Subgroups — C₃×AGL(1,𝔽₅)×M₃₂ #14312587

Order 1920.

392 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	C₃×D₅⋊M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₃×D₅×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₃×#64844	C₂	1	2	yes	maximal, complemented
5	C₃×C₁₆×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
6	AGL(1,𝔽₅)×M₃₂	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₄×AGL(1,𝔽₅)	C₄	1	1	yes	characteristic
8	C₂×C₃×D₅⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
10	C₃×#3998	C₄	1	2	yes	complemented
11	C₃×C₈×AGL(1,𝔽₅)	C₄	1	1	yes	characteristic
12	C₃×C₈×AGL(1,𝔽₅)	C₄	1	1	yes	characteristic
13	C₃×C₈×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
14	C₃×C₈×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
15	C₃×#3990	C₄	1	2	yes	complemented
16	C₃×D₅⋊C₁₆	C₂²	1	2	yes	—
17	C₃×C₅×M₃₂	C₄	1	1	yes	characteristic, complemented
18	C₃×#3944	C₄	1	1	yes	characteristic, complemented
19	C₃×#3930	C₄	1	2	yes	complemented
20	C₃×D₅×C₁₆	C₂²	1	2	yes	—
21	C₃×C₄×M₃₂	—	5	1	no	maximal, Hall
22	C₂×C₈×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic
23	D₅⋊M₃₂	C₂×C₃	1	1	yes	characteristic
24	D₅×M₃₂	C₂×C₃	1	1	yes	characteristic
25	#64844	C₂×C₃	1	2	yes	complemented
26	C₁₆×AGL(1,𝔽₅)	C₂×C₃	1	2	yes	complemented
27	C₂²×C₃×AGL(1,𝔽₅)	C₈	1	2	yes	—
28	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×#170	C₈	1	2	yes	—
30	C₃×C₄×AGL(1,𝔽₅)	C₈	1	2	yes	—
31	C₃×C₄×AGL(1,𝔽₅)	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄×AGL(1,𝔽₅)	C₂×C₄	1	1	yes	characteristic
33	C₃×D₅⋊C₈	C₈	1	2	yes	—
34	C₃×D₅⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₃×D₅⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
38	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
42	C₃×#674	C₂×C₄	1	4	yes	complemented
43	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
44	C₃×#672	C₂×C₄	1	2	yes	complemented
45	C₂×C₃×M₃₂	—	5	1	no	—
46	C₂×C₃×M₃₂	—	5	1	no	—
47	C₂×C₃×C₄×C₈	—	5	1	no	—
48	C₃×#346	—	5	2	no	—
49	C₃×C₄×C₁₆	—	5	2	no	—
50	C₂×C₄×AGL(1,𝔽₅)	C₃×C₄	1	1	yes	characteristic
51	C₂×D₅⋊C₈	C₃×C₄	1	1	yes	characteristic
52	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
53	#3998	C₃×C₄	1	2	yes	complemented
54	C₈×AGL(1,𝔽₅)	C₃×C₄	1	1	yes	characteristic
55	C₈×AGL(1,𝔽₅)	C₃×C₄	1	1	yes	characteristic
56	C₈×AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
57	C₈×AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
58	#3990	C₃×C₄	1	2	yes	complemented
59	D₅⋊C₁₆	C₂²×C₃	1	2	yes	—
60	C₅×M₃₂	C₃×C₄	1	1	yes	characteristic, complemented
61	#3944	C₃×C₄	1	1	yes	characteristic, complemented
62	#3930	C₃×C₄	1	2	yes	complemented
63	D₅×C₁₆	C₂²×C₃	1	2	yes	—
64	C₄×M₃₂	—	5	1	no	Sylow
65	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
66	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
67	C₂×C₃×AGL(1,𝔽₅)	C₂×C₈	1	2	yes	—
68	C₂×C₃×AGL(1,𝔽₅)	C₂×C₈	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
70	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
71	C₃×C₄×D₅	C₄²	1	1	yes	characteristic
72	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
73	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
74	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
75	C₃×#170	C₂×C₈	1	2	yes	—
76	C₃×#170	C₂×C₈	1	2	yes	—
77	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
78	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₃×#168	C₄²	1	1	yes	characteristic
80	C₃×#168	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₃×C₈	—	5	1	no	—
82	C₂²×C₃×C₈	—	5	1	no	—
83	C₂×C₃×C₄²	—	5	1	no	—
84	C₃×M₃₂	—	5	2	no	—
85	C₃×M₃₂	—	5	2	no	—
86	C₃×M₃₂	—	5	1	no	—
87	C₃×M₃₂	—	5	2	no	—
88	C₃×M₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
89	C₂×C₃×C₁₆	—	5	2	no	—
90	C₂×C₃×C₁₆	—	5	2	no	—
91	C₃×C₄×C₈	—	5	1	no	—
92	C₃×C₄×C₈	—	5	1	no	—
93	C₃×C₄×C₈	—	5	1	no	—
94	C₃×C₄×C₈	—	5	1	no	—
95	C₂²×AGL(1,𝔽₅)	C₃×C₈	1	2	yes	—
96	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×#170	C₃×C₈	1	2	yes	—
98	C₄×AGL(1,𝔽₅)	C₃×C₈	1	2	yes	—
99	C₄×AGL(1,𝔽₅)	C₂×C₃×C₄	1	1	yes	characteristic
100	C₄×AGL(1,𝔽₅)	C₂×C₃×C₄	1	1	yes	characteristic
101	D₅⋊C₈	C₃×C₈	1	2	yes	—
102	D₅⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
103	D₅⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
106	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
107	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
108	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
109	C₈×D₅	C₂³×C₃	1	1	yes	characteristic
110	#674	C₂×C₃×C₄	1	4	yes	complemented
111	C₅×C₁₆	C₂×C₃×C₄	1	2	yes	complemented
112	#672	C₂×C₃×C₄	1	2	yes	complemented
113	C₂×M₃₂	—	5	1	no	—
114	C₂×M₃₂	—	5	1	no	—
115	C₂×C₄×C₈	—	5	1	no	—
116	#346	—	5	2	no	—
117	C₄×C₁₆	—	5	2	no	—
118	C₂²×C₃×C₅	C₄×C₈	1	1	yes	characteristic
119	C₂×C₃×D₅	—	2	2	no	—
120	C₂×C₃×D₅	C₄×C₈	1	1	yes	characteristic
121	C₂×C₃×D₅	C₂²×C₈	1	1	yes	characteristic
122	C₃×AGL(1,𝔽₅)	—	2	2	no	—
123	C₃×AGL(1,𝔽₅)	M₃₂	1	4	yes	direct product factor
124	C₃×C₄×C₅	C₄×C₈	1	1	yes	characteristic
125	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
126	C₃×Dic₅	C₄×C₈	1	1	yes	characteristic
127	C₃×Dic₅	C₂²×C₈	1	1	yes	characteristic
128	C₂²×C₃×C₄	—	5	1	no	—
129	C₂²×C₃×C₄	—	5	2	no	—
130	C₂×C₃×C₈	—	5	1	no	—
131	C₂×C₃×C₈	—	5	1	no	—
132	C₂×C₃×C₈	—	5	2	no	—
133	C₂×C₃×C₈	—	5	2	no	—
134	C₂×C₃×C₈	—	5	1	no	—
135	C₂×C₃×C₈	—	5	1	no	—
136	C₂×C₃×C₈	—	5	1	no	—
137	C₂×C₃×C₈	—	5	1	no	—
138	C₂×C₃×C₈	—	5	1	no	—
139	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₃×C₄²	—	5	1	no	—
141	C₃×C₄²	—	5	1	no	—
142	C₃×C₄²	—	5	2	no	—
143	C₃×C₁₆	—	5	4	no	—
144	C₃×C₁₆	—	5	2	no	—
145	C₃×C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
146	C₂²×D₅	C₂×C₃×C₈	1	1	yes	characteristic
147	C₂×AGL(1,𝔽₅)	—	2	4	no	—
148	C₂×AGL(1,𝔽₅)	C₂×C₃×C₈	1	2	yes	—
149	C₂×AGL(1,𝔽₅)	C₂×C₃×C₈	1	2	yes	—
150	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
151	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
152	C₄×D₅	C₃×C₄²	1	1	yes	characteristic
153	C₄×D₅	C₂×C₃×C₈	1	1	yes	characteristic
154	C₄×D₅	C₂×C₃×C₈	1	1	yes	characteristic
155	C₄×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
156	#170	C₂×C₃×C₈	1	2	yes	—
157	#170	C₂×C₃×C₈	1	2	yes	—
158	C₅×C₈	C₃×C₄²	1	1	yes	characteristic
159	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
160	#168	C₃×C₄²	1	1	yes	characteristic
161	#168	C₂²×C₃×C₄	1	1	yes	characteristic
162	C₂²×C₈	—	5	1	no	—
163	C₂²×C₈	—	5	1	no	—
164	C₂×C₄²	—	5	1	no	—
165	M₃₂	—	5	2	no	—
166	M₃₂	—	5	2	no	—
167	M₃₂	—	5	1	no	—
168	M₃₂	—	5	2	no	—
169	M₃₂	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
170	C₂×C₁₆	—	5	2	no	—
171	C₂×C₁₆	—	5	2	no	—
172	C₄×C₈	—	5	1	no	—
173	C₄×C₈	—	5	1	no	—
174	C₄×C₈	—	5	1	no	—
175	C₄×C₈	—	5	1	no	—
176	C₂×C₃×C₅	—	2	1	no	—
177	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
178	C₃×D₅	—	2	1	no	—
179	C₃×D₅	C₂×M₃₂	1	2	yes	—
180	C₂³×C₃	—	5	1	no	—
181	C₂×C₃×C₄	—	5	2	no	—
182	C₂×C₃×C₄	—	5	1	no	—
183	C₂×C₃×C₄	—	5	1	no	—
184	C₂×C₃×C₄	—	5	1	no	—
185	C₂×C₃×C₄	—	5	1	no	—
186	C₂×C₃×C₄	—	5	1	no	—
187	C₂×C₃×C₄	—	5	2	no	—
188	C₂×C₃×C₄	—	10	4	no	—
189	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
190	C₃×C₈	—	5	2	no	—
191	C₃×C₈	—	5	2	no	—
192	C₃×C₈	—	5	1	no	—
193	C₃×C₈	—	5	1	no	—
194	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
195	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
196	C₂²×C₅	C₃×C₄×C₈	1	1	yes	characteristic
197	C₂×D₅	—	2	2	no	—
198	C₂×D₅	C₃×C₄×C₈	1	1	yes	characteristic
199	C₂×D₅	C₂²×C₃×C₈	1	1	yes	characteristic
200	AGL(1,𝔽₅)	—	2	2	no	—
201	AGL(1,𝔽₅)	C₃×M₃₂	1	4	yes	direct product factor
202	C₄×C₅	C₃×C₄×C₈	1	1	yes	characteristic
203	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
204	Dic₅	C₃×C₄×C₈	1	1	yes	characteristic
205	Dic₅	C₂²×C₃×C₈	1	1	yes	characteristic
206	C₂²×C₄	—	5	1	no	—
207	C₂²×C₄	—	5	2	no	—
208	C₂×C₈	—	5	1	no	—
209	C₂×C₈	—	5	1	no	—
210	C₂×C₈	—	5	2	no	—
211	C₂×C₈	—	5	2	no	—
212	C₂×C₈	—	5	1	no	—
213	C₂×C₈	—	5	1	no	—
214	C₂×C₈	—	5	1	no	—
215	C₂×C₈	—	5	1	no	—
216	C₂×C₈	—	5	1	no	—
217	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
218	C₄²	—	5	1	no	—
219	C₄²	—	5	1	no	—
220	C₄²	—	5	2	no	—
221	C₁₆	—	5	4	no	—
222	C₁₆	—	5	2	no	—
223	C₁₆	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
224	C₃×C₅	C₄×M₃₂	1	1	yes	characteristic, complemented, Hall
225	C₂²×C₃	—	5	1	no	—
226	C₂²×C₃	—	5	1	no	—
227	C₂²×C₃	—	10	2	no	—
228	C₂²×C₃	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
229	C₃×C₄	—	5	4	no	—
230	C₃×C₄	—	5	1	no	—
231	C₃×C₄	—	5	1	no	—
232	C₃×C₄	—	10	2	no	—
233	C₃×C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
234	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
235	C₂×C₅	—	2	1	no	—
236	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
237	D₅	—	2	1	no	—
238	D₅	C₂×C₃×M₃₂	1	2	yes	—
239	C₂³	—	5	1	no	—
240	C₂×C₄	—	5	2	no	—
241	C₂×C₄	—	5	1	no	—
242	C₂×C₄	—	5	1	no	—
243	C₂×C₄	—	5	1	no	—
244	C₂×C₄	—	5	1	no	—
245	C₂×C₄	—	5	1	no	—
246	C₂×C₄	—	5	2	no	—
247	C₂×C₄	—	10	4	no	—
248	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
249	C₈	—	5	2	no	—
250	C₈	—	5	2	no	—
251	C₈	—	5	1	no	—
252	C₈	—	5	1	no	—
253	C₈	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
254	C₈	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	5	2	no	—
257	C₂×C₃	—	10	1	no	—
258	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
259	C₅	C₃×C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	5	1	no	—
261	C₂²	—	5	1	no	—
262	C₂²	—	10	2	no	—
263	C₂²	C₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
264	C₄	—	5	4	no	—
265	C₄	—	5	1	no	—
266	C₄	—	5	1	no	—
267	C₄	—	10	2	no	—
268	C₄	C₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
269	C₄	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
270	C₃	AGL(1,𝔽₅)×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
271	C₂	—	2	1	no	—
272	C₂	—	5	2	no	—
273	C₂	—	10	1	no	—
274	C₂	C₂×C₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃×C₄×C₈
Frattini	C₈	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×M₃₂	C₄
socle	C₂×C₃×C₅	C₂×C₄×C₈
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₈×D₅	C₂³×C₃
hypercenter	C₃×M₃₂	AGL(1,𝔽₅)
hypocenter	C₅	C₃×C₄×M₃₂
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₈ ◁ C₃×M₃₂

Complements

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

N	H ≅ G/N	type
4. C₃×#64844	C₂	semidirect
5. C₃×C₁₆×AGL(1,𝔽₅)	C₂	semidirect
6. AGL(1,𝔽₅)×M₃₂	C₃	direct (N × H)
10. C₃×#3998	C₄	semidirect
15. C₃×#3990	C₄	semidirect
17. C₃×C₅×M₃₂	C₄	semidirect
18. C₃×#3944	C₄	semidirect
19. C₃×#3930	C₄	semidirect
25. #64844	C₂×C₃	semidirect
26. C₁₆×AGL(1,𝔽₅)	C₂×C₃	semidirect
42. C₃×#674	C₂×C₄	semidirect
43. C₃×C₅×C₁₆	C₂×C₄	semidirect
44. C₃×#672	C₂×C₄	semidirect
53. #3998	C₃×C₄	semidirect
58. #3990	C₃×C₄	semidirect
60. C₅×M₃₂	C₃×C₄	semidirect
61. #3944	C₃×C₄	semidirect
62. #3930	C₃×C₄	semidirect
88. C₃×M₃₂	AGL(1,𝔽₅)	direct (N × H)
110. #674	C₂×C₃×C₄	semidirect
111. C₅×C₁₆	C₂×C₃×C₄	semidirect
112. #672	C₂×C₃×C₄	semidirect
123. C₃×AGL(1,𝔽₅)	M₃₂	direct (N × H)
145. C₃×C₁₆	C₂×AGL(1,𝔽₅)	semidirect
169. M₃₂	C₃×AGL(1,𝔽₅)	direct (N × H)
201. AGL(1,𝔽₅)	C₃×M₃₂	direct (N × H)
223. C₁₆	C₂×C₃×AGL(1,𝔽₅)	semidirect
224. C₃×C₅	C₄×M₃₂	semidirect
259. C₅	C₃×C₄×M₃₂	semidirect
270. C₃	AGL(1,𝔽₅)×M₃₂	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×C₈×AGL(1,𝔽₅)	2
C₃×D₅⋊M₃₂	2
C₃×D₅×M₃₂	2
C₃×#64844	2
C₃×C₁₆×AGL(1,𝔽₅)	2
AGL(1,𝔽₅)×M₃₂	3
C₃×C₄×M₃₂	5

Subgroups — C3×AGL(1,𝔽5)×M32 #14312587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×AGL(1,𝔽₅)×M₃₂ #14312587