Subgroups — C₅×#5833 #11756218

Order 960.

588 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#946	C₂	1	2	yes	maximal, complemented
4	C₅×#939	C₂	1	4	yes	maximal, complemented
5	C₅×#938	C₂	1	2	yes	maximal, complemented
6	C₅×#930	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#924	C₂	1	2	yes	maximal, complemented
9	C₅×#537	—	3	1	no	maximal, Hall
10	C₃×C₅×#41	—	2	4	no	—
11	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
14	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
15	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₅×#67	C₂²	1	2	yes	—
17	C₂×C₅×#67	C₂²	1	4	yes	—
18	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
19	C₅×#235	—	2	4	no	—
20	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
22	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
23	C₅×#217	C₂²	1	4	yes	—
24	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
25	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
26	C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
27	C₅×Dic₃⋊C₄	C₂²	1	4	yes	—
28	#5833	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₂×C₅×#41	—	3	1	no	—
30	C₂×C₅×#41	D₃	1	1	yes	characteristic, complemented
31	C₅×C₂²⋊Q₈	—	3	2	no	—
32	C₅×C₂²⋊Q₈	—	3	2	no	—
33	C₅×#121	—	3	2	no	—
34	C₅×#121	—	3	4	no	—
35	C₅×#121	—	3	2	no	—
36	C₂×C₅×C₄⋊C₄	—	3	1	no	—
37	C₂³×C₃×C₅	C₂³	1	2	yes	—
38	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₅×D₃	C₂³	1	2	yes	—
40	C₃×C₅×Q₈	—	2	2	no	—
41	C₃×C₅×D₄	—	2	4	no	—
42	C₃×C₅×D₄	—	2	2	no	—
43	C₂×C₃×C₄×C₅	—	2	4	no	—
44	C₂×C₃×C₄×C₅	D₄	1	4	yes	complemented
45	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
46	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₅×#67	—	2	4	no	—
49	C₅×#67	—	4	4	no	—
50	C₂×C₅×Dic₃	—	4	2	no	—
51	C₂×C₅×Dic₃	C₂³	1	2	yes	—
52	C₂×C₅×Dic₃	C₂³	1	4	yes	—
53	C₅×D₁₂	—	2	2	no	—
54	C₄×C₅×D₃	—	2	4	no	—
55	C₅×Dic₆	—	2	2	no	—
56	C₂×C₃×#41	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×#235	C₂×C₅	1	1	yes	characteristic, complemented
58	#946	C₂×C₅	1	2	yes	complemented
59	#939	C₂×C₅	1	4	yes	complemented
60	#938	C₂×C₅	1	2	yes	complemented
61	#930	C₂×C₅	1	2	yes	complemented
62	C₂×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
63	#924	C₂×C₅	1	2	yes	complemented
64	C₅×#41	—	2	4	no	—
65	C₅×#41	—	6	4	no	—
66	C₂×C₅×Q₈	—	3	1	no	—
67	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
68	C₂×C₅×D₄	—	3	2	no	—
69	C₂×C₅×D₄	—	3	4	no	—
70	C₂×C₅×D₄	—	3	1	no	—
71	C₂×C₅×D₄	C₂×D₃	1	2	yes	—
72	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
73	C₂²×C₄×C₅	—	3	2	no	—
74	C₂²×C₄×C₅	—	3	2	no	—
75	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
76	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
77	C₅×C₄⋊C₄	—	3	2	no	—
78	C₅×C₄⋊C₄	—	3	2	no	—
79	C₅×C₄⋊C₄	—	3	4	no	—
80	C₅×C₂²⋊C₄	—	3	4	no	—
81	C₅×C₂²⋊C₄	—	3	4	no	—
82	#537	—	3	1	no	Sylow
83	C₂²×C₃×C₅	—	2	1	no	—
84	C₂²×C₃×C₅	—	2	1	no	—
85	C₂²×C₃×C₅	—	2	2	no	—
86	C₂²×C₃×C₅	—	2	2	no	—
87	C₂²×C₃×C₅	—	2	2	no	—
88	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
89	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
90	C₂×C₅×D₃	—	2	2	no	—
91	C₂×C₅×D₃	—	2	2	no	—
92	C₂×C₅×D₃	—	2	2	no	—
93	C₃×C₄×C₅	—	2	2	no	—
94	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
95	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
96	C₅×Dic₃	—	2	2	no	—
97	C₅×Dic₃	—	2	4	no	—
98	C₃×#41	—	2	4	no	—
99	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
101	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
102	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
103	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
104	C₂×#67	C₂²×C₅	1	2	yes	—
105	C₂×#67	C₂²×C₅	1	4	yes	—
106	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
107	#235	—	2	4	no	—
108	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
110	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
111	#217	C₂²×C₅	1	4	yes	—
112	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
113	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
114	C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
115	Dic₃⋊C₄	C₂²×C₅	1	4	yes	—
116	C₂³×C₅	—	3	2	no	—
117	C₂³×C₅	C₂²×D₃	1	2	yes	—
118	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
119	C₅×Q₈	—	2	2	no	—
120	C₅×Q₈	—	6	2	no	—
121	C₅×D₄	—	2	4	no	—
122	C₅×D₄	—	2	2	no	—
123	C₅×D₄	—	6	4	no	—
124	C₅×D₄	—	6	2	no	—
125	C₅×D₄	—	12	4	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	3	4	no	—
128	C₂×C₄×C₅	—	3	2	no	—
129	C₂×C₄×C₅	—	6	4	no	—
130	C₂×C₄×C₅	—	12	2	no	—
131	C₂×C₄×C₅	#67	1	4	yes	complemented
132	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
133	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
135	C₂×#41	—	3	1	no	—
136	C₂×#41	C₅×D₃	1	1	yes	characteristic, complemented
137	C₂²⋊Q₈	—	3	2	no	—
138	C₂²⋊Q₈	—	3	2	no	—
139	#121	—	3	2	no	—
140	#121	—	3	4	no	—
141	#121	—	3	2	no	—
142	C₂×C₄⋊C₄	—	3	1	no	—
143	C₂×C₃×C₅	—	2	2	no	—
144	C₂×C₃×C₅	—	4	2	no	—
145	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
148	C₅×D₃	—	4	2	no	—
149	C₂³×C₃	C₂³×C₅	1	2	yes	—
150	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
151	C₂²×D₃	C₂³×C₅	1	2	yes	—
152	C₃×Q₈	—	2	2	no	—
153	C₃×D₄	—	2	4	no	—
154	C₃×D₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	C₅×D₄	1	4	yes	complemented
157	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
158	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
160	#67	—	2	4	no	—
161	#67	—	4	4	no	—
162	C₂×Dic₃	—	4	2	no	—
163	C₂×Dic₃	C₂³×C₅	1	2	yes	—
164	C₂×Dic₃	C₂³×C₅	1	4	yes	—
165	D₁₂	—	2	2	no	—
166	C₄×D₃	—	2	4	no	—
167	Dic₆	—	2	2	no	—
168	C₂²×C₅	—	2	1	no	—
169	C₂²×C₅	—	2	1	no	—
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	—	2	2	no	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	6	2	no	—
174	C₂²×C₅	—	6	2	no	—
175	C₂²×C₅	—	6	2	no	—
176	C₂²×C₅	C₂×#67	1	2	yes	—
177	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	—	6	4	no	—
180	C₄×C₅	—	6	2	no	—
181	C₄×C₅	C₂×#67	1	2	yes	—
182	C₄×C₅	C₂×#67	1	2	yes	—
183	#41	—	2	4	no	—
184	#41	—	6	4	no	—
185	C₂×Q₈	—	3	1	no	—
186	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
187	C₂×D₄	—	3	2	no	—
188	C₂×D₄	—	3	4	no	—
189	C₂×D₄	—	3	1	no	—
190	C₂×D₄	C₂×C₅×D₃	1	2	yes	—
191	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
192	C₂²×C₄	—	3	2	no	—
193	C₂²×C₄	—	3	2	no	—
194	C₂²×C₄	C₂×C₅×D₃	1	2	yes	—
195	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
196	C₄⋊C₄	—	3	2	no	—
197	C₄⋊C₄	—	3	2	no	—
198	C₄⋊C₄	—	3	4	no	—
199	C₂²⋊C₄	—	3	4	no	—
200	C₂²⋊C₄	—	3	4	no	—
201	C₃×C₅	#537	1	1	yes	characteristic, complemented, Hall
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	2	no	—
207	C₂²×C₃	C₂×C₅×D₄	1	2	yes	—
208	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
209	C₂×D₃	—	2	2	no	—
210	C₂×D₃	—	2	2	no	—
211	C₂×D₃	—	2	2	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
214	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
215	Dic₃	—	2	2	no	—
216	Dic₃	—	2	4	no	—
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	—	4	2	no	—
219	C₂×C₅	—	12	2	no	—
220	C₂×C₅	#1009	1	1	yes	characteristic
221	C₂×C₅	#1010	1	1	yes	characteristic
222	C₂×C₅	C₂²×#67	1	1	yes	characteristic
223	C₂³	—	3	2	no	—
224	C₂³	C₂²×C₅×D₃	1	2	yes	—
225	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
226	Q₈	—	2	2	no	—
227	Q₈	—	6	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	2	no	—
230	D₄	—	6	4	no	—
231	D₄	—	6	2	no	—
232	D₄	—	12	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	3	4	no	—
235	C₂×C₄	—	3	2	no	—
236	C₂×C₄	—	6	4	no	—
237	C₂×C₄	—	12	2	no	—
238	C₂×C₄	C₅×#67	1	4	yes	complemented
239	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
240	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
242	C₂×C₃	—	2	2	no	—
243	C₂×C₃	—	4	2	no	—
244	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
245	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
246	C₂×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
247	D₃	—	4	2	no	—
248	C₅	#5833	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	6	2	no	—
255	C₂²	—	6	2	no	—
256	C₂²	—	6	2	no	—
257	C₂²	C₂×C₅×#67	1	2	yes	—
258	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	6	4	no	—
261	C₄	—	6	2	no	—
262	C₄	C₂×C₅×#67	1	2	yes	—
263	C₄	C₂×C₅×#67	1	2	yes	—
264	C₃	C₅×#537	1	1	yes	characteristic, complemented, Sylow
265	C₂	—	2	2	no	—
266	C₂	—	4	2	no	—
267	C₂	—	12	2	no	—
268	C₂	C₅×#1009	1	1	yes	characteristic
269	C₂	C₅×#1010	1	1	yes	characteristic
270	C₂	C₂²×C₅×#67	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×#41	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₂×C₅×#41	D₃
hypocenter	C₃	C₅×#537
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#41	C₂	semidirect
2. C₂×C₅×#235	C₂	semidirect
3. C₅×#946	C₂	semidirect
4. C₅×#939	C₂	semidirect
5. C₅×#938	C₂	semidirect
6. C₅×#930	C₂	semidirect
7. C₂×C₅×C₄⋊Dic₃	C₂	semidirect
8. C₅×#924	C₂	semidirect
28. #5833	C₅	direct (N × H)
30. C₂×C₅×#41	D₃	semidirect
44. C₂×C₃×C₄×C₅	D₄	semidirect
56. C₂×C₃×#41	C₂×C₅	semidirect
57. C₂×#235	C₂×C₅	semidirect
58. #946	C₂×C₅	semidirect
59. #939	C₂×C₅	semidirect
60. #938	C₂×C₅	semidirect
61. #930	C₂×C₅	semidirect
62. C₂×C₄⋊Dic₃	C₂×C₅	semidirect
63. #924	C₂×C₅	semidirect
131. C₂×C₄×C₅	#67	semidirect
136. C₂×#41	C₅×D₃	semidirect
156. C₂×C₃×C₄	C₅×D₄	semidirect
201. C₃×C₅	#537	semidirect
238. C₂×C₄	C₅×#67	semidirect
248. C₅	#5833	direct (N × H)
264. C₃	C₅×#537	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×#41	2
C₂×C₅×#235	2
C₅×#946	2
C₅×#939	2
C₅×#938	2
C₅×#930	2
C₂×C₅×C₄⋊Dic₃	2
C₅×#924	2
C₅×#537	3
#5833	5

Subgroups — C5×#5833 #11756218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5833 #11756218