Subgroups — C₂×C₅×#2163 #11956136

Order 1280.

686 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#2163	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₅×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×C₂²⋊C₁₆	C₂	1	2	yes	maximal
5	C₂×C₅×#346	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₂²	1	4	yes	—
10	C₂×C₄×C₅×C₈	C₄	1	4	yes	—
11	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×C₄⋊C₁₆	C₂²	1	8	yes	complemented
17	C₅×C₂²⋊C₁₆	C₂²	1	8	yes	—
18	C₅×#346	C₂²	1	8	yes	complemented
19	C₂×#2163	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₅	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₅	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	—	2	4	no	—
24	C₂²×C₅×C₈	—	2	4	no	—
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂³	1	2	yes	—
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₈	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₈	1	4	yes	—
34	C₂×C₄²×C₅	C₈	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×C₁₆	—	2	8	no	—
39	C₂×C₅×C₁₆	C₂³	1	8	yes	—
40	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
41	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
43	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
44	C₄×C₅×C₈	C₂³	1	2	yes	—
45	C₄×C₅×C₈	C₂³	1	2	yes	—
46	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
47	#2163	C₂×C₅	1	8	yes	direct product factor
48	C₂×C₄⋊C₁₆	C₂×C₅	1	2	yes	complemented
49	C₂×C₂²⋊C₁₆	C₂×C₅	1	2	yes	—
50	C₂×#346	C₂×C₅	1	2	yes	complemented
51	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅	—	2	4	no	—
53	C₂²×C₄×C₅	—	2	4	no	—
54	C₂²×C₄×C₅	—	2	2	no	—
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	—	2	2	no	—
68	C₂×C₅×C₈	—	2	4	no	—
69	C₂×C₅×C₈	—	2	8	no	—
70	C₂×C₅×C₈	—	2	4	no	—
71	C₂×C₅×C₈	M₁₆	1	4	yes	—
72	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₅×C₈	#41	1	4	yes	—
77	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
78	C₄²×C₅	C₂×C₈	1	4	yes	—
79	C₄²×C₅	C₂×C₈	1	4	yes	—
80	C₄²×C₅	C₂×C₈	1	2	yes	—
81	C₄²×C₅	C₂×C₈	1	2	yes	—
82	C₄²×C₅	C₂²×C₄	1	2	yes	—
83	C₄²×C₅	C₂²×C₄	1	2	yes	—
84	C₅×C₁₆	—	2	8	no	—
85	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
86	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
87	C₂²×C₄²	C₄×C₅	1	1	yes	characteristic
88	C₂²×C₁₆	C₂²×C₅	1	4	yes	—
89	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
90	C₂×C₄×C₈	C₄×C₅	1	4	yes	—
91	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
95	C₄⋊C₁₆	C₂²×C₅	1	8	yes	complemented
96	C₂²⋊C₁₆	C₂²×C₅	1	8	yes	—
97	#346	C₂²×C₅	1	8	yes	complemented
98	C₂³×C₅	—	2	2	no	—
99	C₂³×C₅	—	2	2	no	—
100	C₂³×C₅	M₃₂	1	4	yes	—
101	C₂³×C₅	C₂²×C₈	1	2	yes	—
102	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	8	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	—	2	8	no	—
112	C₂×C₄×C₅	M₃₂	1	4	yes	—
113	C₂×C₄×C₅	#117	1	2	yes	—
114	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
121	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
122	C₅×C₈	—	2	2	no	—
123	C₅×C₈	—	2	2	no	—
124	C₅×C₈	#117	1	2	yes	—
125	C₅×C₈	C₂×M₁₆	1	4	yes	—
126	C₅×C₈	C₂×#41	1	2	yes	—
127	C₂³×C₄	C₅×C₈	1	1	yes	characteristic
128	C₂³×C₄	C₅×C₈	1	1	yes	characteristic
129	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
130	C₂²×C₈	—	2	4	no	—
131	C₂²×C₈	—	2	4	no	—
132	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
133	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
134	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
135	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
136	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
137	C₂²×C₈	C₂³×C₅	1	2	yes	—
138	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
139	C₂×C₄²	C₅×C₈	1	1	yes	characteristic
140	C₂×C₄²	C₅×C₈	1	4	yes	—
141	C₂×C₄²	C₅×C₈	1	1	yes	characteristic
142	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
143	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
144	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
145	C₂×C₁₆	—	2	8	no	—
146	C₂×C₁₆	C₂³×C₅	1	8	yes	—
147	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
148	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
149	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
150	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
151	C₄×C₈	C₂³×C₅	1	2	yes	—
152	C₄×C₈	C₂³×C₅	1	2	yes	—
153	C₂²×C₅	—	2	2	no	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	2	8	no	—
156	C₂²×C₅	#431	1	2	yes	—
157	C₂²×C₅	C₂×M₃₂	1	4	yes	—
158	C₂²×C₅	C₂×M₃₂	1	2	yes	—
159	C₂²×C₅	C₂×M₃₂	1	2	yes	—
160	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	2	2	no	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	#431	1	2	yes	—
166	C₄×C₅	C₂×M₃₂	1	4	yes	—
167	C₄×C₅	C₂×#117	1	1	yes	characteristic
168	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
169	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
176	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
177	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
181	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
182	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
183	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	8	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	C₅×M₁₆	1	4	yes	—
190	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
191	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
192	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
193	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
194	C₂×C₈	C₅×#41	1	4	yes	—
195	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
196	C₄²	C₂×C₅×C₈	1	4	yes	—
197	C₄²	C₂×C₅×C₈	1	4	yes	—
198	C₄²	C₂×C₅×C₈	1	2	yes	—
199	C₄²	C₂×C₅×C₈	1	2	yes	—
200	C₄²	C₂²×C₄×C₅	1	2	yes	—
201	C₄²	C₂²×C₄×C₅	1	2	yes	—
202	C₁₆	—	2	8	no	—
203	C₂×C₅	—	2	4	no	—
204	C₂×C₅	#2163	1	4	yes	direct product factor
205	C₂×C₅	C₂×#431	1	1	yes	characteristic
206	C₂×C₅	C₂²×M₃₂	1	1	yes	characteristic
207	C₂×C₅	C₂²×M₃₂	1	1	yes	characteristic
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	C₅×M₃₂	1	4	yes	—
211	C₂³	C₂²×C₅×C₈	1	2	yes	—
212	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	C₅×M₃₂	1	4	yes	—
223	C₂×C₄	C₅×#117	1	2	yes	—
224	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
225	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
226	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
227	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
228	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
229	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
230	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
231	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	C₅×#117	1	2	yes	—
235	C₈	C₂×C₅×M₁₆	1	4	yes	—
236	C₈	C₂×C₅×#41	1	2	yes	—
237	C₅	C₂×#2163	1	1	yes	characteristic, direct product factor, Sylow
238	C₂²	—	2	8	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	C₅×#431	1	2	yes	—
242	C₂²	C₂×C₅×M₃₂	1	4	yes	—
243	C₂²	C₂×C₅×M₃₂	1	2	yes	—
244	C₂²	C₂×C₅×M₃₂	1	2	yes	—
245	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	C₅×#431	1	2	yes	—
251	C₄	C₂×C₅×M₃₂	1	4	yes	—
252	C₄	C₂×C₅×#117	1	1	yes	characteristic
253	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	C₅×#2163	1	4	yes	direct product factor
256	C₂	C₂×C₅×#431	1	1	yes	characteristic
257	C₂	C₂²×C₅×M₃₂	1	1	yes	characteristic
258	C₂	C₂²×C₅×M₃₂	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₅×#2163	C₂	direct (N × H)
3. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
5. C₂×C₅×#346	C₂	semidirect
16. C₅×C₄⋊C₁₆	C₂²	semidirect
18. C₅×#346	C₂²	semidirect
19. C₂×#2163	C₅	direct (N × H)
47. #2163	C₂×C₅	direct (N × H)
48. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
50. C₂×#346	C₂×C₅	semidirect
95. C₄⋊C₁₆	C₂²×C₅	semidirect
97. #346	C₂²×C₅	semidirect
204. C₂×C₅	#2163	direct (N × H)
237. C₅	C₂×#2163	direct (N × H)
255. C₂	C₅×#2163	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×C₅×C₈	2
C₅×#2163	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×C₂²⋊C₁₆	2
C₂×C₅×#346	2
C₂×#2163	5

Subgroups — C2×C5×#2163 #11956136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2163 #11956136