Subgroups — C₅×#19292 #11942012

Order 1280.

848 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3038	C₂	1	2	yes	maximal, complemented
3	C₅×#2965	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1688	C₂	1	4	yes	maximal, complemented
5	C₅×#1687	C₂	1	4	yes	maximal, complemented
6	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1545	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×AGL(1,ℤ/8)	—	2	2	no	—
9	C₂²×C₅×D₈	—	2	2	no	—
10	C₂×C₅×#127	C₂²	1	2	yes	—
11	C₄×C₅×#41	C₂²	1	1	yes	characteristic
12	C₅×#493	—	2	1	no	—
13	C₅×#493	—	2	1	no	—
14	C₅×#493	C₂²	1	4	yes	—
15	C₅×#460	—	2	2	no	—
16	C₅×#460	C₂²	1	4	yes	complemented
17	C₅×#459	—	2	2	no	—
18	C₅×#459	C₂²	1	4	yes	complemented
19	C₅×#446	C₂²	1	2	yes	complemented
20	C₅×#436	C₂²	1	2	yes	complemented
21	C₅×#435	—	2	2	no	—
22	C₅×#431	—	2	1	no	—
23	C₅×#431	C₂²	1	1	yes	characteristic
24	C₅×#423	C₂²	1	2	yes	—
25	C₅×#418	—	2	2	no	—
26	C₄×C₅×M₁₆	—	2	1	no	—
27	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
28	C₅×#334	C₂²	1	4	yes	complemented
29	C₅×#331	C₂²	1	8	yes	complemented
30	#19292	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₅×#41	D₄	1	1	yes	characteristic
32	C₂²×C₅×D₄	—	2	2	no	—
33	C₂²×C₅×D₄	—	2	2	no	—
34	C₂²×C₅×D₄	—	2	2	no	—
35	C₅×AGL(1,ℤ/8)	—	4	8	no	—
36	C₂×C₅×SD₁₆	—	2	4	no	—
37	C₂×C₅×D₈	—	2	4	no	—
38	C₂×C₅×D₈	—	2	4	no	—
39	C₂×C₅×D₈	—	2	4	no	—
40	C₂×C₅×D₈	—	4	8	no	—
41	C₂×C₅×M₁₆	—	2	2	no	—
42	C₂×C₅×M₁₆	—	2	2	no	—
43	C₂²×C₅×C₈	D₄	1	2	yes	—
44	C₅×#127	—	2	2	no	—
45	C₅×#127	—	2	2	no	—
46	C₅×#127	—	2	2	no	—
47	C₅×#127	C₂³	1	4	yes	—
48	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
49	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
50	C₄×C₅×D₄	—	2	1	no	—
51	C₄×C₅×D₄	—	2	1	no	—
52	C₄×C₅×D₄	D₄	1	1	yes	characteristic
53	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
54	C₅×#117	—	2	1	no	—
55	C₅×#117	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅	—	2	1	no	—
57	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
58	C₅×C₄⋊C₈	—	2	2	no	—
59	C₅×C₄⋊C₈	—	2	2	no	—
60	C₅×C₄⋊C₈	C₂³	1	2	yes	—
61	C₅×C₄⋊C₈	C₂³	1	4	yes	—
62	C₅×#102	—	2	4	no	—
63	C₅×#102	—	2	4	no	—
64	C₅×C₂²⋊C₈	—	2	1	no	—
65	C₅×C₂²⋊C₈	—	2	2	no	—
66	C₅×#97	—	2	2	no	—
67	C₄×C₅×C₈	—	2	1	no	—
68	C₄×C₅×C₈	D₄	1	1	yes	characteristic
69	C₄×C₅×C₈	D₄	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂³	1	2	yes	—
71	C₂×#493	C₂×C₅	1	1	yes	characteristic, complemented
72	#3038	C₂×C₅	1	2	yes	complemented
73	#2965	C₂×C₅	1	1	yes	characteristic, complemented
74	#1688	C₂×C₅	1	4	yes	complemented
75	#1687	C₂×C₅	1	4	yes	complemented
76	#1566	C₂×C₅	1	1	yes	characteristic, complemented
77	#1545	C₂×C₅	1	2	yes	complemented
78	C₂⁴×C₅	—	4	2	no	—
79	C₅×#41	—	2	4	no	—
80	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₅×D₄	—	2	1	no	—
82	C₂×C₅×D₄	—	2	4	no	—
83	C₂×C₅×D₄	—	2	4	no	—
84	C₂×C₅×D₄	—	2	4	no	—
85	C₂×C₅×D₄	—	4	8	no	—
86	C₂×C₅×D₄	—	4	2	no	—
87	C₂×C₅×D₄	—	4	2	no	—
88	C₂×C₅×D₄	—	4	2	no	—
89	C₂×C₅×D₄	—	4	4	no	—
90	C₂×C₅×D₄	—	4	4	no	—
91	C₂×C₅×D₄	—	8	4	no	—
92	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	—	2	1	no	—
94	C₂²×C₄×C₅	—	2	1	no	—
95	C₂²×C₄×C₅	—	2	1	no	—
96	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅	#41	1	1	yes	characteristic
99	C₅×SD₁₆	—	4	8	no	—
100	C₅×D₈	—	4	8	no	—
101	C₅×D₈	—	4	8	no	—
102	C₅×D₈	—	4	8	no	—
103	C₅×M₁₆	—	4	4	no	—
104	C₅×M₁₆	—	4	4	no	—
105	C₂×C₅×C₈	—	2	4	no	—
106	C₂×C₅×C₈	—	2	4	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
109	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
110	C₅×C₄⋊C₄	—	2	1	no	—
111	C₅×C₄⋊C₄	—	2	1	no	—
112	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
113	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
114	C₅×C₂²⋊C₄	—	2	1	no	—
115	C₅×C₂²⋊C₄	—	2	1	no	—
116	C₅×C₂²⋊C₄	—	2	1	no	—
117	C₄²×C₅	—	2	1	no	—
118	C₄²×C₅	—	2	1	no	—
119	C₄²×C₅	—	2	1	no	—
120	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₅	#41	1	1	yes	characteristic
123	C₄²×C₅	C₂⁴	1	1	yes	characteristic
124	C₂×AGL(1,ℤ/8)	—	2	2	no	—
125	C₂²×D₈	—	2	2	no	—
126	C₂×#127	C₂²×C₅	1	2	yes	—
127	C₄×#41	C₂²×C₅	1	1	yes	characteristic
128	#493	—	2	1	no	—
129	#493	—	2	1	no	—
130	#493	C₂²×C₅	1	4	yes	—
131	#460	—	2	2	no	—
132	#460	C₂²×C₅	1	4	yes	complemented
133	#459	—	2	2	no	—
134	#459	C₂²×C₅	1	4	yes	complemented
135	#446	C₂²×C₅	1	2	yes	complemented
136	#436	C₂²×C₅	1	2	yes	complemented
137	#435	—	2	2	no	—
138	#431	—	2	1	no	—
139	#431	C₂²×C₅	1	1	yes	characteristic
140	#423	C₂²×C₅	1	2	yes	—
141	#418	—	2	2	no	—
142	C₄×M₁₆	—	2	1	no	—
143	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
144	#334	C₂²×C₅	1	4	yes	complemented
145	#331	C₂²×C₅	1	8	yes	complemented
146	C₂³×C₅	—	2	1	no	—
147	C₂³×C₅	—	4	4	no	—
148	C₂³×C₅	—	8	4	no	—
149	C₂³×C₅	—	8	4	no	—
150	C₂³×C₅	—	8	4	no	—
151	C₂³×C₅	#121	1	1	yes	characteristic
152	C₅×Q₈	—	2	2	no	—
153	C₅×D₄	—	2	2	no	—
154	C₅×D₄	—	4	2	no	—
155	C₅×D₄	—	4	8	no	—
156	C₅×D₄	—	4	4	no	—
157	C₅×D₄	—	4	4	no	—
158	C₅×D₄	—	8	4	no	—
159	C₅×D₄	—	8	4	no	—
160	C₅×D₄	—	8	4	no	—
161	C₅×D₄	—	8	4	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	1	no	—
165	C₂×C₄×C₅	—	2	1	no	—
166	C₂×C₄×C₅	—	2	1	no	—
167	C₂×C₄×C₅	—	4	2	no	—
168	C₂×C₄×C₅	—	4	2	no	—
169	C₂×C₄×C₅	—	4	2	no	—
170	C₂×C₄×C₅	#121	1	1	yes	characteristic
171	C₂×C₄×C₅	#121	1	1	yes	characteristic
172	C₂×C₄×C₅	#121	1	1	yes	characteristic
173	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
174	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
177	C₅×C₈	—	2	2	no	—
178	C₅×C₈	—	4	2	no	—
179	C₅×C₈	—	4	4	no	—
180	C₅×C₈	AGL(1,ℤ/8)	1	4	yes	complemented
181	C₂×#41	C₅×D₄	1	1	yes	characteristic
182	C₂²×D₄	—	2	2	no	—
183	C₂²×D₄	—	2	2	no	—
184	C₂²×D₄	—	2	2	no	—
185	AGL(1,ℤ/8)	—	4	8	no	—
186	C₂×SD₁₆	—	2	4	no	—
187	C₂×D₈	—	2	4	no	—
188	C₂×D₈	—	2	4	no	—
189	C₂×D₈	—	2	4	no	—
190	C₂×D₈	—	4	8	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂²×C₈	C₅×D₄	1	2	yes	—
194	#127	—	2	2	no	—
195	#127	—	2	2	no	—
196	#127	—	2	2	no	—
197	#127	C₂³×C₅	1	4	yes	—
198	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
199	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
200	C₄×D₄	—	2	1	no	—
201	C₄×D₄	—	2	1	no	—
202	C₄×D₄	C₅×D₄	1	1	yes	characteristic
203	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
204	#117	—	2	1	no	—
205	#117	C₅×D₄	1	1	yes	characteristic
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
208	C₄⋊C₈	—	2	2	no	—
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	C₂³×C₅	1	2	yes	—
211	C₄⋊C₈	C₂³×C₅	1	4	yes	—
212	#102	—	2	4	no	—
213	#102	—	2	4	no	—
214	C₂²⋊C₈	—	2	1	no	—
215	C₂²⋊C₈	—	2	2	no	—
216	#97	—	2	2	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	C₅×D₄	1	1	yes	characteristic
219	C₄×C₈	C₅×D₄	1	1	yes	characteristic
220	C₄×C₈	C₂³×C₅	1	2	yes	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	4	1	no	—
224	C₂²×C₅	—	4	1	no	—
225	C₂²×C₅	—	4	1	no	—
226	C₂²×C₅	—	8	4	no	—
227	C₂²×C₅	—	8	4	no	—
228	C₂²×C₅	—	8	4	no	—
229	C₂²×C₅	—	16	8	no	—
230	C₂²×C₅	#469	1	2	yes	—
231	C₂²×C₅	C₂×#121	1	1	yes	characteristic
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	4	1	no	—
235	C₄×C₅	—	4	1	no	—
236	C₄×C₅	—	4	1	no	—
237	C₄×C₅	#469	1	2	yes	—
238	C₄×C₅	#472	1	2	yes	—
239	C₄×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
240	C₄×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
241	C₂⁴	—	4	2	no	—
242	#41	—	2	4	no	—
243	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	2	4	no	—
248	C₂×D₄	—	4	8	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	4	no	—
253	C₂×D₄	—	4	4	no	—
254	C₂×D₄	—	8	4	no	—
255	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
261	C₂²×C₄	C₅×#41	1	1	yes	characteristic
262	SD₁₆	—	4	8	no	—
263	D₈	—	4	8	no	—
264	D₈	—	4	8	no	—
265	D₈	—	4	8	no	—
266	M₁₆	—	4	4	no	—
267	M₁₆	—	4	4	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
272	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
276	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
284	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
285	C₄²	C₅×#41	1	1	yes	characteristic
286	C₄²	C₂⁴×C₅	1	1	yes	characteristic
287	C₂×C₅	—	2	2	no	—
288	C₂×C₅	—	8	1	no	—
289	C₂×C₅	—	16	4	no	—
290	C₂×C₅	#3038	1	1	yes	characteristic
291	C₂×C₅	C₂×#469	1	1	yes	characteristic
292	C₂×C₅	C₂×#472	1	1	yes	characteristic
293	C₂³	—	2	1	no	—
294	C₂³	—	4	4	no	—
295	C₂³	—	8	4	no	—
296	C₂³	—	8	4	no	—
297	C₂³	—	8	4	no	—
298	C₂³	C₅×#121	1	1	yes	characteristic
299	Q₈	—	2	2	no	—
300	D₄	—	2	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	8	no	—
303	D₄	—	4	4	no	—
304	D₄	—	4	4	no	—
305	D₄	—	8	4	no	—
306	D₄	—	8	4	no	—
307	D₄	—	8	4	no	—
308	D₄	—	8	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	C₅×#121	1	1	yes	characteristic
318	C₂×C₄	C₅×#121	1	1	yes	characteristic
319	C₂×C₄	C₅×#121	1	1	yes	characteristic
320	C₂×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
321	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
323	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	4	4	no	—
326	C₈	—	4	2	no	—
327	C₈	C₅×AGL(1,ℤ/8)	1	4	yes	complemented
328	C₅	#19292	1	1	yes	characteristic, direct product factor, Sylow
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	8	4	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	8	4	no	—
337	C₂²	—	16	8	no	—
338	C₂²	C₅×#469	1	2	yes	—
339	C₂²	C₂×C₅×#121	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	C₅×#469	1	2	yes	—
346	C₄	C₅×#472	1	2	yes	—
347	C₄	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
348	C₄	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
349	C₂	—	2	2	no	—
350	C₂	—	8	1	no	—
351	C₂	—	16	4	no	—
352	C₂	C₅×#3038	1	1	yes	characteristic
353	C₂	C₂×C₅×#469	1	1	yes	characteristic
354	C₂	C₂×C₅×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
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
1. C₂×C₅×#493	C₂	semidirect
2. C₅×#3038	C₂	semidirect
3. C₅×#2965	C₂	semidirect
4. C₅×#1688	C₂	semidirect
5. C₅×#1687	C₂	semidirect
6. C₅×#1566	C₂	semidirect
7. C₅×#1545	C₂	semidirect
16. C₅×#460	C₂²	semidirect
18. C₅×#459	C₂²	semidirect
19. C₅×#446	C₂²	semidirect
20. C₅×#436	C₂²	semidirect
28. C₅×#334	C₂²	semidirect
29. C₅×#331	C₂²	semidirect
30. #19292	C₅	direct (N × H)
71. C₂×#493	C₂×C₅	semidirect
72. #3038	C₂×C₅	semidirect
73. #2965	C₂×C₅	semidirect
74. #1688	C₂×C₅	semidirect
75. #1687	C₂×C₅	semidirect
76. #1566	C₂×C₅	semidirect
77. #1545	C₂×C₅	semidirect
132. #460	C₂²×C₅	semidirect
134. #459	C₂²×C₅	semidirect
135. #446	C₂²×C₅	semidirect
136. #436	C₂²×C₅	semidirect
144. #334	C₂²×C₅	semidirect
145. #331	C₂²×C₅	semidirect
180. C₅×C₈	AGL(1,ℤ/8)	semidirect
327. C₈	C₅×AGL(1,ℤ/8)	semidirect
328. C₅	#19292	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×#493	2
C₅×#3038	2
C₅×#2965	2
C₅×#1688	2
C₅×#1687	2
C₅×#1566	2
C₅×#1545	2
#19292	5

Subgroups — C5×#19292 #11942012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19292 #11942012