Subgroups — C₃×#10617178 #14327124

Order 1920.

1408 conjugacy classes of subgroups.

The subgroup lattice has 552 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	2	yes	maximal
2	C₃×C₄×D₄×D₅	C₂	1	1	yes	maximal, characteristic
3	C₃×#65765	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₃×#65712	C₂	1	2	yes	maximal, complemented
6	C₃×#65699	C₂	1	4	yes	maximal, complemented
7	C₃×#65689	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×C₄⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
9	#10617178	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂³×C₃×AGL(1,𝔽₅)	C₂²	1	4	yes	—
11	C₂×C₃×D₄×D₅	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	—
13	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	C₂²	1	4	yes	—
14	C₃×D₄×AGL(1,𝔽₅)	—	2	8	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	4	yes	—
18	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
19	C₃×C₄×C₅×D₄	C₄	1	1	yes	characteristic, complemented
20	C₃×D₄×Dic₅	C₄	1	1	yes	characteristic, complemented
21	C₃×C₄×C₂²⋊D₅	C₄	1	2	yes	complemented
22	C₃×#4039	C₄	1	1	yes	characteristic, complemented
23	C₃×D₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₃×#4027	C₄	1	2	yes	complemented
25	C₃×D₅×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₃×C₄×D₂₀	C₄	1	1	yes	characteristic, complemented
27	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
28	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	1	yes	characteristic
29	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	1	yes	characteristic
30	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	2	yes	—
31	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	2	yes	—
32	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	4	yes	—
33	C₃×#2349	—	5	1	no	maximal, Hall
34	C₂×D₄×AGL(1,𝔽₅)	C₂×C₃	1	2	yes	—
35	C₄×D₄×D₅	C₂×C₃	1	1	yes	characteristic
36	#65765	C₂×C₃	1	2	yes	complemented
37	C₂×(C₂×AGL(1,𝔽₅))⋊C₄	C₂×C₃	1	2	yes	complemented
38	#65712	C₂×C₃	1	2	yes	complemented
39	#65699	C₂×C₃	1	4	yes	complemented
40	#65689	C₂×C₃	1	1	yes	characteristic, complemented
41	C₄×C₄⋊AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂³×C₃×D₅	C₂³	1	2	yes	—
43	C₂²×C₃×AGL(1,𝔽₅)	—	2	8	no	—
44	C₂²×C₃×AGL(1,𝔽₅)	—	2	8	no	—
45	C₂²×C₃×AGL(1,𝔽₅)	—	4	4	no	—
46	C₂²×C₃×AGL(1,𝔽₅)	C₂³	1	2	yes	—
47	C₂²×C₃×AGL(1,𝔽₅)	C₂³	1	4	yes	—
48	C₂²×C₃×AGL(1,𝔽₅)	C₂³	1	2	yes	—
49	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	complemented
51	C₂×C₃×C₂²⋊D₅	C₂×C₄	1	2	yes	—
52	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	complemented
53	C₃×D₄×D₅	D₄	1	4	yes	—
54	C₃×D₄×D₅	Q₈	1	4	yes	—
55	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
56	C₂×C₃×C₄×D₅	—	4	4	no	—
57	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	complemented
58	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	complemented
59	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
60	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
63	C₃×C₂²⋊AGL(1,𝔽₅)	—	2	8	no	—
64	C₃×C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
65	C₃×C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
66	C₃×C₄⋊AGL(1,𝔽₅)	—	4	4	no	—
67	C₃×C₄×AGL(1,𝔽₅)	—	2	4	no	—
68	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic, complemented
69	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	complemented
70	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic, complemented
71	C₃×C₂²⋊Dic₅	C₂×C₄	1	2	yes	complemented
72	C₃×#685	C₂×C₄	1	2	yes	complemented
73	C₃×#685	C₂×C₄	1	2	yes	complemented
74	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic, complemented
75	C₃×#683	C₂×C₄	1	2	yes	complemented
76	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic, complemented
77	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic, complemented
78	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic, complemented
79	C₂×C₃×C₄×D₄	—	5	1	no	—
80	C₂×C₃×C₄×D₄	—	5	2	no	—
81	C₃×#389	—	5	2	no	—
82	C₃×#385	—	5	4	no	—
83	C₃×#382	—	5	1	no	—
84	C₃×#380	—	5	2	no	—
85	C₃×C₄×C₄⋊C₄	—	5	1	no	—
86	C₂×C₃×#95	—	5	2	no	—
87	C₂³×AGL(1,𝔽₅)	C₂²×C₃	1	4	yes	—
88	C₂×D₄×D₅	C₂²×C₃	1	1	yes	characteristic
89	C₂²×C₄×D₅	C₂²×C₃	1	2	yes	—
90	C₂×C₂²⋊AGL(1,𝔽₅)	C₂²×C₃	1	4	yes	—
91	D₄×AGL(1,𝔽₅)	—	2	8	no	—
92	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
93	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
94	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	4	yes	—
95	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
96	C₄×C₅×D₄	C₃×C₄	1	1	yes	characteristic, complemented
97	D₄×Dic₅	C₃×C₄	1	1	yes	characteristic, complemented
98	C₄×C₂²⋊D₅	C₃×C₄	1	2	yes	complemented
99	#4039	C₃×C₄	1	1	yes	characteristic, complemented
100	D₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
101	#4027	C₃×C₄	1	2	yes	complemented
102	D₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
103	C₄×D₂₀	C₃×C₄	1	1	yes	characteristic, complemented
104	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
105	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	1	yes	characteristic
106	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	1	yes	characteristic
107	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	2	yes	—
108	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	2	yes	—
109	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	4	yes	—
110	#2349	—	5	1	no	Sylow
111	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
112	C₂²×C₃×D₅	—	2	4	no	—
113	C₂²×C₃×D₅	—	2	4	no	—
114	C₂²×C₃×D₅	C₂²×C₄	1	2	yes	—
115	C₂²×C₃×D₅	C₂×D₄	1	4	yes	—
116	C₂²×C₃×D₅	C₂×Q₈	1	4	yes	—
117	C₂²×C₃×D₅	C₂⁴	1	1	yes	characteristic
118	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
119	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
120	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
121	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
122	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
123	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
124	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
125	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
126	C₂×C₃×AGL(1,𝔽₅)	—	4	16	no	—
127	C₂×C₃×AGL(1,𝔽₅)	#41	1	4	yes	—
128	C₃×C₅×D₄	C₄⋊C₄	1	4	yes	complemented
129	C₂×C₃×C₄×C₅	—	4	2	no	—
130	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
133	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
134	C₃×C₂²⋊D₅	C₄⋊C₄	1	8	yes	complemented
135	C₂×C₃×Dic₅	—	4	2	no	—
136	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
139	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
140	C₃×D₂₀	C₄⋊C₄	1	4	yes	complemented
141	C₃×C₄×D₅	—	2	2	no	—
142	C₃×C₄×D₅	—	2	4	no	—
143	C₃×C₄×D₅	—	2	2	no	—
144	C₃×C₄×D₅	—	4	2	no	—
145	C₃×C₄×D₅	—	4	2	no	—
146	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
147	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
148	C₂²×C₃×D₄	—	5	1	no	—
149	C₂³×C₃×C₄	—	5	2	no	—
150	C₂³×C₃×C₄	—	5	4	no	—
151	C₃×C₄×D₄	—	5	1	no	—
152	C₃×C₄×D₄	—	5	2	no	—
153	C₃×C₄×D₄	—	5	1	no	—
154	C₃×C₄×D₄	—	5	2	no	—
155	C₃×C₄×D₄	—	5	1	no	—
156	C₃×C₄×D₄	—	10	8	no	—
157	C₃×C₄×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
158	C₂×C₃×C₄⋊C₄	—	5	1	no	—
159	C₂×C₃×C₄⋊C₄	—	5	2	no	—
160	C₂×C₃×C₄⋊C₄	—	5	2	no	—
161	C₂×C₃×C₄⋊C₄	—	5	4	no	—
162	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
163	C₂×C₃×C₂²⋊C₄	—	5	4	no	—
164	C₂×C₃×C₄²	—	5	1	no	—
165	C₂×C₃×C₄²	—	5	2	no	—
166	C₃×#95	—	5	1	no	—
167	C₃×#95	—	5	2	no	—
168	C₃×#95	—	5	4	no	—
169	C₃×#95	—	5	1	no	—
170	C₃×#95	—	5	2	no	—
171	C₂³×D₅	C₂³×C₃	1	2	yes	—
172	C₂²×AGL(1,𝔽₅)	—	2	8	no	—
173	C₂²×AGL(1,𝔽₅)	—	2	8	no	—
174	C₂²×AGL(1,𝔽₅)	—	4	4	no	—
175	C₂²×AGL(1,𝔽₅)	C₂³×C₃	1	2	yes	—
176	C₂²×AGL(1,𝔽₅)	C₂³×C₃	1	4	yes	—
177	C₂²×AGL(1,𝔽₅)	C₂³×C₃	1	2	yes	—
178	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
179	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	complemented
180	C₂×C₂²⋊D₅	C₂×C₃×C₄	1	2	yes	—
181	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	complemented
182	D₄×D₅	C₃×D₄	1	4	yes	—
183	D₄×D₅	C₃×Q₈	1	4	yes	—
184	C₂×D₂₀	C₂×C₃×C₄	1	1	yes	characteristic
185	C₂×C₄×D₅	—	4	4	no	—
186	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	complemented
187	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	complemented
188	C₂×C₄×D₅	C₂³×C₃	1	2	yes	—
189	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
190	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
191	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
192	C₂²⋊AGL(1,𝔽₅)	—	2	8	no	—
193	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
194	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
195	C₄⋊AGL(1,𝔽₅)	—	4	4	no	—
196	C₄×AGL(1,𝔽₅)	—	2	4	no	—
197	C₅×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic, complemented
198	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	complemented
199	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
200	C₂²⋊Dic₅	C₂×C₃×C₄	1	2	yes	complemented
201	#685	C₂×C₃×C₄	1	2	yes	complemented
202	#685	C₂×C₃×C₄	1	2	yes	complemented
203	C₄⋊Dic₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
204	#683	C₂×C₃×C₄	1	2	yes	complemented
205	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
206	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
207	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
208	C₂×C₄×D₄	—	5	1	no	—
209	C₂×C₄×D₄	—	5	2	no	—
210	#389	—	5	2	no	—
211	#385	—	5	4	no	—
212	#382	—	5	1	no	—
213	#380	—	5	2	no	—
214	C₄×C₄⋊C₄	—	5	1	no	—
215	C₂×#95	—	5	2	no	—
216	C₂²×C₃×C₅	—	2	2	no	—
217	C₂²×C₃×C₅	—	2	2	no	—
218	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	4	yes	—
219	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
220	C₂×C₃×D₅	—	2	2	no	—
221	C₂×C₃×D₅	—	2	2	no	—
222	C₂×C₃×D₅	—	2	8	no	—
223	C₂×C₃×D₅	—	2	8	no	—
224	C₂×C₃×D₅	C₂×C₄⋊C₄	1	4	yes	—
225	C₂×C₃×D₅	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₃×D₅	C₂²×Q₈	1	1	yes	characteristic
227	C₂×C₃×D₅	C₂×#41	1	2	yes	—
228	C₂×C₃×D₅	H(5,𝔽₂)	1	2	yes	—
229	C₃×AGL(1,𝔽₅)	—	2	4	no	—
230	C₃×AGL(1,𝔽₅)	—	4	2	no	—
231	C₃×AGL(1,𝔽₅)	—	4	4	no	—
232	C₃×C₄×C₅	—	2	1	no	—
233	C₃×C₄×C₅	—	2	1	no	—
234	C₃×C₄×C₅	—	2	2	no	—
235	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
236	C₃×Dic₅	—	2	1	no	—
237	C₃×Dic₅	—	2	1	no	—
238	C₃×Dic₅	—	2	2	no	—
239	C₃×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
240	C₂⁴×C₃	—	5	2	no	—
241	C₂×C₃×D₄	—	5	2	no	—
242	C₂×C₃×D₄	—	5	4	no	—
243	C₂×C₃×D₄	—	5	4	no	—
244	C₂×C₃×D₄	—	5	1	no	—
245	C₂×C₃×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
246	C₂²×C₃×C₄	—	5	1	no	—
247	C₂²×C₃×C₄	—	5	2	no	—
248	C₂²×C₃×C₄	—	5	2	no	—
249	C₂²×C₃×C₄	—	5	1	no	—
250	C₂²×C₃×C₄	—	5	1	no	—
251	C₂²×C₃×C₄	—	5	2	no	—
252	C₂²×C₃×C₄	—	5	2	no	—
253	C₂²×C₃×C₄	—	5	2	no	—
254	C₂²×C₃×C₄	—	5	4	no	—
255	C₂²×C₃×C₄	—	5	2	no	—
256	C₂²×C₃×C₄	—	10	8	no	—
257	C₂²×C₃×C₄	—	10	8	no	—
258	C₂²×C₃×C₄	—	20	4	no	—
259	C₂²×C₃×C₄	—	20	4	no	—
260	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
261	C₃×C₄⋊C₄	—	5	2	no	—
262	C₃×C₄⋊C₄	—	5	1	no	—
263	C₃×C₄⋊C₄	—	10	4	no	—
264	C₃×C₄⋊C₄	—	10	4	no	—
265	C₃×C₄⋊C₄	—	20	4	no	—
266	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
267	C₃×C₂²⋊C₄	—	5	2	no	—
268	C₃×C₂²⋊C₄	—	5	2	no	—
269	C₃×C₂²⋊C₄	—	5	2	no	—
270	C₃×C₂²⋊C₄	—	10	8	no	—
271	C₃×C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
272	C₃×C₄²	—	5	1	no	—
273	C₃×C₄²	—	5	1	no	—
274	C₃×C₄²	—	5	1	no	—
275	C₃×C₄²	—	10	4	no	—
276	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
277	C₂³×C₅	C₂²×C₃×C₄	1	2	yes	—
278	C₂²×D₅	—	2	4	no	—
279	C₂²×D₅	—	2	4	no	—
280	C₂²×D₅	C₂²×C₃×C₄	1	2	yes	—
281	C₂²×D₅	C₂×C₃×D₄	1	4	yes	—
282	C₂²×D₅	C₂×C₃×Q₈	1	4	yes	—
283	C₂²×D₅	C₂⁴×C₃	1	1	yes	characteristic
284	C₂×AGL(1,𝔽₅)	—	2	2	no	—
285	C₂×AGL(1,𝔽₅)	—	2	2	no	—
286	C₂×AGL(1,𝔽₅)	—	2	2	no	—
287	C₂×AGL(1,𝔽₅)	—	2	4	no	—
288	C₂×AGL(1,𝔽₅)	—	2	2	no	—
289	C₂×AGL(1,𝔽₅)	—	2	4	no	—
290	C₂×AGL(1,𝔽₅)	—	2	2	no	—
291	C₂×AGL(1,𝔽₅)	—	2	4	no	—
292	C₂×AGL(1,𝔽₅)	—	4	16	no	—
293	C₂×AGL(1,𝔽₅)	C₃×#41	1	4	yes	—
294	C₅×D₄	C₃×C₄⋊C₄	1	4	yes	complemented
295	C₂×C₄×C₅	—	4	2	no	—
296	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
297	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
298	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
299	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
300	C₂²⋊D₅	C₃×C₄⋊C₄	1	8	yes	complemented
301	C₂×Dic₅	—	4	2	no	—
302	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
303	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
304	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
305	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
306	D₂₀	C₃×C₄⋊C₄	1	4	yes	complemented
307	C₄×D₅	—	2	2	no	—
308	C₄×D₅	—	2	4	no	—
309	C₄×D₅	—	2	2	no	—
310	C₄×D₅	—	4	2	no	—
311	C₄×D₅	—	4	2	no	—
312	C₄×D₅	C₂×C₃×D₄	1	2	yes	—
313	C₄×D₅	C₂×C₃×Q₈	1	2	yes	—
314	C₂²×D₄	—	5	1	no	—
315	C₂³×C₄	—	5	2	no	—
316	C₂³×C₄	—	5	4	no	—
317	C₄×D₄	—	5	1	no	—
318	C₄×D₄	—	5	2	no	—
319	C₄×D₄	—	5	1	no	—
320	C₄×D₄	—	5	2	no	—
321	C₄×D₄	—	5	1	no	—
322	C₄×D₄	—	10	8	no	—
323	C₄×D₄	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
324	C₂×C₄⋊C₄	—	5	1	no	—
325	C₂×C₄⋊C₄	—	5	2	no	—
326	C₂×C₄⋊C₄	—	5	2	no	—
327	C₂×C₄⋊C₄	—	5	4	no	—
328	C₂×C₂²⋊C₄	—	5	2	no	—
329	C₂×C₂²⋊C₄	—	5	4	no	—
330	C₂×C₄²	—	5	1	no	—
331	C₂×C₄²	—	5	2	no	—
332	#95	—	5	1	no	—
333	#95	—	5	2	no	—
334	#95	—	5	4	no	—
335	#95	—	5	1	no	—
336	#95	—	5	2	no	—
337	C₂×C₃×C₅	—	2	4	no	—
338	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
339	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
340	C₂×C₃×C₅	#518	1	1	yes	characteristic
341	C₃×D₅	—	2	4	no	—
342	C₃×D₅	#546	1	2	yes	—
343	C₃×D₅	#554	1	2	yes	—
344	C₂³×C₃	—	5	1	no	—
345	C₂³×C₃	—	5	4	no	—
346	C₂³×C₃	—	5	4	no	—
347	C₂³×C₃	—	5	2	no	—
348	C₂³×C₃	—	10	4	no	—
349	C₂³×C₃	—	10	4	no	—
350	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	2	yes	—
351	C₃×D₄	—	5	8	no	—
352	C₃×D₄	—	5	4	no	—
353	C₃×D₄	C₄⋊AGL(1,𝔽₅)	1	4	yes	complemented
354	C₂×C₃×C₄	—	4	2	no	—
355	C₂×C₃×C₄	—	5	1	no	—
356	C₂×C₃×C₄	—	5	2	no	—
357	C₂×C₃×C₄	—	5	2	no	—
358	C₂×C₃×C₄	—	5	4	no	—
359	C₂×C₃×C₄	—	5	1	no	—
360	C₂×C₃×C₄	—	5	1	no	—
361	C₂×C₃×C₄	—	5	2	no	—
362	C₂×C₃×C₄	—	10	2	no	—
363	C₂×C₃×C₄	—	10	4	no	—
364	C₂×C₃×C₄	—	10	2	no	—
365	C₂×C₃×C₄	—	10	2	no	—
366	C₂×C₃×C₄	—	10	2	no	—
367	C₂×C₃×C₄	—	10	2	no	—
368	C₂×C₃×C₄	—	10	4	no	—
369	C₂×C₃×C₄	—	10	2	no	—
370	C₂×C₃×C₄	—	10	2	no	—
371	C₂×C₃×C₄	—	10	4	no	—
372	C₂×C₃×C₄	—	10	4	no	—
373	C₂×C₃×C₄	—	20	2	no	—
374	C₂×C₃×C₄	—	20	2	no	—
375	C₂×C₃×C₄	—	20	2	no	—
376	C₂×C₃×C₄	—	20	16	no	—
377	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
378	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
379	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
380	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
381	C₂²×C₅	—	2	2	no	—
382	C₂²×C₅	—	2	2	no	—
383	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	4	yes	—
384	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
385	C₂×D₅	—	2	2	no	—
386	C₂×D₅	—	2	2	no	—
387	C₂×D₅	—	2	8	no	—
388	C₂×D₅	—	2	8	no	—
389	C₂×D₅	C₂×C₃×C₄⋊C₄	1	4	yes	—
390	C₂×D₅	C₂²×C₃×D₄	1	1	yes	characteristic
391	C₂×D₅	C₂²×C₃×Q₈	1	1	yes	characteristic
392	C₂×D₅	C₂×C₃×#41	1	2	yes	—
393	C₂×D₅	C₃×H(5,𝔽₂)	1	2	yes	—
394	AGL(1,𝔽₅)	—	2	4	no	—
395	AGL(1,𝔽₅)	—	4	2	no	—
396	AGL(1,𝔽₅)	—	4	4	no	—
397	C₄×C₅	—	2	1	no	—
398	C₄×C₅	—	2	1	no	—
399	C₄×C₅	—	2	2	no	—
400	C₄×C₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
401	Dic₅	—	2	1	no	—
402	Dic₅	—	2	1	no	—
403	Dic₅	—	2	2	no	—
404	Dic₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
405	C₂⁴	—	5	2	no	—
406	C₂×D₄	—	5	2	no	—
407	C₂×D₄	—	5	4	no	—
408	C₂×D₄	—	5	4	no	—
409	C₂×D₄	—	5	1	no	—
410	C₂×D₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
411	C₂²×C₄	—	5	1	no	—
412	C₂²×C₄	—	5	2	no	—
413	C₂²×C₄	—	5	2	no	—
414	C₂²×C₄	—	5	1	no	—
415	C₂²×C₄	—	5	1	no	—
416	C₂²×C₄	—	5	2	no	—
417	C₂²×C₄	—	5	2	no	—
418	C₂²×C₄	—	5	2	no	—
419	C₂²×C₄	—	5	4	no	—
420	C₂²×C₄	—	5	2	no	—
421	C₂²×C₄	—	10	8	no	—
422	C₂²×C₄	—	10	8	no	—
423	C₂²×C₄	—	20	4	no	—
424	C₂²×C₄	—	20	4	no	—
425	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
426	C₄⋊C₄	—	5	2	no	—
427	C₄⋊C₄	—	5	1	no	—
428	C₄⋊C₄	—	10	4	no	—
429	C₄⋊C₄	—	10	4	no	—
430	C₄⋊C₄	—	20	4	no	—
431	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
432	C₂²⋊C₄	—	5	2	no	—
433	C₂²⋊C₄	—	5	2	no	—
434	C₂²⋊C₄	—	5	2	no	—
435	C₂²⋊C₄	—	10	8	no	—
436	C₂²⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
437	C₄²	—	5	1	no	—
438	C₄²	—	5	1	no	—
439	C₄²	—	5	1	no	—
440	C₄²	—	10	4	no	—
441	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
442	C₃×C₅	#2349	1	1	yes	characteristic, complemented, Hall
443	C₂²×C₃	—	2	2	no	—
444	C₂²×C₃	—	2	2	no	—
445	C₂²×C₃	—	5	1	no	—
446	C₂²×C₃	—	5	1	no	—
447	C₂²×C₃	—	5	2	no	—
448	C₂²×C₃	—	5	2	no	—
449	C₂²×C₃	—	5	4	no	—
450	C₂²×C₃	—	10	8	no	—
451	C₂²×C₃	—	10	8	no	—
452	C₂²×C₃	—	10	2	no	—
453	C₂²×C₃	—	10	2	no	—
454	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	4	yes	—
455	C₂²×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
456	C₃×C₄	—	2	1	no	—
457	C₃×C₄	—	2	2	no	—
458	C₃×C₄	—	2	1	no	—
459	C₃×C₄	—	5	2	no	—
460	C₃×C₄	—	10	4	no	—
461	C₃×C₄	—	10	2	no	—
462	C₃×C₄	—	10	1	no	—
463	C₃×C₄	—	10	1	no	—
464	C₃×C₄	—	20	2	no	—
465	C₃×C₄	—	20	4	no	—
466	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
467	C₂×C₅	—	2	4	no	—
468	C₂×C₅	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
469	C₂×C₅	C₃×C₄×#41	1	1	yes	characteristic
470	C₂×C₅	C₃×#518	1	1	yes	characteristic
471	D₅	—	2	4	no	—
472	D₅	C₃×#546	1	2	yes	—
473	D₅	C₃×#554	1	2	yes	—
474	C₂³	—	5	1	no	—
475	C₂³	—	5	4	no	—
476	C₂³	—	5	4	no	—
477	C₂³	—	5	2	no	—
478	C₂³	—	10	4	no	—
479	C₂³	—	10	4	no	—
480	C₂³	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
481	D₄	—	5	8	no	—
482	D₄	—	5	4	no	—
483	D₄	C₃×C₄⋊AGL(1,𝔽₅)	1	4	yes	complemented
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	5	1	no	—
486	C₂×C₄	—	5	2	no	—
487	C₂×C₄	—	5	2	no	—
488	C₂×C₄	—	5	4	no	—
489	C₂×C₄	—	5	1	no	—
490	C₂×C₄	—	5	1	no	—
491	C₂×C₄	—	5	2	no	—
492	C₂×C₄	—	10	2	no	—
493	C₂×C₄	—	10	4	no	—
494	C₂×C₄	—	10	2	no	—
495	C₂×C₄	—	10	2	no	—
496	C₂×C₄	—	10	2	no	—
497	C₂×C₄	—	10	2	no	—
498	C₂×C₄	—	10	4	no	—
499	C₂×C₄	—	10	2	no	—
500	C₂×C₄	—	10	2	no	—
501	C₂×C₄	—	10	4	no	—
502	C₂×C₄	—	10	4	no	—
503	C₂×C₄	—	20	2	no	—
504	C₂×C₄	—	20	2	no	—
505	C₂×C₄	—	20	2	no	—
506	C₂×C₄	—	20	16	no	—
507	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
508	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
509	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
510	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
511	C₂×C₃	—	2	4	no	—
512	C₂×C₃	—	5	2	no	—
513	C₂×C₃	—	5	2	no	—
514	C₂×C₃	—	10	4	no	—
515	C₂×C₃	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
516	C₂×C₃	#66258	1	1	yes	characteristic
517	C₂×C₃	#41×AGL(1,𝔽₅)	1	1	yes	characteristic
518	C₅	C₃×#2349	1	1	yes	characteristic, complemented, Sylow
519	C₂²	—	2	2	no	—
520	C₂²	—	2	2	no	—
521	C₂²	—	5	1	no	—
522	C₂²	—	5	1	no	—
523	C₂²	—	5	2	no	—
524	C₂²	—	5	2	no	—
525	C₂²	—	5	4	no	—
526	C₂²	—	10	8	no	—
527	C₂²	—	10	8	no	—
528	C₂²	—	10	2	no	—
529	C₂²	—	10	2	no	—
530	C₂²	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	4	yes	—
531	C₂²	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
532	C₄	—	2	1	no	—
533	C₄	—	2	2	no	—
534	C₄	—	2	1	no	—
535	C₄	—	5	2	no	—
536	C₄	—	10	4	no	—
537	C₄	—	10	1	no	—
538	C₄	—	10	2	no	—
539	C₄	—	10	1	no	—
540	C₄	—	20	4	no	—
541	C₄	—	20	2	no	—
542	C₄	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
543	C₃	#10617178	1	1	yes	characteristic, direct product factor, Sylow
544	C₂	—	2	4	no	—
545	C₂	—	5	2	no	—
546	C₂	—	5	2	no	—
547	C₂	—	10	4	no	—
548	C₂	C₂²×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
549	C₂	C₃×#66258	1	1	yes	characteristic
550	C₂	C₃×#41×AGL(1,𝔽₅)	1	1	yes	characteristic

	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×C₄×C₅×D₄	C₄
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂×C₃×D₄×D₅	C₂²
cosocle	C₂²×D₅	C₂⁴×C₃
hypercenter	C₃×C₄×D₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2349
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
3. C₃×#65765	C₂	semidirect
4. C₂×C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂	semidirect
5. C₃×#65712	C₂	semidirect
6. C₃×#65699	C₂	semidirect
7. C₃×#65689	C₂	semidirect
8. C₃×C₄×C₄⋊AGL(1,𝔽₅)	C₂	semidirect
9. #10617178	C₃	direct (N × H)
19. C₃×C₄×C₅×D₄	C₄	semidirect
20. C₃×D₄×Dic₅	C₄	semidirect
21. C₃×C₄×C₂²⋊D₅	C₄	semidirect
22. C₃×#4039	C₄	semidirect
24. C₃×#4027	C₄	semidirect
26. C₃×C₄×D₂₀	C₄	semidirect
36. #65765	C₂×C₃	semidirect
37. C₂×(C₂×AGL(1,𝔽₅))⋊C₄	C₂×C₃	semidirect
38. #65712	C₂×C₃	semidirect
39. #65699	C₂×C₃	semidirect
40. #65689	C₂×C₃	semidirect
41. C₄×C₄⋊AGL(1,𝔽₅)	C₂×C₃	semidirect
50. C₂²×C₃×C₄×C₅	C₂×C₄	semidirect
52. C₂²×C₃×Dic₅	C₂×C₄	semidirect
57. C₂×C₃×C₄×D₅	C₂×C₄	semidirect
58. C₂×C₃×C₄×D₅	C₂×C₄	semidirect
68. C₃×C₅×C₄⋊C₄	C₂×C₄	semidirect
69. C₃×C₅×C₂²⋊C₄	C₂×C₄	semidirect
70. C₃×C₄²×C₅	C₂×C₄	semidirect
71. C₃×C₂²⋊Dic₅	C₂×C₄	semidirect
72. C₃×#685	C₂×C₄	semidirect
73. C₃×#685	C₂×C₄	semidirect
74. C₃×C₄⋊Dic₅	C₂×C₄	semidirect
75. C₃×#683	C₂×C₄	semidirect
76. C₃×C₄×Dic₅	C₂×C₄	semidirect
77. C₃×C₄×Dic₅	C₂×C₄	semidirect
78. C₃×C₄×Dic₅	C₂×C₄	semidirect
96. C₄×C₅×D₄	C₃×C₄	semidirect
97. D₄×Dic₅	C₃×C₄	semidirect
98. C₄×C₂²⋊D₅	C₃×C₄	semidirect
99. #4039	C₃×C₄	semidirect
101. #4027	C₃×C₄	semidirect
103. C₄×D₂₀	C₃×C₄	semidirect
128. C₃×C₅×D₄	C₄⋊C₄	semidirect
134. C₃×C₂²⋊D₅	C₄⋊C₄	semidirect
140. C₃×D₂₀	C₄⋊C₄	semidirect
157. C₃×C₄×D₄	AGL(1,𝔽₅)	semidirect
179. C₂²×C₄×C₅	C₂×C₃×C₄	semidirect
181. C₂²×Dic₅	C₂×C₃×C₄	semidirect
186. C₂×C₄×D₅	C₂×C₃×C₄	semidirect
187. C₂×C₄×D₅	C₂×C₃×C₄	semidirect
197. C₅×C₄⋊C₄	C₂×C₃×C₄	semidirect
198. C₅×C₂²⋊C₄	C₂×C₃×C₄	semidirect
199. C₄²×C₅	C₂×C₃×C₄	semidirect
200. C₂²⋊Dic₅	C₂×C₃×C₄	semidirect
201. #685	C₂×C₃×C₄	semidirect
202. #685	C₂×C₃×C₄	semidirect
203. C₄⋊Dic₅	C₂×C₃×C₄	semidirect
204. #683	C₂×C₃×C₄	semidirect
205. C₄×Dic₅	C₂×C₃×C₄	semidirect
206. C₄×Dic₅	C₂×C₃×C₄	semidirect
207. C₄×Dic₅	C₂×C₃×C₄	semidirect
260. C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	semidirect
266. C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	semidirect
271. C₃×C₂²⋊C₄	C₂×AGL(1,𝔽₅)	semidirect
276. C₃×C₄²	C₂×AGL(1,𝔽₅)	semidirect
294. C₅×D₄	C₃×C₄⋊C₄	semidirect
300. C₂²⋊D₅	C₃×C₄⋊C₄	semidirect
306. D₂₀	C₃×C₄⋊C₄	semidirect
323. C₄×D₄	C₃×AGL(1,𝔽₅)	semidirect
353. C₃×D₄	C₄⋊AGL(1,𝔽₅)	semidirect
425. C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	semidirect
431. C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	semidirect
436. C₂²⋊C₄	C₂×C₃×AGL(1,𝔽₅)	semidirect
441. C₄²	C₂×C₃×AGL(1,𝔽₅)	semidirect
442. C₃×C₅	#2349	semidirect
483. D₄	C₃×C₄⋊AGL(1,𝔽₅)	semidirect
518. C₅	C₃×#2349	semidirect
543. C₃	#10617178	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×C₄×D₄×D₅	2
C₃×#65765	2
C₂×C₃×(C₂×AGL(1,𝔽₅))⋊C₄	2
C₃×#65712	2
C₃×#65699	2
C₃×#65689	2
C₃×C₄×C₄⋊AGL(1,𝔽₅)	2
#10617178	3
C₃×#2349	5

Subgroups — C3×#10617178 #14327124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10617178 #14327124