Subgroups — C₅×#5568 #11755953

Order 960.

424 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#947	C₂	1	1	yes	maximal, characteristic
3	C₅×#944	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#898	C₂	1	2	yes	maximal, complemented
5	C₅×#896	C₂	1	2	yes	maximal, complemented
6	C₅×#893	C₂	1	1	yes	maximal, characteristic
7	C₅×#887	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#876	C₂	1	2	yes	maximal, complemented
9	C₅×#875	C₂	1	1	yes	maximal, characteristic
10	C₅×#872	C₂	1	2	yes	maximal, complemented
11	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#556	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
16	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
17	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₅	C₂²	1	2	yes	—
22	C₅×(C₂×D₃)⋊C₄	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×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
29	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
30	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
31	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
32	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
33	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
34	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
35	C₄×C₅×Dic₃	C₂²	1	2	yes	—
36	#5568	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×#124	—	3	1	no	—
41	C₅×#124	—	3	1	no	—
42	C₅×C₂²⋊Q₈	—	3	2	no	—
43	C₄×C₅×Q₈	—	3	1	no	—
44	C₄×C₅×Q₈	—	3	1	no	—
45	C₄×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
46	C₄×C₅×D₄	—	3	1	no	—
47	C₅×#117	—	3	2	no	—
48	C₂²×C₅×D₃	C₂³	1	2	yes	—
49	C₃×C₅×Q₈	—	2	2	no	—
50	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
60	C₅×D₁₂	—	2	2	no	—
61	C₄×C₅×D₃	—	2	4	no	—
62	C₅×Dic₆	—	2	2	no	—
63	C₅×Dic₆	—	2	2	no	—
64	C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
65	#947	C₂×C₅	1	1	yes	characteristic
66	#944	C₂×C₅	1	1	yes	characteristic, complemented
67	#898	C₂×C₅	1	2	yes	complemented
68	#896	C₂×C₅	1	2	yes	complemented
69	#893	C₂×C₅	1	1	yes	characteristic
70	#887	C₂×C₅	1	1	yes	characteristic, complemented
71	#876	C₂×C₅	1	2	yes	complemented
72	#875	C₂×C₅	1	1	yes	characteristic
73	#872	C₂×C₅	1	2	yes	complemented
74	C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
75	C₂×C₅×Q₈	—	3	1	no	—
76	C₂×C₅×Q₈	—	3	1	no	—
77	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
78	C₂×C₅×D₄	—	3	1	no	—
79	C₂²×C₄×C₅	—	3	2	no	—
80	C₅×C₄⋊C₄	—	3	1	no	—
81	C₅×C₄⋊C₄	—	3	2	no	—
82	C₅×C₄⋊C₄	—	3	2	no	—
83	C₅×C₄⋊C₄	—	3	1	no	—
84	C₅×C₄⋊C₄	—	3	1	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	2	no	—
89	C₅×C₂²⋊C₄	—	3	2	no	—
90	C₅×C₂²⋊C₄	—	3	2	no	—
91	C₅×C₂²⋊C₄	—	3	2	no	—
92	C₅×C₂²⋊C₄	—	3	2	no	—
93	C₄²×C₅	—	3	2	no	—
94	C₄²×C₅	—	3	1	no	—
95	C₄²×C₅	—	3	1	no	—
96	C₄²×C₅	C₂×D₃	1	2	yes	—
97	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
98	#556	—	3	1	no	Sylow
99	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
100	C₂×C₅×D₃	—	2	2	no	—
101	C₂×C₅×D₃	—	2	2	no	—
102	C₂×C₅×D₃	—	2	2	no	—
103	C₃×C₄×C₅	—	2	1	no	—
104	C₃×C₄×C₅	—	2	2	no	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	—	2	1	no	—
107	C₃×C₄×C₅	#41	1	2	yes	—
108	C₅×Dic₃	—	2	2	no	—
109	C₅×Dic₃	—	2	1	no	—
110	C₅×Dic₃	—	2	1	no	—
111	C₅×Dic₃	—	2	1	no	—
112	C₅×Dic₃	#41	1	2	yes	—
113	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
114	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
115	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
116	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
117	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
118	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
119	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
120	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
121	C₃×C₄²	C₂²×C₅	1	2	yes	—
122	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
123	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
124	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
125	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
126	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
127	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
128	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
129	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
130	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
131	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
132	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
133	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
134	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
135	C₄×Dic₃	C₂²×C₅	1	2	yes	—
136	C₂³×C₅	—	3	2	no	—
137	C₅×Q₈	—	2	2	no	—
138	C₅×Q₈	—	6	2	no	—
139	C₅×Q₈	—	6	2	no	—
140	C₅×D₄	—	6	2	no	—
141	C₂×C₄×C₅	—	3	1	no	—
142	C₂×C₄×C₅	—	3	1	no	—
143	C₂×C₄×C₅	—	3	1	no	—
144	C₂×C₄×C₅	—	3	2	no	—
145	C₂×C₄×C₅	—	3	1	no	—
146	C₂×C₄×C₅	—	6	4	no	—
147	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
148	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
150	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
152	C₄⋊Q₈	—	3	1	no	—
153	#126	—	3	2	no	—
154	#126	—	3	2	no	—
155	#124	—	3	1	no	—
156	#124	—	3	1	no	—
157	C₂²⋊Q₈	—	3	2	no	—
158	C₄×Q₈	—	3	1	no	—
159	C₄×Q₈	—	3	1	no	—
160	C₄×Q₈	C₅×D₃	1	1	yes	characteristic, complemented
161	C₄×D₄	—	3	1	no	—
162	#117	—	3	2	no	—
163	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
165	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
166	C₅×D₃	—	4	2	no	—
167	C₂²×D₃	C₂³×C₅	1	2	yes	—
168	C₃×Q₈	—	2	2	no	—
169	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
170	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
172	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
174	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
175	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
176	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
177	C₂×Dic₃	C₂³×C₅	1	2	yes	—
178	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
179	D₁₂	—	2	2	no	—
180	C₄×D₃	—	2	4	no	—
181	Dic₆	—	2	2	no	—
182	Dic₆	—	2	2	no	—
183	C₂²×C₅	—	6	2	no	—
184	C₂²×C₅	—	6	2	no	—
185	C₂²×C₅	—	6	2	no	—
186	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	—	3	2	no	—
192	C₄×C₅	—	6	2	no	—
193	C₄×C₅	—	6	1	no	—
194	C₄×C₅	—	6	1	no	—
195	C₄×C₅	—	6	1	no	—
196	C₄×C₅	#235	1	2	yes	—
197	C₂×Q₈	—	3	1	no	—
198	C₂×Q₈	—	3	1	no	—
199	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
200	C₂×D₄	—	3	1	no	—
201	C₂²×C₄	—	3	2	no	—
202	C₄⋊C₄	—	3	1	no	—
203	C₄⋊C₄	—	3	2	no	—
204	C₄⋊C₄	—	3	2	no	—
205	C₄⋊C₄	—	3	1	no	—
206	C₄⋊C₄	—	3	1	no	—
207	C₄⋊C₄	—	3	2	no	—
208	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
209	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
210	C₂²⋊C₄	—	3	2	no	—
211	C₂²⋊C₄	—	3	2	no	—
212	C₂²⋊C₄	—	3	2	no	—
213	C₂²⋊C₄	—	3	2	no	—
214	C₂²⋊C₄	—	3	2	no	—
215	C₄²	—	3	2	no	—
216	C₄²	—	3	1	no	—
217	C₄²	—	3	1	no	—
218	C₄²	C₂×C₅×D₃	1	2	yes	—
219	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
220	C₃×C₅	#556	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
222	C₂×D₃	—	2	2	no	—
223	C₂×D₃	—	2	2	no	—
224	C₂×D₃	—	2	2	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	C₅×#41	1	2	yes	—
230	Dic₃	—	2	2	no	—
231	Dic₃	—	2	1	no	—
232	Dic₃	—	2	1	no	—
233	Dic₃	—	2	1	no	—
234	Dic₃	C₅×#41	1	2	yes	—
235	C₂×C₅	—	12	2	no	—
236	C₂×C₅	C₂×#235	1	1	yes	characteristic
237	C₂×C₅	#1007	1	1	yes	characteristic
238	C₂×C₅	D₃×#41	1	1	yes	characteristic
239	C₂³	—	3	2	no	—
240	Q₈	—	2	2	no	—
241	Q₈	—	6	2	no	—
242	Q₈	—	6	2	no	—
243	D₄	—	6	2	no	—
244	C₂×C₄	—	3	1	no	—
245	C₂×C₄	—	3	1	no	—
246	C₂×C₄	—	3	1	no	—
247	C₂×C₄	—	3	2	no	—
248	C₂×C₄	—	3	1	no	—
249	C₂×C₄	—	6	4	no	—
250	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
251	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
253	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
255	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
256	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
257	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
258	D₃	—	4	2	no	—
259	C₅	#5568	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	6	2	no	—
261	C₂²	—	6	2	no	—
262	C₂²	—	6	2	no	—
263	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	3	2	no	—
269	C₄	—	6	2	no	—
270	C₄	—	6	1	no	—
271	C₄	—	6	1	no	—
272	C₄	—	6	1	no	—
273	C₄	C₅×#235	1	2	yes	—
274	C₃	C₅×#556	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	12	2	no	—
276	C₂	C₂×C₅×#235	1	1	yes	characteristic
277	C₂	C₅×#1007	1	1	yes	characteristic
278	C₂	C₅×D₃×#41	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×C₄×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₅×D₁₂	C₂²
hypercenter	C₄×C₅×Q₈	D₃
hypocenter	C₃	C₅×#556
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₅×Q₈	C₂	semidirect
3. C₅×#944	C₂	semidirect
4. C₅×#898	C₂	semidirect
5. C₅×#896	C₂	semidirect
7. C₅×#887	C₂	semidirect
8. C₅×#876	C₂	semidirect
10. C₅×#872	C₂	semidirect
11. C₄×C₅×Dic₆	C₂	semidirect
36. #5568	C₅	direct (N × H)
45. C₄×C₅×Q₈	D₃	semidirect
64. C₃×C₄×Q₈	C₂×C₅	semidirect
66. #944	C₂×C₅	semidirect
67. #898	C₂×C₅	semidirect
68. #896	C₂×C₅	semidirect
70. #887	C₂×C₅	semidirect
71. #876	C₂×C₅	semidirect
73. #872	C₂×C₅	semidirect
74. C₄×Dic₆	C₂×C₅	semidirect
160. C₄×Q₈	C₅×D₃	semidirect
220. C₃×C₅	#556	semidirect
259. C₅	#5568	direct (N × H)
274. C₃	C₅×#556	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×Q₈	2
C₅×#947	2
C₅×#944	2
C₅×#898	2
C₅×#896	2
C₅×#893	2
C₅×#887	2
C₅×#876	2
C₅×#875	2
C₅×#872	2
C₄×C₅×Dic₆	2
C₅×#556	3
#5568	5

Subgroups — C5×#5568 #11755953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5568 #11755953