Subgroups — C₃×#65858 #11755535

Order 960.

660 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#709	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4043	C₂	1	2	yes	maximal, complemented
5	C₃×#4041	C₂	1	2	yes	maximal, complemented
6	C₃×#4032	C₂	1	2	yes	maximal, complemented
7	C₃×#4028	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×D₂₀	C₂	1	4	yes	maximal, complemented
9	#65858	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂³×C₃×D₅	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
13	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
14	C₃×#709	—	2	4	no	—
15	C₂×C₃×D₂₀	C₂²	1	4	yes	—
16	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃×C₄×D₅	—	2	4	no	—
18	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
19	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
20	C₂×C₃×C₄×D₅	C₂²	1	2	yes	complemented
21	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
22	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
23	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×C₄²×C₅	C₂²	1	2	yes	complemented
25	C₃×#685	C₂²	1	4	yes	—
26	C₃×#685	C₂²	1	4	yes	complemented
27	C₃×C₄⋊Dic₅	C₂²	1	4	yes	complemented
28	C₃×#525	—	5	1	no	maximal, Hall
29	C₂×#709	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂²×C₄×D₅	C₂×C₃	1	1	yes	characteristic, complemented
31	C₅×#117	C₂×C₃	1	1	yes	characteristic, complemented
32	#4043	C₂×C₃	1	2	yes	complemented
33	#4041	C₂×C₃	1	2	yes	complemented
34	#4032	C₂×C₃	1	2	yes	complemented
35	#4028	C₂×C₃	1	2	yes	complemented
36	C₄×D₂₀	C₂×C₃	1	4	yes	complemented
37	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
38	C₂²×C₃×D₅	—	2	2	no	—
39	C₂²×C₃×D₅	—	2	4	no	—
40	C₂²×C₃×D₅	C₂³	1	2	yes	—
41	C₂²×C₃×D₅	C₂³	1	2	yes	—
42	C₂×C₃×C₄×C₅	D₄	1	4	yes	complemented
43	C₂×C₃×C₄×C₅	C₂³	1	4	yes	—
44	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₃×C₂²⋊D₅	—	2	4	no	—
47	C₂×C₃×Dic₅	—	2	2	no	—
48	C₂×C₃×Dic₅	C₂³	1	2	yes	—
49	C₂×C₃×Dic₅	C₂³	1	2	yes	—
50	C₃×D₂₀	—	2	8	no	—
51	C₃×D₂₀	—	2	2	no	—
52	C₃×C₄×D₅	—	2	4	no	—
53	C₃×C₄×D₅	—	2	4	no	—
54	C₃×C₄×D₅	—	2	4	no	—
55	C₃×Dic₁₀	—	2	2	no	—
56	C₂×C₃×#41	—	5	1	no	—
57	C₂³×C₃×C₄	—	5	1	no	—
58	C₃×#123	—	5	2	no	—
59	C₃×C₂²⋊Q₈	—	5	2	no	—
60	C₃×#121	—	5	2	no	—
61	C₃×#120	—	5	2	no	—
62	C₃×C₄×D₄	—	5	4	no	—
63	C₃×#117	D₅	1	1	yes	characteristic, complemented
64	C₂³×D₅	C₂²×C₃	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
67	C₂²×Dic₅	C₂²×C₃	1	1	yes	characteristic
68	#709	—	2	4	no	—
69	C₂×D₂₀	C₂²×C₃	1	4	yes	—
70	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic, complemented
71	C₂×C₄×D₅	—	2	4	no	—
72	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
73	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
74	C₂×C₄×D₅	C₂²×C₃	1	2	yes	complemented
75	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic, complemented
76	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
77	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₄²×C₅	C₂²×C₃	1	2	yes	complemented
79	#685	C₂²×C₃	1	4	yes	—
80	#685	C₂²×C₃	1	4	yes	complemented
81	C₄⋊Dic₅	C₂²×C₃	1	4	yes	complemented
82	#525	—	5	1	no	Sylow
83	C₂²×C₃×C₅	—	2	2	no	—
84	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
85	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
86	C₂×C₃×D₅	—	2	8	no	—
87	C₂×C₃×D₅	—	2	2	no	—
88	C₂×C₃×D₅	—	2	2	no	—
89	C₂×C₃×D₅	—	2	2	no	—
90	C₂×C₃×D₅	—	2	4	no	—
91	C₂×C₃×D₅	#41	1	4	yes	complemented
92	C₃×C₄×C₅	—	2	4	no	—
93	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
94	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
95	C₃×Dic₅	—	2	2	no	—
96	C₃×Dic₅	—	2	2	no	—
97	C₂⁴×C₃	—	5	1	no	—
98	C₃×#41	—	10	4	no	—
99	C₂×C₃×Q₈	—	5	1	no	—
100	C₂×C₃×D₄	—	5	2	no	—
101	C₂×C₃×D₄	—	5	4	no	—
102	C₂×C₃×D₄	—	5	1	no	—
103	C₂²×C₃×C₄	—	5	1	no	—
104	C₂²×C₃×C₄	—	5	2	no	—
105	C₂²×C₃×C₄	—	5	2	no	—
106	C₂²×C₃×C₄	—	5	2	no	—
107	C₂²×C₃×C₄	—	10	4	no	—
108	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
109	C₃×C₄⋊C₄	—	5	4	no	—
110	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	complemented
111	C₃×C₂²⋊C₄	—	5	4	no	—
112	C₃×C₂²⋊C₄	—	5	4	no	—
113	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
114	C₃×C₄²	C₂×D₅	1	2	yes	complemented
115	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
116	C₂²×D₅	—	2	2	no	—
117	C₂²×D₅	—	2	4	no	—
118	C₂²×D₅	C₂³×C₃	1	2	yes	—
119	C₂²×D₅	C₂³×C₃	1	2	yes	—
120	C₂×C₄×C₅	C₃×D₄	1	4	yes	complemented
121	C₂×C₄×C₅	C₂³×C₃	1	4	yes	—
122	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
124	C₂²⋊D₅	—	2	4	no	—
125	C₂×Dic₅	—	2	2	no	—
126	C₂×Dic₅	C₂³×C₃	1	2	yes	—
127	C₂×Dic₅	C₂³×C₃	1	2	yes	—
128	D₂₀	—	2	8	no	—
129	D₂₀	—	2	2	no	—
130	C₄×D₅	—	2	4	no	—
131	C₄×D₅	—	2	4	no	—
132	C₄×D₅	—	2	4	no	—
133	Dic₁₀	—	2	2	no	—
134	C₂×#41	—	5	1	no	—
135	C₂³×C₄	—	5	1	no	—
136	#123	—	5	2	no	—
137	C₂²⋊Q₈	—	5	2	no	—
138	#121	—	5	2	no	—
139	#120	—	5	2	no	—
140	C₄×D₄	—	5	4	no	—
141	#117	C₃×D₅	1	1	yes	characteristic, complemented
142	C₂×C₃×C₅	—	2	2	no	—
143	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₃×C₅	C₂×#41	1	2	yes	—
145	C₃×D₅	—	2	4	no	—
146	C₃×D₅	—	4	2	no	—
147	C₂³×C₃	—	5	2	no	—
148	C₂³×C₃	—	5	2	no	—
149	C₂³×C₃	—	10	4	no	—
150	C₂³×C₃	—	10	2	no	—
151	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
152	C₃×Q₈	—	10	2	no	—
153	C₃×D₄	—	10	8	no	—
154	C₃×D₄	—	10	4	no	—
155	C₃×D₄	—	10	2	no	—
156	C₂×C₃×C₄	—	5	2	no	—
157	C₂×C₃×C₄	—	5	2	no	—
158	C₂×C₃×C₄	—	10	2	no	—
159	C₂×C₃×C₄	—	10	4	no	—
160	C₂×C₃×C₄	—	10	4	no	—
161	C₂×C₃×C₄	—	10	4	no	—
162	C₂×C₃×C₄	D₂₀	1	4	yes	complemented
163	C₂×C₃×C₄	C₂²×D₅	1	4	yes	—
164	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
168	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
169	C₂×D₅	—	2	8	no	—
170	C₂×D₅	—	2	2	no	—
171	C₂×D₅	—	2	2	no	—
172	C₂×D₅	—	2	2	no	—
173	C₂×D₅	—	2	4	no	—
174	C₂×D₅	C₃×#41	1	4	yes	complemented
175	C₄×C₅	—	2	4	no	—
176	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
177	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
178	Dic₅	—	2	2	no	—
179	Dic₅	—	2	2	no	—
180	C₂⁴	—	5	1	no	—
181	#41	—	10	4	no	—
182	C₂×Q₈	—	5	1	no	—
183	C₂×D₄	—	5	2	no	—
184	C₂×D₄	—	5	4	no	—
185	C₂×D₄	—	5	1	no	—
186	C₂²×C₄	—	5	1	no	—
187	C₂²×C₄	—	5	2	no	—
188	C₂²×C₄	—	5	2	no	—
189	C₂²×C₄	—	5	2	no	—
190	C₂²×C₄	—	10	4	no	—
191	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
192	C₄⋊C₄	—	5	4	no	—
193	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	complemented
194	C₂²⋊C₄	—	5	4	no	—
195	C₂²⋊C₄	—	5	4	no	—
196	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
197	C₄²	C₂×C₃×D₅	1	2	yes	complemented
198	C₃×C₅	#525	1	1	yes	characteristic, complemented, Hall
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	5	4	no	—
201	C₂²×C₃	—	10	8	no	—
202	C₂²×C₃	—	10	2	no	—
203	C₂²×C₃	—	10	2	no	—
204	C₂²×C₃	—	10	2	no	—
205	C₂²×C₃	—	10	4	no	—
206	C₂²×C₃	C₂×D₂₀	1	2	yes	—
207	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
208	C₃×C₄	—	2	4	no	—
209	C₃×C₄	—	10	2	no	—
210	C₃×C₄	—	10	2	no	—
211	C₃×C₄	C₂×D₂₀	1	2	yes	—
212	C₃×C₄	C₂×D₂₀	1	2	yes	—
213	C₂×C₅	—	2	2	no	—
214	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
215	C₂×C₅	C₂×C₃×#41	1	2	yes	—
216	D₅	—	2	4	no	—
217	D₅	—	4	2	no	—
218	C₂³	—	5	2	no	—
219	C₂³	—	5	2	no	—
220	C₂³	—	10	4	no	—
221	C₂³	—	10	2	no	—
222	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
223	Q₈	—	10	2	no	—
224	D₄	—	10	8	no	—
225	D₄	—	10	4	no	—
226	D₄	—	10	2	no	—
227	C₂×C₄	—	5	2	no	—
228	C₂×C₄	—	5	2	no	—
229	C₂×C₄	—	10	2	no	—
230	C₂×C₄	—	10	4	no	—
231	C₂×C₄	—	10	4	no	—
232	C₂×C₄	—	10	4	no	—
233	C₂×C₄	C₃×D₂₀	1	4	yes	complemented
234	C₂×C₄	C₂²×C₃×D₅	1	4	yes	—
235	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
237	C₂×C₃	—	2	2	no	—
238	C₂×C₃	—	10	4	no	—
239	C₂×C₃	—	20	2	no	—
240	C₂×C₃	C₂²×D₂₀	1	1	yes	characteristic
241	C₂×C₃	D₅×#41	1	2	yes	—
242	C₅	C₃×#525	1	1	yes	characteristic, complemented, Sylow
243	C₂²	—	2	2	no	—
244	C₂²	—	5	4	no	—
245	C₂²	—	10	8	no	—
246	C₂²	—	10	2	no	—
247	C₂²	—	10	2	no	—
248	C₂²	—	10	2	no	—
249	C₂²	—	10	4	no	—
250	C₂²	C₂×C₃×D₂₀	1	2	yes	—
251	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	10	2	no	—
254	C₄	—	10	2	no	—
255	C₄	C₂×C₃×D₂₀	1	2	yes	—
256	C₄	C₂×C₃×D₂₀	1	2	yes	—
257	C₃	#65858	1	1	yes	characteristic, direct product factor, Sylow
258	C₂	—	2	2	no	—
259	C₂	—	10	4	no	—
260	C₂	—	20	2	no	—
261	C₂	C₂²×C₃×D₂₀	1	1	yes	characteristic
262	C₂	C₃×D₅×#41	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₅×#117	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₃×#117	D₅
hypocenter	C₅	C₃×#525
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×#117

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#709	C₂	semidirect
2. C₂²×C₃×C₄×D₅	C₂	semidirect
3. C₃×C₅×#117	C₂	semidirect
4. C₃×#4043	C₂	semidirect
5. C₃×#4041	C₂	semidirect
6. C₃×#4032	C₂	semidirect
7. C₃×#4028	C₂	semidirect
8. C₃×C₄×D₂₀	C₂	semidirect
9. #65858	C₃	direct (N × H)
16. C₂×C₃×D₂₀	C₂²	semidirect
20. C₂×C₃×C₄×D₅	C₂²	semidirect
21. C₂×C₃×Dic₁₀	C₂²	semidirect
22. C₃×C₅×C₄⋊C₄	C₂²	semidirect
24. C₃×C₄²×C₅	C₂²	semidirect
26. C₃×#685	C₂²	semidirect
27. C₃×C₄⋊Dic₅	C₂²	semidirect
29. C₂×#709	C₂×C₃	semidirect
30. C₂²×C₄×D₅	C₂×C₃	semidirect
31. C₅×#117	C₂×C₃	semidirect
32. #4043	C₂×C₃	semidirect
33. #4041	C₂×C₃	semidirect
34. #4032	C₂×C₃	semidirect
35. #4028	C₂×C₃	semidirect
36. C₄×D₂₀	C₂×C₃	semidirect
42. C₂×C₃×C₄×C₅	D₄	semidirect
63. C₃×#117	D₅	semidirect
70. C₂×D₂₀	C₂²×C₃	semidirect
74. C₂×C₄×D₅	C₂²×C₃	semidirect
75. C₂×Dic₁₀	C₂²×C₃	semidirect
76. C₅×C₄⋊C₄	C₂²×C₃	semidirect
78. C₄²×C₅	C₂²×C₃	semidirect
80. #685	C₂²×C₃	semidirect
81. C₄⋊Dic₅	C₂²×C₃	semidirect
91. C₂×C₃×D₅	#41	semidirect
110. C₃×C₄⋊C₄	C₂×D₅	semidirect
114. C₃×C₄²	C₂×D₅	semidirect
120. C₂×C₄×C₅	C₃×D₄	semidirect
141. #117	C₃×D₅	semidirect
162. C₂×C₃×C₄	D₂₀	semidirect
174. C₂×D₅	C₃×#41	semidirect
193. C₄⋊C₄	C₂×C₃×D₅	semidirect
197. C₄²	C₂×C₃×D₅	semidirect
198. C₃×C₅	#525	semidirect
233. C₂×C₄	C₃×D₂₀	semidirect
242. C₅	C₃×#525	semidirect
257. C₃	#65858	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×#709	2
C₂²×C₃×C₄×D₅	2
C₃×C₅×#117	2
C₃×#4043	2
C₃×#4041	2
C₃×#4032	2
C₃×#4028	2
C₃×C₄×D₂₀	2
#65858	3
C₃×#525	5

Subgroups — C3×#65858 #11755535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65858 #11755535