Subgroups — C₅×#5684 #11756069

Order 960.

440 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	C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#946	C₂	1	2	yes	maximal, complemented
3	C₅×#939	C₂	1	2	yes	maximal, complemented
4	C₅×#886	C₂	1	2	yes	maximal, complemented
5	C₅×#884	C₂	1	2	yes	maximal, complemented
6	C₅×#883	C₂	1	2	yes	maximal, complemented
7	C₅×#878	C₂	1	2	yes	maximal, complemented
8	C₅×#875	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#870	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#562	—	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×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
29	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
30	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
31	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
32	#5684	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₅×#125	—	3	1	no	—
34	C₅×#124	—	3	1	no	—
35	C₅×#124	D₃	1	1	yes	characteristic, complemented
36	C₅×#123	—	3	2	no	—
37	C₅×#123	—	3	2	no	—
38	C₅×C₂²⋊Q₈	—	3	2	no	—
39	C₅×C₂²⋊Q₈	—	3	2	no	—
40	C₅×#121	—	3	2	no	—
41	C₅×#121	—	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	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₅×#67	—	4	2	no	—
51	C₅×#67	—	4	2	no	—
52	C₂×C₅×Dic₃	—	4	2	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	—	4	2	no	—
58	C₅×Dic₆	—	4	1	no	—
59	C₃×#124	C₂×C₅	1	1	yes	characteristic, complemented
60	#946	C₂×C₅	1	2	yes	complemented
61	#939	C₂×C₅	1	2	yes	complemented
62	#886	C₂×C₅	1	2	yes	complemented
63	#884	C₂×C₅	1	2	yes	complemented
64	#883	C₂×C₅	1	2	yes	complemented
65	#878	C₂×C₅	1	2	yes	complemented
66	#875	C₂×C₅	1	1	yes	characteristic, complemented
67	#870	C₂×C₅	1	1	yes	characteristic, 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×C₄⋊C₄	—	3	2	no	—
80	C₅×C₄⋊C₄	—	3	1	no	—
81	C₅×C₄⋊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	—	3	2	no	—
86	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
87	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
88	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
89	#562	—	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	1	no	—
98	C₃×C₄×C₅	—	2	2	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	—	2	2	no	—
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	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
120	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
121	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
122	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
123	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
124	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
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	—	12	1	no	—
131	C₅×D₄	—	12	2	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	—	12	2	no	—
137	C₂×C₄×C₅	—	12	2	no	—
138	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
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	2	yes	—
142	#125	—	3	1	no	—
143	#124	—	3	1	no	—
144	#124	C₅×D₃	1	1	yes	characteristic, complemented
145	#123	—	3	2	no	—
146	#123	—	3	2	no	—
147	C₂²⋊Q₈	—	3	2	no	—
148	C₂²⋊Q₈	—	3	2	no	—
149	#121	—	3	2	no	—
150	#121	—	3	2	no	—
151	C₂×C₃×C₅	—	4	2	no	—
152	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
155	C₅×D₃	—	4	2	no	—
156	C₂³×C₃	C₂³×C₅	1	2	yes	—
157	C₂²×D₃	C₂³×C₅	1	2	yes	—
158	C₃×Q₈	—	4	1	no	—
159	C₃×D₄	—	4	1	no	—
160	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
163	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
164	#67	—	4	2	no	—
165	#67	—	4	2	no	—
166	C₂×Dic₃	—	4	2	no	—
167	C₂×Dic₃	C₂³×C₅	1	2	yes	—
168	C₂×Dic₃	C₂³×C₅	1	2	yes	—
169	C₂×Dic₃	C₂³×C₅	1	2	yes	—
170	D₁₂	—	4	1	no	—
171	C₄×D₃	—	4	2	no	—
172	Dic₆	—	4	1	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	6	2	no	—
177	C₂²×C₅	—	6	2	no	—
178	C₂²×C₅	—	6	2	no	—
179	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	—	6	2	no	—
185	C₄×C₅	—	6	2	no	—
186	C₄×C₅	—	6	2	no	—
187	C₂×Q₈	—	3	1	no	—
188	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
189	C₂×D₄	—	3	2	no	—
190	C₂×D₄	—	3	2	no	—
191	C₂×D₄	—	3	1	no	—
192	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
193	C₂²×C₄	—	3	2	no	—
194	C₂²×C₄	—	3	2	no	—
195	C₄⋊C₄	—	3	2	no	—
196	C₄⋊C₄	—	3	2	no	—
197	C₄⋊C₄	—	3	2	no	—
198	C₄⋊C₄	—	3	2	no	—
199	C₄⋊C₄	—	3	1	no	—
200	C₄⋊C₄	—	3	1	no	—
201	C₂²⋊C₄	—	3	2	no	—
202	C₂²⋊C₄	—	3	2	no	—
203	C₂²⋊C₄	—	3	2	no	—
204	C₂²⋊C₄	—	3	2	no	—
205	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
206	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
207	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
208	C₃×C₅	#562	1	1	yes	characteristic, complemented, Hall
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
213	C₂×D₃	—	2	2	no	—
214	C₂×D₃	—	2	2	no	—
215	C₂×D₃	—	2	2	no	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	1	no	—
220	Dic₃	—	2	2	no	—
221	Dic₃	—	2	2	no	—
222	Dic₃	—	2	2	no	—
223	C₂×C₅	—	4	2	no	—
224	C₂×C₅	—	12	2	no	—
225	C₂×C₅	#1004	1	1	yes	characteristic
226	C₂×C₅	#1009	1	1	yes	characteristic
227	C₂×C₅	#1010	1	1	yes	characteristic
228	C₂³	—	3	2	no	—
229	C₂³	C₂²×C₅×D₃	1	2	yes	—
230	Q₈	—	4	1	no	—
231	Q₈	—	12	1	no	—
232	D₄	—	4	1	no	—
233	D₄	—	12	1	no	—
234	D₄	—	12	2	no	—
235	D₄	—	12	2	no	—
236	C₂×C₄	—	3	2	no	—
237	C₂×C₄	—	3	2	no	—
238	C₂×C₄	—	3	2	no	—
239	C₂×C₄	—	12	2	no	—
240	C₂×C₄	—	12	2	no	—
241	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
245	C₂×C₃	—	4	2	no	—
246	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
247	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
248	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
249	D₃	—	4	2	no	—
250	C₅	#5684	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	6	2	no	—
255	C₂²	—	6	2	no	—
256	C₂²	—	6	2	no	—
257	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	6	2	no	—
263	C₄	—	6	2	no	—
264	C₄	—	6	2	no	—
265	C₃	C₅×#562	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	4	2	no	—
267	C₂	—	12	2	no	—
268	C₂	C₅×#1004	1	1	yes	characteristic
269	C₂	C₅×#1009	1	1	yes	characteristic
270	C₂	C₅×#1010	1	1	yes	characteristic

	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×#562
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×#946	C₂	semidirect
3. C₅×#939	C₂	semidirect
4. C₅×#886	C₂	semidirect
5. C₅×#884	C₂	semidirect
6. C₅×#883	C₂	semidirect
7. C₅×#878	C₂	semidirect
8. C₅×#875	C₂	semidirect
9. C₅×#870	C₂	semidirect
32. #5684	C₅	direct (N × H)
35. C₅×#124	D₃	semidirect
59. C₃×#124	C₂×C₅	semidirect
60. #946	C₂×C₅	semidirect
61. #939	C₂×C₅	semidirect
62. #886	C₂×C₅	semidirect
63. #884	C₂×C₅	semidirect
64. #883	C₂×C₅	semidirect
65. #878	C₂×C₅	semidirect
66. #875	C₂×C₅	semidirect
67. #870	C₂×C₅	semidirect
144. #124	C₅×D₃	semidirect
208. C₃×C₅	#562	semidirect
250. C₅	#5684	direct (N × H)
265. C₃	C₅×#562	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×#946	2
C₅×#939	2
C₅×#886	2
C₅×#884	2
C₅×#883	2
C₅×#878	2
C₅×#875	2
C₅×#870	2
C₅×#562	3
#5684	5

Subgroups — C5×#5684 #11756069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5684 #11756069