Subgroups — C₂×C₅×#2178 #11956181

Order 1280.

662 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#508	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#443	C₂	1	2	yes	maximal, complemented
3	C₅×#2178	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#359	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#135	C₂²	1	2	yes	—
8	C₂×C₅×#131	—	2	2	no	—
9	C₅×#508	C₄	1	4	yes	complemented
10	C₅×#508	C₂²	1	4	yes	—
11	C₂×C₅×Q₃₂	C₄	1	1	yes	characteristic, complemented
12	C₂×C₅×SD₃₂	C₄	1	2	yes	complemented
13	C₂×C₅×D₁₆	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₅×#443	—	2	4	no	—
17	C₅×#443	C₂²	1	8	yes	complemented
18	C₂×C₅×#108	C₂²	1	2	yes	—
19	C₂×C₅×#104	—	2	2	no	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#368	C₂²	1	4	yes	complemented
22	C₅×#359	C₂²	1	8	yes	complemented
23	C₄×C₅×C₁₆	C₂²	1	4	yes	complemented
24	C₂×#2178	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₅×#41	—	2	2	no	—
26	C₅×#135	—	2	2	no	—
27	C₅×#135	—	2	2	no	—
28	C₅×#135	C₂×C₄	1	4	yes	—
29	C₅×#135	C₂³	1	4	yes	—
30	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₅×SD₁₆	—	2	2	no	—
32	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
33	C₅×#131	—	2	8	no	—
34	C₅×#131	—	4	4	no	—
35	C₂×C₅×M₁₆	—	2	2	no	—
36	C₂×C₅×M₁₆	—	4	2	no	—
37	C₂²×C₅×C₈	—	4	2	no	—
38	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₅×Q₃₂	C₂×C₄	1	4	yes	complemented
42	C₅×SD₃₂	C₂×C₄	1	8	yes	complemented
43	C₅×D₁₆	C₂×C₄	1	4	yes	complemented
44	C₂×C₅×C₁₆	—	2	1	no	—
45	C₂×C₅×C₁₆	—	2	1	no	—
46	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
47	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
48	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
49	C₂×C₅×C₁₆	C₂³	1	2	yes	—
50	C₂×C₅×C₁₆	C₂³	1	2	yes	—
51	C₅×#108	—	2	2	no	—
52	C₅×#108	C₂³	1	4	yes	—
53	C₅×#104	—	2	8	no	—
54	C₄×C₅×C₈	D₄	1	2	yes	—
55	C₄×C₅×C₈	C₂³	1	2	yes	—
56	C₂×#508	C₂×C₅	1	1	yes	characteristic
57	C₂×#443	C₂×C₅	1	2	yes	complemented
58	#2178	C₂×C₅	1	8	yes	direct product factor
59	C₂×#368	C₂×C₅	1	1	yes	characteristic, complemented
60	C₂×#359	C₂×C₅	1	2	yes	complemented
61	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
62	C₅×#41	—	2	4	no	—
63	C₅×#41	—	2	4	no	—
64	C₅×#41	—	4	2	no	—
65	C₅×#41	—	4	2	no	—
66	C₂×C₅×Q₈	—	2	2	no	—
67	C₂×C₅×D₄	—	2	2	no	—
68	C₂×C₅×D₄	—	4	2	no	—
69	C₂²×C₄×C₅	—	2	1	no	—
70	C₂²×C₄×C₅	—	4	2	no	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₅×Q₁₆	—	2	2	no	—
73	C₅×Q₁₆	C₂²×C₄	1	4	yes	—
74	C₅×SD₁₆	—	2	4	no	—
75	C₅×SD₁₆	—	2	4	no	—
76	C₅×D₈	—	2	2	no	—
77	C₅×D₈	C₂²×C₄	1	4	yes	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	4	2	no	—
80	C₅×M₁₆	—	4	8	no	—
81	C₂×C₅×C₈	—	4	2	no	—
82	C₂×C₅×C₈	—	4	8	no	—
83	C₂×C₅×C₈	D₈	1	1	yes	characteristic
84	C₂×C₅×C₈	D₈	1	2	yes	—
85	C₂×C₅×C₈	D₈	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₅×C₈	#41	1	1	yes	characteristic
91	C₂×C₅×C₈	#41	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
93	C₄²×C₅	—	2	1	no	—
94	C₄²×C₅	C₂×D₄	1	2	yes	—
95	C₅×C₁₆	—	2	2	no	—
96	C₅×C₁₆	C₂²×C₄	1	2	yes	—
97	C₅×C₁₆	C₂²×C₄	1	2	yes	—
98	C₂×#135	C₂²×C₅	1	2	yes	—
99	C₂×#131	—	2	2	no	—
100	#508	C₄×C₅	1	4	yes	complemented
101	#508	C₂²×C₅	1	4	yes	—
102	C₂×Q₃₂	C₄×C₅	1	1	yes	characteristic, complemented
103	C₂×SD₃₂	C₄×C₅	1	2	yes	complemented
104	C₂×D₁₆	C₄×C₅	1	1	yes	characteristic, complemented
105	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
106	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
107	#443	—	2	4	no	—
108	#443	C₂²×C₅	1	8	yes	complemented
109	C₂×#108	C₂²×C₅	1	2	yes	—
110	C₂×#104	—	2	2	no	—
111	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
112	#368	C₂²×C₅	1	4	yes	complemented
113	#359	C₂²×C₅	1	8	yes	complemented
114	C₄×C₁₆	C₂²×C₅	1	4	yes	complemented
115	C₂³×C₅	—	4	2	no	—
116	C₂³×C₅	#135	1	1	yes	characteristic
117	C₅×Q₈	—	2	4	no	—
118	C₅×Q₈	—	4	2	no	—
119	C₅×D₄	—	2	4	no	—
120	C₅×D₄	—	4	4	no	—
121	C₅×D₄	—	4	4	no	—
122	C₅×D₄	—	4	2	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	—	4	4	no	—
128	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅	#135	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
134	C₅×C₈	—	4	4	no	—
135	C₅×C₈	C₄×D₄	1	1	yes	characteristic
136	C₅×C₈	C₄×D₄	1	1	yes	characteristic
137	C₅×C₈	C₂×D₈	1	2	yes	—
138	C₅×C₈	C₂×D₈	1	2	yes	—
139	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
140	C₅×C₈	C₂×#41	1	1	yes	characteristic
141	C₂×#41	—	2	2	no	—
142	#135	—	2	2	no	—
143	#135	—	2	2	no	—
144	#135	C₂×C₄×C₅	1	4	yes	—
145	#135	C₂³×C₅	1	4	yes	—
146	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
147	C₂×SD₁₆	—	2	2	no	—
148	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
149	#131	—	2	8	no	—
150	#131	—	4	4	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	4	2	no	—
153	C₂²×C₈	—	4	2	no	—
154	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
156	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
157	Q₃₂	C₂×C₄×C₅	1	4	yes	complemented
158	SD₃₂	C₂×C₄×C₅	1	8	yes	complemented
159	D₁₆	C₂×C₄×C₅	1	4	yes	complemented
160	C₂×C₁₆	—	2	1	no	—
161	C₂×C₁₆	—	2	1	no	—
162	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
163	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
164	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
165	C₂×C₁₆	C₂³×C₅	1	2	yes	—
166	C₂×C₁₆	C₂³×C₅	1	2	yes	—
167	#108	—	2	2	no	—
168	#108	C₂³×C₅	1	4	yes	—
169	#104	—	2	8	no	—
170	C₄×C₈	C₅×D₄	1	2	yes	—
171	C₄×C₈	C₂³×C₅	1	2	yes	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	4	4	no	—
174	C₂²×C₅	—	8	4	no	—
175	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
176	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
177	C₂²×C₅	C₂×#135	1	1	yes	characteristic
178	C₄×C₅	—	2	4	no	—
179	C₄×C₅	—	4	4	no	—
180	C₄×C₅	C₄×D₈	1	1	yes	characteristic
181	C₄×C₅	C₄×D₈	1	1	yes	characteristic
182	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
183	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
184	#41	—	2	4	no	—
185	#41	—	2	4	no	—
186	#41	—	4	2	no	—
187	#41	—	4	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	Q₁₆	—	2	2	no	—
195	Q₁₆	C₂²×C₄×C₅	1	4	yes	—
196	SD₁₆	—	2	4	no	—
197	SD₁₆	—	2	4	no	—
198	D₈	—	2	2	no	—
199	D₈	C₂²×C₄×C₅	1	4	yes	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	4	2	no	—
202	M₁₆	—	4	8	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	8	no	—
205	C₂×C₈	C₅×D₈	1	1	yes	characteristic
206	C₂×C₈	C₅×D₈	1	1	yes	characteristic
207	C₂×C₈	C₅×D₈	1	2	yes	—
208	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
209	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
210	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
211	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
212	C₂×C₈	C₅×#41	1	1	yes	characteristic
213	C₂×C₈	C₅×#41	1	1	yes	characteristic
214	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
215	C₄²	—	2	1	no	—
216	C₄²	C₂×C₅×D₄	1	2	yes	—
217	C₁₆	—	2	2	no	—
218	C₁₆	C₂²×C₄×C₅	1	2	yes	—
219	C₁₆	C₂²×C₄×C₅	1	2	yes	—
220	C₂×C₅	—	2	1	no	—
221	C₂×C₅	—	2	1	no	—
222	C₂×C₅	—	8	4	no	—
223	C₂×C₅	#2178	1	2	yes	direct product factor
224	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
225	C₂³	—	4	2	no	—
226	C₂³	C₅×#135	1	1	yes	characteristic
227	Q₈	—	2	4	no	—
228	Q₈	—	4	2	no	—
229	D₄	—	2	4	no	—
230	D₄	—	4	4	no	—
231	D₄	—	4	4	no	—
232	D₄	—	4	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
239	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
240	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
241	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
242	C₂×C₄	C₅×#135	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
244	C₈	—	4	4	no	—
245	C₈	C₄×C₅×D₄	1	1	yes	characteristic
246	C₈	C₄×C₅×D₄	1	1	yes	characteristic
247	C₈	C₂×C₅×D₈	1	2	yes	—
248	C₈	C₂×C₅×D₈	1	2	yes	—
249	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
250	C₈	C₂×C₅×#41	1	1	yes	characteristic
251	C₅	C₂×#2178	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	2	2	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	8	4	no	—
255	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
256	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
257	C₂²	C₂×C₅×#135	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	4	4	no	—
260	C₄	C₄×C₅×D₈	1	1	yes	characteristic
261	C₄	C₄×C₅×D₈	1	1	yes	characteristic
262	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
263	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	8	4	no	—
267	C₂	C₅×#2178	1	2	yes	direct product factor
268	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	D₈
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂²×C₅	C₄×D₈
prime radical	C₂×C₅×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂²×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#443	C₂	semidirect
3. C₅×#2178	C₂	direct (N × H)
4. C₂×C₅×#368	C₂	semidirect
5. C₂×C₅×#359	C₂	semidirect
6. C₂×C₄×C₅×C₁₆	C₂	semidirect
9. C₅×#508	C₄	semidirect
11. C₂×C₅×Q₃₂	C₄	semidirect
12. C₂×C₅×SD₃₂	C₄	semidirect
13. C₂×C₅×D₁₆	C₄	semidirect
17. C₅×#443	C₂²	semidirect
21. C₅×#368	C₂²	semidirect
22. C₅×#359	C₂²	semidirect
23. C₄×C₅×C₁₆	C₂²	semidirect
24. C₂×#2178	C₅	direct (N × H)
41. C₅×Q₃₂	C₂×C₄	semidirect
42. C₅×SD₃₂	C₂×C₄	semidirect
43. C₅×D₁₆	C₂×C₄	semidirect
57. C₂×#443	C₂×C₅	semidirect
58. #2178	C₂×C₅	direct (N × H)
59. C₂×#368	C₂×C₅	semidirect
60. C₂×#359	C₂×C₅	semidirect
61. C₂×C₄×C₁₆	C₂×C₅	semidirect
100. #508	C₄×C₅	semidirect
102. C₂×Q₃₂	C₄×C₅	semidirect
103. C₂×SD₃₂	C₄×C₅	semidirect
104. C₂×D₁₆	C₄×C₅	semidirect
108. #443	C₂²×C₅	semidirect
112. #368	C₂²×C₅	semidirect
113. #359	C₂²×C₅	semidirect
114. C₄×C₁₆	C₂²×C₅	semidirect
157. Q₃₂	C₂×C₄×C₅	semidirect
158. SD₃₂	C₂×C₄×C₅	semidirect
159. D₁₆	C₂×C₄×C₅	semidirect
223. C₂×C₅	#2178	direct (N × H)
251. C₅	C₂×#2178	direct (N × H)
267. C₂	C₅×#2178	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×#508	2
C₂×C₅×#443	2
C₅×#2178	2
C₂×C₅×#368	2
C₂×C₅×#359	2
C₂×C₄×C₅×C₁₆	2
C₂×#2178	5

Subgroups — C2×C5×#2178 #11956181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2178 #11956181