Subgroups — C₂×C₃×#11716761 #72277285

Order 4800.

864 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×#87862	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#87767	C₂	1	1	yes	maximal, characteristic
3	C₃×#11716761	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₄×#1043	C₂	1	1	yes	maximal, characteristic
5	C₂×#11716761	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×#1043	C₄	1	2	yes	—
7	C₂×C₃×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×#6034	C₄	1	2	yes	complemented
9	C₃×#87862	C₄	1	2	yes	complemented
10	C₃×#87862	C₂²	1	2	yes	—
11	C₃×C₄×#1043	C₄	1	2	yes	—
12	C₃×C₄×#1043	C₂²	1	2	yes	—
13	C₂×C₃×C₅×#168	C₄	1	2	yes	complemented
14	C₃×#87767	C₄	1	2	yes	complemented
15	C₃×#87767	C₂²	1	2	yes	—
16	C₂×C₃×C₈×AGL(1,𝔽₅)	—	5	2	no	maximal
17	C₂×#87862	C₂×C₃	1	1	yes	characteristic
18	C₂×C₄×#1043	C₂×C₃	1	1	yes	characteristic
19	C₂×#87767	C₂×C₃	1	1	yes	characteristic
20	#11716761	C₂×C₃	1	4	yes	direct product factor
21	C₂²×C₃×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×#1043	C₈	1	8	yes	complemented
23	C₂×C₃×#1043	C₂×C₄	1	4	yes	—
24	C₂×C₃×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×#1039	C₂×C₄	1	1	yes	characteristic
26	C₃×C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄×C₅⋊D₅	C₂³	1	1	yes	characteristic
30	C₃×#6034	C₂×C₄	1	4	yes	complemented
31	C₃×C₅×#168	C₂×C₄	1	4	yes	complemented
32	C₂×C₃×C₄×AGL(1,𝔽₅)	—	5	2	no	—
33	C₂×C₃×C₄×AGL(1,𝔽₅)	—	10	2	no	—
34	C₂×C₃×D₅⋊C₈	—	5	2	no	—
35	C₂×C₃×C₈×D₅	—	5	2	no	—
36	C₃×C₈×AGL(1,𝔽₅)	—	5	8	no	—
37	C₂²×#1043	C₃×C₄	1	2	yes	—
38	C₂×C₄×C₅⋊D₅	C₂²×C₃	1	1	yes	characteristic
39	C₂×#6034	C₃×C₄	1	2	yes	complemented
40	#87862	C₃×C₄	1	2	yes	complemented
41	#87862	C₂²×C₃	1	2	yes	—
42	C₄×#1043	C₃×C₄	1	2	yes	—
43	C₄×#1043	C₂²×C₃	1	2	yes	—
44	C₂×C₅×#168	C₃×C₄	1	2	yes	complemented
45	#87767	C₃×C₄	1	2	yes	complemented
46	#87767	C₂²×C₃	1	2	yes	—
47	C₂×C₈×AGL(1,𝔽₅)	—	5	2	no	—
48	C₂²×C₃×C₅²	C₄²	1	1	yes	characteristic
49	C₂×C₃×C₅⋊D₅	C₄²	1	1	yes	characteristic
50	C₂×C₃×C₅⋊D₅	C₂×C₈	1	4	yes	—
51	C₂×C₃×C₅⋊D₅	C₂²×C₄	1	1	yes	characteristic
52	C₃×#1043	C₂×C₈	1	8	yes	complemented
53	C₃×C₄×C₅²	C₄²	1	1	yes	characteristic
54	C₃×C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
55	C₃×#1039	C₄²	1	1	yes	characteristic
56	C₃×#1039	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₃×AGL(1,𝔽₅)	—	5	4	no	—
58	C₂²×C₃×AGL(1,𝔽₅)	—	10	4	no	—
59	C₂×C₃×C₄×D₅	—	5	2	no	—
60	C₂×C₃×C₄×D₅	—	10	2	no	—
61	C₂×C₃×#170	—	5	4	no	—
62	C₃×C₄×AGL(1,𝔽₅)	—	5	4	no	—
63	C₃×C₄×AGL(1,𝔽₅)	—	5	4	no	—
64	C₃×C₄×AGL(1,𝔽₅)	—	10	4	no	—
65	C₃×C₄×AGL(1,𝔽₅)	—	10	4	no	—
66	C₃×D₅⋊C₈	—	5	4	no	—
67	C₃×D₅⋊C₈	—	5	4	no	—
68	C₂×C₃×C₅×C₈	—	5	2	no	—
69	C₂×C₃×#168	AGL(1,𝔽₅)	1	2	yes	complemented
70	C₃×C₈×D₅	—	5	4	no	—
71	C₃×C₈×D₅	—	5	4	no	—
72	C₂²×C₅⋊D₅	C₂×C₃×C₄	1	1	yes	characteristic
73	C₂×#1043	C₃×C₈	1	8	yes	complemented
74	C₂×#1043	C₂×C₃×C₄	1	4	yes	—
75	C₂×C₄×C₅²	C₂×C₃×C₄	1	1	yes	characteristic
76	C₂×#1039	C₂×C₃×C₄	1	1	yes	characteristic
77	C₄×C₅⋊D₅	C₂×C₃×C₄	1	1	yes	characteristic
78	C₄×C₅⋊D₅	C₂×C₃×C₄	1	1	yes	characteristic
79	C₄×C₅⋊D₅	C₂×C₃×C₄	1	1	yes	characteristic
80	C₄×C₅⋊D₅	C₂³×C₃	1	1	yes	characteristic
81	#6034	C₂×C₃×C₄	1	4	yes	complemented
82	C₅×#168	C₂×C₃×C₄	1	4	yes	complemented
83	C₂×C₃×C₄×C₈	—	25	1	no	Hall
84	C₂×C₄×AGL(1,𝔽₅)	—	5	2	no	—
85	C₂×C₄×AGL(1,𝔽₅)	—	10	2	no	—
86	C₂×D₅⋊C₈	—	5	2	no	—
87	C₂×C₈×D₅	—	5	2	no	—
88	C₈×AGL(1,𝔽₅)	—	5	8	no	—
89	C₂×C₃×C₅²	C₄×C₈	1	2	yes	complemented
90	C₂×C₃×C₅²	C₂×C₄²	1	1	yes	characteristic
91	C₃×C₅⋊D₅	C₄×C₈	1	2	yes	complemented
92	C₃×C₅⋊D₅	C₂²×C₈	1	2	yes	—
93	C₂²×C₃×D₅	—	5	2	no	—
94	C₂²×C₃×D₅	—	10	2	no	—
95	C₂×C₃×AGL(1,𝔽₅)	—	5	16	no	—
96	C₂×C₃×AGL(1,𝔽₅)	—	5	8	no	—
97	C₂×C₃×AGL(1,𝔽₅)	—	10	16	no	—
98	C₂×C₃×AGL(1,𝔽₅)	—	10	8	no	—
99	C₂×C₃×C₄×C₅	—	2	2	no	—
100	C₂×C₃×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
101	C₂×C₃×Dic₅	—	5	2	no	—
102	C₂×C₃×Dic₅	—	10	2	no	—
103	C₃×C₄×D₅	—	5	2	no	—
104	C₃×C₄×D₅	—	5	2	no	—
105	C₃×C₄×D₅	—	5	2	no	—
106	C₃×C₄×D₅	—	5	2	no	—
107	C₃×C₄×D₅	—	10	2	no	—
108	C₃×C₄×D₅	—	10	2	no	—
109	C₃×C₄×D₅	—	10	2	no	—
110	C₃×C₄×D₅	—	10	2	no	—
111	C₃×#170	—	5	8	no	—
112	C₃×C₅×C₈	—	5	4	no	—
113	C₃×#168	C₂×AGL(1,𝔽₅)	1	4	yes	complemented
114	C₂²×C₅²	C₃×C₄²	1	1	yes	characteristic
115	C₂×C₅⋊D₅	C₃×C₄²	1	1	yes	characteristic
116	C₂×C₅⋊D₅	C₂×C₃×C₈	1	4	yes	—
117	C₂×C₅⋊D₅	C₂²×C₃×C₄	1	1	yes	characteristic
118	#1043	C₂×C₃×C₈	1	8	yes	complemented
119	C₄×C₅²	C₃×C₄²	1	1	yes	characteristic
120	C₄×C₅²	C₂²×C₃×C₄	1	1	yes	characteristic
121	#1039	C₃×C₄²	1	1	yes	characteristic
122	#1039	C₂²×C₃×C₄	1	1	yes	characteristic
123	C₂²×C₃×C₈	—	25	1	no	—
124	C₂²×C₃×C₈	—	25	1	no	—
125	C₂×C₃×C₄²	—	25	1	no	—
126	C₃×C₄×C₈	—	25	4	no	—
127	C₂²×AGL(1,𝔽₅)	—	5	4	no	—
128	C₂²×AGL(1,𝔽₅)	—	10	4	no	—
129	C₂×C₄×D₅	—	5	2	no	—
130	C₂×C₄×D₅	—	10	2	no	—
131	C₂×#170	—	5	4	no	—
132	C₄×AGL(1,𝔽₅)	—	5	4	no	—
133	C₄×AGL(1,𝔽₅)	—	5	4	no	—
134	C₄×AGL(1,𝔽₅)	—	10	4	no	—
135	C₄×AGL(1,𝔽₅)	—	10	4	no	—
136	D₅⋊C₈	—	5	4	no	—
137	D₅⋊C₈	—	5	4	no	—
138	C₂×C₅×C₈	—	5	2	no	—
139	C₂×#168	C₃×AGL(1,𝔽₅)	1	2	yes	complemented
140	C₈×D₅	—	5	4	no	—
141	C₈×D₅	—	5	4	no	—
142	C₃×C₅²	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Hall
143	C₂×C₄×C₈	—	25	1	no	Sylow
144	C₂²×C₃×C₅	—	2	2	no	—
145	C₂²×C₃×C₅	C₄×AGL(1,𝔽₅)	1	2	yes	—
146	C₂×C₃×D₅	—	5	2	no	—
147	C₂×C₃×D₅	—	5	2	no	—
148	C₂×C₃×D₅	—	5	8	no	—
149	C₂×C₃×D₅	—	10	2	no	—
150	C₂×C₃×D₅	—	10	2	no	—
151	C₂×C₃×D₅	—	10	8	no	—
152	C₃×AGL(1,𝔽₅)	—	5	16	no	—
153	C₃×AGL(1,𝔽₅)	—	10	16	no	—
154	C₃×C₄×C₅	—	2	2	no	—
155	C₃×C₄×C₅	—	2	2	no	—
156	C₃×C₄×C₅	C₄×AGL(1,𝔽₅)	1	2	yes	—
157	C₃×C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
158	C₃×Dic₅	—	5	2	no	—
159	C₃×Dic₅	—	5	2	no	—
160	C₃×Dic₅	—	10	2	no	—
161	C₃×Dic₅	—	10	2	no	—
162	C₂×C₅²	C₃×C₄×C₈	1	2	yes	complemented
163	C₂×C₅²	C₂×C₃×C₄²	1	1	yes	characteristic
164	C₅⋊D₅	C₃×C₄×C₈	1	2	yes	complemented
165	C₅⋊D₅	C₂²×C₃×C₈	1	2	yes	—
166	C₂²×C₃×C₄	—	25	2	no	—
167	C₂²×C₃×C₄	—	25	1	no	—
168	C₂×C₃×C₈	—	5	2	no	—
169	C₂×C₃×C₈	—	25	2	no	—
170	C₂×C₃×C₈	—	25	2	no	—
171	C₂×C₃×C₈	—	25	2	no	—
172	C₂×C₃×C₈	—	25	2	no	—
173	C₂×C₃×C₈	—	25	2	no	—
174	C₃×C₄²	—	25	2	no	—
175	C₃×C₄²	—	25	2	no	—
176	C₂²×D₅	—	5	2	no	—
177	C₂²×D₅	—	10	2	no	—
178	C₂×AGL(1,𝔽₅)	—	5	16	no	—
179	C₂×AGL(1,𝔽₅)	—	5	8	no	—
180	C₂×AGL(1,𝔽₅)	—	10	16	no	—
181	C₂×AGL(1,𝔽₅)	—	10	8	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
184	C₂×Dic₅	—	5	2	no	—
185	C₂×Dic₅	—	10	2	no	—
186	C₄×D₅	—	5	2	no	—
187	C₄×D₅	—	5	2	no	—
188	C₄×D₅	—	5	2	no	—
189	C₄×D₅	—	5	2	no	—
190	C₄×D₅	—	10	2	no	—
191	C₄×D₅	—	10	2	no	—
192	C₄×D₅	—	10	2	no	—
193	C₄×D₅	—	10	2	no	—
194	#170	—	5	8	no	—
195	C₅×C₈	—	5	4	no	—
196	#168	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	complemented
197	C₂²×C₈	—	25	1	no	—
198	C₂²×C₈	—	25	1	no	—
199	C₂×C₄²	—	25	1	no	—
200	C₄×C₈	—	25	4	no	—
201	C₂×C₃×C₅	—	2	4	no	—
202	C₂×C₃×C₅	—	2	2	no	—
203	C₂×C₃×C₅	C₈×AGL(1,𝔽₅)	1	4	yes	complemented
204	C₂×C₃×C₅	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
205	C₃×D₅	—	5	4	no	—
206	C₃×D₅	—	5	4	no	—
207	C₃×D₅	—	10	4	no	—
208	C₃×D₅	—	10	4	no	—
209	C₅²	C₂×C₃×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
210	C₂³×C₃	—	25	1	no	—
211	C₂×C₃×C₄	—	25	4	no	—
212	C₂×C₃×C₄	—	25	8	no	—
213	C₂×C₃×C₄	—	25	1	no	—
214	C₂×C₃×C₄	—	25	1	no	—
215	C₂×C₃×C₄	—	25	1	no	—
216	C₂×C₃×C₄	—	25	1	no	—
217	C₂×C₃×C₄	—	25	1	no	—
218	C₂×C₃×C₄	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
219	C₃×C₈	—	5	4	no	—
220	C₃×C₈	—	25	4	no	—
221	C₂²×C₅	—	2	2	no	—
222	C₂²×C₅	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
223	C₂×D₅	—	5	8	no	—
224	C₂×D₅	—	5	2	no	—
225	C₂×D₅	—	5	2	no	—
226	C₂×D₅	—	10	2	no	—
227	C₂×D₅	—	10	8	no	—
228	C₂×D₅	—	10	2	no	—
229	AGL(1,𝔽₅)	—	5	16	no	—
230	AGL(1,𝔽₅)	—	10	16	no	—
231	C₄×C₅	—	2	2	no	—
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
234	C₄×C₅	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
235	Dic₅	—	5	2	no	—
236	Dic₅	—	5	2	no	—
237	Dic₅	—	10	2	no	—
238	Dic₅	—	10	2	no	—
239	C₂²×C₄	—	25	2	no	—
240	C₂²×C₄	—	25	1	no	—
241	C₂×C₈	—	5	2	no	—
242	C₂×C₈	—	25	2	no	—
243	C₂×C₈	—	25	2	no	—
244	C₂×C₈	—	25	2	no	—
245	C₂×C₈	—	25	2	no	—
246	C₂×C₈	—	25	2	no	—
247	C₄²	—	25	2	no	—
248	C₄²	—	25	2	no	—
249	C₃×C₅	—	2	2	no	—
250	C₃×C₅	C₂×C₈×AGL(1,𝔽₅)	1	2	yes	complemented
251	C₂²×C₃	—	25	4	no	—
252	C₂²×C₃	—	25	1	no	—
253	C₂²×C₃	—	25	1	no	—
254	C₂²×C₃	#87830	1	1	yes	characteristic
255	C₃×C₄	—	25	8	no	—
256	C₃×C₄	—	25	1	no	—
257	C₃×C₄	—	25	1	no	—
258	C₃×C₄	#87830	1	1	yes	characteristic
259	C₃×C₄	C₂×D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
260	C₂×C₅	—	2	4	no	—
261	C₂×C₅	—	2	2	no	—
262	C₂×C₅	C₃×C₈×AGL(1,𝔽₅)	1	4	yes	complemented
263	C₂×C₅	C₂×C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
264	D₅	—	5	4	no	—
265	D₅	—	5	4	no	—
266	D₅	—	10	4	no	—
267	D₅	—	10	4	no	—
268	C₂³	—	25	1	no	—
269	C₂×C₄	—	25	4	no	—
270	C₂×C₄	—	25	8	no	—
271	C₂×C₄	—	25	1	no	—
272	C₂×C₄	—	25	1	no	—
273	C₂×C₄	—	25	1	no	—
274	C₂×C₄	—	25	1	no	—
275	C₂×C₄	—	25	1	no	—
276	C₂×C₄	C₃×D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
277	C₈	—	5	4	no	—
278	C₈	—	25	4	no	—
279	C₂×C₃	—	25	2	no	—
280	C₂×C₃	—	25	2	no	—
281	C₂×C₃	#11716761	1	2	yes	direct product factor
282	C₂×C₃	C₂×#87830	1	1	yes	characteristic
283	C₅	—	2	2	no	—
284	C₅	C₂×C₃×C₈×AGL(1,𝔽₅)	1	2	yes	complemented
285	C₂²	—	25	4	no	—
286	C₂²	—	25	1	no	—
287	C₂²	—	25	1	no	—
288	C₂²	C₃×#87830	1	1	yes	characteristic
289	C₄	—	25	8	no	—
290	C₄	—	25	1	no	—
291	C₄	—	25	1	no	—
292	C₄	C₃×#87830	1	1	yes	characteristic
293	C₄	C₂×C₃×D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
294	C₃	C₂×#11716761	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	—	25	2	no	—
296	C₂	—	25	2	no	—
297	C₂	C₃×#11716761	1	2	yes	direct product factor
298	C₂	C₂×C₃×#87830	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	D₅⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅²	C₂×C₃×C₄×C₈
Frattini	C₄	C₂×C₃×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₃×C₄×C₅²	C₂×C₄
socle	C₂²×C₃×C₅²	C₄²
prime radical	C₂²×C₃×C₅⋊D₅	C₂×C₄
cosocle	C₄×C₅⋊D₅	C₂³×C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
3. C₃×#11716761	C₂	direct (N × H)
5. C₂×#11716761	C₃	direct (N × H)
8. C₂×C₃×#6034	C₄	semidirect
9. C₃×#87862	C₄	semidirect
13. C₂×C₃×C₅×#168	C₄	semidirect
14. C₃×#87767	C₄	semidirect
20. #11716761	C₂×C₃	direct (N × H)
22. C₂×C₃×#1043	C₈	semidirect
30. C₃×#6034	C₂×C₄	semidirect
31. C₃×C₅×#168	C₂×C₄	semidirect
39. C₂×#6034	C₃×C₄	semidirect
40. #87862	C₃×C₄	semidirect
44. C₂×C₅×#168	C₃×C₄	semidirect
45. #87767	C₃×C₄	semidirect
52. C₃×#1043	C₂×C₈	semidirect
69. C₂×C₃×#168	AGL(1,𝔽₅)	semidirect
73. C₂×#1043	C₃×C₈	semidirect
81. #6034	C₂×C₃×C₄	semidirect
82. C₅×#168	C₂×C₃×C₄	semidirect
89. C₂×C₃×C₅²	C₄×C₈	semidirect
91. C₃×C₅⋊D₅	C₄×C₈	semidirect
113. C₃×#168	C₂×AGL(1,𝔽₅)	semidirect
118. #1043	C₂×C₃×C₈	semidirect
139. C₂×#168	C₃×AGL(1,𝔽₅)	semidirect
142. C₃×C₅²	C₂×C₄×C₈	semidirect
162. C₂×C₅²	C₃×C₄×C₈	semidirect
164. C₅⋊D₅	C₃×C₄×C₈	semidirect
196. #168	C₂×C₃×AGL(1,𝔽₅)	semidirect
203. C₂×C₃×C₅	C₈×AGL(1,𝔽₅)	semidirect
209. C₅²	C₂×C₃×C₄×C₈	semidirect
250. C₃×C₅	C₂×C₈×AGL(1,𝔽₅)	semidirect
262. C₂×C₅	C₃×C₈×AGL(1,𝔽₅)	semidirect
281. C₂×C₃	#11716761	direct (N × H)
284. C₅	C₂×C₃×C₈×AGL(1,𝔽₅)	semidirect
294. C₃	C₂×#11716761	direct (N × H)
297. C₂	C₃×#11716761	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×#87862	2
C₂×C₃×#87767	2
C₃×#11716761	2
C₂×C₃×C₄×#1043	2
C₂×#11716761	3
C₂×C₃×C₈×AGL(1,𝔽₅)	5

Subgroups — C2×C3×#11716761 #72277285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#11716761 #72277285