Subgroups — C₃×#66138 #11755815

Order 960.

692 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4099	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4085	C₂	1	4	yes	maximal, complemented
6	C₃×#4083	C₂	1	2	yes	maximal, complemented
7	C₃×#4081	C₂	1	2	yes	maximal, complemented
8	C₃×#4075	C₂	1	2	yes	maximal, complemented
9	#66138	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂⁴×C₃×C₅	C₂²	1	1	yes	characteristic
11	C₂⁴×C₃×C₅	C₂²	1	1	yes	characteristic, complemented
12	C₂³×C₃×D₅	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₃×C₅×D₄	—	2	4	no	—
14	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
15	C₂×C₃×C₅×D₄	C₂²	1	2	yes	complemented
16	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃×C₂²⋊D₅	—	2	4	no	—
18	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
19	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	complemented
20	C₂×C₃×C₂²⋊D₅	C₂²	1	4	yes	complemented
21	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic, complemented
22	C₂²×C₃×Dic₅	C₂²	1	2	yes	complemented
23	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	complemented
24	C₃×C₂²⋊Dic₅	C₂²	1	4	yes	complemented
25	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	complemented
26	C₃×#685	C₂²	1	4	yes	complemented
27	C₃×#683	C₂²	1	4	yes	complemented
28	C₃×#534	—	5	1	no	maximal, Hall
29	C₂²×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂²×C₂²⋊D₅	C₂×C₃	1	1	yes	characteristic, complemented
31	#4099	C₂×C₃	1	2	yes	complemented
32	C₂×C₂²⋊Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
33	#4085	C₂×C₃	1	4	yes	complemented
34	#4083	C₂×C₃	1	2	yes	complemented
35	#4081	C₂×C₃	1	2	yes	complemented
36	#4075	C₂×C₃	1	2	yes	complemented
37	C₂³×C₃×C₅	—	2	2	no	—
38	C₂³×C₃×C₅	—	4	2	no	—
39	C₂³×C₃×C₅	—	4	2	no	—
40	C₂³×C₃×C₅	D₄	1	4	yes	complemented
41	C₂³×C₃×C₅	C₂³	1	2	yes	—
42	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
43	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
44	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
45	C₂²×C₃×D₅	—	2	2	no	—
46	C₂²×C₃×D₅	—	4	2	no	—
47	C₂²×C₃×D₅	C₂³	1	2	yes	—
48	C₃×C₅×D₄	—	2	4	no	—
49	C₃×C₅×D₄	—	2	4	no	—
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₃×C₂²⋊D₅	—	2	4	no	—
53	C₃×C₂²⋊D₅	—	2	4	no	—
54	C₃×C₂²⋊D₅	—	4	4	no	—
55	C₂×C₃×Dic₅	—	2	2	no	—
56	C₂×C₃×Dic₅	—	4	2	no	—
57	C₂×C₃×Dic₅	C₂³	1	2	yes	—
58	C₂×C₃×Dic₅	C₂³	1	4	yes	—
59	C₂²×C₃×D₄	—	5	1	no	—
60	C₂²×C₃×D₄	D₅	1	1	yes	characteristic, complemented
61	C₃×#123	—	5	2	no	—
62	C₃×#123	—	5	2	no	—
63	C₃×#121	—	5	4	no	—
64	C₃×#120	—	5	2	no	—
65	C₃×#120	—	5	2	no	—
66	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
67	C₂⁴×C₅	C₂²×C₃	1	1	yes	characteristic
68	C₂⁴×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
69	C₂³×D₅	C₂²×C₃	1	1	yes	characteristic, complemented
70	C₂×C₅×D₄	—	2	4	no	—
71	C₂×C₅×D₄	C₂²×C₃	1	2	yes	—
72	C₂×C₅×D₄	C₂²×C₃	1	2	yes	complemented
73	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
74	C₂×C₂²⋊D₅	—	2	4	no	—
75	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
76	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	complemented
77	C₂×C₂²⋊D₅	C₂²×C₃	1	4	yes	complemented
78	C₂²×Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
79	C₂²×Dic₅	C₂²×C₃	1	2	yes	complemented
80	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	complemented
81	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	complemented
82	C₂²⋊Dic₅	C₂²×C₃	1	4	yes	complemented
83	#685	C₂²×C₃	1	4	yes	complemented
84	#683	C₂²×C₃	1	4	yes	complemented
85	#534	—	5	1	no	Sylow
86	C₂²×C₃×C₅	—	2	2	no	—
87	C₂²×C₃×C₅	—	2	2	no	—
88	C₂²×C₃×C₅	—	2	2	no	—
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	—	2	4	no	—
91	C₂²×C₃×C₅	—	4	4	no	—
92	C₂²×C₃×C₅	—	4	4	no	—
93	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
94	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
95	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
96	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
97	C₂×C₃×D₅	—	2	2	no	—
98	C₂×C₃×D₅	—	2	4	no	—
99	C₂×C₃×D₅	—	4	4	no	—
100	C₃×C₄×C₅	—	2	2	no	—
101	C₃×Dic₅	—	2	2	no	—
102	C₃×Dic₅	—	2	4	no	—
103	C₂⁴×C₃	—	5	1	no	—
104	C₂⁴×C₃	C₂×D₅	1	1	yes	characteristic
105	C₂⁴×C₃	C₂×D₅	1	1	yes	characteristic, complemented
106	C₂×C₃×D₄	—	2	4	no	—
107	C₂×C₃×D₄	—	5	2	no	—
108	C₂×C₃×D₄	—	5	2	no	—
109	C₂×C₃×D₄	—	5	4	no	—
110	C₂×C₃×D₄	—	10	4	no	—
111	C₂×C₃×D₄	C₂×D₅	1	2	yes	—
112	C₂×C₃×D₄	C₂×D₅	1	2	yes	complemented
113	C₂²×C₃×C₄	—	5	1	no	—
114	C₂²×C₃×C₄	—	5	2	no	—
115	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic, complemented
116	C₃×C₄⋊C₄	—	5	4	no	—
117	C₃×C₂²⋊C₄	—	5	2	no	—
118	C₃×C₂²⋊C₄	—	5	2	no	—
119	C₃×C₂²⋊C₄	—	5	4	no	—
120	C₃×C₂²⋊C₄	—	5	4	no	—
121	C₂³×C₅	—	2	2	no	—
122	C₂³×C₅	—	4	2	no	—
123	C₂³×C₅	—	4	2	no	—
124	C₂³×C₅	C₃×D₄	1	4	yes	complemented
125	C₂³×C₅	C₂³×C₃	1	2	yes	—
126	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
127	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
128	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
129	C₂²×D₅	—	2	2	no	—
130	C₂²×D₅	—	4	2	no	—
131	C₂²×D₅	C₂³×C₃	1	2	yes	—
132	C₅×D₄	—	2	4	no	—
133	C₅×D₄	—	2	4	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
136	C₂²⋊D₅	—	2	4	no	—
137	C₂²⋊D₅	—	2	4	no	—
138	C₂²⋊D₅	—	4	4	no	—
139	C₂×Dic₅	—	2	2	no	—
140	C₂×Dic₅	—	4	2	no	—
141	C₂×Dic₅	C₂³×C₃	1	2	yes	—
142	C₂×Dic₅	C₂³×C₃	1	4	yes	—
143	C₂²×D₄	—	5	1	no	—
144	C₂²×D₄	C₃×D₅	1	1	yes	characteristic, complemented
145	#123	—	5	2	no	—
146	#123	—	5	2	no	—
147	#121	—	5	4	no	—
148	#120	—	5	2	no	—
149	#120	—	5	2	no	—
150	C₂×C₂²⋊C₄	—	5	1	no	—
151	C₂×C₃×C₅	—	2	2	no	—
152	C₂×C₃×C₅	—	2	2	no	—
153	C₂×C₃×C₅	—	2	2	no	—
154	C₂×C₃×C₅	—	4	2	no	—
155	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
157	C₃×D₅	—	4	2	no	—
158	C₂³×C₃	—	2	2	no	—
159	C₂³×C₃	—	4	2	no	—
160	C₂³×C₃	—	4	2	no	—
161	C₂³×C₃	—	5	2	no	—
162	C₂³×C₃	—	10	2	no	—
163	C₂³×C₃	—	20	2	no	—
164	C₂³×C₃	C₂²⋊D₅	1	4	yes	complemented
165	C₂³×C₃	C₂²×D₅	1	2	yes	—
166	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
167	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
168	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
169	C₃×D₄	—	2	4	no	—
170	C₃×D₄	—	2	4	no	—
171	C₃×D₄	—	10	4	no	—
172	C₃×D₄	—	10	4	no	—
173	C₃×D₄	—	20	4	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	5	4	no	—
176	C₂×C₃×C₄	—	5	2	no	—
177	C₂×C₃×C₄	—	10	2	no	—
178	C₂×C₃×C₄	—	20	2	no	—
179	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	2	4	no	—
185	C₂²×C₅	—	4	4	no	—
186	C₂²×C₅	—	4	4	no	—
187	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
188	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
189	C₂²×C₅	C₂×C₃×D₄	1	2	yes	—
190	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
191	C₂×D₅	—	2	2	no	—
192	C₂×D₅	—	2	4	no	—
193	C₂×D₅	—	4	4	no	—
194	C₄×C₅	—	2	2	no	—
195	Dic₅	—	2	2	no	—
196	Dic₅	—	2	4	no	—
197	C₂⁴	—	5	1	no	—
198	C₂⁴	C₂×C₃×D₅	1	1	yes	characteristic
199	C₂⁴	C₂×C₃×D₅	1	1	yes	characteristic, complemented
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	5	2	no	—
202	C₂×D₄	—	5	2	no	—
203	C₂×D₄	—	5	4	no	—
204	C₂×D₄	—	10	4	no	—
205	C₂×D₄	C₂×C₃×D₅	1	2	yes	—
206	C₂×D₄	C₂×C₃×D₅	1	2	yes	complemented
207	C₂²×C₄	—	5	1	no	—
208	C₂²×C₄	—	5	2	no	—
209	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic, complemented
210	C₄⋊C₄	—	5	4	no	—
211	C₂²⋊C₄	—	5	2	no	—
212	C₂²⋊C₄	—	5	2	no	—
213	C₂²⋊C₄	—	5	4	no	—
214	C₂²⋊C₄	—	5	4	no	—
215	C₃×C₅	#534	1	1	yes	characteristic, complemented, Hall
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	—	2	4	no	—
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	—	4	4	no	—
222	C₂²×C₃	—	4	4	no	—
223	C₂²×C₃	—	10	4	no	—
224	C₂²×C₃	—	10	2	no	—
225	C₂²×C₃	—	20	4	no	—
226	C₂²×C₃	C₂×C₂²⋊D₅	1	2	yes	—
227	C₂²×C₃	C₂×C₂²⋊D₅	1	2	yes	—
228	C₂²×C₃	C₂×C₂²⋊D₅	1	2	yes	—
229	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	10	4	no	—
232	C₃×C₄	—	10	2	no	—
233	C₂×C₅	—	2	2	no	—
234	C₂×C₅	—	2	2	no	—
235	C₂×C₅	—	2	2	no	—
236	C₂×C₅	—	4	2	no	—
237	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
238	C₂×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
239	D₅	—	4	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	5	2	no	—
244	C₂³	—	10	2	no	—
245	C₂³	—	20	2	no	—
246	C₂³	C₃×C₂²⋊D₅	1	4	yes	complemented
247	C₂³	C₂²×C₃×D₅	1	2	yes	—
248	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
249	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
250	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
251	D₄	—	2	4	no	—
252	D₄	—	2	4	no	—
253	D₄	—	10	4	no	—
254	D₄	—	10	4	no	—
255	D₄	—	20	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	5	4	no	—
258	C₂×C₄	—	5	2	no	—
259	C₂×C₄	—	10	2	no	—
260	C₂×C₄	—	20	2	no	—
261	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
262	C₂×C₃	—	2	2	no	—
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	2	2	no	—
265	C₂×C₃	—	4	2	no	—
266	C₂×C₃	—	20	2	no	—
267	C₂×C₃	#4144	1	2	yes	—
268	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
269	C₅	C₃×#534	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	10	4	no	—
278	C₂²	—	10	2	no	—
279	C₂²	—	20	4	no	—
280	C₂²	C₂×C₃×C₂²⋊D₅	1	2	yes	—
281	C₂²	C₂×C₃×C₂²⋊D₅	1	2	yes	—
282	C₂²	C₂×C₃×C₂²⋊D₅	1	2	yes	—
283	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	10	4	no	—
286	C₄	—	10	2	no	—
287	C₃	#66138	1	1	yes	characteristic, direct product factor, Sylow
288	C₂	—	2	2	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	—	4	2	no	—
292	C₂	—	20	2	no	—
293	C₂	C₃×#4144	1	2	yes	—
294	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	C₂²×C₃×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₂²×C₃×D₄	D₅
hypocenter	C₅	C₃×#534
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×D₄	C₂	semidirect
2. C₂²×C₃×C₂²⋊D₅	C₂	semidirect
3. C₃×#4099	C₂	semidirect
4. C₂×C₃×C₂²⋊Dic₅	C₂	semidirect
5. C₃×#4085	C₂	semidirect
6. C₃×#4083	C₂	semidirect
7. C₃×#4081	C₂	semidirect
8. C₃×#4075	C₂	semidirect
9. #66138	C₃	direct (N × H)
11. C₂⁴×C₃×C₅	C₂²	semidirect
12. C₂³×C₃×D₅	C₂²	semidirect
15. C₂×C₃×C₅×D₄	C₂²	semidirect
16. C₂²×C₃×C₄×C₅	C₂²	semidirect
19. C₂×C₃×C₂²⋊D₅	C₂²	semidirect
20. C₂×C₃×C₂²⋊D₅	C₂²	semidirect
21. C₂²×C₃×Dic₅	C₂²	semidirect
22. C₂²×C₃×Dic₅	C₂²	semidirect
23. C₃×C₂²⋊Dic₅	C₂²	semidirect
24. C₃×C₂²⋊Dic₅	C₂²	semidirect
25. C₃×C₂²⋊Dic₅	C₂²	semidirect
26. C₃×#685	C₂²	semidirect
27. C₃×#683	C₂²	semidirect
29. C₂²×C₅×D₄	C₂×C₃	semidirect
30. C₂²×C₂²⋊D₅	C₂×C₃	semidirect
31. #4099	C₂×C₃	semidirect
32. C₂×C₂²⋊Dic₅	C₂×C₃	semidirect
33. #4085	C₂×C₃	semidirect
34. #4083	C₂×C₃	semidirect
35. #4081	C₂×C₃	semidirect
36. #4075	C₂×C₃	semidirect
40. C₂³×C₃×C₅	D₄	semidirect
60. C₂²×C₃×D₄	D₅	semidirect
68. C₂⁴×C₅	C₂²×C₃	semidirect
69. C₂³×D₅	C₂²×C₃	semidirect
72. C₂×C₅×D₄	C₂²×C₃	semidirect
73. C₂²×C₄×C₅	C₂²×C₃	semidirect
76. C₂×C₂²⋊D₅	C₂²×C₃	semidirect
77. C₂×C₂²⋊D₅	C₂²×C₃	semidirect
78. C₂²×Dic₅	C₂²×C₃	semidirect
79. C₂²×Dic₅	C₂²×C₃	semidirect
80. C₂²⋊Dic₅	C₂²×C₃	semidirect
81. C₂²⋊Dic₅	C₂²×C₃	semidirect
82. C₂²⋊Dic₅	C₂²×C₃	semidirect
83. #685	C₂²×C₃	semidirect
84. #683	C₂²×C₃	semidirect
105. C₂⁴×C₃	C₂×D₅	semidirect
112. C₂×C₃×D₄	C₂×D₅	semidirect
115. C₂²×C₃×C₄	C₂×D₅	semidirect
124. C₂³×C₅	C₃×D₄	semidirect
144. C₂²×D₄	C₃×D₅	semidirect
164. C₂³×C₃	C₂²⋊D₅	semidirect
199. C₂⁴	C₂×C₃×D₅	semidirect
206. C₂×D₄	C₂×C₃×D₅	semidirect
209. C₂²×C₄	C₂×C₃×D₅	semidirect
215. C₃×C₅	#534	semidirect
246. C₂³	C₃×C₂²⋊D₅	semidirect
269. C₅	C₃×#534	semidirect
287. C₃	#66138	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₅×D₄	2
C₂²×C₃×C₂²⋊D₅	2
C₃×#4099	2
C₂×C₃×C₂²⋊Dic₅	2
C₃×#4085	2
C₃×#4083	2
C₃×#4081	2
C₃×#4075	2
#66138	3
C₃×#534	5

Subgroups — C3×#66138 #11755815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66138 #11755815