C₅×C₄⋊C₄	—	2	1	no	—
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	—	2	2	no	—
110	C₅×C₄⋊C₄	—	4	1	no	—
111	C₅×C₄⋊C₄	—	4	1	no	—
112	C₅×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
113	C₅×C₂²⋊C₄	—	2	2	no	—
114	C₄²×C₅	—	2	2	no	—
115	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₅	#41	1	1	yes	characteristic
119	#533	C₂²×C₅	1	1	yes	characteristic
120	C₂×#127	C₂²×C₅	1	1	yes	characteristic
121	#492	C₄×C₅	1	2	yes	complemented
122	#486	C₄×C₅	1	2	yes	complemented
123	#460	—	2	2	no	—
124	#450	—	2	2	no	—
125	#446	—	2	2	no	—
126	#446	—	2	2	no	—
127	#431	C₂²×C₅	1	1	yes	characteristic
128	#431	C₂²×C₅	1	1	yes	characteristic
129	#424	—	2	1	no	—
130	#423	C₂²×C₅	1	1	yes	characteristic
131	C₂×#102	—	2	1	no	—
132	C₂×#102	—	2	1	no	—
133	#405	—	2	1	no	—
134	C₂×#97	C₂²×C₅	1	1	yes	characteristic
135	#382	C₂²×C₅	1	1	yes	characteristic
136	#331	—	2	2	no	—
137	#328	—	2	2	no	—
138	C₂³×C₅	—	4	2	no	—
139	C₂³×C₅	—	8	2	no	—
140	C₂³×C₅	—	8	2	no	—
141	C₂³×C₅	—	8	1	no	—
142	C₂³×C₅	—	8	1	no	—
143	C₂³×C₅	#121	1	1	yes	characteristic
144	C₅×Q₈	—	4	2	no	—
145	C₅×Q₈	—	8	1	no	—
146	C₅×D₄	—	4	4	no	—
147	C₅×D₄	—	4	2	no	—
148	C₅×D₄	—	4	2	no	—
149	C₅×D₄	—	8	2	no	—
150	C₅×D₄	—	8	2	no	—
151	C₅×D₄	—	8	2	no	—
152	C₅×D₄	—	8	1	no	—
153	C₅×D₄	—	8	1	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	4	1	no	—
157	C₂×C₄×C₅	—	4	1	no	—
158	C₂×C₄×C₅	—	8	1	no	—
159	C₂×C₄×C₅	—	8	1	no	—
160	C₂×C₄×C₅	—	8	1	no	—
161	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅	#120	1	1	yes	characteristic
165	C₂×C₄×C₅	#121	1	1	yes	characteristic
166	C₂×C₄×C₅	#123	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
168	C₂×C₄×C₅	#135	1	1	yes	characteristic
169	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×C₄×C₅	#137	1	1	yes	characteristic
171	C₅×C₈	—	2	2	no	—
172	C₅×C₈	—	4	2	no	—
173	C₅×C₈	—	4	1	no	—
174	C₅×C₈	—	4	2	no	—
175	C₅×C₈	—	8	1	no	—
176	C₂²×D₄	—	2	1	no	—
177	C₂²×D₄	—	2	1	no	—
178	C₂²×D₄	—	2	1	no	—
179	C₂×SD₁₆	—	2	4	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
182	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
183	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
184	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
185	#127	—	2	1	no	—
186	#127	—	2	1	no	—
187	#127	—	2	1	no	—
188	#127	C₂×C₄×C₅	1	2	yes	—
189	#125	—	2	1	no	—
190	C₂²⋊Q₈	—	2	1	no	—
191	C₂²⋊Q₈	C₅×D₄	1	2	yes	—
192	C₄×Q₈	—	2	1	no	—
193	C₄×Q₈	C₅×D₄	1	2	yes	—
194	#117	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
197	C₄⋊C₈	—	2	4	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	C₅×D₄	1	2	yes	—
202	#102	—	2	4	no	—
203	#102	—	4	2	no	—
204	#102	—	4	2	no	—
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	#97	—	2	2	no	—
208	#97	—	2	1	no	—
209	#97	—	2	1	no	—
210	C₄×C₈	—	2	2	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
213	#95	—	2	1	no	—
214	#95	—	2	1	no	—
215	C₂²×C₅	—	2	1	no	—
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	8	2	no	—
218	C₂²×C₅	—	8	2	no	—
219	C₂²×C₅	—	8	2	no	—
220	C₂²×C₅	—	16	2	no	—
221	C₂²×C₅	—	16	2	no	—
222	C₂²×C₅	#386	1	1	yes	characteristic
223	C₂²×C₅	#460	1	1	yes	characteristic
224	C₂²×C₅	#464	1	1	yes	characteristic
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	4	2	no	—
230	C₄×C₅	—	4	2	no	—
231	C₄×C₅	—	16	1	no	—
232	C₄×C₅	C₄×SD₁₆	1	1	yes	characteristic
233	C₄×C₅	#440	1	1	yes	characteristic
234	C₄×C₅	#448	1	1	yes	characteristic
235	C₄×C₅	#449	1	1	yes	characteristic
236	C₂⁴	—	4	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	C₅×C₂²⋊C₄	1	2	yes	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	8	2	no	—
250	C₂×D₄	—	8	2	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
254	C₂²×C₄	C₅×#41	1	1	yes	characteristic
255	SD₁₆	—	8	4	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	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	C₅×SD₁₆	1	2	yes	complemented
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
278	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
279	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
280	C₄²	C₅×#41	1	1	yes	characteristic
281	C₂×C₅	—	2	1	no	—
282	C₂×C₅	—	2	1	no	—
283	C₂×C₅	—	16	2	no	—
284	C₂×C₅	#1880	1	1	yes	characteristic
285	C₂×C₅	#1903	1	1	yes	characteristic
286	C₂×C₅	#1910	1	1	yes	characteristic
287	C₂³	—	4	2	no	—
288	C₂³	—	8	2	no	—
289	C₂³	—	8	2	no	—
290	C₂³	—	8	1	no	—
291	C₂³	—	8	1	no	—
292	C₂³	C₅×#121	1	1	yes	characteristic
293	Q₈	—	4	2	no	—
294	Q₈	—	8	1	no	—
295	D₄	—	4	4	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	8	2	no	—
299	D₄	—	8	2	no	—
300	D₄	—	8	2	no	—
301	D₄	—	8	1	no	—
302	D₄	—	8	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	—	8	1	no	—
310	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
311	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
312	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
313	C₂×C₄	C₅×#120	1	1	yes	characteristic
314	C₂×C₄	C₅×#121	1	1	yes	characteristic
315	C₂×C₄	C₅×#123	1	1	yes	characteristic
316	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
317	C₂×C₄	C₅×#135	1	1	yes	characteristic
318	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
319	C₂×C₄	C₅×#137	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	4	2	no	—
322	C₈	—	4	1	no	—
323	C₈	—	4	2	no	—
324	C₈	—	8	1	no	—
325	C₅	#9294	1	1	yes	characteristic, direct product factor, Sylow
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	16	2	no	—
332	C₂²	—	16	2	no	—
333	C₂²	C₅×#386	1	1	yes	characteristic
334	C₂²	C₅×#460	1	1	yes	characteristic
335	C₂²	C₅×#464	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	16	1	no	—
343	C₄	C₄×C₅×SD₁₆	1	1	yes	characteristic
344	C₄	C₅×#440	1	1	yes	characteristic
345	C₄	C₅×#448	1	1	yes	characteristic
346	C₄	C₅×#449	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	16	2	no	—
350	C₂	C₅×#1880	1	1	yes	characteristic
351	C₂	C₅×#1903	1	1	yes	characteristic
352	C₂	C₅×#1910	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂×C₅×#127	C₂²
LCS	G ▷ 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
2. C₅×#2961	C₂	semidirect
5. C₅×#1518	C₂	semidirect
6. C₅×#1286	C₂	semidirect
7. C₅×#1275	C₂	semidirect
10. C₅×#492	C₄	semidirect
11. C₅×#486	C₄	semidirect
27. #9294	C₅	direct (N × H)
68. #2961	C₂×C₅	semidirect
71. #1518	C₂×C₅	semidirect
72. #1286	C₂×C₅	semidirect
73. #1275	C₂×C₅	semidirect
104. C₂×C₅×C₈	SD₁₆	semidirect
121. #492	C₄×C₅	semidirect
122. #486	C₄×C₅	semidirect
266. C₂×C₈	C₅×SD₁₆	semidirect
325. C₅	#9294	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×#3208	2
C₅×#2961	2
C₅×#1961	2
C₅×#1545	2
C₅×#1518	2
C₅×#1286	2
C₅×#1275	2
#9294	5

Subgroups — C5×#9294 #11932014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9294 #11932014