Subgroups — C₅×#5625 #11756010

Order 960.

384 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×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×Q₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#944	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#926	C₂	1	1	yes	maximal, characteristic
5	C₅×#924	C₂	1	1	yes	maximal, characteristic
6	C₅×#890	C₂	1	2	yes	maximal, complemented
7	C₅×#890	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#889	C₂	1	2	yes	maximal, complemented
9	C₅×#887	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#879	C₂	1	2	yes	maximal
11	C₅×#879	C₂	1	2	yes	maximal
12	C₅×#541	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₅×#217	C₂²	1	2	yes	—
20	C₅×#217	C₂²	1	1	yes	characteristic
21	C₅×#217	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
23	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
24	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
28	C₅×Dic₃⋊C₄	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	C₄×C₅×Dic₃	C₂²	1	2	yes	—
33	C₄×C₅×Dic₃	C₂²	1	2	yes	—
34	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
35	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
36	#5625	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₅×C₄⋊Q₈	—	3	1	no	—
38	C₅×#126	—	3	2	no	—
39	C₅×#126	—	3	2	no	—
40	C₅×#125	—	3	2	no	—
41	C₅×#125	—	3	1	no	—
42	C₅×#125	—	3	2	no	—
43	C₅×C₂²⋊Q₈	—	3	1	no	—
44	C₅×C₂²⋊Q₈	D₃	1	1	yes	characteristic
45	C₄×C₅×Q₈	—	3	1	no	—
46	C₄×C₅×Q₈	—	3	1	no	—
47	C₅×#117	—	3	1	no	—
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₃×C₅×Q₈	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
56	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂³	1	2	yes	—
59	C₂×C₅×Dic₃	C₂³	1	2	yes	—
60	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₅×Dic₆	—	2	2	no	—
62	C₃×C₂²⋊Q₈	C₂×C₅	1	1	yes	characteristic
63	Q₈×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
64	#944	C₂×C₅	1	1	yes	characteristic, complemented
65	#926	C₂×C₅	1	1	yes	characteristic
66	#924	C₂×C₅	1	1	yes	characteristic
67	#890	C₂×C₅	1	2	yes	complemented
68	#890	C₂×C₅	1	1	yes	characteristic, complemented
69	#889	C₂×C₅	1	2	yes	complemented
70	#887	C₂×C₅	1	1	yes	characteristic, complemented
71	#879	C₂×C₅	1	2	yes	—
72	#879	C₂×C₅	1	2	yes	—
73	C₂×C₅×Q₈	—	3	1	no	—
74	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
76	C₅×C₄⋊C₄	—	3	1	no	—
77	C₅×C₄⋊C₄	—	3	1	no	—
78	C₅×C₄⋊C₄	—	3	1	no	—
79	C₅×C₄⋊C₄	—	3	2	no	—
80	C₅×C₄⋊C₄	—	3	2	no	—
81	C₅×C₄⋊C₄	—	3	2	no	—
82	C₅×C₄⋊C₄	—	3	2	no	—
83	C₅×C₄⋊C₄	—	3	2	no	—
84	C₅×C₄⋊C₄	—	3	2	no	—
85	C₅×C₄⋊C₄	—	3	2	no	—
86	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
87	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
88	C₅×C₂²⋊C₄	—	3	1	no	—
89	C₅×C₂²⋊C₄	—	3	1	no	—
90	C₅×C₂²⋊C₄	—	3	2	no	—
91	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
92	C₄²×C₅	—	3	1	no	—
93	C₄²×C₅	—	3	1	no	—
94	C₄²×C₅	—	3	2	no	—
95	C₄²×C₅	—	3	2	no	—
96	#541	—	3	1	no	Sylow
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	—	2	1	no	—
99	C₂²×C₃×C₅	—	2	1	no	—
100	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	—	2	2	no	—
103	C₃×C₄×C₅	—	2	2	no	—
104	C₃×C₄×C₅	#41	1	2	yes	—
105	C₅×Dic₃	—	2	2	no	—
106	C₅×Dic₃	—	2	1	no	—
107	C₅×Dic₃	—	2	2	no	—
108	C₅×Dic₃	—	2	1	no	—
109	C₅×Dic₃	—	2	1	no	—
110	C₅×Dic₃	—	2	1	no	—
111	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
113	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
114	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
115	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
116	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
117	#217	C₂²×C₅	1	2	yes	—
118	#217	C₂²×C₅	1	1	yes	characteristic
119	#217	C₂²×C₅	1	1	yes	characteristic
120	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
121	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
122	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
123	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
124	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
125	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
126	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
127	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
128	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
129	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
130	C₄×Dic₃	C₂²×C₅	1	2	yes	—
131	C₄×Dic₃	C₂²×C₅	1	2	yes	—
132	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
133	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
134	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
135	C₅×Q₈	—	2	2	no	—
136	C₅×Q₈	—	6	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	3	1	no	—
139	C₂×C₄×C₅	—	3	1	no	—
140	C₂×C₄×C₅	—	3	1	no	—
141	C₂×C₄×C₅	—	3	2	no	—
142	C₂×C₄×C₅	—	3	1	no	—
143	C₂×C₄×C₅	—	3	2	no	—
144	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
145	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
146	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
148	C₄⋊Q₈	—	3	1	no	—
149	#126	—	3	2	no	—
150	#126	—	3	2	no	—
151	#125	—	3	2	no	—
152	#125	—	3	1	no	—
153	#125	—	3	2	no	—
154	C₂²⋊Q₈	—	3	1	no	—
155	C₂²⋊Q₈	C₅×D₃	1	1	yes	characteristic
156	C₄×Q₈	—	3	1	no	—
157	C₄×Q₈	—	3	1	no	—
158	#117	—	3	1	no	—
159	C₂×C₃×C₅	—	4	1	no	—
160	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
161	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
162	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
163	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
164	C₃×Q₈	—	2	2	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
167	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
168	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
170	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
171	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
172	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
173	C₂×Dic₃	C₂³×C₅	1	2	yes	—
174	C₂×Dic₃	C₂³×C₅	1	2	yes	—
175	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
176	Dic₆	—	2	2	no	—
177	C₂²×C₅	—	2	1	no	—
178	C₂²×C₅	—	2	1	no	—
179	C₂²×C₅	—	2	1	no	—
180	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	6	2	no	—
185	C₄×C₅	—	6	1	no	—
186	C₄×C₅	—	6	1	no	—
187	C₄×C₅	—	6	2	no	—
188	C₄×C₅	—	6	1	no	—
189	C₄×C₅	—	6	1	no	—
190	C₄×C₅	#237	1	2	yes	—
191	C₂×Q₈	—	3	1	no	—
192	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
194	C₄⋊C₄	—	3	1	no	—
195	C₄⋊C₄	—	3	1	no	—
196	C₄⋊C₄	—	3	1	no	—
197	C₄⋊C₄	—	3	2	no	—
198	C₄⋊C₄	—	3	2	no	—
199	C₄⋊C₄	—	3	2	no	—
200	C₄⋊C₄	—	3	2	no	—
201	C₄⋊C₄	—	3	2	no	—
202	C₄⋊C₄	—	3	2	no	—
203	C₄⋊C₄	—	3	2	no	—
204	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
205	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
206	C₂²⋊C₄	—	3	1	no	—
207	C₂²⋊C₄	—	3	1	no	—
208	C₂²⋊C₄	—	3	2	no	—
209	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
210	C₄²	—	3	1	no	—
211	C₄²	—	3	1	no	—
212	C₄²	—	3	2	no	—
213	C₄²	—	3	2	no	—
214	C₃×C₅	#541	1	1	yes	characteristic, complemented, Hall
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	C₅×#41	1	2	yes	—
223	Dic₃	—	2	2	no	—
224	Dic₃	—	2	1	no	—
225	Dic₃	—	2	2	no	—
226	Dic₃	—	2	1	no	—
227	Dic₃	—	2	1	no	—
228	Dic₃	—	2	1	no	—
229	C₂×C₅	—	4	1	no	—
230	C₂×C₅	C₂×#237	1	1	yes	characteristic
231	C₂×C₅	#1007	1	1	yes	characteristic
232	C₂×C₅	#1010	1	1	yes	characteristic
233	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	Q₈	—	6	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	3	1	no	—
238	C₂×C₄	—	3	1	no	—
239	C₂×C₄	—	3	1	no	—
240	C₂×C₄	—	3	2	no	—
241	C₂×C₄	—	3	1	no	—
242	C₂×C₄	—	3	2	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×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
247	C₂×C₃	—	4	1	no	—
248	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
249	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
250	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
251	C₅	#5625	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	6	2	no	—
260	C₄	—	6	1	no	—
261	C₄	—	6	1	no	—
262	C₄	—	6	2	no	—
263	C₄	—	6	1	no	—
264	C₄	—	6	1	no	—
265	C₄	C₅×#237	1	2	yes	—
266	C₃	C₅×#541	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	4	1	no	—
268	C₂	C₂×C₅×#237	1	1	yes	characteristic
269	C₂	C₅×#1007	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×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂³×C₃×C₅	C₂³
hypercenter	C₅×C₂²⋊Q₈	D₃
hypocenter	C₃	C₅×#541
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₅×Q₈×Dic₃	C₂	semidirect
3. C₅×#944	C₂	semidirect
6. C₅×#890	C₂	semidirect
7. C₅×#890	C₂	semidirect
8. C₅×#889	C₂	semidirect
9. C₅×#887	C₂	semidirect
36. #5625	C₅	direct (N × H)
63. Q₈×Dic₃	C₂×C₅	semidirect
64. #944	C₂×C₅	semidirect
67. #890	C₂×C₅	semidirect
68. #890	C₂×C₅	semidirect
69. #889	C₂×C₅	semidirect
70. #887	C₂×C₅	semidirect
214. C₃×C₅	#541	semidirect
251. C₅	#5625	direct (N × H)
266. C₃	C₅×#541	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₂²⋊Q₈	2
C₅×Q₈×Dic₃	2
C₅×#944	2
C₅×#926	2
C₅×#924	2
C₅×#890	2
C₅×#889	2
C₅×#887	2
C₅×#879	2
C₅×#541	3
#5625	5

Subgroups — C5×#5625 #11756010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5625 #11756010