Subgroups — C₂×C₅×#2233 #11956363

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#466	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2233	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#367	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#357	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#121	—	2	1	no	—
10	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
11	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
12	C₅×#466	—	2	2	no	—
13	C₅×#466	C₂²	1	4	yes	—
14	C₂×C₅×#107	—	2	1	no	—
15	C₂×C₅×#107	—	2	2	no	—
16	C₂×C₅×#107	C₂²	1	1	yes	characteristic
17	C₂×C₅×#107	C₂²	1	2	yes	—
18	C₂×C₅×#107	C₂²	1	4	yes	complemented
19	C₂×C₅×#102	—	2	1	no	—
20	C₅×#367	C₂²	1	8	yes	complemented
21	C₅×#357	C₂²	1	8	yes	complemented
22	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
23	C₂×#2233	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	—	4	1	no	—
27	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂×C₅×D₈	—	2	1	no	—
29	C₂×C₅×D₈	—	2	4	no	—
30	C₂×C₅×D₈	C₂³	1	2	yes	—
31	C₂²×C₅×C₈	—	2	4	no	—
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	D₄	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×#121	—	2	4	no	—
37	C₅×#121	—	4	2	no	—
38	C₂×C₅×C₄⋊C₄	—	2	1	no	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₄⋊C₄	—	2	4	no	—
41	C₂×C₅×C₄⋊C₄	—	4	2	no	—
42	C₂×C₅×C₄⋊C₄	—	4	1	no	—
43	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
44	C₂×C₅×C₁₆	—	2	4	no	—
45	C₂×C₅×C₁₆	C₂³	1	4	yes	—
46	C₅×#107	—	2	2	no	—
47	C₅×#107	—	2	4	no	—
48	C₅×#107	—	2	1	no	—
49	C₅×#107	C₂³	1	4	yes	—
50	C₅×#107	C₂³	1	2	yes	—
51	C₅×#102	—	2	4	no	—
52	C₂×#466	C₂×C₅	1	1	yes	characteristic
53	C₂²×#107	C₂×C₅	1	1	yes	characteristic, complemented
54	#2233	C₂×C₅	1	8	yes	direct product factor
55	C₂×#367	C₂×C₅	1	2	yes	complemented
56	C₂×#357	C₂×C₅	1	2	yes	complemented
57	C₂×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂⁴×C₅	—	4	1	no	—
59	C₂⁴×C₅	D₈	1	1	yes	characteristic
60	C₂×C₅×D₄	—	2	2	no	—
61	C₂×C₅×D₄	—	4	4	no	—
62	C₂×C₅×D₄	—	4	2	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂×C₅×D₄	—	4	2	no	—
65	C₂×C₅×D₄	—	8	4	no	—
66	C₂²×C₄×C₅	—	2	2	no	—
67	C₂²×C₄×C₅	—	2	2	no	—
68	C₂²×C₄×C₅	—	4	4	no	—
69	C₂²×C₄×C₅	—	4	1	no	—
70	C₂²×C₄×C₅	—	4	2	no	—
71	C₂²×C₄×C₅	—	8	4	no	—
72	C₂²×C₄×C₅	D₈	1	2	yes	—
73	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₅×D₈	—	2	4	no	—
77	C₅×D₈	—	4	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
82	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×C₈	#41	1	4	yes	—
84	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
85	C₅×C₄⋊C₄	—	2	4	no	—
86	C₅×C₄⋊C₄	—	2	2	no	—
87	C₅×C₄⋊C₄	—	4	4	no	—
88	C₅×C₄⋊C₄	—	4	2	no	—
89	C₅×C₄⋊C₄	—	4	1	no	—
90	C₅×C₂²⋊C₄	—	4	4	no	—
91	C₅×C₁₆	—	2	4	no	—
92	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
93	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
94	C₂×#121	—	2	1	no	—
95	C₂²×C₄⋊C₄	—	2	1	no	—
96	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
97	#466	—	2	2	no	—
98	#466	C₂²×C₅	1	4	yes	—
99	C₂×#107	—	2	2	no	—
100	C₂×#107	—	2	1	no	—
101	C₂×#107	C₂²×C₅	1	2	yes	—
102	C₂×#107	C₂²×C₅	1	1	yes	characteristic
103	C₂×#107	C₂²×C₅	1	4	yes	complemented
104	C₂×#102	—	2	1	no	—
105	#367	C₂²×C₅	1	8	yes	complemented
106	#357	C₂²×C₅	1	8	yes	complemented
107	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	complemented
108	C₂³×C₅	—	2	2	no	—
109	C₂³×C₅	—	2	2	no	—
110	C₂³×C₅	—	4	2	no	—
111	C₂³×C₅	—	8	2	no	—
112	C₂³×C₅	—	8	2	no	—
113	C₂³×C₅	—	8	2	no	—
114	C₂³×C₅	SD₃₂	1	4	yes	—
115	C₂³×C₅	C₂×D₈	1	2	yes	—
116	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
117	C₅×D₄	—	4	2	no	—
118	C₅×D₄	—	4	2	no	—
119	C₅×D₄	—	8	8	no	—
120	C₅×D₄	—	8	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	—	8	8	no	—
128	C₂×C₄×C₅	—	8	2	no	—
129	C₂×C₄×C₅	—	8	4	no	—
130	C₂×C₄×C₅	#123	1	2	yes	—
131	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅	#137	1	2	yes	—
134	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
135	C₅×C₈	—	2	2	no	—
136	C₅×C₈	#123	1	2	yes	—
137	C₅×C₈	C₂×#41	1	2	yes	—
138	C₂²×D₄	—	2	1	no	—
139	C₂²×D₄	—	4	1	no	—
140	C₂³×C₄	—	4	1	no	—
141	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
142	C₂×D₈	—	2	1	no	—
143	C₂×D₈	—	2	4	no	—
144	C₂×D₈	C₂³×C₅	1	2	yes	—
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
147	C₂²×C₈	C₅×D₄	1	2	yes	—
148	C₂²×C₈	C₂³×C₅	1	2	yes	—
149	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
150	#121	—	2	4	no	—
151	#121	—	4	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	4	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	4	1	no	—
156	C₂×C₄⋊C₄	—	4	2	no	—
157	C₂×C₂²⋊C₄	—	4	1	no	—
158	C₂×C₁₆	—	2	4	no	—
159	C₂×C₁₆	C₂³×C₅	1	4	yes	—
160	#107	—	2	2	no	—
161	#107	—	2	4	no	—
162	#107	—	2	1	no	—
163	#107	C₂³×C₅	1	4	yes	—
164	#107	C₂³×C₅	1	2	yes	—
165	#102	—	2	4	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	—	2	8	no	—
169	C₂²×C₅	—	8	2	no	—
170	C₂²×C₅	—	8	2	no	—
171	C₂²×C₅	—	8	2	no	—
172	C₂²×C₅	—	16	4	no	—
173	C₂²×C₅	#480	1	2	yes	—
174	C₂²×C₅	C₂×SD₃₂	1	4	yes	—
175	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
176	C₂²×C₅	#509	1	2	yes	—
177	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	—	8	2	no	—
180	C₄×C₅	—	8	4	no	—
181	C₄×C₅	#480	1	2	yes	—
182	C₄×C₅	C₂×#123	1	1	yes	characteristic
183	C₄×C₅	C₂×#137	1	1	yes	characteristic
184	C₂⁴	—	4	1	no	—
185	C₂⁴	C₅×D₈	1	1	yes	characteristic
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	4	4	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	8	4	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	8	4	no	—
198	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
199	C₂²×C₄	C₅×D₈	1	2	yes	—
200	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
201	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
202	D₈	—	2	4	no	—
203	D₈	—	4	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
208	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂×C₈	C₅×#41	1	4	yes	—
210	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₂²⋊C₄	—	4	4	no	—
217	C₁₆	—	2	4	no	—
218	C₂×C₅	—	2	4	no	—
219	C₂×C₅	—	16	2	no	—
220	C₂×C₅	#2233	1	4	yes	direct product factor
221	C₂×C₅	C₂×#480	1	1	yes	characteristic
222	C₂×C₅	C₂²×SD₃₂	1	1	yes	characteristic
223	C₂×C₅	C₂×#509	1	1	yes	characteristic
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	C₅×SD₃₂	1	4	yes	—
231	C₂³	C₂×C₅×D₈	1	2	yes	—
232	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	8	no	—
236	D₄	—	8	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	8	8	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	C₅×#123	1	2	yes	—
247	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
248	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
249	C₂×C₄	C₅×#137	1	2	yes	—
250	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	C₅×#123	1	2	yes	—
253	C₈	C₂×C₅×#41	1	2	yes	—
254	C₅	C₂×#2233	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	16	4	no	—
262	C₂²	C₅×#480	1	2	yes	—
263	C₂²	C₂×C₅×SD₃₂	1	4	yes	—
264	C₂²	C₂×C₅×SD₃₂	1	2	yes	—
265	C₂²	C₅×#509	1	2	yes	—
266	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	8	4	no	—
270	C₄	C₅×#480	1	2	yes	—
271	C₄	C₂×C₅×#123	1	1	yes	characteristic
272	C₄	C₂×C₅×#137	1	1	yes	characteristic
273	C₂	—	2	4	no	—
274	C₂	—	16	2	no	—
275	C₂	C₅×#2233	1	4	yes	direct product factor
276	C₂	C₂×C₅×#480	1	1	yes	characteristic
277	C₂	C₂²×C₅×SD₃₂	1	1	yes	characteristic
278	C₂	C₂×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×#107	C₂	semidirect
3. C₅×#2233	C₂	direct (N × H)
4. C₂×C₅×#367	C₂	semidirect
5. C₂×C₅×#357	C₂	semidirect
6. C₂×C₅×C₂²⋊C₁₆	C₂	semidirect
18. C₂×C₅×#107	C₂²	semidirect
20. C₅×#367	C₂²	semidirect
21. C₅×#357	C₂²	semidirect
22. C₅×C₂²⋊C₁₆	C₂²	semidirect
23. C₂×#2233	C₅	direct (N × H)
53. C₂²×#107	C₂×C₅	semidirect
54. #2233	C₂×C₅	direct (N × H)
55. C₂×#367	C₂×C₅	semidirect
56. C₂×#357	C₂×C₅	semidirect
57. C₂×C₂²⋊C₁₆	C₂×C₅	semidirect
103. C₂×#107	C₂²×C₅	semidirect
105. #367	C₂²×C₅	semidirect
106. #357	C₂²×C₅	semidirect
107. C₂²⋊C₁₆	C₂²×C₅	semidirect
220. C₂×C₅	#2233	direct (N × H)
254. C₅	C₂×#2233	direct (N × H)
275. C₂	C₅×#2233	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×#466	2
C₂²×C₅×#107	2
C₅×#2233	2
C₂×C₅×#367	2
C₂×C₅×#357	2
C₂×C₅×C₂²⋊C₁₆	2
C₂×#2233	5

Subgroups — C2×C5×#2233 #11956363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2233 #11956363