Subgroups — C₅×#12869 #11935589

Order 1280.

404 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#1810	C₂	1	1	yes	maximal, characteristic
4	C₅×#1382	C₂	1	4	yes	maximal, complemented
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₅×#117	—	2	1	no	—
7	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
8	C₂×C₅×#108	C₂²	1	2	yes	—
9	C₅×#427	—	2	1	no	—
10	C₅×#427	—	2	1	no	—
11	C₅×#427	C₄	1	2	yes	—
12	C₅×#427	C₂²	1	2	yes	—
13	C₂×C₅×#107	C₄	1	1	yes	characteristic
14	C₂×C₅×#107	C₄	1	1	yes	characteristic
15	C₂×C₅×#106	—	2	1	no	—
16	C₅×#407	—	2	1	no	—
17	C₅×C₂²⋊C₁₆	—	2	1	no	—
18	C₅×C₂²⋊C₁₆	—	2	1	no	—
19	C₅×#341	—	2	1	no	—
20	#12869	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
22	C₂×C₅×M₁₆	—	2	2	no	—
23	C₂²×C₅×C₈	—	2	4	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₅×#117	—	2	2	no	—
31	C₅×#117	—	2	2	no	—
32	C₅×#117	—	4	1	no	—
33	C₅×#117	—	4	1	no	—
34	C₂×C₅×C₄⋊C₄	—	2	1	no	—
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
37	C₂×C₄²×C₅	—	4	1	no	—
38	C₂×C₅×C₁₆	—	2	2	no	—
39	C₂×C₅×C₁₆	—	2	4	no	—
40	C₅×#108	—	2	4	no	—
41	C₅×#107	—	2	1	no	—
42	C₅×#107	—	2	1	no	—
43	C₅×#107	C₂×C₄	1	2	yes	—
44	C₅×#107	C₂×C₄	1	2	yes	—
45	C₅×#106	—	2	2	no	—
46	C₅×#106	—	2	2	no	—
47	C₅×C₂²⋊C₈	—	4	1	no	—
48	C₅×C₂²⋊C₈	—	4	1	no	—
49	C₂×#427	C₂×C₅	1	1	yes	characteristic
50	C₂×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic
51	#1810	C₂×C₅	1	1	yes	characteristic
52	#1382	C₂×C₅	1	4	yes	complemented
53	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅	—	2	2	no	—
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	—	4	1	no	—
57	C₂²×C₄×C₅	—	4	1	no	—
58	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₅×M₁₆	—	4	2	no	—
65	C₅×M₁₆	—	4	2	no	—
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	—	2	2	no	—
70	C₂×C₅×C₈	—	2	2	no	—
71	C₂×C₅×C₈	—	4	2	no	—
72	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₅×C₈	#41	1	4	yes	—
77	C₅×C₄⋊C₄	—	2	2	no	—
78	C₅×C₄⋊C₄	—	2	2	no	—
79	C₅×C₄⋊C₄	—	4	1	no	—
80	C₅×C₄⋊C₄	—	4	1	no	—
81	C₅×C₂²⋊C₄	—	4	2	no	—
82	C₅×C₂²⋊C₄	—	4	2	no	—
83	C₄²×C₅	—	4	2	no	—
84	C₄²×C₅	—	4	2	no	—
85	C₅×C₁₆	—	4	2	no	—
86	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
87	C₂×#117	—	2	1	no	—
88	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
89	C₂×#108	C₂²×C₅	1	2	yes	—
90	#427	—	2	1	no	—
91	#427	—	2	1	no	—
92	#427	C₄×C₅	1	2	yes	—
93	#427	C₂²×C₅	1	2	yes	—
94	C₂×#107	C₄×C₅	1	1	yes	characteristic
95	C₂×#107	C₄×C₅	1	1	yes	characteristic
96	C₂×#106	—	2	1	no	—
97	#407	—	2	1	no	—
98	C₂²⋊C₁₆	—	2	1	no	—
99	C₂²⋊C₁₆	—	2	1	no	—
100	#341	—	2	1	no	—
101	C₂³×C₅	—	2	2	no	—
102	C₂³×C₅	—	2	2	no	—
103	C₂³×C₅	—	4	1	no	—
104	C₂³×C₅	#102	1	1	yes	characteristic
105	C₂³×C₅	#102	1	1	yes	characteristic
106	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	—	8	2	no	—
117	C₂×C₄×C₅	—	8	2	no	—
118	C₂×C₄×C₅	#102	1	1	yes	characteristic
119	C₂×C₄×C₅	#102	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	#123	1	2	yes	—
122	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅	#137	1	2	yes	—
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	—	2	1	no	—
126	C₅×C₈	—	8	2	no	—
127	C₅×C₈	#117	1	2	yes	—
128	C₅×C₈	#123	1	2	yes	—
129	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
130	C₂×M₁₆	—	2	2	no	—
131	C₂²×C₈	—	2	4	no	—
132	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
133	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
134	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
135	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
136	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
137	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
138	#117	—	2	2	no	—
139	#117	—	2	2	no	—
140	#117	—	4	1	no	—
141	#117	—	4	1	no	—
142	C₂×C₄⋊C₄	—	2	1	no	—
143	C₂×C₄⋊C₄	—	2	1	no	—
144	C₂×C₂²⋊C₄	—	4	1	no	—
145	C₂×C₄²	—	4	1	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	2	4	no	—
148	#108	—	2	4	no	—
149	#107	—	2	1	no	—
150	#107	—	2	1	no	—
151	#107	C₂×C₄×C₅	1	2	yes	—
152	#107	C₂×C₄×C₅	1	2	yes	—
153	#106	—	2	2	no	—
154	#106	—	2	2	no	—
155	C₂²⋊C₈	—	4	1	no	—
156	C₂²⋊C₈	—	4	1	no	—
157	C₂²×C₅	—	2	1	no	—
158	C₂²×C₅	—	2	1	no	—
159	C₂²×C₅	—	2	1	no	—
160	C₂²×C₅	—	2	1	no	—
161	C₂²×C₅	—	2	1	no	—
162	C₂²×C₅	—	2	1	no	—
163	C₂²×C₅	—	2	1	no	—
164	C₂²×C₅	—	4	4	no	—
165	C₂²×C₅	#359	1	2	yes	—
166	C₂²×C₅	C₂×#102	1	1	yes	characteristic
167	C₂²×C₅	#483	1	2	yes	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	—	8	2	no	—
171	C₄×C₅	—	8	2	no	—
172	C₄×C₅	#381	1	1	yes	characteristic
173	C₄×C₅	#419	1	1	yes	characteristic
174	C₄×C₅	#480	1	2	yes	—
175	C₂⁴	C₅×SD₁₆	1	1	yes	characteristic
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	4	1	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
183	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
184	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
186	M₁₆	—	4	2	no	—
187	M₁₆	—	4	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
197	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂×C₈	C₅×#41	1	4	yes	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₄⋊C₄	—	4	1	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₂²⋊C₄	—	4	2	no	—
205	C₄²	—	4	2	no	—
206	C₄²	—	4	2	no	—
207	C₁₆	—	4	2	no	—
208	C₂×C₅	—	2	2	no	—
209	C₂×C₅	—	2	2	no	—
210	C₂×C₅	—	4	1	no	—
211	C₂×C₅	#1823	1	1	yes	characteristic
212	C₂×C₅	C₂×#359	1	1	yes	characteristic
213	C₂×C₅	#2143	1	1	yes	characteristic
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	4	1	no	—
217	C₂³	C₅×#102	1	1	yes	characteristic
218	C₂³	C₅×#102	1	1	yes	characteristic
219	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₅×#102	1	1	yes	characteristic
232	C₂×C₄	C₅×#102	1	1	yes	characteristic
233	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₄	C₅×#123	1	2	yes	—
235	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
236	C₂×C₄	C₅×#137	1	2	yes	—
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	8	2	no	—
240	C₈	C₅×#117	1	2	yes	—
241	C₈	C₅×#123	1	2	yes	—
242	C₅	#12869	1	1	yes	characteristic, direct product factor, Sylow
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	C₅×#359	1	2	yes	—
252	C₂²	C₂×C₅×#102	1	1	yes	characteristic
253	C₂²	C₅×#483	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₄	C₅×#381	1	1	yes	characteristic
259	C₄	C₅×#419	1	1	yes	characteristic
260	C₄	C₅×#480	1	2	yes	—
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	—	4	1	no	—
264	C₂	C₅×#1823	1	1	yes	characteristic
265	C₂	C₂×C₅×#359	1	1	yes	characteristic
266	C₂	C₅×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#102
prime radical	C₂⁴×C₅	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1382	C₂	semidirect
20. #12869	C₅	direct (N × H)
52. #1382	C₂×C₅	semidirect
242. C₅	#12869	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×#427	2
C₂×C₅×C₂²⋊C₁₆	2
C₅×#1810	2
C₅×#1382	2
#12869	5

Subgroups — C5×#12869 #11935589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12869 #11935589