Subgroups — C₅×#5511 #11755896

Order 960.

456 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	C₂×C₃×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×#67	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#67	C₂	1	2	yes	maximal
4	C₅×#924	C₂	1	2	yes	maximal, complemented
5	C₅×#924	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#897	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#896	C₂	1	2	yes	maximal, complemented
8	C₅×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#889	C₂	1	2	yes	maximal, complemented
10	C₅×#888	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#887	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#554	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
15	C₂×C₅×#67	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
18	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
19	C₂×C₅×Dic₆	C₂²	1	2	yes	—
20	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
23	C₅×#217	C₂²	1	2	yes	—
24	C₅×#217	C₂²	1	1	yes	characteristic
25	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
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×Dic₃⋊C₄	C₂²	1	2	yes	—
30	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
31	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
33	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
34	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
35	C₄×C₅×Dic₃	C₂²	1	2	yes	—
36	#5511	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₅×C₄⋊Q₈	—	3	1	no	—
38	C₅×#125	—	3	2	no	—
39	C₅×C₂²⋊Q₈	—	3	2	no	—
40	C₅×C₂²⋊Q₈	—	3	1	no	—
41	C₅×C₂²⋊Q₈	—	3	2	no	—
42	C₅×C₂²⋊Q₈	—	3	1	no	—
43	C₄×C₅×Q₈	—	3	1	no	—
44	C₄×C₅×D₄	—	3	2	no	—
45	C₄×C₅×D₄	—	3	1	no	—
46	C₂×C₅×C₄⋊C₄	—	3	1	no	—
47	C₂×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	—	4	2	no	—
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	2	yes	—
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₅×#67	Q₈	1	4	yes	complemented
57	C₂×C₅×Dic₃	C₂³	1	2	yes	—
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	1	yes	characteristic
62	C₄×C₅×D₃	—	2	2	no	—
63	C₄×C₅×D₃	—	4	2	no	—
64	C₅×Dic₆	—	2	2	no	—
65	C₅×Dic₆	—	4	2	no	—
66	C₂×C₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
67	C₄×#67	C₂×C₅	1	1	yes	characteristic
68	C₄×#67	C₂×C₅	1	2	yes	—
69	#924	C₂×C₅	1	2	yes	complemented
70	#924	C₂×C₅	1	1	yes	characteristic, complemented
71	#897	C₂×C₅	1	1	yes	characteristic, complemented
72	#896	C₂×C₅	1	2	yes	complemented
73	D₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
74	#889	C₂×C₅	1	2	yes	complemented
75	#888	C₂×C₅	1	1	yes	characteristic, complemented
76	#887	C₂×C₅	1	1	yes	characteristic, complemented
77	C₂×C₅×Q₈	—	3	1	no	—
78	C₂×C₅×Q₈	—	3	2	no	—
79	C₂×C₅×D₄	—	3	1	no	—
80	C₂²×C₄×C₅	—	3	1	no	—
81	C₂²×C₄×C₅	—	3	2	no	—
82	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
84	C₅×C₄⋊C₄	—	3	2	no	—
85	C₅×C₄⋊C₄	—	3	1	no	—
86	C₅×C₄⋊C₄	—	3	1	no	—
87	C₅×C₄⋊C₄	—	3	2	no	—
88	C₅×C₄⋊C₄	—	3	2	no	—
89	C₅×C₄⋊C₄	—	3	2	no	—
90	C₅×C₄⋊C₄	—	3	2	no	—
91	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
92	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
93	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
94	C₅×C₂²⋊C₄	—	3	2	no	—
95	C₅×C₂²⋊C₄	—	3	1	no	—
96	C₅×C₂²⋊C₄	—	3	2	no	—
97	C₅×C₂²⋊C₄	—	3	1	no	—
98	C₄²×C₅	—	3	2	no	—
99	C₄²×C₅	—	3	1	no	—
100	#554	—	3	1	no	Sylow
101	C₂²×C₃×C₅	—	2	1	no	—
102	C₂²×C₃×C₅	—	2	1	no	—
103	C₂²×C₃×C₅	C₂×Q₈	1	2	yes	—
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	C₂×Q₈	1	2	yes	—
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	—	2	2	no	—
110	C₃×C₄×C₅	—	2	2	no	—
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	2	no	—
116	C₅×Dic₃	C₂×Q₈	1	2	yes	—
117	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
119	C₂×#67	C₂²×C₅	1	1	yes	characteristic
120	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
122	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
123	C₂×Dic₆	C₂²×C₅	1	2	yes	—
124	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
125	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
126	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
127	#217	C₂²×C₅	1	2	yes	—
128	#217	C₂²×C₅	1	1	yes	characteristic
129	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
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	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
134	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
135	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
136	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
137	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
138	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
139	C₄×Dic₃	C₂²×C₅	1	2	yes	—
140	C₂³×C₅	—	3	1	no	—
141	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₅×Q₈	—	6	2	no	—
143	C₅×Q₈	—	12	2	no	—
144	C₅×D₄	—	3	4	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	3	1	no	—
147	C₂×C₄×C₅	—	3	2	no	—
148	C₂×C₄×C₅	—	3	1	no	—
149	C₂×C₄×C₅	—	3	1	no	—
150	C₂×C₄×C₅	—	3	2	no	—
151	C₂×C₄×C₅	—	4	2	no	—
152	C₂×C₄×C₅	—	6	2	no	—
153	C₂×C₄×C₅	—	12	2	no	—
154	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
155	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
157	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
158	C₄⋊Q₈	—	3	1	no	—
159	#125	—	3	2	no	—
160	C₂²⋊Q₈	—	3	2	no	—
161	C₂²⋊Q₈	—	3	1	no	—
162	C₂²⋊Q₈	—	3	2	no	—
163	C₂²⋊Q₈	—	3	1	no	—
164	C₄×Q₈	—	3	1	no	—
165	C₄×D₄	—	3	2	no	—
166	C₄×D₄	—	3	1	no	—
167	C₂×C₄⋊C₄	—	3	1	no	—
168	C₂×C₄⋊C₄	C₅×D₃	1	1	yes	characteristic, complemented
169	C₂×C₃×C₅	—	2	2	no	—
170	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
172	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
173	C₅×D₃	—	2	2	no	—
174	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
175	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	4	2	no	—
178	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
179	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
181	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
182	#67	C₅×Q₈	1	4	yes	complemented
183	C₂×Dic₃	C₂³×C₅	1	2	yes	—
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	1	yes	characteristic
188	C₄×D₃	—	2	2	no	—
189	C₄×D₃	—	4	2	no	—
190	Dic₆	—	2	2	no	—
191	Dic₆	—	4	2	no	—
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	—	2	1	no	—
194	C₂²×C₅	—	3	2	no	—
195	C₂²×C₅	—	6	1	no	—
196	C₂²×C₅	—	6	1	no	—
197	C₂²×C₅	D₃×Q₈	1	2	yes	—
198	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	3	2	no	—
203	C₄×C₅	—	6	1	no	—
204	C₄×C₅	—	6	2	no	—
205	C₄×C₅	—	6	2	no	—
206	C₄×C₅	—	6	1	no	—
207	C₄×C₅	#235	1	2	yes	—
208	C₂×Q₈	—	3	1	no	—
209	C₂×Q₈	—	3	2	no	—
210	C₂×D₄	—	3	1	no	—
211	C₂²×C₄	—	3	1	no	—
212	C₂²×C₄	—	3	2	no	—
213	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₅×D₃	1	2	yes	—
215	C₄⋊C₄	—	3	2	no	—
216	C₄⋊C₄	—	3	1	no	—
217	C₄⋊C₄	—	3	1	no	—
218	C₄⋊C₄	—	3	2	no	—
219	C₄⋊C₄	—	3	2	no	—
220	C₄⋊C₄	—	3	2	no	—
221	C₄⋊C₄	—	3	2	no	—
222	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
223	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
224	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
225	C₂²⋊C₄	—	3	2	no	—
226	C₂²⋊C₄	—	3	1	no	—
227	C₂²⋊C₄	—	3	2	no	—
228	C₂²⋊C₄	—	3	1	no	—
229	C₄²	—	3	2	no	—
230	C₄²	—	3	1	no	—
231	C₃×C₅	#554	1	1	yes	characteristic, complemented, Hall
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	C₂×C₅×Q₈	1	2	yes	—
235	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
236	C₂×D₃	—	2	1	no	—
237	C₂×D₃	—	2	1	no	—
238	C₂×D₃	C₂×C₅×Q₈	1	2	yes	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	2	2	no	—
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	2	no	—
247	Dic₃	C₂×C₅×Q₈	1	2	yes	—
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	—	6	2	no	—
250	C₂×C₅	C₂×#235	1	1	yes	characteristic
251	C₂×C₅	#1004	1	1	yes	characteristic
252	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
253	C₂³	—	3	1	no	—
254	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
255	Q₈	—	6	2	no	—
256	Q₈	—	12	2	no	—
257	D₄	—	3	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	3	2	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	3	1	no	—
263	C₂×C₄	—	3	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	6	2	no	—
266	C₂×C₄	—	12	2	no	—
267	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
268	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
270	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
273	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
274	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
275	D₃	—	2	2	no	—
276	C₅	#5511	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	3	2	no	—
280	C₂²	—	6	1	no	—
281	C₂²	—	6	1	no	—
282	C₂²	C₅×D₃×Q₈	1	2	yes	—
283	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	3	2	no	—
288	C₄	—	6	1	no	—
289	C₄	—	6	2	no	—
290	C₄	—	6	2	no	—
291	C₄	—	6	1	no	—
292	C₄	C₅×#235	1	2	yes	—
293	C₃	C₅×#554	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	2	2	no	—
295	C₂	—	6	2	no	—
296	C₂	C₂×C₅×#235	1	1	yes	characteristic
297	C₂	C₅×#1004	1	1	yes	characteristic
298	C₂	C₂×C₅×D₃×Q₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×C₄⋊C₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₅×#67	C₂²
hypercenter	C₂×C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₅×#554
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₄⋊C₄	C₂	semidirect
4. C₅×#924	C₂	semidirect
5. C₅×#924	C₂	semidirect
6. C₅×#897	C₂	semidirect
7. C₅×#896	C₂	semidirect
8. C₅×D₃×C₄⋊C₄	C₂	semidirect
9. C₅×#889	C₂	semidirect
10. C₅×#888	C₂	semidirect
11. C₅×#887	C₂	semidirect
36. #5511	C₅	direct (N × H)
47. C₂×C₅×C₄⋊C₄	D₃	semidirect
56. C₅×#67	Q₈	semidirect
66. C₂×C₃×C₄⋊C₄	C₂×C₅	semidirect
69. #924	C₂×C₅	semidirect
70. #924	C₂×C₅	semidirect
71. #897	C₂×C₅	semidirect
72. #896	C₂×C₅	semidirect
73. D₃×C₄⋊C₄	C₂×C₅	semidirect
74. #889	C₂×C₅	semidirect
75. #888	C₂×C₅	semidirect
76. #887	C₂×C₅	semidirect
168. C₂×C₄⋊C₄	C₅×D₃	semidirect
182. #67	C₅×Q₈	semidirect
231. C₃×C₅	#554	semidirect
276. C₅	#5511	direct (N × H)
293. C₃	C₅×#554	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊C₄	2
C₄×C₅×#67	2
C₅×#924	2
C₅×#897	2
C₅×#896	2
C₅×D₃×C₄⋊C₄	2
C₅×#889	2
C₅×#888	2
C₅×#887	2
C₅×#554	3
#5511	5

Subgroups — C5×#5511 #11755896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5511 #11755896