Subgroups — C₂×C₃×#64998 #14413833

Order 1920.

824 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	C₂×C₃×#4018	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3952	C₂	1	2	yes	maximal
4	C₂×C₃×#3946	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64998	C₂	1	8	yes	maximal, direct product factor
7	C₂×#64998	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×#685	C₄	1	2	yes	—
12	C₂×C₃×#683	C₄	1	2	yes	—
13	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
14	C₂²×C₃×#168	C₂²	1	2	yes	—
15	C₃×#4018	C₄	1	4	yes	—
16	C₃×#4018	C₂²	1	4	yes	—
17	C₃×C₅×#97	C₂²	1	4	yes	complemented
18	C₃×#3952	C₂²	1	8	yes	—
19	C₃×#3946	C₂²	1	8	yes	complemented
20	C₃×C₄×#168	C₂²	1	4	yes	complemented
21	C₂×C₃×#433	—	5	1	no	maximal, Hall
22	C₂×#4018	C₂×C₃	1	1	yes	characteristic
23	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#3952	C₂×C₃	1	2	yes	—
25	C₂×#3946	C₂×C₃	1	2	yes	complemented
26	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic, complemented
27	#64998	C₂×C₃	1	8	yes	direct product factor
28	C₂³×C₃×D₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×D₅	—	2	4	no	—
34	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
36	C₂×C₃×C₅×C₈	—	2	4	no	—
37	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
38	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂³	1	2	yes	—
40	C₃×#685	C₂×C₄	1	8	yes	—
41	C₃×#683	C₂×C₄	1	8	yes	—
42	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
43	C₃×C₄×Dic₅	C₂³	1	2	yes	—
44	C₂×C₃×#168	—	2	4	no	—
45	C₂×C₃×#168	C₂³	1	4	yes	—
46	C₂×C₃×#117	—	5	1	no	—
47	C₃×#433	—	5	8	no	—
48	C₂×C₃×C₄⋊C₈	—	5	2	no	—
49	C₂×C₃×C₂²⋊C₈	—	5	2	no	—
50	C₂×C₃×#97	D₅	1	1	yes	characteristic, complemented
51	C₂×C₃×C₄×C₈	—	5	1	no	—
52	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
54	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
55	C₂×#685	C₃×C₄	1	2	yes	—
56	C₂×#683	C₃×C₄	1	2	yes	—
57	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
58	C₂²×#168	C₂²×C₃	1	2	yes	—
59	#4018	C₃×C₄	1	4	yes	—
60	#4018	C₂²×C₃	1	4	yes	—
61	C₅×#97	C₂²×C₃	1	4	yes	complemented
62	#3952	C₂²×C₃	1	8	yes	—
63	#3946	C₂²×C₃	1	8	yes	complemented
64	C₄×#168	C₂²×C₃	1	4	yes	complemented
65	C₂×#433	—	5	1	no	Sylow
66	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₃×D₅	—	2	2	no	—
68	C₂²×C₃×D₅	—	2	4	no	—
69	C₂²×C₃×D₅	C₂²×C₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	—	2	4	no	—
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	#41	1	4	yes	—
74	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₂×C₃×Dic₅	—	2	2	no	—
76	C₂×C₃×Dic₅	—	2	4	no	—
77	C₂×C₃×Dic₅	C₂²×C₄	1	4	yes	—
78	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
79	C₃×C₄×D₅	—	2	4	no	—
80	C₃×C₄×D₅	—	2	4	no	—
81	C₃×C₅×C₈	—	2	4	no	—
82	C₃×#168	—	2	4	no	—
83	C₂³×C₃×C₄	—	5	1	no	—
84	C₂²×C₃×C₈	—	5	2	no	—
85	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
86	C₃×#117	—	5	4	no	—
87	C₃×#117	—	5	4	no	—
88	C₂×C₃×C₄⋊C₄	—	5	2	no	—
89	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
90	C₂×C₃×C₄²	—	5	1	no	—
91	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
92	C₃×C₄⋊C₈	—	5	8	no	—
93	C₃×C₂²⋊C₈	—	5	8	no	—
94	C₃×#97	C₂×D₅	1	4	yes	complemented
95	C₃×C₄×C₈	—	5	4	no	—
96	C₂³×D₅	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
98	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
99	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
100	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
101	C₂×C₄×D₅	—	2	4	no	—
102	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	—
103	C₂×C₄×D₅	C₂³×C₃	1	2	yes	—
104	C₂×C₅×C₈	—	2	4	no	—
105	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
106	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
107	C₄²×C₅	C₂³×C₃	1	2	yes	—
108	#685	C₂×C₃×C₄	1	8	yes	—
109	#683	C₂×C₃×C₄	1	8	yes	—
110	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
111	C₄×Dic₅	C₂³×C₃	1	2	yes	—
112	C₂×#168	—	2	4	no	—
113	C₂×#168	C₂³×C₃	1	4	yes	—
114	C₂×#117	—	5	1	no	—
115	#433	—	5	8	no	—
116	C₂×C₄⋊C₈	—	5	2	no	—
117	C₂×C₂²⋊C₈	—	5	2	no	—
118	C₂×#97	C₃×D₅	1	1	yes	characteristic, complemented
119	C₂×C₄×C₈	—	5	1	no	—
120	C₂²×C₃×C₅	#117	1	2	yes	—
121	C₂²×C₃×C₅	#131	1	4	yes	—
122	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₃×D₅	—	2	2	no	—
124	C₂×C₃×D₅	—	2	4	no	—
125	C₂×C₃×D₅	—	4	4	no	—
126	C₃×C₄×C₅	—	2	4	no	—
127	C₃×C₄×C₅	#117	1	2	yes	—
128	C₃×C₄×C₅	C₂×#41	1	2	yes	—
129	C₃×Dic₅	—	2	2	no	—
130	C₃×Dic₅	—	2	4	no	—
131	C₂⁴×C₃	—	5	1	no	—
132	C₂²×C₃×C₄	—	5	2	no	—
133	C₂²×C₃×C₄	—	5	1	no	—
134	C₂²×C₃×C₄	—	5	2	no	—
135	C₂²×C₃×C₄	—	5	2	no	—
136	C₂²×C₃×C₄	—	10	4	no	—
137	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
138	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
139	C₂×C₃×C₈	—	2	4	no	—
140	C₂×C₃×C₈	—	5	4	no	—
141	C₂×C₃×C₈	—	10	4	no	—
142	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
143	C₃×C₄⋊C₄	—	5	8	no	—
144	C₃×C₂²⋊C₄	—	5	8	no	—
145	C₃×C₄²	—	5	2	no	—
146	C₃×C₄²	—	5	2	no	—
147	C₃×C₄²	C₄×D₅	1	2	yes	—
148	C₃×C₄²	C₂²×D₅	1	2	yes	—
149	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
150	C₂²×D₅	—	2	2	no	—
151	C₂²×D₅	—	2	4	no	—
152	C₂²×D₅	C₂²×C₃×C₄	1	2	yes	—
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
155	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
156	C₂×C₄×C₅	C₃×#41	1	4	yes	—
157	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
158	C₂×Dic₅	—	2	2	no	—
159	C₂×Dic₅	—	2	4	no	—
160	C₂×Dic₅	C₂²×C₃×C₄	1	4	yes	—
161	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
162	C₄×D₅	—	2	4	no	—
163	C₄×D₅	—	2	4	no	—
164	C₅×C₈	—	2	4	no	—
165	#168	—	2	4	no	—
166	C₂³×C₄	—	5	1	no	—
167	C₂²×C₈	—	5	2	no	—
168	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
169	#117	—	5	4	no	—
170	#117	—	5	4	no	—
171	C₂×C₄⋊C₄	—	5	2	no	—
172	C₂×C₂²⋊C₄	—	5	2	no	—
173	C₂×C₄²	—	5	1	no	—
174	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
175	C₄⋊C₈	—	5	8	no	—
176	C₂²⋊C₈	—	5	8	no	—
177	#97	C₂×C₃×D₅	1	4	yes	complemented
178	C₄×C₈	—	5	4	no	—
179	C₂×C₃×C₅	#433	1	4	yes	complemented
180	C₂×C₃×C₅	C₂×#117	1	1	yes	characteristic
181	C₂×C₃×C₅	C₂×#131	1	2	yes	—
182	C₃×D₅	—	4	2	no	—
183	C₂³×C₃	—	5	2	no	—
184	C₂³×C₃	—	10	2	no	—
185	C₂³×C₃	—	10	4	no	—
186	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	5	4	no	—
189	C₂×C₃×C₄	—	5	2	no	—
190	C₂×C₃×C₄	—	10	2	no	—
191	C₂×C₃×C₄	—	10	4	no	—
192	C₂×C₃×C₄	—	10	4	no	—
193	C₂×C₃×C₄	—	10	4	no	—
194	C₂×C₃×C₄	C₂×C₄×D₅	1	4	yes	—
195	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
196	C₂×C₃×C₄	#709	1	4	yes	—
197	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
198	C₃×C₈	—	2	4	no	—
199	C₃×C₈	—	10	4	no	—
200	C₂²×C₅	C₃×#117	1	2	yes	—
201	C₂²×C₅	C₃×#131	1	4	yes	—
202	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
203	C₂×D₅	—	2	4	no	—
204	C₂×D₅	—	2	2	no	—
205	C₂×D₅	—	4	4	no	—
206	C₄×C₅	—	2	4	no	—
207	C₄×C₅	C₃×#117	1	2	yes	—
208	C₄×C₅	C₂×C₃×#41	1	2	yes	—
209	Dic₅	—	2	4	no	—
210	Dic₅	—	2	2	no	—
211	C₂⁴	—	5	1	no	—
212	C₂²×C₄	—	5	2	no	—
213	C₂²×C₄	—	5	1	no	—
214	C₂²×C₄	—	5	2	no	—
215	C₂²×C₄	—	5	2	no	—
216	C₂²×C₄	—	10	4	no	—
217	C₂²×C₄	C₃×C₄×D₅	1	2	yes	—
218	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	5	4	no	—
221	C₂×C₈	—	10	4	no	—
222	C₂×C₈	C₂²×C₃×D₅	1	4	yes	—
223	C₄⋊C₄	—	5	8	no	—
224	C₂²⋊C₄	—	5	8	no	—
225	C₄²	—	5	2	no	—
226	C₄²	—	5	2	no	—
227	C₄²	C₃×C₄×D₅	1	2	yes	—
228	C₄²	C₂²×C₃×D₅	1	2	yes	—
229	C₃×C₅	C₂×#433	1	1	yes	characteristic, complemented, Hall
230	C₂²×C₃	—	10	2	no	—
231	C₂²×C₃	—	10	4	no	—
232	C₂²×C₃	—	20	4	no	—
233	C₂²×C₃	#4018	1	2	yes	—
234	C₂²×C₃	#4053	1	4	yes	—
235	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
236	C₃×C₄	—	2	4	no	—
237	C₃×C₄	—	10	2	no	—
238	C₃×C₄	—	10	4	no	—
239	C₃×C₄	#4018	1	2	yes	—
240	C₃×C₄	C₂×#709	1	2	yes	—
241	C₂×C₅	C₃×#433	1	4	yes	complemented
242	C₂×C₅	C₂×C₃×#117	1	1	yes	characteristic
243	C₂×C₅	C₂×C₃×#131	1	2	yes	—
244	D₅	—	4	2	no	—
245	C₂³	—	5	2	no	—
246	C₂³	—	10	2	no	—
247	C₂³	—	10	4	no	—
248	C₂³	C₂×C₃×C₄×D₅	1	1	yes	characteristic
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	5	4	no	—
251	C₂×C₄	—	5	2	no	—
252	C₂×C₄	—	10	2	no	—
253	C₂×C₄	—	10	4	no	—
254	C₂×C₄	—	10	4	no	—
255	C₂×C₄	—	10	4	no	—
256	C₂×C₄	C₂×C₃×C₄×D₅	1	4	yes	—
257	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
258	C₂×C₄	C₃×#709	1	4	yes	—
259	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	—	10	4	no	—
262	C₂×C₃	—	20	2	no	—
263	C₂×C₃	#64998	1	4	yes	direct product factor
264	C₂×C₃	C₂×#4018	1	1	yes	characteristic
265	C₂×C₃	C₂×#4053	1	2	yes	—
266	C₅	C₂×C₃×#433	1	1	yes	characteristic, complemented, Sylow
267	C₂²	—	10	2	no	—
268	C₂²	—	10	4	no	—
269	C₂²	—	20	4	no	—
270	C₂²	C₃×#4018	1	2	yes	—
271	C₂²	C₃×#4053	1	4	yes	—
272	C₂²	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	10	2	no	—
275	C₄	—	10	4	no	—
276	C₄	C₃×#4018	1	2	yes	—
277	C₄	C₂×C₃×#709	1	2	yes	—
278	C₃	C₂×#64998	1	1	yes	characteristic, direct product factor, Sylow
279	C₂	—	20	2	no	—
280	C₂	C₃×#64998	1	4	yes	direct product factor
281	C₂	C₂×C₃×#4018	1	1	yes	characteristic
282	C₂	C₂×C₃×#4053	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₅
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×#97	C₂
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₃×D₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂⁴×C₃
hypercenter	C₂×C₃×#97	D₅
hypocenter	C₅	C₂×C₃×#433
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#97	C₂	semidirect
4. C₂×C₃×#3946	C₂	semidirect
5. C₂×C₃×C₄×#168	C₂	semidirect
6. C₃×#64998	C₂	direct (N × H)
7. C₂×#64998	C₃	direct (N × H)
17. C₃×C₅×#97	C₂²	semidirect
19. C₃×#3946	C₂²	semidirect
20. C₃×C₄×#168	C₂²	semidirect
23. C₂×C₅×#97	C₂×C₃	semidirect
25. C₂×#3946	C₂×C₃	semidirect
26. C₂×C₄×#168	C₂×C₃	semidirect
27. #64998	C₂×C₃	direct (N × H)
50. C₂×C₃×#97	D₅	semidirect
61. C₅×#97	C₂²×C₃	semidirect
63. #3946	C₂²×C₃	semidirect
64. C₄×#168	C₂²×C₃	semidirect
94. C₃×#97	C₂×D₅	semidirect
118. C₂×#97	C₃×D₅	semidirect
177. #97	C₂×C₃×D₅	semidirect
179. C₂×C₃×C₅	#433	semidirect
229. C₃×C₅	C₂×#433	semidirect
241. C₂×C₅	C₃×#433	semidirect
263. C₂×C₃	#64998	direct (N × H)
266. C₅	C₂×C₃×#433	semidirect
278. C₃	C₂×#64998	direct (N × H)
280. C₂	C₃×#64998	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×#4018	2
C₂×C₃×C₅×#97	2
C₂×C₃×#3952	2
C₂×C₃×#3946	2
C₂×C₃×C₄×#168	2
C₃×#64998	2
C₂×#64998	3
C₂×C₃×#433	5

Subgroups — C2×C3×#64998 #14413833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#64998 #14413833