Subgroups — C₅×#11634 #11934354

Order 1280.

386 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#436	C₂	1	1	yes	maximal, characteristic
2	C₅×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1330	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	—	2	1	no	—
6	C₂×C₄×C₅×D₄	C₄	1	1	yes	characteristic
7	C₅×#436	—	2	1	no	—
8	C₅×#436	—	2	1	no	—
9	C₅×#436	C₄	1	2	yes	—
10	C₅×#436	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×C₄⋊C₁₆	—	2	1	no	—
15	C₅×C₄⋊C₁₆	C₂²	1	2	yes	complemented
16	C₅×C₂²⋊C₁₆	—	2	1	no	—
17	C₅×C₂²⋊C₁₆	—	2	1	no	—
18	C₅×#346	C₂²	1	2	yes	complemented
19	C₄×C₅×C₁₆	—	2	1	no	—
20	#11634	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×D₄	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₅	—	2	1	no	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	4	4	no	—
26	C₂²×C₅×C₈	—	2	1	no	—
27	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₄×C₅×D₄	—	2	1	no	—
30	C₄×C₅×D₄	—	2	1	no	—
31	C₄×C₅×D₄	C₈	1	2	yes	—
32	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
33	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
34	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₅×C₁₆	—	2	2	no	—
37	C₂×C₅×C₁₆	—	2	2	no	—
38	C₅×C₄⋊C₈	—	2	1	no	—
39	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₅×C₂²⋊C₈	—	2	2	no	—
41	C₅×C₂²⋊C₈	—	2	2	no	—
42	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×C₈	D₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₂×#436	C₂×C₅	1	1	yes	characteristic
47	#2163	C₂×C₅	1	1	yes	characteristic, complemented
48	#2162	C₂×C₅	1	1	yes	characteristic, complemented
49	#1330	C₂×C₅	1	4	yes	complemented
50	C₂⁴×C₅	—	2	1	no	—
51	C₂×C₅×D₄	—	2	1	no	—
52	C₂×C₅×D₄	—	4	2	no	—
53	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅	—	2	1	no	—
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	4	2	no	—
58	C₂²×C₄×C₅	—	4	2	no	—
59	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
62	C₅×M₁₆	—	4	4	no	—
63	C₅×M₁₆	—	4	4	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	4	1	no	—
67	C₂×C₅×C₈	—	4	2	no	—
68	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₅×C₄⋊C₄	—	2	1	no	—
73	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
74	C₅×C₂²⋊C₄	—	2	2	no	—
75	C₅×C₂²⋊C₄	—	2	2	no	—
76	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
78	C₄²×C₅	M₁₆	1	1	yes	characteristic
79	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₅×C₁₆	—	4	2	no	—
81	C₅×C₁₆	—	4	2	no	—
82	C₂²×M₁₆	—	2	1	no	—
83	C₂×C₄×D₄	C₄×C₅	1	1	yes	characteristic
84	#436	—	2	1	no	—
85	#436	—	2	1	no	—
86	#436	C₄×C₅	1	2	yes	—
87	#436	C₂²×C₅	1	2	yes	—
88	C₂×C₄⋊C₈	C₄×C₅	1	1	yes	characteristic
89	C₂×C₂²⋊C₈	—	2	1	no	—
90	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
91	C₄⋊C₁₆	—	2	1	no	—
92	C₄⋊C₁₆	C₂²×C₅	1	2	yes	complemented
93	C₂²⋊C₁₆	—	2	1	no	—
94	C₂²⋊C₁₆	—	2	1	no	—
95	#346	C₂²×C₅	1	2	yes	complemented
96	C₄×C₁₆	—	2	1	no	—
97	C₂³×C₅	—	2	2	no	—
98	C₂³×C₅	—	4	2	no	—
99	C₂³×C₅	—	4	2	no	—
100	C₂³×C₅	—	4	2	no	—
101	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
102	C₅×D₄	—	4	2	no	—
103	C₅×D₄	—	4	2	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	4	4	no	—
110	C₂×C₄×C₅	—	4	2	no	—
111	C₂×C₄×C₅	—	4	2	no	—
112	C₂×C₄×C₅	—	4	2	no	—
113	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	—	2	1	no	—
121	C₅×C₈	—	2	1	no	—
122	C₅×C₈	—	4	2	no	—
123	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
124	C₂²×D₄	C₅×C₈	1	1	yes	characteristic
125	C₂³×C₄	—	2	1	no	—
126	C₂×M₁₆	—	2	2	no	—
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	—	4	4	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
131	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
132	C₄×D₄	—	2	1	no	—
133	C₄×D₄	—	2	1	no	—
134	C₄×D₄	C₅×C₈	1	2	yes	—
135	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
136	C₂×C₄⋊C₄	C₅×C₈	1	1	yes	characteristic
137	C₂×C₂²⋊C₄	—	2	1	no	—
138	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	—	2	2	no	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
143	C₂²⋊C₈	—	2	2	no	—
144	C₂²⋊C₈	—	2	2	no	—
145	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
146	C₄×C₈	C₅×D₄	1	1	yes	characteristic
147	C₄×C₈	C₅×D₄	1	1	yes	characteristic
148	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
149	C₂²×C₅	—	2	1	no	—
150	C₂²×C₅	—	2	1	no	—
151	C₂²×C₅	—	4	2	no	—
152	C₂²×C₅	—	4	2	no	—
153	C₂²×C₅	—	4	2	no	—
154	C₂²×C₅	—	8	4	no	—
155	C₂²×C₅	#350	1	2	yes	—
156	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
157	C₄×C₅	—	2	1	no	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	—	4	2	no	—
162	C₄×C₅	#349	1	2	yes	—
163	C₄×C₅	#418	1	1	yes	characteristic
164	C₄×C₅	#421	1	1	yes	characteristic
165	C₂⁴	—	2	1	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
176	C₂²×C₄	C₅×M₁₆	1	1	yes	characteristic
177	M₁₆	—	4	4	no	—
178	M₁₆	—	4	4	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
186	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	C₂×C₅×C₈	1	2	yes	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
192	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
193	C₄²	C₅×M₁₆	1	1	yes	characteristic
194	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
195	C₁₆	—	4	2	no	—
196	C₁₆	—	4	2	no	—
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	—	8	2	no	—
199	C₂×C₅	#1478	1	1	yes	characteristic
200	C₂×C₅	C₂×#349	1	1	yes	characteristic
201	C₂×C₅	C₂×#350	1	1	yes	characteristic
202	C₂³	—	2	2	no	—
203	C₂³	—	4	2	no	—
204	C₂³	—	4	2	no	—
205	C₂³	—	4	2	no	—
206	C₂³	C₅×C₂²⋊C₈	1	1	yes	characteristic
207	D₄	—	4	2	no	—
208	D₄	—	4	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
219	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
220	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
221	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
223	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	4	2	no	—
228	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
229	C₅	#11634	1	1	yes	characteristic, direct product factor, Sylow
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	8	4	no	—
236	C₂²	C₅×#350	1	2	yes	—
237	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	4	2	no	—
243	C₄	C₅×#349	1	2	yes	—
244	C₄	C₅×#418	1	1	yes	characteristic
245	C₄	C₅×#421	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	8	2	no	—
248	C₂	C₅×#1478	1	1	yes	characteristic
249	C₂	C₂×C₅×#349	1	1	yes	characteristic
250	C₂	C₂×C₅×#350	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₅×#2163	C₂	semidirect
3. C₅×#2162	C₂	semidirect
4. C₅×#1330	C₂	semidirect
15. C₅×C₄⋊C₁₆	C₂²	semidirect
18. C₅×#346	C₂²	semidirect
20. #11634	C₅	direct (N × H)
47. #2163	C₂×C₅	semidirect
48. #2162	C₂×C₅	semidirect
49. #1330	C₂×C₅	semidirect
92. C₄⋊C₁₆	C₂²×C₅	semidirect
95. #346	C₂²×C₅	semidirect
229. C₅	#11634	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×#436	2
C₅×#2163	2
C₅×#2162	2
C₅×#1330	2
#11634	5

Subgroups — C5×#11634 #11934354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11634 #11934354