Subgroups — C₅×#5674 #11756059

Order 960.

668 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	C₂×C₅×#239	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×D₃×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#947	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#940	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#884	C₂	1	4	yes	maximal, complemented
7	C₅×#882	C₂	1	4	yes	maximal, complemented
8	C₅×#874	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#872	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#535	—	3	1	no	maximal, Hall
11	C₂³×C₅×D₃	C₂²	1	2	yes	—
12	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₅×#67	C₂²	1	2	yes	—
15	C₂×C₅×#67	C₂²	1	4	yes	complemented
16	C₅×#239	—	2	4	no	—
17	C₅×D₃×D₄	—	2	4	no	—
18	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
19	C₂×C₅×D₁₂	C₂²	1	2	yes	—
20	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₅×D₁₂	C₂²	1	4	yes	complemented
22	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₅×D₃	C₂²	1	2	yes	complemented
24	C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
25	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
26	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	complemented
28	C₅×Dic₃⋊C₄	C₂²	1	2	yes	complemented
29	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
30	#5674	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₂×C₅×#41	—	3	1	no	—
32	C₂²×C₅×D₄	—	3	1	no	—
33	C₅×#127	—	3	1	no	—
34	C₅×#127	—	3	1	no	—
35	C₅×#124	—	3	1	no	—
36	C₅×#124	D₃	1	1	yes	characteristic, complemented
37	C₅×#121	—	3	4	no	—
38	C₅×#120	—	3	4	no	—
39	C₅×#117	—	3	1	no	—
40	C₂³×C₃×C₅	C₂³	1	2	yes	—
41	C₂²×C₅×D₃	—	2	4	no	—
42	C₂²×C₅×D₃	—	4	4	no	—
43	C₂²×C₅×D₃	C₂³	1	2	yes	—
44	C₂²×C₅×D₃	C₂³	1	2	yes	—
45	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
46	C₃×C₅×Q₈	—	2	2	no	—
47	C₃×C₅×D₄	—	2	2	no	—
48	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₅×#67	—	2	4	no	—
52	C₅×#67	—	4	4	no	—
53	C₂×C₅×Dic₃	C₂³	1	2	yes	—
54	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
55	C₅×D₁₂	—	2	4	no	—
56	C₅×D₁₂	—	2	2	no	—
57	C₅×D₁₂	—	2	2	no	—
58	C₅×D₁₂	—	4	4	no	—
59	C₄×C₅×D₃	—	2	4	no	—
60	C₄×C₅×D₃	D₄	1	4	yes	complemented
61	C₂×#239	C₂×C₅	1	1	yes	characteristic, complemented
62	C₂×D₃×D₄	C₂×C₅	1	1	yes	characteristic, complemented
63	C₃×#124	C₂×C₅	1	1	yes	characteristic, complemented
64	#947	C₂×C₅	1	1	yes	characteristic, complemented
65	#940	C₂×C₅	1	1	yes	characteristic, complemented
66	#884	C₂×C₅	1	4	yes	complemented
67	#882	C₂×C₅	1	4	yes	complemented
68	#874	C₂×C₅	1	1	yes	characteristic, complemented
69	#872	C₂×C₅	1	1	yes	characteristic, complemented
70	C₂⁴×C₅	—	3	2	no	—
71	C₅×#41	—	6	4	no	—
72	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
73	C₂×C₅×D₄	—	3	2	no	—
74	C₂×C₅×D₄	—	3	4	no	—
75	C₂×C₅×D₄	—	3	2	no	—
76	C₂×C₅×D₄	—	3	1	no	—
77	C₂×C₅×D₄	—	3	1	no	—
78	C₂×C₅×D₄	—	3	4	no	—
79	C₂×C₅×D₄	—	6	4	no	—
80	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
81	C₂²×C₄×C₅	—	3	1	no	—
82	C₂²×C₄×C₅	—	3	2	no	—
83	C₅×C₄⋊C₄	—	3	2	no	—
84	C₅×C₂²⋊C₄	—	3	2	no	—
85	C₅×C₂²⋊C₄	—	3	4	no	—
86	C₅×C₂²⋊C₄	C₂×D₃	1	4	yes	complemented
87	C₄²×C₅	—	3	1	no	—
88	C₄²×C₅	C₂×D₃	1	1	yes	characteristic, complemented
89	#535	—	3	1	no	Sylow
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	2	4	no	—
92	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
93	C₂×C₅×D₃	—	2	2	no	—
94	C₂×C₅×D₃	—	2	2	no	—
95	C₂×C₅×D₃	—	2	2	no	—
96	C₂×C₅×D₃	—	2	4	no	—
97	C₂×C₅×D₃	—	2	4	no	—
98	C₂×C₅×D₃	—	4	8	no	—
99	C₂×C₅×D₃	C₂×D₄	1	2	yes	—
100	C₃×C₄×C₅	—	2	2	no	—
101	C₃×C₄×C₅	—	2	2	no	—
102	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
103	C₅×Dic₃	—	2	2	no	—
104	C₅×Dic₃	C₂×D₄	1	2	yes	—
105	C₂³×D₃	C₂²×C₅	1	2	yes	—
106	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
107	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
108	C₂×#67	C₂²×C₅	1	2	yes	—
109	C₂×#67	C₂²×C₅	1	4	yes	complemented
110	#239	—	2	4	no	—
111	D₃×D₄	—	2	4	no	—
112	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
113	C₂×D₁₂	C₂²×C₅	1	2	yes	—
114	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic, complemented
115	C₂×D₁₂	C₂²×C₅	1	4	yes	complemented
116	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₄×D₃	C₂²×C₅	1	2	yes	complemented
118	C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
119	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic, complemented
120	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
121	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	complemented
122	Dic₃⋊C₄	C₂²×C₅	1	2	yes	complemented
123	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
124	C₂³×C₅	—	3	1	no	—
125	C₂³×C₅	—	3	2	no	—
126	C₂³×C₅	—	3	2	no	—
127	C₂³×C₅	—	6	4	no	—
128	C₂³×C₅	—	12	4	no	—
129	C₂³×C₅	C₂²×D₃	1	2	yes	—
130	C₅×Q₈	—	2	2	no	—
131	C₅×D₄	—	2	2	no	—
132	C₅×D₄	—	6	4	no	—
133	C₅×D₄	—	6	4	no	—
134	C₅×D₄	—	6	2	no	—
135	C₅×D₄	—	6	2	no	—
136	C₅×D₄	—	12	4	no	—
137	C₅×D₄	—	12	4	no	—
138	C₂×C₄×C₅	—	3	1	no	—
139	C₂×C₄×C₅	—	3	4	no	—
140	C₂×C₄×C₅	—	3	2	no	—
141	C₂×C₄×C₅	—	6	4	no	—
142	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
143	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
144	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
145	C₂×#41	—	3	1	no	—
146	C₂²×D₄	—	3	1	no	—
147	#127	—	3	1	no	—
148	#127	—	3	1	no	—
149	#124	—	3	1	no	—
150	#124	C₅×D₃	1	1	yes	characteristic, complemented
151	#121	—	3	4	no	—
152	#120	—	3	4	no	—
153	#117	—	3	1	no	—
154	C₂×C₃×C₅	—	4	2	no	—
155	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
157	C₅×D₃	—	2	2	no	—
158	C₅×D₃	—	4	2	no	—
159	C₅×D₃	—	4	2	no	—
160	C₂³×C₃	C₂³×C₅	1	2	yes	—
161	C₂²×D₃	—	2	4	no	—
162	C₂²×D₃	—	4	4	no	—
163	C₂²×D₃	C₂³×C₅	1	2	yes	—
164	C₂²×D₃	C₂³×C₅	1	2	yes	—
165	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
166	C₃×Q₈	—	2	2	no	—
167	C₃×D₄	—	2	2	no	—
168	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
169	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
170	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
171	#67	—	2	4	no	—
172	#67	—	4	4	no	—
173	C₂×Dic₃	C₂³×C₅	1	2	yes	—
174	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
175	D₁₂	—	2	4	no	—
176	D₁₂	—	2	2	no	—
177	D₁₂	—	2	2	no	—
178	D₁₂	—	4	4	no	—
179	C₄×D₃	—	2	4	no	—
180	C₄×D₃	C₅×D₄	1	4	yes	complemented
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	2	4	no	—
183	C₂²×C₅	—	3	2	no	—
184	C₂²×C₅	—	6	2	no	—
185	C₂²×C₅	—	6	2	no	—
186	C₂²×C₅	—	6	2	no	—
187	C₂²×C₅	—	6	4	no	—
188	C₂²×C₅	—	6	4	no	—
189	C₂²×C₅	—	12	8	no	—
190	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	3	2	no	—
194	C₄×C₅	—	6	2	no	—
195	C₄×C₅	D₃×D₄	1	2	yes	—
196	C₂⁴	—	3	2	no	—
197	#41	—	6	4	no	—
198	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
199	C₂×D₄	—	3	2	no	—
200	C₂×D₄	—	3	4	no	—
201	C₂×D₄	—	3	2	no	—
202	C₂×D₄	—	3	1	no	—
203	C₂×D₄	—	3	1	no	—
204	C₂×D₄	—	3	4	no	—
205	C₂×D₄	—	6	4	no	—
206	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
207	C₂²×C₄	—	3	1	no	—
208	C₂²×C₄	—	3	2	no	—
209	C₄⋊C₄	—	3	2	no	—
210	C₂²⋊C₄	—	3	2	no	—
211	C₂²⋊C₄	—	3	4	no	—
212	C₂²⋊C₄	C₂×C₅×D₃	1	4	yes	complemented
213	C₄²	—	3	1	no	—
214	C₄²	C₂×C₅×D₃	1	1	yes	characteristic, complemented
215	C₃×C₅	#535	1	1	yes	characteristic, complemented, Hall
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	4	no	—
218	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
219	C₂×D₃	—	2	2	no	—
220	C₂×D₃	—	2	2	no	—
221	C₂×D₃	—	2	2	no	—
222	C₂×D₃	—	2	4	no	—
223	C₂×D₃	—	2	4	no	—
224	C₂×D₃	—	4	8	no	—
225	C₂×D₃	C₂×C₅×D₄	1	2	yes	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
229	Dic₃	—	2	2	no	—
230	Dic₃	C₂×C₅×D₄	1	2	yes	—
231	C₂×C₅	—	4	2	no	—
232	C₂×C₅	—	6	2	no	—
233	C₂×C₅	—	12	2	no	—
234	C₂×C₅	—	12	2	no	—
235	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
236	C₂×C₅	#1009	1	2	yes	—
237	C₂³	—	3	1	no	—
238	C₂³	—	3	2	no	—
239	C₂³	—	3	2	no	—
240	C₂³	—	6	4	no	—
241	C₂³	—	12	4	no	—
242	C₂³	C₂²×C₅×D₃	1	2	yes	—
243	Q₈	—	2	2	no	—
244	D₄	—	2	2	no	—
245	D₄	—	6	4	no	—
246	D₄	—	6	4	no	—
247	D₄	—	6	2	no	—
248	D₄	—	6	2	no	—
249	D₄	—	12	4	no	—
250	D₄	—	12	4	no	—
251	C₂×C₄	—	3	1	no	—
252	C₂×C₄	—	3	4	no	—
253	C₂×C₄	—	3	2	no	—
254	C₂×C₄	—	6	4	no	—
255	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
256	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
257	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
258	C₂×C₃	—	4	2	no	—
259	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
260	C₂×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
261	D₃	—	2	2	no	—
262	D₃	—	4	2	no	—
263	D₃	—	4	2	no	—
264	C₅	#5674	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	3	2	no	—
268	C₂²	—	6	2	no	—
269	C₂²	—	6	2	no	—
270	C₂²	—	6	2	no	—
271	C₂²	—	6	4	no	—
272	C₂²	—	6	4	no	—
273	C₂²	—	12	8	no	—
274	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	3	2	no	—
278	C₄	—	6	2	no	—
279	C₄	C₅×D₃×D₄	1	2	yes	—
280	C₃	C₅×#535	1	1	yes	characteristic, complemented, Sylow
281	C₂	—	4	2	no	—
282	C₂	—	6	2	no	—
283	C₂	—	12	2	no	—
284	C₂	—	12	2	no	—
285	C₂	C₂×C₅×D₃×D₄	1	1	yes	characteristic
286	C₂	C₅×#1009	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×#124	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₅×#124	D₃
hypocenter	C₃	C₅×#535
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#124

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#239	C₂	semidirect
2. C₂×C₅×D₃×D₄	C₂	semidirect
3. C₃×C₅×#124	C₂	semidirect
4. C₅×#947	C₂	semidirect
5. C₅×#940	C₂	semidirect
6. C₅×#884	C₂	semidirect
7. C₅×#882	C₂	semidirect
8. C₅×#874	C₂	semidirect
9. C₅×#872	C₂	semidirect
12. C₂×C₃×C₅×Q₈	C₂²	semidirect
15. C₂×C₅×#67	C₂²	semidirect
20. C₂×C₅×D₁₂	C₂²	semidirect
21. C₂×C₅×D₁₂	C₂²	semidirect
23. C₂×C₄×C₅×D₃	C₂²	semidirect
24. C₃×C₅×C₂²⋊C₄	C₂²	semidirect
25. C₃×C₄²×C₅	C₂²	semidirect
27. C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
28. C₅×Dic₃⋊C₄	C₂²	semidirect
29. C₄×C₅×Dic₃	C₂²	semidirect
30. #5674	C₅	direct (N × H)
36. C₅×#124	D₃	semidirect
60. C₄×C₅×D₃	D₄	semidirect
61. C₂×#239	C₂×C₅	semidirect
62. C₂×D₃×D₄	C₂×C₅	semidirect
63. C₃×#124	C₂×C₅	semidirect
64. #947	C₂×C₅	semidirect
65. #940	C₂×C₅	semidirect
66. #884	C₂×C₅	semidirect
67. #882	C₂×C₅	semidirect
68. #874	C₂×C₅	semidirect
69. #872	C₂×C₅	semidirect
72. C₂×C₅×Q₈	C₂×D₃	semidirect
86. C₅×C₂²⋊C₄	C₂×D₃	semidirect
88. C₄²×C₅	C₂×D₃	semidirect
106. C₂×C₃×Q₈	C₂²×C₅	semidirect
109. C₂×#67	C₂²×C₅	semidirect
114. C₂×D₁₂	C₂²×C₅	semidirect
115. C₂×D₁₂	C₂²×C₅	semidirect
117. C₂×C₄×D₃	C₂²×C₅	semidirect
118. C₃×C₂²⋊C₄	C₂²×C₅	semidirect
119. C₃×C₄²	C₂²×C₅	semidirect
121. (C₂×D₃)⋊C₄	C₂²×C₅	semidirect
122. Dic₃⋊C₄	C₂²×C₅	semidirect
123. C₄×Dic₃	C₂²×C₅	semidirect
150. #124	C₅×D₃	semidirect
180. C₄×D₃	C₅×D₄	semidirect
198. C₂×Q₈	C₂×C₅×D₃	semidirect
212. C₂²⋊C₄	C₂×C₅×D₃	semidirect
214. C₄²	C₂×C₅×D₃	semidirect
215. C₃×C₅	#535	semidirect
264. C₅	#5674	direct (N × H)
280. C₃	C₅×#535	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#239	2
C₂×C₅×D₃×D₄	2
C₃×C₅×#124	2
C₅×#947	2
C₅×#940	2
C₅×#884	2
C₅×#882	2
C₅×#874	2
C₅×#872	2
C₅×#535	3
#5674	5

Subgroups — C5×#5674 #11756059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5674 #11756059