Subgroups — C₅×#33617 #11956337

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	C₂×C₅×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3067	C₂	1	2	yes	maximal, complemented
4	C₅×#2223	C₂	1	4	yes	maximal, complemented
5	C₅×#2222	C₂	1	2	yes	maximal, complemented
6	C₅×#2221	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2148	C₂	1	2	yes	maximal, complemented
9	C₅×#575	—	2	2	no	—
10	C₂×C₅×#135	C₂²	1	2	yes	—
11	C₂×C₅×#131	C₂²	1	1	yes	characteristic
12	C₅×#508	—	2	1	no	—
13	C₅×#508	—	2	1	no	—
14	C₅×#508	—	2	1	no	—
15	C₅×#508	—	2	1	no	—
16	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₅×SD₃₂	C₂²	1	2	yes	complemented
18	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic, complemented
19	C₅×#504	—	2	1	no	—
20	C₅×#504	—	2	1	no	—
21	C₅×#504	C₂²	1	2	yes	—
22	C₅×#504	C₂²	1	2	yes	—
23	C₂×C₅×M₃₂	—	2	1	no	—
24	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
25	C₂²×C₅×C₁₆	—	2	1	no	—
26	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
27	C₅×#473	C₂²	1	4	yes	—
28	C₅×#472	C₂²	1	4	yes	—
29	C₅×#471	—	2	4	no	—
30	C₂×C₅×#108	C₂²	1	2	yes	—
31	C₅×#413	—	2	2	no	—
32	C₅×#368	C₂²	1	2	yes	complemented
33	C₅×#368	C₂²	1	2	yes	complemented
34	C₅×#362	C₂²	1	4	yes	complemented
35	C₅×#361	C₂²	1	4	yes	complemented
36	#33617	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₅×#41	—	2	2	no	—
38	C₂×C₅×#41	D₄	1	1	yes	characteristic
39	C₅×#137	—	2	4	no	—
40	C₅×#137	—	4	2	no	—
41	C₅×AGL(1,ℤ/8)	—	2	4	no	—
42	C₅×AGL(1,ℤ/8)	—	4	2	no	—
43	C₅×#135	—	2	2	no	—
44	C₅×#135	—	2	2	no	—
45	C₅×#135	—	2	2	no	—
46	C₅×#135	—	2	2	no	—
47	C₅×#135	—	4	4	no	—
48	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
49	C₂×C₅×SD₁₆	—	2	2	no	—
50	C₂×C₅×D₈	C₂³	1	2	yes	—
51	C₅×#131	—	2	1	no	—
52	C₅×#131	—	2	1	no	—
53	C₅×#131	D₄	1	2	yes	—
54	C₅×#131	C₂³	1	2	yes	—
55	C₂×C₅×M₁₆	—	2	1	no	—
56	C₂×C₅×M₁₆	—	2	2	no	—
57	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
58	C₂²×C₅×C₈	—	2	1	no	—
59	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
60	C₅×Q₃₂	—	2	1	no	—
61	C₅×Q₃₂	—	2	1	no	—
62	C₅×SD₃₂	—	2	2	no	—
63	C₅×SD₃₂	—	2	2	no	—
64	C₅×D₁₆	—	2	1	no	—
65	C₅×D₁₆	—	2	1	no	—
66	C₅×M₃₂	—	2	1	no	—
67	C₅×M₃₂	—	2	2	no	—
68	C₅×M₃₂	—	2	2	no	—
69	C₅×M₃₂	C₂³	1	2	yes	—
70	C₂×C₅×C₁₆	—	2	1	no	—
71	C₂×C₅×C₁₆	—	2	2	no	—
72	C₂×C₅×C₁₆	—	2	2	no	—
73	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
74	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
75	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
76	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
77	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
78	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
79	C₅×#108	—	2	2	no	—
80	C₅×#108	C₂³	1	4	yes	—
81	C₅×#101	—	2	4	no	—
82	C₅×#100	—	2	4	no	—
83	C₂×#508	C₂×C₅	1	1	yes	characteristic, complemented
84	C₂×#504	C₂×C₅	1	1	yes	characteristic, complemented
85	#3067	C₂×C₅	1	2	yes	complemented
86	#2223	C₂×C₅	1	4	yes	complemented
87	#2222	C₂×C₅	1	2	yes	complemented
88	#2221	C₂×C₅	1	2	yes	complemented
89	C₂×#368	C₂×C₅	1	1	yes	characteristic, complemented
90	#2148	C₂×C₅	1	2	yes	complemented
91	C₅×#41	—	2	1	no	—
92	C₅×#41	—	2	1	no	—
93	C₅×#41	—	4	2	no	—
94	C₅×#41	—	4	2	no	—
95	C₅×#41	—	4	2	no	—
96	C₅×#41	—	4	2	no	—
97	C₅×#41	D₈	1	2	yes	—
98	C₅×#41	C₂×D₄	1	2	yes	—
99	C₂×C₅×Q₈	—	2	2	no	—
100	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
101	C₂×C₅×D₄	—	2	1	no	—
102	C₂×C₅×D₄	—	2	2	no	—
103	C₂×C₅×D₄	—	4	2	no	—
104	C₂×C₅×D₄	D₈	1	1	yes	characteristic
105	C₂²×C₄×C₅	—	2	1	no	—
106	C₂²×C₄×C₅	—	4	2	no	—
107	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
108	C₅×Q₁₆	—	2	2	no	—
109	C₅×Q₁₆	—	2	2	no	—
110	C₅×Q₁₆	—	4	4	no	—
111	C₅×SD₁₆	—	4	4	no	—
112	C₅×SD₁₆	—	4	4	no	—
113	C₅×SD₁₆	—	4	2	no	—
114	C₅×SD₁₆	—	4	2	no	—
115	C₅×D₈	—	2	2	no	—
116	C₅×D₈	—	2	2	no	—
117	C₅×D₈	—	4	4	no	—
118	C₅×M₁₆	—	2	1	no	—
119	C₅×M₁₆	—	2	2	no	—
120	C₅×M₁₆	—	2	2	no	—
121	C₅×M₁₆	—	2	4	no	—
122	C₅×M₁₆	—	4	2	no	—
123	C₅×M₁₆	C₂×D₄	1	2	yes	—
124	C₂×C₅×C₈	—	2	1	no	—
125	C₂×C₅×C₈	—	2	2	no	—
126	C₂×C₅×C₈	—	2	2	no	—
127	C₂×C₅×C₈	—	4	2	no	—
128	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
129	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₅×C₈	#41	1	1	yes	characteristic
132	C₂×C₅×C₈	#41	1	1	yes	characteristic
133	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
134	C₅×C₁₆	—	2	2	no	—
135	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
136	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
137	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
138	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
139	#575	—	2	2	no	—
140	C₂×#135	C₂²×C₅	1	2	yes	—
141	C₂×#131	C₂²×C₅	1	1	yes	characteristic
142	#508	—	2	1	no	—
143	#508	—	2	1	no	—
144	#508	—	2	1	no	—
145	#508	—	2	1	no	—
146	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic, complemented
147	C₂×SD₃₂	C₂²×C₅	1	2	yes	complemented
148	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
149	#504	—	2	1	no	—
150	#504	—	2	1	no	—
151	#504	C₂²×C₅	1	2	yes	—
152	#504	C₂²×C₅	1	2	yes	—
153	C₂×M₃₂	—	2	1	no	—
154	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
155	C₂²×C₁₆	—	2	1	no	—
156	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
157	#473	C₂²×C₅	1	4	yes	—
158	#472	C₂²×C₅	1	4	yes	—
159	#471	—	2	4	no	—
160	C₂×#108	C₂²×C₅	1	2	yes	—
161	#413	—	2	2	no	—
162	#368	C₂²×C₅	1	2	yes	complemented
163	#368	C₂²×C₅	1	2	yes	complemented
164	#362	C₂²×C₅	1	4	yes	complemented
165	#361	C₂²×C₅	1	4	yes	complemented
166	C₂³×C₅	—	2	1	no	—
167	C₂³×C₅	—	4	2	no	—
168	C₂³×C₅	#135	1	1	yes	characteristic
169	C₅×Q₈	—	2	1	no	—
170	C₅×Q₈	—	4	2	no	—
171	C₅×Q₈	—	4	2	no	—
172	C₅×Q₈	C₂×D₈	1	2	yes	—
173	C₅×D₄	—	2	2	no	—
174	C₅×D₄	—	2	2	no	—
175	C₅×D₄	—	2	1	no	—
176	C₅×D₄	—	4	2	no	—
177	C₅×D₄	—	4	2	no	—
178	C₅×D₄	—	8	2	no	—
179	C₅×D₄	—	8	2	no	—
180	C₅×D₄	C₂×D₈	1	2	yes	—
181	C₂×C₄×C₅	—	2	1	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	2	2	no	—
184	C₂×C₄×C₅	—	4	2	no	—
185	C₂×C₄×C₅	—	8	2	no	—
186	C₂×C₄×C₅	—	8	2	no	—
187	C₂×C₄×C₅	#121	1	1	yes	characteristic
188	C₂×C₄×C₅	#121	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
191	C₂×C₄×C₅	#135	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
193	C₅×C₈	—	2	2	no	—
194	C₅×C₈	—	4	4	no	—
195	C₅×C₈	#121	1	1	yes	characteristic
196	C₅×C₈	#121	1	1	yes	characteristic
197	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
198	C₅×C₈	C₂×#41	1	1	yes	characteristic
199	C₂×#41	—	2	2	no	—
200	C₂×#41	C₅×D₄	1	1	yes	characteristic
201	#137	—	2	4	no	—
202	#137	—	4	2	no	—
203	AGL(1,ℤ/8)	—	2	4	no	—
204	AGL(1,ℤ/8)	—	4	2	no	—
205	#135	—	2	2	no	—
206	#135	—	2	2	no	—
207	#135	—	2	2	no	—
208	#135	—	2	2	no	—
209	#135	—	4	4	no	—
210	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×D₈	C₂³×C₅	1	2	yes	—
213	#131	—	2	1	no	—
214	#131	—	2	1	no	—
215	#131	C₅×D₄	1	2	yes	—
216	#131	C₂³×C₅	1	2	yes	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	2	2	no	—
219	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
220	C₂²×C₈	—	2	1	no	—
221	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
222	Q₃₂	—	2	1	no	—
223	Q₃₂	—	2	1	no	—
224	SD₃₂	—	2	2	no	—
225	SD₃₂	—	2	2	no	—
226	D₁₆	—	2	1	no	—
227	D₁₆	—	2	1	no	—
228	M₃₂	—	2	1	no	—
229	M₃₂	—	2	2	no	—
230	M₃₂	—	2	2	no	—
231	M₃₂	C₂³×C₅	1	2	yes	—
232	C₂×C₁₆	—	2	1	no	—
233	C₂×C₁₆	—	2	2	no	—
234	C₂×C₁₆	—	2	2	no	—
235	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
236	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
237	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
238	C₂×C₁₆	C₅×D₄	1	1	yes	characteristic
239	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
240	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
241	#108	—	2	2	no	—
242	#108	C₂³×C₅	1	4	yes	—
243	#101	—	2	4	no	—
244	#100	—	2	4	no	—
245	C₂²×C₅	—	2	2	no	—
246	C₂²×C₅	—	4	1	no	—
247	C₂²×C₅	—	4	2	no	—
248	C₂²×C₅	—	8	2	no	—
249	C₂²×C₅	—	16	2	no	—
250	C₂²×C₅	#466	1	1	yes	characteristic
251	C₂²×C₅	#466	1	1	yes	characteristic
252	C₂²×C₅	C₂×#135	1	1	yes	characteristic
253	C₄×C₅	—	2	2	no	—
254	C₄×C₅	—	8	2	no	—
255	C₄×C₅	#466	1	1	yes	characteristic
256	C₄×C₅	#466	1	1	yes	characteristic
257	C₄×C₅	C₂×#121	1	1	yes	characteristic
258	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
259	#41	—	2	1	no	—
260	#41	—	2	1	no	—
261	#41	—	4	2	no	—
262	#41	—	4	2	no	—
263	#41	—	4	2	no	—
264	#41	—	4	2	no	—
265	#41	C₅×D₈	1	2	yes	—
266	#41	C₂×C₅×D₄	1	2	yes	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	C₅×D₈	1	1	yes	characteristic
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂×D₄	C₅×D₈	1	1	yes	characteristic
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	4	2	no	—
275	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
276	Q₁₆	—	2	2	no	—
277	Q₁₆	—	2	2	no	—
278	Q₁₆	—	4	4	no	—
279	SD₁₆	—	4	4	no	—
280	SD₁₆	—	4	4	no	—
281	SD₁₆	—	4	2	no	—
282	SD₁₆	—	4	2	no	—
283	D₈	—	2	2	no	—
284	D₈	—	2	2	no	—
285	D₈	—	4	4	no	—
286	M₁₆	—	2	1	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	2	4	no	—
290	M₁₆	—	4	2	no	—
291	M₁₆	C₂×C₅×D₄	1	2	yes	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	4	2	no	—
296	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
297	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
298	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
299	C₂×C₈	C₅×#41	1	1	yes	characteristic
300	C₂×C₈	C₅×#41	1	1	yes	characteristic
301	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
302	C₁₆	—	2	2	no	—
303	C₁₆	C₂×C₅×D₄	1	1	yes	characteristic
304	C₁₆	C₂×C₅×D₄	1	1	yes	characteristic
305	C₁₆	C₂×C₅×D₄	1	1	yes	characteristic
306	C₁₆	C₂×C₅×D₄	1	1	yes	characteristic
307	C₂×C₅	—	2	1	no	—
308	C₂×C₅	—	2	1	no	—
309	C₂×C₅	—	2	1	no	—
310	C₂×C₅	—	4	2	no	—
311	C₂×C₅	—	16	2	no	—
312	C₂×C₅	C₂×#466	1	1	yes	characteristic
313	C₂³	—	2	1	no	—
314	C₂³	—	4	2	no	—
315	C₂³	C₅×#135	1	1	yes	characteristic
316	Q₈	—	2	1	no	—
317	Q₈	—	4	2	no	—
318	Q₈	—	4	2	no	—
319	Q₈	C₂×C₅×D₈	1	2	yes	—
320	D₄	—	2	2	no	—
321	D₄	—	2	2	no	—
322	D₄	—	2	1	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	2	no	—
325	D₄	—	8	2	no	—
326	D₄	—	8	2	no	—
327	D₄	C₂×C₅×D₈	1	2	yes	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	8	2	no	—
333	C₂×C₄	—	8	2	no	—
334	C₂×C₄	C₅×#121	1	1	yes	characteristic
335	C₂×C₄	C₅×#121	1	1	yes	characteristic
336	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
337	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
338	C₂×C₄	C₅×#135	1	1	yes	characteristic
339	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	4	4	no	—
342	C₈	C₅×#121	1	1	yes	characteristic
343	C₈	C₅×#121	1	1	yes	characteristic
344	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
345	C₈	C₂×C₅×#41	1	1	yes	characteristic
346	C₅	#33617	1	1	yes	characteristic, direct product factor, Sylow
347	C₂²	—	2	2	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	2	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	16	2	no	—
352	C₂²	C₅×#466	1	1	yes	characteristic
353	C₂²	C₅×#466	1	1	yes	characteristic
354	C₂²	C₂×C₅×#135	1	1	yes	characteristic
355	C₄	—	2	2	no	—
356	C₄	—	8	2	no	—
357	C₄	C₅×#466	1	1	yes	characteristic
358	C₄	C₅×#466	1	1	yes	characteristic
359	C₄	C₂×C₅×#121	1	1	yes	characteristic
360	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	4	2	no	—
365	C₂	—	16	2	no	—
366	C₂	C₂×C₅×#466	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#466
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂×C₅	C₂×#466
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#508	C₂	semidirect
2. C₂×C₅×#504	C₂	semidirect
3. C₅×#3067	C₂	semidirect
4. C₅×#2223	C₂	semidirect
5. C₅×#2222	C₂	semidirect
6. C₅×#2221	C₂	semidirect
7. C₂×C₅×#368	C₂	semidirect
8. C₅×#2148	C₂	semidirect
16. C₂×C₅×Q₃₂	C₂²	semidirect
17. C₂×C₅×SD₃₂	C₂²	semidirect
18. C₂×C₅×D₁₆	C₂²	semidirect
32. C₅×#368	C₂²	semidirect
33. C₅×#368	C₂²	semidirect
34. C₅×#362	C₂²	semidirect
35. C₅×#361	C₂²	semidirect
36. #33617	C₅	direct (N × H)
83. C₂×#508	C₂×C₅	semidirect
84. C₂×#504	C₂×C₅	semidirect
85. #3067	C₂×C₅	semidirect
86. #2223	C₂×C₅	semidirect
87. #2222	C₂×C₅	semidirect
88. #2221	C₂×C₅	semidirect
89. C₂×#368	C₂×C₅	semidirect
90. #2148	C₂×C₅	semidirect
146. C₂×Q₃₂	C₂²×C₅	semidirect
147. C₂×SD₃₂	C₂²×C₅	semidirect
148. C₂×D₁₆	C₂²×C₅	semidirect
162. #368	C₂²×C₅	semidirect
163. #368	C₂²×C₅	semidirect
164. #362	C₂²×C₅	semidirect
165. #361	C₂²×C₅	semidirect
346. C₅	#33617	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×#508	2
C₂×C₅×#504	2
C₅×#3067	2
C₅×#2223	2
C₅×#2222	2
C₅×#2221	2
C₂×C₅×#368	2
C₅×#2148	2
#33617	5

Subgroups — C5×#33617 #11956337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33617 #11956337