Subgroups — C₅×#12934 #11935654

Order 1280.

394 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2912	C₂	1	1	yes	maximal, characteristic
2	C₅×#2155	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#365	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2117	C₂	1	1	yes	maximal, characteristic
5	C₅×#1814	C₂	1	1	yes	maximal, characteristic
6	C₅×#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#131	C₂²	1	1	yes	characteristic
9	C₅×#520	—	2	1	no	—
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₅×#108	C₂²	1	1	yes	characteristic
13	C₅×#428	—	2	2	no	—
14	C₅×#428	—	2	1	no	—
15	C₅×#427	—	2	1	no	—
16	C₅×#427	C₂²	1	1	yes	characteristic
17	C₅×#427	C₂²	1	1	yes	characteristic
18	C₂×C₅×#106	—	2	1	no	—
19	C₂×C₅×#106	C₂²	1	1	yes	characteristic
20	C₅×#421	—	2	1	no	—
21	C₅×#365	—	2	2	no	—
22	C₅×#365	—	2	1	no	—
23	C₅×#365	—	2	1	no	—
24	C₅×#350	C₄	1	4	yes	complemented
25	C₅×#344	—	2	1	no	—
26	#12934	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	D₄	1	1	yes	characteristic
28	C₅×#131	—	2	1	no	—
29	C₅×#131	—	2	1	no	—
30	C₅×#131	C₂×C₄	1	2	yes	—
31	C₅×#131	Q₈	1	2	yes	—
32	C₂×C₅×M₁₆	—	2	1	no	—
33	C₂×C₅×M₁₆	—	2	1	no	—
34	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	—	2	1	no	—
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₄×C₅×Q₈	—	4	1	no	—
38	C₄×C₅×D₄	—	4	2	no	—
39	C₄×C₅×D₄	—	4	1	no	—
40	C₅×#117	—	2	2	no	—
41	C₅×#117	—	2	1	no	—
42	C₅×#117	—	2	1	no	—
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	—	2	1	no	—
45	C₅×M₃₂	—	2	1	no	—
46	C₅×M₃₂	—	2	1	no	—
47	C₅×M₃₂	—	2	1	no	—
48	C₅×M₃₂	C₂×C₄	1	2	yes	complemented
49	C₂×C₅×C₁₆	—	2	1	no	—
50	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
51	C₅×#108	—	2	1	no	—
52	C₅×#108	—	2	1	no	—
53	C₅×#107	—	2	2	no	—
54	C₅×#107	—	2	1	no	—
55	C₅×#107	—	2	1	no	—
56	C₅×#106	—	2	2	no	—
57	C₅×#106	—	2	1	no	—
58	C₅×#106	—	2	1	no	—
59	C₅×#106	D₄	1	1	yes	characteristic
60	C₅×#106	D₄	1	2	yes	—
61	C₅×#106	D₄	1	1	yes	characteristic
62	C₅×#104	—	4	2	no	—
63	#2912	C₂×C₅	1	1	yes	characteristic
64	#2155	C₂×C₅	1	1	yes	characteristic, complemented
65	C₂×#365	C₂×C₅	1	1	yes	characteristic, complemented
66	#2117	C₂×C₅	1	1	yes	characteristic
67	#1814	C₂×C₅	1	1	yes	characteristic
68	#1386	C₂×C₅	1	1	yes	characteristic, complemented
69	#1384	C₂×C₅	1	1	yes	characteristic, complemented
70	C₅×#41	—	2	1	no	—
71	C₅×#41	—	2	1	no	—
72	C₅×#41	C₄⋊C₄	1	2	yes	—
73	C₅×#41	SD₁₆	1	2	yes	—
74	C₂×C₅×Q₈	SD₁₆	1	1	yes	characteristic
75	C₂×C₅×D₄	—	2	1	no	—
76	C₂×C₅×D₄	SD₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₅	—	2	1	no	—
78	C₂²×C₄×C₅	—	4	2	no	—
79	C₂²×C₄×C₅	—	4	1	no	—
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₅×M₁₆	—	2	1	no	—
82	C₅×M₁₆	—	2	2	no	—
83	C₅×M₁₆	—	2	2	no	—
84	C₅×M₁₆	—	4	1	no	—
85	C₅×M₁₆	—	4	1	no	—
86	C₅×M₁₆	C₄⋊C₄	1	2	yes	—
87	C₂×C₅×C₈	—	2	1	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	4	1	no	—
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₅×C₈	#41	1	1	yes	characteristic
96	C₂×C₅×C₈	#41	1	1	yes	characteristic
97	C₅×C₄⋊C₄	—	2	1	no	—
98	C₅×C₄⋊C₄	—	2	2	no	—
99	C₅×C₄⋊C₄	—	2	1	no	—
100	C₅×C₄⋊C₄	—	4	2	no	—
101	C₅×C₄⋊C₄	—	4	1	no	—
102	C₅×C₄⋊C₄	—	4	1	no	—
103	C₅×C₂²⋊C₄	—	4	2	no	—
104	C₅×C₂²⋊C₄	—	4	1	no	—
105	C₅×C₂²⋊C₄	—	4	1	no	—
106	C₄²×C₅	—	4	2	no	—
107	C₄²×C₅	—	4	1	no	—
108	C₄²×C₅	—	4	1	no	—
109	C₅×C₁₆	—	2	2	no	—
110	C₅×C₁₆	—	2	2	no	—
111	C₂×#131	C₂²×C₅	1	1	yes	characteristic
112	#520	—	2	1	no	—
113	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
114	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
115	C₂×#108	C₂²×C₅	1	1	yes	characteristic
116	#428	—	2	1	no	—
117	#428	—	2	2	no	—
118	#427	—	2	1	no	—
119	#427	C₂²×C₅	1	1	yes	characteristic
120	#427	C₂²×C₅	1	1	yes	characteristic
121	C₂×#106	—	2	1	no	—
122	C₂×#106	C₂²×C₅	1	1	yes	characteristic
123	#421	—	2	1	no	—
124	#365	—	2	2	no	—
125	#365	—	2	1	no	—
126	#365	—	2	1	no	—
127	#350	C₄×C₅	1	4	yes	complemented
128	#344	—	2	1	no	—
129	C₂³×C₅	—	2	1	no	—
130	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₅×Q₈	—	2	1	no	—
132	C₅×Q₈	#106	1	2	yes	—
133	C₅×D₄	—	2	2	no	—
134	C₅×D₄	—	2	2	no	—
135	C₅×D₄	—	2	1	no	—
136	C₅×D₄	#106	1	2	yes	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	4	1	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	—	4	1	no	—
143	C₂×C₄×C₅	—	8	1	no	—
144	C₂×C₄×C₅	—	16	1	no	—
145	C₂×C₄×C₅	—	16	2	no	—
146	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	#120	1	1	yes	characteristic
148	C₂×C₄×C₅	#123	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
151	C₂×C₄×C₅	#137	1	1	yes	characteristic
152	C₅×C₈	—	2	2	no	—
153	C₅×C₈	—	8	1	no	—
154	C₅×C₈	C₄×D₄	1	1	yes	characteristic
155	C₅×C₈	C₄×D₄	1	1	yes	characteristic
156	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
157	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×#41	C₅×D₄	1	1	yes	characteristic
159	#131	—	2	1	no	—
160	#131	—	2	1	no	—
161	#131	C₂×C₄×C₅	1	2	yes	—
162	#131	C₅×Q₈	1	2	yes	—
163	C₂×M₁₆	—	2	1	no	—
164	C₂×M₁₆	—	2	1	no	—
165	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
168	C₄×Q₈	—	4	1	no	—
169	C₄×D₄	—	4	2	no	—
170	C₄×D₄	—	4	1	no	—
171	#117	—	2	2	no	—
172	#117	—	2	1	no	—
173	#117	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	M₃₂	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	C₂×C₄×C₅	1	2	yes	complemented
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	complemented
182	#108	—	2	1	no	—
183	#108	—	2	1	no	—
184	#107	—	2	2	no	—
185	#107	—	2	1	no	—
186	#107	—	2	1	no	—
187	#106	—	2	2	no	—
188	#106	—	2	1	no	—
189	#106	—	2	1	no	—
190	#106	C₅×D₄	1	1	yes	characteristic
191	#106	C₅×D₄	1	2	yes	—
192	#106	C₅×D₄	1	1	yes	characteristic
193	#104	—	4	2	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	4	1	no	—
196	C₂²×C₅	—	4	2	no	—
197	C₂²×C₅	#428	1	1	yes	characteristic
198	C₂²×C₅	#450	1	1	yes	characteristic
199	C₂²×C₅	#481	1	1	yes	characteristic
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	8	1	no	—
202	C₄×C₅	—	8	2	no	—
203	C₄×C₅	—	8	1	no	—
204	C₄×C₅	—	16	1	no	—
205	C₄×C₅	#385	1	1	yes	characteristic
206	C₄×C₅	C₂×#106	1	1	yes	characteristic
207	C₄×C₅	#448	1	1	yes	characteristic
208	C₄×C₅	#483	1	1	yes	characteristic
209	#41	—	2	1	no	—
210	#41	—	2	1	no	—
211	#41	C₅×C₄⋊C₄	1	2	yes	—
212	#41	C₅×SD₁₆	1	2	yes	—
213	C₂×Q₈	C₅×SD₁₆	1	1	yes	characteristic
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	C₅×SD₁₆	1	1	yes	characteristic
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	M₁₆	—	2	1	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	C₅×C₄⋊C₄	1	2	yes	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
231	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
233	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
234	C₂×C₈	C₅×#41	1	1	yes	characteristic
235	C₂×C₈	C₅×#41	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	4	1	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₄²	—	4	2	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	1	no	—
248	C₁₆	—	2	2	no	—
249	C₁₆	—	2	2	no	—
250	C₂×C₅	—	2	1	no	—
251	C₂×C₅	—	2	1	no	—
252	C₂×C₅	—	2	1	no	—
253	C₂×C₅	—	4	2	no	—
254	C₂×C₅	#1853	1	1	yes	characteristic
255	C₂³	—	2	1	no	—
256	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
257	Q₈	—	2	1	no	—
258	Q₈	C₅×#106	1	2	yes	—
259	D₄	—	2	2	no	—
260	D₄	—	2	2	no	—
261	D₄	—	2	1	no	—
262	D₄	C₅×#106	1	2	yes	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	16	1	no	—
271	C₂×C₄	—	16	2	no	—
272	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₅×#120	1	1	yes	characteristic
274	C₂×C₄	C₅×#123	1	1	yes	characteristic
275	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
276	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
277	C₂×C₄	C₅×#137	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	8	1	no	—
280	C₈	C₄×C₅×D₄	1	1	yes	characteristic
281	C₈	C₄×C₅×D₄	1	1	yes	characteristic
282	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
283	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
284	C₅	#12934	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²	—	2	2	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	2	no	—
288	C₂²	C₅×#428	1	1	yes	characteristic
289	C₂²	C₅×#450	1	1	yes	characteristic
290	C₂²	C₅×#481	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	1	no	—
294	C₄	—	8	1	no	—
295	C₄	—	16	1	no	—
296	C₄	C₅×#385	1	1	yes	characteristic
297	C₄	C₂×C₅×#106	1	1	yes	characteristic
298	C₄	C₅×#448	1	1	yes	characteristic
299	C₄	C₅×#483	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	2	no	—
304	C₂	C₅×#1853	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	#1853
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2155	C₂	semidirect
3. C₂×C₅×#365	C₂	semidirect
6. C₅×#1386	C₂	semidirect
7. C₅×#1384	C₂	semidirect
24. C₅×#350	C₄	semidirect
26. #12934	C₅	direct (N × H)
48. C₅×M₃₂	C₂×C₄	semidirect
50. C₂×C₅×C₁₆	C₂×C₄	semidirect
64. #2155	C₂×C₅	semidirect
65. C₂×#365	C₂×C₅	semidirect
68. #1386	C₂×C₅	semidirect
69. #1384	C₂×C₅	semidirect
127. #350	C₄×C₅	semidirect
179. M₃₂	C₂×C₄×C₅	semidirect
181. C₂×C₁₆	C₂×C₄×C₅	semidirect
284. C₅	#12934	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×#2912	2
C₅×#2155	2
C₂×C₅×#365	2
C₅×#2117	2
C₅×#1814	2
C₅×#1386	2
C₅×#1384	2
#12934	5

Subgroups — C5×#12934 #11935654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12934 #11935654