Subgroups — C₅×#13254 #11935974

Order 1280.

514 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#358	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#357	C₂	1	2	yes	maximal
5	C₄×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅×SD₁₆	—	2	1	no	—
8	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#124	—	2	1	no	—
10	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
11	C₅×#495	—	2	2	no	—
12	C₅×#495	C₄	1	4	yes	complemented
13	C₂×C₅×#107	C₂²	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₅×C₄⋊C₄	—	2	1	no	—
16	C₅×C₄⋊C₁₆	—	2	2	no	—
17	C₅×#358	—	2	4	no	—
18	C₅×#357	—	2	4	no	—
19	C₅×#340	—	2	1	no	—
20	#13254	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂²×C₅×D₄	—	2	1	no	—
23	C₂×C₅×Q₁₆	—	2	2	no	—
24	C₂×C₅×Q₁₆	—	2	1	no	—
25	C₂×C₅×Q₁₆	—	2	2	no	—
26	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₅×SD₁₆	—	2	4	no	—
28	C₂×C₅×SD₁₆	—	4	4	no	—
29	C₂×C₅×D₈	—	2	2	no	—
30	C₂×C₅×D₈	—	2	1	no	—
31	C₂×C₅×D₈	—	2	2	no	—
32	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₅×#124	—	2	4	no	—
36	C₅×#124	—	4	2	no	—
37	C₂×C₅×C₄⋊C₄	—	2	2	no	—
38	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
39	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
40	C₂×C₄²×C₅	—	4	1	no	—
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₅×C₁₆	—	2	2	no	—
43	C₂×C₅×C₁₆	—	2	2	no	—
44	C₂×C₅×C₁₆	—	2	2	no	—
45	C₅×#107	—	2	2	no	—
46	C₅×#107	D₄	1	4	yes	—
47	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
48	C₄×C₅×C₈	D₄	1	2	yes	—
49	C₅×#95	—	4	1	no	—
50	C₂×#495	C₂×C₅	1	1	yes	characteristic
51	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×#358	C₂×C₅	1	2	yes	complemented
53	C₂×#357	C₂×C₅	1	2	yes	—
54	C₄×#107	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂⁴×C₅	—	4	1	no	—
56	C₂×C₅×Q₈	—	2	2	no	—
57	C₂×C₅×Q₈	—	4	1	no	—
58	C₂×C₅×Q₈	—	4	2	no	—
59	C₂×C₅×Q₈	—	4	2	no	—
60	C₂×C₅×D₄	—	2	2	no	—
61	C₂×C₅×D₄	—	4	2	no	—
62	C₂×C₅×D₄	—	4	1	no	—
63	C₂×C₅×D₄	—	4	2	no	—
64	C₂²×C₄×C₅	—	4	2	no	—
65	C₂²×C₄×C₅	—	4	1	no	—
66	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₅×Q₁₆	—	2	4	no	—
70	C₅×Q₁₆	—	4	1	no	—
71	C₅×Q₁₆	—	4	1	no	—
72	C₅×SD₁₆	—	4	8	no	—
73	C₅×D₈	—	2	4	no	—
74	C₅×D₈	—	4	1	no	—
75	C₅×D₈	—	4	1	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
80	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₅×C₈	#41	1	2	yes	—
82	C₅×C₄⋊C₄	—	2	4	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₅×C₂²⋊C₄	—	4	4	no	—
85	C₅×C₂²⋊C₄	—	4	4	no	—
86	C₄²×C₅	—	8	2	no	—
87	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₅	SD₁₆	1	2	yes	—
89	C₅×C₁₆	—	4	2	no	—
90	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
91	C₂²×SD₁₆	—	2	1	no	—
92	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
93	C₂×#124	—	2	1	no	—
94	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
95	#495	—	2	2	no	—
96	#495	C₄×C₅	1	4	yes	complemented
97	C₂×#107	C₂²×C₅	1	2	yes	—
98	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
99	C₄×C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₁₆	—	2	2	no	—
101	#358	—	2	4	no	—
102	#357	—	2	4	no	—
103	#340	—	2	1	no	—
104	C₂³×C₅	—	4	2	no	—
105	C₂³×C₅	—	8	2	no	—
106	C₂³×C₅	—	8	2	no	—
107	C₂³×C₅	—	8	1	no	—
108	C₂³×C₅	—	8	1	no	—
109	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
110	C₅×Q₈	—	4	2	no	—
111	C₅×Q₈	—	4	2	no	—
112	C₅×Q₈	—	8	2	no	—
113	C₅×D₄	—	4	2	no	—
114	C₅×D₄	—	4	2	no	—
115	C₅×D₄	—	8	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	—	4	4	no	—
121	C₂×C₄×C₅	—	8	2	no	—
122	C₂×C₄×C₅	—	8	2	no	—
123	C₂×C₄×C₅	—	8	2	no	—
124	C₂×C₄×C₅	#102	1	2	yes	—
125	C₂×C₄×C₅	#102	1	2	yes	—
126	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅	#121	1	2	yes	—
128	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
130	C₅×C₈	—	2	2	no	—
131	C₅×C₈	C₄×D₄	1	2	yes	—
132	C₅×C₈	#124	1	1	yes	characteristic
133	C₅×C₈	#127	1	1	yes	characteristic
134	C₂²×Q₈	—	2	1	no	—
135	C₂²×D₄	—	2	1	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	2	1	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
140	C₂×SD₁₆	—	2	4	no	—
141	C₂×SD₁₆	—	4	4	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×D₈	—	2	1	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
146	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
148	#124	—	2	4	no	—
149	#124	—	4	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₂²⋊C₄	—	4	1	no	—
152	C₂×C₂²⋊C₄	—	4	1	no	—
153	C₂×C₄²	—	4	1	no	—
154	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	#107	—	2	2	no	—
159	#107	C₅×D₄	1	4	yes	—
160	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
161	C₄×C₈	C₅×D₄	1	2	yes	—
162	#95	—	4	1	no	—
163	C₂²×C₅	—	8	2	no	—
164	C₂²×C₅	—	8	2	no	—
165	C₂²×C₅	—	8	2	no	—
166	C₂²×C₅	—	16	2	no	—
167	C₂²×C₅	—	16	2	no	—
168	C₂²×C₅	C₂×#102	1	1	yes	characteristic
169	C₂²×C₅	#459	1	2	yes	—
170	C₂²×C₅	#508	1	2	yes	—
171	C₂²×C₅	#509	1	1	yes	characteristic
172	C₂²×C₅	#510	1	1	yes	characteristic
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	2	2	no	—
175	C₄×C₅	—	8	4	no	—
176	C₄×C₅	—	8	2	no	—
177	C₄×C₅	#390	1	1	yes	characteristic
178	C₄×C₅	#418	1	1	yes	characteristic
179	C₄×C₅	#460	1	2	yes	—
180	C₂⁴	—	4	1	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	4	1	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	1	no	—
191	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
192	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	Q₁₆	—	2	4	no	—
195	Q₁₆	—	4	1	no	—
196	Q₁₆	—	4	1	no	—
197	SD₁₆	—	4	8	no	—
198	D₈	—	2	4	no	—
199	D₈	—	4	1	no	—
200	D₈	—	4	1	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
203	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
204	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
205	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂×C₈	C₅×#41	1	2	yes	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₂²⋊C₄	—	4	4	no	—
211	C₄²	—	8	2	no	—
212	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
213	C₄²	C₅×SD₁₆	1	2	yes	—
214	C₁₆	—	4	2	no	—
215	C₂×C₅	—	16	2	no	—
216	C₂×C₅	#1982	1	1	yes	characteristic
217	C₂×C₅	#2138	1	1	yes	characteristic
218	C₂×C₅	#2139	1	1	yes	characteristic
219	C₂×C₅	#2210	1	2	yes	—
220	C₂×C₅	#2211	1	2	yes	—
221	C₂³	—	4	2	no	—
222	C₂³	—	8	2	no	—
223	C₂³	—	8	2	no	—
224	C₂³	—	8	1	no	—
225	C₂³	—	8	1	no	—
226	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
227	Q₈	—	4	2	no	—
228	Q₈	—	4	2	no	—
229	Q₈	—	8	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	8	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	C₅×#102	1	2	yes	—
242	C₂×C₄	C₅×#102	1	2	yes	—
243	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄	C₅×#121	1	2	yes	—
245	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
246	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
247	C₈	—	2	2	no	—
248	C₈	C₄×C₅×D₄	1	2	yes	—
249	C₈	C₅×#124	1	1	yes	characteristic
250	C₈	C₅×#127	1	1	yes	characteristic
251	C₅	#13254	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	—	16	2	no	—
257	C₂²	C₂×C₅×#102	1	1	yes	characteristic
258	C₂²	C₅×#459	1	2	yes	—
259	C₂²	C₅×#508	1	2	yes	—
260	C₂²	C₅×#509	1	1	yes	characteristic
261	C₂²	C₅×#510	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	8	4	no	—
265	C₄	—	8	2	no	—
266	C₄	C₅×#390	1	1	yes	characteristic
267	C₄	C₅×#418	1	1	yes	characteristic
268	C₄	C₅×#460	1	2	yes	—
269	C₂	—	16	2	no	—
270	C₂	C₅×#1982	1	1	yes	characteristic
271	C₂	C₅×#2138	1	1	yes	characteristic
272	C₂	C₅×#2139	1	1	yes	characteristic
273	C₂	C₅×#2210	1	2	yes	—
274	C₂	C₅×#2211	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₅×#358	C₂	semidirect
5. C₄×C₅×#107	C₂	semidirect
12. C₅×#495	C₄	semidirect
20. #13254	C₅	direct (N × H)
51. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
52. C₂×#358	C₂×C₅	semidirect
54. C₄×#107	C₂×C₅	semidirect
96. #495	C₄×C₅	semidirect
251. C₅	#13254	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×#495	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#358	2
C₂×C₅×#357	2
C₄×C₅×#107	2
#13254	5

Subgroups — C5×#13254 #11935974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13254 #11935974