Subgroups — C₅×#5733 #11756118

Order 960.

432 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#947	C₂	1	2	yes	maximal, complemented
3	C₅×#946	C₂	1	2	yes	maximal, complemented
4	C₅×#898	C₂	1	2	yes	maximal, complemented
5	C₅×#897	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#895	C₂	1	1	yes	maximal, characteristic
7	C₅×#894	C₂	1	2	yes	maximal, complemented
8	C₅×#893	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#887	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#875	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#872	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#542	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
14	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
15	C₂×C₅×D₁₂	C₂²	1	2	yes	—
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	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
23	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
24	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
25	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
28	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
29	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
30	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
31	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
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	#5733	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
38	C₅×#126	—	3	2	no	—
39	C₅×#124	—	3	2	no	—
40	C₅×#124	—	3	1	no	—
41	C₅×#123	—	3	2	no	—
42	C₅×C₂²⋊Q₈	—	3	2	no	—
43	C₅×C₂²⋊Q₈	—	3	1	no	—
44	C₅×#121	—	3	1	no	—
45	C₄×C₅×Q₈	—	3	1	no	—
46	C₄×C₅×D₄	—	3	1	no	—
47	C₅×#117	—	3	1	no	—
48	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
49	C₂²×C₅×D₃	C₂³	1	2	yes	—
50	C₃×C₅×Q₈	—	4	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
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	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	C₅×D₁₂	—	4	2	no	—
62	C₄×C₅×D₃	—	2	2	no	—
63	C₄×C₅×D₃	—	4	2	no	—
64	C₅×Dic₆	—	2	2	no	—
65	C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
66	#947	C₂×C₅	1	2	yes	complemented
67	#946	C₂×C₅	1	2	yes	complemented
68	#898	C₂×C₅	1	2	yes	complemented
69	#897	C₂×C₅	1	1	yes	characteristic, complemented
70	#895	C₂×C₅	1	1	yes	characteristic
71	#894	C₂×C₅	1	2	yes	complemented
72	#893	C₂×C₅	1	1	yes	characteristic, complemented
73	#887	C₂×C₅	1	1	yes	characteristic, complemented
74	#875	C₂×C₅	1	1	yes	characteristic, complemented
75	#872	C₂×C₅	1	1	yes	characteristic, complemented
76	C₂×C₅×Q₈	—	3	1	no	—
77	C₂×C₅×Q₈	C₂×D₃	1	2	yes	—
78	C₂×C₅×D₄	—	3	1	no	—
79	C₂×C₅×D₄	—	3	2	no	—
80	C₂²×C₄×C₅	—	3	1	no	—
81	C₂²×C₄×C₅	—	3	2	no	—
82	C₅×C₄⋊C₄	—	3	1	no	—
83	C₅×C₄⋊C₄	—	3	1	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	1	yes	characteristic
88	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
89	C₅×C₂²⋊C₄	—	3	1	no	—
90	C₅×C₂²⋊C₄	—	3	1	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	—	3	2	no	—
95	C₅×C₂²⋊C₄	—	3	2	no	—
96	C₄²×C₅	—	3	1	no	—
97	C₄²×C₅	—	3	2	no	—
98	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
99	#542	—	3	1	no	Sylow
100	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
101	C₂×C₅×D₃	—	2	1	no	—
102	C₂×C₅×D₃	—	2	2	no	—
103	C₂×C₅×D₃	—	2	2	no	—
104	C₂×C₅×D₃	—	2	1	no	—
105	C₂×C₅×D₃	—	2	1	no	—
106	C₂×C₅×D₃	—	2	2	no	—
107	C₃×C₄×C₅	—	2	2	no	—
108	C₃×C₄×C₅	—	2	2	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	2	1	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	1	no	—
116	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
117	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
118	C₂×D₁₂	C₂²×C₅	1	2	yes	—
119	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
120	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
121	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
122	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
123	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
124	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
125	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
126	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
127	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
128	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
129	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
130	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
131	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
132	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
133	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
134	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
135	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
136	Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
137	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
138	C₄×Dic₃	C₂²×C₅	1	2	yes	—
139	C₂³×C₅	—	3	2	no	—
140	C₂³×C₅	—	3	1	no	—
141	C₅×Q₈	—	4	2	no	—
142	C₅×Q₈	—	6	2	no	—
143	C₅×D₄	—	6	2	no	—
144	C₅×D₄	—	12	2	no	—
145	C₂×C₄×C₅	—	3	1	no	—
146	C₂×C₄×C₅	—	3	2	no	—
147	C₂×C₄×C₅	—	3	1	no	—
148	C₂×C₄×C₅	—	3	1	no	—
149	C₂×C₄×C₅	—	6	2	no	—
150	C₂×C₄×C₅	—	12	2	no	—
151	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
152	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
155	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
156	C₄⋊Q₈	C₅×D₃	1	1	yes	characteristic, complemented
157	#126	—	3	2	no	—
158	#124	—	3	2	no	—
159	#124	—	3	1	no	—
160	#123	—	3	2	no	—
161	C₂²⋊Q₈	—	3	2	no	—
162	C₂²⋊Q₈	—	3	1	no	—
163	#121	—	3	1	no	—
164	C₄×Q₈	—	3	1	no	—
165	C₄×D₄	—	3	1	no	—
166	#117	—	3	1	no	—
167	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
168	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
170	C₅×D₃	—	4	1	no	—
171	C₅×D₃	—	4	2	no	—
172	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
173	C₂²×D₃	C₂³×C₅	1	2	yes	—
174	C₃×Q₈	—	4	2	no	—
175	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
176	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
179	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
180	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
181	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
182	C₂×Dic₃	C₂³×C₅	1	2	yes	—
183	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
184	D₁₂	—	2	2	no	—
185	D₁₂	—	4	2	no	—
186	C₄×D₃	—	2	2	no	—
187	C₄×D₃	—	4	2	no	—
188	Dic₆	—	2	2	no	—
189	C₂²×C₅	—	6	2	no	—
190	C₂²×C₅	—	6	1	no	—
191	C₂²×C₅	—	6	1	no	—
192	C₂²×C₅	—	6	2	no	—
193	C₂²×C₅	—	6	2	no	—
194	C₂²×C₅	—	6	1	no	—
195	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	6	1	no	—
201	C₄×C₅	—	6	1	no	—
202	C₄×C₅	—	6	2	no	—
203	C₄×C₅	—	6	1	no	—
204	C₄×C₅	#239	1	2	yes	—
205	C₂×Q₈	—	3	1	no	—
206	C₂×Q₈	C₂×C₅×D₃	1	2	yes	—
207	C₂×D₄	—	3	1	no	—
208	C₂×D₄	—	3	2	no	—
209	C₂²×C₄	—	3	1	no	—
210	C₂²×C₄	—	3	2	no	—
211	C₄⋊C₄	—	3	1	no	—
212	C₄⋊C₄	—	3	1	no	—
213	C₄⋊C₄	—	3	2	no	—
214	C₄⋊C₄	—	3	2	no	—
215	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
216	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
217	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
218	C₂²⋊C₄	—	3	1	no	—
219	C₂²⋊C₄	—	3	1	no	—
220	C₂²⋊C₄	—	3	2	no	—
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	C₄²	—	3	1	no	—
226	C₄²	—	3	2	no	—
227	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
228	C₃×C₅	#542	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
230	C₂×D₃	—	2	1	no	—
231	C₂×D₃	—	2	2	no	—
232	C₂×D₃	—	2	2	no	—
233	C₂×D₃	—	2	1	no	—
234	C₂×D₃	—	2	1	no	—
235	C₂×D₃	—	2	2	no	—
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	C₅×#41	1	2	yes	—
241	Dic₃	—	2	2	no	—
242	Dic₃	—	2	1	no	—
243	Dic₃	—	2	1	no	—
244	Dic₃	—	2	1	no	—
245	C₂×C₅	—	12	2	no	—
246	C₂×C₅	—	12	1	no	—
247	C₂×C₅	#1004	1	1	yes	characteristic
248	C₂×C₅	C₂×#239	1	1	yes	characteristic
249	C₂×C₅	#1007	1	1	yes	characteristic
250	C₂³	—	3	2	no	—
251	C₂³	—	3	1	no	—
252	Q₈	—	4	2	no	—
253	Q₈	—	6	2	no	—
254	D₄	—	6	2	no	—
255	D₄	—	12	2	no	—
256	C₂×C₄	—	3	1	no	—
257	C₂×C₄	—	3	2	no	—
258	C₂×C₄	—	3	1	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	6	2	no	—
261	C₂×C₄	—	12	2	no	—
262	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
263	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
266	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
267	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
268	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
269	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
270	D₃	—	4	1	no	—
271	D₃	—	4	2	no	—
272	C₅	#5733	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²	—	6	2	no	—
274	C₂²	—	6	1	no	—
275	C₂²	—	6	1	no	—
276	C₂²	—	6	2	no	—
277	C₂²	—	6	2	no	—
278	C₂²	—	6	1	no	—
279	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	6	1	no	—
285	C₄	—	6	1	no	—
286	C₄	—	6	2	no	—
287	C₄	—	6	1	no	—
288	C₄	C₅×#239	1	2	yes	—
289	C₃	C₅×#542	1	1	yes	characteristic, complemented, Sylow
290	C₂	—	12	2	no	—
291	C₂	—	12	1	no	—
292	C₂	C₅×#1004	1	1	yes	characteristic
293	C₂	C₂×C₅×#239	1	1	yes	characteristic
294	C₂	C₅×#1007	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×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₅×#895	C₂
hypercenter	C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₅×#542
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₄⋊Q₈	C₂	semidirect
2. C₅×#947	C₂	semidirect
3. C₅×#946	C₂	semidirect
4. C₅×#898	C₂	semidirect
5. C₅×#897	C₂	semidirect
7. C₅×#894	C₂	semidirect
8. C₅×#893	C₂	semidirect
9. C₅×#887	C₂	semidirect
10. C₅×#875	C₂	semidirect
11. C₅×#872	C₂	semidirect
36. #5733	C₅	direct (N × H)
37. C₅×C₄⋊Q₈	D₃	semidirect
65. C₃×C₄⋊Q₈	C₂×C₅	semidirect
66. #947	C₂×C₅	semidirect
67. #946	C₂×C₅	semidirect
68. #898	C₂×C₅	semidirect
69. #897	C₂×C₅	semidirect
71. #894	C₂×C₅	semidirect
72. #893	C₂×C₅	semidirect
73. #887	C₂×C₅	semidirect
74. #875	C₂×C₅	semidirect
75. #872	C₂×C₅	semidirect
156. C₄⋊Q₈	C₅×D₃	semidirect
228. C₃×C₅	#542	semidirect
272. C₅	#5733	direct (N × H)
289. C₃	C₅×#542	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₄⋊Q₈	2
C₅×#947	2
C₅×#946	2
C₅×#898	2
C₅×#897	2
C₅×#895	2
C₅×#894	2
C₅×#893	2
C₅×#887	2
C₅×#875	2
C₅×#872	2
C₅×#542	3
#5733	5

Subgroups — C5×#5733 #11756118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5733 #11756118