Subgroups — C₅×#13231 #11935951

Order 1280.

468 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3058	C₂	1	1	yes	maximal, characteristic
2	C₅×#2128	C₂	1	2	yes	maximal, complemented
3	C₅×#2122	C₂	1	2	yes	maximal, complemented
4	C₅×#1812	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1757	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×D₈	C₄	1	1	yes	characteristic, complemented
7	C₂×C₅×#131	C₂²	1	1	yes	characteristic
8	C₅×#535	C₂²	1	2	yes	—
9	C₅×#469	—	2	1	no	—
10	C₅×#469	—	2	1	no	—
11	C₅×#466	—	2	1	no	—
12	C₅×#466	—	2	1	no	—
13	C₅×#427	—	2	1	no	—
14	C₅×#427	C₄	1	1	yes	characteristic, complemented
15	C₅×#418	C₄	1	2	yes	complemented
16	C₅×#413	C₂²	1	1	yes	characteristic
17	C₅×#410	C₂²	1	2	yes	—
18	C₅×#410	C₂²	1	1	yes	characteristic
19	C₅×#405	—	2	1	no	—
20	C₅×#353	—	2	4	no	—
21	C₅×AΣL(1,𝔽₁₆)	—	2	4	no	—
22	C₅×#344	—	2	1	no	—
23	C₅×H(3,ℤ/4)	—	2	1	no	—
24	#13231	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₅×#41	C₂³	1	1	yes	characteristic
26	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
27	C₂×C₅×D₈	—	2	1	no	—
28	C₂×C₅×D₈	—	2	1	no	—
29	C₂×C₅×D₈	—	4	1	no	—
30	C₂×C₅×D₈	—	4	1	no	—
31	C₅×#131	—	4	1	no	—
32	C₅×#131	—	4	1	no	—
33	C₂×C₅×M₁₆	—	2	1	no	—
34	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
36	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₅×#127	—	2	2	no	—
41	C₅×#124	—	2	2	no	—
42	C₅×#121	—	2	2	no	—
43	C₅×#121	—	2	2	no	—
44	C₅×#120	—	2	2	no	—
45	C₅×#120	—	2	2	no	—
46	C₅×#117	—	2	1	no	—
47	C₅×#117	—	2	2	no	—
48	C₅×#117	C₂×C₄	1	2	yes	—
49	C₅×#107	—	2	1	no	—
50	C₅×#107	—	4	1	no	—
51	C₅×#106	—	2	1	no	—
52	C₅×#106	—	4	1	no	—
53	C₅×#102	—	2	2	no	—
54	C₅×#102	—	2	2	no	—
55	C₅×#101	—	2	1	no	—
56	C₅×#100	D₄	1	2	yes	—
57	C₅×#99	—	2	1	no	—
58	C₅×#99	—	2	2	no	—
59	C₅×#99	—	2	2	no	—
60	C₅×#99	D₄	1	2	yes	—
61	C₅×#97	—	4	1	no	—
62	C₄×C₅×C₈	—	4	1	no	—
63	#3058	C₂×C₅	1	1	yes	characteristic
64	#2128	C₂×C₅	1	2	yes	complemented
65	#2122	C₂×C₅	1	2	yes	complemented
66	#1812	C₂×C₅	1	1	yes	characteristic, complemented
67	#1757	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂⁴×C₅	—	2	2	no	—
69	C₅×#41	—	4	1	no	—
70	C₅×#41	—	4	1	no	—
71	C₂×C₅×Q₈	#41	1	1	yes	characteristic
72	C₂×C₅×D₄	—	2	2	no	—
73	C₂×C₅×D₄	—	2	2	no	—
74	C₂×C₅×D₄	—	4	2	no	—
75	C₂×C₅×D₄	—	4	2	no	—
76	C₂×C₅×D₄	—	4	2	no	—
77	C₂×C₅×D₄	—	4	2	no	—
78	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₅×D₄	#41	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₅×D₈	—	4	2	no	—
85	C₅×D₈	—	4	1	no	—
86	C₅×D₈	—	4	1	no	—
87	C₅×M₁₆	—	2	2	no	—
88	C₅×M₁₆	—	4	1	no	—
89	C₅×M₁₆	—	4	1	no	—
90	C₅×M₁₆	—	4	1	no	—
91	C₅×M₁₆	—	4	1	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	—	2	1	no	—
94	C₂×C₅×C₈	—	2	1	no	—
95	C₂×C₅×C₈	—	2	1	no	—
96	C₂×C₅×C₈	—	2	1	no	—
97	C₂×C₅×C₈	—	2	1	no	—
98	C₂×C₅×C₈	—	4	1	no	—
99	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	—	4	1	no	—
105	C₅×C₄⋊C₄	—	4	2	no	—
106	C₅×C₂²⋊C₄	—	2	2	no	—
107	C₅×C₂²⋊C₄	—	2	2	no	—
108	C₅×C₂²⋊C₄	—	4	2	no	—
109	C₅×C₂²⋊C₄	—	4	2	no	—
110	C₅×C₂²⋊C₄	—	4	2	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	—	4	1	no	—
113	C₄²×C₅	—	4	2	no	—
114	C₂²×D₈	C₄×C₅	1	1	yes	characteristic, complemented
115	C₂×#131	C₂²×C₅	1	1	yes	characteristic
116	#535	C₂²×C₅	1	2	yes	—
117	#469	—	2	1	no	—
118	#469	—	2	1	no	—
119	#466	—	2	1	no	—
120	#466	—	2	1	no	—
121	#427	—	2	1	no	—
122	#427	C₄×C₅	1	1	yes	characteristic, complemented
123	#418	C₄×C₅	1	2	yes	complemented
124	#413	C₂²×C₅	1	1	yes	characteristic
125	#410	C₂²×C₅	1	2	yes	—
126	#410	C₂²×C₅	1	1	yes	characteristic
127	#405	—	2	1	no	—
128	#353	—	2	4	no	—
129	AΣL(1,𝔽₁₆)	—	2	4	no	—
130	#344	—	2	1	no	—
131	H(3,ℤ/4)	—	2	1	no	—
132	C₂³×C₅	—	4	2	no	—
133	C₂³×C₅	—	8	2	no	—
134	C₂³×C₅	—	16	2	no	—
135	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
136	C₂³×C₅	#121	1	1	yes	characteristic
137	C₂³×C₅	#127	1	1	yes	characteristic
138	C₅×Q₈	—	4	1	no	—
139	C₅×D₄	—	4	1	no	—
140	C₅×D₄	—	4	1	no	—
141	C₅×D₄	—	4	1	no	—
142	C₅×D₄	—	4	2	no	—
143	C₅×D₄	—	4	2	no	—
144	C₅×D₄	—	8	2	no	—
145	C₅×D₄	—	16	2	no	—
146	C₅×D₄	—	16	2	no	—
147	C₂×C₄×C₅	—	2	1	no	—
148	C₂×C₄×C₅	—	2	1	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	—	8	2	no	—
156	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅	#121	1	1	yes	characteristic
159	C₂×C₄×C₅	#124	1	1	yes	characteristic
160	C₅×C₈	—	2	1	no	—
161	C₅×C₈	—	2	1	no	—
162	C₅×C₈	—	2	1	no	—
163	C₅×C₈	—	4	2	no	—
164	C₅×C₈	#99	1	1	yes	characteristic
165	C₅×C₈	#99	1	1	yes	characteristic
166	C₂×#41	C₂³×C₅	1	1	yes	characteristic
167	C₂²×D₄	C₂×C₄×C₅	1	2	yes	—
168	C₂×D₈	—	2	1	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	4	1	no	—
171	C₂×D₈	—	4	1	no	—
172	#131	—	4	1	no	—
173	#131	—	4	1	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
176	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
177	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
178	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
179	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
180	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
181	#127	—	2	2	no	—
182	#124	—	2	2	no	—
183	#121	—	2	2	no	—
184	#121	—	2	2	no	—
185	#120	—	2	2	no	—
186	#120	—	2	2	no	—
187	#117	—	2	1	no	—
188	#117	—	2	2	no	—
189	#117	C₂×C₄×C₅	1	2	yes	—
190	#107	—	2	1	no	—
191	#107	—	4	1	no	—
192	#106	—	2	1	no	—
193	#106	—	4	1	no	—
194	#102	—	2	2	no	—
195	#102	—	2	2	no	—
196	#101	—	2	1	no	—
197	#100	C₅×D₄	1	2	yes	—
198	#99	—	2	1	no	—
199	#99	—	2	2	no	—
200	#99	—	2	2	no	—
201	#99	C₅×D₄	1	2	yes	—
202	#97	—	4	1	no	—
203	C₄×C₈	—	4	1	no	—
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	—	2	1	no	—
206	C₂²×C₅	—	2	1	no	—
207	C₂²×C₅	—	4	1	no	—
208	C₂²×C₅	—	4	1	no	—
209	C₂²×C₅	—	4	1	no	—
210	C₂²×C₅	—	8	2	no	—
211	C₂²×C₅	—	8	2	no	—
212	C₂²×C₅	—	8	2	no	—
213	C₂²×C₅	—	16	4	no	—
214	C₂²×C₅	#390	1	1	yes	characteristic
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	8	2	no	—
219	C₄×C₅	—	8	2	no	—
220	C₄×C₅	—	16	2	no	—
221	C₄×C₅	C₂×#99	1	1	yes	characteristic
222	C₄×C₅	#410	1	1	yes	characteristic
223	C₂⁴	—	2	2	no	—
224	#41	—	4	1	no	—
225	#41	—	4	1	no	—
226	C₂×Q₈	C₅×#41	1	1	yes	characteristic
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
234	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
235	C₂×D₄	C₅×#41	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
239	D₈	—	4	2	no	—
240	D₈	—	4	1	no	—
241	D₈	—	4	1	no	—
242	M₁₆	—	2	2	no	—
243	M₁₆	—	4	1	no	—
244	M₁₆	—	4	1	no	—
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₂²⋊C₄	—	2	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	4	2	no	—
269	C₂×C₅	—	2	1	no	—
270	C₂×C₅	—	4	1	no	—
271	C₂×C₅	—	4	1	no	—
272	C₂×C₅	—	4	1	no	—
273	C₂×C₅	—	16	2	no	—
274	C₂×C₅	#1964	1	1	yes	characteristic
275	C₂³	—	4	2	no	—
276	C₂³	—	8	2	no	—
277	C₂³	—	16	2	no	—
278	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
279	C₂³	C₅×#121	1	1	yes	characteristic
280	C₂³	C₅×#127	1	1	yes	characteristic
281	Q₈	—	4	1	no	—
282	D₄	—	4	1	no	—
283	D₄	—	4	1	no	—
284	D₄	—	4	1	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	8	2	no	—
288	D₄	—	16	2	no	—
289	D₄	—	16	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
300	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
301	C₂×C₄	C₅×#121	1	1	yes	characteristic
302	C₂×C₄	C₅×#124	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	4	2	no	—
307	C₈	C₅×#99	1	1	yes	characteristic
308	C₈	C₅×#99	1	1	yes	characteristic
309	C₅	#13231	1	1	yes	characteristic, direct product factor, Sylow
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	16	4	no	—
320	C₂²	C₅×#390	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	16	2	no	—
327	C₄	C₂×C₅×#99	1	1	yes	characteristic
328	C₄	C₅×#410	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	4	1	no	—
331	C₂	—	4	1	no	—
332	C₂	—	4	1	no	—
333	C₂	—	16	2	no	—
334	C₂	C₅×#1964	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	#1964
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×#41	C₂³×C₅
prime radical	C₅×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2128	C₂	semidirect
3. C₅×#2122	C₂	semidirect
4. C₅×#1812	C₂	semidirect
5. C₅×#1757	C₂	semidirect
6. C₂²×C₅×D₈	C₄	semidirect
14. C₅×#427	C₄	semidirect
15. C₅×#418	C₄	semidirect
24. #13231	C₅	direct (N × H)
64. #2128	C₂×C₅	semidirect
65. #2122	C₂×C₅	semidirect
66. #1812	C₂×C₅	semidirect
67. #1757	C₂×C₅	semidirect
114. C₂²×D₈	C₄×C₅	semidirect
122. #427	C₄×C₅	semidirect
123. #418	C₄×C₅	semidirect
309. C₅	#13231	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×#3058	2
C₅×#2128	2
C₅×#2122	2
C₅×#1812	2
C₅×#1757	2
#13231	5

Subgroups — C5×#13231 #11935951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13231 #11935951