Subgroups — C₅×#82466 #14329508

Order 1920.

1264 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#947	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#893	C₂	1	1	yes	maximal, characteristic
4	C₅×#5229	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4990	C₂	1	2	yes	maximal, complemented
6	C₅×#4989	C₂	1	2	yes	maximal, complemented
7	C₅×Dic₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4940	C₂	1	2	yes	maximal, complemented
9	C₅×#4939	C₂	1	1	yes	maximal, characteristic
10	C₄×C₅×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal, complemented
11	C₄×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#2367	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
16	C₃×C₄×C₅×Q₈	—	2	4	no	—
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
21	C₅×#947	C₄	1	8	yes	complemented
22	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
24	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
32	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
33	C₅×#893	—	2	4	no	—
34	C₅×C₃⋊#95	C₂²	1	1	yes	characteristic
35	C₅×C₃⋊#95	C₂²	1	1	yes	characteristic
36	C₅×C₃⋊#95	C₂²	1	2	yes	—
37	C₅×C₃⋊#95	C₂²	1	2	yes	—
38	C₅×C₃⋊#95	C₂²	1	2	yes	—
39	#82466	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
40	C₂×C₅×#124	—	3	1	no	—
41	C₂×C₄×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
42	C₂×C₄×C₅×D₄	—	3	1	no	—
43	C₅×#391	—	3	1	no	—
44	C₅×#390	—	3	1	no	—
45	C₅×#388	—	3	2	no	—
46	C₅×#388	—	3	2	no	—
47	C₅×#385	—	3	2	no	—
48	C₄×C₅×C₄⋊C₄	—	3	1	no	—
49	C₄×C₅×C₄⋊C₄	—	3	1	no	—
50	C₄×C₅×C₂²⋊C₄	—	3	2	no	—
51	C₂³×C₅×D₃	C₂³	1	2	yes	—
52	C₂×C₃×C₅×Q₈	—	2	2	no	—
53	C₂×C₃×C₅×Q₈	—	2	2	no	—
54	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	complemented
55	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
57	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
58	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
62	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
63	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
64	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
65	C₂×C₅×D₁₂	—	2	2	no	—
66	C₂×C₅×D₁₂	—	2	2	no	—
67	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
68	C₂×C₄×C₅×D₃	—	2	4	no	—
69	C₂×C₄×C₅×D₃	—	2	4	no	—
70	C₂×C₄×C₅×D₃	—	4	2	no	—
71	C₃×C₅×C₄⋊C₄	—	2	4	no	—
72	C₃×C₅×C₄⋊C₄	D₄	1	4	yes	—
73	C₃×C₄²×C₅	—	2	4	no	—
74	C₃×C₄²×C₅	—	2	2	no	—
75	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
76	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
77	C₅×(C₂×D₃)⋊C₄	—	4	2	no	—
78	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	complemented
79	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	complemented
80	C₅×C₄⋊Dic₃	—	2	2	no	—
81	C₅×Dic₃⋊C₄	—	2	2	no	—
82	C₅×Dic₃⋊C₄	—	2	2	no	—
83	C₄×C₅×Dic₃	—	2	2	no	—
84	C₄×C₅×Dic₃	—	4	2	no	—
85	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	complemented
86	C₂×C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
87	C₂×#947	C₂×C₅	1	1	yes	characteristic
88	C₂×#893	C₂×C₅	1	1	yes	characteristic
89	#5229	C₂×C₅	1	1	yes	characteristic, complemented
90	#4990	C₂×C₅	1	2	yes	complemented
91	#4989	C₂×C₅	1	2	yes	complemented
92	Dic₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
93	#4940	C₂×C₅	1	2	yes	complemented
94	#4939	C₂×C₅	1	1	yes	characteristic
95	C₄×(C₂×D₃)⋊C₄	C₂×C₅	1	2	yes	complemented
96	C₄×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
97	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
98	C₂²×C₅×D₄	—	3	1	no	—
99	C₂³×C₄×C₅	—	3	2	no	—
100	C₅×#124	—	3	8	no	—
101	C₄×C₅×Q₈	—	2	4	no	—
102	C₄×C₅×D₄	—	6	4	no	—
103	C₂×C₅×C₄⋊C₄	—	3	1	no	—
104	C₂×C₅×C₄⋊C₄	—	3	1	no	—
105	C₂×C₅×C₄⋊C₄	—	3	1	no	—
106	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
107	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
108	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
109	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
110	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
111	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
112	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
113	C₂×C₄²×C₅	—	3	1	no	—
114	C₂×C₄²×C₅	—	3	1	no	—
115	C₂×C₄²×C₅	—	3	2	no	—
116	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
117	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
118	C₅×#95	—	3	1	no	—
119	C₅×#95	—	3	2	no	—
120	C₅×#95	—	3	2	no	—
121	C₅×#95	—	3	1	no	—
122	C₅×#95	—	3	2	no	—
123	#2367	—	3	1	no	Sylow
124	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
125	C₂²×C₅×D₃	—	2	2	no	—
126	C₂²×C₅×D₃	—	2	2	no	—
127	C₂²×C₅×D₃	—	2	2	no	—
128	C₂²×C₅×D₃	—	2	2	no	—
129	C₂²×C₅×D₃	—	2	2	no	—
130	C₂²×C₅×D₃	—	2	2	no	—
131	C₂²×C₅×D₃	C₂²×C₄	1	4	yes	—
132	C₃×C₅×Q₈	—	2	8	no	—
133	C₂×C₃×C₄×C₅	—	2	1	no	—
134	C₂×C₃×C₄×C₅	—	2	2	no	—
135	C₂×C₃×C₄×C₅	—	2	2	no	—
136	C₂×C₃×C₄×C₅	—	2	2	no	—
137	C₂×C₃×C₄×C₅	—	2	2	no	—
138	C₂×C₃×C₄×C₅	—	2	1	no	—
139	C₂×C₃×C₄×C₅	—	2	1	no	—
140	C₂×C₃×C₄×C₅	—	2	1	no	—
141	C₂×C₃×C₄×C₅	—	2	1	no	—
142	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
143	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
144	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
145	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
146	C₂×C₃×C₄×C₅	#41	1	2	yes	—
147	C₂×C₃×C₄×C₅	#41	1	2	yes	—
148	C₂×C₅×Dic₃	—	2	1	no	—
149	C₂×C₅×Dic₃	—	2	1	no	—
150	C₂×C₅×Dic₃	—	2	1	no	—
151	C₂×C₅×Dic₃	—	2	1	no	—
152	C₂×C₅×Dic₃	—	2	1	no	—
153	C₂×C₅×Dic₃	—	2	1	no	—
154	C₂×C₅×Dic₃	—	2	1	no	—
155	C₂×C₅×Dic₃	—	2	1	no	—
156	C₂×C₅×Dic₃	—	2	2	no	—
157	C₂×C₅×Dic₃	—	2	2	no	—
158	C₂×C₅×Dic₃	—	2	2	no	—
159	C₂×C₅×Dic₃	—	2	1	no	—
160	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
161	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
162	C₂×C₅×Dic₃	#41	1	2	yes	—
163	C₅×D₁₂	—	2	8	no	—
164	C₄×C₅×D₃	—	4	8	no	—
165	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
166	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
167	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
168	C₃×C₄×Q₈	—	2	4	no	—
169	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
170	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
171	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
172	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
173	#947	C₄×C₅	1	8	yes	complemented
174	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
175	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
176	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
177	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
178	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
179	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
180	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
181	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
182	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
183	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
184	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
185	#893	—	2	4	no	—
186	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
187	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
188	C₃⋊#95	C₂²×C₅	1	2	yes	—
189	C₃⋊#95	C₂²×C₅	1	2	yes	—
190	C₃⋊#95	C₂²×C₅	1	2	yes	—
191	C₂⁴×C₅	—	3	2	no	—
192	C₂×C₅×Q₈	—	2	2	no	—
193	C₂×C₅×Q₈	—	2	2	no	—
194	C₂×C₅×Q₈	C₄×D₃	1	4	yes	complemented
195	C₂×C₅×D₄	—	3	4	no	—
196	C₂×C₅×D₄	—	6	2	no	—
197	C₂×C₅×D₄	—	6	2	no	—
198	C₂²×C₄×C₅	—	3	1	no	—
199	C₂²×C₄×C₅	—	3	1	no	—
200	C₂²×C₄×C₅	—	3	1	no	—
201	C₂²×C₄×C₅	—	3	2	no	—
202	C₂²×C₄×C₅	—	3	1	no	—
203	C₂²×C₄×C₅	—	6	4	no	—
204	C₂²×C₄×C₅	—	6	4	no	—
205	C₂²×C₄×C₅	—	12	2	no	—
206	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
207	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
208	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
209	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
210	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
211	C₅×C₄⋊C₄	—	2	4	no	—
212	C₅×C₄⋊C₄	—	6	2	no	—
213	C₅×C₄⋊C₄	—	6	2	no	—
214	C₅×C₄⋊C₄	—	6	2	no	—
215	C₅×C₄⋊C₄	#67	1	4	yes	—
216	C₅×C₂²⋊C₄	—	3	8	no	—
217	C₅×C₂²⋊C₄	—	3	8	no	—
218	C₅×C₂²⋊C₄	—	6	4	no	—
219	C₅×C₂²⋊C₄	—	6	4	no	—
220	C₅×C₂²⋊C₄	—	12	2	no	—
221	C₄²×C₅	—	2	4	no	—
222	C₄²×C₅	—	2	2	no	—
223	C₄²×C₅	—	3	4	no	—
224	C₄²×C₅	—	6	2	no	—
225	C₄²×C₅	—	12	2	no	—
226	C₂×#124	—	3	1	no	—
227	C₂×C₄×Q₈	C₅×D₃	1	1	yes	characteristic, complemented
228	C₂×C₄×D₄	—	3	1	no	—
229	#391	—	3	1	no	—
230	#390	—	3	1	no	—
231	#388	—	3	2	no	—
232	#388	—	3	2	no	—
233	#385	—	3	2	no	—
234	C₄×C₄⋊C₄	—	3	1	no	—
235	C₄×C₄⋊C₄	—	3	1	no	—
236	C₄×C₂²⋊C₄	—	3	2	no	—
237	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
238	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
239	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
240	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
241	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
242	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
243	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
244	C₂×C₅×D₃	—	2	4	no	—
245	C₂×C₅×D₃	—	2	4	no	—
246	C₂×C₅×D₃	—	2	4	no	—
247	C₂×C₅×D₃	—	4	2	no	—
248	C₂×C₅×D₃	—	4	2	no	—
249	C₂×C₅×D₃	—	4	2	no	—
250	C₂×C₅×D₃	—	4	2	no	—
251	C₃×C₄×C₅	—	2	4	no	—
252	C₃×C₄×C₅	—	2	4	no	—
253	C₃×C₄×C₅	—	2	2	no	—
254	C₃×C₄×C₅	—	4	1	no	—
255	C₃×C₄×C₅	C₄×D₄	1	4	yes	—
256	C₅×Dic₃	—	2	2	no	—
257	C₅×Dic₃	—	2	2	no	—
258	C₅×Dic₃	—	2	2	no	—
259	C₅×Dic₃	—	4	1	no	—
260	C₅×Dic₃	—	4	2	no	—
261	C₂³×D₃	C₂³×C₅	1	2	yes	—
262	C₂×C₃×Q₈	—	2	2	no	—
263	C₂×C₃×Q₈	—	2	2	no	—
264	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	complemented
265	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
266	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
267	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
268	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
269	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
270	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
271	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
272	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
273	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
274	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
275	C₂×D₁₂	—	2	2	no	—
276	C₂×D₁₂	—	2	2	no	—
277	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	—
278	C₂×C₄×D₃	—	2	4	no	—
279	C₂×C₄×D₃	—	2	4	no	—
280	C₂×C₄×D₃	—	4	2	no	—
281	C₃×C₄⋊C₄	—	2	4	no	—
282	C₃×C₄⋊C₄	C₅×D₄	1	4	yes	—
283	C₃×C₄²	—	2	4	no	—
284	C₃×C₄²	—	2	2	no	—
285	(C₂×D₃)⋊C₄	—	2	4	no	—
286	(C₂×D₃)⋊C₄	—	2	4	no	—
287	(C₂×D₃)⋊C₄	—	4	2	no	—
288	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	8	yes	complemented
289	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	8	yes	complemented
290	C₄⋊Dic₃	—	2	2	no	—
291	Dic₃⋊C₄	—	2	2	no	—
292	Dic₃⋊C₄	—	2	2	no	—
293	C₄×Dic₃	—	2	2	no	—
294	C₄×Dic₃	—	4	2	no	—
295	C₄×Dic₃	C₂×C₄×C₅	1	4	yes	complemented
296	C₂³×C₅	—	3	4	no	—
297	C₂³×C₅	—	6	2	no	—
298	C₂³×C₅	—	6	2	no	—
299	C₂³×C₅	—	6	2	no	—
300	C₂³×C₅	—	6	2	no	—
301	C₂³×C₅	—	6	2	no	—
302	C₂³×C₅	—	6	2	no	—
303	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
304	C₅×Q₈	—	2	8	no	—
305	C₅×D₄	—	6	8	no	—
306	C₂×C₄×C₅	—	2	2	no	—
307	C₂×C₄×C₅	—	2	1	no	—
308	C₂×C₄×C₅	—	2	2	no	—
309	C₂×C₄×C₅	—	2	2	no	—
310	C₂×C₄×C₅	—	2	1	no	—
311	C₂×C₄×C₅	—	2	2	no	—
312	C₂×C₄×C₅	—	2	1	no	—
313	C₂×C₄×C₅	—	2	1	no	—
314	C₂×C₄×C₅	—	2	1	no	—
315	C₂×C₄×C₅	—	3	2	no	—
316	C₂×C₄×C₅	—	3	2	no	—
317	C₂×C₄×C₅	—	3	2	no	—
318	C₂×C₄×C₅	—	6	1	no	—
319	C₂×C₄×C₅	—	6	1	no	—
320	C₂×C₄×C₅	—	6	1	no	—
321	C₂×C₄×C₅	—	6	1	no	—
322	C₂×C₄×C₅	—	6	1	no	—
323	C₂×C₄×C₅	—	6	1	no	—
324	C₂×C₄×C₅	—	6	2	no	—
325	C₂×C₄×C₅	—	6	2	no	—
326	C₂×C₄×C₅	—	6	1	no	—
327	C₂×C₄×C₅	—	6	1	no	—
328	C₂×C₄×C₅	—	6	2	no	—
329	C₂×C₄×C₅	—	6	1	no	—
330	C₂×C₄×C₅	—	12	8	no	—
331	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
332	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
333	C₂×C₄×C₅	#235	1	2	yes	—
334	C₂×C₄×C₅	#239	1	2	yes	—
335	C₂×C₄×C₅	C₂×#67	1	4	yes	—
336	C₂×C₄×C₅	C₂×#67	1	2	yes	—
337	C₂²×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
338	C₂²×D₄	—	3	1	no	—
339	C₂³×C₄	—	3	2	no	—
340	#124	—	3	8	no	—
341	C₄×Q₈	—	2	4	no	—
342	C₄×D₄	—	6	4	no	—
343	C₂×C₄⋊C₄	—	3	1	no	—
344	C₂×C₄⋊C₄	—	3	1	no	—
345	C₂×C₄⋊C₄	—	3	1	no	—
346	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
347	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
348	C₂×C₂²⋊C₄	—	3	2	no	—
349	C₂×C₂²⋊C₄	—	3	2	no	—
350	C₂×C₂²⋊C₄	—	3	2	no	—
351	C₂×C₂²⋊C₄	—	3	2	no	—
352	C₂×C₂²⋊C₄	—	3	2	no	—
353	C₂×C₄²	—	3	1	no	—
354	C₂×C₄²	—	3	1	no	—
355	C₂×C₄²	—	3	2	no	—
356	C₂×C₄²	C₂×C₅×D₃	1	2	yes	—
357	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
358	#95	—	3	1	no	—
359	#95	—	3	2	no	—
360	#95	—	3	2	no	—
361	#95	—	3	1	no	—
362	#95	—	3	2	no	—
363	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
364	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
365	C₂×C₃×C₅	#520	1	1	yes	characteristic
366	C₂×C₃×C₅	#546	1	1	yes	characteristic
367	C₂×C₃×C₅	#548	1	1	yes	characteristic
368	C₂×C₃×C₅	#556	1	1	yes	characteristic
369	C₂×C₃×C₅	#559	1	1	yes	characteristic
370	C₅×D₃	—	4	4	no	—
371	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
372	C₂²×D₃	—	2	2	no	—
373	C₂²×D₃	—	2	2	no	—
374	C₂²×D₃	—	2	2	no	—
375	C₂²×D₃	—	2	2	no	—
376	C₂²×D₃	—	2	2	no	—
377	C₂²×D₃	—	2	2	no	—
378	C₂²×D₃	C₂²×C₄×C₅	1	4	yes	—
379	C₃×Q₈	—	2	8	no	—
380	C₂×C₃×C₄	—	2	1	no	—
381	C₂×C₃×C₄	—	2	2	no	—
382	C₂×C₃×C₄	—	2	2	no	—
383	C₂×C₃×C₄	—	2	2	no	—
384	C₂×C₃×C₄	—	2	2	no	—
385	C₂×C₃×C₄	—	2	1	no	—
386	C₂×C₃×C₄	—	2	1	no	—
387	C₂×C₃×C₄	—	2	1	no	—
388	C₂×C₃×C₄	—	2	1	no	—
389	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
390	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
391	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
392	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
393	C₂×C₃×C₄	C₅×#41	1	2	yes	—
394	C₂×C₃×C₄	C₅×#41	1	2	yes	—
395	C₂×Dic₃	—	2	1	no	—
396	C₂×Dic₃	—	2	1	no	—
397	C₂×Dic₃	—	2	1	no	—
398	C₂×Dic₃	—	2	1	no	—
399	C₂×Dic₃	—	2	1	no	—
400	C₂×Dic₃	—	2	1	no	—
401	C₂×Dic₃	—	2	1	no	—
402	C₂×Dic₃	—	2	1	no	—
403	C₂×Dic₃	—	2	2	no	—
404	C₂×Dic₃	—	2	2	no	—
405	C₂×Dic₃	—	2	2	no	—
406	C₂×Dic₃	—	2	1	no	—
407	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
408	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
409	C₂×Dic₃	C₅×#41	1	2	yes	—
410	D₁₂	—	2	8	no	—
411	C₄×D₃	—	4	8	no	—
412	C₂²×C₅	—	6	4	no	—
413	C₂²×C₅	—	6	4	no	—
414	C₂²×C₅	—	6	4	no	—
415	C₂²×C₅	—	12	2	no	—
416	C₂²×C₅	—	12	2	no	—
417	C₂²×C₅	—	12	2	no	—
418	C₂²×C₅	—	12	2	no	—
419	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
420	C₂²×C₅	C₂×#235	1	1	yes	characteristic
421	C₂²×C₅	C₂×#239	1	1	yes	characteristic
422	C₂²×C₅	#1007	1	1	yes	characteristic
423	C₂²×C₅	D₃×#41	1	1	yes	characteristic
424	C₂²×C₅	#1009	1	1	yes	characteristic
425	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
426	C₄×C₅	—	2	4	no	—
427	C₄×C₅	—	2	2	no	—
428	C₄×C₅	—	2	4	no	—
429	C₄×C₅	—	4	1	no	—
430	C₄×C₅	—	6	2	no	—
431	C₄×C₅	—	6	2	no	—
432	C₄×C₅	—	6	2	no	—
433	C₄×C₅	—	12	1	no	—
434	C₄×C₅	—	12	2	no	—
435	C₄×C₅	C₄×#67	1	4	yes	—
436	C₂⁴	—	3	2	no	—
437	C₂×Q₈	—	2	2	no	—
438	C₂×Q₈	—	2	2	no	—
439	C₂×Q₈	C₄×C₅×D₃	1	4	yes	complemented
440	C₂×D₄	—	3	4	no	—
441	C₂×D₄	—	6	2	no	—
442	C₂×D₄	—	6	2	no	—
443	C₂²×C₄	—	3	1	no	—
444	C₂²×C₄	—	3	1	no	—
445	C₂²×C₄	—	3	1	no	—
446	C₂²×C₄	—	3	2	no	—
447	C₂²×C₄	—	3	1	no	—
448	C₂²×C₄	—	6	4	no	—
449	C₂²×C₄	—	6	4	no	—
450	C₂²×C₄	—	12	2	no	—
451	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
452	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
453	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
454	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
455	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	6	2	no	—
458	C₄⋊C₄	—	6	2	no	—
459	C₄⋊C₄	—	6	2	no	—
460	C₄⋊C₄	C₅×#67	1	4	yes	—
461	C₂²⋊C₄	—	3	8	no	—
462	C₂²⋊C₄	—	3	8	no	—
463	C₂²⋊C₄	—	6	4	no	—
464	C₂²⋊C₄	—	6	4	no	—
465	C₂²⋊C₄	—	12	2	no	—
466	C₄²	—	2	4	no	—
467	C₄²	—	2	2	no	—
468	C₄²	—	3	4	no	—
469	C₄²	—	6	2	no	—
470	C₄²	—	12	2	no	—
471	C₃×C₅	#2367	1	1	yes	characteristic, complemented, Hall
472	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
473	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
474	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
475	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
476	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
477	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
478	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
479	C₂×D₃	—	2	4	no	—
480	C₂×D₃	—	2	4	no	—
481	C₂×D₃	—	2	4	no	—
482	C₂×D₃	—	4	2	no	—
483	C₂×D₃	—	4	2	no	—
484	C₂×D₃	—	4	2	no	—
485	C₂×D₃	—	4	2	no	—
486	C₃×C₄	—	2	4	no	—
487	C₃×C₄	—	2	4	no	—
488	C₃×C₄	—	2	2	no	—
489	C₃×C₄	—	4	1	no	—
490	C₃×C₄	C₄×C₅×D₄	1	4	yes	—
491	Dic₃	—	2	2	no	—
492	Dic₃	—	2	2	no	—
493	Dic₃	—	2	2	no	—
494	Dic₃	—	4	1	no	—
495	Dic₃	—	4	2	no	—
496	C₂×C₅	—	12	4	no	—
497	C₂×C₅	#5568	1	1	yes	characteristic
498	C₂×C₅	C₄×#239	1	1	yes	characteristic
499	C₂×C₅	#5574	1	1	yes	characteristic
500	C₂×C₅	#5585	1	1	yes	characteristic
501	C₂×C₅	C₂×C₄×#67	1	1	yes	characteristic
502	C₂×C₅	#5817	1	1	yes	characteristic
503	C₂×C₅	#5829	1	1	yes	characteristic
504	C₂³	—	3	4	no	—
505	C₂³	—	6	2	no	—
506	C₂³	—	6	2	no	—
507	C₂³	—	6	2	no	—
508	C₂³	—	6	2	no	—
509	C₂³	—	6	2	no	—
510	C₂³	—	6	2	no	—
511	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
512	Q₈	—	2	8	no	—
513	D₄	—	6	8	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	3	2	no	—
524	C₂×C₄	—	3	2	no	—
525	C₂×C₄	—	3	2	no	—
526	C₂×C₄	—	6	1	no	—
527	C₂×C₄	—	6	1	no	—
528	C₂×C₄	—	6	1	no	—
529	C₂×C₄	—	6	1	no	—
530	C₂×C₄	—	6	1	no	—
531	C₂×C₄	—	6	1	no	—
532	C₂×C₄	—	6	2	no	—
533	C₂×C₄	—	6	2	no	—
534	C₂×C₄	—	6	1	no	—
535	C₂×C₄	—	6	1	no	—
536	C₂×C₄	—	6	2	no	—
537	C₂×C₄	—	6	1	no	—
538	C₂×C₄	—	12	8	no	—
539	C₂×C₄	C₂×C₄×C₅×D₃	1	4	yes	—
540	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
541	C₂×C₄	C₅×#235	1	2	yes	—
542	C₂×C₄	C₅×#239	1	2	yes	—
543	C₂×C₄	C₂×C₅×#67	1	4	yes	—
544	C₂×C₄	C₂×C₅×#67	1	2	yes	—
545	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
546	C₂×C₃	C₄×C₅×#41	1	1	yes	characteristic
547	C₂×C₃	C₅×#520	1	1	yes	characteristic
548	C₂×C₃	C₅×#546	1	1	yes	characteristic
549	C₂×C₃	C₅×#548	1	1	yes	characteristic
550	C₂×C₃	C₅×#556	1	1	yes	characteristic
551	C₂×C₃	C₅×#559	1	1	yes	characteristic
552	D₃	—	4	4	no	—
553	C₅	#82466	1	1	yes	characteristic, direct product factor, Sylow
554	C₂²	—	6	4	no	—
555	C₂²	—	6	4	no	—
556	C₂²	—	6	4	no	—
557	C₂²	—	12	2	no	—
558	C₂²	—	12	2	no	—
559	C₂²	—	12	2	no	—
560	C₂²	—	12	2	no	—
561	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
562	C₂²	C₂×C₅×#235	1	1	yes	characteristic
563	C₂²	C₂×C₅×#239	1	1	yes	characteristic
564	C₂²	C₅×#1007	1	1	yes	characteristic
565	C₂²	C₅×D₃×#41	1	1	yes	characteristic
566	C₂²	C₅×#1009	1	1	yes	characteristic
567	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
568	C₄	—	2	4	no	—
569	C₄	—	2	2	no	—
570	C₄	—	2	4	no	—
571	C₄	—	4	1	no	—
572	C₄	—	6	2	no	—
573	C₄	—	6	2	no	—
574	C₄	—	6	2	no	—
575	C₄	—	12	1	no	—
576	C₄	—	12	2	no	—
577	C₄	C₄×C₅×#67	1	4	yes	—
578	C₃	C₅×#2367	1	1	yes	characteristic, complemented, Sylow
579	C₂	—	12	4	no	—
580	C₂	C₅×#5568	1	1	yes	characteristic
581	C₂	C₄×C₅×#239	1	1	yes	characteristic
582	C₂	C₅×#5574	1	1	yes	characteristic
583	C₂	C₅×#5585	1	1	yes	characteristic
584	C₂	C₂×C₄×C₅×#67	1	1	yes	characteristic
585	C₂	C₅×#5817	1	1	yes	characteristic
586	C₂	C₅×#5829	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×Q₈	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂²×C₅×D₁₂	C₂²
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₂×C₄×C₅×Q₈	D₃
hypocenter	C₃	C₅×#2367
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₅×Q₈	C₂	semidirect
4. C₅×#5229	C₂	semidirect
5. C₅×#4990	C₂	semidirect
6. C₅×#4989	C₂	semidirect
7. C₅×Dic₃×C₄⋊C₄	C₂	semidirect
8. C₅×#4940	C₂	semidirect
10. C₄×C₅×(C₂×D₃)⋊C₄	C₂	semidirect
11. C₄×C₅×Dic₃⋊C₄	C₂	semidirect
21. C₅×#947	C₄	semidirect
39. #82466	C₅	direct (N × H)
41. C₂×C₄×C₅×Q₈	D₃	semidirect
54. C₂×C₃×C₅×Q₈	C₂×C₄	semidirect
78. C₅×(C₂×D₃)⋊C₄	C₂×C₄	semidirect
79. C₅×(C₂×D₃)⋊C₄	C₂×C₄	semidirect
85. C₄×C₅×Dic₃	C₂×C₄	semidirect
86. C₂×C₃×C₄×Q₈	C₂×C₅	semidirect
89. #5229	C₂×C₅	semidirect
90. #4990	C₂×C₅	semidirect
91. #4989	C₂×C₅	semidirect
92. Dic₃×C₄⋊C₄	C₂×C₅	semidirect
93. #4940	C₂×C₅	semidirect
95. C₄×(C₂×D₃)⋊C₄	C₂×C₅	semidirect
96. C₄×Dic₃⋊C₄	C₂×C₅	semidirect
173. #947	C₄×C₅	semidirect
194. C₂×C₅×Q₈	C₄×D₃	semidirect
227. C₂×C₄×Q₈	C₅×D₃	semidirect
264. C₂×C₃×Q₈	C₂×C₄×C₅	semidirect
288. (C₂×D₃)⋊C₄	C₂×C₄×C₅	semidirect
289. (C₂×D₃)⋊C₄	C₂×C₄×C₅	semidirect
295. C₄×Dic₃	C₂×C₄×C₅	semidirect
439. C₂×Q₈	C₄×C₅×D₃	semidirect
471. C₃×C₅	#2367	semidirect
553. C₅	#82466	direct (N × H)
578. C₃	C₅×#2367	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×Q₈	2
C₂×C₅×#947	2
C₂×C₅×#893	2
C₅×#5229	2
C₅×#4990	2
C₅×#4989	2
C₅×Dic₃×C₄⋊C₄	2
C₅×#4940	2
C₅×#4939	2
C₄×C₅×(C₂×D₃)⋊C₄	2
C₄×C₅×Dic₃⋊C₄	2
C₅×#2367	3
#82466	5

Subgroups — C5×#82466 #14329508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#82466 #14329508