Subgroups — C₅×#5670 #11756055

Order 960.

472 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#947	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#944	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#939	C₂	1	2	yes	maximal, complemented
5	C₅×#885	C₂	1	2	yes	maximal, complemented
6	C₅×#884	C₂	1	2	yes	maximal, complemented
7	C₅×#67⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₅×#877	C₂	1	2	yes	maximal, complemented
9	C₅×#872	C₂	1	2	yes	maximal, complemented
10	C₅×#555	—	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	C₂×C₅×#67	C₂²	1	2	yes	—
15	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
16	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
18	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₅×#217	C₂²	1	2	yes	—
23	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
24	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
27	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
28	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
29	C₄×C₅×Dic₃	C₂²	1	2	yes	—
30	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
31	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
32	#5670	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₅×C₄⋊Q₈	—	3	1	no	—
34	C₅×#124	—	3	1	no	—
35	C₅×#124	—	3	2	no	—
36	C₅×#124	D₃	1	1	yes	characteristic, complemented
37	C₅×#121	—	3	2	no	—
38	C₅×#121	—	3	2	no	—
39	C₄×C₅×D₄	—	3	2	no	—
40	C₅×#117	—	3	2	no	—
41	C₅×#117	—	3	2	no	—
42	C₂³×C₃×C₅	C₂³	1	2	yes	—
43	C₂²×C₅×D₃	C₂³	1	2	yes	—
44	C₃×C₅×Q₈	—	4	1	no	—
45	C₃×C₅×D₄	—	4	1	no	—
46	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₅×#67	—	2	4	no	—
51	C₅×#67	—	4	2	no	—
52	C₂×C₅×Dic₃	—	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	C₅×D₁₂	—	4	1	no	—
57	C₄×C₅×D₃	—	2	4	no	—
58	C₅×Dic₆	—	4	1	no	—
59	C₃×#124	C₂×C₅	1	1	yes	characteristic, complemented
60	#947	C₂×C₅	1	1	yes	characteristic, complemented
61	#944	C₂×C₅	1	1	yes	characteristic, complemented
62	#939	C₂×C₅	1	2	yes	complemented
63	#885	C₂×C₅	1	2	yes	complemented
64	#884	C₂×C₅	1	2	yes	complemented
65	#67⋊C₄	C₂×C₅	1	2	yes	complemented
66	#877	C₂×C₅	1	2	yes	complemented
67	#872	C₂×C₅	1	2	yes	complemented
68	C₂×C₅×Q₈	—	3	1	no	—
69	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
70	C₂×C₅×D₄	—	3	2	no	—
71	C₂×C₅×D₄	—	3	2	no	—
72	C₂×C₅×D₄	—	3	1	no	—
73	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
74	C₂²×C₄×C₅	—	3	2	no	—
75	C₂²×C₄×C₅	—	3	2	no	—
76	C₅×C₄⋊C₄	—	3	2	no	—
77	C₅×C₄⋊C₄	—	3	2	no	—
78	C₅×C₄⋊C₄	—	3	2	no	—
79	C₅×C₂²⋊C₄	—	3	2	no	—
80	C₅×C₂²⋊C₄	—	3	2	no	—
81	C₅×C₂²⋊C₄	—	3	2	no	—
82	C₅×C₂²⋊C₄	—	3	2	no	—
83	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
84	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
85	C₄²×C₅	—	3	1	no	—
86	C₄²×C₅	—	3	2	no	—
87	C₄²×C₅	—	3	1	no	—
88	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
89	#555	—	3	1	no	Sylow
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	2	2	no	—
92	C₂²×C₃×C₅	—	2	2	no	—
93	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
94	C₂×C₅×D₃	—	2	2	no	—
95	C₂×C₅×D₃	—	2	2	no	—
96	C₂×C₅×D₃	—	2	2	no	—
97	C₃×C₄×C₅	—	2	2	no	—
98	C₃×C₄×C₅	—	2	1	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	2	1	no	—
101	C₅×Dic₃	—	2	2	no	—
102	C₅×Dic₃	—	2	2	no	—
103	C₅×Dic₃	#41	1	4	yes	—
104	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
106	C₂×#67	C₂²×C₅	1	2	yes	—
107	C₂×#67	C₂²×C₅	1	2	yes	—
108	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
109	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
111	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
112	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
113	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
114	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
115	#217	C₂²×C₅	1	2	yes	—
116	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
117	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
118	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
119	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
120	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
121	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
122	C₄×Dic₃	C₂²×C₅	1	2	yes	—
123	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
124	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
125	C₂³×C₅	—	3	2	no	—
126	C₂³×C₅	C₂²×D₃	1	2	yes	—
127	C₅×Q₈	—	4	1	no	—
128	C₅×Q₈	—	12	1	no	—
129	C₅×D₄	—	4	1	no	—
130	C₅×D₄	—	6	4	no	—
131	C₅×D₄	—	12	1	no	—
132	C₅×D₄	—	12	2	no	—
133	C₂×C₄×C₅	—	3	2	no	—
134	C₂×C₄×C₅	—	3	2	no	—
135	C₂×C₄×C₅	—	3	2	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	1	yes	characteristic
140	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₄⋊Q₈	—	3	1	no	—
143	#124	—	3	1	no	—
144	#124	—	3	2	no	—
145	#124	C₅×D₃	1	1	yes	characteristic, complemented
146	#121	—	3	2	no	—
147	#121	—	3	2	no	—
148	C₄×D₄	—	3	2	no	—
149	#117	—	3	2	no	—
150	#117	—	3	2	no	—
151	C₂×C₃×C₅	—	4	2	no	—
152	C₂×C₃×C₅	C₂×#41	1	2	yes	—
153	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
154	C₅×D₃	—	4	2	no	—
155	C₂³×C₃	C₂³×C₅	1	2	yes	—
156	C₂²×D₃	C₂³×C₅	1	2	yes	—
157	C₃×Q₈	—	4	1	no	—
158	C₃×D₄	—	4	1	no	—
159	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
161	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
163	#67	—	2	4	no	—
164	#67	—	4	2	no	—
165	C₂×Dic₃	—	2	4	no	—
166	C₂×Dic₃	C₂³×C₅	1	2	yes	—
167	C₂×Dic₃	C₂³×C₅	1	2	yes	—
168	C₂×Dic₃	C₂³×C₅	1	2	yes	—
169	D₁₂	—	4	1	no	—
170	C₄×D₃	—	2	4	no	—
171	Dic₆	—	4	1	no	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	6	2	no	—
176	C₂²×C₅	—	6	2	no	—
177	C₂²×C₅	—	6	2	no	—
178	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	3	4	no	—
184	C₄×C₅	—	6	2	no	—
185	C₄×C₅	—	6	2	no	—
186	C₂×Q₈	—	3	1	no	—
187	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
188	C₂×D₄	—	3	2	no	—
189	C₂×D₄	—	3	2	no	—
190	C₂×D₄	—	3	1	no	—
191	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
192	C₂²×C₄	—	3	2	no	—
193	C₂²×C₄	—	3	2	no	—
194	C₄⋊C₄	—	3	2	no	—
195	C₄⋊C₄	—	3	2	no	—
196	C₄⋊C₄	—	3	2	no	—
197	C₂²⋊C₄	—	3	2	no	—
198	C₂²⋊C₄	—	3	2	no	—
199	C₂²⋊C₄	—	3	2	no	—
200	C₂²⋊C₄	—	3	2	no	—
201	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
202	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
203	C₄²	—	3	1	no	—
204	C₄²	—	3	2	no	—
205	C₄²	—	3	1	no	—
206	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
207	C₃×C₅	#555	1	1	yes	characteristic, complemented, Hall
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
212	C₂×D₃	—	2	2	no	—
213	C₂×D₃	—	2	2	no	—
214	C₂×D₃	—	2	2	no	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	Dic₃	—	2	2	no	—
220	Dic₃	—	2	2	no	—
221	Dic₃	C₅×#41	1	4	yes	—
222	C₂×C₅	—	4	2	no	—
223	C₂×C₅	—	12	2	no	—
224	C₂×C₅	#1004	1	1	yes	characteristic
225	C₂×C₅	D₃×#41	1	2	yes	—
226	C₂³	—	3	2	no	—
227	C₂³	C₂²×C₅×D₃	1	2	yes	—
228	Q₈	—	4	1	no	—
229	Q₈	—	12	1	no	—
230	D₄	—	4	1	no	—
231	D₄	—	6	4	no	—
232	D₄	—	12	1	no	—
233	D₄	—	12	2	no	—
234	C₂×C₄	—	3	2	no	—
235	C₂×C₄	—	3	2	no	—
236	C₂×C₄	—	3	2	no	—
237	C₂×C₄	—	6	4	no	—
238	C₂×C₄	—	6	4	no	—
239	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
240	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
243	C₂×C₃	—	4	2	no	—
244	C₂×C₃	C₂×C₅×#41	1	2	yes	—
245	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
246	D₃	—	4	2	no	—
247	C₅	#5670	1	1	yes	characteristic, direct product factor, Sylow
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	6	2	no	—
252	C₂²	—	6	2	no	—
253	C₂²	—	6	2	no	—
254	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	2	no	—
259	C₄	—	3	4	no	—
260	C₄	—	6	2	no	—
261	C₄	—	6	2	no	—
262	C₃	C₅×#555	1	1	yes	characteristic, complemented, Sylow
263	C₂	—	4	2	no	—
264	C₂	—	12	2	no	—
265	C₂	C₅×#1004	1	1	yes	characteristic
266	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×#555
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. C₃×C₅×#124	C₂	semidirect
2. C₅×#947	C₂	semidirect
3. C₅×#944	C₂	semidirect
4. C₅×#939	C₂	semidirect
5. C₅×#885	C₂	semidirect
6. C₅×#884	C₂	semidirect
7. C₅×#67⋊C₄	C₂	semidirect
8. C₅×#877	C₂	semidirect
9. C₅×#872	C₂	semidirect
32. #5670	C₅	direct (N × H)
36. C₅×#124	D₃	semidirect
59. C₃×#124	C₂×C₅	semidirect
60. #947	C₂×C₅	semidirect
61. #944	C₂×C₅	semidirect
62. #939	C₂×C₅	semidirect
63. #885	C₂×C₅	semidirect
64. #884	C₂×C₅	semidirect
65. #67⋊C₄	C₂×C₅	semidirect
66. #877	C₂×C₅	semidirect
67. #872	C₂×C₅	semidirect
145. #124	C₅×D₃	semidirect
207. C₃×C₅	#555	semidirect
247. C₅	#5670	direct (N × H)
262. C₃	C₅×#555	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#124	2
C₅×#947	2
C₅×#944	2
C₅×#939	2
C₅×#885	2
C₅×#884	2
C₅×#67⋊C₄	2
C₅×#877	2
C₅×#872	2
C₅×#555	3
#5670	5

Subgroups — C5×#5670 #11756055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5670 #11756055