Subgroups — C₅×#5659 #11756044

Order 960.

496 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#937	C₂	1	1	yes	maximal, characteristic
3	C₅×#936	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×#67	C₂	1	2	yes	maximal, complemented
5	C₅×#896	C₂	1	2	yes	maximal, complemented
6	C₅×#892	C₂	1	2	yes	maximal, complemented
7	C₅×#883	C₂	1	2	yes	maximal, complemented
8	C₅×D₃×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₅×#879	C₂	1	2	yes	maximal, complemented
10	C₅×#551	—	3	1	no	maximal, Hall
11	C₂³×C₅×D₃	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₅×#67	C₂²	1	2	yes	—
15	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
16	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
17	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic, complemented
18	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
19	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
21	C₅×#217	C₂²	1	1	yes	characteristic
22	C₅×#217	C₂²	1	2	yes	—
23	C₅×#217	C₂²	1	2	yes	complemented
24	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₅×C₄⋊Dic₃	C₂²	1	2	yes	complemented
28	C₅×Dic₃⋊C₄	C₂²	1	2	yes	complemented
29	C₅×Dic₃⋊C₄	C₂²	1	2	yes	complemented
30	C₄×C₅×Dic₃	C₂²	1	2	yes	complemented
31	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
32	#5659	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₅×#126	—	3	2	no	—
34	C₅×#124	—	3	1	no	—
35	C₅×#123	—	3	2	no	—
36	C₅×#123	D₃	1	1	yes	characteristic, complemented
37	C₅×C₂²⋊Q₈	—	3	2	no	—
38	C₅×#120	—	3	1	no	—
39	C₄×C₅×D₄	—	3	2	no	—
40	C₅×#117	—	3	2	no	—
41	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
42	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
43	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
44	C₂²×C₅×D₃	—	2	4	no	—
45	C₂²×C₅×D₃	—	4	1	no	—
46	C₂²×C₅×D₃	C₂³	1	2	yes	—
47	C₃×C₅×D₄	—	4	1	no	—
48	C₂×C₃×C₄×C₅	—	4	1	no	—
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₅×#67	—	2	4	no	—
53	C₂×C₅×Dic₃	C₂³	1	2	yes	—
54	C₂×C₅×Dic₃	C₂³	1	2	yes	—
55	C₂×C₅×Dic₃	C₂³	1	2	yes	—
56	C₄×C₅×D₃	—	2	4	no	—
57	C₄×C₅×D₃	—	4	2	no	—
58	C₅×Dic₆	—	4	1	no	—
59	C₃×#123	C₂×C₅	1	1	yes	characteristic, complemented
60	#937	C₂×C₅	1	1	yes	characteristic
61	#936	C₂×C₅	1	1	yes	characteristic, complemented
62	C₄×#67	C₂×C₅	1	2	yes	complemented
63	#896	C₂×C₅	1	2	yes	complemented
64	#892	C₂×C₅	1	2	yes	complemented
65	#883	C₂×C₅	1	2	yes	complemented
66	D₃×C₂²⋊C₄	C₂×C₅	1	2	yes	complemented
67	#879	C₂×C₅	1	2	yes	complemented
68	C₂⁴×C₅	—	3	1	no	—
69	C₂×C₅×Q₈	—	3	1	no	—
70	C₂×C₅×D₄	—	3	2	no	—
71	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
72	C₂²×C₄×C₅	—	3	2	no	—
73	C₂²×C₄×C₅	—	3	2	no	—
74	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic, complemented
75	C₅×C₄⋊C₄	—	3	2	no	—
76	C₅×C₄⋊C₄	—	3	2	no	—
77	C₅×C₄⋊C₄	—	3	2	no	—
78	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	complemented
79	C₅×C₂²⋊C₄	—	3	2	no	—
80	C₅×C₂²⋊C₄	—	3	2	no	—
81	C₅×C₂²⋊C₄	—	3	1	no	—
82	C₅×C₂²⋊C₄	—	3	2	no	—
83	C₅×C₂²⋊C₄	—	3	2	no	—
84	C₅×C₂²⋊C₄	—	3	2	no	—
85	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
86	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
87	C₄²×C₅	—	3	1	no	—
88	C₄²×C₅	—	3	2	no	—
89	#551	—	3	1	no	Sylow
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	2	2	no	—
92	C₂²×C₃×C₅	—	2	1	no	—
93	C₂²×C₃×C₅	—	2	1	no	—
94	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
95	C₂×C₅×D₃	—	2	2	no	—
96	C₂×C₅×D₃	—	2	2	no	—
97	C₂×C₅×D₃	—	4	4	no	—
98	C₂×C₅×D₃	#41	1	4	yes	—
99	C₃×C₄×C₅	—	2	2	no	—
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	—	2	2	no	—
102	C₅×Dic₃	—	2	2	no	—
103	C₅×Dic₃	—	2	2	no	—
104	C₅×Dic₃	—	2	2	no	—
105	C₂³×D₃	C₂²×C₅	1	1	yes	characteristic
106	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic, complemented
108	C₂×#67	C₂²×C₅	1	2	yes	—
109	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
110	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
111	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic, complemented
112	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
113	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
114	C₃×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
115	#217	C₂²×C₅	1	1	yes	characteristic
116	#217	C₂²×C₅	1	2	yes	—
117	#217	C₂²×C₅	1	2	yes	complemented
118	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
119	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
120	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
121	C₄⋊Dic₃	C₂²×C₅	1	2	yes	complemented
122	Dic₃⋊C₄	C₂²×C₅	1	2	yes	complemented
123	Dic₃⋊C₄	C₂²×C₅	1	2	yes	complemented
124	C₄×Dic₃	C₂²×C₅	1	2	yes	complemented
125	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
126	C₂³×C₅	—	3	2	no	—
127	C₂³×C₅	—	6	4	no	—
128	C₂³×C₅	—	12	1	no	—
129	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
130	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
131	C₅×Q₈	—	12	1	no	—
132	C₅×D₄	—	4	1	no	—
133	C₅×D₄	—	6	4	no	—
134	C₂×C₄×C₅	—	3	2	no	—
135	C₂×C₄×C₅	—	3	2	no	—
136	C₂×C₄×C₅	—	3	2	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	—	6	4	no	—
139	C₂×C₄×C₅	—	12	2	no	—
140	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
142	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
143	#126	—	3	2	no	—
144	#124	—	3	1	no	—
145	#123	—	3	2	no	—
146	#123	C₅×D₃	1	1	yes	characteristic, complemented
147	C₂²⋊Q₈	—	3	2	no	—
148	#120	—	3	1	no	—
149	C₄×D₄	—	3	2	no	—
150	#117	—	3	2	no	—
151	C₂×C₂²⋊C₄	—	3	2	no	—
152	C₂×C₃×C₅	—	4	1	no	—
153	C₂×C₃×C₅	—	4	1	no	—
154	C₂×C₃×C₅	C₂×#41	1	2	yes	—
155	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
156	C₅×D₃	—	2	4	no	—
157	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
158	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
159	C₂²×D₃	—	2	4	no	—
160	C₂²×D₃	—	4	1	no	—
161	C₂²×D₃	C₂³×C₅	1	2	yes	—
162	C₃×D₄	—	4	1	no	—
163	C₂×C₃×C₄	—	4	1	no	—
164	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
165	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
167	#67	—	2	4	no	—
168	C₂×Dic₃	C₂³×C₅	1	2	yes	—
169	C₂×Dic₃	C₂³×C₅	1	2	yes	—
170	C₂×Dic₃	C₂³×C₅	1	2	yes	—
171	C₄×D₃	—	2	4	no	—
172	C₄×D₃	—	4	2	no	—
173	Dic₆	—	4	1	no	—
174	C₂²×C₅	—	2	1	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	3	4	no	—
179	C₂²×C₅	—	6	2	no	—
180	C₂²×C₅	—	6	2	no	—
181	C₂²×C₅	—	12	4	no	—
182	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	6	2	no	—
187	C₄×C₅	—	6	2	no	—
188	C₄×C₅	—	6	2	no	—
189	C₂⁴	—	3	1	no	—
190	C₂×Q₈	—	3	1	no	—
191	C₂×D₄	—	3	2	no	—
192	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
193	C₂²×C₄	—	3	2	no	—
194	C₂²×C₄	—	3	2	no	—
195	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic, complemented
196	C₄⋊C₄	—	3	2	no	—
197	C₄⋊C₄	—	3	2	no	—
198	C₄⋊C₄	—	3	2	no	—
199	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	complemented
200	C₂²⋊C₄	—	3	2	no	—
201	C₂²⋊C₄	—	3	2	no	—
202	C₂²⋊C₄	—	3	1	no	—
203	C₂²⋊C₄	—	3	2	no	—
204	C₂²⋊C₄	—	3	2	no	—
205	C₂²⋊C₄	—	3	2	no	—
206	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
207	C₂²⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic, complemented
208	C₄²	—	3	1	no	—
209	C₄²	—	3	2	no	—
210	C₃×C₅	#551	1	1	yes	characteristic, complemented, Hall
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
216	C₂×D₃	—	2	2	no	—
217	C₂×D₃	—	2	2	no	—
218	C₂×D₃	—	4	4	no	—
219	C₂×D₃	C₅×#41	1	4	yes	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	2	no	—
223	Dic₃	—	2	2	no	—
224	Dic₃	—	2	2	no	—
225	Dic₃	—	2	2	no	—
226	C₂×C₅	—	4	1	no	—
227	C₂×C₅	—	4	1	no	—
228	C₂×C₅	—	6	4	no	—
229	C₂×C₅	#1004	1	1	yes	characteristic
230	C₂×C₅	D₃×#41	1	2	yes	—
231	C₂³	—	3	2	no	—
232	C₂³	—	6	4	no	—
233	C₂³	—	12	1	no	—
234	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
235	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
236	Q₈	—	12	1	no	—
237	D₄	—	4	1	no	—
238	D₄	—	6	4	no	—
239	C₂×C₄	—	3	2	no	—
240	C₂×C₄	—	3	2	no	—
241	C₂×C₄	—	3	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	6	4	no	—
244	C₂×C₄	—	12	2	no	—
245	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
247	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
248	C₂×C₃	—	4	1	no	—
249	C₂×C₃	—	4	1	no	—
250	C₂×C₃	C₂×C₅×#41	1	2	yes	—
251	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
252	D₃	—	2	4	no	—
253	C₅	#5659	1	1	yes	characteristic, direct product factor, Sylow
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	3	4	no	—
259	C₂²	—	6	2	no	—
260	C₂²	—	6	2	no	—
261	C₂²	—	12	4	no	—
262	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	2	no	—
266	C₄	—	6	2	no	—
267	C₄	—	6	2	no	—
268	C₄	—	6	2	no	—
269	C₃	C₅×#551	1	1	yes	characteristic, complemented, Sylow
270	C₂	—	4	1	no	—
271	C₂	—	4	1	no	—
272	C₂	—	6	4	no	—
273	C₂	C₅×#1004	1	1	yes	characteristic
274	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×#123	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₅×#937	C₂
hypercenter	C₅×#123	D₃
hypocenter	C₃	C₅×#551
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#123

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#123	C₂	semidirect
3. C₅×#936	C₂	semidirect
4. C₄×C₅×#67	C₂	semidirect
5. C₅×#896	C₂	semidirect
6. C₅×#892	C₂	semidirect
7. C₅×#883	C₂	semidirect
8. C₅×D₃×C₂²⋊C₄	C₂	semidirect
9. C₅×#879	C₂	semidirect
13. C₂²×C₃×C₄×C₅	C₂²	semidirect
17. C₂×C₅×Dic₆	C₂²	semidirect
18. C₃×C₅×C₄⋊C₄	C₂²	semidirect
20. C₃×C₅×C₂²⋊C₄	C₂²	semidirect
23. C₅×#217	C₂²	semidirect
27. C₅×C₄⋊Dic₃	C₂²	semidirect
28. C₅×Dic₃⋊C₄	C₂²	semidirect
29. C₅×Dic₃⋊C₄	C₂²	semidirect
30. C₄×C₅×Dic₃	C₂²	semidirect
31. C₄×C₅×Dic₃	C₂²	semidirect
32. #5659	C₅	direct (N × H)
36. C₅×#123	D₃	semidirect
59. C₃×#123	C₂×C₅	semidirect
61. #936	C₂×C₅	semidirect
62. C₄×#67	C₂×C₅	semidirect
63. #896	C₂×C₅	semidirect
64. #892	C₂×C₅	semidirect
65. #883	C₂×C₅	semidirect
66. D₃×C₂²⋊C₄	C₂×C₅	semidirect
67. #879	C₂×C₅	semidirect
74. C₂²×C₄×C₅	C₂×D₃	semidirect
78. C₅×C₄⋊C₄	C₂×D₃	semidirect
86. C₅×C₂²⋊C₄	C₂×D₃	semidirect
107. C₂²×C₃×C₄	C₂²×C₅	semidirect
111. C₂×Dic₆	C₂²×C₅	semidirect
112. C₃×C₄⋊C₄	C₂²×C₅	semidirect
114. C₃×C₂²⋊C₄	C₂²×C₅	semidirect
117. #217	C₂²×C₅	semidirect
121. C₄⋊Dic₃	C₂²×C₅	semidirect
122. Dic₃⋊C₄	C₂²×C₅	semidirect
123. Dic₃⋊C₄	C₂²×C₅	semidirect
124. C₄×Dic₃	C₂²×C₅	semidirect
125. C₄×Dic₃	C₂²×C₅	semidirect
146. #123	C₅×D₃	semidirect
195. C₂²×C₄	C₂×C₅×D₃	semidirect
199. C₄⋊C₄	C₂×C₅×D₃	semidirect
207. C₂²⋊C₄	C₂×C₅×D₃	semidirect
210. C₃×C₅	#551	semidirect
253. C₅	#5659	direct (N × H)
269. C₃	C₅×#551	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#123	2
C₅×#937	2
C₅×#936	2
C₄×C₅×#67	2
C₅×#896	2
C₅×#892	2
C₅×#883	2
C₅×D₃×C₂²⋊C₄	2
C₅×#879	2
C₅×#551	3
#5659	5

Subgroups — C5×#5659 #11756044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5659 #11756044