Subgroups — C₅×#10172 #11932892

Order 1280.

562 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3227	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3152	C₂	1	1	yes	maximal, characteristic
4	C₅×#1997	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1546	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1492	C₂	1	1	yes	maximal, characteristic
7	C₅×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#137	—	2	2	no	—
9	C₅×#532	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₅×#498	—	2	2	no	—
13	C₅×#498	—	2	2	no	—
14	C₅×#498	—	2	1	no	—
15	C₅×#497	—	2	2	no	—
16	C₅×#494	—	2	1	no	—
17	C₅×#492	—	2	1	no	—
18	C₅×#486	—	2	1	no	—
19	C₅×#481	—	2	2	no	—
20	C₅×#476	—	2	2	no	—
21	C₅×#431	C₂²	1	1	yes	characteristic
22	C₂×C₅×#106	—	2	2	no	—
23	C₅×#423	C₂²	1	1	yes	characteristic
24	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
26	C₅×#343	—	2	2	no	—
27	C₅×#332	—	2	2	no	—
28	C₅×D₄⋊C₈	—	2	2	no	—
29	#10172	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×#41	—	2	1	no	—
31	C₂²×C₅×Q₈	—	2	1	no	—
32	C₂²×C₅×Q₈	—	2	1	no	—
33	C₅×#137	—	4	4	no	—
34	C₅×#137	—	4	4	no	—
35	C₂×C₅×Q₁₆	—	2	4	no	—
36	C₂×C₅×SD₁₆	—	2	4	no	—
37	C₂×C₅×M₁₆	—	2	1	no	—
38	C₂×C₅×M₁₆	D₄	1	2	yes	—
39	C₂²×C₅×C₈	—	2	1	no	—
40	C₅×C₄⋊Q₈	—	2	1	no	—
41	C₅×C₄⋊Q₈	—	2	1	no	—
42	C₅×C₄⋊Q₈	—	2	1	no	—
43	C₅×C₄⋊Q₈	—	2	1	no	—
44	C₅×C₄⋊Q₈	—	2	1	no	—
45	C₅×C₄⋊Q₈	—	2	1	no	—
46	C₅×C₄⋊Q₈	—	2	1	no	—
47	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
48	C₅×C₄⋊Q₈	D₄	1	2	yes	—
49	C₅×#127	D₄	1	1	yes	characteristic
50	C₅×#124	—	2	1	no	—
51	C₅×#121	—	2	1	no	—
52	C₅×#121	—	2	1	no	—
53	C₄×C₅×D₄	—	2	1	no	—
54	C₄×C₅×D₄	—	2	1	no	—
55	C₂×C₅×C₄⋊C₄	—	2	1	no	—
56	C₂×C₅×C₄⋊C₄	—	2	1	no	—
57	C₂×C₅×C₄⋊C₄	—	2	1	no	—
58	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
59	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
60	C₅×#106	—	2	8	no	—
61	C₅×#106	—	2	4	no	—
62	C₅×C₄⋊C₈	—	2	1	no	—
63	C₅×C₄⋊C₈	—	2	1	no	—
64	C₅×C₄⋊C₈	D₄	1	2	yes	—
65	C₅×#102	—	2	4	no	—
66	C₅×C₂²⋊C₈	—	2	1	no	—
67	C₅×#97	—	2	1	no	—
68	C₄×C₅×C₈	—	2	1	no	—
69	C₄×C₅×C₈	—	2	1	no	—
70	C₄×C₅×C₈	—	2	1	no	—
71	C₄×C₅×C₈	D₄	1	2	yes	—
72	#3227	C₂×C₅	1	1	yes	characteristic
73	C₂×#498	C₂×C₅	1	1	yes	characteristic, complemented
74	#3152	C₂×C₅	1	1	yes	characteristic
75	#1997	C₂×C₅	1	1	yes	characteristic, complemented
76	#1546	C₂×C₅	1	1	yes	characteristic, complemented
77	#1492	C₂×C₅	1	1	yes	characteristic
78	#1278	C₂×C₅	1	1	yes	characteristic, complemented
79	C₅×#41	—	4	2	no	—
80	C₅×#41	—	4	2	no	—
81	C₂×C₅×Q₈	—	2	1	no	—
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	4	1	no	—
84	C₂×C₅×Q₈	—	4	1	no	—
85	C₂×C₅×Q₈	—	4	1	no	—
86	C₂×C₅×Q₈	—	4	2	no	—
87	C₂×C₅×Q₈	—	4	2	no	—
88	C₂×C₅×Q₈	—	8	2	no	—
89	C₂×C₅×D₄	—	2	1	no	—
90	C₂×C₅×D₄	—	2	1	no	—
91	C₂×C₅×D₄	—	2	1	no	—
92	C₂×C₅×D₄	—	4	1	no	—
93	C₂²×C₄×C₅	—	2	2	no	—
94	C₂²×C₄×C₅	—	4	1	no	—
95	C₂²×C₄×C₅	—	4	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×Q₁₆	—	4	8	no	—
100	C₅×SD₁₆	—	4	8	no	—
101	C₅×M₁₆	—	2	2	no	—
102	C₅×M₁₆	—	4	2	no	—
103	C₅×M₁₆	SD₁₆	1	4	yes	—
104	C₂×C₅×C₈	—	2	4	no	—
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	—	4	1	no	—
109	C₅×C₄⋊C₄	—	2	2	no	—
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	—	2	2	no	—
112	C₅×C₄⋊C₄	—	2	1	no	—
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	—	4	1	no	—
116	C₅×C₄⋊C₄	—	4	1	no	—
117	C₅×C₄⋊C₄	—	4	1	no	—
118	C₅×C₄⋊C₄	—	4	1	no	—
119	C₅×C₂²⋊C₄	—	4	1	no	—
120	C₅×C₂²⋊C₄	—	4	1	no	—
121	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
125	C₂×#137	—	2	2	no	—
126	#532	C₂²×C₅	1	1	yes	characteristic
127	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
128	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
129	#498	—	2	2	no	—
130	#498	—	2	1	no	—
131	#498	—	2	2	no	—
132	#497	—	2	2	no	—
133	#494	—	2	1	no	—
134	#492	—	2	1	no	—
135	#486	—	2	1	no	—
136	#481	—	2	2	no	—
137	#476	—	2	2	no	—
138	#431	C₂²×C₅	1	1	yes	characteristic
139	C₂×#106	—	2	2	no	—
140	#423	C₂²×C₅	1	1	yes	characteristic
141	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
143	#343	—	2	2	no	—
144	#332	—	2	2	no	—
145	D₄⋊C₈	—	2	2	no	—
146	C₂³×C₅	—	4	1	no	—
147	C₂³×C₅	#121	1	1	yes	characteristic
148	C₅×Q₈	—	4	4	no	—
149	C₅×Q₈	—	4	2	no	—
150	C₅×Q₈	—	8	2	no	—
151	C₅×Q₈	—	8	2	no	—
152	C₅×Q₈	—	8	1	no	—
153	C₅×Q₈	—	8	1	no	—
154	C₅×D₄	—	4	2	no	—
155	C₅×D₄	—	4	2	no	—
156	C₅×D₄	—	4	2	no	—
157	C₅×D₄	—	8	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	—	4	2	no	—
163	C₂×C₄×C₅	—	8	2	no	—
164	C₂×C₄×C₅	—	8	1	no	—
165	C₂×C₄×C₅	—	8	1	no	—
166	C₂×C₄×C₅	—	8	2	no	—
167	C₂×C₄×C₅	#120	1	1	yes	characteristic
168	C₂×C₄×C₅	#120	1	1	yes	characteristic
169	C₂×C₄×C₅	#120	1	1	yes	characteristic
170	C₂×C₄×C₅	#121	1	1	yes	characteristic
171	C₂×C₄×C₅	#121	1	1	yes	characteristic
172	C₂×C₄×C₅	#127	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
174	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
175	C₅×C₈	—	2	4	no	—
176	C₅×C₈	—	2	4	no	—
177	C₅×C₈	—	4	2	no	—
178	C₅×C₈	—	8	1	no	—
179	C₂×#41	—	2	1	no	—
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×Q₈	—	2	1	no	—
182	#137	—	4	4	no	—
183	#137	—	4	4	no	—
184	C₂×Q₁₆	—	2	4	no	—
185	C₂×SD₁₆	—	2	4	no	—
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	C₅×D₄	1	2	yes	—
188	C₂²×C₈	—	2	1	no	—
189	C₄⋊Q₈	—	2	1	no	—
190	C₄⋊Q₈	—	2	1	no	—
191	C₄⋊Q₈	—	2	1	no	—
192	C₄⋊Q₈	—	2	1	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	2	1	no	—
195	C₄⋊Q₈	—	2	1	no	—
196	C₄⋊Q₈	C₅×D₄	1	2	yes	—
197	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
198	#127	C₅×D₄	1	1	yes	characteristic
199	#124	—	2	1	no	—
200	#121	—	2	1	no	—
201	#121	—	2	1	no	—
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
208	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
209	#106	—	2	8	no	—
210	#106	—	2	4	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	C₅×D₄	1	2	yes	—
214	#102	—	2	4	no	—
215	C₂²⋊C₈	—	2	1	no	—
216	#97	—	2	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	C₅×D₄	1	2	yes	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	8	1	no	—
224	C₂²×C₅	—	8	1	no	—
225	C₂²×C₅	—	8	1	no	—
226	C₂²×C₅	#392	1	1	yes	characteristic
227	C₂²×C₅	#460	1	2	yes	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	8	1	no	—
231	C₄×C₅	—	8	2	no	—
232	C₄×C₅	—	8	2	no	—
233	C₄×C₅	#450	1	2	yes	—
234	C₄×C₅	#456	1	2	yes	—
235	C₄×C₅	#471	1	2	yes	—
236	C₄×C₅	#492	1	1	yes	characteristic
237	C₄×C₅	#496	1	1	yes	characteristic
238	#41	—	4	2	no	—
239	#41	—	4	2	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	8	2	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
257	C₂²×C₄	C₅×#41	1	1	yes	characteristic
258	Q₁₆	—	4	8	no	—
259	SD₁₆	—	4	8	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	4	2	no	—
262	M₁₆	C₅×SD₁₆	1	4	yes	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₂²⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
281	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
282	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
283	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
284	C₂×C₅	—	2	2	no	—
285	C₂×C₅	—	16	1	no	—
286	C₂×C₅	#2005	1	1	yes	characteristic
287	C₂×C₅	#2014	1	1	yes	characteristic
288	C₂×C₅	#2019	1	1	yes	characteristic
289	C₂³	—	4	1	no	—
290	C₂³	C₅×#121	1	1	yes	characteristic
291	Q₈	—	4	4	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	8	2	no	—
294	Q₈	—	8	2	no	—
295	Q₈	—	8	1	no	—
296	Q₈	—	8	1	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	8	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	C₅×#120	1	1	yes	characteristic
311	C₂×C₄	C₅×#120	1	1	yes	characteristic
312	C₂×C₄	C₅×#120	1	1	yes	characteristic
313	C₂×C₄	C₅×#121	1	1	yes	characteristic
314	C₂×C₄	C₅×#121	1	1	yes	characteristic
315	C₂×C₄	C₅×#127	1	1	yes	characteristic
316	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
317	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
318	C₈	—	2	4	no	—
319	C₈	—	2	4	no	—
320	C₈	—	4	2	no	—
321	C₈	—	8	1	no	—
322	C₅	#10172	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	C₅×#392	1	1	yes	characteristic
329	C₂²	C₅×#460	1	2	yes	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	2	no	—
335	C₄	C₅×#450	1	2	yes	—
336	C₄	C₅×#456	1	2	yes	—
337	C₄	C₅×#471	1	2	yes	—
338	C₄	C₅×#492	1	1	yes	characteristic
339	C₄	C₅×#496	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	16	1	no	—
342	C₂	C₅×#2005	1	1	yes	characteristic
343	C₂	C₅×#2014	1	1	yes	characteristic
344	C₂	C₅×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×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. C₂×C₅×#498	C₂	semidirect
4. C₅×#1997	C₂	semidirect
5. C₅×#1546	C₂	semidirect
7. C₅×#1278	C₂	semidirect
29. #10172	C₅	direct (N × H)
73. C₂×#498	C₂×C₅	semidirect
75. #1997	C₂×C₅	semidirect
76. #1546	C₂×C₅	semidirect
78. #1278	C₂×C₅	semidirect
322. C₅	#10172	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×#3227	2
C₂×C₅×#498	2
C₅×#3152	2
C₅×#1997	2
C₅×#1546	2
C₅×#1492	2
C₅×#1278	2
#10172	5

Subgroups — C5×#10172 #11932892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10172 #11932892