Subgroups — C₂×C₅×#3150 #11983355

Order 1280.

3326 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#135	C₂	1	4	yes	maximal, complemented
2	C₂×C₅×#532	C₂	1	2	yes	maximal, complemented
3	C₅×#3150	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅×#497	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#496	C₂	1	4	yes	maximal, complemented
6	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×#41	C₂²	1	4	yes	—
8	C₂×C₅×#135	—	2	32	no	—
9	C₂×C₅×#135	C₂²	1	32	yes	complemented
10	C₂²×C₅×Q₁₆	C₂²	1	4	yes	—
11	C₂²×C₅×SD₁₆	C₂²	1	8	yes	—
12	C₂²×C₅×D₈	C₂²	1	4	yes	—
13	C₂³×C₅×C₈	C₂²	1	2	yes	—
14	C₅×#532	—	2	8	no	—
15	C₅×#532	C₂²	1	16	yes	complemented
16	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₅×#127	C₂²	1	2	yes	—
18	C₂×C₅×#124	C₂²	1	4	yes	—
19	C₂×C₅×#121	—	2	4	no	—
20	C₂×C₄×C₅×D₄	—	2	4	no	—
21	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₅×#497	C₂²	1	32	yes	complemented
23	C₅×#496	C₂²	1	32	yes	complemented
24	C₂×C₅×#97	C₂²	1	4	yes	—
25	C₂×C₅×#97	C₂²	1	8	yes	complemented
26	C₂×#3150	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	—	2	8	no	—
28	C₂×C₅×#41	—	2	32	no	—
29	C₂×C₅×#41	C₂³	1	16	yes	—
30	C₂²×C₅×Q₈	C₂³	1	4	yes	—
31	C₂²×C₅×D₄	—	2	4	no	—
32	C₂²×C₅×D₄	—	2	4	no	—
33	C₂²×C₅×D₄	C₂³	1	4	yes	—
34	C₂³×C₄×C₅	—	2	4	no	—
35	C₂³×C₄×C₅	D₄	1	2	yes	—
36	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₅×#135	—	2	128	no	—
38	C₂×C₅×Q₁₆	—	2	16	no	—
39	C₂×C₅×Q₁₆	C₂³	1	16	yes	—
40	C₂×C₅×SD₁₆	—	2	32	no	—
41	C₂×C₅×SD₁₆	C₂³	1	32	yes	—
42	C₂×C₅×D₈	—	2	16	no	—
43	C₂×C₅×D₈	C₂³	1	16	yes	—
44	C₂²×C₅×C₈	D₄	1	16	yes	complemented
45	C₂²×C₅×C₈	C₂³	1	4	yes	—
46	C₂²×C₅×C₈	C₂³	1	8	yes	—
47	C₅×C₄⋊Q₈	—	2	4	no	—
48	C₅×C₄⋊Q₈	C₂³	1	8	yes	—
49	C₅×#127	—	2	4	no	—
50	C₅×#127	C₂³	1	8	yes	—
51	C₅×#124	—	2	8	no	—
52	C₅×#124	C₂³	1	16	yes	—
53	C₅×#121	—	2	32	no	—
54	C₄×C₅×D₄	—	2	32	no	—
55	C₂×C₅×C₄⋊C₄	—	2	4	no	—
56	C₂×C₅×C₂²⋊C₄	—	2	8	no	—
57	C₂×C₄²×C₅	D₄	1	4	yes	—
58	C₂×C₄²×C₅	D₄	1	2	yes	—
59	C₂×C₄²×C₅	C₂³	1	2	yes	—
60	C₂×C₄²×C₅	C₂³	1	4	yes	—
61	C₅×#97	C₂³	1	16	yes	—
62	C₂²×#135	C₂×C₅	1	4	yes	complemented
63	C₂×#532	C₂×C₅	1	2	yes	complemented
64	#3150	C₂×C₅	1	16	yes	direct product factor
65	C₂×#497	C₂×C₅	1	4	yes	complemented
66	C₂×#496	C₂×C₅	1	4	yes	complemented
67	C₂²×#97	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂⁴×C₅	—	2	4	no	—
69	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
70	C₅×#41	—	2	64	no	—
71	C₅×#41	—	4	32	no	—
72	C₂×C₅×Q₈	—	2	4	no	—
73	C₂×C₅×Q₈	—	2	16	no	—
74	C₂×C₅×Q₈	C₂⁴	1	8	yes	—
75	C₂×C₅×D₄	—	2	16	no	—
76	C₂×C₅×D₄	—	2	16	no	—
77	C₂×C₅×D₄	—	2	16	no	—
78	C₂×C₅×D₄	—	2	4	no	—
79	C₂×C₅×D₄	—	4	16	no	—
80	C₂×C₅×D₄	—	4	16	no	—
81	C₂×C₅×D₄	C₂⁴	1	8	yes	—
82	C₂²×C₄×C₅	—	2	4	no	—
83	C₂²×C₄×C₅	—	2	16	no	—
84	C₂²×C₄×C₅	—	4	4	no	—
85	C₂²×C₄×C₅	—	4	16	no	—
86	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
88	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
93	C₅×Q₁₆	—	2	32	no	—
94	C₅×SD₁₆	—	2	64	no	—
95	C₅×D₈	—	2	32	no	—
96	C₂×C₅×C₈	C₂×D₄	1	16	yes	—
97	C₂×C₅×C₈	C₂×D₄	1	32	yes	complemented
98	C₂×C₅×C₈	C₂⁴	1	8	yes	—
99	C₅×C₄⋊C₄	—	2	16	no	—
100	C₅×C₂²⋊C₄	—	2	32	no	—
101	C₄²×C₅	C₂×D₄	1	8	yes	—
102	C₄²×C₅	C₂×D₄	1	4	yes	—
103	C₄²×C₅	C₂⁴	1	4	yes	—
104	C₂²×#41	C₂²×C₅	1	4	yes	—
105	C₂×#135	—	2	32	no	—
106	C₂×#135	C₂²×C₅	1	32	yes	complemented
107	C₂²×Q₁₆	C₂²×C₅	1	4	yes	—
108	C₂²×SD₁₆	C₂²×C₅	1	8	yes	—
109	C₂²×D₈	C₂²×C₅	1	4	yes	—
110	C₂³×C₈	C₂²×C₅	1	2	yes	—
111	#532	—	2	8	no	—
112	#532	C₂²×C₅	1	16	yes	complemented
113	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
114	C₂×#127	C₂²×C₅	1	2	yes	—
115	C₂×#124	C₂²×C₅	1	4	yes	—
116	C₂×#121	—	2	4	no	—
117	C₂×C₄×D₄	—	2	4	no	—
118	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
119	#497	C₂²×C₅	1	32	yes	complemented
120	#496	C₂²×C₅	1	32	yes	complemented
121	C₂×#97	C₂²×C₅	1	4	yes	—
122	C₂×#97	C₂²×C₅	1	8	yes	complemented
123	C₂³×C₅	—	2	4	no	—
124	C₂³×C₅	—	2	8	no	—
125	C₂³×C₅	—	4	8	no	—
126	C₂³×C₅	—	4	16	no	—
127	C₂³×C₅	#127	1	4	yes	—
128	C₂³×C₅	C₂²×D₄	1	2	yes	—
129	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
130	C₅×Q₈	—	2	16	no	—
131	C₅×Q₈	—	4	8	no	—
132	C₅×D₄	—	2	16	no	—
133	C₅×D₄	—	4	32	no	—
134	C₅×D₄	—	4	32	no	—
135	C₅×D₄	—	4	8	no	—
136	C₂×C₄×C₅	—	2	8	no	—
137	C₂×C₄×C₅	—	2	8	no	—
138	C₂×C₄×C₅	—	4	8	no	—
139	C₂×C₄×C₅	—	4	8	no	—
140	C₂×C₄×C₅	—	4	32	no	—
141	C₂×C₄×C₅	#127	1	4	yes	—
142	C₂×C₄×C₅	#127	1	8	yes	—
143	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
144	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
145	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
147	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
148	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
149	C₅×C₈	C₂²×D₄	1	16	yes	—
150	C₂×#41	—	2	8	no	—
151	C₂×#41	—	2	32	no	—
152	C₂×#41	C₂³×C₅	1	16	yes	—
153	C₂²×Q₈	C₂³×C₅	1	4	yes	—
154	C₂²×D₄	—	2	4	no	—
155	C₂²×D₄	—	2	4	no	—
156	C₂²×D₄	C₂³×C₅	1	4	yes	—
157	C₂³×C₄	—	2	4	no	—
158	C₂³×C₄	C₅×D₄	1	2	yes	—
159	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
160	#135	—	2	128	no	—
161	C₂×Q₁₆	—	2	16	no	—
162	C₂×Q₁₆	C₂³×C₅	1	16	yes	—
163	C₂×SD₁₆	—	2	32	no	—
164	C₂×SD₁₆	C₂³×C₅	1	32	yes	—
165	C₂×D₈	—	2	16	no	—
166	C₂×D₈	C₂³×C₅	1	16	yes	—
167	C₂²×C₈	C₅×D₄	1	16	yes	complemented
168	C₂²×C₈	C₂³×C₅	1	4	yes	—
169	C₂²×C₈	C₂³×C₅	1	8	yes	—
170	C₄⋊Q₈	—	2	4	no	—
171	C₄⋊Q₈	C₂³×C₅	1	8	yes	—
172	#127	—	2	4	no	—
173	#127	C₂³×C₅	1	8	yes	—
174	#124	—	2	8	no	—
175	#124	C₂³×C₅	1	16	yes	—
176	#121	—	2	32	no	—
177	C₄×D₄	—	2	32	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₂²⋊C₄	—	2	8	no	—
180	C₂×C₄²	C₅×D₄	1	2	yes	—
181	C₂×C₄²	C₅×D₄	1	4	yes	—
182	C₂×C₄²	C₂³×C₅	1	2	yes	—
183	C₂×C₄²	C₂³×C₅	1	4	yes	—
184	#97	C₂³×C₅	1	16	yes	—
185	C₂²×C₅	—	2	8	no	—
186	C₂²×C₅	—	2	4	no	—
187	C₂²×C₅	—	4	8	no	—
188	C₂²×C₅	—	4	16	no	—
189	C₂²×C₅	—	8	16	no	—
190	C₂²×C₅	C₂×#127	1	4	yes	—
191	C₂²×C₅	C₂×#127	1	2	yes	—
192	C₂²×C₅	#575	1	4	yes	—
193	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
194	C₄×C₅	—	2	8	no	—
195	C₄×C₅	—	4	8	no	—
196	C₄×C₅	C₂×#127	1	4	yes	—
197	C₄×C₅	C₂×#127	1	2	yes	—
198	C₄×C₅	C₂³×D₄	1	2	yes	—
199	C₂⁴	—	2	4	no	—
200	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
201	#41	—	2	64	no	—
202	#41	—	4	32	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	2	16	no	—
205	C₂×Q₈	C₂⁴×C₅	1	8	yes	—
206	C₂×D₄	—	2	16	no	—
207	C₂×D₄	—	2	16	no	—
208	C₂×D₄	—	2	16	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	4	16	no	—
211	C₂×D₄	—	4	16	no	—
212	C₂×D₄	C₂⁴×C₅	1	8	yes	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	16	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	—	4	16	no	—
217	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
218	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
219	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
221	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
222	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
223	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
224	Q₁₆	—	2	32	no	—
225	SD₁₆	—	2	64	no	—
226	D₈	—	2	32	no	—
227	C₂×C₈	C₂×C₅×D₄	1	16	yes	—
228	C₂×C₈	C₂×C₅×D₄	1	32	yes	complemented
229	C₂×C₈	C₂⁴×C₅	1	8	yes	—
230	C₄⋊C₄	—	2	16	no	—
231	C₂²⋊C₄	—	2	32	no	—
232	C₄²	C₂×C₅×D₄	1	8	yes	—
233	C₄²	C₂×C₅×D₄	1	4	yes	—
234	C₄²	C₂⁴×C₅	1	4	yes	—
235	C₂×C₅	—	2	4	no	—
236	C₂×C₅	—	8	8	no	—
237	C₂×C₅	#3150	1	4	yes	direct product factor
238	C₂×C₅	C₂²×#127	1	1	yes	characteristic
239	C₂×C₅	C₂×#575	1	2	yes	—
240	C₂³	—	2	4	no	—
241	C₂³	—	2	8	no	—
242	C₂³	—	4	8	no	—
243	C₂³	—	4	16	no	—
244	C₂³	C₅×#127	1	4	yes	—
245	C₂³	C₂²×C₅×D₄	1	2	yes	—
246	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
247	Q₈	—	2	16	no	—
248	Q₈	—	4	8	no	—
249	D₄	—	2	16	no	—
250	D₄	—	4	32	no	—
251	D₄	—	4	32	no	—
252	D₄	—	4	8	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	4	32	no	—
258	C₂×C₄	C₅×#127	1	4	yes	—
259	C₂×C₄	C₅×#127	1	8	yes	—
260	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
261	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
262	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
264	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
265	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
266	C₈	C₂²×C₅×D₄	1	16	yes	—
267	C₅	C₂×#3150	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	—	2	8	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	4	8	no	—
271	C₂²	—	4	16	no	—
272	C₂²	—	8	16	no	—
273	C₂²	C₂×C₅×#127	1	4	yes	—
274	C₂²	C₂×C₅×#127	1	2	yes	—
275	C₂²	C₅×#575	1	4	yes	—
276	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
277	C₄	—	2	8	no	—
278	C₄	—	4	8	no	—
279	C₄	C₂×C₅×#127	1	4	yes	—
280	C₄	C₂×C₅×#127	1	2	yes	—
281	C₄	C₂³×C₅×D₄	1	2	yes	—
282	C₂	—	2	4	no	—
283	C₂	—	8	8	no	—
284	C₂	C₅×#3150	1	4	yes	direct product factor
285	C₂	C₂²×C₅×#127	1	1	yes	characteristic
286	C₂	C₂×C₅×#575	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
socle	C₂³×C₅	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#135	C₂	semidirect
2. C₂×C₅×#532	C₂	semidirect
3. C₅×#3150	C₂	direct (N × H)
4. C₂×C₅×#497	C₂	semidirect
5. C₂×C₅×#496	C₂	semidirect
6. C₂²×C₅×#97	C₂	semidirect
9. C₂×C₅×#135	C₂²	semidirect
15. C₅×#532	C₂²	semidirect
22. C₅×#497	C₂²	semidirect
23. C₅×#496	C₂²	semidirect
25. C₂×C₅×#97	C₂²	semidirect
26. C₂×#3150	C₅	direct (N × H)
44. C₂²×C₅×C₈	D₄	semidirect
62. C₂²×#135	C₂×C₅	semidirect
63. C₂×#532	C₂×C₅	semidirect
64. #3150	C₂×C₅	direct (N × H)
65. C₂×#497	C₂×C₅	semidirect
66. C₂×#496	C₂×C₅	semidirect
67. C₂²×#97	C₂×C₅	semidirect
97. C₂×C₅×C₈	C₂×D₄	semidirect
106. C₂×#135	C₂²×C₅	semidirect
112. #532	C₂²×C₅	semidirect
119. #497	C₂²×C₅	semidirect
120. #496	C₂²×C₅	semidirect
122. C₂×#97	C₂²×C₅	semidirect
167. C₂²×C₈	C₅×D₄	semidirect
228. C₂×C₈	C₂×C₅×D₄	semidirect
237. C₂×C₅	#3150	direct (N × H)
267. C₅	C₂×#3150	direct (N × H)
284. C₂	C₅×#3150	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅×#135	2
C₂×C₅×#532	2
C₅×#3150	2
C₂×C₅×#497	2
C₂×C₅×#496	2
C₂²×C₅×#97	2
C₂×#3150	5

Subgroups — C2×C5×#3150 #11983355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3150 #11983355