Subgroups — C₅×#33779 #11956499

Order 1280.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#495	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2250	C₂	1	2	yes	maximal, complemented
4	C₅×#2243	C₂	1	2	yes	maximal, complemented
5	C₅×#2210	C₂	1	4	yes	maximal, complemented
6	C₅×#2185	C₂	1	2	yes	maximal, complemented
7	C₅×#2184	C₂	1	2	yes	maximal, complemented
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×#532	—	2	1	no	—
13	C₂×C₅×#124	—	2	1	no	—
14	C₂×C₅×SD₃₂	C₂²	1	4	yes	complemented
15	C₂×C₅×D₁₆	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×#493	C₂²	1	1	yes	characteristic, complemented
22	C₅×#486	—	2	1	no	—
23	C₅×#466	—	2	1	no	—
24	C₅×#466	C₂²	1	2	yes	—
25	C₄×C₅×D₈	—	2	1	no	—
26	C₄×C₅×D₈	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	#33779	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₅×#41	—	4	1	no	—
35	C₂²×C₅×Q₈	—	2	1	no	—
36	C₂²×C₅×D₄	—	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×SD₁₆	—	2	2	no	—
41	C₂×C₅×SD₁₆	—	2	2	no	—
42	C₂×C₅×SD₁₆	—	4	4	no	—
43	C₂×C₅×D₈	—	2	1	no	—
44	C₂×C₅×D₈	—	2	1	no	—
45	C₂×C₅×D₈	—	4	2	no	—
46	C₂×C₅×D₈	C₂³	1	2	yes	—
47	C₂×C₅×D₈	C₂³	1	2	yes	—
48	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
49	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₅×C₄⋊Q₈	—	2	1	no	—
51	C₅×#127	—	2	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×#124	—	4	1	no	—
57	C₅×#121	—	2	2	no	—
58	C₅×#121	—	4	1	no	—
59	C₄×C₅×D₄	—	2	2	no	—
60	C₄×C₅×D₄	—	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×SD₃₂	—	4	4	no	—
65	C₅×D₁₆	—	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×#102	—	2	2	no	—
70	C₅×#102	—	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	#3169	C₂×C₅	1	1	yes	characteristic, complemented
76	C₂×#495	C₂×C₅	1	1	yes	characteristic, complemented
77	#2250	C₂×C₅	1	2	yes	complemented
78	#2243	C₂×C₅	1	2	yes	complemented
79	#2210	C₂×C₅	1	4	yes	complemented
80	#2185	C₂×C₅	1	2	yes	complemented
81	#2184	C₂×C₅	1	2	yes	complemented
82	#2163	C₂×C₅	1	1	yes	characteristic, complemented
83	C₂⁴×C₅	—	4	1	no	—
84	C₅×#41	—	8	4	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	—	2	2	no	—
92	C₂×C₅×D₄	—	4	2	no	—
93	C₂×C₅×D₄	—	4	2	no	—
94	C₂×C₅×D₄	—	4	2	no	—
95	C₂×C₅×D₄	—	4	1	no	—
96	C₂×C₅×D₄	—	4	1	no	—
97	C₂×C₅×D₄	—	4	1	no	—
98	C₂²×C₄×C₅	—	4	1	no	—
99	C₂²×C₄×C₅	—	4	2	no	—
100	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
101	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
103	C₅×Q₁₆	—	2	4	no	—
104	C₅×Q₁₆	—	4	2	no	—
105	C₅×SD₁₆	—	4	4	no	—
106	C₅×SD₁₆	—	4	4	no	—
107	C₅×D₈	—	2	4	no	—
108	C₅×D₈	—	2	4	no	—
109	C₅×D₈	—	4	2	no	—
110	C₅×D₈	—	4	2	no	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	—	4	1	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	C₅×C₂²⋊C₄	—	4	2	no	—
124	C₅×C₂²⋊C₄	—	4	2	no	—
125	C₄²×C₅	D₈	1	1	yes	characteristic
126	C₄²×C₅	D₈	1	1	yes	characteristic
127	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
129	C₅×C₁₆	—	2	4	no	—
130	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
131	C₂²×SD₁₆	—	2	1	no	—
132	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
133	#532	—	2	1	no	—
134	C₂×#124	—	2	1	no	—
135	C₂×SD₃₂	C₂²×C₅	1	4	yes	complemented
136	C₂×D₁₆	C₂²×C₅	1	4	yes	—
137	#500	C₂²×C₅	1	1	yes	characteristic, complemented
138	#495	—	2	1	no	—
139	#495	—	2	1	no	—
140	#495	C₂²×C₅	1	2	yes	—
141	#495	C₂²×C₅	1	2	yes	—
142	#493	C₂²×C₅	1	1	yes	characteristic, complemented
143	#486	—	2	1	no	—
144	#466	—	2	1	no	—
145	#466	C₂²×C₅	1	2	yes	—
146	C₄×D₈	—	2	1	no	—
147	C₄×D₈	C₂²×C₅	1	2	yes	complemented
148	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
149	C₄⋊C₁₆	C₂²×C₅	1	2	yes	complemented
150	#358	C₂²×C₅	1	4	yes	complemented
151	#357	C₂²×C₅	1	4	yes	—
152	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
153	#346	C₂²×C₅	1	2	yes	complemented
154	C₂³×C₅	—	4	2	no	—
155	C₂³×C₅	—	4	2	no	—
156	C₂³×C₅	—	8	2	no	—
157	C₂³×C₅	—	8	2	no	—
158	C₂³×C₅	—	8	2	no	—
159	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
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	—	4	2	no	—
167	C₅×D₄	—	4	2	no	—
168	C₅×D₄	—	8	2	no	—
169	C₅×D₄	—	8	2	no	—
170	C₅×D₄	—	8	2	no	—
171	C₅×D₄	—	8	4	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	1	no	—
174	C₂×C₄×C₅	—	2	1	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	—	4	2	no	—
178	C₂×C₄×C₅	—	8	4	no	—
179	C₂×C₄×C₅	—	8	2	no	—
180	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
186	C₅×C₈	—	2	1	no	—
187	C₅×C₈	—	2	1	no	—
188	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
189	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
190	C₂×#41	—	4	1	no	—
191	C₂²×Q₈	—	2	1	no	—
192	C₂²×D₄	—	2	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	—	4	2	no	—
195	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
196	C₂×SD₁₆	—	2	2	no	—
197	C₂×SD₁₆	—	2	2	no	—
198	C₂×SD₁₆	—	4	4	no	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	4	2	no	—
202	C₂×D₈	C₂³×C₅	1	2	yes	—
203	C₂×D₈	C₂³×C₅	1	2	yes	—
204	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
206	C₄⋊Q₈	—	2	1	no	—
207	#127	—	2	1	no	—
208	#124	—	2	2	no	—
209	#124	—	2	2	no	—
210	#124	—	4	1	no	—
211	#124	—	4	1	no	—
212	#124	—	4	1	no	—
213	#121	—	2	2	no	—
214	#121	—	4	1	no	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	—	4	1	no	—
217	C₂×C₂²⋊C₄	—	4	1	no	—
218	C₂×C₂²⋊C₄	—	4	1	no	—
219	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
220	SD₃₂	—	4	4	no	—
221	D₁₆	—	4	4	no	—
222	C₂×C₁₆	C₂³×C₅	1	4	yes	—
223	#107	—	2	1	no	—
224	#107	C₂³×C₅	1	2	yes	—
225	#102	—	2	2	no	—
226	#102	—	2	2	no	—
227	C₄×C₈	C₅×D₄	1	1	yes	characteristic
228	C₄×C₈	C₅×D₄	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
231	C₂²×C₅	—	2	1	no	—
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	—	8	2	no	—
234	C₂²×C₅	—	8	2	no	—
235	C₂²×C₅	—	8	2	no	—
236	C₂²×C₅	—	8	2	no	—
237	C₂²×C₅	—	8	2	no	—
238	C₂²×C₅	—	8	2	no	—
239	C₂²×C₅	—	16	4	no	—
240	C₂²×C₅	#509	1	2	yes	—
241	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	2	1	no	—
245	C₄×C₅	—	2	1	no	—
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	—	8	2	no	—
248	C₄×C₅	—	8	2	no	—
249	C₄×C₅	#535	1	1	yes	characteristic
250	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₂⁴	—	4	1	no	—
252	#41	—	8	4	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂²×C₄	—	4	1	no	—
267	C₂²×C₄	—	4	2	no	—
268	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
269	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
271	Q₁₆	—	2	4	no	—
272	Q₁₆	—	4	2	no	—
273	SD₁₆	—	4	4	no	—
274	SD₁₆	—	4	4	no	—
275	D₈	—	2	4	no	—
276	D₈	—	2	4	no	—
277	D₈	—	4	2	no	—
278	D₈	—	4	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
282	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
283	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
284	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₄²	C₅×D₈	1	1	yes	characteristic
294	C₄²	C₅×D₈	1	1	yes	characteristic
295	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
296	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
297	C₁₆	—	2	4	no	—
298	C₂×C₅	—	2	2	no	—
299	C₂×C₅	—	16	2	no	—
300	C₂×C₅	—	16	2	no	—
301	C₂×C₅	#3205	1	1	yes	characteristic
302	C₂×C₅	C₂×#509	1	1	yes	characteristic
303	C₂×C₅	#3414	1	1	yes	characteristic
304	C₂³	—	4	2	no	—
305	C₂³	—	4	2	no	—
306	C₂³	—	8	2	no	—
307	C₂³	—	8	2	no	—
308	C₂³	—	8	2	no	—
309	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	8	2	no	—
313	Q₈	—	8	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	8	2	no	—
319	D₄	—	8	2	no	—
320	D₄	—	8	2	no	—
321	D₄	—	8	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	4	no	—
329	C₂×C₄	—	8	2	no	—
330	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
331	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
332	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
333	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
334	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
335	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
339	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
340	C₅	#33779	1	1	yes	characteristic, direct product factor, Sylow
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	16	4	no	—
350	C₂²	C₅×#509	1	2	yes	—
351	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	8	2	no	—
358	C₄	—	8	2	no	—
359	C₄	C₅×#535	1	1	yes	characteristic
360	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
361	C₂	—	2	2	no	—
362	C₂	—	16	2	no	—
363	C₂	—	16	2	no	—
364	C₂	C₅×#3205	1	1	yes	characteristic
365	C₂	C₂×C₅×#509	1	1	yes	characteristic
366	C₂	C₅×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
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×#3169	C₂	semidirect
2. C₂×C₅×#495	C₂	semidirect
3. C₅×#2250	C₂	semidirect
4. C₅×#2243	C₂	semidirect
5. C₅×#2210	C₂	semidirect
6. C₅×#2185	C₂	semidirect
7. C₅×#2184	C₂	semidirect
8. C₅×#2163	C₂	semidirect
14. C₂×C₅×SD₃₂	C₂²	semidirect
16. C₅×#500	C₂²	semidirect
21. C₅×#493	C₂²	semidirect
26. C₄×C₅×D₈	C₂²	semidirect
28. C₅×C₄⋊C₁₆	C₂²	semidirect
29. C₅×#358	C₂²	semidirect
32. C₅×#346	C₂²	semidirect
33. #33779	C₅	direct (N × H)
75. #3169	C₂×C₅	semidirect
76. C₂×#495	C₂×C₅	semidirect
77. #2250	C₂×C₅	semidirect
78. #2243	C₂×C₅	semidirect
79. #2210	C₂×C₅	semidirect
80. #2185	C₂×C₅	semidirect
81. #2184	C₂×C₅	semidirect
82. #2163	C₂×C₅	semidirect
135. C₂×SD₃₂	C₂²×C₅	semidirect
137. #500	C₂²×C₅	semidirect
142. #493	C₂²×C₅	semidirect
147. C₄×D₈	C₂²×C₅	semidirect
149. C₄⋊C₁₆	C₂²×C₅	semidirect
150. #358	C₂²×C₅	semidirect
153. #346	C₂²×C₅	semidirect
340. C₅	#33779	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×#3169	2
C₂×C₅×#495	2
C₅×#2250	2
C₅×#2243	2
C₅×#2210	2
C₅×#2185	2
C₅×#2184	2
C₅×#2163	2
#33779	5

Subgroups — C5×#33779 #11956499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33779 #11956499