Subgroups — C₅×#19432 #11942152

Order 1280.

666 conjugacy classes of subgroups.

The subgroup lattice has 388 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×#3078	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#460	C₂	1	1	yes	maximal, characteristic
4	C₅×#2983	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1698	C₂	1	4	yes	maximal, complemented
6	C₅×#1686	C₂	1	4	yes	maximal, complemented
7	C₅×#1571	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1545	C₂	1	1	yes	maximal, characteristic
9	C₅×#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₅×SD₁₆	—	2	1	no	—
11	C₅×#533	C₂²	1	1	yes	characteristic
12	C₂×C₅×#127	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₅×#492	C₂²	1	2	yes	—
15	C₅×#486	C₂²	1	2	yes	—
16	C₅×#479	—	2	2	no	—
17	C₅×#477	C₂²	1	2	yes	complemented
18	C₅×#475	C₂²	1	2	yes	complemented
19	C₅×#460	—	2	2	no	—
20	C₅×#460	—	2	2	no	—
21	C₅×#460	C₂²	1	4	yes	—
22	C₅×#450	—	2	2	no	—
23	C₅×#446	—	2	2	no	—
24	C₅×#446	C₂²	1	4	yes	complemented
25	C₅×#432	—	2	2	no	—
26	C₅×#431	C₂²	1	1	yes	characteristic
27	C₅×#431	C₂²	1	1	yes	characteristic
28	C₅×#428	—	2	1	no	—
29	C₅×#423	C₂²	1	1	yes	characteristic
30	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₅×#419	—	2	1	no	—
32	C₂×C₅×#102	—	2	1	no	—
33	C₅×#335	C₂²	1	4	yes	complemented
34	C₅×#333	C₂²	1	2	yes	complemented
35	C₅×#332	C₂²	1	2	yes	complemented
36	C₅×#331	C₂²	1	4	yes	—
37	#19432	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
39	C₂²×C₅×D₄	—	2	1	no	—
40	C₂²×C₅×D₄	—	2	1	no	—
41	C₂²×C₅×D₄	—	2	1	no	—
42	C₂×C₅×SD₁₆	—	2	4	no	—
43	C₂×C₅×SD₁₆	—	2	4	no	—
44	C₂×C₅×SD₁₆	—	4	4	no	—
45	C₂×C₅×M₁₆	—	2	1	no	—
46	C₂²×C₅×C₈	—	2	1	no	—
47	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
48	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
49	C₅×#127	—	2	1	no	—
50	C₅×#127	—	2	1	no	—
51	C₅×#127	—	2	1	no	—
52	C₅×#127	C₂³	1	2	yes	—
53	C₅×#125	—	2	1	no	—
54	C₅×C₂²⋊Q₈	—	2	1	no	—
55	C₅×C₂²⋊Q₈	—	2	1	no	—
56	C₄×C₅×Q₈	—	2	1	no	—
57	C₄×C₅×Q₈	—	2	1	no	—
58	C₄×C₅×Q₈	—	2	1	no	—
59	C₄×C₅×Q₈	—	2	1	no	—
60	C₄×C₅×Q₈	D₄	1	2	yes	—
61	C₄×C₅×Q₈	C₂³	1	2	yes	—
62	C₅×#117	—	2	1	no	—
63	C₂×C₅×C₄⋊C₄	—	2	1	no	—
64	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₅	—	2	1	no	—
66	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
67	C₅×#107	—	2	2	no	—
68	C₅×#106	—	2	2	no	—
69	C₅×C₄⋊C₈	—	2	4	no	—
70	C₅×C₄⋊C₈	—	2	1	no	—
71	C₅×C₄⋊C₈	—	2	1	no	—
72	C₅×C₄⋊C₈	D₄	1	2	yes	—
73	C₅×C₄⋊C₈	C₂³	1	2	yes	—
74	C₅×C₄⋊C₈	C₂³	1	2	yes	—
75	C₅×C₄⋊C₈	C₂³	1	2	yes	—
76	C₅×#103	—	2	2	no	—
77	C₅×#103	—	2	2	no	—
78	C₅×#102	—	2	4	no	—
79	C₅×#102	—	2	4	no	—
80	C₅×C₂²⋊C₈	—	2	1	no	—
81	C₅×C₂²⋊C₈	D₄	1	2	yes	—
82	C₅×#97	—	2	4	no	—
83	C₄×C₅×C₈	—	2	1	no	—
84	C₄×C₅×C₈	C₂³	1	2	yes	—
85	#3208	C₂×C₅	1	1	yes	characteristic
86	#3078	C₂×C₅	1	1	yes	characteristic, complemented
87	C₂×#460	C₂×C₅	1	1	yes	characteristic
88	#2983	C₂×C₅	1	1	yes	characteristic, complemented
89	#1698	C₂×C₅	1	4	yes	complemented
90	#1686	C₂×C₅	1	4	yes	complemented
91	#1571	C₂×C₅	1	1	yes	characteristic, complemented
92	#1545	C₂×C₅	1	1	yes	characteristic
93	#1542	C₂×C₅	1	1	yes	characteristic, complemented
94	C₂⁴×C₅	—	4	1	no	—
95	C₂×C₅×Q₈	—	2	1	no	—
96	C₂×C₅×Q₈	—	2	2	no	—
97	C₂×C₅×Q₈	—	2	2	no	—
98	C₂×C₅×Q₈	—	2	1	no	—
99	C₂×C₅×Q₈	—	2	1	no	—
100	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
101	C₂×C₅×D₄	—	2	2	no	—
102	C₂×C₅×D₄	—	2	2	no	—
103	C₂×C₅×D₄	—	2	2	no	—
104	C₂×C₅×D₄	—	4	1	no	—
105	C₂×C₅×D₄	—	4	1	no	—
106	C₂×C₅×D₄	—	4	1	no	—
107	C₂×C₅×D₄	—	4	2	no	—
108	C₂×C₅×D₄	—	4	2	no	—
109	C₂×C₅×D₄	—	8	2	no	—
110	C₂×C₅×D₄	—	8	2	no	—
111	C₂²×C₄×C₅	—	2	1	no	—
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₅	#41	1	1	yes	characteristic
116	C₅×SD₁₆	—	4	8	no	—
117	C₅×SD₁₆	—	8	4	no	—
118	C₅×M₁₆	—	4	2	no	—
119	C₂×C₅×C₈	—	2	2	no	—
120	C₂×C₅×C₈	—	2	2	no	—
121	C₂×C₅×C₈	—	2	2	no	—
122	C₂×C₅×C₈	—	4	2	no	—
123	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
124	C₅×C₄⋊C₄	—	2	2	no	—
125	C₅×C₄⋊C₄	—	2	1	no	—
126	C₅×C₄⋊C₄	—	2	2	no	—
127	C₅×C₄⋊C₄	—	2	1	no	—
128	C₅×C₄⋊C₄	—	2	1	no	—
129	C₅×C₄⋊C₄	—	2	1	no	—
130	C₅×C₄⋊C₄	—	2	1	no	—
131	C₅×C₄⋊C₄	—	4	1	no	—
132	C₅×C₄⋊C₄	—	4	1	no	—
133	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
134	C₅×C₂²⋊C₄	—	4	1	no	—
135	C₄²×C₅	—	2	2	no	—
136	C₄²×C₅	—	2	2	no	—
137	C₄²×C₅	—	4	1	no	—
138	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₅	#41	1	1	yes	characteristic
141	C₄²×C₅	C₂⁴	1	1	yes	characteristic
142	C₂²×SD₁₆	—	2	1	no	—
143	#533	C₂²×C₅	1	1	yes	characteristic
144	C₂×#127	C₂²×C₅	1	1	yes	characteristic
145	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
146	#492	C₂²×C₅	1	2	yes	—
147	#486	C₂²×C₅	1	2	yes	—
148	#479	—	2	2	no	—
149	#477	C₂²×C₅	1	2	yes	complemented
150	#475	C₂²×C₅	1	2	yes	complemented
151	#460	—	2	2	no	—
152	#460	—	2	2	no	—
153	#460	C₂²×C₅	1	4	yes	—
154	#450	—	2	2	no	—
155	#446	—	2	2	no	—
156	#446	C₂²×C₅	1	4	yes	complemented
157	#432	—	2	2	no	—
158	#431	C₂²×C₅	1	1	yes	characteristic
159	#431	C₂²×C₅	1	1	yes	characteristic
160	#428	—	2	1	no	—
161	#423	C₂²×C₅	1	1	yes	characteristic
162	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
163	#419	—	2	1	no	—
164	C₂×#102	—	2	1	no	—
165	#335	C₂²×C₅	1	4	yes	complemented
166	#333	C₂²×C₅	1	2	yes	complemented
167	#332	C₂²×C₅	1	2	yes	complemented
168	#331	C₂²×C₅	1	4	yes	—
169	C₂³×C₅	—	4	2	no	—
170	C₂³×C₅	—	8	2	no	—
171	C₂³×C₅	—	8	2	no	—
172	C₂³×C₅	—	8	2	no	—
173	C₂³×C₅	#121	1	1	yes	characteristic
174	C₅×Q₈	—	2	2	no	—
175	C₅×Q₈	—	2	2	no	—
176	C₅×Q₈	—	4	2	no	—
177	C₅×Q₈	—	4	2	no	—
178	C₅×Q₈	—	8	1	no	—
179	C₅×D₄	—	4	2	no	—
180	C₅×D₄	—	4	2	no	—
181	C₅×D₄	—	4	4	no	—
182	C₅×D₄	—	8	2	no	—
183	C₅×D₄	—	8	2	no	—
184	C₅×D₄	—	8	2	no	—
185	C₅×D₄	—	8	2	no	—
186	C₂×C₄×C₅	—	2	2	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	4	1	no	—
190	C₂×C₄×C₅	—	4	2	no	—
191	C₂×C₄×C₅	—	4	1	no	—
192	C₂×C₄×C₅	—	4	1	no	—
193	C₂×C₄×C₅	—	8	1	no	—
194	C₂×C₄×C₅	#121	1	1	yes	characteristic
195	C₂×C₄×C₅	#121	1	1	yes	characteristic
196	C₂×C₄×C₅	#121	1	1	yes	characteristic
197	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
198	C₂×C₄×C₅	#137	1	2	yes	—
199	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
202	C₅×C₈	—	2	4	no	—
203	C₅×C₈	—	4	2	no	—
204	C₅×C₈	—	4	2	no	—
205	C₅×C₈	—	4	2	no	—
206	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
207	C₂²×D₄	—	2	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	C₂²×D₄	—	2	1	no	—
210	C₂×SD₁₆	—	2	4	no	—
211	C₂×SD₁₆	—	2	4	no	—
212	C₂×SD₁₆	—	4	4	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂²×C₈	—	2	1	no	—
215	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
216	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
217	#127	—	2	1	no	—
218	#127	—	2	1	no	—
219	#127	—	2	1	no	—
220	#127	C₂³×C₅	1	2	yes	—
221	#125	—	2	1	no	—
222	C₂²⋊Q₈	—	2	1	no	—
223	C₂²⋊Q₈	—	2	1	no	—
224	C₄×Q₈	—	2	1	no	—
225	C₄×Q₈	—	2	1	no	—
226	C₄×Q₈	—	2	1	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	C₅×D₄	1	2	yes	—
229	C₄×Q₈	C₂³×C₅	1	2	yes	—
230	#117	—	2	1	no	—
231	C₂×C₄⋊C₄	—	2	1	no	—
232	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
233	C₂×C₄²	—	2	1	no	—
234	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
235	#107	—	2	2	no	—
236	#106	—	2	2	no	—
237	C₄⋊C₈	—	2	4	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	C₅×D₄	1	2	yes	—
241	C₄⋊C₈	C₂³×C₅	1	2	yes	—
242	C₄⋊C₈	C₂³×C₅	1	2	yes	—
243	C₄⋊C₈	C₂³×C₅	1	2	yes	—
244	#103	—	2	2	no	—
245	#103	—	2	2	no	—
246	#102	—	2	4	no	—
247	#102	—	2	4	no	—
248	C₂²⋊C₈	—	2	1	no	—
249	C₂²⋊C₈	C₅×D₄	1	2	yes	—
250	#97	—	2	4	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	C₂³×C₅	1	2	yes	—
253	C₂²×C₅	—	2	1	no	—
254	C₂²×C₅	—	2	1	no	—
255	C₂²×C₅	—	8	2	no	—
256	C₂²×C₅	—	8	2	no	—
257	C₂²×C₅	—	8	2	no	—
258	C₂²×C₅	—	16	4	no	—
259	C₂²×C₅	#468	1	2	yes	—
260	C₂²×C₅	C₂×#121	1	1	yes	characteristic
261	C₄×C₅	—	2	1	no	—
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	—	2	1	no	—
264	C₄×C₅	—	4	2	no	—
265	C₄×C₅	—	4	2	no	—
266	C₄×C₅	—	8	1	no	—
267	C₄×C₅	—	8	1	no	—
268	C₄×C₅	#468	1	2	yes	—
269	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
270	C₄×C₅	C₂×#137	1	1	yes	characteristic
271	C₄×C₅	#577	1	2	yes	—
272	C₂⁴	—	4	1	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	4	1	no	—
283	C₂×D₄	—	4	1	no	—
284	C₂×D₄	—	4	1	no	—
285	C₂×D₄	—	4	2	no	—
286	C₂×D₄	—	4	2	no	—
287	C₂×D₄	—	8	2	no	—
288	C₂×D₄	—	8	2	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
292	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
293	C₂²×C₄	C₅×#41	1	1	yes	characteristic
294	SD₁₆	—	4	8	no	—
295	SD₁₆	—	8	4	no	—
296	M₁₆	—	4	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	1	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	1	no	—
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
312	C₂²⋊C₄	—	4	1	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	4	1	no	—
316	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
317	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
318	C₄²	C₅×#41	1	1	yes	characteristic
319	C₄²	C₂⁴×C₅	1	1	yes	characteristic
320	C₂×C₅	—	2	2	no	—
321	C₂×C₅	—	16	2	no	—
322	C₂×C₅	#3043	1	1	yes	characteristic
323	C₂×C₅	C₂×#468	1	1	yes	characteristic
324	C₂×C₅	#3068	1	1	yes	characteristic
325	C₂³	—	4	2	no	—
326	C₂³	—	8	2	no	—
327	C₂³	—	8	2	no	—
328	C₂³	—	8	2	no	—
329	C₂³	C₅×#121	1	1	yes	characteristic
330	Q₈	—	2	2	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	8	1	no	—
335	D₄	—	4	2	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	4	no	—
338	D₄	—	8	2	no	—
339	D₄	—	8	2	no	—
340	D₄	—	8	2	no	—
341	D₄	—	8	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	8	1	no	—
350	C₂×C₄	C₅×#121	1	1	yes	characteristic
351	C₂×C₄	C₅×#121	1	1	yes	characteristic
352	C₂×C₄	C₅×#121	1	1	yes	characteristic
353	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
354	C₂×C₄	C₅×#137	1	2	yes	—
355	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
356	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
357	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
358	C₈	—	2	4	no	—
359	C₈	—	4	2	no	—
360	C₈	—	4	2	no	—
361	C₈	—	4	2	no	—
362	C₅	#19432	1	1	yes	characteristic, direct product factor, Sylow
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	16	4	no	—
369	C₂²	C₅×#468	1	2	yes	—
370	C₂²	C₂×C₅×#121	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	4	2	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	1	no	—
378	C₄	C₅×#468	1	2	yes	—
379	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
380	C₄	C₂×C₅×#137	1	1	yes	characteristic
381	C₄	C₅×#577	1	2	yes	—
382	C₂	—	2	2	no	—
383	C₂	—	16	2	no	—
384	C₂	C₅×#3043	1	1	yes	characteristic
385	C₂	C₂×C₅×#468	1	1	yes	characteristic
386	C₂	C₅×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×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×#3078	C₂	semidirect
4. C₅×#2983	C₂	semidirect
5. C₅×#1698	C₂	semidirect
6. C₅×#1686	C₂	semidirect
7. C₅×#1571	C₂	semidirect
9. C₅×#1542	C₂	semidirect
17. C₅×#477	C₂²	semidirect
18. C₅×#475	C₂²	semidirect
24. C₅×#446	C₂²	semidirect
33. C₅×#335	C₂²	semidirect
34. C₅×#333	C₂²	semidirect
35. C₅×#332	C₂²	semidirect
37. #19432	C₅	direct (N × H)
86. #3078	C₂×C₅	semidirect
88. #2983	C₂×C₅	semidirect
89. #1698	C₂×C₅	semidirect
90. #1686	C₂×C₅	semidirect
91. #1571	C₂×C₅	semidirect
93. #1542	C₂×C₅	semidirect
149. #477	C₂²×C₅	semidirect
150. #475	C₂²×C₅	semidirect
156. #446	C₂²×C₅	semidirect
165. #335	C₂²×C₅	semidirect
166. #333	C₂²×C₅	semidirect
167. #332	C₂²×C₅	semidirect
362. C₅	#19432	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×#3078	2
C₂×C₅×#460	2
C₅×#2983	2
C₅×#1698	2
C₅×#1686	2
C₅×#1571	2
C₅×#1545	2
C₅×#1542	2
#19432	5

Subgroups — C5×#19432 #11942152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19432 #11942152