Subgroups — C₃×#10601400 #14299103

Order 1920.

496 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×C₅×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3950	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₃×#65266	C₂	1	2	yes	maximal
5	C₂×C₃×#3940	C₂	1	2	yes	maximal
6	#10601400	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
10	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
11	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
12	C₃×C₅×#107	—	2	2	no	—
13	C₃×#3963	—	2	4	no	—
14	C₃×#3950	—	2	2	no	—
15	C₃×C₈×Dic₅	C₄	1	4	yes	complemented
16	C₃×#3940	—	2	4	no	—
17	C₃×#1846	—	5	1	no	maximal, Hall
18	C₂×C₅×#107	C₂×C₃	1	1	yes	characteristic
19	C₂×#3950	C₂×C₃	1	1	yes	characteristic
20	C₂×C₈×Dic₅	C₂×C₃	1	1	yes	characteristic
21	#65266	C₂×C₃	1	2	yes	—
22	C₂×#3940	C₂×C₃	1	2	yes	—
23	C₂²×C₃×C₄×C₅	—	2	2	no	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₃×Dic₅	—	2	2	no	—
26	C₂²×C₃×Dic₅	D₄	1	2	yes	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₃×C₄⋊Dic₅	—	2	2	no	—
33	C₃×C₄⋊Dic₅	—	2	2	no	—
34	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
35	C₃×C₄×Dic₅	Q₈	1	2	yes	—
36	C₂×C₃×#168	C₂×C₄	1	2	yes	—
37	C₂×C₃×#168	D₄	1	2	yes	—
38	C₂×C₃×#168	Q₈	1	2	yes	—
39	C₂×C₃×#107	—	5	1	no	—
40	C₂×C₃×#107	D₅	1	1	yes	characteristic
41	C₂×C₃×#106	—	5	2	no	—
42	C₂×C₃×C₄×C₈	—	5	1	no	—
43	C₃×#382	—	5	2	no	—
44	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
46	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
47	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
48	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
49	C₅×#107	—	2	2	no	—
50	#3963	—	2	4	no	—
51	#3950	—	2	2	no	—
52	C₈×Dic₅	C₃×C₄	1	4	yes	complemented
53	#3940	—	2	4	no	—
54	#1846	—	5	1	no	Sylow
55	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	—	4	2	no	—
57	C₂×C₃×C₄×C₅	—	4	4	no	—
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃×Dic₅	—	2	2	no	—
65	C₂×C₃×Dic₅	—	2	2	no	—
66	C₂×C₃×Dic₅	—	4	2	no	—
67	C₂×C₃×Dic₅	—	4	4	no	—
68	C₂×C₃×Dic₅	C₄⋊C₄	1	4	yes	—
69	C₃×C₅×C₈	C₄⋊C₄	1	4	yes	—
70	C₃×#168	C₄⋊C₄	1	4	yes	—
71	C₂²×C₃×C₈	—	5	1	no	—
72	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
73	C₂×C₃×C₄⋊C₄	—	5	2	no	—
74	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
75	C₂×C₃×C₄²	—	5	1	no	—
76	C₃×#107	—	2	2	no	—
77	C₃×#107	—	10	2	no	—
78	C₃×#106	—	10	4	no	—
79	C₃×C₄×C₈	—	5	4	no	—
80	C₃×#95	—	10	4	no	—
81	C₂²×C₄×C₅	—	2	2	no	—
82	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
83	C₂²×Dic₅	—	2	2	no	—
84	C₂²×Dic₅	C₃×D₄	1	2	yes	—
85	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
86	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
87	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₄⋊Dic₅	—	2	2	no	—
91	C₄⋊Dic₅	—	2	2	no	—
92	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
93	C₄×Dic₅	C₃×Q₈	1	2	yes	—
94	C₂×#168	C₂×C₃×C₄	1	2	yes	—
95	C₂×#168	C₃×D₄	1	2	yes	—
96	C₂×#168	C₃×Q₈	1	2	yes	—
97	C₂×#107	—	5	1	no	—
98	C₂×#107	C₃×D₅	1	1	yes	characteristic
99	C₂×#106	—	5	2	no	—
100	C₂×C₄×C₈	—	5	1	no	—
101	#382	—	5	2	no	—
102	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₃×C₅	#127	1	1	yes	characteristic
104	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
105	C₂²×C₃×C₅	#135	1	2	yes	—
106	C₂²×C₃×C₅	#135	1	2	yes	—
107	C₃×C₄×C₅	—	8	2	no	—
108	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
111	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
112	C₃×Dic₅	—	2	4	no	—
113	C₃×Dic₅	—	8	2	no	—
114	C₂²×C₃×C₄	—	2	2	no	—
115	C₂²×C₃×C₄	—	5	2	no	—
116	C₂²×C₃×C₄	—	10	2	no	—
117	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
118	C₂×C₃×C₈	—	5	2	no	—
119	C₂×C₃×C₈	—	5	2	no	—
120	C₂×C₃×C₈	—	5	2	no	—
121	C₂×C₃×C₈	Dic₁₀	1	2	yes	—
122	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
123	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
124	C₃×C₄⋊C₄	—	2	2	no	—
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₄⋊C₄	—	10	2	no	—
127	C₃×C₄⋊C₄	—	10	2	no	—
128	C₃×C₄²	—	5	2	no	—
129	C₃×C₄²	—	5	2	no	—
130	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
139	C₂×Dic₅	—	2	2	no	—
140	C₂×Dic₅	—	2	2	no	—
141	C₂×Dic₅	—	4	2	no	—
142	C₂×Dic₅	—	4	4	no	—
143	C₂×Dic₅	C₃×C₄⋊C₄	1	4	yes	—
144	C₅×C₈	C₃×C₄⋊C₄	1	4	yes	—
145	#168	C₃×C₄⋊C₄	1	4	yes	—
146	C₂²×C₈	—	5	1	no	—
147	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
148	C₂×C₄⋊C₄	—	5	2	no	—
149	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
150	C₂×C₄²	—	5	1	no	—
151	#107	—	2	2	no	—
152	#107	—	10	2	no	—
153	#106	—	10	4	no	—
154	C₄×C₈	—	5	4	no	—
155	#95	—	10	4	no	—
156	C₂×C₃×C₅	#384	1	1	yes	characteristic
157	C₂×C₃×C₅	#427	1	2	yes	—
158	C₂×C₃×C₅	#495	1	1	yes	characteristic
159	C₂×C₃×C₅	#495	1	1	yes	characteristic
160	C₂×C₃×C₅	#499	1	2	yes	—
161	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
162	C₂×C₃×C₄	—	4	4	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	—	5	4	no	—
165	C₂×C₃×C₄	—	10	2	no	—
166	C₂×C₃×C₄	—	10	2	no	—
167	C₂×C₃×C₄	—	20	4	no	—
168	C₂×C₃×C₄	—	20	2	no	—
169	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
171	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
172	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
173	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
175	C₃×C₈	—	5	4	no	—
176	C₃×C₈	#683	1	4	yes	—
177	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₅	C₃×#127	1	1	yes	characteristic
179	C₂²×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
180	C₂²×C₅	C₃×#135	1	2	yes	—
181	C₂²×C₅	C₃×#135	1	2	yes	—
182	C₄×C₅	—	8	2	no	—
183	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
184	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
185	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
186	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
187	Dic₅	—	2	4	no	—
188	Dic₅	—	8	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	5	2	no	—
191	C₂²×C₄	—	10	2	no	—
192	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
193	C₂×C₈	—	5	2	no	—
194	C₂×C₈	—	5	2	no	—
195	C₂×C₈	—	5	2	no	—
196	C₂×C₈	C₃×Dic₁₀	1	2	yes	—
197	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
198	C₂×C₈	C₃×C₂²⋊D₅	1	2	yes	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	10	2	no	—
202	C₄⋊C₄	—	10	2	no	—
203	C₄²	—	5	2	no	—
204	C₄²	—	5	2	no	—
205	C₃×C₅	#1846	1	1	yes	characteristic, complemented, Hall
206	C₂²×C₃	#4034	1	1	yes	characteristic
207	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
208	C₂²×C₃	#4058	1	2	yes	—
209	C₂²×C₃	#4065	1	2	yes	—
210	C₂²×C₃	#4086	1	1	yes	characteristic
211	C₃×C₄	—	8	2	no	—
212	C₃×C₄	—	10	4	no	—
213	C₃×C₄	—	40	2	no	—
214	C₃×C₄	#4034	1	1	yes	characteristic
215	C₃×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
216	C₃×C₄	C₂×#683	1	1	yes	characteristic
217	C₃×C₄	#4090	1	1	yes	characteristic
218	C₂×C₅	C₃×#384	1	1	yes	characteristic
219	C₂×C₅	C₃×#427	1	2	yes	—
220	C₂×C₅	C₃×#495	1	1	yes	characteristic
221	C₂×C₅	C₃×#495	1	1	yes	characteristic
222	C₂×C₅	C₃×#499	1	2	yes	—
223	C₂³	C₃×D₄×D₅	1	1	yes	characteristic
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	5	4	no	—
227	C₂×C₄	—	10	2	no	—
228	C₂×C₄	—	10	2	no	—
229	C₂×C₄	—	20	2	no	—
230	C₂×C₄	—	20	4	no	—
231	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
232	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
233	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
234	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
235	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
236	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
237	C₈	—	5	4	no	—
238	C₈	C₃×#683	1	4	yes	—
239	C₂×C₃	#65167	1	2	yes	—
240	C₂×C₃	#65169	1	2	yes	—
241	C₂×C₃	#65265	1	1	yes	characteristic
242	C₂×C₃	#65444	1	1	yes	characteristic
243	C₂×C₃	#65480	1	1	yes	characteristic
244	C₅	C₃×#1846	1	1	yes	characteristic, complemented, Sylow
245	C₂²	C₃×#4034	1	1	yes	characteristic
246	C₂²	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
247	C₂²	C₃×#4058	1	2	yes	—
248	C₂²	C₃×#4065	1	2	yes	—
249	C₂²	C₃×#4086	1	1	yes	characteristic
250	C₄	—	8	2	no	—
251	C₄	—	10	4	no	—
252	C₄	—	40	2	no	—
253	C₄	C₃×#4034	1	1	yes	characteristic
254	C₄	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
255	C₄	C₂×C₃×#683	1	1	yes	characteristic
256	C₄	C₃×#4090	1	1	yes	characteristic
257	C₃	#10601400	1	1	yes	characteristic, direct product factor, Sylow
258	C₂	C₃×#65167	1	2	yes	—
259	C₂	C₃×#65169	1	2	yes	—
260	C₂	C₃×#65265	1	1	yes	characteristic
261	C₂	C₃×#65444	1	1	yes	characteristic
262	C₂	C₃×#65480	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#107	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#107	D₅
hypocenter	C₅	C₃×#1846
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#107

Complements

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

N	H ≅ G/N	type
6. #10601400	C₃	direct (N × H)
15. C₃×C₈×Dic₅	C₄	semidirect
52. C₈×Dic₅	C₃×C₄	semidirect
205. C₃×C₅	#1846	semidirect
244. C₅	C₃×#1846	semidirect
257. C₃	#10601400	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×C₅×#107	2
C₂×C₃×#3950	2
C₂×C₃×C₈×Dic₅	2
C₃×#65266	2
C₂×C₃×#3940	2
#10601400	3
C₃×#1846	5

Subgroups — C3×#10601400 #14299103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601400 #14299103