Subgroups — C₅×#5832 #11756217

Order 960.

620 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×#947	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#944	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#940	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#939	C₂	1	4	yes	maximal, complemented
7	C₅×#936	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×#67	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#532	—	3	1	no	maximal, Hall
11	C₃×C₅×#41	—	2	4	no	—
12	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
16	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
17	C₂×C₅×#67	C₂²	1	4	yes	—
18	C₂×C₅×#67	C₂²	1	2	yes	—
19	C₂²×C₅×Dic₃	C₂²	1	2	yes	—
20	C₅×#235	—	2	4	no	—
21	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
23	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
24	C₅×#217	C₂²	1	4	yes	—
25	C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
26	C₅×Dic₃⋊C₄	C₂²	1	4	yes	—
27	C₄×C₅×Dic₃	C₂²	1	2	yes	—
28	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
29	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
30	#5832	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₂×C₅×#41	—	3	1	no	—
32	C₂×C₅×#41	D₃	1	1	yes	characteristic, complemented
33	C₅×C₄⋊Q₈	—	3	1	no	—
34	C₅×#127	—	3	1	no	—
35	C₅×#124	—	3	1	no	—
36	C₅×#124	—	3	1	no	—
37	C₅×#121	—	3	4	no	—
38	C₄×C₅×D₄	—	3	4	no	—
39	C₂×C₄²×C₅	—	3	1	no	—
40	C₂³×C₃×C₅	C₂³	1	2	yes	—
41	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
42	C₂²×C₅×D₃	C₂³	1	2	yes	—
43	C₃×C₅×Q₈	—	2	2	no	—
44	C₃×C₅×D₄	—	2	4	no	—
45	C₃×C₅×D₄	—	2	2	no	—
46	C₂×C₃×C₄×C₅	—	2	4	no	—
47	C₂×C₃×C₄×C₅	D₄	1	4	yes	complemented
48	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₅×#67	—	2	4	no	—
52	C₅×#67	—	2	8	no	—
53	C₂×C₅×Dic₃	—	2	4	no	—
54	C₂×C₅×Dic₃	C₂³	1	2	yes	—
55	C₂×C₅×Dic₃	C₂³	1	2	yes	—
56	C₂×C₅×Dic₃	C₂³	1	2	yes	—
57	C₅×D₁₂	—	2	2	no	—
58	C₄×C₅×D₃	—	2	4	no	—
59	C₅×Dic₆	—	2	2	no	—
60	C₂×C₃×#41	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂×#235	C₂×C₅	1	1	yes	characteristic, complemented
62	#947	C₂×C₅	1	1	yes	characteristic, complemented
63	#944	C₂×C₅	1	1	yes	characteristic, complemented
64	#940	C₂×C₅	1	1	yes	characteristic, complemented
65	#939	C₂×C₅	1	4	yes	complemented
66	#936	C₂×C₅	1	1	yes	characteristic, complemented
67	C₄×#67	C₂×C₅	1	4	yes	complemented
68	C₂×C₄×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
69	C₅×#41	—	2	4	no	—
70	C₅×#41	—	6	4	no	—
71	C₂×C₅×Q₈	—	3	1	no	—
72	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
73	C₂×C₅×D₄	—	3	4	no	—
74	C₂×C₅×D₄	—	3	2	no	—
75	C₂×C₅×D₄	—	3	1	no	—
76	C₂×C₅×D₄	C₂×D₃	1	2	yes	—
77	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
78	C₂²×C₄×C₅	—	3	2	no	—
79	C₂²×C₄×C₅	—	3	2	no	—
80	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
81	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
82	C₅×C₄⋊C₄	—	3	4	no	—
83	C₅×C₂²⋊C₄	—	3	4	no	—
84	C₅×C₂²⋊C₄	—	3	4	no	—
85	C₄²×C₅	—	3	1	no	—
86	C₄²×C₅	—	3	1	no	—
87	C₄²×C₅	—	3	2	no	—
88	#532	—	3	1	no	Sylow
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	2	4	no	—
92	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
93	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
94	C₂×C₅×D₃	—	2	2	no	—
95	C₂×C₅×D₃	—	2	4	no	—
96	C₃×C₄×C₅	—	2	2	no	—
97	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
98	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
99	C₅×Dic₃	—	2	2	no	—
100	C₅×Dic₃	—	2	2	no	—
101	C₅×Dic₃	#41	1	4	yes	complemented
102	C₃×#41	—	2	4	no	—
103	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
104	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₃×D₄	C₂²×C₅	1	2	yes	—
106	C₂²×C₃×C₄	C₂²×C₅	1	2	yes	—
107	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
108	C₂×#67	C₂²×C₅	1	4	yes	—
109	C₂×#67	C₂²×C₅	1	2	yes	—
110	C₂²×Dic₃	C₂²×C₅	1	2	yes	—
111	#235	—	2	4	no	—
112	C₂×D₁₂	C₂²×C₅	1	1	yes	characteristic
113	C₂×C₄×D₃	C₂²×C₅	1	2	yes	—
114	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
115	#217	C₂²×C₅	1	4	yes	—
116	(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
117	Dic₃⋊C₄	C₂²×C₅	1	4	yes	—
118	C₄×Dic₃	C₂²×C₅	1	2	yes	—
119	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
120	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
121	C₂³×C₅	—	3	2	no	—
122	C₂³×C₅	C₂²×D₃	1	2	yes	—
123	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
124	C₅×Q₈	—	2	2	no	—
125	C₅×Q₈	—	6	2	no	—
126	C₅×D₄	—	2	4	no	—
127	C₅×D₄	—	2	2	no	—
128	C₅×D₄	—	6	8	no	—
129	C₅×D₄	—	6	4	no	—
130	C₅×D₄	—	6	2	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	3	2	no	—
133	C₂×C₄×C₅	—	3	2	no	—
134	C₂×C₄×C₅	—	3	2	no	—
135	C₂×C₄×C₅	—	6	4	no	—
136	C₂×C₄×C₅	—	6	4	no	—
137	C₂×C₄×C₅	#67	1	4	yes	complemented
138	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
139	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
141	C₂×#41	—	3	1	no	—
142	C₂×#41	C₅×D₃	1	1	yes	characteristic, complemented
143	C₄⋊Q₈	—	3	1	no	—
144	#127	—	3	1	no	—
145	#124	—	3	1	no	—
146	#124	—	3	1	no	—
147	#121	—	3	4	no	—
148	C₄×D₄	—	3	4	no	—
149	C₂×C₄²	—	3	1	no	—
150	C₂×C₃×C₅	—	2	2	no	—
151	C₂×C₃×C₅	—	4	2	no	—
152	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₃×C₅	C₂×#41	1	2	yes	—
154	C₅×D₃	—	4	2	no	—
155	C₂³×C₃	C₂³×C₅	1	2	yes	—
156	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
157	C₂²×D₃	C₂³×C₅	1	2	yes	—
158	C₃×Q₈	—	2	2	no	—
159	C₃×D₄	—	2	4	no	—
160	C₃×D₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	C₅×D₄	1	4	yes	complemented
163	C₂×C₃×C₄	C₂³×C₅	1	2	yes	—
164	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
166	#67	—	2	4	no	—
167	#67	—	2	8	no	—
168	C₂×Dic₃	—	2	4	no	—
169	C₂×Dic₃	C₂³×C₅	1	2	yes	—
170	C₂×Dic₃	C₂³×C₅	1	2	yes	—
171	C₂×Dic₃	C₂³×C₅	1	2	yes	—
172	D₁₂	—	2	2	no	—
173	C₄×D₃	—	2	4	no	—
174	Dic₆	—	2	2	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	4	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	6	4	no	—
179	C₂²×C₅	—	6	2	no	—
180	C₂²×C₅	C₂×#67	1	2	yes	—
181	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	3	4	no	—
184	C₄×C₅	—	6	2	no	—
185	C₄×C₅	—	6	2	no	—
186	C₄×C₅	C₂×#67	1	2	yes	—
187	C₄×C₅	C₂×#67	1	2	yes	—
188	#41	—	2	4	no	—
189	#41	—	6	4	no	—
190	C₂×Q₈	—	3	1	no	—
191	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
192	C₂×D₄	—	3	4	no	—
193	C₂×D₄	—	3	2	no	—
194	C₂×D₄	—	3	1	no	—
195	C₂×D₄	C₂×C₅×D₃	1	2	yes	—
196	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
197	C₂²×C₄	—	3	2	no	—
198	C₂²×C₄	—	3	2	no	—
199	C₂²×C₄	C₂×C₅×D₃	1	2	yes	—
200	C₂²×C₄	C₂×C₅×D₃	1	1	yes	characteristic
201	C₄⋊C₄	—	3	4	no	—
202	C₂²⋊C₄	—	3	4	no	—
203	C₂²⋊C₄	—	3	4	no	—
204	C₄²	—	3	2	no	—
205	C₄²	—	3	1	no	—
206	C₄²	—	3	1	no	—
207	C₃×C₅	#532	1	1	yes	characteristic, complemented, Hall
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	4	no	—
211	C₂²×C₃	C₂×C₅×D₄	1	2	yes	—
212	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
213	C₂×D₃	—	2	2	no	—
214	C₂×D₃	—	2	4	no	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
217	C₃×C₄	C₂×C₅×D₄	1	2	yes	—
218	Dic₃	—	2	2	no	—
219	Dic₃	—	2	2	no	—
220	Dic₃	C₅×#41	1	4	yes	complemented
221	C₂×C₅	—	2	2	no	—
222	C₂×C₅	—	4	2	no	—
223	C₂×C₅	—	12	2	no	—
224	C₂×C₅	D₃×#41	1	2	yes	—
225	C₂×C₅	C₂²×#67	1	1	yes	characteristic
226	C₂³	—	3	2	no	—
227	C₂³	C₂²×C₅×D₃	1	2	yes	—
228	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	Q₈	—	6	2	no	—
231	D₄	—	2	4	no	—
232	D₄	—	2	2	no	—
233	D₄	—	6	8	no	—
234	D₄	—	6	4	no	—
235	D₄	—	6	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	3	2	no	—
238	C₂×C₄	—	3	2	no	—
239	C₂×C₄	—	3	2	no	—
240	C₂×C₄	—	6	4	no	—
241	C₂×C₄	—	6	4	no	—
242	C₂×C₄	C₅×#67	1	4	yes	complemented
243	C₂×C₄	C₂²×C₅×D₃	1	2	yes	—
244	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	—	4	2	no	—
248	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
249	C₂×C₃	C₂×C₅×#41	1	2	yes	—
250	D₃	—	4	2	no	—
251	C₅	#5832	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	6	4	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	—	3	4	no	—
261	C₄	—	6	2	no	—
262	C₄	—	6	2	no	—
263	C₄	C₂×C₅×#67	1	2	yes	—
264	C₄	C₂×C₅×#67	1	2	yes	—
265	C₃	C₅×#532	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	2	2	no	—
267	C₂	—	4	2	no	—
268	C₂	—	12	2	no	—
269	C₂	C₅×D₃×#41	1	2	yes	—
270	C₂	C₂²×C₅×#67	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	C₂×C₃×C₅×#41	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₂×C₅×#41	D₃
hypocenter	C₃	C₅×#532
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×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×#947	C₂	semidirect
4. C₅×#944	C₂	semidirect
5. C₅×#940	C₂	semidirect
6. C₅×#939	C₂	semidirect
7. C₅×#936	C₂	semidirect
8. C₄×C₅×#67	C₂	semidirect
9. C₂×C₄×C₅×Dic₃	C₂	semidirect
30. #5832	C₅	direct (N × H)
32. C₂×C₅×#41	D₃	semidirect
47. C₂×C₃×C₄×C₅	D₄	semidirect
60. C₂×C₃×#41	C₂×C₅	semidirect
61. C₂×#235	C₂×C₅	semidirect
62. #947	C₂×C₅	semidirect
63. #944	C₂×C₅	semidirect
64. #940	C₂×C₅	semidirect
65. #939	C₂×C₅	semidirect
66. #936	C₂×C₅	semidirect
67. C₄×#67	C₂×C₅	semidirect
68. C₂×C₄×Dic₃	C₂×C₅	semidirect
101. C₅×Dic₃	#41	semidirect
137. C₂×C₄×C₅	#67	semidirect
142. C₂×#41	C₅×D₃	semidirect
162. C₂×C₃×C₄	C₅×D₄	semidirect
207. C₃×C₅	#532	semidirect
220. Dic₃	C₅×#41	semidirect
242. C₂×C₄	C₅×#67	semidirect
251. C₅	#5832	direct (N × H)
265. C₃	C₅×#532	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×#947	2
C₅×#944	2
C₅×#940	2
C₅×#939	2
C₅×#936	2
C₄×C₅×#67	2
C₂×C₄×C₅×Dic₃	2
C₅×#532	3
#5832	5

Subgroups — C5×#5832 #11756217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5832 #11756217