Subgroups — C₃×#65873 #11755550

Order 960.

456 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₃×D₄×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₃×D₄×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₃×#4070	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#683	C₂	1	2	yes	maximal, complemented
6	C₃×#4036	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4034	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4024	C₂	1	2	yes	maximal, complemented
9	C₃×#4024	C₂	1	2	yes	maximal, complemented
10	C₃×#4016	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#65873	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
15	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
16	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
17	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
23	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
24	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
28	C₃×#683	C₂²	1	1	yes	characteristic
29	C₃×#683	C₂²	1	1	yes	characteristic
30	C₃×#683	C₂²	1	2	yes	—
31	C₃×#683	C₂²	1	2	yes	—
32	C₃×#683	C₂²	1	2	yes	—
33	C₃×#683	C₂²	1	2	yes	—
34	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
36	C₃×#554	—	5	1	no	maximal, Hall
37	C₄×C₅×D₄	C₂×C₃	1	1	yes	characteristic
38	D₄×Dic₅	C₂×C₃	1	1	yes	characteristic
39	D₄×Dic₅	C₂×C₃	1	1	yes	characteristic
40	#4070	C₂×C₃	1	2	yes	complemented
41	C₂×#683	C₂×C₃	1	2	yes	complemented
42	#4036	C₂×C₃	1	1	yes	characteristic, complemented
43	#4034	C₂×C₃	1	1	yes	characteristic, complemented
44	#4024	C₂×C₃	1	2	yes	complemented
45	#4024	C₂×C₃	1	2	yes	complemented
46	#4016	C₂×C₃	1	1	yes	characteristic, complemented
47	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂³×C₃×C₅	C₂³	1	2	yes	—
49	C₃×C₅×D₄	Q₈	1	4	yes	complemented
50	C₂×C₃×C₄×C₅	—	4	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₃×Dic₅	—	2	4	no	—
56	C₂×C₃×Dic₅	—	4	2	no	—
57	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
58	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
59	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₅	C₂³	1	2	yes	—
61	C₂×C₃×Dic₅	C₂³	1	2	yes	—
62	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
63	C₃×Dic₁₀	—	2	2	no	—
64	C₃×Dic₁₀	—	4	2	no	—
65	C₃×C₄⋊Q₈	—	5	1	no	—
66	C₃×#125	—	5	1	no	—
67	C₃×#125	—	5	1	no	—
68	C₃×C₂²⋊Q₈	—	5	2	no	—
69	C₃×C₂²⋊Q₈	—	5	2	no	—
70	C₃×C₂²⋊Q₈	—	5	2	no	—
71	C₃×C₄×Q₈	—	5	1	no	—
72	C₃×C₄×D₄	—	5	1	no	—
73	C₃×C₄×D₄	—	5	1	no	—
74	C₃×C₄×D₄	D₅	1	1	yes	characteristic
75	C₂×C₃×C₄⋊C₄	—	5	2	no	—
76	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
78	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
79	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
80	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
81	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
82	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
84	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
85	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
86	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
87	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
88	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
89	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
90	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
91	#683	C₂²×C₃	1	1	yes	characteristic
92	#683	C₂²×C₃	1	1	yes	characteristic
93	#683	C₂²×C₃	1	2	yes	—
94	#683	C₂²×C₃	1	2	yes	—
95	#683	C₂²×C₃	1	2	yes	—
96	#683	C₂²×C₃	1	2	yes	—
97	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
98	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
99	#554	—	5	1	no	Sylow
100	C₂²×C₃×C₅	—	2	2	no	—
101	C₂²×C₃×C₅	—	2	2	no	—
102	C₂²×C₃×C₅	C₂×Q₈	1	4	yes	—
103	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
104	C₃×C₄×C₅	—	2	1	no	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	—	2	1	no	—
107	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
108	C₃×Dic₅	—	2	1	no	—
109	C₃×Dic₅	—	2	2	no	—
110	C₃×Dic₅	—	2	1	no	—
111	C₃×Dic₅	—	2	2	no	—
112	C₃×Dic₅	—	2	1	no	—
113	C₃×Dic₅	#41	1	2	yes	—
114	C₂×C₃×Q₈	—	5	1	no	—
115	C₂×C₃×Q₈	—	5	2	no	—
116	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
117	C₂²×C₃×C₄	—	5	2	no	—
118	C₂²×C₃×C₄	—	5	2	no	—
119	C₂²×C₃×C₄	C₂×D₅	1	2	yes	—
120	C₃×C₄⋊C₄	—	5	1	no	—
121	C₃×C₄⋊C₄	—	5	2	no	—
122	C₃×C₄⋊C₄	—	5	2	no	—
123	C₃×C₄⋊C₄	—	5	2	no	—
124	C₃×C₄⋊C₄	—	5	1	no	—
125	C₃×C₄⋊C₄	—	5	2	no	—
126	C₃×C₄⋊C₄	—	5	1	no	—
127	C₃×C₄⋊C₄	—	5	1	no	—
128	C₃×C₄⋊C₄	—	5	1	no	—
129	C₃×C₄⋊C₄	—	5	2	no	—
130	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
131	C₃×C₂²⋊C₄	—	5	2	no	—
132	C₃×C₂²⋊C₄	—	5	2	no	—
133	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
134	C₃×C₄²	—	5	1	no	—
135	C₃×C₄²	—	5	1	no	—
136	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
137	C₂³×C₅	C₂³×C₃	1	2	yes	—
138	C₅×D₄	C₃×Q₈	1	4	yes	complemented
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
143	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
144	C₂×Dic₅	—	2	4	no	—
145	C₂×Dic₅	—	4	2	no	—
146	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
147	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
148	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
149	C₂×Dic₅	C₂³×C₃	1	2	yes	—
150	C₂×Dic₅	C₂³×C₃	1	2	yes	—
151	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
152	Dic₁₀	—	2	2	no	—
153	Dic₁₀	—	4	2	no	—
154	C₄⋊Q₈	—	5	1	no	—
155	#125	—	5	1	no	—
156	#125	—	5	1	no	—
157	C₂²⋊Q₈	—	5	2	no	—
158	C₂²⋊Q₈	—	5	2	no	—
159	C₂²⋊Q₈	—	5	2	no	—
160	C₄×Q₈	—	5	1	no	—
161	C₄×D₄	—	5	1	no	—
162	C₄×D₄	—	5	1	no	—
163	C₄×D₄	C₃×D₅	1	1	yes	characteristic
164	C₂×C₄⋊C₄	—	5	2	no	—
165	C₂×C₃×C₅	—	2	4	no	—
166	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
168	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂³×C₃	C₂²×D₅	1	2	yes	—
170	C₃×Q₈	—	10	2	no	—
171	C₃×Q₈	—	20	2	no	—
172	C₃×D₄	Dic₁₀	1	4	yes	complemented
173	C₂×C₃×C₄	—	4	2	no	—
174	C₂×C₃×C₄	—	5	1	no	—
175	C₂×C₃×C₄	—	5	1	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	5	1	no	—
178	C₂×C₃×C₄	—	5	2	no	—
179	C₂×C₃×C₄	—	5	2	no	—
180	C₂×C₃×C₄	—	10	4	no	—
181	C₂×C₃×C₄	—	20	2	no	—
182	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
185	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	C₂×C₃×Q₈	1	4	yes	—
189	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
194	Dic₅	—	2	1	no	—
195	Dic₅	—	2	2	no	—
196	Dic₅	—	2	1	no	—
197	Dic₅	—	2	2	no	—
198	Dic₅	—	2	1	no	—
199	Dic₅	C₃×#41	1	2	yes	—
200	C₂×Q₈	—	5	1	no	—
201	C₂×Q₈	—	5	2	no	—
202	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
203	C₂²×C₄	—	5	2	no	—
204	C₂²×C₄	—	5	2	no	—
205	C₂²×C₄	C₂×C₃×D₅	1	2	yes	—
206	C₄⋊C₄	—	5	1	no	—
207	C₄⋊C₄	—	5	2	no	—
208	C₄⋊C₄	—	5	2	no	—
209	C₄⋊C₄	—	5	2	no	—
210	C₄⋊C₄	—	5	1	no	—
211	C₄⋊C₄	—	5	1	no	—
212	C₄⋊C₄	—	5	1	no	—
213	C₄⋊C₄	—	5	1	no	—
214	C₄⋊C₄	—	5	2	no	—
215	C₄⋊C₄	—	5	2	no	—
216	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
217	C₂²⋊C₄	—	5	2	no	—
218	C₂²⋊C₄	—	5	2	no	—
219	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
220	C₄²	—	5	1	no	—
221	C₄²	—	5	1	no	—
222	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
223	C₃×C₅	#554	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	C₂×Dic₁₀	1	4	yes	—
227	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	5	2	no	—
232	C₃×C₄	—	10	1	no	—
233	C₃×C₄	—	10	2	no	—
234	C₃×C₄	—	10	1	no	—
235	C₃×C₄	—	10	2	no	—
236	C₃×C₄	—	10	1	no	—
237	C₃×C₄	C₂×Dic₁₀	1	2	yes	—
238	C₂×C₅	—	2	4	no	—
239	C₂×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
240	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
241	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
242	C₂³	C₂²×C₃×D₅	1	2	yes	—
243	Q₈	—	10	2	no	—
244	Q₈	—	20	2	no	—
245	D₄	C₃×Dic₁₀	1	4	yes	complemented
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	5	1	no	—
248	C₂×C₄	—	5	1	no	—
249	C₂×C₄	—	5	1	no	—
250	C₂×C₄	—	5	1	no	—
251	C₂×C₄	—	5	2	no	—
252	C₂×C₄	—	5	2	no	—
253	C₂×C₄	—	10	4	no	—
254	C₂×C₄	—	20	2	no	—
255	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
258	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
259	C₂×C₃	—	2	4	no	—
260	C₂×C₃	C₂²×Dic₁₀	1	1	yes	characteristic
261	C₂×C₃	#4144	1	1	yes	characteristic
262	C₂×C₃	D₅×#41	1	1	yes	characteristic
263	C₅	C₃×#554	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	C₂×C₃×Dic₁₀	1	4	yes	—
267	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	1	no	—
271	C₄	—	5	2	no	—
272	C₄	—	10	1	no	—
273	C₄	—	10	2	no	—
274	C₄	—	10	1	no	—
275	C₄	—	10	2	no	—
276	C₄	—	10	1	no	—
277	C₄	C₂×C₃×Dic₁₀	1	2	yes	—
278	C₃	#65873	1	1	yes	characteristic, direct product factor, Sylow
279	C₂	—	2	4	no	—
280	C₂	C₂²×C₃×Dic₁₀	1	1	yes	characteristic
281	C₂	C₃×#4144	1	1	yes	characteristic
282	C₂	C₃×D₅×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×C₅×D₄	C₂²
hypercenter	C₃×C₄×D₄	D₅
hypocenter	C₅	C₃×#554
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
4. C₃×#4070	C₂	semidirect
5. C₂×C₃×#683	C₂	semidirect
6. C₃×#4036	C₂	semidirect
7. C₃×#4034	C₂	semidirect
8. C₃×#4024	C₂	semidirect
9. C₃×#4024	C₂	semidirect
10. C₃×#4016	C₂	semidirect
11. C₃×C₄×Dic₁₀	C₂	semidirect
12. #65873	C₃	direct (N × H)
40. #4070	C₂×C₃	semidirect
41. C₂×#683	C₂×C₃	semidirect
42. #4036	C₂×C₃	semidirect
43. #4034	C₂×C₃	semidirect
44. #4024	C₂×C₃	semidirect
45. #4024	C₂×C₃	semidirect
46. #4016	C₂×C₃	semidirect
47. C₄×Dic₁₀	C₂×C₃	semidirect
49. C₃×C₅×D₄	Q₈	semidirect
138. C₅×D₄	C₃×Q₈	semidirect
172. C₃×D₄	Dic₁₀	semidirect
223. C₃×C₅	#554	semidirect
245. D₄	C₃×Dic₁₀	semidirect
263. C₅	C₃×#554	semidirect
278. C₃	#65873	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×D₄	2
C₃×D₄×Dic₅	2
C₃×#4070	2
C₂×C₃×#683	2
C₃×#4036	2
C₃×#4034	2
C₃×#4024	2
C₃×#4016	2
C₃×C₄×Dic₁₀	2
#65873	3
C₃×#554	5

Subgroups — C3×#65873 #11755550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65873 #11755550