Subgroups — C₅×#5729 #11756114

Order 960.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 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×#946	C₂	1	2	yes	maximal
3	C₅×#944	C₂	1	2	yes	maximal, complemented
4	C₅×#896	C₂	1	2	yes	maximal
5	C₅×#892	C₂	1	1	yes	maximal, characteristic
6	C₅×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₅×#890	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#889	C₂	1	2	yes	maximal, complemented
9	C₅×#888	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#872	C₂	1	1	yes	maximal, characteristic
11	C₅×#870	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#544	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
14	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
15	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
16	C₂×C₅×Dic₆	C₂²	1	2	yes	—
17	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
22	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
23	C₅×(C₂×D₃)⋊C₄	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	2	yes	—
27	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
28	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
29	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
30	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
31	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
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	#5729	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₅×C₄⋊Q₈	—	3	2	no	—
38	C₅×C₄⋊Q₈	—	3	1	no	—
39	C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
40	C₅×#125	—	3	2	no	—
41	C₅×#125	—	3	1	no	—
42	C₅×#125	—	3	1	no	—
43	C₅×C₂²⋊Q₈	—	3	2	no	—
44	C₅×C₂²⋊Q₈	—	3	2	no	—
45	C₅×#117	—	3	1	no	—
46	C₅×#117	—	3	1	no	—
47	C₂×C₅×C₄⋊C₄	—	3	1	no	—
48	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
49	C₃×C₅×Q₈	—	4	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	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
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	2	yes	—
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	C₄×C₅×D₃	—	4	2	no	—
61	C₄×C₅×D₃	Q₈	1	4	yes	complemented
62	C₅×Dic₆	—	4	2	no	—
63	C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
64	#946	C₂×C₅	1	2	yes	—
65	#944	C₂×C₅	1	2	yes	complemented
66	#896	C₂×C₅	1	2	yes	—
67	#892	C₂×C₅	1	1	yes	characteristic
68	D₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
69	#890	C₂×C₅	1	1	yes	characteristic, complemented
70	#889	C₂×C₅	1	2	yes	complemented
71	#888	C₂×C₅	1	1	yes	characteristic, complemented
72	#872	C₂×C₅	1	1	yes	characteristic
73	#870	C₂×C₅	1	1	yes	characteristic, complemented
74	C₂×C₅×Q₈	—	3	2	no	—
75	C₂×C₅×Q₈	C₂×D₃	1	2	yes	—
76	C₂²×C₄×C₅	—	3	1	no	—
77	C₂²×C₄×C₅	—	3	2	no	—
78	C₅×C₄⋊C₄	—	3	2	no	—
79	C₅×C₄⋊C₄	—	3	1	no	—
80	C₅×C₄⋊C₄	—	3	1	no	—
81	C₅×C₄⋊C₄	—	3	1	no	—
82	C₅×C₄⋊C₄	—	3	1	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	—	3	2	no	—
87	C₅×C₄⋊C₄	—	3	2	no	—
88	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
89	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
90	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
91	C₅×C₂²⋊C₄	—	3	2	no	—
92	C₅×C₂²⋊C₄	—	3	1	no	—
93	C₅×C₂²⋊C₄	—	3	1	no	—
94	C₄²×C₅	—	3	2	no	—
95	C₄²×C₅	—	3	1	no	—
96	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
97	#544	—	3	1	no	Sylow
98	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
99	C₂×C₅×D₃	—	2	1	no	—
100	C₂×C₅×D₃	—	2	1	no	—
101	C₂×C₅×D₃	C₂×Q₈	1	2	yes	—
102	C₃×C₄×C₅	—	2	2	no	—
103	C₃×C₄×C₅	—	2	2	no	—
104	C₃×C₄×C₅	—	2	1	no	—
105	C₃×C₄×C₅	—	2	1	no	—
106	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
107	C₅×Dic₃	—	2	1	no	—
108	C₅×Dic₃	—	2	2	no	—
109	C₅×Dic₃	—	2	1	no	—
110	C₅×Dic₃	—	2	2	no	—
111	C₅×Dic₃	C₂×Q₈	1	2	yes	—
112	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
113	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
114	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
115	C₂×Dic₆	C₂²×C₅	1	2	yes	—
116	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
117	C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
118	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
119	C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
120	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
121	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
122	(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
123	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
124	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
125	C₄⋊Dic₃	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	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	2	yes	—
135	C₂³×C₅	—	3	1	no	—
136	C₅×Q₈	—	4	2	no	—
137	C₅×Q₈	—	12	2	no	—
138	C₂×C₄×C₅	—	3	1	no	—
139	C₂×C₄×C₅	—	3	4	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	—	12	2	no	—
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	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
149	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
150	C₄⋊Q₈	—	3	2	no	—
151	C₄⋊Q₈	—	3	1	no	—
152	C₄⋊Q₈	C₅×D₃	1	1	yes	characteristic, complemented
153	#125	—	3	2	no	—
154	#125	—	3	1	no	—
155	#125	—	3	1	no	—
156	C₂²⋊Q₈	—	3	2	no	—
157	C₂²⋊Q₈	—	3	2	no	—
158	#117	—	3	1	no	—
159	#117	—	3	1	no	—
160	C₂×C₄⋊C₄	—	3	1	no	—
161	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
164	C₅×D₃	—	2	2	no	—
165	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
166	C₃×Q₈	—	4	2	no	—
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×C₃×C₄	C₂³×C₅	1	2	yes	—
171	C₂×C₃×C₄	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	1	yes	characteristic
175	C₂×Dic₃	C₂³×C₅	1	2	yes	—
176	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
177	C₄×D₃	—	4	2	no	—
178	C₄×D₃	C₅×Q₈	1	4	yes	complemented
179	Dic₆	—	4	2	no	—
180	C₂²×C₅	—	3	2	no	—
181	C₂²×C₅	—	6	1	no	—
182	C₂²×C₅	—	6	1	no	—
183	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	2	1	no	—
186	C₄×C₅	—	2	2	no	—
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	—	3	2	no	—
189	C₄×C₅	—	6	2	no	—
190	C₄×C₅	—	6	1	no	—
191	C₄×C₅	—	6	1	no	—
192	C₄×C₅	—	6	2	no	—
193	C₄×C₅	D₃×Q₈	1	2	yes	—
194	C₂×Q₈	—	3	2	no	—
195	C₂×Q₈	C₂×C₅×D₃	1	2	yes	—
196	C₂²×C₄	—	3	1	no	—
197	C₂²×C₄	—	3	2	no	—
198	C₄⋊C₄	—	3	2	no	—
199	C₄⋊C₄	—	3	1	no	—
200	C₄⋊C₄	—	3	1	no	—
201	C₄⋊C₄	—	3	1	no	—
202	C₄⋊C₄	—	3	1	no	—
203	C₄⋊C₄	—	3	2	no	—
204	C₄⋊C₄	—	3	2	no	—
205	C₄⋊C₄	—	3	2	no	—
206	C₄⋊C₄	—	3	2	no	—
207	C₄⋊C₄	—	3	2	no	—
208	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
209	C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
210	C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
211	C₂²⋊C₄	—	3	2	no	—
212	C₂²⋊C₄	—	3	1	no	—
213	C₂²⋊C₄	—	3	1	no	—
214	C₄²	—	3	2	no	—
215	C₄²	—	3	1	no	—
216	C₄²	C₂×C₅×D₃	1	1	yes	characteristic
217	C₃×C₅	#544	1	1	yes	characteristic, complemented, Hall
218	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
219	C₂×D₃	—	2	1	no	—
220	C₂×D₃	—	2	1	no	—
221	C₂×D₃	C₂×C₅×Q₈	1	2	yes	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
227	Dic₃	—	2	1	no	—
228	Dic₃	—	2	2	no	—
229	Dic₃	—	2	1	no	—
230	Dic₃	—	2	2	no	—
231	Dic₃	C₂×C₅×Q₈	1	2	yes	—
232	C₂×C₅	—	6	2	no	—
233	C₂×C₅	#1004	1	1	yes	characteristic
234	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
235	C₂×C₅	#1007	1	1	yes	characteristic
236	C₂³	—	3	1	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	12	2	no	—
239	C₂×C₄	—	3	1	no	—
240	C₂×C₄	—	3	4	no	—
241	C₂×C₄	—	3	1	no	—
242	C₂×C₄	—	3	2	no	—
243	C₂×C₄	—	3	1	no	—
244	C₂×C₄	—	3	2	no	—
245	C₂×C₄	—	12	2	no	—
246	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
247	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
250	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
251	C₂×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
252	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
253	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
254	D₃	—	2	2	no	—
255	C₅	#5729	1	1	yes	characteristic, direct product factor, Sylow
256	C₂²	—	3	2	no	—
257	C₂²	—	6	1	no	—
258	C₂²	—	6	1	no	—
259	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	1	no	—
264	C₄	—	3	2	no	—
265	C₄	—	6	2	no	—
266	C₄	—	6	1	no	—
267	C₄	—	6	1	no	—
268	C₄	—	6	2	no	—
269	C₄	C₅×D₃×Q₈	1	2	yes	—
270	C₃	C₅×#544	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	6	2	no	—
272	C₂	C₅×#1004	1	1	yes	characteristic
273	C₂	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
274	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	C₂²×C₅×D₃	C₂³
hypercenter	C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₅×#544
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
3. C₅×#944	C₂	semidirect
7. C₅×#890	C₂	semidirect
8. C₅×#889	C₂	semidirect
9. C₅×#888	C₂	semidirect
11. C₅×#870	C₂	semidirect
36. #5729	C₅	direct (N × H)
39. C₅×C₄⋊Q₈	D₃	semidirect
61. C₄×C₅×D₃	Q₈	semidirect
63. C₃×C₄⋊Q₈	C₂×C₅	semidirect
65. #944	C₂×C₅	semidirect
69. #890	C₂×C₅	semidirect
70. #889	C₂×C₅	semidirect
71. #888	C₂×C₅	semidirect
73. #870	C₂×C₅	semidirect
152. C₄⋊Q₈	C₅×D₃	semidirect
178. C₄×D₃	C₅×Q₈	semidirect
217. C₃×C₅	#544	semidirect
255. C₅	#5729	direct (N × H)
270. C₃	C₅×#544	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×#946	2
C₅×#944	2
C₅×#896	2
C₅×#892	2
C₅×D₃×C₄⋊C₄	2
C₅×#890	2
C₅×#889	2
C₅×#888	2
C₅×#872	2
C₅×#870	2
C₅×#544	3
#5729	5

Subgroups — C5×#5729 #11756114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5729 #11756114