Subgroups — C₂×C₅×#2231 #11956356

Order 1280.

526 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#443	C₂	1	1	yes	maximal, characteristic
3	C₅×#2231	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×#125	—	2	1	no	—
10	C₂×C₅×M₃₂	C₂²	1	2	yes	—
11	C₅×#499	—	2	2	no	—
12	C₅×#499	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×#107	—	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×#2231	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×#41	—	2	1	no	—
25	C₅×#135	—	2	1	no	—
26	C₅×#135	—	2	1	no	—
27	C₅×#135	Q₈	1	2	yes	—
28	C₅×#135	C₂³	1	2	yes	—
29	C₂×C₅×Q₁₆	Q₈	1	1	yes	characteristic
30	C₂×C₅×SD₁₆	—	2	1	no	—
31	C₂×C₅×D₈	Q₈	1	1	yes	characteristic
32	C₅×#131	—	2	4	no	—
33	C₅×#131	—	4	2	no	—
34	C₂×C₅×M₁₆	—	2	1	no	—
35	C₂×C₅×M₁₆	—	4	1	no	—
36	C₂²×C₅×C₈	—	4	1	no	—
37	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₅×#125	—	2	4	no	—
40	C₅×#125	—	4	2	no	—
41	C₂×C₅×C₄⋊C₄	—	2	2	no	—
42	C₂×C₅×C₄⋊C₄	—	4	2	no	—
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₅×M₃₂	—	2	2	no	—
45	C₅×M₃₂	C₂³	1	4	yes	—
46	C₂×C₅×C₁₆	—	2	2	no	—
47	C₅×#108	—	2	1	no	—
48	C₅×#108	C₂³	1	2	yes	—
49	C₅×#107	—	2	4	no	—
50	C₅×#106	—	2	2	no	—
51	C₅×#106	C₂³	1	4	yes	—
52	C₅×#104	—	2	4	no	—
53	C₄×C₅×C₈	D₄	1	2	yes	—
54	C₄×C₅×C₈	C₂³	1	2	yes	—
55	C₂×#499	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂×#443	C₂×C₅	1	1	yes	characteristic
57	#2231	C₂×C₅	1	8	yes	direct product factor
58	C₂×#365	C₂×C₅	1	2	yes	complemented
59	C₂×#364	C₂×C₅	1	1	yes	characteristic, complemented
60	C₂×#360	C₂×C₅	1	2	yes	—
61	C₅×#41	—	2	2	no	—
62	C₅×#41	—	2	2	no	—
63	C₅×#41	—	4	1	no	—
64	C₅×#41	—	4	1	no	—
65	C₂×C₅×Q₈	—	2	1	no	—
66	C₂×C₅×D₄	—	2	1	no	—
67	C₂×C₅×D₄	—	4	1	no	—
68	C₂²×C₄×C₅	—	2	1	no	—
69	C₂²×C₄×C₅	—	4	1	no	—
70	C₂²×C₄×C₅	—	4	2	no	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₅×Q₁₆	—	2	1	no	—
73	C₅×Q₁₆	C₂×Q₈	1	2	yes	—
74	C₅×SD₁₆	—	2	2	no	—
75	C₅×SD₁₆	—	2	2	no	—
76	C₅×D₈	—	2	1	no	—
77	C₅×D₈	C₂×Q₈	1	2	yes	—
78	C₅×M₁₆	—	2	2	no	—
79	C₅×M₁₆	—	4	1	no	—
80	C₅×M₁₆	—	4	4	no	—
81	C₂×C₅×C₈	—	4	1	no	—
82	C₂×C₅×C₈	—	4	4	no	—
83	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
84	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
85	C₂×C₅×C₈	SD₁₆	1	2	yes	—
86	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
88	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
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	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₄⋊C₄	—	4	8	no	—
95	C₅×C₄⋊C₄	—	4	2	no	—
96	C₄²×C₅	—	2	1	no	—
97	C₄²×C₅	C₂×D₄	1	2	yes	—
98	C₅×C₁₆	—	2	4	no	—
99	C₂×#135	C₂²×C₅	1	1	yes	characteristic
100	C₂×#131	—	2	1	no	—
101	C₂×#125	—	2	1	no	—
102	C₂×M₃₂	C₂²×C₅	1	2	yes	—
103	#499	—	2	2	no	—
104	#499	C₂²×C₅	1	4	yes	complemented
105	#443	—	2	2	no	—
106	#443	C₂²×C₅	1	4	yes	—
107	C₂×#108	C₂²×C₅	1	1	yes	characteristic
108	C₂×#107	—	2	1	no	—
109	C₂×#106	C₂²×C₅	1	2	yes	—
110	C₂×#104	—	2	1	no	—
111	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
112	#365	C₂²×C₅	1	8	yes	complemented
113	#364	C₂²×C₅	1	4	yes	complemented
114	#360	C₂²×C₅	1	8	yes	—
115	C₂³×C₅	—	4	1	no	—
116	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₅×Q₈	—	2	2	no	—
118	C₅×Q₈	—	4	1	no	—
119	C₅×D₄	—	2	2	no	—
120	C₅×D₄	—	4	2	no	—
121	C₅×D₄	—	4	2	no	—
122	C₅×D₄	—	4	1	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	4	2	no	—
125	C₂×C₄×C₅	—	4	1	no	—
126	C₂×C₄×C₅	—	4	2	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	—	8	4	no	—
130	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₅×C₈	—	4	2	no	—
137	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
138	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
139	C₅×C₈	C₂×SD₁₆	1	2	yes	—
140	C₅×C₈	C₂×SD₁₆	1	2	yes	—
141	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
142	C₅×C₈	C₂×#41	1	1	yes	characteristic
143	C₂×#41	—	2	1	no	—
144	#135	—	2	1	no	—
145	#135	—	2	1	no	—
146	#135	C₅×Q₈	1	2	yes	—
147	#135	C₂³×C₅	1	2	yes	—
148	C₂×Q₁₆	C₅×Q₈	1	1	yes	characteristic
149	C₂×SD₁₆	—	2	1	no	—
150	C₂×D₈	C₅×Q₈	1	1	yes	characteristic
151	#131	—	2	4	no	—
152	#131	—	4	2	no	—
153	C₂×M₁₆	—	2	1	no	—
154	C₂×M₁₆	—	4	1	no	—
155	C₂²×C₈	—	4	1	no	—
156	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
158	#125	—	2	4	no	—
159	#125	—	4	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	4	2	no	—
162	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
163	M₃₂	—	2	2	no	—
164	M₃₂	C₂³×C₅	1	4	yes	—
165	C₂×C₁₆	—	2	2	no	—
166	#108	—	2	1	no	—
167	#108	C₂³×C₅	1	2	yes	—
168	#107	—	2	4	no	—
169	#106	—	2	2	no	—
170	#106	C₂³×C₅	1	4	yes	—
171	#104	—	2	4	no	—
172	C₄×C₈	C₅×D₄	1	2	yes	—
173	C₄×C₈	C₂³×C₅	1	2	yes	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	4	2	no	—
176	C₂²×C₅	—	8	2	no	—
177	C₂²×C₅	#476	1	1	yes	characteristic
178	C₂²×C₅	#476	1	1	yes	characteristic
179	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₄×C₅	—	4	2	no	—
181	C₄×C₅	—	4	2	no	—
182	C₄×C₅	—	8	4	no	—
183	C₄×C₅	#476	1	1	yes	characteristic
184	C₄×C₅	#476	1	1	yes	characteristic
185	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
186	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	#41	—	4	1	no	—
190	#41	—	4	1	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
198	Q₁₆	—	2	1	no	—
199	Q₁₆	C₂×C₅×Q₈	1	2	yes	—
200	SD₁₆	—	2	2	no	—
201	SD₁₆	—	2	2	no	—
202	D₈	—	2	1	no	—
203	D₈	C₂×C₅×Q₈	1	2	yes	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	4	1	no	—
206	M₁₆	—	4	4	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	4	no	—
209	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
210	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
211	C₂×C₈	C₅×SD₁₆	1	2	yes	—
212	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
214	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
215	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
216	C₂×C₈	C₅×#41	1	1	yes	characteristic
217	C₂×C₈	C₅×#41	1	1	yes	characteristic
218	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	8	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	C₂×C₅×D₄	1	2	yes	—
224	C₁₆	—	2	4	no	—
225	C₂×C₅	—	2	1	no	—
226	C₂×C₅	—	2	1	no	—
227	C₂×C₅	—	8	2	no	—
228	C₂×C₅	#2231	1	2	yes	direct product factor
229	C₂×C₅	C₂×#476	1	1	yes	characteristic
230	C₂³	—	4	1	no	—
231	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	Q₈	—	4	1	no	—
234	D₄	—	2	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
246	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
248	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
249	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
250	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
251	C₈	—	4	2	no	—
252	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
253	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
254	C₈	C₂×C₅×SD₁₆	1	2	yes	—
255	C₈	C₂×C₅×SD₁₆	1	2	yes	—
256	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
257	C₈	C₂×C₅×#41	1	1	yes	characteristic
258	C₅	C₂×#2231	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	C₅×#476	1	1	yes	characteristic
263	C₂²	C₅×#476	1	1	yes	characteristic
264	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	8	4	no	—
268	C₄	C₅×#476	1	1	yes	characteristic
269	C₄	C₅×#476	1	1	yes	characteristic
270	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
271	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	8	2	no	—
275	C₂	C₅×#2231	1	2	yes	direct product factor
276	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×#499	C₂	semidirect
3. C₅×#2231	C₂	direct (N × H)
4. C₂×C₅×#365	C₂	semidirect
5. C₂×C₅×#364	C₂	semidirect
12. C₅×#499	C₂²	semidirect
20. C₅×#365	C₂²	semidirect
21. C₅×#364	C₂²	semidirect
23. C₂×#2231	C₅	direct (N × H)
55. C₂×#499	C₂×C₅	semidirect
57. #2231	C₂×C₅	direct (N × H)
58. C₂×#365	C₂×C₅	semidirect
59. C₂×#364	C₂×C₅	semidirect
104. #499	C₂²×C₅	semidirect
112. #365	C₂²×C₅	semidirect
113. #364	C₂²×C₅	semidirect
228. C₂×C₅	#2231	direct (N × H)
258. C₅	C₂×#2231	direct (N × H)
275. C₂	C₅×#2231	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×#499	2
C₂×C₅×#443	2
C₅×#2231	2
C₂×C₅×#365	2
C₂×C₅×#364	2
C₂×C₅×#360	2
C₂×#2231	5

Subgroups — C2×C5×#2231 #11956356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2231 #11956356