Subgroups — C₃×AGL(1,𝔽₅)×#127 #14322257

Order 1920.

1848 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×D₄×AGL(1,𝔽₅)	C₂	1	6	yes	maximal, complemented
2	C₃×D₅×#127	C₂	1	1	yes	maximal, characteristic
3	C₃×#65780	C₂	1	6	yes	maximal, complemented
4	C₃×#65688	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄²×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
6	AGL(1,𝔽₅)×#127	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₃×AGL(1,𝔽₅)	C₂²	1	4	yes	—
8	C₂×C₃×D₄×D₅	C₂²	1	6	yes	—
9	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	C₂²	1	12	yes	—
10	C₃×D₄×AGL(1,𝔽₅)	—	2	24	no	—
11	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	6	yes	—
12	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	3	yes	—
13	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	3	yes	—
14	C₃×C₅×#127	C₄	1	1	yes	characteristic, complemented
15	C₃×#4086	C₄	1	6	yes	complemented
16	C₃×#4020	C₄	1	1	yes	characteristic, complemented
17	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
18	C₃×C₄×#127	—	5	1	no	maximal, Hall
19	C₂×D₄×AGL(1,𝔽₅)	C₂×C₃	1	6	yes	complemented
20	D₅×#127	C₂×C₃	1	1	yes	characteristic
21	#65780	C₂×C₃	1	6	yes	complemented
22	#65688	C₂×C₃	1	1	yes	characteristic, complemented
23	C₄²×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂³×C₃×D₅	C₂³	1	4	yes	—
25	C₂²×C₃×AGL(1,𝔽₅)	—	2	24	no	—
26	C₂²×C₃×AGL(1,𝔽₅)	C₂³	1	4	yes	—
27	C₂²×C₃×AGL(1,𝔽₅)	C₂³	1	3	yes	—
28	C₂²×C₃×AGL(1,𝔽₅)	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₅×D₄	C₂×C₄	1	6	yes	complemented
30	C₂×C₃×C₂²⋊D₅	C₂×C₄	1	12	yes	complemented
31	C₃×D₄×D₅	—	2	12	no	—
32	C₃×D₄×D₅	—	2	12	no	—
33	C₂×C₃×D₂₀	C₂×C₄	1	6	yes	complemented
34	C₂×C₃×C₄×D₅	C₂³	1	3	yes	—
35	C₃×C₂²⋊AGL(1,𝔽₅)	—	2	24	no	—
36	C₃×C₄⋊AGL(1,𝔽₅)	—	2	12	no	—
37	C₃×C₄×AGL(1,𝔽₅)	—	2	6	no	—
38	C₃×C₄×AGL(1,𝔽₅)	D₄	1	12	yes	complemented
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic, complemented
40	C₃×C₄×Dic₅	C₂×C₄	1	3	yes	complemented
41	C₂×C₃×#127	—	5	1	no	—
42	C₂×C₃×C₄×D₄	—	5	6	no	—
43	C₃×#390	—	5	6	no	—
44	C₃×#384	—	5	1	no	—
45	C₃×C₄³	—	5	1	no	—
46	C₂³×AGL(1,𝔽₅)	C₂²×C₃	1	4	yes	—
47	C₂×D₄×D₅	C₂²×C₃	1	6	yes	—
48	C₂×C₂²⋊AGL(1,𝔽₅)	C₂²×C₃	1	12	yes	—
49	D₄×AGL(1,𝔽₅)	—	2	24	no	—
50	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	6	yes	—
51	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	3	yes	—
52	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	3	yes	—
53	C₅×#127	C₃×C₄	1	1	yes	characteristic, complemented
54	#4086	C₃×C₄	1	6	yes	complemented
55	#4020	C₃×C₄	1	1	yes	characteristic, complemented
56	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
57	C₄×#127	—	5	1	no	Sylow
58	C₂³×C₃×C₅	C₂²×C₄	1	4	yes	—
59	C₂²×C₃×D₅	—	2	12	no	—
60	C₂²×C₃×D₅	—	2	12	no	—
61	C₂²×C₃×D₅	C₂²×C₄	1	4	yes	—
62	C₂²×C₃×D₅	C₂⁴	1	1	yes	characteristic
63	C₂×C₃×AGL(1,𝔽₅)	—	2	6	no	—
64	C₂×C₃×AGL(1,𝔽₅)	—	2	12	no	—
65	C₂×C₃×AGL(1,𝔽₅)	—	4	16	no	—
66	C₂×C₃×AGL(1,𝔽₅)	C₂×D₄	1	6	yes	—
67	C₂×C₃×AGL(1,𝔽₅)	C₂×D₄	1	6	yes	—
68	C₃×C₅×D₄	—	2	12	no	—
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
70	C₃×C₂²⋊D₅	—	2	24	no	—
71	C₂×C₃×Dic₅	C₂²×C₄	1	3	yes	—
72	C₃×D₂₀	—	2	12	no	—
73	C₃×C₄×D₅	C₂×D₄	1	6	yes	—
74	C₃×C₄×D₅	#41	1	6	yes	—
75	C₂²×C₃×D₄	—	5	6	no	—
76	C₂³×C₃×C₄	—	5	4	no	—
77	C₃×#127	—	5	6	no	—
78	C₃×#127	—	5	1	no	—
79	C₃×#127	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
80	C₃×C₄×D₄	—	10	24	no	—
81	C₂×C₃×C₄⋊C₄	—	5	6	no	—
82	C₂×C₃×C₂²⋊C₄	—	5	12	no	—
83	C₂×C₃×C₄²	—	5	3	no	—
84	C₂×C₃×C₄²	—	5	1	no	—
85	C₂×C₃×C₄²	—	5	3	no	—
86	C₂³×D₅	C₂³×C₃	1	4	yes	—
87	C₂²×AGL(1,𝔽₅)	—	2	24	no	—
88	C₂²×AGL(1,𝔽₅)	C₂³×C₃	1	4	yes	—
89	C₂²×AGL(1,𝔽₅)	C₂³×C₃	1	3	yes	—
90	C₂²×AGL(1,𝔽₅)	C₂³×C₃	1	1	yes	characteristic
91	C₂×C₅×D₄	C₂×C₃×C₄	1	6	yes	complemented
92	C₂×C₂²⋊D₅	C₂×C₃×C₄	1	12	yes	complemented
93	D₄×D₅	—	2	12	no	—
94	D₄×D₅	—	2	12	no	—
95	C₂×D₂₀	C₂×C₃×C₄	1	6	yes	complemented
96	C₂×C₄×D₅	C₂³×C₃	1	3	yes	—
97	C₂²⋊AGL(1,𝔽₅)	—	2	24	no	—
98	C₄⋊AGL(1,𝔽₅)	—	2	12	no	—
99	C₄×AGL(1,𝔽₅)	—	2	6	no	—
100	C₄×AGL(1,𝔽₅)	C₃×D₄	1	12	yes	complemented
101	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
102	C₄×Dic₅	C₂×C₃×C₄	1	3	yes	complemented
103	C₂×#127	—	5	1	no	—
104	C₂×C₄×D₄	—	5	6	no	—
105	#390	—	5	6	no	—
106	#384	—	5	1	no	—
107	C₄³	—	5	1	no	—
108	C₂²×C₃×C₅	—	2	12	no	—
109	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₃×D₅	—	2	12	no	—
111	C₂×C₃×D₅	—	4	4	no	—
112	C₂×C₃×D₅	—	4	12	no	—
113	C₂×C₃×D₅	C₂²×D₄	1	3	yes	—
114	C₂×C₃×D₅	C₂×#41	1	3	yes	—
115	C₃×AGL(1,𝔽₅)	—	2	6	no	—
116	C₃×AGL(1,𝔽₅)	—	4	4	no	—
117	C₃×AGL(1,𝔽₅)	#127	1	4	yes	direct product factor
118	C₃×C₄×C₅	C₄×D₄	1	6	yes	complemented
119	C₃×Dic₅	C₄×D₄	1	6	yes	complemented
120	C₂⁴×C₃	—	5	4	no	—
121	C₂×C₃×D₄	—	5	12	no	—
122	C₂×C₃×D₄	—	5	6	no	—
123	C₂×C₃×D₄	—	10	12	no	—
124	C₂×C₃×D₄	—	10	12	no	—
125	C₂×C₃×D₄	C₂×AGL(1,𝔽₅)	1	6	yes	complemented
126	C₂²×C₃×C₄	—	5	1	no	—
127	C₂²×C₃×C₄	—	5	3	no	—
128	C₂²×C₃×C₄	—	5	3	no	—
129	C₂²×C₃×C₄	—	5	4	no	—
130	C₂²×C₃×C₄	—	10	24	no	—
131	C₃×C₄⋊C₄	—	10	12	no	—
132	C₃×C₂²⋊C₄	—	10	24	no	—
133	C₃×C₄²	—	5	12	no	—
134	C₃×C₄²	—	5	3	no	—
135	C₃×C₄²	—	10	6	no	—
136	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
137	C₂³×C₅	C₂²×C₃×C₄	1	4	yes	—
138	C₂²×D₅	—	2	12	no	—
139	C₂²×D₅	—	2	12	no	—
140	C₂²×D₅	C₂²×C₃×C₄	1	4	yes	—
141	C₂²×D₅	C₂⁴×C₃	1	1	yes	characteristic
142	C₂×AGL(1,𝔽₅)	—	2	6	no	—
143	C₂×AGL(1,𝔽₅)	—	2	12	no	—
144	C₂×AGL(1,𝔽₅)	—	4	16	no	—
145	C₂×AGL(1,𝔽₅)	C₂×C₃×D₄	1	6	yes	—
146	C₂×AGL(1,𝔽₅)	C₂×C₃×D₄	1	6	yes	—
147	C₅×D₄	—	2	12	no	—
148	C₂×C₄×C₅	C₂²×C₃×C₄	1	3	yes	—
149	C₂²⋊D₅	—	2	24	no	—
150	C₂×Dic₅	C₂²×C₃×C₄	1	3	yes	—
151	D₂₀	—	2	12	no	—
152	C₄×D₅	C₂×C₃×D₄	1	6	yes	—
153	C₄×D₅	C₃×#41	1	6	yes	—
154	C₂²×D₄	—	5	6	no	—
155	C₂³×C₄	—	5	4	no	—
156	#127	—	5	6	no	—
157	#127	—	5	1	no	—
158	#127	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
159	C₄×D₄	—	10	24	no	—
160	C₂×C₄⋊C₄	—	5	6	no	—
161	C₂×C₂²⋊C₄	—	5	12	no	—
162	C₂×C₄²	—	5	3	no	—
163	C₂×C₄²	—	5	1	no	—
164	C₂×C₄²	—	5	3	no	—
165	C₂×C₃×C₅	—	4	4	no	—
166	C₂×C₃×C₅	C₂×C₄×D₄	1	3	yes	—
167	C₃×D₅	—	4	4	no	—
168	C₃×D₅	C₂×#127	1	1	yes	characteristic
169	C₃×D₅	#532	1	3	yes	—
170	C₂³×C₃	—	5	1	no	—
171	C₂³×C₃	—	5	4	no	—
172	C₂³×C₃	—	10	12	no	—
173	C₂³×C₃	—	10	12	no	—
174	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	4	yes	—
175	C₃×D₄	—	2	12	no	—
176	C₃×D₄	—	10	24	no	—
177	C₃×D₄	—	10	12	no	—
178	C₂×C₃×C₄	—	5	6	no	—
179	C₂×C₃×C₄	—	5	3	no	—
180	C₂×C₃×C₄	—	5	6	no	—
181	C₂×C₃×C₄	—	5	6	no	—
182	C₂×C₃×C₄	—	5	6	no	—
183	C₂×C₃×C₄	—	10	6	no	—
184	C₂×C₃×C₄	—	10	12	no	—
185	C₂×C₃×C₄	—	20	16	no	—
186	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
187	C₂²×C₅	—	2	12	no	—
188	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
189	C₂×D₅	—	2	12	no	—
190	C₂×D₅	—	4	4	no	—
191	C₂×D₅	—	4	12	no	—
192	C₂×D₅	C₂²×C₃×D₄	1	3	yes	—
193	C₂×D₅	C₂×C₃×#41	1	3	yes	—
194	AGL(1,𝔽₅)	—	2	6	no	—
195	AGL(1,𝔽₅)	—	4	4	no	—
196	AGL(1,𝔽₅)	C₃×#127	1	4	yes	direct product factor
197	C₄×C₅	C₃×C₄×D₄	1	6	yes	complemented
198	Dic₅	C₃×C₄×D₄	1	6	yes	complemented
199	C₂⁴	—	5	4	no	—
200	C₂×D₄	—	5	12	no	—
201	C₂×D₄	—	5	6	no	—
202	C₂×D₄	—	10	12	no	—
203	C₂×D₄	—	10	12	no	—
204	C₂×D₄	C₂×C₃×AGL(1,𝔽₅)	1	6	yes	complemented
205	C₂²×C₄	—	5	1	no	—
206	C₂²×C₄	—	5	3	no	—
207	C₂²×C₄	—	5	3	no	—
208	C₂²×C₄	—	5	4	no	—
209	C₂²×C₄	—	10	24	no	—
210	C₄⋊C₄	—	10	12	no	—
211	C₂²⋊C₄	—	10	24	no	—
212	C₄²	—	5	12	no	—
213	C₄²	—	5	3	no	—
214	C₄²	—	10	6	no	—
215	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
216	C₃×C₅	C₄×#127	1	1	yes	characteristic, complemented, Hall
217	C₂²×C₃	—	2	12	no	—
218	C₂²×C₃	—	5	3	no	—
219	C₂²×C₃	—	5	3	no	—
220	C₂²×C₃	—	10	12	no	—
221	C₂²×C₃	—	20	4	no	—
222	C₂²×C₃	—	20	12	no	—
223	C₂²×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
224	C₃×C₄	—	5	4	no	—
225	C₃×C₄	—	5	6	no	—
226	C₃×C₄	—	10	6	no	—
227	C₃×C₄	—	20	4	no	—
228	C₃×C₄	D₄×AGL(1,𝔽₅)	1	6	yes	complemented
229	C₂×C₅	—	4	4	no	—
230	C₂×C₅	C₂×C₃×C₄×D₄	1	3	yes	—
231	D₅	—	4	4	no	—
232	D₅	C₂×C₃×#127	1	1	yes	characteristic
233	D₅	C₃×#532	1	3	yes	—
234	C₂³	—	5	1	no	—
235	C₂³	—	5	4	no	—
236	C₂³	—	10	12	no	—
237	C₂³	—	10	12	no	—
238	C₂³	C₂²×C₃×AGL(1,𝔽₅)	1	4	yes	—
239	D₄	—	2	12	no	—
240	D₄	—	10	24	no	—
241	D₄	—	10	12	no	—
242	C₂×C₄	—	5	6	no	—
243	C₂×C₄	—	5	3	no	—
244	C₂×C₄	—	5	6	no	—
245	C₂×C₄	—	5	6	no	—
246	C₂×C₄	—	5	6	no	—
247	C₂×C₄	—	10	6	no	—
248	C₂×C₄	—	10	12	no	—
249	C₂×C₄	—	20	16	no	—
250	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	3	yes	—
251	C₂×C₃	—	4	4	no	—
252	C₂×C₃	—	5	1	no	—
253	C₂×C₃	—	5	3	no	—
254	C₂×C₃	—	20	4	no	—
255	C₂×C₃	C₂×D₄×AGL(1,𝔽₅)	1	3	yes	—
256	C₅	C₃×C₄×#127	1	1	yes	characteristic, complemented, Sylow
257	C₂²	—	2	12	no	—
258	C₂²	—	5	3	no	—
259	C₂²	—	5	3	no	—
260	C₂²	—	10	12	no	—
261	C₂²	—	20	4	no	—
262	C₂²	—	20	12	no	—
263	C₂²	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
264	C₄	—	5	4	no	—
265	C₄	—	5	6	no	—
266	C₄	—	10	6	no	—
267	C₄	—	20	4	no	—
268	C₄	C₃×D₄×AGL(1,𝔽₅)	1	6	yes	complemented
269	C₃	AGL(1,𝔽₅)×#127	1	1	yes	characteristic, direct product factor, Sylow
270	C₂	—	4	4	no	—
271	C₂	—	5	1	no	—
272	C₂	—	5	3	no	—
273	C₂	—	20	4	no	—
274	C₂	C₂×C₃×D₄×AGL(1,𝔽₅)	1	3	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#127	C₄
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₃×D₅×#127	C₂
cosocle	C₂²×D₅	C₂⁴×C₃
hypercenter	C₃×#127	AGL(1,𝔽₅)
hypocenter	C₅	C₃×C₄×#127
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₄×AGL(1,𝔽₅)	C₂	semidirect
3. C₃×#65780	C₂	semidirect
4. C₃×#65688	C₂	semidirect
5. C₃×C₄²×AGL(1,𝔽₅)	C₂	semidirect
6. AGL(1,𝔽₅)×#127	C₃	direct (N × H)
14. C₃×C₅×#127	C₄	semidirect
15. C₃×#4086	C₄	semidirect
16. C₃×#4020	C₄	semidirect
19. C₂×D₄×AGL(1,𝔽₅)	C₂×C₃	semidirect
21. #65780	C₂×C₃	semidirect
22. #65688	C₂×C₃	semidirect
23. C₄²×AGL(1,𝔽₅)	C₂×C₃	semidirect
29. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
30. C₂×C₃×C₂²⋊D₅	C₂×C₄	semidirect
33. C₂×C₃×D₂₀	C₂×C₄	semidirect
38. C₃×C₄×AGL(1,𝔽₅)	D₄	semidirect
39. C₃×C₄²×C₅	C₂×C₄	semidirect
40. C₃×C₄×Dic₅	C₂×C₄	semidirect
53. C₅×#127	C₃×C₄	semidirect
54. #4086	C₃×C₄	semidirect
55. #4020	C₃×C₄	semidirect
79. C₃×#127	AGL(1,𝔽₅)	direct (N × H)
91. C₂×C₅×D₄	C₂×C₃×C₄	semidirect
92. C₂×C₂²⋊D₅	C₂×C₃×C₄	semidirect
95. C₂×D₂₀	C₂×C₃×C₄	semidirect
100. C₄×AGL(1,𝔽₅)	C₃×D₄	semidirect
101. C₄²×C₅	C₂×C₃×C₄	semidirect
102. C₄×Dic₅	C₂×C₃×C₄	semidirect
117. C₃×AGL(1,𝔽₅)	#127	direct (N × H)
118. C₃×C₄×C₅	C₄×D₄	semidirect
119. C₃×Dic₅	C₄×D₄	semidirect
125. C₂×C₃×D₄	C₂×AGL(1,𝔽₅)	semidirect
136. C₃×C₄²	C₂×AGL(1,𝔽₅)	semidirect
158. #127	C₃×AGL(1,𝔽₅)	direct (N × H)
196. AGL(1,𝔽₅)	C₃×#127	direct (N × H)
197. C₄×C₅	C₃×C₄×D₄	semidirect
198. Dic₅	C₃×C₄×D₄	semidirect
204. C₂×D₄	C₂×C₃×AGL(1,𝔽₅)	semidirect
215. C₄²	C₂×C₃×AGL(1,𝔽₅)	semidirect
216. C₃×C₅	C₄×#127	semidirect
228. C₃×C₄	D₄×AGL(1,𝔽₅)	semidirect
256. C₅	C₃×C₄×#127	semidirect
268. C₄	C₃×D₄×AGL(1,𝔽₅)	semidirect
269. C₃	AGL(1,𝔽₅)×#127	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×D₄×AGL(1,𝔽₅)	2
C₃×D₅×#127	2
C₃×#65780	2
C₃×#65688	2
C₃×C₄²×AGL(1,𝔽₅)	2
AGL(1,𝔽₅)×#127	3
C₃×C₄×#127	5

Subgroups — C3×AGL(1,𝔽5)×#127 #14322257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×AGL(1,𝔽₅)×#127 #14322257