Subgroups — C₅×#33781 #11956501

Order 1280.

598 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#495	C₂	1	1	yes	maximal, characteristic
3	C₅×#2251	C₂	1	2	yes	maximal, complemented
4	C₅×#2243	C₂	1	2	yes	maximal, complemented
5	C₅×#2211	C₂	1	4	yes	maximal, complemented
6	C₅×#2188	C₂	1	2	yes	maximal, complemented
7	C₅×#2187	C₂	1	2	yes	maximal
8	C₅×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅×SD₁₆	—	2	1	no	—
11	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
12	C₅×#533	—	2	1	no	—
13	C₂×C₅×#124	—	2	1	no	—
14	C₂×C₅×Q₃₂	C₂²	1	4	yes	complemented
15	C₂×C₅×SD₃₂	C₂²	1	4	yes	—
16	C₅×#500	C₂²	1	1	yes	characteristic, complemented
17	C₅×#495	—	2	1	no	—
18	C₅×#495	—	2	1	no	—
19	C₅×#495	C₂²	1	2	yes	—
20	C₅×#495	C₂²	1	2	yes	—
21	C₅×#494	C₂²	1	1	yes	characteristic, complemented
22	C₅×#487	—	2	1	no	—
23	C₅×#467	—	2	1	no	—
24	C₅×#467	C₂²	1	2	yes	—
25	C₄×C₅×Q₁₆	—	2	1	no	—
26	C₄×C₅×Q₁₆	C₂²	1	2	yes	complemented
27	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
28	C₅×C₄⋊C₁₆	C₂²	1	2	yes	complemented
29	C₅×#358	C₂²	1	4	yes	complemented
30	C₅×#357	C₂²	1	4	yes	—
31	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₅×#346	C₂²	1	2	yes	complemented
33	#33781	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₅×Q₈	—	2	1	no	—
35	C₂²×C₅×D₄	—	2	1	no	—
36	C₂×C₅×Q₁₆	—	2	1	no	—
37	C₂×C₅×Q₁₆	—	2	1	no	—
38	C₂×C₅×Q₁₆	—	4	2	no	—
39	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
40	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
41	C₂×C₅×SD₁₆	—	2	2	no	—
42	C₂×C₅×SD₁₆	—	2	2	no	—
43	C₂×C₅×SD₁₆	—	4	4	no	—
44	C₂×C₅×D₈	—	2	1	no	—
45	C₂×C₅×D₈	—	4	2	no	—
46	C₂×C₅×D₈	C₂³	1	2	yes	—
47	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₅×C₄⋊Q₈	—	2	1	no	—
50	C₅×C₄⋊Q₈	—	2	1	no	—
51	C₅×#125	—	4	1	no	—
52	C₅×#124	—	2	2	no	—
53	C₅×#124	—	2	2	no	—
54	C₅×#124	—	4	1	no	—
55	C₅×#124	—	4	1	no	—
56	C₅×C₂²⋊Q₈	—	2	2	no	—
57	C₅×C₂²⋊Q₈	—	4	1	no	—
58	C₄×C₅×Q₈	—	2	2	no	—
59	C₄×C₅×Q₈	—	4	1	no	—
60	C₅×#117	—	4	1	no	—
61	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
62	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
63	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
64	C₅×Q₃₂	—	4	4	no	—
65	C₅×SD₃₂	—	4	4	no	—
66	C₂×C₅×C₁₆	C₂³	1	4	yes	—
67	C₅×#107	—	2	1	no	—
68	C₅×#107	C₂³	1	2	yes	—
69	C₅×#103	—	2	2	no	—
70	C₅×#103	—	2	2	no	—
71	C₄×C₅×C₈	D₄	1	1	yes	characteristic
72	C₄×C₅×C₈	D₄	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
75	#3170	C₂×C₅	1	1	yes	characteristic, complemented
76	C₂×#495	C₂×C₅	1	1	yes	characteristic
77	#2251	C₂×C₅	1	2	yes	complemented
78	#2243	C₂×C₅	1	2	yes	complemented
79	#2211	C₂×C₅	1	4	yes	complemented
80	#2188	C₂×C₅	1	2	yes	complemented
81	#2187	C₂×C₅	1	2	yes	—
82	#2163	C₂×C₅	1	1	yes	characteristic, complemented
83	C₂⁴×C₅	—	4	1	no	—
84	C₂×C₅×Q₈	—	2	2	no	—
85	C₂×C₅×Q₈	—	2	2	no	—
86	C₂×C₅×Q₈	—	4	2	no	—
87	C₂×C₅×Q₈	—	4	2	no	—
88	C₂×C₅×Q₈	—	4	1	no	—
89	C₂×C₅×Q₈	—	4	1	no	—
90	C₂×C₅×D₄	—	2	2	no	—
91	C₂×C₅×D₄	—	4	2	no	—
92	C₂×C₅×D₄	—	4	2	no	—
93	C₂×C₅×D₄	—	4	1	no	—
94	C₂²×C₄×C₅	—	4	1	no	—
95	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
96	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₅×Q₁₆	—	2	4	no	—
99	C₅×Q₁₆	—	2	4	no	—
100	C₅×Q₁₆	—	4	2	no	—
101	C₅×Q₁₆	—	4	2	no	—
102	C₅×SD₁₆	—	4	4	no	—
103	C₅×SD₁₆	—	4	4	no	—
104	C₅×D₈	—	2	4	no	—
105	C₅×D₈	—	4	2	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
112	C₅×C₄⋊C₄	—	2	2	no	—
113	C₅×C₄⋊C₄	—	4	1	no	—
114	C₅×C₄⋊C₄	—	4	1	no	—
115	C₅×C₄⋊C₄	—	4	2	no	—
116	C₅×C₄⋊C₄	—	4	2	no	—
117	C₅×C₄⋊C₄	—	4	1	no	—
118	C₅×C₂²⋊C₄	—	4	2	no	—
119	C₅×C₂²⋊C₄	—	4	2	no	—
120	C₅×C₂²⋊C₄	—	4	2	no	—
121	C₅×C₂²⋊C₄	—	4	2	no	—
122	C₅×C₂²⋊C₄	—	4	2	no	—
123	C₄²×C₅	—	4	2	no	—
124	C₄²×C₅	D₈	1	1	yes	characteristic
125	C₄²×C₅	D₈	1	1	yes	characteristic
126	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
128	C₅×C₁₆	—	2	4	no	—
129	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
130	C₂²×SD₁₆	—	2	1	no	—
131	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
132	#533	—	2	1	no	—
133	C₂×#124	—	2	1	no	—
134	C₂×Q₃₂	C₂²×C₅	1	4	yes	complemented
135	C₂×SD₃₂	C₂²×C₅	1	4	yes	—
136	#500	C₂²×C₅	1	1	yes	characteristic, complemented
137	#495	—	2	1	no	—
138	#495	—	2	1	no	—
139	#495	C₂²×C₅	1	2	yes	—
140	#495	C₂²×C₅	1	2	yes	—
141	#494	C₂²×C₅	1	1	yes	characteristic, complemented
142	#487	—	2	1	no	—
143	#467	—	2	1	no	—
144	#467	C₂²×C₅	1	2	yes	—
145	C₄×Q₁₆	—	2	1	no	—
146	C₄×Q₁₆	C₂²×C₅	1	2	yes	complemented
147	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
148	C₄⋊C₁₆	C₂²×C₅	1	2	yes	complemented
149	#358	C₂²×C₅	1	4	yes	complemented
150	#357	C₂²×C₅	1	4	yes	—
151	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
152	#346	C₂²×C₅	1	2	yes	complemented
153	C₂³×C₅	—	4	2	no	—
154	C₂³×C₅	—	8	2	no	—
155	C₂³×C₅	—	8	2	no	—
156	C₂³×C₅	—	8	2	no	—
157	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
158	C₅×Q₈	—	4	2	no	—
159	C₅×Q₈	—	4	2	no	—
160	C₅×Q₈	—	4	2	no	—
161	C₅×Q₈	—	4	2	no	—
162	C₅×Q₈	—	8	2	no	—
163	C₅×Q₈	—	8	2	no	—
164	C₅×D₄	—	4	2	no	—
165	C₅×D₄	—	4	2	no	—
166	C₅×D₄	—	8	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	4	1	no	—
171	C₂×C₄×C₅	—	4	2	no	—
172	C₂×C₄×C₅	—	4	2	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	—	8	2	no	—
175	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
181	C₅×C₈	—	2	1	no	—
182	C₅×C₈	—	2	1	no	—
183	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
184	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
185	C₂²×Q₈	—	2	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	—	4	2	no	—
190	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
191	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
192	C₂×SD₁₆	—	2	2	no	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×SD₁₆	—	4	4	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	4	2	no	—
197	C₂×D₈	C₂³×C₅	1	2	yes	—
198	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	1	no	—
201	C₄⋊Q₈	—	2	1	no	—
202	#125	—	4	1	no	—
203	#124	—	2	2	no	—
204	#124	—	2	2	no	—
205	#124	—	4	1	no	—
206	#124	—	4	1	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	C₂²⋊Q₈	—	4	1	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	4	1	no	—
211	#117	—	4	1	no	—
212	C₂×C₂²⋊C₄	—	4	1	no	—
213	C₂×C₂²⋊C₄	—	4	1	no	—
214	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
215	Q₃₂	—	4	4	no	—
216	SD₃₂	—	4	4	no	—
217	C₂×C₁₆	C₂³×C₅	1	4	yes	—
218	#107	—	2	1	no	—
219	#107	C₂³×C₅	1	2	yes	—
220	#103	—	2	2	no	—
221	#103	—	2	2	no	—
222	C₄×C₈	C₅×D₄	1	1	yes	characteristic
223	C₄×C₈	C₅×D₄	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	—	8	2	no	—
229	C₂²×C₅	—	8	2	no	—
230	C₂²×C₅	—	8	2	no	—
231	C₂²×C₅	—	16	4	no	—
232	C₂²×C₅	#510	1	2	yes	—
233	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	8	2	no	—
240	C₄×C₅	—	8	2	no	—
241	C₄×C₅	—	8	2	no	—
242	C₄×C₅	#535	1	1	yes	characteristic
243	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₂⁴	—	4	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	—	4	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
257	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
259	Q₁₆	—	2	4	no	—
260	Q₁₆	—	2	4	no	—
261	Q₁₆	—	4	2	no	—
262	Q₁₆	—	4	2	no	—
263	SD₁₆	—	4	4	no	—
264	SD₁₆	—	4	4	no	—
265	D₈	—	2	4	no	—
266	D₈	—	4	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
270	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
271	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
272	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₄²	—	4	2	no	—
285	C₄²	C₅×D₈	1	1	yes	characteristic
286	C₄²	C₅×D₈	1	1	yes	characteristic
287	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
288	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
289	C₁₆	—	2	4	no	—
290	C₂×C₅	—	2	2	no	—
291	C₂×C₅	—	16	2	no	—
292	C₂×C₅	#3205	1	1	yes	characteristic
293	C₂×C₅	C₂×#510	1	1	yes	characteristic
294	C₂×C₅	#3414	1	1	yes	characteristic
295	C₂³	—	4	2	no	—
296	C₂³	—	8	2	no	—
297	C₂³	—	8	2	no	—
298	C₂³	—	8	2	no	—
299	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	8	2	no	—
305	Q₈	—	8	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
318	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
319	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
320	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
321	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
326	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
327	C₅	#33781	1	1	yes	characteristic, direct product factor, Sylow
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	16	4	no	—
334	C₂²	C₅×#510	1	2	yes	—
335	C₂²	C₂²×C₅×D₈	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	—	2	1	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	2	no	—
344	C₄	C₅×#535	1	1	yes	characteristic
345	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	—	16	2	no	—
348	C₂	C₅×#3205	1	1	yes	characteristic
349	C₂	C₂×C₅×#510	1	1	yes	characteristic
350	C₂	C₅×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3170	C₂	semidirect
3. C₅×#2251	C₂	semidirect
4. C₅×#2243	C₂	semidirect
5. C₅×#2211	C₂	semidirect
6. C₅×#2188	C₂	semidirect
8. C₅×#2163	C₂	semidirect
14. C₂×C₅×Q₃₂	C₂²	semidirect
16. C₅×#500	C₂²	semidirect
21. C₅×#494	C₂²	semidirect
26. C₄×C₅×Q₁₆	C₂²	semidirect
28. C₅×C₄⋊C₁₆	C₂²	semidirect
29. C₅×#358	C₂²	semidirect
32. C₅×#346	C₂²	semidirect
33. #33781	C₅	direct (N × H)
75. #3170	C₂×C₅	semidirect
77. #2251	C₂×C₅	semidirect
78. #2243	C₂×C₅	semidirect
79. #2211	C₂×C₅	semidirect
80. #2188	C₂×C₅	semidirect
82. #2163	C₂×C₅	semidirect
134. C₂×Q₃₂	C₂²×C₅	semidirect
136. #500	C₂²×C₅	semidirect
141. #494	C₂²×C₅	semidirect
146. C₄×Q₁₆	C₂²×C₅	semidirect
148. C₄⋊C₁₆	C₂²×C₅	semidirect
149. #358	C₂²×C₅	semidirect
152. #346	C₂²×C₅	semidirect
327. C₅	#33781	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×#3170	2
C₂×C₅×#495	2
C₅×#2251	2
C₅×#2243	2
C₅×#2211	2
C₅×#2188	2
C₅×#2187	2
C₅×#2163	2
#33781	5

Subgroups — C5×#33781 #11956501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33781 #11956501