Subgroups — C₃×#10617128 #14321093

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4130	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₄×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	C₃×D₅×#117	C₂	1	1	yes	maximal, characteristic
4	C₃×#65712	C₂	1	4	yes	maximal, complemented
5	C₃×#65699	C₂	1	4	yes	maximal
6	C₃×C₄×C₄⋊AGL(1,𝔽₅)	C₂	1	4	yes	maximal, complemented
7	#10617128	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂³×C₃×AGL(1,𝔽₅)	C₂²	1	2	yes	—
9	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
11	C₃×#4130	—	2	4	no	—
12	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	8	yes	—
13	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
14	C₂×C₃×C₄×AGL(1,𝔽₅)	—	2	4	no	—
15	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
16	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
17	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
18	C₃×C₅×#117	C₄	1	1	yes	characteristic, complemented
19	C₃×#4072	C₄	1	1	yes	characteristic, complemented
20	C₃×#4038	C₄	1	2	yes	complemented
21	C₃×D₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×D₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₃×#4023	C₄	1	2	yes	complemented
24	C₃×#4018	C₄	1	2	yes	complemented
25	C₃×C₄²×D₅	C₂²	1	2	yes	—
26	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	8	yes	—
27	C₃×#2296	—	5	1	no	maximal, Hall
28	C₂×#4130	C₂×C₃	1	1	yes	characteristic
29	C₂²×C₄×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic
30	D₅×#117	C₂×C₃	1	1	yes	characteristic
31	#65712	C₂×C₃	1	4	yes	complemented
32	#65699	C₂×C₃	1	4	yes	—
33	C₄×C₄⋊AGL(1,𝔽₅)	C₂×C₃	1	4	yes	complemented
34	C₂³×C₃×D₅	C₂³	1	1	yes	characteristic
35	C₂²×C₃×AGL(1,𝔽₅)	—	2	4	no	—
36	C₂²×C₃×AGL(1,𝔽₅)	—	2	8	no	—
37	C₂²×C₃×AGL(1,𝔽₅)	C₂³	1	4	yes	—
38	C₂²×C₃×AGL(1,𝔽₅)	C₂³	1	4	yes	—
39	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₄×D₅	D₄	1	4	yes	—
44	C₂×C₃×C₄×D₅	Q₈	1	4	yes	—
45	C₂×C₃×C₄×D₅	C₂³	1	4	yes	—
46	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
48	C₃×C₂²⋊AGL(1,𝔽₅)	—	2	4	no	—
49	C₃×C₄⋊AGL(1,𝔽₅)	—	2	16	no	—
50	C₃×C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
51	C₃×C₄×AGL(1,𝔽₅)	—	2	4	no	—
52	C₃×C₄×AGL(1,𝔽₅)	—	2	4	no	—
53	C₃×C₄×AGL(1,𝔽₅)	—	2	4	no	—
54	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	complemented
55	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
56	C₃×C₄²×C₅	C₂×C₄	1	2	yes	complemented
57	C₃×C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
58	C₃×#685	C₂×C₄	1	4	yes	—
59	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	complemented
60	C₃×#683	C₂×C₄	1	4	yes	complemented
61	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	complemented
62	C₃×C₄×Dic₅	C₂×C₄	1	4	yes	complemented
63	C₂×C₃×#117	—	5	1	no	—
64	C₂×C₃×#117	—	5	1	no	—
65	C₂²×C₃×C₄²	—	5	1	no	—
66	C₃×#389	—	5	4	no	—
67	C₃×#385	—	5	4	no	—
68	C₃×C₄×C₄⋊C₄	—	5	4	no	—
69	C₂³×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
70	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₂²⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
72	#4130	—	2	4	no	—
73	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	8	yes	—
74	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
75	C₂×C₄×AGL(1,𝔽₅)	—	2	4	no	—
76	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
77	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
78	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
79	C₅×#117	C₃×C₄	1	1	yes	characteristic, complemented
80	#4072	C₃×C₄	1	1	yes	characteristic, complemented
81	#4038	C₃×C₄	1	2	yes	complemented
82	D₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
83	D₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
84	#4023	C₃×C₄	1	2	yes	complemented
85	#4018	C₃×C₄	1	2	yes	complemented
86	C₄²×D₅	C₂²×C₃	1	2	yes	—
87	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	8	yes	—
88	#2296	—	5	1	no	Sylow
89	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₃×D₅	—	2	4	no	—
91	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₃×D₅	C₂×D₄	1	2	yes	—
93	C₂²×C₃×D₅	C₂×Q₈	1	2	yes	—
94	C₂²×C₃×D₅	C₂⁴	1	1	yes	characteristic
95	C₂×C₃×AGL(1,𝔽₅)	—	2	16	no	—
96	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
97	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
98	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
99	C₂×C₃×AGL(1,𝔽₅)	—	2	8	no	—
100	C₂×C₃×AGL(1,𝔽₅)	#41	1	8	yes	—
101	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	complemented
102	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
103	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₃×Dic₅	C₄⋊C₄	1	4	yes	complemented
106	C₂×C₃×Dic₅	C₂²×C₄	1	4	yes	—
107	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
109	C₃×C₄×D₅	—	2	8	no	—
110	C₃×C₄×D₅	C₄⋊C₄	1	4	yes	complemented
111	C₃×C₄×D₅	C₄⋊C₄	1	4	yes	complemented
112	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
113	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
114	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
115	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
116	C₂³×C₃×C₄	—	5	1	no	—
117	C₂³×C₃×C₄	—	5	2	no	—
118	C₃×#117	—	5	1	no	—
119	C₃×#117	—	5	2	no	—
120	C₃×#117	—	5	2	no	—
121	C₃×#117	—	5	2	no	—
122	C₃×#117	—	10	4	no	—
123	C₃×#117	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
124	C₂×C₃×C₄⋊C₄	—	5	2	no	—
125	C₂×C₃×C₄⋊C₄	—	5	8	no	—
126	C₂×C₃×C₄⋊C₄	—	5	2	no	—
127	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
128	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
129	C₂×C₃×C₄²	—	5	2	no	—
130	C₂×C₃×C₄²	—	5	2	no	—
131	C₂×C₃×C₄²	—	5	2	no	—
132	C₂×C₃×C₄²	—	5	2	no	—
133	C₂×C₃×C₄²	—	10	4	no	—
134	C₃×#95	—	5	8	no	—
135	C₂³×D₅	C₂³×C₃	1	1	yes	characteristic
136	C₂²×AGL(1,𝔽₅)	—	2	4	no	—
137	C₂²×AGL(1,𝔽₅)	—	2	8	no	—
138	C₂²×AGL(1,𝔽₅)	C₂³×C₃	1	4	yes	—
139	C₂²×AGL(1,𝔽₅)	C₂³×C₃	1	4	yes	—
140	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
141	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
142	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
143	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
144	C₂×C₄×D₅	C₃×D₄	1	4	yes	—
145	C₂×C₄×D₅	C₃×Q₈	1	4	yes	—
146	C₂×C₄×D₅	C₂³×C₃	1	4	yes	—
147	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
148	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
149	C₂²⋊AGL(1,𝔽₅)	—	2	4	no	—
150	C₄⋊AGL(1,𝔽₅)	—	2	16	no	—
151	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
152	C₄×AGL(1,𝔽₅)	—	2	4	no	—
153	C₄×AGL(1,𝔽₅)	—	2	4	no	—
154	C₄×AGL(1,𝔽₅)	—	2	4	no	—
155	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	complemented
156	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
157	C₄²×C₅	C₂×C₃×C₄	1	2	yes	complemented
158	C₂²⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
159	#685	C₂×C₃×C₄	1	4	yes	—
160	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	complemented
161	#683	C₂×C₃×C₄	1	4	yes	complemented
162	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	complemented
163	C₄×Dic₅	C₂×C₃×C₄	1	4	yes	complemented
164	C₂×#117	—	5	1	no	—
165	C₂×#117	—	5	1	no	—
166	C₂²×C₄²	—	5	1	no	—
167	#389	—	5	4	no	—
168	#385	—	5	4	no	—
169	C₄×C₄⋊C₄	—	5	4	no	—
170	C₂²×C₃×C₅	—	2	2	no	—
171	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
172	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₃×D₅	—	2	2	no	—
174	C₂×C₃×D₅	—	2	4	no	—
175	C₂×C₃×D₅	—	2	4	no	—
176	C₂×C₃×D₅	C₂×C₄⋊C₄	1	2	yes	—
177	C₂×C₃×D₅	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₃×D₅	C₂²×Q₈	1	1	yes	characteristic
179	C₂×C₃×D₅	C₂×#41	1	4	yes	—
180	C₃×AGL(1,𝔽₅)	—	2	8	no	—
181	C₃×AGL(1,𝔽₅)	—	4	4	no	—
182	C₃×C₄×C₅	—	2	4	no	—
183	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
184	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
185	C₃×Dic₅	—	2	4	no	—
186	C₃×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
187	C₃×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
188	C₂⁴×C₃	—	5	1	no	—
189	C₂²×C₃×C₄	—	5	1	no	—
190	C₂²×C₃×C₄	—	5	4	no	—
191	C₂²×C₃×C₄	—	5	4	no	—
192	C₂²×C₃×C₄	—	5	1	no	—
193	C₂²×C₃×C₄	—	5	1	no	—
194	C₂²×C₃×C₄	—	5	1	no	—
195	C₂²×C₃×C₄	—	5	1	no	—
196	C₂²×C₃×C₄	—	5	4	no	—
197	C₂²×C₃×C₄	—	5	4	no	—
198	C₂²×C₃×C₄	—	5	4	no	—
199	C₂²×C₃×C₄	—	10	8	no	—
200	C₂²×C₃×C₄	—	10	4	no	—
201	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
202	C₃×C₄⋊C₄	—	5	4	no	—
203	C₃×C₄⋊C₄	—	5	2	no	—
204	C₃×C₄⋊C₄	—	10	16	no	—
205	C₃×C₄⋊C₄	—	10	4	no	—
206	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
207	C₃×C₂²⋊C₄	—	5	2	no	—
208	C₃×C₂²⋊C₄	—	5	4	no	—
209	C₃×C₂²⋊C₄	—	10	4	no	—
210	C₃×C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
211	C₃×C₄²	—	5	4	no	—
212	C₃×C₄²	—	5	2	no	—
213	C₃×C₄²	—	10	4	no	—
214	C₃×C₄²	—	10	4	no	—
215	C₃×C₄²	—	10	4	no	—
216	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
217	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
218	C₂²×D₅	—	2	4	no	—
219	C₂²×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
220	C₂²×D₅	C₂×C₃×D₄	1	2	yes	—
221	C₂²×D₅	C₂×C₃×Q₈	1	2	yes	—
222	C₂²×D₅	C₂⁴×C₃	1	1	yes	characteristic
223	C₂×AGL(1,𝔽₅)	—	2	16	no	—
224	C₂×AGL(1,𝔽₅)	—	2	4	no	—
225	C₂×AGL(1,𝔽₅)	—	2	4	no	—
226	C₂×AGL(1,𝔽₅)	—	2	4	no	—
227	C₂×AGL(1,𝔽₅)	—	2	8	no	—
228	C₂×AGL(1,𝔽₅)	C₃×#41	1	8	yes	—
229	C₂×C₄×C₅	C₃×C₄⋊C₄	1	4	yes	complemented
230	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
231	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
232	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
233	C₂×Dic₅	C₃×C₄⋊C₄	1	4	yes	complemented
234	C₂×Dic₅	C₂²×C₃×C₄	1	4	yes	—
235	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
236	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
237	C₄×D₅	—	2	8	no	—
238	C₄×D₅	C₃×C₄⋊C₄	1	4	yes	complemented
239	C₄×D₅	C₃×C₄⋊C₄	1	4	yes	complemented
240	C₄×D₅	C₂×C₃×D₄	1	2	yes	—
241	C₄×D₅	C₂×C₃×D₄	1	2	yes	—
242	C₄×D₅	C₂×C₃×Q₈	1	2	yes	—
243	C₄×D₅	C₂×C₃×Q₈	1	2	yes	—
244	C₂³×C₄	—	5	1	no	—
245	C₂³×C₄	—	5	2	no	—
246	#117	—	5	1	no	—
247	#117	—	5	2	no	—
248	#117	—	5	2	no	—
249	#117	—	5	2	no	—
250	#117	—	10	4	no	—
251	#117	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
252	C₂×C₄⋊C₄	—	5	2	no	—
253	C₂×C₄⋊C₄	—	5	2	no	—
254	C₂×C₄⋊C₄	—	5	8	no	—
255	C₂×C₂²⋊C₄	—	5	2	no	—
256	C₂×C₂²⋊C₄	—	5	2	no	—
257	C₂×C₄²	—	5	2	no	—
258	C₂×C₄²	—	5	2	no	—
259	C₂×C₄²	—	5	2	no	—
260	C₂×C₄²	—	5	2	no	—
261	C₂×C₄²	—	10	4	no	—
262	#95	—	5	8	no	—
263	C₂×C₃×C₅	—	2	2	no	—
264	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
265	C₂×C₃×C₅	C₄×#41	1	2	yes	—
266	C₃×D₅	—	2	2	no	—
267	C₃×D₅	#525	1	2	yes	—
268	C₃×D₅	#533	1	2	yes	—
269	C₂³×C₃	—	5	1	no	—
270	C₂³×C₃	—	5	2	no	—
271	C₂³×C₃	—	5	2	no	—
272	C₂³×C₃	—	5	1	no	—
273	C₂³×C₃	—	10	4	no	—
274	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
275	C₂×C₃×C₄	—	5	4	no	—
276	C₂×C₃×C₄	—	5	1	no	—
277	C₂×C₃×C₄	—	5	1	no	—
278	C₂×C₃×C₄	—	5	4	no	—
279	C₂×C₃×C₄	—	5	4	no	—
280	C₂×C₃×C₄	—	5	2	no	—
281	C₂×C₃×C₄	—	5	2	no	—
282	C₂×C₃×C₄	—	5	2	no	—
283	C₂×C₃×C₄	—	5	2	no	—
284	C₂×C₃×C₄	—	5	8	no	—
285	C₂×C₃×C₄	—	5	4	no	—
286	C₂×C₃×C₄	—	10	8	no	—
287	C₂×C₃×C₄	—	10	4	no	—
288	C₂×C₃×C₄	—	10	4	no	—
289	C₂×C₃×C₄	—	10	16	no	—
290	C₂×C₃×C₄	—	10	4	no	—
291	C₂×C₃×C₄	—	10	8	no	—
292	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	4	yes	complemented
293	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
294	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
295	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
296	C₂²×C₅	—	2	2	no	—
297	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
298	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
299	C₂×D₅	—	2	2	no	—
300	C₂×D₅	—	2	4	no	—
301	C₂×D₅	—	2	4	no	—
302	C₂×D₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
303	C₂×D₅	C₂²×C₃×D₄	1	1	yes	characteristic
304	C₂×D₅	C₂²×C₃×Q₈	1	1	yes	characteristic
305	C₂×D₅	C₂×C₃×#41	1	4	yes	—
306	AGL(1,𝔽₅)	—	2	8	no	—
307	AGL(1,𝔽₅)	—	4	4	no	—
308	C₄×C₅	—	2	4	no	—
309	C₄×C₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
310	C₄×C₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
311	Dic₅	—	2	4	no	—
312	Dic₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
313	Dic₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
314	C₂⁴	—	5	1	no	—
315	C₂²×C₄	—	5	1	no	—
316	C₂²×C₄	—	5	4	no	—
317	C₂²×C₄	—	5	4	no	—
318	C₂²×C₄	—	5	1	no	—
319	C₂²×C₄	—	5	1	no	—
320	C₂²×C₄	—	5	1	no	—
321	C₂²×C₄	—	5	1	no	—
322	C₂²×C₄	—	5	4	no	—
323	C₂²×C₄	—	5	4	no	—
324	C₂²×C₄	—	5	4	no	—
325	C₂²×C₄	—	10	8	no	—
326	C₂²×C₄	—	10	4	no	—
327	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
328	C₄⋊C₄	—	5	4	no	—
329	C₄⋊C₄	—	5	2	no	—
330	C₄⋊C₄	—	10	4	no	—
331	C₄⋊C₄	—	10	16	no	—
332	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
333	C₂²⋊C₄	—	5	2	no	—
334	C₂²⋊C₄	—	5	4	no	—
335	C₂²⋊C₄	—	10	4	no	—
336	C₂²⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
337	C₄²	—	5	4	no	—
338	C₄²	—	5	2	no	—
339	C₄²	—	10	4	no	—
340	C₄²	—	10	4	no	—
341	C₄²	—	10	4	no	—
342	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
343	C₃×C₅	#2296	1	1	yes	characteristic, complemented, Hall
344	C₂²×C₃	—	2	2	no	—
345	C₂²×C₃	—	5	1	no	—
346	C₂²×C₃	—	5	1	no	—
347	C₂²×C₃	—	5	4	no	—
348	C₂²×C₃	—	5	2	no	—
349	C₂²×C₃	—	10	4	no	—
350	C₂²×C₃	—	10	4	no	—
351	C₂²×C₃	—	10	2	no	—
352	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
353	C₂²×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
354	C₃×C₄	—	2	4	no	—
355	C₃×C₄	—	5	2	no	—
356	C₃×C₄	—	5	2	no	—
357	C₃×C₄	—	10	8	no	—
358	C₃×C₄	—	10	4	no	—
359	C₃×C₄	—	20	4	no	—
360	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
361	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
362	C₂×C₅	—	2	2	no	—
363	C₂×C₅	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
364	C₂×C₅	C₃×C₄×#41	1	2	yes	—
365	D₅	—	2	2	no	—
366	D₅	C₃×#525	1	2	yes	—
367	D₅	C₃×#533	1	2	yes	—
368	C₂³	—	5	1	no	—
369	C₂³	—	5	2	no	—
370	C₂³	—	5	2	no	—
371	C₂³	—	5	1	no	—
372	C₂³	—	10	4	no	—
373	C₂³	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
374	C₂×C₄	—	5	4	no	—
375	C₂×C₄	—	5	1	no	—
376	C₂×C₄	—	5	1	no	—
377	C₂×C₄	—	5	2	no	—
378	C₂×C₄	—	5	2	no	—
379	C₂×C₄	—	5	4	no	—
380	C₂×C₄	—	5	4	no	—
381	C₂×C₄	—	5	2	no	—
382	C₂×C₄	—	5	2	no	—
383	C₂×C₄	—	5	8	no	—
384	C₂×C₄	—	5	4	no	—
385	C₂×C₄	—	10	8	no	—
386	C₂×C₄	—	10	4	no	—
387	C₂×C₄	—	10	4	no	—
388	C₂×C₄	—	10	16	no	—
389	C₂×C₄	—	10	4	no	—
390	C₂×C₄	—	10	8	no	—
391	C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	1	4	yes	complemented
392	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	4	yes	—
393	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
394	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
395	C₂×C₃	—	2	2	no	—
396	C₂×C₃	—	5	2	no	—
397	C₂×C₃	—	5	2	no	—
398	C₂×C₃	—	10	2	no	—
399	C₂×C₃	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
400	C₂×C₃	#41×AGL(1,𝔽₅)	1	2	yes	—
401	C₅	C₃×#2296	1	1	yes	characteristic, complemented, Sylow
402	C₂²	—	2	2	no	—
403	C₂²	—	5	1	no	—
404	C₂²	—	5	1	no	—
405	C₂²	—	5	4	no	—
406	C₂²	—	5	2	no	—
407	C₂²	—	10	4	no	—
408	C₂²	—	10	4	no	—
409	C₂²	—	10	2	no	—
410	C₂²	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
411	C₂²	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
412	C₄	—	2	4	no	—
413	C₄	—	5	2	no	—
414	C₄	—	5	2	no	—
415	C₄	—	10	8	no	—
416	C₄	—	10	4	no	—
417	C₄	—	20	4	no	—
418	C₄	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
419	C₄	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
420	C₃	#10617128	1	1	yes	characteristic, direct product factor, Sylow
421	C₂	—	2	2	no	—
422	C₂	—	5	2	no	—
423	C₂	—	5	2	no	—
424	C₂	—	10	2	no	—
425	C₂	C₂²×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
426	C₂	C₃×#41×AGL(1,𝔽₅)	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#117	C₄
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂³×C₃×D₅	C₂³
cosocle	C₂²×D₅	C₂⁴×C₃
hypercenter	C₃×#117	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2296
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×#117

Complements

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

N	H ≅ G/N	type
4. C₃×#65712	C₂	semidirect
6. C₃×C₄×C₄⋊AGL(1,𝔽₅)	C₂	semidirect
7. #10617128	C₃	direct (N × H)
18. C₃×C₅×#117	C₄	semidirect
19. C₃×#4072	C₄	semidirect
20. C₃×#4038	C₄	semidirect
23. C₃×#4023	C₄	semidirect
24. C₃×#4018	C₄	semidirect
31. #65712	C₂×C₃	semidirect
33. C₄×C₄⋊AGL(1,𝔽₅)	C₂×C₃	semidirect
54. C₃×C₅×C₄⋊C₄	C₂×C₄	semidirect
56. C₃×C₄²×C₅	C₂×C₄	semidirect
59. C₃×C₄⋊Dic₅	C₂×C₄	semidirect
60. C₃×#683	C₂×C₄	semidirect
61. C₃×C₄×Dic₅	C₂×C₄	semidirect
62. C₃×C₄×Dic₅	C₂×C₄	semidirect
79. C₅×#117	C₃×C₄	semidirect
80. #4072	C₃×C₄	semidirect
81. #4038	C₃×C₄	semidirect
84. #4023	C₃×C₄	semidirect
85. #4018	C₃×C₄	semidirect
101. C₂×C₃×C₄×C₅	C₄⋊C₄	semidirect
105. C₂×C₃×Dic₅	C₄⋊C₄	semidirect
110. C₃×C₄×D₅	C₄⋊C₄	semidirect
111. C₃×C₄×D₅	C₄⋊C₄	semidirect
123. C₃×#117	AGL(1,𝔽₅)	semidirect
155. C₅×C₄⋊C₄	C₂×C₃×C₄	semidirect
157. C₄²×C₅	C₂×C₃×C₄	semidirect
160. C₄⋊Dic₅	C₂×C₃×C₄	semidirect
161. #683	C₂×C₃×C₄	semidirect
162. C₄×Dic₅	C₂×C₃×C₄	semidirect
163. C₄×Dic₅	C₂×C₃×C₄	semidirect
206. C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	semidirect
216. C₃×C₄²	C₂×AGL(1,𝔽₅)	semidirect
229. C₂×C₄×C₅	C₃×C₄⋊C₄	semidirect
233. C₂×Dic₅	C₃×C₄⋊C₄	semidirect
238. C₄×D₅	C₃×C₄⋊C₄	semidirect
239. C₄×D₅	C₃×C₄⋊C₄	semidirect
251. #117	C₃×AGL(1,𝔽₅)	semidirect
292. C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	semidirect
332. C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	semidirect
342. C₄²	C₂×C₃×AGL(1,𝔽₅)	semidirect
343. C₃×C₅	#2296	semidirect
391. C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	semidirect
401. C₅	C₃×#2296	semidirect
420. C₃	#10617128	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×#4130	2
C₂²×C₃×C₄×AGL(1,𝔽₅)	2
C₃×D₅×#117	2
C₃×#65712	2
C₃×#65699	2
C₃×C₄×C₄⋊AGL(1,𝔽₅)	2
#10617128	3
C₃×#2296	5

Subgroups — C3×#10617128 #14321093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10617128 #14321093