Subgroups — C₂×C₅×#1594 #11940609

Order 1280.

806 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#436	C₂	1	2	yes	maximal, complemented
3	C₅×#1594	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×D₈	C₄	1	1	yes	characteristic
8	C₂²×C₅×M₁₆	—	2	2	no	—
9	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
10	C₄×C₅×D₈	C₄	1	4	yes	—
11	C₄×C₅×D₈	C₂²	1	4	yes	—
12	C₅×#436	—	2	4	no	—
13	C₅×#436	C₂²	1	8	yes	complemented
14	C₂×C₅×#107	C₄	1	1	yes	characteristic
15	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₅×#102	C₄	1	2	yes	—
17	C₂×C₅×C₂²⋊C₈	—	2	2	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#334	C₂²	1	4	yes	complemented
22	C₅×D₄⋊C₈	C₂²	1	8	yes	complemented
23	C₅×#322	C₂²	1	4	yes	complemented
24	C₂×#1594	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₅	—	2	2	no	—
27	C₂×C₅×D₈	—	2	4	no	—
28	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
29	C₂×C₅×M₁₆	—	2	8	no	—
30	C₂×C₅×M₁₆	—	4	8	no	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	—	2	1	no	—
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	D₄	1	2	yes	—
36	C₄×C₅×D₄	—	2	2	no	—
37	C₄×C₅×D₄	—	2	2	no	—
38	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
39	C₄×C₅×D₄	C₂³	1	4	yes	—
40	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×#107	C₂×C₄	1	4	yes	—
44	C₅×C₄⋊C₈	—	2	2	no	—
45	C₅×C₄⋊C₈	C₂³	1	4	yes	—
46	C₅×#102	C₂×C₄	1	8	yes	—
47	C₅×C₂²⋊C₈	—	2	8	no	—
48	C₄×C₅×C₈	—	2	1	no	—
49	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
50	C₄×C₅×C₈	D₄	1	2	yes	—
51	C₄×C₅×C₈	C₂³	1	2	yes	—
52	C₄×C₅×C₈	C₂³	1	2	yes	—
53	C₄×C₅×C₈	C₂³	1	2	yes	—
54	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic
55	C₂×#436	C₂×C₅	1	2	yes	complemented
56	#1594	C₂×C₅	1	8	yes	direct product factor
57	C₂×#334	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×D₄⋊C₈	C₂×C₅	1	2	yes	complemented
59	C₂×#322	C₂×C₅	1	1	yes	characteristic, complemented
60	C₂⁴×C₅	—	2	2	no	—
61	C₂×C₅×D₄	—	2	2	no	—
62	C₂×C₅×D₄	—	2	4	no	—
63	C₂×C₅×D₄	—	2	4	no	—
64	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	—	2	4	no	—
68	C₂²×C₄×C₅	—	4	4	no	—
69	C₂²×C₄×C₅	—	4	4	no	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	1	yes	characteristic
73	C₅×D₈	—	2	8	no	—
74	C₅×M₁₆	—	4	16	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	4	no	—
77	C₂×C₅×C₈	—	2	4	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	4	2	no	—
80	C₂×C₅×C₈	—	4	4	no	—
81	C₂×C₅×C₈	M₁₆	1	4	yes	complemented
82	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
85	C₅×C₄⋊C₄	—	2	2	no	—
86	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
87	C₅×C₂²⋊C₄	—	2	4	no	—
88	C₅×C₂²⋊C₄	—	2	4	no	—
89	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅	#41	1	1	yes	characteristic
92	C₄²×C₅	C₂⁴	1	1	yes	characteristic
93	C₂²×D₈	C₄×C₅	1	1	yes	characteristic
94	C₂²×M₁₆	—	2	2	no	—
95	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
96	C₄×D₈	C₄×C₅	1	4	yes	—
97	C₄×D₈	C₂²×C₅	1	4	yes	—
98	#436	—	2	4	no	—
99	#436	C₂²×C₅	1	8	yes	complemented
100	C₂×#107	C₄×C₅	1	1	yes	characteristic
101	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
102	C₂×#102	C₄×C₅	1	2	yes	—
103	C₂×C₂²⋊C₈	—	2	2	no	—
104	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
107	#334	C₂²×C₅	1	4	yes	complemented
108	D₄⋊C₈	C₂²×C₅	1	8	yes	complemented
109	#322	C₂²×C₅	1	4	yes	complemented
110	C₂³×C₅	—	2	4	no	—
111	C₂³×C₅	—	4	4	no	—
112	C₂³×C₅	—	4	4	no	—
113	C₂³×C₅	—	4	4	no	—
114	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
115	C₅×D₄	—	2	4	no	—
116	C₅×D₄	—	2	4	no	—
117	C₅×D₄	—	4	4	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	—	4	4	no	—
121	C₂×C₄×C₅	—	4	4	no	—
122	C₂×C₄×C₅	—	4	4	no	—
123	C₂×C₄×C₅	—	4	8	no	—
124	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₅	#131	1	2	yes	—
129	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
130	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
133	C₅×C₈	—	2	2	no	—
134	C₅×C₈	—	2	4	no	—
135	C₅×C₈	—	4	2	no	—
136	C₅×C₈	—	4	4	no	—
137	C₅×C₈	C₂×M₁₆	1	4	yes	complemented
138	C₂²×D₄	C₂×C₄×C₅	1	2	yes	—
139	C₂³×C₄	—	2	2	no	—
140	C₂×D₈	—	2	4	no	—
141	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
142	C₂×M₁₆	—	2	8	no	—
143	C₂×M₁₆	—	4	8	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	1	no	—
146	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂²×C₈	C₅×D₄	1	2	yes	—
149	C₄×D₄	—	2	2	no	—
150	C₄×D₄	—	2	2	no	—
151	C₄×D₄	C₂×C₄×C₅	1	4	yes	—
152	C₄×D₄	C₂³×C₅	1	4	yes	—
153	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
154	C₂×C₂²⋊C₄	—	2	2	no	—
155	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
156	#107	C₂×C₄×C₅	1	4	yes	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	C₂³×C₅	1	4	yes	—
159	#102	C₂×C₄×C₅	1	8	yes	—
160	C₂²⋊C₈	—	2	8	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
163	C₄×C₈	C₅×D₄	1	2	yes	—
164	C₄×C₈	C₂³×C₅	1	2	yes	—
165	C₄×C₈	C₂³×C₅	1	2	yes	—
166	C₄×C₈	C₂³×C₅	1	2	yes	—
167	C₂²×C₅	—	4	4	no	—
168	C₂²×C₅	—	4	4	no	—
169	C₂²×C₅	—	4	4	no	—
170	C₂²×C₅	—	8	8	no	—
171	C₂²×C₅	#436	1	2	yes	—
172	C₂²×C₅	#442	1	2	yes	—
173	C₂²×C₅	#444	1	2	yes	—
174	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	—	4	4	no	—
178	C₄×C₅	#436	1	2	yes	—
179	C₄×C₅	#442	1	2	yes	—
180	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
181	C₄×C₅	C₂×#131	1	1	yes	characteristic
182	C₄×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
183	C₂⁴	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂²×C₄	C₅×#41	1	1	yes	characteristic
196	D₈	—	2	8	no	—
197	M₁₆	—	4	16	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	4	no	—
204	C₂×C₈	C₅×M₁₆	1	4	yes	complemented
205	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
207	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
213	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
214	C₄²	C₅×#41	1	1	yes	characteristic
215	C₄²	C₂⁴×C₅	1	1	yes	characteristic
216	C₂×C₅	—	8	4	no	—
217	C₂×C₅	#1594	1	4	yes	direct product factor
218	C₂×C₅	C₂×#436	1	1	yes	characteristic
219	C₂×C₅	C₂×#442	1	1	yes	characteristic
220	C₂×C₅	C₂×#444	1	1	yes	characteristic
221	C₂³	—	2	4	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	4	4	no	—
224	C₂³	—	4	4	no	—
225	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
226	D₄	—	2	4	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
236	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
237	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
238	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
239	C₂×C₄	C₅×#131	1	2	yes	—
240	C₂×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
241	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	4	no	—
246	C₈	—	4	2	no	—
247	C₈	—	4	4	no	—
248	C₈	C₂×C₅×M₁₆	1	4	yes	complemented
249	C₅	C₂×#1594	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	4	4	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	8	8	no	—
254	C₂²	C₅×#436	1	2	yes	—
255	C₂²	C₅×#442	1	2	yes	—
256	C₂²	C₅×#444	1	2	yes	—
257	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	4	no	—
261	C₄	C₅×#436	1	2	yes	—
262	C₄	C₅×#442	1	2	yes	—
263	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
264	C₄	C₂×C₅×#131	1	1	yes	characteristic
265	C₄	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
266	C₂	—	8	4	no	—
267	C₂	C₅×#1594	1	4	yes	direct product factor
268	C₂	C₂×C₅×#436	1	1	yes	characteristic
269	C₂	C₂×C₅×#442	1	1	yes	characteristic
270	C₂	C₂×C₅×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	C₂⁴×C₅
socle	C₂³×C₅	C₄×D₄
prime radical	C₂²×C₅×D₈	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×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×#436	C₂	semidirect
3. C₅×#1594	C₂	direct (N × H)
4. C₂×C₅×#334	C₂	semidirect
5. C₂×C₅×D₄⋊C₈	C₂	semidirect
6. C₂×C₅×#322	C₂	semidirect
13. C₅×#436	C₂²	semidirect
21. C₅×#334	C₂²	semidirect
22. C₅×D₄⋊C₈	C₂²	semidirect
23. C₅×#322	C₂²	semidirect
24. C₂×#1594	C₅	direct (N × H)
55. C₂×#436	C₂×C₅	semidirect
56. #1594	C₂×C₅	direct (N × H)
57. C₂×#334	C₂×C₅	semidirect
58. C₂×D₄⋊C₈	C₂×C₅	semidirect
59. C₂×#322	C₂×C₅	semidirect
81. C₂×C₅×C₈	M₁₆	semidirect
99. #436	C₂²×C₅	semidirect
107. #334	C₂²×C₅	semidirect
108. D₄⋊C₈	C₂²×C₅	semidirect
109. #322	C₂²×C₅	semidirect
137. C₅×C₈	C₂×M₁₆	semidirect
204. C₂×C₈	C₅×M₁₆	semidirect
217. C₂×C₅	#1594	direct (N × H)
248. C₈	C₂×C₅×M₁₆	semidirect
249. C₅	C₂×#1594	direct (N × H)
267. C₂	C₅×#1594	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×C₄×C₅×D₈	2
C₂×C₅×#436	2
C₅×#1594	2
C₂×C₅×#334	2
C₂×C₅×D₄⋊C₈	2
C₂×C₅×#322	2
C₂×#1594	5

Subgroups — C2×C5×#1594 #11940609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1594 #11940609