Subgroups — C₃×#10615546 #14412205

Order 1920.

1408 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4150	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#66153	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#3956	C₂	1	6	yes	maximal, complemented
5	C₃×#65421	C₂	1	6	yes	maximal, complemented
6	#10615546	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×#4150	—	2	4	no	—
9	C₃×#4150	—	2	4	no	—
10	C₂×C₃×Q₈×D₅	C₄	1	2	yes	—
11	C₂×C₃×#711	C₄	1	6	yes	—
12	C₂×C₃×#709	C₂²	1	3	yes	—
13	C₂²×C₃×Dic₁₀	C₂²	1	3	yes	—
14	C₃×C₅×#131	—	2	4	no	—
15	C₂×C₃×C₅×M₁₆	C₂²	1	6	yes	—
16	C₃×#4094	—	2	4	no	—
17	C₂×C₃×#681	C₂²	1	6	yes	—
18	C₃×#3956	C₂²	1	12	yes	—
19	C₃×#3955	C₂²	1	4	yes	—
20	C₃×#2889	—	5	1	no	maximal, Hall
21	C₂×#4150	C₂×C₃	1	1	yes	characteristic
22	C₅×#568	C₂×C₃	1	1	yes	characteristic, complemented
23	#66153	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#3956	C₂×C₃	1	6	yes	complemented
25	#65421	C₂×C₃	1	6	yes	complemented
26	C₃×C₅×#41	D₄	1	4	yes	—
27	C₃×C₅×#41	D₄	1	4	yes	—
28	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₅×D₄	C₂×C₄	1	3	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
31	C₂×C₃×C₂²⋊D₅	C₂×C₄	1	6	yes	—
32	C₂²×C₃×Dic₅	C₂×C₄	1	6	yes	—
33	C₃×Q₈×D₅	—	2	8	no	—
34	C₃×#711	—	2	24	no	—
35	C₃×#709	—	2	6	no	—
36	C₃×#709	—	2	6	no	—
37	C₂×C₃×D₂₀	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄×D₅	C₂×C₄	1	6	yes	—
39	C₂×C₃×Dic₁₀	—	2	6	no	—
40	C₂×C₃×Dic₁₀	—	2	6	no	—
41	C₂×C₃×Dic₁₀	C₂×C₄	1	6	yes	—
42	C₂×C₃×Dic₁₀	C₂³	1	3	yes	—
43	C₃×C₅×M₁₆	—	2	6	no	—
44	C₃×C₅×M₁₆	C₂³	1	4	yes	—
45	C₂×C₃×C₅×C₈	—	2	6	no	—
46	C₃×#681	—	2	6	no	—
47	C₃×#681	C₂³	1	4	yes	—
48	C₂×C₃×#168	—	2	6	no	—
49	C₂×C₃×2₋¹⁺⁴	—	5	1	no	—
50	C₃×#568	—	5	1	no	—
51	C₃×#568	D₅	1	1	yes	characteristic, complemented
52	C₃×#413	—	5	6	no	—
53	C₂×C₃×#101	—	5	6	no	—
54	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
55	#4150	—	2	4	no	—
56	#4150	—	2	4	no	—
57	C₂×Q₈×D₅	C₃×C₄	1	2	yes	—
58	C₂×#711	C₃×C₄	1	6	yes	—
59	C₂×#709	C₂²×C₃	1	3	yes	—
60	C₂²×Dic₁₀	C₂²×C₃	1	3	yes	—
61	C₅×#131	—	2	4	no	—
62	C₂×C₅×M₁₆	C₂²×C₃	1	6	yes	—
63	#4094	—	2	4	no	—
64	C₂×#681	C₂²×C₃	1	6	yes	—
65	#3956	C₂²×C₃	1	12	yes	—
66	#3955	C₂²×C₃	1	4	yes	—
67	#2889	—	5	1	no	Sylow
68	C₂³×C₃×C₅	C₂²×C₄	1	3	yes	—
69	C₂²×C₃×D₅	C₂²×C₄	1	2	yes	—
70	C₃×C₅×Q₈	C₂²⋊C₄	1	4	yes	—
71	C₃×C₅×D₄	C₂²⋊C₄	1	12	yes	—
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
74	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
75	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
76	C₃×C₂²⋊D₅	—	2	12	no	—
77	C₂×C₃×Dic₅	—	2	12	no	—
78	C₂×C₃×Dic₅	C₂²×C₄	1	6	yes	—
79	C₃×D₂₀	—	2	1	no	—
80	C₃×D₂₀	—	2	1	no	—
81	C₃×C₄×D₅	—	2	12	no	—
82	C₃×Dic₁₀	—	2	12	no	—
83	C₃×Dic₁₀	—	2	3	no	—
84	C₃×Dic₁₀	—	2	3	no	—
85	C₃×C₅×C₈	—	2	4	no	—
86	C₃×#168	—	2	4	no	—
87	C₃×2₋¹⁺⁴	—	10	4	no	—
88	C₃×2₋¹⁺⁴	—	10	4	no	—
89	C₂×C₃×#41	—	5	6	no	—
90	C₂×C₃×#41	—	5	3	no	—
91	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
92	C₂²×C₃×Q₈	—	5	2	no	—
93	C₂²×C₃×Q₈	—	5	3	no	—
94	C₃×#131	—	2	4	no	—
95	C₃×#131	—	10	4	no	—
96	C₂×C₃×M₁₆	—	5	6	no	—
97	C₂×C₃×M₁₆	C₂×D₅	1	6	yes	—
98	C₃×#101	—	5	12	no	—
99	C₃×#100	—	5	4	no	—
100	C₅×#41	C₃×D₄	1	4	yes	—
101	C₅×#41	C₃×D₄	1	4	yes	—
102	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
103	C₂×C₅×D₄	C₂×C₃×C₄	1	3	yes	—
104	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
105	C₂×C₂²⋊D₅	C₂×C₃×C₄	1	6	yes	—
106	C₂²×Dic₅	C₂×C₃×C₄	1	6	yes	—
107	Q₈×D₅	—	2	8	no	—
108	#711	—	2	24	no	—
109	#709	—	2	6	no	—
110	#709	—	2	6	no	—
111	C₂×D₂₀	C₂³×C₃	1	1	yes	characteristic
112	C₂×C₄×D₅	C₂×C₃×C₄	1	6	yes	—
113	C₂×Dic₁₀	—	2	6	no	—
114	C₂×Dic₁₀	—	2	6	no	—
115	C₂×Dic₁₀	C₂×C₃×C₄	1	6	yes	—
116	C₂×Dic₁₀	C₂³×C₃	1	3	yes	—
117	C₅×M₁₆	—	2	6	no	—
118	C₅×M₁₆	C₂³×C₃	1	4	yes	—
119	C₂×C₅×C₈	—	2	6	no	—
120	#681	—	2	6	no	—
121	#681	C₂³×C₃	1	4	yes	—
122	C₂×#168	—	2	6	no	—
123	C₂×2₋¹⁺⁴	—	5	1	no	—
124	#568	—	5	1	no	—
125	#568	C₃×D₅	1	1	yes	characteristic, complemented
126	#413	—	5	6	no	—
127	C₂×#101	—	5	6	no	—
128	C₂²×C₃×C₅	—	4	3	no	—
129	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	6	yes	—
130	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₃×D₅	—	2	2	no	—
132	C₂×C₃×D₅	—	4	2	no	—
133	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
134	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
135	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₃×Dic₅	—	2	6	no	—
137	C₃×#41	—	10	24	no	—
138	C₃×#41	—	10	6	no	—
139	C₃×#41	—	10	6	no	—
140	C₃×#41	D₂₀	1	4	yes	—
141	C₃×#41	C₂²⋊D₅	1	4	yes	—
142	C₂×C₃×Q₈	—	5	6	no	—
143	C₂×C₃×Q₈	—	5	3	no	—
144	C₂×C₃×Q₈	—	10	8	no	—
145	C₂×C₃×Q₈	—	10	6	no	—
146	C₂×C₃×Q₈	—	10	6	no	—
147	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
148	C₂×C₃×D₄	—	5	6	no	—
149	C₂×C₃×D₄	—	5	1	no	—
150	C₂×C₃×D₄	C₄×D₅	1	3	yes	—
151	C₂²×C₃×C₄	—	5	6	no	—
152	C₂²×C₃×C₄	—	5	6	no	—
153	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
154	C₃×M₁₆	—	2	6	no	—
155	C₃×M₁₆	—	5	4	no	—
156	C₃×M₁₆	—	10	6	no	—
157	C₃×M₁₆	C₂²×D₅	1	4	yes	—
158	C₂×C₃×C₈	—	2	6	no	—
159	C₂×C₃×C₈	—	10	6	no	—
160	C₂³×C₅	C₂²×C₃×C₄	1	3	yes	—
161	C₂²×D₅	C₂²×C₃×C₄	1	2	yes	—
162	C₅×Q₈	C₃×C₂²⋊C₄	1	4	yes	—
163	C₅×D₄	C₃×C₂²⋊C₄	1	12	yes	—
164	C₂×C₄×C₅	C₂²×C₃×C₄	1	3	yes	—
165	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
166	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
167	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
168	C₂²⋊D₅	—	2	12	no	—
169	C₂×Dic₅	—	2	12	no	—
170	C₂×Dic₅	C₂²×C₃×C₄	1	6	yes	—
171	D₂₀	—	2	1	no	—
172	D₂₀	—	2	1	no	—
173	C₄×D₅	—	2	12	no	—
174	Dic₁₀	—	2	12	no	—
175	Dic₁₀	—	2	3	no	—
176	Dic₁₀	—	2	3	no	—
177	C₅×C₈	—	2	4	no	—
178	#168	—	2	4	no	—
179	2₋¹⁺⁴	—	10	4	no	—
180	2₋¹⁺⁴	—	10	4	no	—
181	C₂×#41	—	5	6	no	—
182	C₂×#41	—	5	3	no	—
183	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
184	C₂²×Q₈	—	5	2	no	—
185	C₂²×Q₈	—	5	3	no	—
186	#131	—	2	4	no	—
187	#131	—	10	4	no	—
188	C₂×M₁₆	—	5	6	no	—
189	C₂×M₁₆	C₂×C₃×D₅	1	6	yes	—
190	#101	—	5	12	no	—
191	#100	—	5	4	no	—
192	C₂×C₃×C₅	—	2	1	no	—
193	C₂×C₃×C₅	—	2	6	no	—
194	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
195	C₃×D₅	—	4	2	no	—
196	C₂³×C₃	—	5	2	no	—
197	C₂³×C₃	C₂×C₄×D₅	1	3	yes	—
198	C₃×Q₈	—	10	3	no	—
199	C₃×Q₈	—	10	3	no	—
200	C₃×Q₈	—	10	12	no	—
201	C₃×Q₈	#685	1	4	yes	—
202	C₃×D₄	—	10	12	no	—
203	C₃×D₄	—	10	1	no	—
204	C₃×D₄	—	10	1	no	—
205	C₃×D₄	#685	1	12	yes	—
206	C₂×C₃×C₄	—	5	6	no	—
207	C₂×C₃×C₄	—	10	12	no	—
208	C₂×C₃×C₄	—	10	12	no	—
209	C₂×C₃×C₄	C₂×C₄×D₅	1	3	yes	—
210	C₂×C₃×C₄	C₂×D₂₀	1	6	yes	—
211	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	6	yes	—
212	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
213	C₃×C₈	—	2	4	no	—
214	C₃×C₈	—	10	4	no	—
215	C₂²×C₅	—	4	3	no	—
216	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	6	yes	—
217	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
218	C₂×D₅	—	2	2	no	—
219	C₂×D₅	—	4	2	no	—
220	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	6	yes	—
221	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
222	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
223	Dic₅	—	2	6	no	—
224	#41	—	10	24	no	—
225	#41	—	10	6	no	—
226	#41	—	10	6	no	—
227	#41	C₃×D₂₀	1	4	yes	—
228	#41	C₃×C₂²⋊D₅	1	4	yes	—
229	C₂×Q₈	—	5	6	no	—
230	C₂×Q₈	—	5	3	no	—
231	C₂×Q₈	—	10	8	no	—
232	C₂×Q₈	—	10	6	no	—
233	C₂×Q₈	—	10	6	no	—
234	C₂×Q₈	C₃×C₄×D₅	1	1	yes	characteristic
235	C₂×D₄	—	5	6	no	—
236	C₂×D₄	—	5	1	no	—
237	C₂×D₄	C₃×C₄×D₅	1	3	yes	—
238	C₂²×C₄	—	5	6	no	—
239	C₂²×C₄	—	5	6	no	—
240	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
241	M₁₆	—	2	6	no	—
242	M₁₆	—	5	4	no	—
243	M₁₆	—	10	6	no	—
244	M₁₆	C₂²×C₃×D₅	1	4	yes	—
245	C₂×C₈	—	2	6	no	—
246	C₂×C₈	—	10	6	no	—
247	C₃×C₅	#2889	1	1	yes	characteristic, complemented, Hall
248	C₂²×C₃	—	4	3	no	—
249	C₂²×C₃	—	10	2	no	—
250	C₂²×C₃	—	20	2	no	—
251	C₂²×C₃	C₂×#685	1	6	yes	—
252	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
253	C₃×C₄	—	10	6	no	—
254	C₃×C₄	C₂×#685	1	6	yes	—
255	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
256	C₃×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
257	C₂×C₅	—	2	1	no	—
258	C₂×C₅	—	2	6	no	—
259	C₂×C₅	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
260	D₅	—	4	2	no	—
261	C₂³	—	5	2	no	—
262	C₂³	C₂×C₃×C₄×D₅	1	3	yes	—
263	Q₈	—	10	3	no	—
264	Q₈	—	10	3	no	—
265	Q₈	—	10	12	no	—
266	Q₈	C₃×#685	1	4	yes	—
267	D₄	—	10	12	no	—
268	D₄	—	10	1	no	—
269	D₄	—	10	1	no	—
270	D₄	C₃×#685	1	12	yes	—
271	C₂×C₄	—	5	6	no	—
272	C₂×C₄	—	10	12	no	—
273	C₂×C₄	—	10	12	no	—
274	C₂×C₄	C₂×C₃×C₄×D₅	1	3	yes	—
275	C₂×C₄	C₂×C₃×D₂₀	1	6	yes	—
276	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	6	yes	—
277	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
278	C₈	—	2	4	no	—
279	C₈	—	10	4	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	2	6	no	—
282	C₂×C₃	—	20	2	no	—
283	C₂×C₃	C₂²×#685	1	1	yes	characteristic
284	C₅	C₃×#2889	1	1	yes	characteristic, complemented, Sylow
285	C₂²	—	4	3	no	—
286	C₂²	—	10	2	no	—
287	C₂²	—	20	2	no	—
288	C₂²	C₂×C₃×#685	1	6	yes	—
289	C₂²	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
290	C₄	—	10	6	no	—
291	C₄	C₂×C₃×#685	1	6	yes	—
292	C₄	C₂²×C₃×D₂₀	1	1	yes	characteristic
293	C₄	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
294	C₃	#10615546	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	—	2	1	no	—
296	C₂	—	2	6	no	—
297	C₂	—	20	2	no	—
298	C₂	C₂²×C₃×#685	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×#685
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#568	C₂
socle	C₂×C₃×C₅	C₂²×C₂²⋊C₄
prime radical	C₂×C₃×#4150	C₂
cosocle	C₂×C₄×C₅	C₂⁴×C₃
hypercenter	C₃×#568	D₅
hypocenter	C₅	C₃×#2889
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×#41 ◁ C₃×#568

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#568	C₂	semidirect
3. C₃×#66153	C₂	semidirect
4. C₂×C₃×#3956	C₂	semidirect
5. C₃×#65421	C₂	semidirect
6. #10615546	C₃	direct (N × H)
22. C₅×#568	C₂×C₃	semidirect
23. #66153	C₂×C₃	semidirect
24. C₂×#3956	C₂×C₃	semidirect
25. #65421	C₂×C₃	semidirect
51. C₃×#568	D₅	semidirect
125. #568	C₃×D₅	semidirect
247. C₃×C₅	#2889	semidirect
284. C₅	C₃×#2889	semidirect
294. C₃	#10615546	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×#4150	2
C₃×C₅×#568	2
C₃×#66153	2
C₂×C₃×#3956	2
C₃×#65421	2
#10615546	3
C₃×#2889	5

Subgroups — C3×#10615546 #14412205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10615546 #14412205