Subgroups — C₂×C₅×#2230 #11956355

Order 1280.

558 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#443	C₂	1	1	yes	maximal, characteristic
3	C₅×#2230	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#365	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#360	C₂	1	2	yes	maximal
7	C₂×C₅×#135	C₂²	1	1	yes	characteristic
8	C₂×C₅×#131	—	2	1	no	—
9	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₅×M₃₂	C₂²	1	2	yes	—
11	C₅×#498	—	2	2	no	—
12	C₅×#498	C₂²	1	4	yes	complemented
13	C₅×#443	—	2	2	no	—
14	C₅×#443	C₂²	1	4	yes	—
15	C₂×C₅×#108	C₂²	1	1	yes	characteristic
16	C₂×C₅×#106	—	2	1	no	—
17	C₂×C₅×#106	C₂²	1	2	yes	—
18	C₂×C₅×#104	—	2	1	no	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#365	C₂²	1	8	yes	complemented
21	C₅×#364	C₂²	1	4	yes	complemented
22	C₅×#360	C₂²	1	8	yes	—
23	C₂×#2230	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×#41	—	2	1	no	—
25	C₂²×C₅×Q₈	—	4	1	no	—
26	C₅×#135	—	2	1	no	—
27	C₅×#135	—	2	1	no	—
28	C₅×#135	Q₈	1	2	yes	complemented
29	C₅×#135	C₂³	1	2	yes	—
30	C₂×C₅×Q₁₆	Q₈	1	1	yes	characteristic, complemented
31	C₂×C₅×SD₁₆	—	2	1	no	—
32	C₂×C₅×D₈	Q₈	1	1	yes	characteristic, complemented
33	C₅×#131	—	2	4	no	—
34	C₅×#131	—	4	2	no	—
35	C₂×C₅×M₁₆	—	2	1	no	—
36	C₂×C₅×M₁₆	—	4	1	no	—
37	C₂²×C₅×C₈	—	4	1	no	—
38	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₅×C₄⋊Q₈	—	2	4	no	—
41	C₅×C₄⋊Q₈	—	4	2	no	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₄⋊C₄	—	4	1	no	—
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₅×M₃₂	—	2	2	no	—
46	C₅×M₃₂	C₂³	1	4	yes	—
47	C₂×C₅×C₁₆	—	2	2	no	—
48	C₅×#108	—	2	1	no	—
49	C₅×#108	C₂³	1	2	yes	—
50	C₅×#106	—	2	4	no	—
51	C₅×#106	—	2	2	no	—
52	C₅×#106	C₂³	1	4	yes	—
53	C₅×#104	—	2	4	no	—
54	C₄×C₅×C₈	D₄	1	2	yes	—
55	C₄×C₅×C₈	C₂³	1	2	yes	—
56	C₂×#498	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×#443	C₂×C₅	1	1	yes	characteristic
58	#2230	C₂×C₅	1	8	yes	direct product factor
59	C₂×#365	C₂×C₅	1	2	yes	complemented
60	C₂×#364	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂×#360	C₂×C₅	1	2	yes	—
62	C₅×#41	—	2	2	no	—
63	C₅×#41	—	2	2	no	—
64	C₅×#41	—	4	1	no	—
65	C₅×#41	—	4	1	no	—
66	C₂×C₅×Q₈	—	2	1	no	—
67	C₂×C₅×Q₈	—	4	4	no	—
68	C₂×C₅×Q₈	—	8	4	no	—
69	C₂×C₅×D₄	—	2	1	no	—
70	C₂×C₅×D₄	—	4	1	no	—
71	C₂²×C₄×C₅	—	2	1	no	—
72	C₂²×C₄×C₅	—	4	1	no	—
73	C₂²×C₄×C₅	—	4	2	no	—
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₅×Q₁₆	—	2	1	no	—
76	C₅×Q₁₆	C₂×Q₈	1	2	yes	complemented
77	C₅×SD₁₆	—	2	2	no	—
78	C₅×SD₁₆	—	2	2	no	—
79	C₅×D₈	—	2	1	no	—
80	C₅×D₈	C₂×Q₈	1	2	yes	complemented
81	C₅×M₁₆	—	2	2	no	—
82	C₅×M₁₆	—	4	1	no	—
83	C₅×M₁₆	—	4	4	no	—
84	C₂×C₅×C₈	—	4	1	no	—
85	C₂×C₅×C₈	—	4	4	no	—
86	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
87	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
88	C₂×C₅×C₈	SD₁₆	1	2	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
91	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₅×C₈	#41	1	1	yes	characteristic
94	C₂×C₅×C₈	#41	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
96	C₅×C₄⋊C₄	—	2	4	no	—
97	C₅×C₄⋊C₄	—	4	4	no	—
98	C₅×C₄⋊C₄	—	4	2	no	—
99	C₄²×C₅	—	2	1	no	—
100	C₄²×C₅	C₂×D₄	1	2	yes	—
101	C₅×C₁₆	—	2	4	no	—
102	C₂×#135	C₂²×C₅	1	1	yes	characteristic
103	C₂×#131	—	2	1	no	—
104	C₂×C₄⋊Q₈	—	2	1	no	—
105	C₂×M₃₂	C₂²×C₅	1	2	yes	—
106	#498	—	2	2	no	—
107	#498	C₂²×C₅	1	4	yes	complemented
108	#443	—	2	2	no	—
109	#443	C₂²×C₅	1	4	yes	—
110	C₂×#108	C₂²×C₅	1	1	yes	characteristic
111	C₂×#106	—	2	1	no	—
112	C₂×#106	C₂²×C₅	1	2	yes	—
113	C₂×#104	—	2	1	no	—
114	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
115	#365	C₂²×C₅	1	8	yes	complemented
116	#364	C₂²×C₅	1	4	yes	complemented
117	#360	C₂²×C₅	1	8	yes	—
118	C₂³×C₅	—	4	1	no	—
119	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₅×Q₈	—	2	2	no	—
121	C₅×Q₈	—	4	1	no	—
122	C₅×Q₈	—	8	8	no	—
123	C₅×D₄	—	2	2	no	—
124	C₅×D₄	—	4	2	no	—
125	C₅×D₄	—	4	2	no	—
126	C₅×D₄	—	4	1	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	—	4	2	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	—	8	4	no	—
134	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
140	C₅×C₈	—	4	2	no	—
141	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
142	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
143	C₅×C₈	C₂×SD₁₆	1	2	yes	—
144	C₅×C₈	C₂×SD₁₆	1	2	yes	—
145	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
146	C₅×C₈	C₂×#41	1	1	yes	characteristic
147	C₂×#41	—	2	1	no	—
148	C₂²×Q₈	—	4	1	no	—
149	#135	—	2	1	no	—
150	#135	—	2	1	no	—
151	#135	C₅×Q₈	1	2	yes	complemented
152	#135	C₂³×C₅	1	2	yes	—
153	C₂×Q₁₆	C₅×Q₈	1	1	yes	characteristic, complemented
154	C₂×SD₁₆	—	2	1	no	—
155	C₂×D₈	C₅×Q₈	1	1	yes	characteristic, complemented
156	#131	—	2	4	no	—
157	#131	—	4	2	no	—
158	C₂×M₁₆	—	2	1	no	—
159	C₂×M₁₆	—	4	1	no	—
160	C₂²×C₈	—	4	1	no	—
161	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
163	C₄⋊Q₈	—	2	4	no	—
164	C₄⋊Q₈	—	4	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	4	1	no	—
167	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
168	M₃₂	—	2	2	no	—
169	M₃₂	C₂³×C₅	1	4	yes	—
170	C₂×C₁₆	—	2	2	no	—
171	#108	—	2	1	no	—
172	#108	C₂³×C₅	1	2	yes	—
173	#106	—	2	4	no	—
174	#106	—	2	2	no	—
175	#106	C₂³×C₅	1	4	yes	—
176	#104	—	2	4	no	—
177	C₄×C₈	C₅×D₄	1	2	yes	—
178	C₄×C₈	C₂³×C₅	1	2	yes	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	4	2	no	—
181	C₂²×C₅	—	8	2	no	—
182	C₂²×C₅	#476	1	1	yes	characteristic
183	C₂²×C₅	#476	1	1	yes	characteristic
184	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₄×C₅	—	2	4	no	—
186	C₄×C₅	—	4	2	no	—
187	C₄×C₅	—	8	4	no	—
188	C₄×C₅	#476	1	1	yes	characteristic
189	C₄×C₅	#476	1	1	yes	characteristic
190	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
191	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	#41	—	4	1	no	—
195	#41	—	4	1	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×Q₈	—	8	4	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	Q₁₆	—	2	1	no	—
206	Q₁₆	C₂×C₅×Q₈	1	2	yes	complemented
207	SD₁₆	—	2	2	no	—
208	SD₁₆	—	2	2	no	—
209	D₈	—	2	1	no	—
210	D₈	C₂×C₅×Q₈	1	2	yes	complemented
211	M₁₆	—	2	2	no	—
212	M₁₆	—	4	1	no	—
213	M₁₆	—	4	4	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
217	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
218	C₂×C₈	C₅×SD₁₆	1	2	yes	—
219	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
220	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
221	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
222	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
223	C₂×C₈	C₅×#41	1	1	yes	characteristic
224	C₂×C₈	C₅×#41	1	1	yes	characteristic
225	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄²	—	2	1	no	—
230	C₄²	C₂×C₅×D₄	1	2	yes	—
231	C₁₆	—	2	4	no	—
232	C₂×C₅	—	2	1	no	—
233	C₂×C₅	—	2	1	no	—
234	C₂×C₅	—	8	2	no	—
235	C₂×C₅	#2230	1	2	yes	direct product factor
236	C₂×C₅	C₂×#476	1	1	yes	characteristic
237	C₂³	—	4	1	no	—
238	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	Q₈	—	4	1	no	—
241	Q₈	—	8	8	no	—
242	D₄	—	2	2	no	—
243	D₄	—	4	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	1	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	8	4	no	—
253	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
254	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
256	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
257	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
259	C₈	—	4	2	no	—
260	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
261	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
262	C₈	C₂×C₅×SD₁₆	1	2	yes	—
263	C₈	C₂×C₅×SD₁₆	1	2	yes	—
264	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
265	C₈	C₂×C₅×#41	1	1	yes	characteristic
266	C₅	C₂×#2230	1	1	yes	characteristic, direct product factor, Sylow
267	C₂²	—	2	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	C₅×#476	1	1	yes	characteristic
271	C₂²	C₅×#476	1	1	yes	characteristic
272	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	8	4	no	—
276	C₄	C₅×#476	1	1	yes	characteristic
277	C₄	C₅×#476	1	1	yes	characteristic
278	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
279	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	8	2	no	—
283	C₂	C₅×#2230	1	2	yes	direct product factor
284	C₂	C₂×C₅×#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#498	C₂	semidirect
3. C₅×#2230	C₂	direct (N × H)
4. C₂×C₅×#365	C₂	semidirect
5. C₂×C₅×#364	C₂	semidirect
12. C₅×#498	C₂²	semidirect
20. C₅×#365	C₂²	semidirect
21. C₅×#364	C₂²	semidirect
23. C₂×#2230	C₅	direct (N × H)
28. C₅×#135	Q₈	semidirect
30. C₂×C₅×Q₁₆	Q₈	semidirect
32. C₂×C₅×D₈	Q₈	semidirect
56. C₂×#498	C₂×C₅	semidirect
58. #2230	C₂×C₅	direct (N × H)
59. C₂×#365	C₂×C₅	semidirect
60. C₂×#364	C₂×C₅	semidirect
76. C₅×Q₁₆	C₂×Q₈	semidirect
80. C₅×D₈	C₂×Q₈	semidirect
107. #498	C₂²×C₅	semidirect
115. #365	C₂²×C₅	semidirect
116. #364	C₂²×C₅	semidirect
151. #135	C₅×Q₈	semidirect
153. C₂×Q₁₆	C₅×Q₈	semidirect
155. C₂×D₈	C₅×Q₈	semidirect
206. Q₁₆	C₂×C₅×Q₈	semidirect
210. D₈	C₂×C₅×Q₈	semidirect
235. C₂×C₅	#2230	direct (N × H)
266. C₅	C₂×#2230	direct (N × H)
283. C₂	C₅×#2230	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×#498	2
C₂×C₅×#443	2
C₅×#2230	2
C₂×C₅×#365	2
C₂×C₅×#364	2
C₂×C₅×#360	2
C₂×#2230	5

Subgroups — C2×C5×#2230 #11956355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2230 #11956355