Subgroups — C₅×AGL(1,𝔽₅)×#104 #72275718

Order 3200.

742 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#41×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	C₅×D₅⋊#104	C₂	1	1	yes	maximal, characteristic
3	C₅×M₁₆×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×C₅×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×D₅×#104	C₂	1	1	yes	maximal, characteristic
6	C₅×#64900	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#64862	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×D₅×#41	C₂²	1	1	yes	characteristic
9	C₅×Q₈×AGL(1,𝔽₅)	C₄	1	2	yes	complemented
10	C₅×D₄×AGL(1,𝔽₅)	—	2	2	no	—
11	C₅×D₄×AGL(1,𝔽₅)	C₄	1	2	yes	complemented
12	C₅×#4130	—	2	1	no	—
13	C₅×#4130	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×AGL(1,𝔽₅)	—	2	2	no	—
15	C₂×C₄×C₅×AGL(1,𝔽₅)	—	2	1	no	—
16	C₂×C₄×C₅×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
18	C₅×D₅⋊M₁₆	C₂²	1	1	yes	characteristic
19	C₅×D₅×M₁₆	C₂²	1	1	yes	characteristic
20	C₄²×C₅×D₅	C₂²	1	1	yes	characteristic
21	C₅×#4010	C₄	1	2	yes	complemented
22	C₅×#4008	C₄	1	2	yes	complemented
23	C₅×#3992	—	2	1	no	—
24	C₅×C₈×AGL(1,𝔽₅)	—	2	1	no	—
25	C₅²×#104	C₄	1	1	yes	characteristic, complemented
26	C₅×#3969	C₄	1	1	yes	characteristic, complemented
27	C₅×#3957	C₄	1	1	yes	characteristic, complemented
28	C₅×#3937	C₄	1	1	yes	characteristic, complemented
29	AGL(1,𝔽₅)×#104	C₅	1	1	yes	maximal, characteristic, direct product factor
30	C₄×C₅×#104	—	5	1	no	maximal
31	C₂²×C₅×AGL(1,𝔽₅)	—	2	1	no	—
32	C₂²×C₅×AGL(1,𝔽₅)	—	2	2	no	—
33	C₂²×C₅×AGL(1,𝔽₅)	D₄	1	2	yes	—
34	C₅²×#41	C₂×C₄	1	1	yes	characteristic
35	C₅×#713	C₂×C₄	1	1	yes	characteristic
36	C₅×Q₈×D₅	C₂×C₄	1	1	yes	characteristic
37	C₅×#711	—	2	1	no	—
38	C₅×#711	C₂×C₄	1	1	yes	characteristic
39	C₅×D₄×D₅	—	2	1	no	—
40	C₅×D₄×D₅	C₂×C₄	1	1	yes	characteristic
41	C₅×#709	—	2	1	no	—
42	C₅×#709	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅×D₅	—	2	1	no	—
44	C₂×C₄×C₅×D₅	—	2	1	no	—
45	C₂×C₄×C₅×D₅	C₂³	1	1	yes	characteristic
46	C₅×C₂²⋊AGL(1,𝔽₅)	—	2	1	no	—
47	C₅×C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
48	C₅×#704	—	2	1	no	—
49	C₅×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
50	C₅×C₄⋊AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
51	C₄×C₅×AGL(1,𝔽₅)	—	2	8	no	—
52	C₄×C₅×AGL(1,𝔽₅)	—	2	2	no	—
53	C₄×C₅×AGL(1,𝔽₅)	—	2	1	no	—
54	C₄×C₅×AGL(1,𝔽₅)	—	2	1	no	—
55	C₄×C₅×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
56	C₄×C₅×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	complemented
57	C₄×C₅×AGL(1,𝔽₅)	D₄	1	1	yes	characteristic
58	C₄×C₅×AGL(1,𝔽₅)	D₄	1	1	yes	characteristic
59	C₅×#700	C₂×C₄	1	2	yes	complemented
60	C₅×D₅⋊C₈	—	2	1	no	—
61	C₅²×M₁₆	C₂×C₄	1	1	yes	characteristic, complemented
62	C₄²×C₅²	C₂×C₄	1	1	yes	characteristic, complemented
63	C₄×C₅×Dic₅	—	2	1	no	—
64	C₄×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic, complemented
65	C₅×#681	C₂×C₄	1	1	yes	characteristic, complemented
66	C₅×#676	—	2	1	no	—
67	C₅×C₈×D₅	—	2	1	no	—
68	#41×AGL(1,𝔽₅)	C₂×C₅	1	1	yes	characteristic
69	C₄×C₅×#41	—	5	1	no	—
70	D₅⋊#104	C₂×C₅	1	1	yes	characteristic
71	M₁₆×AGL(1,𝔽₅)	C₂×C₅	1	1	yes	characteristic, complemented
72	C₄²×AGL(1,𝔽₅)	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂×C₅×#104	—	5	1	no	—
74	C₂×C₅×#104	—	5	1	no	—
75	C₄×C₅×M₁₆	—	5	1	no	—
76	C₄³×C₅	—	5	1	no	—
77	D₅×#104	C₂×C₅	1	1	yes	characteristic
78	#64900	C₂×C₅	1	1	yes	characteristic, complemented
79	#64862	C₂×C₅	1	1	yes	characteristic, complemented
80	C₅×#339	—	5	1	no	—
81	C₅×#339	—	5	1	no	—
82	C₂²×C₅×D₅	—	2	1	no	—
83	C₂²×C₅×D₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₅×AGL(1,𝔽₅)	—	2	4	no	—
85	C₂×C₅×AGL(1,𝔽₅)	—	2	2	no	—
86	C₂×C₅×AGL(1,𝔽₅)	—	2	4	no	—
87	C₂×C₅×AGL(1,𝔽₅)	—	2	2	no	—
88	C₂×C₅×AGL(1,𝔽₅)	—	4	4	no	—
89	C₂×C₅×AGL(1,𝔽₅)	C₂²⋊C₄	1	2	yes	—
90	C₂×C₅×AGL(1,𝔽₅)	C₂²⋊C₄	1	2	yes	—
91	C₅²×Q₈	C₄²	1	1	yes	characteristic, complemented
92	C₅²×D₄	—	2	1	no	—
93	C₅²×D₄	C₄²	1	1	yes	characteristic, complemented
94	C₂×C₄×C₅²	—	2	1	no	—
95	C₂×C₄×C₅²	—	2	1	no	—
96	C₂×C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
97	C₅×C₂²⋊D₅	—	2	1	no	—
98	C₅×C₂²⋊D₅	—	2	1	no	—
99	C₅×C₂²⋊D₅	—	2	1	no	—
100	C₂×C₅×Dic₅	—	2	1	no	—
101	C₂×C₅×Dic₅	—	2	1	no	—
102	C₂×C₅×Dic₅	#41	1	1	yes	characteristic
103	C₅×D₂₀	—	2	1	no	—
104	C₅×D₂₀	C₄²	1	1	yes	characteristic, complemented
105	C₄×C₅×D₅	—	2	4	no	—
106	C₄×C₅×D₅	—	2	1	no	—
107	C₄×C₅×D₅	—	2	1	no	—
108	C₄×C₅×D₅	—	2	1	no	—
109	C₄×C₅×D₅	C₂²×C₄	1	1	yes	characteristic
110	C₄×C₅×D₅	C₂²×C₄	1	1	yes	characteristic
111	C₄×C₅×D₅	C₂×D₄	1	1	yes	characteristic
112	C₄×C₅×D₅	#41	1	1	yes	characteristic
113	C₅×Dic₁₀	—	2	1	no	—
114	C₅×Dic₁₀	C₄²	1	1	yes	characteristic, complemented
115	C₅×#170	—	2	2	no	—
116	C₅²×C₈	—	2	1	no	—
117	C₅×#168	—	2	1	no	—
118	C₂×C₅×#41	—	5	1	no	—
119	D₅×#41	C₂²×C₅	1	1	yes	characteristic
120	Q₈×AGL(1,𝔽₅)	C₄×C₅	1	2	yes	complemented
121	D₄×AGL(1,𝔽₅)	—	2	2	no	—
122	D₄×AGL(1,𝔽₅)	C₄×C₅	1	2	yes	complemented
123	#4130	—	2	1	no	—
124	#4130	C₂²×C₅	1	1	yes	characteristic
125	C₂×C₄×AGL(1,𝔽₅)	—	2	2	no	—
126	C₂×C₄×AGL(1,𝔽₅)	—	2	1	no	—
127	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₅	1	1	yes	characteristic
128	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₅	1	1	yes	characteristic
129	D₅⋊M₁₆	C₂²×C₅	1	1	yes	characteristic
130	C₂×C₅×M₁₆	—	5	1	no	—
131	C₂×C₅×M₁₆	—	5	1	no	—
132	C₄×C₅×Q₈	—	5	2	no	—
133	C₄×C₅×D₄	—	5	2	no	—
134	C₄×C₅×D₄	—	10	2	no	—
135	C₅×#117	—	5	1	no	—
136	C₅×#117	—	10	1	no	—
137	C₂×C₄²×C₅	—	5	1	no	—
138	C₂×C₄²×C₅	—	5	1	no	—
139	C₂×C₄²×C₅	—	5	1	no	—
140	C₂×C₄²×C₅	—	10	1	no	—
141	C₂×C₄²×C₅	—	10	2	no	—
142	D₅×M₁₆	C₂²×C₅	1	1	yes	characteristic
143	C₄²×D₅	C₂²×C₅	1	1	yes	characteristic
144	#4010	C₄×C₅	1	2	yes	complemented
145	#4008	C₄×C₅	1	2	yes	complemented
146	#3992	—	2	1	no	—
147	C₈×AGL(1,𝔽₅)	—	2	1	no	—
148	C₅×#104	—	4	1	no	—
149	C₅×#104	—	5	2	no	—
150	C₅×#104	—	5	2	no	—
151	C₅×#104	—	5	1	no	—
152	C₅×#104	—	5	1	no	—
153	C₅×#104	—	5	1	no	—
154	C₅×#104	C₄×C₅	1	1	yes	characteristic, complemented
155	C₅×#104	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
156	C₅×#97	—	10	1	no	—
157	C₄×C₅×C₈	—	10	1	no	—
158	#3969	C₄×C₅	1	1	yes	characteristic, complemented
159	#3957	C₄×C₅	1	1	yes	characteristic, complemented
160	#3937	C₄×C₅	1	1	yes	characteristic, complemented
161	C₄×#104	—	5	1	no	Sylow
162	C₂²×C₅²	—	2	1	no	—
163	C₂²×C₅²	C₄×D₄	1	1	yes	characteristic
164	C₂×C₅×D₅	—	2	2	no	—
165	C₂×C₅×D₅	—	2	1	no	—
166	C₂×C₅×D₅	—	4	2	no	—
167	C₂×C₅×D₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₅×D₅	C₄×D₄	1	1	yes	characteristic
169	C₅×AGL(1,𝔽₅)	—	2	4	no	—
170	C₅×AGL(1,𝔽₅)	—	2	2	no	—
171	C₅×AGL(1,𝔽₅)	—	4	2	no	—
172	C₅×AGL(1,𝔽₅)	#104	1	4	yes	direct product factor
173	C₄×C₅²	—	2	2	no	—
174	C₄×C₅²	—	2	1	no	—
175	C₄×C₅²	C₂×C₄²	1	1	yes	characteristic
176	C₄×C₅²	C₄×D₄	1	1	yes	characteristic
177	C₅×Dic₅	—	2	2	no	—
178	C₅×Dic₅	—	2	1	no	—
179	C₅×Dic₅	#117	1	1	yes	characteristic
180	C₅×Dic₅	C₄×D₄	1	1	yes	characteristic
181	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
182	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
183	C₂²×AGL(1,𝔽₅)	C₅×D₄	1	2	yes	—
184	C₅×#41	—	4	1	no	—
185	C₅×#41	—	5	1	no	—
186	C₅×#41	—	5	1	no	—
187	C₅×#41	—	5	1	no	—
188	C₅×#41	—	10	1	no	—
189	C₅×#41	—	10	1	no	—
190	C₅×#41	C₂×C₄×C₅	1	1	yes	characteristic
191	C₅×#41	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
192	C₂×C₅×Q₈	—	5	1	no	—
193	C₂×C₅×D₄	—	5	1	no	—
194	C₂×C₅×D₄	—	10	1	no	—
195	C₂²×C₄×C₅	—	5	1	no	—
196	C₂²×C₄×C₅	—	5	2	no	—
197	C₂²×C₄×C₅	—	10	1	no	—
198	C₂²×C₄×C₅	—	10	1	no	—
199	C₂²×C₄×C₅	—	10	2	no	—
200	C₂²×C₄×C₅	—	10	1	no	—
201	#713	C₂×C₄×C₅	1	1	yes	characteristic
202	Q₈×D₅	C₂×C₄×C₅	1	1	yes	characteristic
203	#711	—	2	1	no	—
204	#711	C₂×C₄×C₅	1	1	yes	characteristic
205	D₄×D₅	—	2	1	no	—
206	D₄×D₅	C₂×C₄×C₅	1	1	yes	characteristic
207	#709	—	2	1	no	—
208	#709	C₂×C₄×C₅	1	1	yes	characteristic
209	C₂×C₄×D₅	—	2	1	no	—
210	C₂×C₄×D₅	—	2	1	no	—
211	C₂×C₄×D₅	C₂³×C₅	1	1	yes	characteristic
212	C₂²⋊AGL(1,𝔽₅)	—	2	1	no	—
213	C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
214	#704	—	2	1	no	—
215	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
216	C₄⋊AGL(1,𝔽₅)	C₂×C₄×C₅	1	2	yes	—
217	C₄×AGL(1,𝔽₅)	—	2	8	no	—
218	C₄×AGL(1,𝔽₅)	—	2	2	no	—
219	C₄×AGL(1,𝔽₅)	—	2	1	no	—
220	C₄×AGL(1,𝔽₅)	—	2	1	no	—
221	C₄×AGL(1,𝔽₅)	C₂×C₄×C₅	1	2	yes	—
222	C₄×AGL(1,𝔽₅)	C₂×C₄×C₅	1	2	yes	complemented
223	C₄×AGL(1,𝔽₅)	C₅×D₄	1	1	yes	characteristic
224	C₄×AGL(1,𝔽₅)	C₅×D₄	1	1	yes	characteristic
225	#700	C₂×C₄×C₅	1	2	yes	complemented
226	D₅⋊C₈	—	2	1	no	—
227	C₅×M₁₆	—	4	1	no	—
228	C₅×M₁₆	—	5	2	no	—
229	C₅×M₁₆	—	5	1	no	—
230	C₅×M₁₆	—	10	1	no	—
231	C₅×M₁₆	—	10	1	no	—
232	C₅×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic, complemented
233	C₅×M₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
234	C₂×C₅×C₈	—	10	1	no	—
235	C₂×C₅×C₈	—	10	1	no	—
236	C₅×C₄⋊C₄	—	5	2	no	—
237	C₅×C₄⋊C₄	—	10	2	no	—
238	C₅×C₂²⋊C₄	—	10	2	no	—
239	C₅×C₂²⋊C₄	—	10	1	no	—
240	C₄²×C₅	—	4	1	no	—
241	C₄²×C₅	—	5	1	no	—
242	C₄²×C₅	—	5	1	no	—
243	C₄²×C₅	—	5	2	no	—
244	C₄²×C₅	—	5	2	no	—
245	C₄²×C₅	—	5	1	no	—
246	C₄²×C₅	—	10	1	no	—
247	C₄²×C₅	—	10	1	no	—
248	C₄²×C₅	—	10	1	no	—
249	C₄²×C₅	—	10	2	no	—
250	C₄²×C₅	—	10	8	no	—
251	C₄²×C₅	C₂×C₄×C₅	1	1	yes	characteristic, complemented
252	C₄²×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
253	C₄×Dic₅	—	2	1	no	—
254	C₄×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic, complemented
255	#681	C₂×C₄×C₅	1	1	yes	characteristic, complemented
256	#676	—	2	1	no	—
257	C₈×D₅	—	2	1	no	—
258	C₄×#41	—	5	1	no	—
259	C₂×#104	—	5	1	no	—
260	C₂×#104	—	5	1	no	—
261	C₄×M₁₆	—	5	1	no	—
262	C₄³	—	5	1	no	—
263	#339	—	5	1	no	—
264	#339	—	5	1	no	—
265	C₂×C₅²	—	2	1	no	—
266	C₂×C₅²	—	4	1	no	—
267	C₂×C₅²	C₄×C₂²⋊C₄	1	1	yes	characteristic
268	C₅×D₅	—	2	1	no	—
269	C₅×D₅	—	4	1	no	—
270	C₅×D₅	C₂×#104	1	2	yes	—
271	C₂³×C₅	—	5	1	no	—
272	C₂³×C₅	—	10	1	no	—
273	C₂²×D₅	—	2	1	no	—
274	C₂²×D₅	C₂×C₅×D₄	1	1	yes	characteristic
275	C₂×AGL(1,𝔽₅)	—	2	4	no	—
276	C₂×AGL(1,𝔽₅)	—	2	2	no	—
277	C₂×AGL(1,𝔽₅)	—	2	4	no	—
278	C₂×AGL(1,𝔽₅)	—	2	2	no	—
279	C₂×AGL(1,𝔽₅)	—	4	4	no	—
280	C₂×AGL(1,𝔽₅)	C₅×C₂²⋊C₄	1	2	yes	—
281	C₂×AGL(1,𝔽₅)	C₅×C₂²⋊C₄	1	2	yes	—
282	C₅×Q₈	—	4	1	no	—
283	C₅×Q₈	—	5	1	no	—
284	C₅×Q₈	—	10	1	no	—
285	C₅×Q₈	C₄²×C₅	1	1	yes	characteristic, complemented
286	C₅×Q₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
287	C₅×D₄	—	2	1	no	—
288	C₅×D₄	—	2	1	no	—
289	C₅×D₄	—	4	1	no	—
290	C₅×D₄	—	5	1	no	—
291	C₅×D₄	—	8	1	no	—
292	C₅×D₄	—	10	1	no	—
293	C₅×D₄	—	10	1	no	—
294	C₅×D₄	—	10	1	no	—
295	C₅×D₄	—	10	1	no	—
296	C₅×D₄	C₄²×C₅	1	1	yes	characteristic, complemented
297	C₅×D₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
298	C₂×C₄×C₅	—	2	1	no	—
299	C₂×C₄×C₅	—	2	1	no	—
300	C₂×C₄×C₅	—	2	1	no	—
301	C₂×C₄×C₅	—	2	1	no	—
302	C₂×C₄×C₅	—	4	1	no	—
303	C₂×C₄×C₅	—	5	2	no	—
304	C₂×C₄×C₅	—	5	1	no	—
305	C₂×C₄×C₅	—	5	1	no	—
306	C₂×C₄×C₅	—	5	1	no	—
307	C₂×C₄×C₅	—	5	1	no	—
308	C₂×C₄×C₅	—	5	1	no	—
309	C₂×C₄×C₅	—	5	2	no	—
310	C₂×C₄×C₅	—	8	1	no	—
311	C₂×C₄×C₅	—	8	1	no	—
312	C₂×C₄×C₅	—	10	1	no	—
313	C₂×C₄×C₅	—	10	1	no	—
314	C₂×C₄×C₅	—	10	4	no	—
315	C₂×C₄×C₅	—	10	1	no	—
316	C₂×C₄×C₅	—	10	4	no	—
317	C₂×C₄×C₅	—	10	2	no	—
318	C₂×C₄×C₅	—	10	4	no	—
319	C₂×C₄×C₅	—	10	1	no	—
320	C₂×C₄×C₅	—	10	1	no	—
321	C₂×C₄×C₅	—	10	2	no	—
322	C₂×C₄×C₅	—	20	4	no	—
323	C₂×C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
324	C₂×C₄×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
325	C₂²⋊D₅	—	2	1	no	—
326	C₂²⋊D₅	—	2	1	no	—
327	C₂²⋊D₅	—	2	1	no	—
328	C₂×Dic₅	—	2	1	no	—
329	C₂×Dic₅	—	2	1	no	—
330	C₂×Dic₅	C₅×#41	1	1	yes	characteristic
331	D₂₀	—	2	1	no	—
332	D₂₀	C₄²×C₅	1	1	yes	characteristic, complemented
333	C₄×D₅	—	2	4	no	—
334	C₄×D₅	—	2	1	no	—
335	C₄×D₅	—	2	1	no	—
336	C₄×D₅	—	2	1	no	—
337	C₄×D₅	C₂²×C₄×C₅	1	1	yes	characteristic
338	C₄×D₅	C₂²×C₄×C₅	1	1	yes	characteristic
339	C₄×D₅	C₂×C₅×D₄	1	1	yes	characteristic
340	C₄×D₅	C₅×#41	1	1	yes	characteristic
341	Dic₁₀	—	2	1	no	—
342	Dic₁₀	C₄²×C₅	1	1	yes	characteristic, complemented
343	#170	—	2	2	no	—
344	C₅×C₈	—	2	1	no	—
345	C₅×C₈	—	2	1	no	—
346	C₅×C₈	—	8	1	no	—
347	C₅×C₈	—	10	2	no	—
348	C₅×C₈	—	10	1	no	—
349	#168	—	2	1	no	—
350	C₂×#41	—	5	1	no	—
351	C₂×M₁₆	—	5	1	no	—
352	C₂×M₁₆	—	5	1	no	—
353	C₄×Q₈	—	5	2	no	—
354	C₄×D₄	—	5	2	no	—
355	C₄×D₄	—	10	2	no	—
356	#117	—	5	1	no	—
357	#117	—	10	1	no	—
358	C₂×C₄²	—	5	1	no	—
359	C₂×C₄²	—	5	1	no	—
360	C₂×C₄²	—	5	1	no	—
361	C₂×C₄²	—	10	1	no	—
362	C₂×C₄²	—	10	2	no	—
363	#104	—	5	2	no	—
364	#104	—	5	2	no	—
365	#104	—	5	1	no	—
366	#104	—	5	1	no	—
367	#104	—	5	1	no	—
368	#104	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
369	#97	—	10	1	no	—
370	C₄×C₈	—	10	1	no	—
371	C₅²	C₄×#104	1	1	yes	characteristic, complemented, Sylow
372	C₂²×C₅	—	2	1	no	—
373	C₂²×C₅	—	2	1	no	—
374	C₂²×C₅	—	4	1	no	—
375	C₂²×C₅	—	5	1	no	—
376	C₂²×C₅	—	5	1	no	—
377	C₂²×C₅	—	8	1	no	—
378	C₂²×C₅	—	10	2	no	—
379	C₂²×C₅	—	10	1	no	—
380	C₂²×C₅	—	20	2	no	—
381	C₂²×C₅	C₄×C₅×D₄	1	1	yes	characteristic
382	C₂²×C₅	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
383	C₂×D₅	—	2	2	no	—
384	C₂×D₅	—	2	1	no	—
385	C₂×D₅	—	4	2	no	—
386	C₂×D₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
387	C₂×D₅	C₄×C₅×D₄	1	1	yes	characteristic
388	AGL(1,𝔽₅)	—	2	4	no	—
389	AGL(1,𝔽₅)	—	2	2	no	—
390	AGL(1,𝔽₅)	—	4	2	no	—
391	AGL(1,𝔽₅)	C₅×#104	1	4	yes	direct product factor
392	C₄×C₅	—	2	2	no	—
393	C₄×C₅	—	2	2	no	—
394	C₄×C₅	—	2	1	no	—
395	C₄×C₅	—	2	1	no	—
396	C₄×C₅	—	4	1	no	—
397	C₄×C₅	—	4	1	no	—
398	C₄×C₅	—	5	4	no	—
399	C₄×C₅	—	5	1	no	—
400	C₄×C₅	—	5	1	no	—
401	C₄×C₅	—	8	1	no	—
402	C₄×C₅	—	8	2	no	—
403	C₄×C₅	—	10	2	no	—
404	C₄×C₅	—	10	2	no	—
405	C₄×C₅	—	10	4	no	—
406	C₄×C₅	—	10	1	no	—
407	C₄×C₅	—	20	2	no	—
408	C₄×C₅	C₂×C₄²×C₅	1	1	yes	characteristic
409	C₄×C₅	C₄×C₅×D₄	1	1	yes	characteristic
410	C₄×C₅	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
411	C₄×C₅	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
412	Dic₅	—	2	2	no	—
413	Dic₅	—	2	1	no	—
414	Dic₅	C₅×#117	1	1	yes	characteristic
415	Dic₅	C₄×C₅×D₄	1	1	yes	characteristic
416	#41	—	5	1	no	—
417	#41	—	5	1	no	—
418	#41	—	5	1	no	—
419	#41	—	10	1	no	—
420	#41	—	10	1	no	—
421	#41	C₂×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
422	C₂×Q₈	—	5	1	no	—
423	C₂×D₄	—	5	1	no	—
424	C₂×D₄	—	10	1	no	—
425	C₂²×C₄	—	5	1	no	—
426	C₂²×C₄	—	5	2	no	—
427	C₂²×C₄	—	10	1	no	—
428	C₂²×C₄	—	10	1	no	—
429	C₂²×C₄	—	10	2	no	—
430	C₂²×C₄	—	10	1	no	—
431	M₁₆	—	5	2	no	—
432	M₁₆	—	5	1	no	—
433	M₁₆	—	10	1	no	—
434	M₁₆	—	10	1	no	—
435	M₁₆	C₂×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
436	C₂×C₈	—	10	1	no	—
437	C₂×C₈	—	10	1	no	—
438	C₄⋊C₄	—	5	2	no	—
439	C₄⋊C₄	—	10	2	no	—
440	C₂²⋊C₄	—	10	2	no	—
441	C₂²⋊C₄	—	10	1	no	—
442	C₄²	—	5	1	no	—
443	C₄²	—	5	1	no	—
444	C₄²	—	5	2	no	—
445	C₄²	—	5	2	no	—
446	C₄²	—	5	1	no	—
447	C₄²	—	10	1	no	—
448	C₄²	—	10	1	no	—
449	C₄²	—	10	1	no	—
450	C₄²	—	10	2	no	—
451	C₄²	—	10	8	no	—
452	C₄²	C₂×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
453	C₂×C₅	—	2	1	no	—
454	C₂×C₅	—	2	1	no	—
455	C₂×C₅	—	4	1	no	—
456	C₂×C₅	—	4	1	no	—
457	C₂×C₅	—	4	1	no	—
458	C₂×C₅	—	5	2	no	—
459	C₂×C₅	—	8	1	no	—
460	C₂×C₅	—	10	1	no	—
461	C₂×C₅	—	16	1	no	—
462	C₂×C₅	—	20	1	no	—
463	C₂×C₅	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
464	C₂×C₅	C₂²⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
465	D₅	—	2	1	no	—
466	D₅	—	4	1	no	—
467	D₅	C₂×C₅×#104	1	2	yes	—
468	C₂³	—	5	1	no	—
469	C₂³	—	10	1	no	—
470	Q₈	—	5	1	no	—
471	Q₈	—	10	1	no	—
472	Q₈	C₄×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
473	D₄	—	2	1	no	—
474	D₄	—	5	1	no	—
475	D₄	—	10	1	no	—
476	D₄	—	10	1	no	—
477	D₄	—	10	1	no	—
478	D₄	—	10	1	no	—
479	D₄	C₄×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	5	2	no	—
483	C₂×C₄	—	5	1	no	—
484	C₂×C₄	—	5	1	no	—
485	C₂×C₄	—	5	1	no	—
486	C₂×C₄	—	5	1	no	—
487	C₂×C₄	—	5	1	no	—
488	C₂×C₄	—	5	2	no	—
489	C₂×C₄	—	10	1	no	—
490	C₂×C₄	—	10	1	no	—
491	C₂×C₄	—	10	4	no	—
492	C₂×C₄	—	10	1	no	—
493	C₂×C₄	—	10	4	no	—
494	C₂×C₄	—	10	2	no	—
495	C₂×C₄	—	10	4	no	—
496	C₂×C₄	—	10	1	no	—
497	C₂×C₄	—	10	1	no	—
498	C₂×C₄	—	10	2	no	—
499	C₂×C₄	—	20	4	no	—
500	C₂×C₄	C₂²×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
501	C₈	—	2	1	no	—
502	C₈	—	10	2	no	—
503	C₈	—	10	1	no	—
504	C₅	—	4	1	no	—
505	C₅	C₄×C₅×#104	1	1	yes	characteristic, complemented
506	C₅	AGL(1,𝔽₅)×#104	1	1	yes	characteristic, direct product factor
507	C₂²	—	2	1	no	—
508	C₂²	—	5	1	no	—
509	C₂²	—	5	1	no	—
510	C₂²	—	10	2	no	—
511	C₂²	—	10	1	no	—
512	C₂²	—	20	2	no	—
513	C₂²	C₅×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
514	C₄	—	2	2	no	—
515	C₄	—	2	1	no	—
516	C₄	—	5	4	no	—
517	C₄	—	5	1	no	—
518	C₄	—	5	1	no	—
519	C₄	—	10	2	no	—
520	C₄	—	10	2	no	—
521	C₄	—	10	4	no	—
522	C₄	—	10	1	no	—
523	C₄	—	20	2	no	—
524	C₄	C₂×C₄×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
525	C₄	C₅×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
526	C₂	—	2	1	no	—
527	C₂	—	4	1	no	—
528	C₂	—	5	2	no	—
529	C₂	—	10	1	no	—
530	C₂	—	20	1	no	—
531	C₂	C₅×C₂²⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₄²×C₅
Frattini	C₂×C₄	C₂²×C₅×AGL(1,𝔽₅)
Fitting	C₅²×#104	C₄
socle	C₂×C₅²	C₄×C₂²⋊C₄
prime radical	C₅×D₅×#41	C₂²
cosocle	C₂×C₄×D₅	C₂³×C₅
hypercenter	C₅×#104	AGL(1,𝔽₅)
hypocenter	C₅	C₄×C₅×#104
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#104

Complements

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

N	H ≅ G/N	type
3. C₅×M₁₆×AGL(1,𝔽₅)	C₂	semidirect
4. C₄²×C₅×AGL(1,𝔽₅)	C₂	semidirect
6. C₅×#64900	C₂	semidirect
7. C₅×#64862	C₂	semidirect
9. C₅×Q₈×AGL(1,𝔽₅)	C₄	semidirect
11. C₅×D₄×AGL(1,𝔽₅)	C₄	semidirect
21. C₅×#4010	C₄	semidirect
22. C₅×#4008	C₄	semidirect
25. C₅²×#104	C₄	semidirect
26. C₅×#3969	C₄	semidirect
27. C₅×#3957	C₄	semidirect
28. C₅×#3937	C₄	semidirect
29. AGL(1,𝔽₅)×#104	C₅	direct (N × H)
56. C₄×C₅×AGL(1,𝔽₅)	C₂×C₄	semidirect
59. C₅×#700	C₂×C₄	semidirect
61. C₅²×M₁₆	C₂×C₄	semidirect
62. C₄²×C₅²	C₂×C₄	semidirect
64. C₄×C₅×Dic₅	C₂×C₄	semidirect
65. C₅×#681	C₂×C₄	semidirect
71. M₁₆×AGL(1,𝔽₅)	C₂×C₅	semidirect
72. C₄²×AGL(1,𝔽₅)	C₂×C₅	semidirect
78. #64900	C₂×C₅	semidirect
79. #64862	C₂×C₅	semidirect
91. C₅²×Q₈	C₄²	semidirect
93. C₅²×D₄	C₄²	semidirect
104. C₅×D₂₀	C₄²	semidirect
114. C₅×Dic₁₀	C₄²	semidirect
120. Q₈×AGL(1,𝔽₅)	C₄×C₅	semidirect
122. D₄×AGL(1,𝔽₅)	C₄×C₅	semidirect
144. #4010	C₄×C₅	semidirect
145. #4008	C₄×C₅	semidirect
154. C₅×#104	C₄×C₅	semidirect
155. C₅×#104	AGL(1,𝔽₅)	direct (N × H)
158. #3969	C₄×C₅	semidirect
159. #3957	C₄×C₅	semidirect
160. #3937	C₄×C₅	semidirect
172. C₅×AGL(1,𝔽₅)	#104	direct (N × H)
222. C₄×AGL(1,𝔽₅)	C₂×C₄×C₅	semidirect
225. #700	C₂×C₄×C₅	semidirect
232. C₅×M₁₆	C₂×C₄×C₅	semidirect
233. C₅×M₁₆	C₂×AGL(1,𝔽₅)	semidirect
251. C₄²×C₅	C₂×C₄×C₅	semidirect
252. C₄²×C₅	C₂×AGL(1,𝔽₅)	semidirect
254. C₄×Dic₅	C₂×C₄×C₅	semidirect
255. #681	C₂×C₄×C₅	semidirect
285. C₅×Q₈	C₄²×C₅	semidirect
286. C₅×Q₈	C₄×AGL(1,𝔽₅)	semidirect
296. C₅×D₄	C₄²×C₅	semidirect
297. C₅×D₄	C₄×AGL(1,𝔽₅)	semidirect
332. D₂₀	C₄²×C₅	semidirect
342. Dic₁₀	C₄²×C₅	semidirect
368. #104	C₅×AGL(1,𝔽₅)	direct (N × H)
371. C₅²	C₄×#104	semidirect
391. AGL(1,𝔽₅)	C₅×#104	direct (N × H)
435. M₁₆	C₂×C₅×AGL(1,𝔽₅)	semidirect
452. C₄²	C₂×C₅×AGL(1,𝔽₅)	semidirect
472. Q₈	C₄×C₅×AGL(1,𝔽₅)	semidirect
479. D₄	C₄×C₅×AGL(1,𝔽₅)	semidirect
505. C₅	C₄×C₅×#104	semidirect
506. C₅	AGL(1,𝔽₅)×#104	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#41×AGL(1,𝔽₅)	2
C₅×D₅⋊#104	2
C₅×M₁₆×AGL(1,𝔽₅)	2
C₄²×C₅×AGL(1,𝔽₅)	2
C₅×D₅×#104	2
C₅×#64900	2
C₅×#64862	2
AGL(1,𝔽₅)×#104	5
C₄×C₅×#104	5

Subgroups — C5×AGL(1,𝔽5)×#104 #72275718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×AGL(1,𝔽₅)×#104 #72275718