Subgroups — C₃×#10617213 #14327792

Order 1920.

960 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₃×(C₂×AGL(1,𝔽₅))⋊Q₈	C₂	1	2	yes	maximal
3	C₃×#65711	C₂	1	4	yes	maximal
4	C₃×C₄⋊C₄×AGL(1,𝔽₅)	C₂	1	2	yes	maximal
5	C₃×#65689	C₂	1	2	yes	maximal
6	C₃×#65688	C₂	1	1	yes	maximal, characteristic
7	C₃×C₄×C₄⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
8	C₃×C₄×C₄⋊AGL(1,𝔽₅)	C₂	1	2	yes	maximal
9	#10617213	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×Q₈×D₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
17	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	4	yes	—
18	C₃×C₄×C₅×Q₈	C₄	1	1	yes	characteristic, complemented
19	C₃×Q₈×Dic₅	C₄	1	1	yes	characteristic, complemented
20	C₃×D₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×D₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×#4033	C₄	1	2	yes	complemented
23	C₃×#4033	C₄	1	1	yes	characteristic, complemented
24	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
25	C₃×C₄²×D₅	C₂²	1	2	yes	—
26	C₃×C₄×Dic₁₀	C₄	1	1	yes	characteristic, complemented
27	C₃×C₄×Dic₁₀	C₄	1	2	yes	complemented
28	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	2	yes	—
29	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	4	yes	—
30	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	4	yes	—
31	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	4	yes	—
32	C₃×#2355	—	5	1	no	maximal, Hall
33	C₄×Q₈×D₅	C₂×C₃	1	1	yes	characteristic
34	(C₂×AGL(1,𝔽₅))⋊Q₈	C₂×C₃	1	2	yes	—
35	#65711	C₂×C₃	1	4	yes	—
36	C₄⋊C₄×AGL(1,𝔽₅)	C₂×C₃	1	2	yes	—
37	#65689	C₂×C₃	1	2	yes	—
38	#65688	C₂×C₃	1	1	yes	characteristic
39	C₄×C₄⋊AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic
40	C₄×C₄⋊AGL(1,𝔽₅)	C₂×C₃	1	2	yes	—
41	C₂²×C₃×AGL(1,𝔽₅)	C₂³	1	4	yes	—
42	C₂²×C₃×AGL(1,𝔽₅)	C₂³	1	2	yes	—
43	C₂²×C₃×AGL(1,𝔽₅)	C₂³	1	2	yes	—
44	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
45	C₃×Q₈×D₅	—	2	2	no	—
46	C₃×Q₈×D₅	—	2	2	no	—
47	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
48	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
53	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
54	C₃×C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
55	C₃×C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
56	C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
57	C₃×C₄⋊AGL(1,𝔽₅)	—	4	2	no	—
58	C₃×C₄⋊AGL(1,𝔽₅)	Q₈	1	4	yes	complemented
59	C₃×C₄×AGL(1,𝔽₅)	—	2	4	no	—
60	C₃×C₄×AGL(1,𝔽₅)	—	4	4	no	—
61	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
62	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
63	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
64	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
65	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
66	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
67	C₃×#683	C₂×C₄	1	2	yes	—
68	C₃×#683	C₂×C₄	1	2	yes	—
69	C₃×#683	C₂×C₄	1	2	yes	—
70	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
71	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
72	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
73	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
74	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
75	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×Q₈	—	5	1	no	—
77	C₃×#391	—	5	2	no	—
78	C₃×#387	—	5	4	no	—
79	C₃×#384	—	5	1	no	—
80	C₃×#382	—	5	2	no	—
81	C₃×C₄×C₄⋊C₄	—	5	1	no	—
82	C₃×C₄×C₄⋊C₄	—	5	2	no	—
83	C₃×C₄×C₄⋊C₄	—	5	2	no	—
84	C₂×Q₈×D₅	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
86	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
88	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
89	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
91	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	4	yes	—
92	C₄×C₅×Q₈	C₃×C₄	1	1	yes	characteristic, complemented
93	Q₈×Dic₅	C₃×C₄	1	1	yes	characteristic, complemented
94	D₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
95	D₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
96	#4033	C₃×C₄	1	1	yes	characteristic, complemented
97	#4033	C₃×C₄	1	2	yes	complemented
98	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
99	C₄²×D₅	C₂²×C₃	1	2	yes	—
100	C₄×Dic₁₀	C₃×C₄	1	1	yes	characteristic, complemented
101	C₄×Dic₁₀	C₃×C₄	1	2	yes	complemented
102	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	2	yes	—
103	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	4	yes	—
104	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	4	yes	—
105	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	4	yes	—
106	#2355	—	5	1	no	Sylow
107	C₂²×C₃×D₅	C₂⁴	1	1	yes	characteristic
108	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
109	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
110	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
111	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
112	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
113	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
114	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
115	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
116	C₂×C₃×AGL(1,𝔽₅)	C₂×Q₈	1	4	yes	—
117	C₃×C₅×Q₈	—	2	2	no	—
118	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
119	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
121	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
124	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
126	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
128	C₃×C₄×D₅	—	2	2	no	—
129	C₃×C₄×D₅	—	2	2	no	—
130	C₃×C₄×D₅	—	2	2	no	—
131	C₃×C₄×D₅	—	2	1	no	—
132	C₃×C₄×D₅	—	2	2	no	—
133	C₃×C₄×D₅	—	2	1	no	—
134	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
135	C₃×C₄×D₅	#41	1	2	yes	—
136	C₃×C₄×D₅	#41	1	2	yes	—
137	C₃×C₄×D₅	#41	1	2	yes	—
138	C₃×Dic₁₀	—	2	2	no	—
139	C₃×Dic₁₀	—	2	4	no	—
140	C₂²×C₃×Q₈	—	5	1	no	—
141	C₃×C₄×Q₈	—	5	1	no	—
142	C₃×C₄×Q₈	—	5	2	no	—
143	C₃×C₄×Q₈	—	5	1	no	—
144	C₃×C₄×Q₈	—	5	2	no	—
145	C₃×C₄×Q₈	—	5	1	no	—
146	C₃×C₄×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
147	C₂×C₃×C₄⋊C₄	—	5	1	no	—
148	C₂×C₃×C₄⋊C₄	—	5	1	no	—
149	C₂×C₃×C₄⋊C₄	—	5	2	no	—
150	C₂×C₃×C₄⋊C₄	—	5	1	no	—
151	C₂×C₃×C₄⋊C₄	—	5	2	no	—
152	C₂×C₃×C₄⋊C₄	—	5	2	no	—
153	C₂×C₃×C₄⋊C₄	—	5	2	no	—
154	C₂×C₃×C₄²	—	5	2	no	—
155	C₂×C₃×C₄²	—	5	1	no	—
156	C₂×C₃×C₄²	—	5	2	no	—
157	C₂×C₃×C₄²	—	5	4	no	—
158	C₃×#95	—	5	4	no	—
159	C₃×#95	—	5	2	no	—
160	C₃×#95	—	5	4	no	—
161	C₃×#95	—	5	4	no	—
162	C₂²×AGL(1,𝔽₅)	C₂³×C₃	1	4	yes	—
163	C₂²×AGL(1,𝔽₅)	C₂³×C₃	1	2	yes	—
164	C₂²×AGL(1,𝔽₅)	C₂³×C₃	1	2	yes	—
165	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
166	Q₈×D₅	—	2	2	no	—
167	Q₈×D₅	—	2	2	no	—
168	C₂×C₄×D₅	C₂³×C₃	1	2	yes	—
169	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
170	C₂×C₄×D₅	C₂³×C₃	1	2	yes	—
171	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
172	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
173	C₂×Dic₁₀	C₂×C₃×C₄	1	1	yes	characteristic
174	C₂×Dic₁₀	C₂×C₃×C₄	1	2	yes	—
175	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
176	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
177	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
178	C₄⋊AGL(1,𝔽₅)	—	4	2	no	—
179	C₄⋊AGL(1,𝔽₅)	C₃×Q₈	1	4	yes	complemented
180	C₄×AGL(1,𝔽₅)	—	2	4	no	—
181	C₄×AGL(1,𝔽₅)	—	4	4	no	—
182	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
183	C₅×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
184	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
185	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
186	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
187	C₄⋊Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
188	#683	C₂×C₃×C₄	1	2	yes	—
189	#683	C₂×C₃×C₄	1	2	yes	—
190	#683	C₂×C₃×C₄	1	2	yes	—
191	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
192	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
193	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
194	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
195	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
196	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
197	C₂×C₄×Q₈	—	5	1	no	—
198	#391	—	5	2	no	—
199	#387	—	5	4	no	—
200	#384	—	5	1	no	—
201	#382	—	5	2	no	—
202	C₄×C₄⋊C₄	—	5	1	no	—
203	C₄×C₄⋊C₄	—	5	2	no	—
204	C₄×C₄⋊C₄	—	5	2	no	—
205	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
206	C₂×C₃×D₅	C₂²×Q₈	1	1	yes	characteristic
207	C₂×C₃×D₅	C₂×#41	1	1	yes	characteristic
208	C₂×C₃×D₅	C₂×#41	1	1	yes	characteristic
209	C₂×C₃×D₅	C₂×#41	1	1	yes	characteristic
210	C₂×C₃×D₅	H(5,𝔽₂)	1	1	yes	characteristic
211	C₂×C₃×D₅	2₋¹⁺⁴	1	1	yes	characteristic
212	C₃×AGL(1,𝔽₅)	—	2	4	no	—
213	C₃×AGL(1,𝔽₅)	—	4	2	no	—
214	C₃×AGL(1,𝔽₅)	—	4	4	no	—
215	C₃×C₄×C₅	—	2	2	no	—
216	C₃×C₄×C₅	—	2	2	no	—
217	C₃×C₄×C₅	—	2	1	no	—
218	C₃×C₄×C₅	#117	1	2	yes	—
219	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
220	C₃×Dic₅	—	2	2	no	—
221	C₃×Dic₅	—	2	2	no	—
222	C₃×Dic₅	—	2	1	no	—
223	C₃×Dic₅	#117	1	2	yes	—
224	C₃×Dic₅	C₄×Q₈	1	2	yes	—
225	C₂×C₃×Q₈	—	5	2	no	—
226	C₂×C₃×Q₈	—	5	1	no	—
227	C₂×C₃×Q₈	—	10	2	no	—
228	C₂×C₃×Q₈	—	10	2	no	—
229	C₂×C₃×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
230	C₂²×C₃×C₄	—	5	1	no	—
231	C₂²×C₃×C₄	—	5	1	no	—
232	C₂²×C₃×C₄	—	5	2	no	—
233	C₂²×C₃×C₄	—	5	2	no	—
234	C₂²×C₃×C₄	—	5	2	no	—
235	C₂²×C₃×C₄	—	5	1	no	—
236	C₂²×C₃×C₄	—	5	2	no	—
237	C₂²×C₃×C₄	—	5	4	no	—
238	C₃×C₄⋊C₄	—	5	4	no	—
239	C₃×C₄⋊C₄	—	5	2	no	—
240	C₃×C₄⋊C₄	—	5	2	no	—
241	C₃×C₄⋊C₄	—	5	2	no	—
242	C₃×C₄⋊C₄	—	5	2	no	—
243	C₃×C₄⋊C₄	—	5	1	no	—
244	C₃×C₄⋊C₄	—	10	4	no	—
245	C₃×C₄⋊C₄	—	10	2	no	—
246	C₃×C₄⋊C₄	—	10	4	no	—
247	C₃×C₄⋊C₄	—	20	2	no	—
248	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
249	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
250	C₃×C₄²	—	5	1	no	—
251	C₃×C₄²	—	5	2	no	—
252	C₃×C₄²	—	5	1	no	—
253	C₃×C₄²	—	5	2	no	—
254	C₃×C₄²	—	5	1	no	—
255	C₃×C₄²	—	5	2	no	—
256	C₃×C₄²	—	10	4	no	—
257	C₃×C₄²	—	20	4	no	—
258	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	2	yes	—
259	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
260	C₂²×D₅	C₂⁴×C₃	1	1	yes	characteristic
261	C₂×AGL(1,𝔽₅)	—	2	2	no	—
262	C₂×AGL(1,𝔽₅)	—	2	2	no	—
263	C₂×AGL(1,𝔽₅)	—	2	2	no	—
264	C₂×AGL(1,𝔽₅)	—	2	4	no	—
265	C₂×AGL(1,𝔽₅)	—	2	4	no	—
266	C₂×AGL(1,𝔽₅)	—	2	2	no	—
267	C₂×AGL(1,𝔽₅)	—	2	4	no	—
268	C₂×AGL(1,𝔽₅)	—	2	2	no	—
269	C₂×AGL(1,𝔽₅)	C₂×C₃×Q₈	1	4	yes	—
270	C₅×Q₈	—	2	2	no	—
271	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
272	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
273	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
274	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
275	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
276	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
277	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
278	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
279	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
280	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
281	C₄×D₅	—	2	2	no	—
282	C₄×D₅	—	2	2	no	—
283	C₄×D₅	—	2	2	no	—
284	C₄×D₅	—	2	1	no	—
285	C₄×D₅	—	2	2	no	—
286	C₄×D₅	—	2	1	no	—
287	C₄×D₅	C₂×C₃×Q₈	1	2	yes	—
288	C₄×D₅	C₃×#41	1	2	yes	—
289	C₄×D₅	C₃×#41	1	2	yes	—
290	C₄×D₅	C₃×#41	1	2	yes	—
291	Dic₁₀	—	2	2	no	—
292	Dic₁₀	—	2	4	no	—
293	C₂²×Q₈	—	5	1	no	—
294	C₄×Q₈	—	5	1	no	—
295	C₄×Q₈	—	5	2	no	—
296	C₄×Q₈	—	5	1	no	—
297	C₄×Q₈	—	5	2	no	—
298	C₄×Q₈	—	5	1	no	—
299	C₄×Q₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
300	C₂×C₄⋊C₄	—	5	1	no	—
301	C₂×C₄⋊C₄	—	5	1	no	—
302	C₂×C₄⋊C₄	—	5	2	no	—
303	C₂×C₄⋊C₄	—	5	1	no	—
304	C₂×C₄⋊C₄	—	5	2	no	—
305	C₂×C₄⋊C₄	—	5	2	no	—
306	C₂×C₄⋊C₄	—	5	2	no	—
307	C₂×C₄²	—	5	2	no	—
308	C₂×C₄²	—	5	1	no	—
309	C₂×C₄²	—	5	2	no	—
310	C₂×C₄²	—	5	4	no	—
311	#95	—	5	4	no	—
312	#95	—	5	2	no	—
313	#95	—	5	4	no	—
314	#95	—	5	4	no	—
315	C₂×C₃×C₅	C₂×#117	1	1	yes	characteristic
316	C₂×C₃×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
317	C₂×C₃×C₅	#520	1	1	yes	characteristic
318	C₃×D₅	#554	1	1	yes	characteristic
319	C₃×D₅	#556	1	1	yes	characteristic
320	C₃×D₅	#557	1	1	yes	characteristic
321	C₃×D₅	#559	1	1	yes	characteristic
322	C₂³×C₃	—	5	1	no	—
323	C₃×Q₈	—	2	2	no	—
324	C₃×Q₈	—	10	4	no	—
325	C₃×Q₈	—	10	2	no	—
326	C₂×C₃×C₄	—	5	1	no	—
327	C₂×C₃×C₄	—	5	1	no	—
328	C₂×C₃×C₄	—	5	2	no	—
329	C₂×C₃×C₄	—	5	2	no	—
330	C₂×C₃×C₄	—	5	2	no	—
331	C₂×C₃×C₄	—	5	2	no	—
332	C₂×C₃×C₄	—	5	4	no	—
333	C₂×C₃×C₄	—	5	2	no	—
334	C₂×C₃×C₄	—	5	1	no	—
335	C₂×C₃×C₄	—	5	2	no	—
336	C₂×C₃×C₄	—	10	2	no	—
337	C₂×C₃×C₄	—	10	1	no	—
338	C₂×C₃×C₄	—	10	2	no	—
339	C₂×C₃×C₄	—	10	1	no	—
340	C₂×C₃×C₄	—	10	2	no	—
341	C₂×C₃×C₄	—	10	2	no	—
342	C₂×C₃×C₄	—	10	2	no	—
343	C₂×C₃×C₄	—	10	2	no	—
344	C₂×C₃×C₄	—	10	2	no	—
345	C₂×C₃×C₄	—	10	4	no	—
346	C₂×C₃×C₄	—	10	4	no	—
347	C₂×C₃×C₄	—	10	2	no	—
348	C₂×C₃×C₄	—	10	4	no	—
349	C₂×C₃×C₄	—	10	2	no	—
350	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
351	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
352	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
353	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
354	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
355	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
356	C₂×D₅	C₂²×C₃×Q₈	1	1	yes	characteristic
357	C₂×D₅	C₂×C₃×#41	1	1	yes	characteristic
358	C₂×D₅	C₂×C₃×#41	1	1	yes	characteristic
359	C₂×D₅	C₂×C₃×#41	1	1	yes	characteristic
360	C₂×D₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
361	C₂×D₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
362	AGL(1,𝔽₅)	—	2	4	no	—
363	AGL(1,𝔽₅)	—	4	4	no	—
364	AGL(1,𝔽₅)	—	4	2	no	—
365	C₄×C₅	—	2	2	no	—
366	C₄×C₅	—	2	2	no	—
367	C₄×C₅	—	2	1	no	—
368	C₄×C₅	C₃×#117	1	2	yes	—
369	C₄×C₅	C₃×C₄×Q₈	1	2	yes	—
370	Dic₅	—	2	2	no	—
371	Dic₅	—	2	2	no	—
372	Dic₅	—	2	1	no	—
373	Dic₅	C₃×#117	1	2	yes	—
374	Dic₅	C₃×C₄×Q₈	1	2	yes	—
375	C₂×Q₈	—	5	2	no	—
376	C₂×Q₈	—	5	1	no	—
377	C₂×Q₈	—	10	2	no	—
378	C₂×Q₈	—	10	2	no	—
379	C₂×Q₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
380	C₂²×C₄	—	5	1	no	—
381	C₂²×C₄	—	5	1	no	—
382	C₂²×C₄	—	5	2	no	—
383	C₂²×C₄	—	5	2	no	—
384	C₂²×C₄	—	5	2	no	—
385	C₂²×C₄	—	5	1	no	—
386	C₂²×C₄	—	5	2	no	—
387	C₂²×C₄	—	5	4	no	—
388	C₄⋊C₄	—	5	4	no	—
389	C₄⋊C₄	—	5	2	no	—
390	C₄⋊C₄	—	5	2	no	—
391	C₄⋊C₄	—	5	2	no	—
392	C₄⋊C₄	—	5	2	no	—
393	C₄⋊C₄	—	5	1	no	—
394	C₄⋊C₄	—	10	4	no	—
395	C₄⋊C₄	—	10	2	no	—
396	C₄⋊C₄	—	10	4	no	—
397	C₄⋊C₄	—	20	2	no	—
398	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
399	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
400	C₄²	—	5	1	no	—
401	C₄²	—	5	2	no	—
402	C₄²	—	5	1	no	—
403	C₄²	—	5	2	no	—
404	C₄²	—	5	1	no	—
405	C₄²	—	5	2	no	—
406	C₄²	—	10	4	no	—
407	C₄²	—	20	4	no	—
408	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
409	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
410	C₃×C₅	#2355	1	1	yes	characteristic, complemented, Hall
411	C₂²×C₃	—	5	1	no	—
412	C₂²×C₃	—	5	1	no	—
413	C₂²×C₃	—	5	1	no	—
414	C₂²×C₃	—	5	1	no	—
415	C₂²×C₃	—	5	1	no	—
416	C₂²×C₃	—	5	1	no	—
417	C₂²×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
418	C₃×C₄	—	2	2	no	—
419	C₃×C₄	—	2	2	no	—
420	C₃×C₄	—	2	1	no	—
421	C₃×C₄	—	5	2	no	—
422	C₃×C₄	—	5	2	no	—
423	C₃×C₄	—	10	4	no	—
424	C₃×C₄	—	10	2	no	—
425	C₃×C₄	—	10	2	no	—
426	C₃×C₄	—	10	1	no	—
427	C₃×C₄	—	20	4	no	—
428	C₃×C₄	—	20	2	no	—
429	C₃×C₄	#4130	1	2	yes	—
430	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
431	C₂×C₅	C₂×C₃×#117	1	1	yes	characteristic
432	C₂×C₅	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
433	C₂×C₅	C₃×#520	1	1	yes	characteristic
434	D₅	C₃×#554	1	1	yes	characteristic
435	D₅	C₃×#556	1	1	yes	characteristic
436	D₅	C₃×#557	1	1	yes	characteristic
437	D₅	C₃×#559	1	1	yes	characteristic
438	C₂³	—	5	1	no	—
439	Q₈	—	2	2	no	—
440	Q₈	—	10	4	no	—
441	Q₈	—	10	2	no	—
442	C₂×C₄	—	5	1	no	—
443	C₂×C₄	—	5	1	no	—
444	C₂×C₄	—	5	2	no	—
445	C₂×C₄	—	5	2	no	—
446	C₂×C₄	—	5	2	no	—
447	C₂×C₄	—	5	2	no	—
448	C₂×C₄	—	5	4	no	—
449	C₂×C₄	—	5	2	no	—
450	C₂×C₄	—	5	1	no	—
451	C₂×C₄	—	5	2	no	—
452	C₂×C₄	—	10	2	no	—
453	C₂×C₄	—	10	1	no	—
454	C₂×C₄	—	10	2	no	—
455	C₂×C₄	—	10	2	no	—
456	C₂×C₄	—	10	2	no	—
457	C₂×C₄	—	10	1	no	—
458	C₂×C₄	—	10	2	no	—
459	C₂×C₄	—	10	2	no	—
460	C₂×C₄	—	10	2	no	—
461	C₂×C₄	—	10	4	no	—
462	C₂×C₄	—	10	4	no	—
463	C₂×C₄	—	10	2	no	—
464	C₂×C₄	—	10	4	no	—
465	C₂×C₄	—	10	2	no	—
466	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
467	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
468	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
469	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
470	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
471	C₂×C₃	—	5	1	no	—
472	C₂×C₃	—	5	1	no	—
473	C₂×C₃	—	5	1	no	—
474	C₂×C₃	—	5	1	no	—
475	C₂×C₃	C₂×#4130	1	1	yes	characteristic
476	C₂×C₃	C₂×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
477	C₂×C₃	#66266	1	1	yes	characteristic
478	C₅	C₃×#2355	1	1	yes	characteristic, complemented, Sylow
479	C₂²	—	5	1	no	—
480	C₂²	—	5	1	no	—
481	C₂²	—	5	1	no	—
482	C₂²	—	5	1	no	—
483	C₂²	—	5	1	no	—
484	C₂²	—	5	1	no	—
485	C₂²	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
486	C₄	—	2	2	no	—
487	C₄	—	2	2	no	—
488	C₄	—	2	1	no	—
489	C₄	—	5	2	no	—
490	C₄	—	5	2	no	—
491	C₄	—	10	4	no	—
492	C₄	—	10	2	no	—
493	C₄	—	10	2	no	—
494	C₄	—	10	1	no	—
495	C₄	—	20	4	no	—
496	C₄	—	20	2	no	—
497	C₄	C₃×#4130	1	2	yes	—
498	C₄	C₃×Q₈×AGL(1,𝔽₅)	1	2	yes	—
499	C₃	#10617213	1	1	yes	characteristic, direct product factor, Sylow
500	C₂	—	5	1	no	—
501	C₂	—	5	1	no	—
502	C₂	—	5	1	no	—
503	C₂	—	5	1	no	—
504	C₂	C₂×C₃×#4130	1	1	yes	characteristic
505	C₂	C₂×C₃×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
506	C₂	C₃×#66266	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×Q₈	C₄
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂²×C₃×D₅	C₂⁴
cosocle	C₂²×D₅	C₂⁴×C₃
hypercenter	C₃×C₄×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2355
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
9. #10617213	C₃	direct (N × H)
18. C₃×C₄×C₅×Q₈	C₄	semidirect
19. C₃×Q₈×Dic₅	C₄	semidirect
22. C₃×#4033	C₄	semidirect
23. C₃×#4033	C₄	semidirect
26. C₃×C₄×Dic₁₀	C₄	semidirect
27. C₃×C₄×Dic₁₀	C₄	semidirect
58. C₃×C₄⋊AGL(1,𝔽₅)	Q₈	semidirect
92. C₄×C₅×Q₈	C₃×C₄	semidirect
93. Q₈×Dic₅	C₃×C₄	semidirect
96. #4033	C₃×C₄	semidirect
97. #4033	C₃×C₄	semidirect
100. C₄×Dic₁₀	C₃×C₄	semidirect
101. C₄×Dic₁₀	C₃×C₄	semidirect
146. C₃×C₄×Q₈	AGL(1,𝔽₅)	semidirect
179. C₄⋊AGL(1,𝔽₅)	C₃×Q₈	semidirect
299. C₄×Q₈	C₃×AGL(1,𝔽₅)	semidirect
410. C₃×C₅	#2355	semidirect
478. C₅	C₃×#2355	semidirect
499. C₃	#10617213	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₄×Q₈×D₅	2
C₃×(C₂×AGL(1,𝔽₅))⋊Q₈	2
C₃×#65711	2
C₃×C₄⋊C₄×AGL(1,𝔽₅)	2
C₃×#65689	2
C₃×#65688	2
C₃×C₄×C₄⋊AGL(1,𝔽₅)	2
#10617213	3
C₃×#2355	5

Subgroups — C3×#10617213 #14327792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10617213 #14327792