Subgroups — C₅×#5668 #11756053

Order 960.

620 conjugacy classes of subgroups.

The subgroup lattice has 272 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×#237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#944	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×#67⋊C₄	C₂	1	4	yes	maximal, complemented
8	C₅×#875	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄²×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#532	—	3	1	no	maximal, Hall
11	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₅×#67	C₂²	1	4	yes	—
14	C₂×C₅×#67	C₂²	1	2	yes	—
15	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
16	C₅×#239	—	2	4	no	—
17	C₅×#237	—	2	4	no	—
18	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
19	C₂×C₅×D₁₂	C₂²	1	2	yes	—
20	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
21	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
22	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
23	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
24	C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
26	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
27	C₅×Dic₃⋊C₄	C₂²	1	4	yes	—
28	C₄×C₅×Dic₃	C₂²	1	2	yes	—
29	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
30	#5668	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₂×C₅×#41	—	3	1	no	—
32	C₂×C₅×#41	—	3	1	no	—
33	C₅×C₄⋊Q₈	—	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	C₄×C₅×D₄	—	3	4	no	—
39	C₂×C₄²×C₅	—	3	1	no	—
40	C₂³×C₃×C₅	C₂³	1	2	yes	—
41	C₂²×C₅×D₃	C₂³	1	2	yes	—
42	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
43	C₃×C₅×Q₈	—	2	2	no	—
44	C₃×C₅×D₄	—	2	2	no	—
45	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
46	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₅×#67	—	2	4	no	—
49	C₅×#67	—	2	8	no	—
50	C₂×C₅×Dic₃	—	2	4	no	—
51	C₂×C₅×Dic₃	C₂³	1	2	yes	—
52	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
53	C₂×C₅×Dic₃	C₂³	1	2	yes	—
54	C₅×D₁₂	—	2	4	no	—
55	C₅×D₁₂	—	2	2	no	—
56	C₄×C₅×D₃	—	2	4	no	—
57	C₄×C₅×D₃	—	2	4	no	—
58	C₄×C₅×D₃	D₄	1	4	yes	complemented
59	C₅×Dic₆	—	2	2	no	—
60	C₂×#239	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂×#237	C₂×C₅	1	1	yes	characteristic, complemented
62	C₃×#124	C₂×C₅	1	1	yes	characteristic, complemented
63	#944	C₂×C₅	1	1	yes	characteristic, complemented
64	#940	C₂×C₅	1	1	yes	characteristic, complemented
65	#884	C₂×C₅	1	4	yes	complemented
66	#67⋊C₄	C₂×C₅	1	4	yes	complemented
67	#875	C₂×C₅	1	1	yes	characteristic, complemented
68	C₄²×D₃	C₂×C₅	1	1	yes	characteristic, complemented
69	C₅×#41	—	6	4	no	—
70	C₅×#41	—	6	4	no	—
71	C₂×C₅×Q₈	—	3	1	no	—
72	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
73	C₂×C₅×D₄	—	3	4	no	—
74	C₂×C₅×D₄	—	3	2	no	—
75	C₂×C₅×D₄	—	3	2	no	—
76	C₂×C₅×D₄	—	3	1	no	—
77	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
78	C₂²×C₄×C₅	—	3	1	no	—
79	C₂²×C₄×C₅	—	3	2	no	—
80	C₂²×C₄×C₅	—	3	2	no	—
81	C₂²×C₄×C₅	—	3	2	no	—
82	C₅×C₄⋊C₄	—	3	4	no	—
83	C₅×C₂²⋊C₄	—	3	4	no	—
84	C₅×C₂²⋊C₄	C₂×D₃	1	4	yes	—
85	C₄²×C₅	—	3	2	no	—
86	C₄²×C₅	—	3	1	no	—
87	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
88	#532	—	3	1	no	Sylow
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	—	2	4	no	—
91	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
92	C₂×C₅×D₃	—	2	2	no	—
93	C₂×C₅×D₃	—	2	2	no	—
94	C₂×C₅×D₃	—	2	4	no	—
95	C₂×C₅×D₃	C₂×D₄	1	2	yes	—
96	C₃×C₄×C₅	—	2	2	no	—
97	C₃×C₄×C₅	—	2	2	no	—
98	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
99	C₅×Dic₃	—	2	2	no	—
100	C₅×Dic₃	C₂×D₄	1	2	yes	—
101	C₅×Dic₃	#41	1	4	yes	complemented
102	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
104	C₂×#67	C₂²×C₅	1	4	yes	—
105	C₂×#67	C₂²×C₅	1	2	yes	—
106	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
107	#239	—	2	4	no	—
108	#237	—	2	4	no	—
109	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
110	C₂×D₁₂	C₂²×C₅	1	2	yes	—
111	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
112	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
113	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
114	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
115	C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
116	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
117	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
118	Dic₃⋊C₄	C₂²×C₅	1	4	yes	—
119	C₄×Dic₃	C₂²×C₅	1	2	yes	—
120	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
121	C₂³×C₅	—	3	1	no	—
122	C₂³×C₅	—	3	2	no	—
123	C₂³×C₅	C₂²×D₃	1	2	yes	—
124	C₅×Q₈	—	2	2	no	—
125	C₅×Q₈	—	6	2	no	—
126	C₅×D₄	—	2	2	no	—
127	C₅×D₄	—	6	4	no	—
128	C₅×D₄	—	6	8	no	—
129	C₅×D₄	—	6	4	no	—
130	C₅×D₄	—	6	2	no	—
131	C₂×C₄×C₅	—	3	2	no	—
132	C₂×C₄×C₅	—	3	1	no	—
133	C₂×C₄×C₅	—	3	4	no	—
134	C₂×C₄×C₅	—	3	2	no	—
135	C₂×C₄×C₅	—	6	4	no	—
136	C₂×C₄×C₅	—	6	4	no	—
137	C₂×C₄×C₅	—	6	4	no	—
138	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
139	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
140	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
141	C₂×#41	—	3	1	no	—
142	C₂×#41	—	3	1	no	—
143	C₄⋊Q₈	—	3	1	no	—
144	#127	—	3	1	no	—
145	#124	—	3	1	no	—
146	#124	C₅×D₃	1	1	yes	characteristic, complemented
147	#121	—	3	4	no	—
148	C₄×D₄	—	3	4	no	—
149	C₂×C₄²	—	3	1	no	—
150	C₂×C₃×C₅	—	4	2	no	—
151	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₃×C₅	C₂×#41	1	2	yes	—
153	C₅×D₃	—	2	2	no	—
154	C₅×D₃	—	4	2	no	—
155	C₂³×C₃	C₂³×C₅	1	2	yes	—
156	C₂²×D₃	C₂³×C₅	1	2	yes	—
157	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
158	C₃×Q₈	—	2	2	no	—
159	C₃×D₄	—	2	2	no	—
160	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
161	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
162	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
163	#67	—	2	4	no	—
164	#67	—	2	8	no	—
165	C₂×Dic₃	—	2	4	no	—
166	C₂×Dic₃	C₂³×C₅	1	2	yes	—
167	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
168	C₂×Dic₃	C₂³×C₅	1	2	yes	—
169	D₁₂	—	2	4	no	—
170	D₁₂	—	2	2	no	—
171	C₄×D₃	—	2	4	no	—
172	C₄×D₃	—	2	4	no	—
173	C₄×D₃	C₅×D₄	1	4	yes	complemented
174	Dic₆	—	2	2	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	4	no	—
177	C₂²×C₅	—	3	2	no	—
178	C₂²×C₅	—	6	2	no	—
179	C₂²×C₅	—	6	2	no	—
180	C₂²×C₅	—	6	4	no	—
181	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	3	2	no	—
185	C₄×C₅	—	3	4	no	—
186	C₄×C₅	—	6	2	no	—
187	C₄×C₅	D₃×D₄	1	2	yes	—
188	#41	—	6	4	no	—
189	#41	—	6	4	no	—
190	C₂×Q₈	—	3	1	no	—
191	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
192	C₂×D₄	—	3	4	no	—
193	C₂×D₄	—	3	2	no	—
194	C₂×D₄	—	3	2	no	—
195	C₂×D₄	—	3	1	no	—
196	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
197	C₂²×C₄	—	3	1	no	—
198	C₂²×C₄	—	3	2	no	—
199	C₂²×C₄	—	3	2	no	—
200	C₂²×C₄	—	3	2	no	—
201	C₄⋊C₄	—	3	4	no	—
202	C₂²⋊C₄	—	3	4	no	—
203	C₂²⋊C₄	C₂×C₅×D₃	1	4	yes	—
204	C₄²	—	3	2	no	—
205	C₄²	—	3	1	no	—
206	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
207	C₃×C₅	#532	1	1	yes	characteristic, complemented, Hall
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
211	C₂×D₃	—	2	2	no	—
212	C₂×D₃	—	2	2	no	—
213	C₂×D₃	—	2	4	no	—
214	C₂×D₃	C₂×C₅×D₄	1	2	yes	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
218	Dic₃	—	2	2	no	—
219	Dic₃	C₂×C₅×D₄	1	2	yes	—
220	Dic₃	C₅×#41	1	4	yes	complemented
221	C₂×C₅	—	4	2	no	—
222	C₂×C₅	—	6	2	no	—
223	C₂×C₅	—	12	2	no	—
224	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
225	C₂×C₅	D₃×#41	1	2	yes	—
226	C₂³	—	3	1	no	—
227	C₂³	—	3	2	no	—
228	C₂³	C₂²×C₅×D₃	1	2	yes	—
229	Q₈	—	2	2	no	—
230	Q₈	—	6	2	no	—
231	D₄	—	2	2	no	—
232	D₄	—	6	4	no	—
233	D₄	—	6	8	no	—
234	D₄	—	6	4	no	—
235	D₄	—	6	2	no	—
236	C₂×C₄	—	3	2	no	—
237	C₂×C₄	—	3	1	no	—
238	C₂×C₄	—	3	4	no	—
239	C₂×C₄	—	3	2	no	—
240	C₂×C₄	—	6	4	no	—
241	C₂×C₄	—	6	4	no	—
242	C₂×C₄	—	6	4	no	—
243	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
244	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
245	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
246	C₂×C₃	—	4	2	no	—
247	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
248	C₂×C₃	C₂×C₅×#41	1	2	yes	—
249	D₃	—	2	2	no	—
250	D₃	—	4	2	no	—
251	C₅	#5668	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	3	2	no	—
255	C₂²	—	6	2	no	—
256	C₂²	—	6	2	no	—
257	C₂²	—	6	4	no	—
258	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	3	2	no	—
262	C₄	—	3	4	no	—
263	C₄	—	6	2	no	—
264	C₄	C₅×D₃×D₄	1	2	yes	—
265	C₃	C₅×#532	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	4	2	no	—
267	C₂	—	6	2	no	—
268	C₂	—	12	2	no	—
269	C₂	C₂×C₅×D₃×D₄	1	1	yes	characteristic
270	C₂	C₅×D₃×#41	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×#532
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×#237	C₂	semidirect
3. C₃×C₅×#124	C₂	semidirect
4. C₅×#944	C₂	semidirect
5. C₅×#940	C₂	semidirect
6. C₅×#884	C₂	semidirect
7. C₅×#67⋊C₄	C₂	semidirect
8. C₅×#875	C₂	semidirect
9. C₄²×C₅×D₃	C₂	semidirect
30. #5668	C₅	direct (N × H)
36. C₅×#124	D₃	semidirect
58. C₄×C₅×D₃	D₄	semidirect
60. C₂×#239	C₂×C₅	semidirect
61. C₂×#237	C₂×C₅	semidirect
62. C₃×#124	C₂×C₅	semidirect
63. #944	C₂×C₅	semidirect
64. #940	C₂×C₅	semidirect
65. #884	C₂×C₅	semidirect
66. #67⋊C₄	C₂×C₅	semidirect
67. #875	C₂×C₅	semidirect
68. C₄²×D₃	C₂×C₅	semidirect
101. C₅×Dic₃	#41	semidirect
146. #124	C₅×D₃	semidirect
173. C₄×D₃	C₅×D₄	semidirect
207. C₃×C₅	#532	semidirect
220. Dic₃	C₅×#41	semidirect
251. C₅	#5668	direct (N × H)
265. C₃	C₅×#532	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×#237	2
C₃×C₅×#124	2
C₅×#944	2
C₅×#940	2
C₅×#884	2
C₅×#67⋊C₄	2
C₅×#875	2
C₄²×C₅×D₃	2
C₅×#532	3
#5668	5

Subgroups — C5×#5668 #11756053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5668 #11756053