Subgroups — C₃×AGL(1,𝔽₅)×#135 #14416224

Order 1920.

1000 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#41×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
2	C₃×D₅×#135	C₂	1	1	yes	maximal, characteristic
3	C₃×#65794	C₂	1	2	yes	maximal, complemented
4	C₃×Q₁₆×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
5	C₃×SD₁₆×AGL(1,𝔽₅)	C₂	1	4	yes	maximal, complemented
6	C₃×D₈×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
7	C₃×#65721	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×C₈×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
9	AGL(1,𝔽₅)×#135	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×D₅×#41	C₂²	1	2	yes	—
11	C₃×Q₈×AGL(1,𝔽₅)	C₂²	1	4	yes	—
12	C₃×D₄×AGL(1,𝔽₅)	—	2	4	no	—
13	C₃×D₄×AGL(1,𝔽₅)	C₂²	1	4	yes	—
14	C₃×#4130	—	2	2	no	—
15	C₃×#4130	C₂²	1	2	yes	—
16	C₂×C₃×C₄×AGL(1,𝔽₅)	—	2	2	no	—
17	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
18	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×C₅×#135	C₄	1	1	yes	characteristic, complemented
20	C₃×#4096	C₄	1	2	yes	complemented
21	C₃×#4065	C₄	1	1	yes	characteristic, complemented
22	C₃×D₅×Q₁₆	C₂²	1	1	yes	characteristic
23	C₃×#4062	C₄	1	2	yes	complemented
24	C₃×D₅×SD₁₆	C₂²	1	2	yes	—
25	C₃×#4058	C₄	1	1	yes	characteristic, complemented
26	C₃×D₅×D₈	C₂²	1	1	yes	characteristic
27	C₃×#4050	C₄	1	1	yes	characteristic, complemented
28	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
29	C₃×#4009	C₂²	1	4	yes	—
30	C₃×#4007	C₂²	1	4	yes	—
31	C₃×#3994	C₂²	1	2	yes	—
32	C₃×#3993	C₂²	1	2	yes	—
33	C₃×C₈×AGL(1,𝔽₅)	C₂²	1	2	yes	—
34	C₃×C₈×AGL(1,𝔽₅)	C₂²	1	2	yes	—
35	C₃×C₄×#135	—	5	1	no	maximal, Hall
36	#41×AGL(1,𝔽₅)	C₂×C₃	1	2	yes	complemented
37	D₅×#135	C₂×C₃	1	1	yes	characteristic
38	#65794	C₂×C₃	1	2	yes	complemented
39	Q₁₆×AGL(1,𝔽₅)	C₂×C₃	1	2	yes	complemented
40	SD₁₆×AGL(1,𝔽₅)	C₂×C₃	1	4	yes	complemented
41	D₈×AGL(1,𝔽₅)	C₂×C₃	1	2	yes	complemented
42	#65721	C₂×C₃	1	1	yes	characteristic, complemented
43	C₂×C₈×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic, complemented
44	C₂²×C₃×AGL(1,𝔽₅)	—	2	4	no	—
45	C₂²×C₃×AGL(1,𝔽₅)	D₄	1	2	yes	—
46	C₃×C₅×#41	C₂×C₄	1	2	yes	complemented
47	C₃×#713	C₂×C₄	1	2	yes	complemented
48	C₃×Q₈×D₅	C₂³	1	2	yes	—
49	C₃×#711	—	2	2	no	—
50	C₃×#711	C₂×C₄	1	2	yes	complemented
51	C₃×D₄×D₅	—	2	2	no	—
52	C₃×D₄×D₅	C₂³	1	2	yes	—
53	C₃×#709	—	2	2	no	—
54	C₃×#709	C₂×C₄	1	2	yes	complemented
55	C₂×C₃×C₄×D₅	—	2	2	no	—
56	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
57	C₃×C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
58	C₃×C₂²⋊AGL(1,𝔽₅)	—	2	4	no	—
59	C₂×C₃×#170	—	2	1	no	—
60	C₃×C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
61	C₃×C₄⋊AGL(1,𝔽₅)	C₂³	1	4	yes	—
62	C₃×C₄×AGL(1,𝔽₅)	—	2	4	no	—
63	C₃×C₄×AGL(1,𝔽₅)	—	2	2	no	—
64	C₃×C₄×AGL(1,𝔽₅)	D₄	1	1	yes	characteristic
65	C₃×C₄×AGL(1,𝔽₅)	D₄	1	1	yes	characteristic
66	C₃×C₄×AGL(1,𝔽₅)	C₂³	1	2	yes	—
67	C₃×D₅⋊C₈	—	2	1	no	—
68	C₃×D₅⋊C₈	C₂³	1	2	yes	—
69	C₃×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
70	C₃×C₅×SD₁₆	C₂×C₄	1	2	yes	complemented
71	C₃×C₅×D₈	C₂×C₄	1	1	yes	characteristic, complemented
72	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic, complemented
73	C₃×#689	C₂×C₄	1	2	yes	complemented
74	C₃×#688	C₂×C₄	1	2	yes	complemented
75	C₃×#687	C₂×C₄	1	2	yes	complemented
76	C₃×#686	C₂×C₄	1	2	yes	complemented
77	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic, complemented
78	C₃×Dic₂₀	C₂×C₄	1	1	yes	characteristic, complemented
79	C₃×D₄₀	C₂×C₄	1	1	yes	characteristic, complemented
80	C₃×#677	C₂×C₄	1	2	yes	complemented
81	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic, complemented
82	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic, complemented
83	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
84	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
85	C₂×C₃×#135	—	5	1	no	—
86	C₃×C₄×#41	—	5	2	no	—
87	C₃×C₄×Q₁₆	—	5	2	no	—
88	C₃×C₄×SD₁₆	—	5	4	no	—
89	C₃×C₄×D₈	—	5	2	no	—
90	C₃×#427	—	5	1	no	—
91	C₃×#416	—	5	2	no	—
92	C₂×C₃×C₄×C₈	—	5	1	no	—
93	D₅×#41	C₂²×C₃	1	2	yes	—
94	Q₈×AGL(1,𝔽₅)	C₂²×C₃	1	4	yes	—
95	D₄×AGL(1,𝔽₅)	—	2	4	no	—
96	D₄×AGL(1,𝔽₅)	C₂²×C₃	1	4	yes	—
97	#4130	—	2	2	no	—
98	#4130	C₂²×C₃	1	2	yes	—
99	C₂×C₄×AGL(1,𝔽₅)	—	2	2	no	—
100	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
101	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
102	C₅×#135	C₃×C₄	1	1	yes	characteristic, complemented
103	#4096	C₃×C₄	1	2	yes	complemented
104	#4065	C₃×C₄	1	1	yes	characteristic, complemented
105	D₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
106	#4062	C₃×C₄	1	2	yes	complemented
107	D₅×SD₁₆	C₂²×C₃	1	2	yes	—
108	#4058	C₃×C₄	1	1	yes	characteristic, complemented
109	D₅×D₈	C₂²×C₃	1	1	yes	characteristic
110	#4050	C₃×C₄	1	1	yes	characteristic, complemented
111	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
112	#4009	C₂²×C₃	1	4	yes	—
113	#4007	C₂²×C₃	1	4	yes	—
114	#3994	C₂²×C₃	1	2	yes	—
115	#3993	C₂²×C₃	1	2	yes	—
116	C₈×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
117	C₈×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
118	C₄×#135	—	5	1	no	Sylow
119	C₂²×C₃×D₅	—	2	2	no	—
120	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
121	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
122	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
123	C₂×C₃×AGL(1,𝔽₅)	—	4	8	no	—
124	C₂×C₃×AGL(1,𝔽₅)	C₂×D₄	1	2	yes	—
125	C₂×C₃×AGL(1,𝔽₅)	C₂×D₄	1	2	yes	—
126	C₃×C₅×Q₈	C₂²×C₄	1	2	yes	—
127	C₃×C₅×D₄	—	2	2	no	—
128	C₃×C₅×D₄	C₂²×C₄	1	2	yes	—
129	C₂×C₃×C₄×C₅	—	2	2	no	—
130	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
131	C₃×C₂²⋊D₅	—	2	2	no	—
132	C₃×C₂²⋊D₅	—	2	2	no	—
133	C₃×C₂²⋊D₅	—	2	2	no	—
134	C₂×C₃×Dic₅	—	2	2	no	—
135	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
136	C₃×D₂₀	—	2	2	no	—
137	C₃×D₂₀	C₂²×C₄	1	2	yes	—
138	C₃×C₄×D₅	—	2	2	no	—
139	C₃×C₄×D₅	—	2	2	no	—
140	C₃×C₄×D₅	—	2	2	no	—
141	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
142	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
143	C₃×C₄×D₅	#41	1	1	yes	characteristic
144	C₃×C₄×D₅	C₂⁴	1	1	yes	characteristic
145	C₃×Dic₁₀	—	2	2	no	—
146	C₃×Dic₁₀	C₂²×C₄	1	2	yes	—
147	C₃×#170	—	2	2	no	—
148	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
149	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
150	C₃×#168	C₂²×C₄	1	1	yes	characteristic
151	C₃×#168	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₃×#41	—	5	2	no	—
153	C₃×#135	—	5	2	no	—
154	C₃×#135	—	5	1	no	—
155	C₃×#135	—	5	2	no	—
156	C₃×#135	—	5	1	no	—
157	C₃×#135	—	5	1	no	—
158	C₃×#135	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
159	C₂×C₃×Q₁₆	—	5	1	no	—
160	C₂×C₃×SD₁₆	—	5	2	no	—
161	C₂×C₃×D₈	—	5	1	no	—
162	C₂²×C₃×C₈	—	5	1	no	—
163	C₂²×C₃×C₈	—	5	1	no	—
164	C₃×C₄×Q₈	—	5	4	no	—
165	C₃×C₄×D₄	—	5	4	no	—
166	C₃×C₄×D₄	—	10	4	no	—
167	C₃×#117	—	5	2	no	—
168	C₃×#117	—	10	2	no	—
169	C₂×C₃×C₄²	—	5	1	no	—
170	C₂×C₃×C₄²	—	10	2	no	—
171	C₃×#107	—	5	2	no	—
172	C₃×#106	—	5	2	no	—
173	C₃×#103	—	5	4	no	—
174	C₃×#102	—	5	4	no	—
175	C₃×C₄×C₈	—	5	2	no	—
176	C₃×C₄×C₈	—	5	2	no	—
177	C₂²×AGL(1,𝔽₅)	—	2	4	no	—
178	C₂²×AGL(1,𝔽₅)	C₃×D₄	1	2	yes	—
179	C₅×#41	C₂×C₃×C₄	1	2	yes	complemented
180	#713	C₂×C₃×C₄	1	2	yes	complemented
181	Q₈×D₅	C₂³×C₃	1	2	yes	—
182	#711	—	2	2	no	—
183	#711	C₂×C₃×C₄	1	2	yes	complemented
184	D₄×D₅	—	2	2	no	—
185	D₄×D₅	C₂³×C₃	1	2	yes	—
186	#709	—	2	2	no	—
187	#709	C₂×C₃×C₄	1	2	yes	complemented
188	C₂×C₄×D₅	—	2	2	no	—
189	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
190	C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
191	C₂²⋊AGL(1,𝔽₅)	—	2	4	no	—
192	C₂×#170	—	2	1	no	—
193	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
194	C₄⋊AGL(1,𝔽₅)	C₂³×C₃	1	4	yes	—
195	C₄×AGL(1,𝔽₅)	—	2	4	no	—
196	C₄×AGL(1,𝔽₅)	—	2	2	no	—
197	C₄×AGL(1,𝔽₅)	C₃×D₄	1	1	yes	characteristic
198	C₄×AGL(1,𝔽₅)	C₃×D₄	1	1	yes	characteristic
199	C₄×AGL(1,𝔽₅)	C₂³×C₃	1	2	yes	—
200	D₅⋊C₈	—	2	1	no	—
201	D₅⋊C₈	C₂³×C₃	1	2	yes	—
202	C₅×Q₁₆	C₂×C₃×C₄	1	1	yes	characteristic, complemented
203	C₅×SD₁₆	C₂×C₃×C₄	1	2	yes	complemented
204	C₅×D₈	C₂×C₃×C₄	1	1	yes	characteristic, complemented
205	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic, complemented
206	#689	C₂×C₃×C₄	1	2	yes	complemented
207	#688	C₂×C₃×C₄	1	2	yes	complemented
208	#687	C₂×C₃×C₄	1	2	yes	complemented
209	#686	C₂×C₃×C₄	1	2	yes	complemented
210	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic, complemented
211	Dic₂₀	C₂×C₃×C₄	1	1	yes	characteristic, complemented
212	D₄₀	C₂×C₃×C₄	1	1	yes	characteristic, complemented
213	#677	C₂×C₃×C₄	1	2	yes	complemented
214	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
215	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
216	C₈×D₅	C₂³×C₃	1	1	yes	characteristic
217	C₈×D₅	C₂³×C₃	1	1	yes	characteristic
218	C₂×#135	—	5	1	no	—
219	C₄×#41	—	5	2	no	—
220	C₄×Q₁₆	—	5	2	no	—
221	C₄×SD₁₆	—	5	4	no	—
222	C₄×D₈	—	5	2	no	—
223	#427	—	5	1	no	—
224	#416	—	5	2	no	—
225	C₂×C₄×C₈	—	5	1	no	—
226	C₂²×C₃×C₅	—	2	2	no	—
227	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
228	C₂×C₃×D₅	—	2	2	no	—
229	C₂×C₃×D₅	—	2	2	no	—
230	C₂×C₃×D₅	—	4	4	no	—
231	C₂×C₃×D₅	C₄×D₄	1	1	yes	characteristic
232	C₂×C₃×D₅	C₂²×D₄	1	1	yes	characteristic
233	C₃×AGL(1,𝔽₅)	—	2	2	no	—
234	C₃×AGL(1,𝔽₅)	—	4	4	no	—
235	C₃×AGL(1,𝔽₅)	#135	1	4	yes	direct product factor
236	C₃×C₄×C₅	—	2	2	no	—
237	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
238	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
239	C₃×Dic₅	—	2	2	no	—
240	C₃×Dic₅	C₄×D₄	1	1	yes	characteristic
241	C₃×Dic₅	C₂×#41	1	1	yes	characteristic
242	C₃×#41	—	5	2	no	—
243	C₃×#41	—	5	2	no	—
244	C₃×#41	—	5	2	no	—
245	C₃×#41	—	10	2	no	—
246	C₃×#41	—	10	2	no	—
247	C₃×#41	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
248	C₂×C₃×Q₈	—	5	2	no	—
249	C₂×C₃×D₄	—	5	2	no	—
250	C₂×C₃×D₄	—	10	2	no	—
251	C₂²×C₃×C₄	—	5	1	no	—
252	C₂²×C₃×C₄	—	5	2	no	—
253	C₂²×C₃×C₄	—	10	2	no	—
254	C₂²×C₃×C₄	—	10	4	no	—
255	C₃×Q₁₆	—	5	2	no	—
256	C₃×Q₁₆	—	5	1	no	—
257	C₃×Q₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
258	C₃×SD₁₆	—	5	2	no	—
259	C₃×SD₁₆	—	5	2	no	—
260	C₃×SD₁₆	—	5	2	no	—
261	C₃×SD₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
262	C₃×D₈	—	5	2	no	—
263	C₃×D₈	—	5	1	no	—
264	C₃×D₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
265	C₂×C₃×C₈	—	5	1	no	—
266	C₂×C₃×C₈	—	5	1	no	—
267	C₂×C₃×C₈	—	5	1	no	—
268	C₂×C₃×C₈	—	5	1	no	—
269	C₂×C₃×C₈	—	5	1	no	—
270	C₂×C₃×C₈	—	5	2	no	—
271	C₂×C₃×C₈	—	10	1	no	—
272	C₂×C₃×C₈	—	10	1	no	—
273	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
274	C₃×C₄⋊C₄	—	5	4	no	—
275	C₃×C₄⋊C₄	—	10	4	no	—
276	C₃×C₂²⋊C₄	—	10	4	no	—
277	C₃×C₂²⋊C₄	—	10	2	no	—
278	C₃×C₄²	—	5	2	no	—
279	C₃×C₄²	—	5	1	no	—
280	C₃×C₄²	—	5	1	no	—
281	C₃×C₄²	—	10	2	no	—
282	C₃×C₄²	—	10	4	no	—
283	C₂²×D₅	—	2	2	no	—
284	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
285	C₂×AGL(1,𝔽₅)	—	2	4	no	—
286	C₂×AGL(1,𝔽₅)	—	2	4	no	—
287	C₂×AGL(1,𝔽₅)	—	4	8	no	—
288	C₂×AGL(1,𝔽₅)	C₂×C₃×D₄	1	2	yes	—
289	C₂×AGL(1,𝔽₅)	C₂×C₃×D₄	1	2	yes	—
290	C₅×Q₈	C₂²×C₃×C₄	1	2	yes	—
291	C₅×D₄	—	2	2	no	—
292	C₅×D₄	C₂²×C₃×C₄	1	2	yes	—
293	C₂×C₄×C₅	—	2	2	no	—
294	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
295	C₂²⋊D₅	—	2	2	no	—
296	C₂²⋊D₅	—	2	2	no	—
297	C₂²⋊D₅	—	2	2	no	—
298	C₂×Dic₅	—	2	2	no	—
299	C₂×Dic₅	C₃×#41	1	1	yes	characteristic
300	D₂₀	—	2	2	no	—
301	D₂₀	C₂²×C₃×C₄	1	2	yes	—
302	C₄×D₅	—	2	2	no	—
303	C₄×D₅	—	2	2	no	—
304	C₄×D₅	—	2	2	no	—
305	C₄×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
306	C₄×D₅	C₂×C₃×D₄	1	1	yes	characteristic
307	C₄×D₅	C₃×#41	1	1	yes	characteristic
308	C₄×D₅	C₂⁴×C₃	1	1	yes	characteristic
309	Dic₁₀	—	2	2	no	—
310	Dic₁₀	C₂²×C₃×C₄	1	2	yes	—
311	#170	—	2	2	no	—
312	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
313	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
314	#168	C₂²×C₃×C₄	1	1	yes	characteristic
315	#168	C₂²×C₃×C₄	1	1	yes	characteristic
316	C₂×#41	—	5	2	no	—
317	#135	—	5	2	no	—
318	#135	—	5	1	no	—
319	#135	—	5	2	no	—
320	#135	—	5	1	no	—
321	#135	—	5	1	no	—
322	#135	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
323	C₂×Q₁₆	—	5	1	no	—
324	C₂×SD₁₆	—	5	2	no	—
325	C₂×D₈	—	5	1	no	—
326	C₂²×C₈	—	5	1	no	—
327	C₂²×C₈	—	5	1	no	—
328	C₄×Q₈	—	5	4	no	—
329	C₄×D₄	—	5	4	no	—
330	C₄×D₄	—	10	4	no	—
331	#117	—	5	2	no	—
332	#117	—	10	2	no	—
333	C₂×C₄²	—	5	1	no	—
334	C₂×C₄²	—	10	2	no	—
335	#107	—	5	2	no	—
336	#106	—	5	2	no	—
337	#103	—	5	4	no	—
338	#102	—	5	4	no	—
339	C₄×C₈	—	5	2	no	—
340	C₄×C₈	—	5	2	no	—
341	C₂×C₃×C₅	—	2	1	no	—
342	C₂×C₃×C₅	—	4	2	no	—
343	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
344	C₃×D₅	—	2	1	no	—
345	C₃×D₅	—	4	2	no	—
346	C₃×D₅	C₂×#135	1	2	yes	—
347	C₂³×C₃	—	5	1	no	—
348	C₂³×C₃	—	10	2	no	—
349	C₃×Q₈	—	5	2	no	—
350	C₃×Q₈	—	10	2	no	—
351	C₃×Q₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
352	C₃×D₄	—	2	2	no	—
353	C₃×D₄	—	5	2	no	—
354	C₃×D₄	—	10	2	no	—
355	C₃×D₄	—	10	2	no	—
356	C₃×D₄	—	10	2	no	—
357	C₃×D₄	—	10	2	no	—
358	C₃×D₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
359	C₂×C₃×C₄	—	2	2	no	—
360	C₂×C₃×C₄	—	5	2	no	—
361	C₂×C₃×C₄	—	5	1	no	—
362	C₂×C₃×C₄	—	5	1	no	—
363	C₂×C₃×C₄	—	5	1	no	—
364	C₂×C₃×C₄	—	5	1	no	—
365	C₂×C₃×C₄	—	5	1	no	—
366	C₂×C₃×C₄	—	5	2	no	—
367	C₂×C₃×C₄	—	10	2	no	—
368	C₂×C₃×C₄	—	10	2	no	—
369	C₂×C₃×C₄	—	10	2	no	—
370	C₂×C₃×C₄	—	10	4	no	—
371	C₂×C₃×C₄	—	10	2	no	—
372	C₂×C₃×C₄	—	10	4	no	—
373	C₂×C₃×C₄	—	20	8	no	—
374	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
375	C₃×C₈	—	5	1	no	—
376	C₃×C₈	—	5	1	no	—
377	C₃×C₈	—	10	2	no	—
378	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
379	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
380	C₂²×C₅	—	2	2	no	—
381	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
382	C₂×D₅	—	2	2	no	—
383	C₂×D₅	—	2	2	no	—
384	C₂×D₅	—	4	4	no	—
385	C₂×D₅	C₃×C₄×D₄	1	1	yes	characteristic
386	C₂×D₅	C₂²×C₃×D₄	1	1	yes	characteristic
387	AGL(1,𝔽₅)	—	2	2	no	—
388	AGL(1,𝔽₅)	—	4	4	no	—
389	AGL(1,𝔽₅)	C₃×#135	1	4	yes	direct product factor
390	C₄×C₅	—	2	2	no	—
391	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
392	C₄×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
393	Dic₅	—	2	2	no	—
394	Dic₅	C₃×C₄×D₄	1	1	yes	characteristic
395	Dic₅	C₂×C₃×#41	1	1	yes	characteristic
396	#41	—	5	2	no	—
397	#41	—	5	2	no	—
398	#41	—	5	2	no	—
399	#41	—	10	2	no	—
400	#41	—	10	2	no	—
401	#41	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
402	C₂×Q₈	—	5	2	no	—
403	C₂×D₄	—	5	2	no	—
404	C₂×D₄	—	10	2	no	—
405	C₂²×C₄	—	5	1	no	—
406	C₂²×C₄	—	5	2	no	—
407	C₂²×C₄	—	10	2	no	—
408	C₂²×C₄	—	10	4	no	—
409	Q₁₆	—	5	2	no	—
410	Q₁₆	—	5	1	no	—
411	Q₁₆	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
412	SD₁₆	—	5	2	no	—
413	SD₁₆	—	5	2	no	—
414	SD₁₆	—	5	2	no	—
415	SD₁₆	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
416	D₈	—	5	2	no	—
417	D₈	—	5	1	no	—
418	D₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
419	C₂×C₈	—	5	1	no	—
420	C₂×C₈	—	5	1	no	—
421	C₂×C₈	—	5	1	no	—
422	C₂×C₈	—	5	1	no	—
423	C₂×C₈	—	5	1	no	—
424	C₂×C₈	—	5	2	no	—
425	C₂×C₈	—	10	1	no	—
426	C₂×C₈	—	10	1	no	—
427	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
428	C₄⋊C₄	—	5	4	no	—
429	C₄⋊C₄	—	10	4	no	—
430	C₂²⋊C₄	—	10	4	no	—
431	C₂²⋊C₄	—	10	2	no	—
432	C₄²	—	5	2	no	—
433	C₄²	—	5	1	no	—
434	C₄²	—	5	1	no	—
435	C₄²	—	10	2	no	—
436	C₄²	—	10	4	no	—
437	C₃×C₅	C₄×#135	1	1	yes	characteristic, complemented, Hall
438	C₂²×C₃	—	2	2	no	—
439	C₂²×C₃	—	5	1	no	—
440	C₂²×C₃	—	5	1	no	—
441	C₂²×C₃	—	10	2	no	—
442	C₂²×C₃	—	10	2	no	—
443	C₂²×C₃	—	20	4	no	—
444	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
445	C₃×C₄	—	2	2	no	—
446	C₃×C₄	—	5	4	no	—
447	C₃×C₄	—	5	1	no	—
448	C₃×C₄	—	5	1	no	—
449	C₃×C₄	—	10	2	no	—
450	C₃×C₄	—	10	2	no	—
451	C₃×C₄	—	20	4	no	—
452	C₃×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
453	C₃×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
454	C₂×C₅	—	2	1	no	—
455	C₂×C₅	—	4	2	no	—
456	C₂×C₅	C₂×C₃×C₄×D₄	1	1	yes	characteristic
457	D₅	—	2	1	no	—
458	D₅	—	4	2	no	—
459	D₅	C₂×C₃×#135	1	2	yes	—
460	C₂³	—	5	1	no	—
461	C₂³	—	10	2	no	—
462	Q₈	—	5	2	no	—
463	Q₈	—	10	2	no	—
464	Q₈	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
465	D₄	—	2	2	no	—
466	D₄	—	5	2	no	—
467	D₄	—	10	2	no	—
468	D₄	—	10	2	no	—
469	D₄	—	10	2	no	—
470	D₄	—	10	2	no	—
471	D₄	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	5	2	no	—
474	C₂×C₄	—	5	1	no	—
475	C₂×C₄	—	5	1	no	—
476	C₂×C₄	—	5	1	no	—
477	C₂×C₄	—	5	1	no	—
478	C₂×C₄	—	5	1	no	—
479	C₂×C₄	—	5	2	no	—
480	C₂×C₄	—	10	2	no	—
481	C₂×C₄	—	10	2	no	—
482	C₂×C₄	—	10	2	no	—
483	C₂×C₄	—	10	4	no	—
484	C₂×C₄	—	10	2	no	—
485	C₂×C₄	—	10	4	no	—
486	C₂×C₄	—	20	8	no	—
487	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
488	C₈	—	5	1	no	—
489	C₈	—	5	1	no	—
490	C₈	—	10	2	no	—
491	C₈	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
492	C₈	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
493	C₂×C₃	—	2	1	no	—
494	C₂×C₃	—	4	2	no	—
495	C₂×C₃	—	5	2	no	—
496	C₂×C₃	—	10	1	no	—
497	C₂×C₃	—	20	2	no	—
498	C₂×C₃	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
499	C₅	C₃×C₄×#135	1	1	yes	characteristic, complemented, Sylow
500	C₂²	—	2	2	no	—
501	C₂²	—	5	1	no	—
502	C₂²	—	5	1	no	—
503	C₂²	—	10	2	no	—
504	C₂²	—	10	2	no	—
505	C₂²	—	20	4	no	—
506	C₂²	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
507	C₄	—	2	2	no	—
508	C₄	—	5	4	no	—
509	C₄	—	5	1	no	—
510	C₄	—	5	1	no	—
511	C₄	—	10	2	no	—
512	C₄	—	10	2	no	—
513	C₄	—	20	4	no	—
514	C₄	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
515	C₄	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
516	C₃	AGL(1,𝔽₅)×#135	1	1	yes	characteristic, direct product factor, Sylow
517	C₂	—	2	1	no	—
518	C₂	—	4	2	no	—
519	C₂	—	5	2	no	—
520	C₂	—	10	1	no	—
521	C₂	—	20	2	no	—
522	C₂	C₂×C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂³×C₃×C₄
Frattini	C₄	C₂³×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#135	C₄
socle	C₂×C₃×C₅	C₂×C₄×D₄
prime radical	C₃×D₅×#135	C₂
cosocle	C₄×D₅	C₂⁴×C₃
hypercenter	C₃×#135	AGL(1,𝔽₅)
hypocenter	C₅	C₃×C₄×#135
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₃×#135

Complements

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

N	H ≅ G/N	type
1. C₃×#41×AGL(1,𝔽₅)	C₂	semidirect
3. C₃×#65794	C₂	semidirect
4. C₃×Q₁₆×AGL(1,𝔽₅)	C₂	semidirect
5. C₃×SD₁₆×AGL(1,𝔽₅)	C₂	semidirect
6. C₃×D₈×AGL(1,𝔽₅)	C₂	semidirect
7. C₃×#65721	C₂	semidirect
8. C₂×C₃×C₈×AGL(1,𝔽₅)	C₂	semidirect
9. AGL(1,𝔽₅)×#135	C₃	direct (N × H)
19. C₃×C₅×#135	C₄	semidirect
20. C₃×#4096	C₄	semidirect
21. C₃×#4065	C₄	semidirect
23. C₃×#4062	C₄	semidirect
25. C₃×#4058	C₄	semidirect
27. C₃×#4050	C₄	semidirect
36. #41×AGL(1,𝔽₅)	C₂×C₃	semidirect
38. #65794	C₂×C₃	semidirect
39. Q₁₆×AGL(1,𝔽₅)	C₂×C₃	semidirect
40. SD₁₆×AGL(1,𝔽₅)	C₂×C₃	semidirect
41. D₈×AGL(1,𝔽₅)	C₂×C₃	semidirect
42. #65721	C₂×C₃	semidirect
43. C₂×C₈×AGL(1,𝔽₅)	C₂×C₃	semidirect
46. C₃×C₅×#41	C₂×C₄	semidirect
47. C₃×#713	C₂×C₄	semidirect
50. C₃×#711	C₂×C₄	semidirect
54. C₃×#709	C₂×C₄	semidirect
69. C₃×C₅×Q₁₆	C₂×C₄	semidirect
70. C₃×C₅×SD₁₆	C₂×C₄	semidirect
71. C₃×C₅×D₈	C₂×C₄	semidirect
72. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
73. C₃×#689	C₂×C₄	semidirect
74. C₃×#688	C₂×C₄	semidirect
75. C₃×#687	C₂×C₄	semidirect
76. C₃×#686	C₂×C₄	semidirect
77. C₂×C₃×#168	C₂×C₄	semidirect
78. C₃×Dic₂₀	C₂×C₄	semidirect
79. C₃×D₄₀	C₂×C₄	semidirect
80. C₃×#677	C₂×C₄	semidirect
81. C₃×C₈×D₅	C₂×C₄	semidirect
82. C₃×C₈×D₅	C₂×C₄	semidirect
102. C₅×#135	C₃×C₄	semidirect
103. #4096	C₃×C₄	semidirect
104. #4065	C₃×C₄	semidirect
106. #4062	C₃×C₄	semidirect
108. #4058	C₃×C₄	semidirect
110. #4050	C₃×C₄	semidirect
158. C₃×#135	AGL(1,𝔽₅)	direct (N × H)
179. C₅×#41	C₂×C₃×C₄	semidirect
180. #713	C₂×C₃×C₄	semidirect
183. #711	C₂×C₃×C₄	semidirect
187. #709	C₂×C₃×C₄	semidirect
202. C₅×Q₁₆	C₂×C₃×C₄	semidirect
203. C₅×SD₁₆	C₂×C₃×C₄	semidirect
204. C₅×D₈	C₂×C₃×C₄	semidirect
205. C₂×C₅×C₈	C₂×C₃×C₄	semidirect
206. #689	C₂×C₃×C₄	semidirect
207. #688	C₂×C₃×C₄	semidirect
208. #687	C₂×C₃×C₄	semidirect
209. #686	C₂×C₃×C₄	semidirect
210. C₂×#168	C₂×C₃×C₄	semidirect
211. Dic₂₀	C₂×C₃×C₄	semidirect
212. D₄₀	C₂×C₃×C₄	semidirect
213. #677	C₂×C₃×C₄	semidirect
214. C₈×D₅	C₂×C₃×C₄	semidirect
215. C₈×D₅	C₂×C₃×C₄	semidirect
235. C₃×AGL(1,𝔽₅)	#135	direct (N × H)
247. C₃×#41	C₂×AGL(1,𝔽₅)	semidirect
257. C₃×Q₁₆	C₂×AGL(1,𝔽₅)	semidirect
261. C₃×SD₁₆	C₂×AGL(1,𝔽₅)	semidirect
264. C₃×D₈	C₂×AGL(1,𝔽₅)	semidirect
273. C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	semidirect
322. #135	C₃×AGL(1,𝔽₅)	direct (N × H)
389. AGL(1,𝔽₅)	C₃×#135	direct (N × H)
401. #41	C₂×C₃×AGL(1,𝔽₅)	semidirect
411. Q₁₆	C₂×C₃×AGL(1,𝔽₅)	semidirect
415. SD₁₆	C₂×C₃×AGL(1,𝔽₅)	semidirect
418. D₈	C₂×C₃×AGL(1,𝔽₅)	semidirect
427. C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	semidirect
437. C₃×C₅	C₄×#135	semidirect
499. C₅	C₃×C₄×#135	semidirect
516. C₃	AGL(1,𝔽₅)×#135	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×#41×AGL(1,𝔽₅)	2
C₃×D₅×#135	2
C₃×#65794	2
C₃×Q₁₆×AGL(1,𝔽₅)	2
C₃×SD₁₆×AGL(1,𝔽₅)	2
C₃×D₈×AGL(1,𝔽₅)	2
C₃×#65721	2
C₂×C₃×C₈×AGL(1,𝔽₅)	2
AGL(1,𝔽₅)×#135	3
C₃×C₄×#135	5

Subgroups — C3×AGL(1,𝔽5)×#135 #14416224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×AGL(1,𝔽₅)×#135 #14416224