Subgroups — C₃×#66167 #11755844

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×C₃×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4092	C₂	1	2	yes	maximal, complemented
4	C₃×#4085	C₂	1	4	yes	maximal, complemented
5	C₃×#4084	C₂	1	2	yes	maximal, complemented
6	C₃×#4076	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4070	C₂	1	2	yes	maximal, complemented
9	#66167	C₃	1	1	yes	maximal, characteristic, direct product factor, 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×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂×C₃×C₂²⋊D₅	C₂²	1	4	yes	—
18	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
19	C₃×#709	—	2	4	no	—
20	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
22	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
23	C₃×C₂²⋊Dic₅	C₂²	1	4	yes	—
24	C₃×#685	C₂²	1	4	yes	—
25	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
27	C₃×#683	C₂²	1	4	yes	—
28	C₃×#537	—	5	1	no	maximal, Hall
29	C₂×C₅×#41	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×#709	C₂×C₃	1	1	yes	characteristic, complemented
31	#4092	C₂×C₃	1	2	yes	complemented
32	#4085	C₂×C₃	1	4	yes	complemented
33	#4084	C₂×C₃	1	2	yes	complemented
34	#4076	C₂×C₃	1	2	yes	complemented
35	C₂×C₄⋊Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
36	#4070	C₂×C₃	1	2	yes	complemented
37	C₂³×C₃×C₅	C₂³	1	2	yes	—
38	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₃×D₅	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	C₃×C₂²⋊D₅	—	2	4	no	—
49	C₃×C₂²⋊D₅	—	4	4	no	—
50	C₂×C₃×Dic₅	—	4	2	no	—
51	C₂×C₃×Dic₅	C₂³	1	2	yes	—
52	C₂×C₃×Dic₅	C₂³	1	4	yes	—
53	C₃×D₂₀	—	2	2	no	—
54	C₃×C₄×D₅	—	2	4	no	—
55	C₃×Dic₁₀	—	2	2	no	—
56	C₂×C₃×#41	—	5	1	no	—
57	C₂×C₃×#41	D₅	1	1	yes	characteristic, complemented
58	C₃×C₂²⋊Q₈	—	5	2	no	—
59	C₃×C₂²⋊Q₈	—	5	2	no	—
60	C₃×#121	—	5	2	no	—
61	C₃×#121	—	5	4	no	—
62	C₃×#121	—	5	2	no	—
63	C₂×C₃×C₄⋊C₄	—	5	1	no	—
64	C₅×#41	—	2	4	no	—
65	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
69	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
71	C₂×C₂²⋊D₅	C₂²×C₃	1	4	yes	—
72	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
73	#709	—	2	4	no	—
74	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
76	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
77	C₂²⋊Dic₅	C₂²×C₃	1	4	yes	—
78	#685	C₂²×C₃	1	4	yes	—
79	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
80	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
81	#683	C₂²×C₃	1	4	yes	—
82	#537	—	5	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×C₃×D₅	—	2	2	no	—
91	C₂×C₃×D₅	—	2	2	no	—
92	C₂×C₃×D₅	—	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	C₃×Dic₅	—	2	2	no	—
97	C₃×Dic₅	—	2	4	no	—
98	C₃×#41	—	2	4	no	—
99	C₃×#41	—	10	4	no	—
100	C₂×C₃×Q₈	—	5	1	no	—
101	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
102	C₂×C₃×D₄	—	5	2	no	—
103	C₂×C₃×D₄	—	5	4	no	—
104	C₂×C₃×D₄	—	5	1	no	—
105	C₂×C₃×D₄	C₂×D₅	1	2	yes	—
106	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
107	C₂²×C₃×C₄	—	5	2	no	—
108	C₂²×C₃×C₄	—	5	2	no	—
109	C₂²×C₃×C₄	C₂×D₅	1	2	yes	—
110	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
111	C₃×C₄⋊C₄	—	5	4	no	—
112	C₃×C₄⋊C₄	—	5	2	no	—
113	C₃×C₄⋊C₄	—	5	2	no	—
114	C₃×C₂²⋊C₄	—	5	4	no	—
115	C₃×C₂²⋊C₄	—	5	4	no	—
116	C₂³×C₅	C₂³×C₃	1	2	yes	—
117	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
118	C₂²×D₅	C₂³×C₃	1	2	yes	—
119	C₅×Q₈	—	2	2	no	—
120	C₅×D₄	—	2	4	no	—
121	C₅×D₄	—	2	2	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	C₃×D₄	1	4	yes	complemented
124	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
125	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
127	C₂²⋊D₅	—	2	4	no	—
128	C₂²⋊D₅	—	4	4	no	—
129	C₂×Dic₅	—	4	2	no	—
130	C₂×Dic₅	C₂³×C₃	1	2	yes	—
131	C₂×Dic₅	C₂³×C₃	1	4	yes	—
132	D₂₀	—	2	2	no	—
133	C₄×D₅	—	2	4	no	—
134	Dic₁₀	—	2	2	no	—
135	C₂×#41	—	5	1	no	—
136	C₂×#41	C₃×D₅	1	1	yes	characteristic, complemented
137	C₂²⋊Q₈	—	5	2	no	—
138	C₂²⋊Q₈	—	5	2	no	—
139	#121	—	5	2	no	—
140	#121	—	5	4	no	—
141	#121	—	5	2	no	—
142	C₂×C₄⋊C₄	—	5	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	C₃×D₅	—	4	2	no	—
149	C₂³×C₃	—	5	2	no	—
150	C₂³×C₃	C₂²×D₅	1	2	yes	—
151	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
152	C₃×Q₈	—	2	2	no	—
153	C₃×Q₈	—	10	2	no	—
154	C₃×D₄	—	2	4	no	—
155	C₃×D₄	—	2	2	no	—
156	C₃×D₄	—	10	4	no	—
157	C₃×D₄	—	10	2	no	—
158	C₃×D₄	—	20	4	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	5	4	no	—
161	C₂×C₃×C₄	—	5	2	no	—
162	C₂×C₃×C₄	—	10	4	no	—
163	C₂×C₃×C₄	—	20	2	no	—
164	C₂×C₃×C₄	C₂²⋊D₅	1	4	yes	complemented
165	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
166	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
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	C₂×C₃×D₄	1	2	yes	—
174	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
175	C₂×D₅	—	2	2	no	—
176	C₂×D₅	—	2	2	no	—
177	C₂×D₅	—	2	2	no	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
180	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
181	Dic₅	—	2	2	no	—
182	Dic₅	—	2	4	no	—
183	#41	—	2	4	no	—
184	#41	—	10	4	no	—
185	C₂×Q₈	—	5	1	no	—
186	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
187	C₂×D₄	—	5	2	no	—
188	C₂×D₄	—	5	4	no	—
189	C₂×D₄	—	5	1	no	—
190	C₂×D₄	C₂×C₃×D₅	1	2	yes	—
191	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
192	C₂²×C₄	—	5	2	no	—
193	C₂²×C₄	—	5	2	no	—
194	C₂²×C₄	C₂×C₃×D₅	1	2	yes	—
195	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
196	C₄⋊C₄	—	5	2	no	—
197	C₄⋊C₄	—	5	2	no	—
198	C₄⋊C₄	—	5	4	no	—
199	C₂²⋊C₄	—	5	4	no	—
200	C₂²⋊C₄	—	5	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	—	10	2	no	—
208	C₂²×C₃	—	10	2	no	—
209	C₂²×C₃	—	10	2	no	—
210	C₂²×C₃	C₂×C₂²⋊D₅	1	2	yes	—
211	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	10	4	no	—
214	C₃×C₄	—	10	2	no	—
215	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
216	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	—	4	2	no	—
219	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
220	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
221	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
222	D₅	—	4	2	no	—
223	C₂³	—	5	2	no	—
224	C₂³	C₂²×C₃×D₅	1	2	yes	—
225	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
226	Q₈	—	2	2	no	—
227	Q₈	—	10	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	2	no	—
230	D₄	—	10	4	no	—
231	D₄	—	10	2	no	—
232	D₄	—	20	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	5	4	no	—
235	C₂×C₄	—	5	2	no	—
236	C₂×C₄	—	10	4	no	—
237	C₂×C₄	—	20	2	no	—
238	C₂×C₄	C₃×C₂²⋊D₅	1	4	yes	complemented
239	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
240	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
242	C₂×C₃	—	2	2	no	—
243	C₂×C₃	—	4	2	no	—
244	C₂×C₃	—	20	2	no	—
245	C₂×C₃	#4149	1	1	yes	characteristic
246	C₂×C₃	#4150	1	1	yes	characteristic
247	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
248	C₅	C₃×#537	1	1	yes	characteristic, complemented, 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	—	10	2	no	—
255	C₂²	—	10	2	no	—
256	C₂²	—	10	2	no	—
257	C₂²	C₂×C₃×C₂²⋊D₅	1	2	yes	—
258	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	10	4	no	—
261	C₄	—	10	2	no	—
262	C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
263	C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
264	C₃	#66167	1	1	yes	characteristic, direct product factor, Sylow
265	C₂	—	2	2	no	—
266	C₂	—	4	2	no	—
267	C₂	—	20	2	no	—
268	C₂	C₃×#4149	1	1	yes	characteristic
269	C₂	C₃×#4150	1	1	yes	characteristic
270	C₂	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×#41	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₂×C₃×#41	D₅
hypocenter	C₅	C₃×#537
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#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×C₃×#709	C₂	semidirect
3. C₃×#4092	C₂	semidirect
4. C₃×#4085	C₂	semidirect
5. C₃×#4084	C₂	semidirect
6. C₃×#4076	C₂	semidirect
7. C₂×C₃×C₄⋊Dic₅	C₂	semidirect
8. C₃×#4070	C₂	semidirect
9. #66167	C₃	direct (N × H)
29. C₂×C₅×#41	C₂×C₃	semidirect
30. C₂×#709	C₂×C₃	semidirect
31. #4092	C₂×C₃	semidirect
32. #4085	C₂×C₃	semidirect
33. #4084	C₂×C₃	semidirect
34. #4076	C₂×C₃	semidirect
35. C₂×C₄⋊Dic₅	C₂×C₃	semidirect
36. #4070	C₂×C₃	semidirect
44. C₂×C₃×C₄×C₅	D₄	semidirect
57. C₂×C₃×#41	D₅	semidirect
123. C₂×C₄×C₅	C₃×D₄	semidirect
136. C₂×#41	C₃×D₅	semidirect
164. C₂×C₃×C₄	C₂²⋊D₅	semidirect
201. C₃×C₅	#537	semidirect
238. C₂×C₄	C₃×C₂²⋊D₅	semidirect
248. C₅	C₃×#537	semidirect
264. C₃	#66167	direct (N × H)

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×C₃×#709	2
C₃×#4092	2
C₃×#4085	2
C₃×#4084	2
C₃×#4076	2
C₂×C₃×C₄⋊Dic₅	2
C₃×#4070	2
#66167	3
C₃×#537	5

Subgroups — C3×#66167 #11755844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66167 #11755844