Subgroups — C₅×#5557 #11755942

Order 960.

468 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#938	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×#926	C₂	1	2	yes	maximal, complemented
6	C₅×#897	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#890	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#883	C₂	1	2	yes	maximal, complemented
9	C₅×#879	C₂	1	2	yes	maximal, complemented
10	C₄²×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#529	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
15	C₂×C₅×#67	C₂²	1	2	yes	—
16	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
19	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
23	C₅×#217	C₂²	1	2	yes	—
24	C₅×#217	C₂²	1	2	yes	—
25	C₅×#217	C₂²	1	2	yes	—
26	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
29	C₅×C₄⋊Dic₃	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	1	yes	characteristic
34	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
35	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
36	#5557	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₅×#126	—	3	2	no	—
38	C₅×#125	—	3	1	no	—
39	C₅×#124	—	3	1	no	—
40	C₅×#123	—	3	2	no	—
41	C₅×C₂²⋊Q₈	—	3	1	no	—
42	C₅×#121	—	3	1	no	—
43	C₄×C₅×Q₈	—	3	1	no	—
44	C₄×C₅×D₄	—	3	2	no	—
45	C₄×C₅×D₄	D₃	1	1	yes	characteristic, complemented
46	C₅×#117	—	3	2	no	—
47	C₂×C₄²×C₅	—	3	1	no	—
48	C₂³×C₃×C₅	C₂³	1	2	yes	—
49	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
50	C₃×C₅×D₄	—	2	2	no	—
51	C₂×C₃×C₄×C₅	—	2	4	no	—
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₅×#67	—	2	4	no	—
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₂×C₅×Dic₃	C₂³	1	2	yes	—
61	C₂×C₅×Dic₃	C₂³	1	2	yes	—
62	C₄×C₅×D₃	—	2	2	no	—
63	C₄×C₅×D₃	—	2	2	no	—
64	C₄×C₅×D₃	—	2	2	no	—
65	C₅×Dic₆	—	2	2	no	—
66	C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic, complemented
67	#938	C₂×C₅	1	1	yes	characteristic
68	#936	C₂×C₅	1	1	yes	characteristic, complemented
69	C₄×#67	C₂×C₅	1	2	yes	complemented
70	#926	C₂×C₅	1	2	yes	complemented
71	#897	C₂×C₅	1	1	yes	characteristic, complemented
72	#890	C₂×C₅	1	1	yes	characteristic, complemented
73	#883	C₂×C₅	1	2	yes	complemented
74	#879	C₂×C₅	1	2	yes	complemented
75	C₄²×D₃	C₂×C₅	1	1	yes	characteristic, complemented
76	C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
77	C₂×C₅×Q₈	—	3	1	no	—
78	C₂×C₅×D₄	—	3	2	no	—
79	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
80	C₂²×C₄×C₅	—	3	1	no	—
81	C₂²×C₄×C₅	—	3	1	no	—
82	C₂²×C₄×C₅	—	3	1	no	—
83	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
84	C₅×C₄⋊C₄	—	3	2	no	—
85	C₅×C₄⋊C₄	—	3	2	no	—
86	C₅×C₄⋊C₄	—	3	2	no	—
87	C₅×C₄⋊C₄	—	3	1	no	—
88	C₅×C₄⋊C₄	—	3	2	no	—
89	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
90	C₅×C₂²⋊C₄	—	3	2	no	—
91	C₅×C₂²⋊C₄	—	3	2	no	—
92	C₅×C₂²⋊C₄	—	3	2	no	—
93	C₅×C₂²⋊C₄	—	3	2	no	—
94	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
95	C₄²×C₅	—	3	1	no	—
96	C₄²×C₅	—	3	1	no	—
97	C₄²×C₅	—	3	1	no	—
98	C₄²×C₅	—	3	2	no	—
99	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
100	#529	—	3	1	no	Sylow
101	C₂²×C₃×C₅	—	2	2	no	—
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	—	2	2	no	—
104	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
105	C₂×C₅×D₃	—	2	1	no	—
106	C₂×C₅×D₃	—	2	1	no	—
107	C₂×C₅×D₃	#41	1	2	yes	—
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	—	2	2	no	—
110	C₃×C₄×C₅	#41	1	2	yes	—
111	C₃×C₄×C₅	#41	1	2	yes	—
112	C₅×Dic₃	—	2	2	no	—
113	C₅×Dic₃	—	2	1	no	—
114	C₅×Dic₃	—	2	1	no	—
115	C₅×Dic₃	—	2	1	no	—
116	C₅×Dic₃	—	2	2	no	—
117	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
119	C₂×#67	C₂²×C₅	1	2	yes	—
120	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
123	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
124	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
125	C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
126	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
127	#217	C₂²×C₅	1	2	yes	—
128	#217	C₂²×C₅	1	2	yes	—
129	#217	C₂²×C₅	1	2	yes	—
130	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
131	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
132	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
133	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
134	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
135	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
136	C₄×Dic₃	C₂²×C₅	1	2	yes	—
137	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
138	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
139	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
140	C₂³×C₅	—	3	1	no	—
141	C₂³×C₅	C₂²×D₃	1	2	yes	—
142	C₅×Q₈	—	6	2	no	—
143	C₅×D₄	—	2	2	no	—
144	C₅×D₄	—	6	4	no	—
145	C₂×C₄×C₅	—	2	4	no	—
146	C₂×C₄×C₅	—	3	1	no	—
147	C₂×C₄×C₅	—	3	1	no	—
148	C₂×C₄×C₅	—	3	1	no	—
149	C₂×C₄×C₅	—	3	2	no	—
150	C₂×C₄×C₅	—	3	2	no	—
151	C₂×C₄×C₅	—	6	2	no	—
152	C₂×C₄×C₅	—	6	2	no	—
153	C₂×C₄×C₅	—	6	2	no	—
154	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
156	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
158	#126	—	3	2	no	—
159	#125	—	3	1	no	—
160	#124	—	3	1	no	—
161	#123	—	3	2	no	—
162	C₂²⋊Q₈	—	3	1	no	—
163	#121	—	3	1	no	—
164	C₄×Q₈	—	3	1	no	—
165	C₄×D₄	—	3	2	no	—
166	C₄×D₄	C₅×D₃	1	1	yes	characteristic, complemented
167	#117	—	3	2	no	—
168	C₂×C₄²	—	3	1	no	—
169	C₂×C₃×C₅	—	4	2	no	—
170	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
172	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
173	C₅×D₃	—	2	2	no	—
174	C₂³×C₃	C₂³×C₅	1	2	yes	—
175	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
176	C₃×D₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
180	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
182	#67	—	2	4	no	—
183	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
184	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
185	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
186	C₂×Dic₃	C₂³×C₅	1	2	yes	—
187	C₂×Dic₃	C₂³×C₅	1	2	yes	—
188	C₄×D₃	—	2	2	no	—
189	C₄×D₃	—	2	2	no	—
190	C₄×D₃	—	2	2	no	—
191	Dic₆	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	3	2	no	—
196	C₂²×C₅	—	6	1	no	—
197	C₂²×C₅	—	6	1	no	—
198	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	2	1	no	—
201	C₄×C₅	—	6	1	no	—
202	C₄×C₅	—	6	2	no	—
203	C₄×C₅	—	6	1	no	—
204	C₄×C₅	—	6	1	no	—
205	C₄×C₅	—	6	2	no	—
206	C₄×C₅	#235	1	2	yes	—
207	C₄×C₅	#237	1	2	yes	—
208	C₂×Q₈	—	3	1	no	—
209	C₂×D₄	—	3	2	no	—
210	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
211	C₂²×C₄	—	3	1	no	—
212	C₂²×C₄	—	3	1	no	—
213	C₂²×C₄	—	3	1	no	—
214	C₂²×C₄	C₂×C₅×D₃	1	2	yes	—
215	C₄⋊C₄	—	3	2	no	—
216	C₄⋊C₄	—	3	2	no	—
217	C₄⋊C₄	—	3	2	no	—
218	C₄⋊C₄	—	3	1	no	—
219	C₄⋊C₄	—	3	2	no	—
220	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
221	C₂²⋊C₄	—	3	2	no	—
222	C₂²⋊C₄	—	3	2	no	—
223	C₂²⋊C₄	—	3	2	no	—
224	C₂²⋊C₄	—	3	2	no	—
225	C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
226	C₄²	—	3	1	no	—
227	C₄²	—	3	1	no	—
228	C₄²	—	3	1	no	—
229	C₄²	—	3	2	no	—
230	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
231	C₃×C₅	#529	1	1	yes	characteristic, complemented, Hall
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
236	C₂×D₃	—	2	1	no	—
237	C₂×D₃	—	2	1	no	—
238	C₂×D₃	C₅×#41	1	2	yes	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	C₅×#41	1	2	yes	—
242	C₃×C₄	C₅×#41	1	2	yes	—
243	Dic₃	—	2	2	no	—
244	Dic₃	—	2	1	no	—
245	Dic₃	—	2	1	no	—
246	Dic₃	—	2	1	no	—
247	Dic₃	—	2	2	no	—
248	C₂×C₅	—	4	2	no	—
249	C₂×C₅	—	6	2	no	—
250	C₂×C₅	C₂×#235	1	1	yes	characteristic
251	C₂×C₅	C₂×#237	1	1	yes	characteristic
252	C₂×C₅	D₃×#41	1	1	yes	characteristic
253	C₂³	—	3	1	no	—
254	C₂³	C₂²×C₅×D₃	1	2	yes	—
255	Q₈	—	6	2	no	—
256	D₄	—	2	2	no	—
257	D₄	—	6	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	3	2	no	—
263	C₂×C₄	—	3	2	no	—
264	C₂×C₄	—	6	2	no	—
265	C₂×C₄	—	6	2	no	—
266	C₂×C₄	—	6	2	no	—
267	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
269	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
271	C₂×C₃	—	4	2	no	—
272	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
273	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
274	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
275	D₃	—	2	2	no	—
276	C₅	#5557	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	3	2	no	—
281	C₂²	—	6	1	no	—
282	C₂²	—	6	1	no	—
283	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	6	1	no	—
287	C₄	—	6	2	no	—
288	C₄	—	6	1	no	—
289	C₄	—	6	1	no	—
290	C₄	—	6	2	no	—
291	C₄	C₅×#235	1	2	yes	—
292	C₄	C₅×#237	1	2	yes	—
293	C₃	C₅×#529	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	4	2	no	—
295	C₂	—	6	2	no	—
296	C₂	C₂×C₅×#235	1	1	yes	characteristic
297	C₂	C₂×C₅×#237	1	1	yes	characteristic
298	C₂	C₅×D₃×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₅×#938	C₂
hypercenter	C₄×C₅×D₄	D₃
hypocenter	C₃	C₅×#529
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₅×D₄	C₂	semidirect
3. C₅×#936	C₂	semidirect
4. C₄×C₅×#67	C₂	semidirect
5. C₅×#926	C₂	semidirect
6. C₅×#897	C₂	semidirect
7. C₅×#890	C₂	semidirect
8. C₅×#883	C₂	semidirect
9. C₅×#879	C₂	semidirect
10. C₄²×C₅×D₃	C₂	semidirect
11. C₄×C₅×Dic₆	C₂	semidirect
36. #5557	C₅	direct (N × H)
45. C₄×C₅×D₄	D₃	semidirect
66. C₃×C₄×D₄	C₂×C₅	semidirect
68. #936	C₂×C₅	semidirect
69. C₄×#67	C₂×C₅	semidirect
70. #926	C₂×C₅	semidirect
71. #897	C₂×C₅	semidirect
72. #890	C₂×C₅	semidirect
73. #883	C₂×C₅	semidirect
74. #879	C₂×C₅	semidirect
75. C₄²×D₃	C₂×C₅	semidirect
76. C₄×Dic₆	C₂×C₅	semidirect
166. C₄×D₄	C₅×D₃	semidirect
231. C₃×C₅	#529	semidirect
276. C₅	#5557	direct (N × H)
293. C₃	C₅×#529	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×D₄	2
C₅×#938	2
C₅×#936	2
C₄×C₅×#67	2
C₅×#926	2
C₅×#897	2
C₅×#890	2
C₅×#883	2
C₅×#879	2
C₄²×C₅×D₃	2
C₄×C₅×Dic₆	2
C₅×#529	3
#5557	5

Subgroups — C5×#5557 #11755942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5557 #11755942