Subgroups — C₅×#5675 #11756060

Order 960.

540 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#237	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#944	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#936	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#883	C₂	1	4	yes	maximal, complemented
7	C₅×#878	C₂	1	4	yes	maximal, complemented
8	C₅×#872	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#869	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#536	—	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	2	yes	—
14	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
15	C₅×D₃×Q₈	—	2	4	no	—
16	C₅×#237	—	2	4	no	—
17	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
18	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
19	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
20	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
21	C₂×C₅×Dic₆	C₂²	1	2	yes	—
22	C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₅×#217	C₂²	1	4	yes	—
25	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₅×C₄⋊Dic₃	C₂²	1	4	yes	—
27	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
28	C₅×Dic₃⋊C₄	C₂²	1	4	yes	—
29	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
30	#5675	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₂×C₅×#41	—	3	1	no	—
32	C₂²×C₅×Q₈	—	3	1	no	—
33	C₅×C₄⋊Q₈	—	3	1	no	—
34	C₅×C₄⋊Q₈	—	3	1	no	—
35	C₅×#124	—	3	1	no	—
36	C₅×#124	D₃	1	1	yes	characteristic, complemented
37	C₅×#123	—	3	4	no	—
38	C₅×C₂²⋊Q₈	—	3	4	no	—
39	C₅×#117	—	3	1	no	—
40	C₂³×C₃×C₅	C₂³	1	2	yes	—
41	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
42	C₃×C₅×Q₈	—	2	2	no	—
43	C₃×C₅×D₄	—	2	2	no	—
44	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
45	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
46	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₅×#67	—	2	4	no	—
48	C₂×C₅×Dic₃	—	2	4	no	—
49	C₂×C₅×Dic₃	C₂³	1	2	yes	—
50	C₂×C₅×Dic₃	C₂³	1	2	yes	—
51	C₂×C₅×Dic₃	C₂³	1	2	yes	—
52	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
53	C₄×C₅×D₃	—	2	4	no	—
54	C₄×C₅×D₃	D₄	1	4	yes	—
55	C₅×Dic₆	—	2	2	no	—
56	C₅×Dic₆	—	2	2	no	—
57	C₅×Dic₆	—	2	4	no	—
58	C₂×D₃×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂×#237	C₂×C₅	1	1	yes	characteristic
60	C₃×#124	C₂×C₅	1	1	yes	characteristic, complemented
61	#944	C₂×C₅	1	1	yes	characteristic, complemented
62	#936	C₂×C₅	1	1	yes	characteristic, complemented
63	#883	C₂×C₅	1	4	yes	complemented
64	#878	C₂×C₅	1	4	yes	complemented
65	#872	C₂×C₅	1	1	yes	characteristic, complemented
66	#869	C₂×C₅	1	1	yes	characteristic, complemented
67	C₅×#41	—	6	4	no	—
68	C₂×C₅×Q₈	—	3	2	no	—
69	C₂×C₅×Q₈	—	3	1	no	—
70	C₂×C₅×Q₈	—	3	1	no	—
71	C₂×C₅×Q₈	—	6	4	no	—
72	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
73	C₂×C₅×D₄	—	3	2	no	—
74	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
75	C₂²×C₄×C₅	—	3	1	no	—
76	C₂²×C₄×C₅	—	3	2	no	—
77	C₂²×C₄×C₅	—	3	2	no	—
78	C₅×C₄⋊C₄	—	3	2	no	—
79	C₅×C₄⋊C₄	—	3	4	no	—
80	C₅×C₄⋊C₄	—	3	4	no	—
81	C₅×C₂²⋊C₄	—	3	4	no	—
82	C₅×C₂²⋊C₄	—	3	2	no	—
83	C₅×C₂²⋊C₄	C₂×D₃	1	4	yes	—
84	C₄²×C₅	—	3	1	no	—
85	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
86	#536	—	3	1	no	Sylow
87	C₂²×C₃×C₅	—	2	2	no	—
88	C₂²×C₃×C₅	—	2	4	no	—
89	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
90	C₂×C₅×D₃	—	2	2	no	—
91	C₂×C₅×D₃	C₂×D₄	1	2	yes	—
92	C₃×C₄×C₅	—	2	2	no	—
93	C₃×C₄×C₅	—	2	2	no	—
94	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
95	C₅×Dic₃	—	2	2	no	—
96	C₅×Dic₃	—	2	2	no	—
97	C₅×Dic₃	—	2	2	no	—
98	C₅×Dic₃	C₂×D₄	1	2	yes	—
99	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
101	C₂×#67	C₂²×C₅	1	2	yes	—
102	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
103	D₃×Q₈	—	2	4	no	—
104	#237	—	2	4	no	—
105	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
106	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
107	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
108	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
109	C₂×Dic₆	C₂²×C₅	1	2	yes	—
110	C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
111	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
112	#217	C₂²×C₅	1	4	yes	—
113	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
114	C₄⋊Dic₃	C₂²×C₅	1	4	yes	—
115	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
116	Dic₃⋊C₄	C₂²×C₅	1	4	yes	—
117	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
118	C₂³×C₅	—	3	1	no	—
119	C₂³×C₅	C₂²×D₃	1	2	yes	—
120	C₅×Q₈	—	2	2	no	—
121	C₅×Q₈	—	6	4	no	—
122	C₅×Q₈	—	6	2	no	—
123	C₅×Q₈	—	6	2	no	—
124	C₅×D₄	—	2	2	no	—
125	C₅×D₄	—	6	4	no	—
126	C₂×C₄×C₅	—	3	1	no	—
127	C₂×C₄×C₅	—	3	4	no	—
128	C₂×C₄×C₅	—	3	2	no	—
129	C₂×C₄×C₅	—	3	2	no	—
130	C₂×C₄×C₅	—	3	2	no	—
131	C₂×C₄×C₅	—	6	4	no	—
132	C₂×C₄×C₅	—	6	4	no	—
133	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
134	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
135	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
136	C₂×#41	—	3	1	no	—
137	C₂²×Q₈	—	3	1	no	—
138	C₄⋊Q₈	—	3	1	no	—
139	C₄⋊Q₈	—	3	1	no	—
140	#124	—	3	1	no	—
141	#124	C₅×D₃	1	1	yes	characteristic, complemented
142	#123	—	3	4	no	—
143	C₂²⋊Q₈	—	3	4	no	—
144	#117	—	3	1	no	—
145	C₂×C₃×C₅	—	4	2	no	—
146	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
148	C₅×D₃	—	2	2	no	—
149	C₂³×C₃	C₂³×C₅	1	2	yes	—
150	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
151	C₃×Q₈	—	2	2	no	—
152	C₃×D₄	—	2	2	no	—
153	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
154	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
155	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
156	#67	—	2	4	no	—
157	C₂×Dic₃	—	2	4	no	—
158	C₂×Dic₃	C₂³×C₅	1	2	yes	—
159	C₂×Dic₃	C₂³×C₅	1	2	yes	—
160	C₂×Dic₃	C₂³×C₅	1	2	yes	—
161	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
162	C₄×D₃	—	2	4	no	—
163	C₄×D₃	C₅×D₄	1	4	yes	—
164	Dic₆	—	2	2	no	—
165	Dic₆	—	2	2	no	—
166	Dic₆	—	2	4	no	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	—	2	4	no	—
169	C₂²×C₅	—	3	2	no	—
170	C₂²×C₅	—	6	2	no	—
171	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
172	C₄×C₅	—	2	2	no	—
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	3	2	no	—
175	C₄×C₅	—	6	2	no	—
176	C₄×C₅	—	6	2	no	—
177	C₄×C₅	—	6	2	no	—
178	C₄×C₅	D₃×D₄	1	2	yes	—
179	#41	—	6	4	no	—
180	C₂×Q₈	—	3	2	no	—
181	C₂×Q₈	—	3	1	no	—
182	C₂×Q₈	—	3	1	no	—
183	C₂×Q₈	—	6	4	no	—
184	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
185	C₂×D₄	—	3	2	no	—
186	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
187	C₂²×C₄	—	3	1	no	—
188	C₂²×C₄	—	3	2	no	—
189	C₂²×C₄	—	3	2	no	—
190	C₄⋊C₄	—	3	2	no	—
191	C₄⋊C₄	—	3	4	no	—
192	C₄⋊C₄	—	3	4	no	—
193	C₂²⋊C₄	—	3	4	no	—
194	C₂²⋊C₄	—	3	2	no	—
195	C₂²⋊C₄	C₂×C₅×D₃	1	4	yes	—
196	C₄²	—	3	1	no	—
197	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
198	C₃×C₅	#536	1	1	yes	characteristic, complemented, Hall
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	2	4	no	—
201	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
202	C₂×D₃	—	2	2	no	—
203	C₂×D₃	C₂×C₅×D₄	1	2	yes	—
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
207	Dic₃	—	2	2	no	—
208	Dic₃	—	2	2	no	—
209	Dic₃	—	2	2	no	—
210	Dic₃	C₂×C₅×D₄	1	2	yes	—
211	C₂×C₅	—	4	2	no	—
212	C₂×C₅	—	6	2	no	—
213	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
214	C₂×C₅	#1010	1	2	yes	—
215	C₂³	—	3	1	no	—
216	C₂³	C₂²×C₅×D₃	1	2	yes	—
217	Q₈	—	2	2	no	—
218	Q₈	—	6	4	no	—
219	Q₈	—	6	2	no	—
220	Q₈	—	6	2	no	—
221	D₄	—	2	2	no	—
222	D₄	—	6	4	no	—
223	C₂×C₄	—	3	1	no	—
224	C₂×C₄	—	3	4	no	—
225	C₂×C₄	—	3	2	no	—
226	C₂×C₄	—	3	2	no	—
227	C₂×C₄	—	3	2	no	—
228	C₂×C₄	—	6	4	no	—
229	C₂×C₄	—	6	4	no	—
230	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
231	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
232	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
233	C₂×C₃	—	4	2	no	—
234	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
235	C₂×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
236	D₃	—	2	2	no	—
237	C₅	#5675	1	1	yes	characteristic, direct product factor, Sylow
238	C₂²	—	2	2	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	3	2	no	—
241	C₂²	—	6	2	no	—
242	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	3	2	no	—
246	C₄	—	6	2	no	—
247	C₄	—	6	2	no	—
248	C₄	—	6	2	no	—
249	C₄	C₅×D₃×D₄	1	2	yes	—
250	C₃	C₅×#536	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	4	2	no	—
252	C₂	—	6	2	no	—
253	C₂	C₂×C₅×D₃×D₄	1	1	yes	characteristic
254	C₂	C₅×#1010	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
prime radical	C₂×C₅×#237	C₂
hypercenter	C₅×#124	D₃
hypocenter	C₃	C₅×#536
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×D₃×Q₈	C₂	semidirect
3. C₃×C₅×#124	C₂	semidirect
4. C₅×#944	C₂	semidirect
5. C₅×#936	C₂	semidirect
6. C₅×#883	C₂	semidirect
7. C₅×#878	C₂	semidirect
8. C₅×#872	C₂	semidirect
9. C₅×#869	C₂	semidirect
30. #5675	C₅	direct (N × H)
36. C₅×#124	D₃	semidirect
58. C₂×D₃×Q₈	C₂×C₅	semidirect
60. C₃×#124	C₂×C₅	semidirect
61. #944	C₂×C₅	semidirect
62. #936	C₂×C₅	semidirect
63. #883	C₂×C₅	semidirect
64. #878	C₂×C₅	semidirect
65. #872	C₂×C₅	semidirect
66. #869	C₂×C₅	semidirect
141. #124	C₅×D₃	semidirect
198. C₃×C₅	#536	semidirect
237. C₅	#5675	direct (N × H)
250. C₃	C₅×#536	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₃×Q₈	2
C₂×C₅×#237	2
C₃×C₅×#124	2
C₅×#944	2
C₅×#936	2
C₅×#883	2
C₅×#878	2
C₅×#872	2
C₅×#869	2
C₅×#536	3
#5675	5

Subgroups — C5×#5675 #11756060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5675 #11756060