Subgroups — C₃×#66019 #11755696

Order 960.

400 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	C₃×#4036	C₂	1	1	yes	maximal, characteristic
3	C₃×#4035	C₂	1	2	yes	maximal
4	C₃×#4035	C₂	1	2	yes	maximal
5	C₃×#4034	C₂	1	1	yes	maximal, characteristic
6	C₃×#4034	C₂	1	2	yes	maximal
7	C₃×#4033	C₂	1	1	yes	maximal, characteristic
8	C₃×#4033	C₂	1	2	yes	maximal
9	C₃×#4033	C₂	1	1	yes	maximal, characteristic
10	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
11	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
12	#66019	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
14	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
15	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
22	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
24	C₃×#683	C₂²	1	1	yes	characteristic
25	C₃×#683	C₂²	1	2	yes	—
26	C₃×#683	C₂²	1	1	yes	characteristic
27	C₃×#683	C₂²	1	2	yes	—
28	C₃×#683	C₂²	1	2	yes	—
29	C₃×#683	C₂²	1	2	yes	—
30	C₃×#683	C₂²	1	2	yes	—
31	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
32	C₃×C₄×Dic₅	C₂²	1	2	yes	—
33	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
34	C₃×C₄×Dic₅	C₂²	1	2	yes	—
35	C₃×C₄×Dic₅	C₂²	1	2	yes	—
36	C₃×#557	—	5	1	no	maximal, Hall
37	C₅×#125	C₂×C₃	1	1	yes	characteristic
38	#4036	C₂×C₃	1	1	yes	characteristic
39	#4035	C₂×C₃	1	2	yes	—
40	#4035	C₂×C₃	1	2	yes	—
41	#4034	C₂×C₃	1	1	yes	characteristic
42	#4034	C₂×C₃	1	2	yes	—
43	#4033	C₂×C₃	1	1	yes	characteristic
44	#4033	C₂×C₃	1	2	yes	—
45	#4033	C₂×C₃	1	1	yes	characteristic
46	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic
47	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
55	C₂×C₃×Dic₅	C₂³	1	2	yes	—
56	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
57	C₂×C₃×Dic₅	C₂³	1	2	yes	—
58	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
59	C₃×Dic₁₀	—	2	2	no	—
60	C₃×Dic₁₀	—	2	4	no	—
61	C₃×Dic₁₀	Q₈	1	4	yes	complemented
62	C₃×C₄⋊Q₈	—	5	2	no	—
63	C₃×C₄⋊Q₈	—	5	1	no	—
64	C₃×#125	—	5	2	no	—
65	C₃×#125	—	5	2	no	—
66	C₃×#125	—	5	1	no	—
67	C₃×#125	D₅	1	1	yes	characteristic
68	C₃×C₄×Q₈	—	5	1	no	—
69	C₃×C₄×Q₈	—	5	2	no	—
70	C₃×C₄×Q₈	—	5	1	no	—
71	C₃×C₄×Q₈	—	5	1	no	—
72	C₃×C₄×Q₈	—	5	1	no	—
73	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
74	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
75	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
76	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
81	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
82	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
83	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
84	#683	C₂²×C₃	1	1	yes	characteristic
85	#683	C₂²×C₃	1	2	yes	—
86	#683	C₂²×C₃	1	1	yes	characteristic
87	#683	C₂²×C₃	1	2	yes	—
88	#683	C₂²×C₃	1	2	yes	—
89	#683	C₂²×C₃	1	2	yes	—
90	#683	C₂²×C₃	1	2	yes	—
91	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
92	C₄×Dic₅	C₂²×C₃	1	2	yes	—
93	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
94	C₄×Dic₅	C₂²×C₃	1	2	yes	—
95	C₄×Dic₅	C₂²×C₃	1	2	yes	—
96	#557	—	5	1	no	Sylow
97	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
98	C₃×C₄×C₅	—	2	2	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	—	2	2	no	—
102	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
103	C₃×Dic₅	—	2	1	no	—
104	C₃×Dic₅	—	2	2	no	—
105	C₃×Dic₅	—	2	1	no	—
106	C₃×Dic₅	—	2	1	no	—
107	C₃×Dic₅	C₂×Q₈	1	4	yes	—
108	C₃×Dic₅	#41	1	2	yes	—
109	C₂×C₃×Q₈	—	5	1	no	—
110	C₂×C₃×Q₈	—	5	2	no	—
111	C₂×C₃×Q₈	—	5	1	no	—
112	C₃×C₄⋊C₄	—	5	1	no	—
113	C₃×C₄⋊C₄	—	5	2	no	—
114	C₃×C₄⋊C₄	—	5	2	no	—
115	C₃×C₄⋊C₄	—	5	1	no	—
116	C₃×C₄⋊C₄	—	5	2	no	—
117	C₃×C₄⋊C₄	—	5	2	no	—
118	C₃×C₄⋊C₄	—	5	2	no	—
119	C₃×C₄⋊C₄	—	5	1	no	—
120	C₃×C₄⋊C₄	—	5	1	no	—
121	C₃×C₄⋊C₄	—	5	2	no	—
122	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
123	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
124	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
125	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
126	C₃×C₄²	—	5	2	no	—
127	C₃×C₄²	—	5	2	no	—
128	C₃×C₄²	—	5	1	no	—
129	C₃×C₄²	—	5	2	no	—
130	C₃×C₄²	—	5	1	no	—
131	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
134	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
135	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
137	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
138	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
139	C₂×Dic₅	C₂³×C₃	1	2	yes	—
140	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
141	C₂×Dic₅	C₂³×C₃	1	2	yes	—
142	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
143	Dic₁₀	—	2	2	no	—
144	Dic₁₀	—	2	4	no	—
145	Dic₁₀	C₃×Q₈	1	4	yes	complemented
146	C₄⋊Q₈	—	5	2	no	—
147	C₄⋊Q₈	—	5	1	no	—
148	#125	—	5	2	no	—
149	#125	—	5	2	no	—
150	#125	—	5	1	no	—
151	#125	C₃×D₅	1	1	yes	characteristic
152	C₄×Q₈	—	5	1	no	—
153	C₄×Q₈	—	5	2	no	—
154	C₄×Q₈	—	5	1	no	—
155	C₄×Q₈	—	5	1	no	—
156	C₄×Q₈	—	5	1	no	—
157	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
159	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
160	C₃×Q₈	—	5	4	no	—
161	C₃×Q₈	—	10	4	no	—
162	C₃×Q₈	—	10	2	no	—
163	C₂×C₃×C₄	—	5	2	no	—
164	C₂×C₃×C₄	—	5	1	no	—
165	C₂×C₃×C₄	—	5	1	no	—
166	C₂×C₃×C₄	—	5	1	no	—
167	C₂×C₃×C₄	—	5	1	no	—
168	C₂×C₃×C₄	—	5	2	no	—
169	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
171	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
172	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
175	C₄×C₅	—	2	2	no	—
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	—	2	1	no	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
180	Dic₅	—	2	1	no	—
181	Dic₅	—	2	2	no	—
182	Dic₅	—	2	1	no	—
183	Dic₅	—	2	1	no	—
184	Dic₅	C₂×C₃×Q₈	1	4	yes	—
185	Dic₅	C₃×#41	1	2	yes	—
186	C₂×Q₈	—	5	1	no	—
187	C₂×Q₈	—	5	2	no	—
188	C₂×Q₈	—	5	1	no	—
189	C₄⋊C₄	—	5	1	no	—
190	C₄⋊C₄	—	5	2	no	—
191	C₄⋊C₄	—	5	2	no	—
192	C₄⋊C₄	—	5	1	no	—
193	C₄⋊C₄	—	5	1	no	—
194	C₄⋊C₄	—	5	2	no	—
195	C₄⋊C₄	—	5	2	no	—
196	C₄⋊C₄	—	5	1	no	—
197	C₄⋊C₄	—	5	2	no	—
198	C₄⋊C₄	—	5	2	no	—
199	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
200	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
201	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
202	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
203	C₄²	—	5	2	no	—
204	C₄²	—	5	2	no	—
205	C₄²	—	5	1	no	—
206	C₄²	—	5	2	no	—
207	C₄²	—	5	1	no	—
208	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
209	C₃×C₅	#557	1	1	yes	characteristic, complemented, Hall
210	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	5	4	no	—
216	C₃×C₄	—	5	2	no	—
217	C₃×C₄	—	10	1	no	—
218	C₃×C₄	—	10	2	no	—
219	C₃×C₄	—	10	1	no	—
220	C₃×C₄	—	10	1	no	—
221	C₃×C₄	Q₈×D₅	1	2	yes	—
222	C₂×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
223	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
224	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
225	Q₈	—	5	4	no	—
226	Q₈	—	10	4	no	—
227	Q₈	—	10	2	no	—
228	C₂×C₄	—	5	2	no	—
229	C₂×C₄	—	5	1	no	—
230	C₂×C₄	—	5	1	no	—
231	C₂×C₄	—	5	1	no	—
232	C₂×C₄	—	5	1	no	—
233	C₂×C₄	—	5	2	no	—
234	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
236	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
237	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
239	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
240	C₂×C₃	D₅×#41	1	1	yes	characteristic
241	C₂×C₃	#4150	1	1	yes	characteristic
242	C₅	C₃×#557	1	1	yes	characteristic, complemented, Sylow
243	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	2	no	—
248	C₄	—	5	4	no	—
249	C₄	—	5	2	no	—
250	C₄	—	10	1	no	—
251	C₄	—	10	2	no	—
252	C₄	—	10	1	no	—
253	C₄	—	10	1	no	—
254	C₄	C₃×Q₈×D₅	1	2	yes	—
255	C₃	#66019	1	1	yes	characteristic, direct product factor, Sylow
256	C₂	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
257	C₂	C₃×D₅×#41	1	1	yes	characteristic
258	C₂	C₃×#4150	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×C₅×#125	C₂
socle, prime radical	C₂²×C₃×C₅	C₂⁴
hypercenter	C₃×#125	D₅
hypocenter	C₅	C₃×#557
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#125

Complements

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

N	H ≅ G/N	type
12. #66019	C₃	direct (N × H)
61. C₃×Dic₁₀	Q₈	semidirect
145. Dic₁₀	C₃×Q₈	semidirect
209. C₃×C₅	#557	semidirect
242. C₅	C₃×#557	semidirect
255. C₃	#66019	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×C₅×#125	2
C₃×#4036	2
C₃×#4035	2
C₃×#4034	2
C₃×#4033	2
C₃×C₄×Dic₁₀	2
#66019	3
C₃×#557	5

Subgroups — C3×#66019 #11755696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66019 #11755696