Subgroups — C₂×C₅×#2238 #11956374

Order 1280.

604 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#469	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#430	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2238	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#365	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#361	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#349	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
8	C₂²×C₅×M₁₆	—	2	1	no	—
9	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₅×#121	—	2	1	no	—
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₅×#469	—	2	2	no	—
13	C₅×#469	C₂²	1	4	yes	—
14	C₅×#430	—	2	2	no	—
15	C₅×#430	C₂²	1	4	yes	complemented
16	C₂×C₅×#108	—	2	1	no	—
17	C₂×C₅×#108	C₂²	1	2	yes	—
18	C₂×C₅×#106	C₂²	1	1	yes	characteristic
19	C₂×C₅×#102	—	2	1	no	—
20	C₅×#365	C₂²	1	8	yes	complemented
21	C₅×#361	C₂²	1	8	yes	complemented
22	C₅×#349	C₂²	1	4	yes	complemented
23	C₂×#2238	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×D₄	—	2	1	no	—
25	C₂²×C₅×D₄	—	4	1	no	—
26	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂×C₅×D₈	—	2	1	no	—
28	C₂×C₅×D₈	—	2	2	no	—
29	C₂×C₅×D₈	—	2	2	no	—
30	C₂×C₅×D₈	C₂³	1	2	yes	—
31	C₂×C₅×M₁₆	—	2	2	no	—
32	C₂×C₅×M₁₆	—	2	2	no	—
33	C₂×C₅×M₁₆	—	2	4	no	—
34	C₂×C₅×M₁₆	—	2	2	no	—
35	C₂×C₅×M₁₆	—	4	2	no	—
36	C₂×C₅×M₁₆	—	4	1	no	—
37	C₂×C₅×M₁₆	D₄	1	2	yes	—
38	C₂×C₅×M₁₆	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	—	4	1	no	—
40	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₅×#121	—	2	4	no	—
43	C₅×#121	—	4	2	no	—
44	C₂×C₅×C₄⋊C₄	—	2	1	no	—
45	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
46	C₅×M₃₂	—	2	2	no	—
47	C₅×M₃₂	C₂³	1	4	yes	—
48	C₂×C₅×C₁₆	—	2	2	no	—
49	C₅×#108	—	2	4	no	—
50	C₅×#108	—	2	2	no	—
51	C₅×#108	C₂³	1	4	yes	—
52	C₅×#106	—	2	1	no	—
53	C₅×#106	C₂³	1	2	yes	—
54	C₅×#102	—	2	4	no	—
55	C₂×#469	C₂×C₅	1	1	yes	characteristic
56	C₂×#430	C₂×C₅	1	1	yes	characteristic, complemented
57	#2238	C₂×C₅	1	8	yes	direct product factor
58	C₂×#365	C₂×C₅	1	2	yes	complemented
59	C₂×#361	C₂×C₅	1	2	yes	complemented
60	C₂×#349	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂⁴×C₅	—	4	1	no	—
62	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
63	C₂×C₅×D₄	—	2	2	no	—
64	C₂×C₅×D₄	—	4	4	no	—
65	C₂×C₅×D₄	—	4	1	no	—
66	C₂×C₅×D₄	—	4	1	no	—
67	C₂×C₅×D₄	—	4	1	no	—
68	C₂×C₅×D₄	—	4	1	no	—
69	C₂×C₅×D₄	—	4	1	no	—
70	C₂×C₅×D₄	—	8	4	no	—
71	C₂²×C₄×C₅	—	2	1	no	—
72	C₂²×C₄×C₅	—	2	1	no	—
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	—	2	1	no	—
75	C₂²×C₄×C₅	—	4	1	no	—
76	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
78	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₅×D₈	—	2	4	no	—
81	C₅×D₈	—	4	2	no	—
82	C₅×M₁₆	—	2	4	no	—
83	C₅×M₁₆	—	2	4	no	—
84	C₅×M₁₆	—	2	4	no	—
85	C₅×M₁₆	—	4	4	no	—
86	C₅×M₁₆	—	4	2	no	—
87	C₂×C₅×C₈	—	2	1	no	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	—	4	4	no	—
90	C₂×C₅×C₈	—	4	2	no	—
91	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
92	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₅×C₈	#41	1	2	yes	—
94	C₂×C₅×C₈	#41	1	2	yes	—
95	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
96	C₅×C₄⋊C₄	—	2	2	no	—
97	C₅×C₄⋊C₄	—	4	1	no	—
98	C₅×C₂²⋊C₄	—	4	4	no	—
99	C₅×C₁₆	—	2	4	no	—
100	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
101	C₂²×M₁₆	—	2	1	no	—
102	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
103	C₂×#121	—	2	1	no	—
104	C₂×M₃₂	C₂²×C₅	1	2	yes	—
105	#469	—	2	2	no	—
106	#469	C₂²×C₅	1	4	yes	—
107	#430	—	2	2	no	—
108	#430	C₂²×C₅	1	4	yes	complemented
109	C₂×#108	—	2	1	no	—
110	C₂×#108	C₂²×C₅	1	2	yes	—
111	C₂×#106	C₂²×C₅	1	1	yes	characteristic
112	C₂×#102	—	2	1	no	—
113	#365	C₂²×C₅	1	8	yes	complemented
114	#361	C₂²×C₅	1	8	yes	complemented
115	#349	C₂²×C₅	1	4	yes	complemented
116	C₂³×C₅	—	2	1	no	—
117	C₂³×C₅	—	2	1	no	—
118	C₂³×C₅	—	4	2	no	—
119	C₂³×C₅	—	4	2	no	—
120	C₂³×C₅	—	8	2	no	—
121	C₂³×C₅	—	8	2	no	—
122	C₂³×C₅	—	8	1	no	—
123	C₂³×C₅	—	8	1	no	—
124	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
125	C₂³×C₅	C₂×SD₁₆	1	2	yes	—
126	C₅×D₄	—	4	2	no	—
127	C₅×D₄	—	4	2	no	—
128	C₅×D₄	—	8	8	no	—
129	C₅×D₄	—	8	1	no	—
130	C₅×D₄	—	8	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	—	8	2	no	—
139	C₂×C₄×C₅	#123	1	1	yes	characteristic
140	C₂×C₄×C₅	#123	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
142	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₅×C₈	—	2	2	no	—
147	C₅×C₈	—	4	4	no	—
148	C₅×C₈	#123	1	2	yes	—
149	C₅×C₈	C₂×#41	1	2	yes	—
150	C₂²×D₄	—	2	1	no	—
151	C₂²×D₄	—	4	1	no	—
152	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
153	C₂×D₈	—	2	1	no	—
154	C₂×D₈	—	2	2	no	—
155	C₂×D₈	—	2	2	no	—
156	C₂×D₈	C₂³×C₅	1	2	yes	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	4	no	—
161	C₂×M₁₆	—	4	2	no	—
162	C₂×M₁₆	—	4	1	no	—
163	C₂×M₁₆	C₅×D₄	1	2	yes	—
164	C₂×M₁₆	C₂³×C₅	1	2	yes	—
165	C₂²×C₈	—	4	1	no	—
166	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
168	#121	—	2	4	no	—
169	#121	—	4	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₂²⋊C₄	—	4	1	no	—
172	M₃₂	—	2	2	no	—
173	M₃₂	C₂³×C₅	1	4	yes	—
174	C₂×C₁₆	—	2	2	no	—
175	#108	—	2	4	no	—
176	#108	—	2	2	no	—
177	#108	C₂³×C₅	1	4	yes	—
178	#106	—	2	1	no	—
179	#106	C₂³×C₅	1	2	yes	—
180	#102	—	2	4	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	4	2	no	—
184	C₂²×C₅	—	4	2	no	—
185	C₂²×C₅	—	8	1	no	—
186	C₂²×C₅	—	8	2	no	—
187	C₂²×C₅	—	8	2	no	—
188	C₂²×C₅	—	8	2	no	—
189	C₂²×C₅	—	16	2	no	—
190	C₂²×C₅	—	16	2	no	—
191	C₂²×C₅	#481	1	1	yes	characteristic
192	C₂²×C₅	#481	1	1	yes	characteristic
193	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	8	2	no	—
196	C₄×C₅	#481	1	1	yes	characteristic
197	C₄×C₅	#481	1	1	yes	characteristic
198	C₄×C₅	C₂×#123	1	1	yes	characteristic
199	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₂⁴	—	4	1	no	—
201	C₂⁴	C₅×SD₁₆	1	1	yes	characteristic
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	4	4	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	—	8	4	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
216	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
217	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
218	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
219	D₈	—	2	4	no	—
220	D₈	—	4	2	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	4	4	no	—
225	M₁₆	—	4	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	4	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
231	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂×C₈	C₅×#41	1	2	yes	—
233	C₂×C₈	C₅×#41	1	2	yes	—
234	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₁₆	—	2	4	no	—
239	C₂×C₅	—	2	1	no	—
240	C₂×C₅	—	2	1	no	—
241	C₂×C₅	—	4	2	no	—
242	C₂×C₅	—	16	2	no	—
243	C₂×C₅	#2238	1	2	yes	direct product factor
244	C₂×C₅	C₂×#481	1	1	yes	characteristic
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	8	2	no	—
250	C₂³	—	8	2	no	—
251	C₂³	—	8	1	no	—
252	C₂³	—	8	1	no	—
253	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
254	C₂³	C₂×C₅×SD₁₆	1	2	yes	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	D₄	—	8	8	no	—
258	D₄	—	8	1	no	—
259	D₄	—	8	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	C₅×#123	1	1	yes	characteristic
269	C₂×C₄	C₅×#123	1	1	yes	characteristic
270	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
271	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
272	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	4	4	no	—
277	C₈	C₅×#123	1	2	yes	—
278	C₈	C₂×C₅×#41	1	2	yes	—
279	C₅	C₂×#2238	1	1	yes	characteristic, direct product factor, Sylow
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	8	1	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	2	no	—
288	C₂²	—	16	2	no	—
289	C₂²	—	16	2	no	—
290	C₂²	C₅×#481	1	1	yes	characteristic
291	C₂²	C₅×#481	1	1	yes	characteristic
292	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	8	2	no	—
295	C₄	C₅×#481	1	1	yes	characteristic
296	C₄	C₅×#481	1	1	yes	characteristic
297	C₄	C₂×C₅×#123	1	1	yes	characteristic
298	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	4	2	no	—
302	C₂	—	16	2	no	—
303	C₂	C₅×#2238	1	2	yes	direct product factor
304	C₂	C₂×C₅×#481	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#469	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#430	C₂	semidirect
3. C₅×#2238	C₂	direct (N × H)
4. C₂×C₅×#365	C₂	semidirect
5. C₂×C₅×#361	C₂	semidirect
6. C₂×C₅×#349	C₂	semidirect
15. C₅×#430	C₂²	semidirect
20. C₅×#365	C₂²	semidirect
21. C₅×#361	C₂²	semidirect
22. C₅×#349	C₂²	semidirect
23. C₂×#2238	C₅	direct (N × H)
56. C₂×#430	C₂×C₅	semidirect
57. #2238	C₂×C₅	direct (N × H)
58. C₂×#365	C₂×C₅	semidirect
59. C₂×#361	C₂×C₅	semidirect
60. C₂×#349	C₂×C₅	semidirect
108. #430	C₂²×C₅	semidirect
113. #365	C₂²×C₅	semidirect
114. #361	C₂²×C₅	semidirect
115. #349	C₂²×C₅	semidirect
243. C₂×C₅	#2238	direct (N × H)
279. C₅	C₂×#2238	direct (N × H)
303. C₂	C₅×#2238	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×#469	2
C₂×C₅×#430	2
C₅×#2238	2
C₂×C₅×#365	2
C₂×C₅×#361	2
C₂×C₅×#349	2
C₂×#2238	5

Subgroups — C2×C5×#2238 #11956374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2238 #11956374