Subgroups — C₂×C₃×C₅×#453 #14276048

Order 1920.

1688 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	C₂×C₃×C₅×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#453	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×C₅×#104	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×C₅×#100	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#453	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃×C₅×H(5,𝔽₂)	—	2	4	no	—
9	C₃×C₅×H(5,𝔽₂)	—	2	2	no	—
10	C₃×C₅×H(5,𝔽₂)	C₂²	1	4	yes	complemented
11	C₂×C₃×C₅×#41	—	2	2	no	—
12	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
13	C₂²×C₃×C₅×D₄	—	2	2	no	—
14	C₂²×C₃×C₅×D₄	—	2	2	no	—
15	C₂²×C₃×C₅×D₄	—	2	2	no	—
16	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
18	C₃×C₅×AGL(1,ℤ/8)	—	2	4	no	—
19	C₃×C₅×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
20	C₂×C₃×C₅×SD₁₆	—	2	2	no	—
21	C₂×C₃×C₅×D₈	—	2	2	no	—
22	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	complemented
23	C₃×C₅×#127	—	2	2	no	—
24	C₃×C₅×#127	C₂²	1	4	yes	complemented
25	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
26	C₃×C₅×#104	C₂²	1	8	yes	complemented
27	C₃×C₅×#100	C₂²	1	4	yes	complemented
28	C₂×C₃×#453	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₂×C₅×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	1	2	yes	complemented
31	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic, complemented
32	C₅×#453	C₂×C₃	1	8	yes	direct product factor
33	C₂×C₅×#104	C₂×C₃	1	2	yes	complemented
34	C₂×C₅×#100	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂⁴×C₃×C₅	—	2	2	no	—
36	C₂⁴×C₃×C₅	—	2	1	no	—
37	C₂⁴×C₃×C₅	D₄	1	2	yes	—
38	C₃×C₅×#41	—	2	8	no	—
39	C₃×C₅×#41	—	2	8	no	—
40	C₃×C₅×#41	—	2	2	no	—
41	C₃×C₅×#41	—	2	2	no	—
42	C₃×C₅×#41	D₄	1	4	yes	complemented
43	C₃×C₅×#41	C₂³	1	4	yes	—
44	C₂×C₃×C₅×Q₈	D₄	1	2	yes	complemented
45	C₂×C₃×C₅×D₄	—	2	4	no	—
46	C₂×C₃×C₅×D₄	—	2	8	no	—
47	C₂×C₃×C₅×D₄	—	2	8	no	—
48	C₂×C₃×C₅×D₄	—	2	2	no	—
49	C₂×C₃×C₅×D₄	—	2	8	no	—
50	C₂×C₃×C₅×D₄	—	2	8	no	—
51	C₂×C₃×C₅×D₄	—	2	8	no	—
52	C₂×C₃×C₅×D₄	—	2	2	no	—
53	C₂×C₃×C₅×D₄	—	2	2	no	—
54	C₂×C₃×C₅×D₄	—	2	2	no	—
55	C₂×C₃×C₅×D₄	—	2	2	no	—
56	C₂×C₃×C₅×D₄	—	2	1	no	—
57	C₂×C₃×C₅×D₄	—	4	8	no	—
58	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
59	C₂×C₃×C₅×D₄	D₄	1	2	yes	complemented
60	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
61	C₂×C₃×C₅×D₄	C₂³	1	2	yes	complemented
62	C₂²×C₃×C₄×C₅	—	2	2	no	—
63	C₂²×C₃×C₄×C₅	—	2	1	no	—
64	C₂²×C₃×C₄×C₅	—	2	2	no	—
65	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
66	C₃×C₅×SD₁₆	—	2	8	no	—
67	C₃×C₅×D₈	—	2	8	no	—
68	C₃×C₅×M₁₆	—	2	2	no	—
69	C₃×C₅×M₁₆	C₂³	1	4	yes	complemented
70	C₂×C₃×C₅×C₈	—	2	2	no	—
71	C₃×C₄²×C₅	—	2	1	no	—
72	C₃×C₄²×C₅	C₂³	1	2	yes	complemented
73	C₂×C₃×H(5,𝔽₂)	C₂×C₅	1	1	yes	characteristic, complemented
74	C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	1	2	yes	complemented
75	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic, complemented
76	C₃×#453	C₂×C₅	1	8	yes	direct product factor
77	C₂×C₃×#104	C₂×C₅	1	2	yes	complemented
78	C₂×C₃×#100	C₂×C₅	1	1	yes	characteristic, complemented
79	C₅×H(5,𝔽₂)	—	2	4	no	—
80	C₅×H(5,𝔽₂)	—	2	2	no	—
81	C₅×H(5,𝔽₂)	C₂²×C₃	1	4	yes	complemented
82	C₂×C₅×#41	—	2	2	no	—
83	C₂×C₅×#41	C₂²×C₃	1	2	yes	—
84	C₂²×C₅×D₄	—	2	2	no	—
85	C₂²×C₅×D₄	—	2	2	no	—
86	C₂²×C₅×D₄	—	2	2	no	—
87	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
88	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
89	C₅×AGL(1,ℤ/8)	—	2	4	no	—
90	C₅×AGL(1,ℤ/8)	C₂²×C₃	1	8	yes	complemented
91	C₂×C₅×SD₁₆	—	2	2	no	—
92	C₂×C₅×D₈	—	2	2	no	—
93	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	complemented
94	C₅×#127	—	2	2	no	—
95	C₅×#127	C₂²×C₃	1	4	yes	complemented
96	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
97	C₅×#104	C₂²×C₃	1	8	yes	complemented
98	C₅×#100	C₂²×C₃	1	4	yes	complemented
99	C₂×#453	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
100	C₂³×C₃×C₅	—	2	8	no	—
101	C₂³×C₃×C₅	—	2	2	no	—
102	C₂³×C₃×C₅	—	2	2	no	—
103	C₂³×C₃×C₅	—	2	2	no	—
104	C₂³×C₃×C₅	—	2	2	no	—
105	C₂³×C₃×C₅	—	4	4	no	—
106	C₂³×C₃×C₅	—	4	4	no	—
107	C₂³×C₃×C₅	—	4	1	no	—
108	C₂³×C₃×C₅	—	4	1	no	—
109	C₂³×C₃×C₅	—	8	4	no	—
110	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
111	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
112	C₃×C₅×Q₈	—	2	2	no	—
113	C₃×C₅×Q₈	C₂×D₄	1	4	yes	complemented
114	C₃×C₅×D₄	—	2	16	no	—
115	C₃×C₅×D₄	—	2	4	no	—
116	C₃×C₅×D₄	—	2	4	no	—
117	C₃×C₅×D₄	—	2	4	no	—
118	C₃×C₅×D₄	—	2	2	no	—
119	C₃×C₅×D₄	—	2	4	no	—
120	C₃×C₅×D₄	—	2	4	no	—
121	C₃×C₅×D₄	—	4	16	no	—
122	C₃×C₅×D₄	—	4	2	no	—
123	C₃×C₅×D₄	C₂×D₄	1	4	yes	complemented
124	C₂×C₃×C₄×C₅	—	2	8	no	—
125	C₂×C₃×C₄×C₅	—	2	4	no	—
126	C₂×C₃×C₄×C₅	—	2	2	no	—
127	C₂×C₃×C₄×C₅	—	2	2	no	—
128	C₂×C₃×C₄×C₅	—	2	4	no	—
129	C₂×C₃×C₄×C₅	—	2	4	no	—
130	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
131	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
132	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
133	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
134	C₃×C₅×C₈	—	2	4	no	—
135	C₃×H(5,𝔽₂)	—	2	4	no	—
136	C₃×H(5,𝔽₂)	—	2	2	no	—
137	C₃×H(5,𝔽₂)	C₂²×C₅	1	4	yes	complemented
138	C₂×C₃×#41	—	2	2	no	—
139	C₂×C₃×#41	C₂²×C₅	1	2	yes	—
140	C₂²×C₃×D₄	—	2	2	no	—
141	C₂²×C₃×D₄	—	2	2	no	—
142	C₂²×C₃×D₄	—	2	2	no	—
143	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
144	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
145	C₃×AGL(1,ℤ/8)	—	2	4	no	—
146	C₃×AGL(1,ℤ/8)	C₂²×C₅	1	8	yes	complemented
147	C₂×C₃×SD₁₆	—	2	2	no	—
148	C₂×C₃×D₈	—	2	2	no	—
149	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	complemented
150	C₃×#127	—	2	2	no	—
151	C₃×#127	C₂²×C₅	1	4	yes	complemented
152	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic, complemented
153	C₃×#104	C₂²×C₅	1	8	yes	complemented
154	C₃×#100	C₂²×C₅	1	4	yes	complemented
155	C₂⁴×C₅	—	2	2	no	—
156	C₂⁴×C₅	—	2	1	no	—
157	C₂⁴×C₅	C₃×D₄	1	2	yes	—
158	C₅×#41	—	2	8	no	—
159	C₅×#41	—	2	8	no	—
160	C₅×#41	—	2	2	no	—
161	C₅×#41	—	2	2	no	—
162	C₅×#41	C₃×D₄	1	4	yes	complemented
163	C₅×#41	C₂³×C₃	1	4	yes	—
164	C₂×C₅×Q₈	C₃×D₄	1	2	yes	complemented
165	C₂×C₅×D₄	—	2	4	no	—
166	C₂×C₅×D₄	—	2	8	no	—
167	C₂×C₅×D₄	—	2	8	no	—
168	C₂×C₅×D₄	—	2	2	no	—
169	C₂×C₅×D₄	—	2	8	no	—
170	C₂×C₅×D₄	—	2	8	no	—
171	C₂×C₅×D₄	—	2	8	no	—
172	C₂×C₅×D₄	—	2	2	no	—
173	C₂×C₅×D₄	—	2	2	no	—
174	C₂×C₅×D₄	—	2	2	no	—
175	C₂×C₅×D₄	—	2	2	no	—
176	C₂×C₅×D₄	—	2	1	no	—
177	C₂×C₅×D₄	—	4	8	no	—
178	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
179	C₂×C₅×D₄	C₃×D₄	1	2	yes	complemented
180	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
181	C₂×C₅×D₄	C₂³×C₃	1	2	yes	complemented
182	C₂²×C₄×C₅	—	2	2	no	—
183	C₂²×C₄×C₅	—	2	1	no	—
184	C₂²×C₄×C₅	—	2	2	no	—
185	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
186	C₅×SD₁₆	—	2	8	no	—
187	C₅×D₈	—	2	8	no	—
188	C₅×M₁₆	—	2	2	no	—
189	C₅×M₁₆	C₂³×C₃	1	4	yes	complemented
190	C₂×C₅×C₈	—	2	2	no	—
191	C₄²×C₅	—	2	1	no	—
192	C₄²×C₅	C₂³×C₃	1	2	yes	complemented
193	C₂×H(5,𝔽₂)	C₂×C₃×C₅	1	1	yes	characteristic, complemented
194	C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	1	2	yes	complemented
195	C₂×#127	C₂×C₃×C₅	1	1	yes	characteristic, complemented
196	#453	C₂×C₃×C₅	1	8	yes	direct product factor
197	C₂×#104	C₂×C₃×C₅	1	2	yes	complemented
198	C₂×#100	C₂×C₃×C₅	1	1	yes	characteristic, complemented
199	C₂²×C₃×C₅	—	2	2	no	—
200	C₂²×C₃×C₅	—	2	4	no	—
201	C₂²×C₃×C₅	—	2	4	no	—
202	C₂²×C₃×C₅	—	4	4	no	—
203	C₂²×C₃×C₅	—	4	4	no	—
204	C₂²×C₃×C₅	—	4	4	no	—
205	C₂²×C₃×C₅	—	4	4	no	—
206	C₂²×C₃×C₅	—	4	2	no	—
207	C₂²×C₃×C₅	—	8	2	no	—
208	C₂²×C₃×C₅	—	8	2	no	—
209	C₂²×C₃×C₅	—	8	8	no	—
210	C₂²×C₃×C₅	—	8	2	no	—
211	C₂²×C₃×C₅	#120	1	1	yes	characteristic
212	C₂²×C₃×C₅	#120	1	1	yes	characteristic
213	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
214	C₃×C₄×C₅	—	2	4	no	—
215	C₃×C₄×C₅	—	2	4	no	—
216	C₃×C₄×C₅	#120	1	2	yes	—
217	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
218	C₂⁴×C₃	—	2	2	no	—
219	C₂⁴×C₃	—	2	1	no	—
220	C₂⁴×C₃	C₅×D₄	1	2	yes	—
221	C₃×#41	—	2	8	no	—
222	C₃×#41	—	2	8	no	—
223	C₃×#41	—	2	2	no	—
224	C₃×#41	—	2	2	no	—
225	C₃×#41	C₅×D₄	1	4	yes	complemented
226	C₃×#41	C₂³×C₅	1	4	yes	—
227	C₂×C₃×Q₈	C₅×D₄	1	2	yes	complemented
228	C₂×C₃×D₄	—	2	4	no	—
229	C₂×C₃×D₄	—	2	8	no	—
230	C₂×C₃×D₄	—	2	8	no	—
231	C₂×C₃×D₄	—	2	2	no	—
232	C₂×C₃×D₄	—	2	8	no	—
233	C₂×C₃×D₄	—	2	8	no	—
234	C₂×C₃×D₄	—	2	8	no	—
235	C₂×C₃×D₄	—	2	2	no	—
236	C₂×C₃×D₄	—	2	2	no	—
237	C₂×C₃×D₄	—	2	2	no	—
238	C₂×C₃×D₄	—	2	2	no	—
239	C₂×C₃×D₄	—	2	1	no	—
240	C₂×C₃×D₄	—	4	8	no	—
241	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
242	C₂×C₃×D₄	C₅×D₄	1	2	yes	complemented
243	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
244	C₂×C₃×D₄	C₂³×C₅	1	2	yes	complemented
245	C₂²×C₃×C₄	—	2	2	no	—
246	C₂²×C₃×C₄	—	2	1	no	—
247	C₂²×C₃×C₄	—	2	2	no	—
248	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
249	C₃×SD₁₆	—	2	8	no	—
250	C₃×D₈	—	2	8	no	—
251	C₃×M₁₆	—	2	2	no	—
252	C₃×M₁₆	C₂³×C₅	1	4	yes	complemented
253	C₂×C₃×C₈	—	2	2	no	—
254	C₃×C₄²	—	2	1	no	—
255	C₃×C₄²	C₂³×C₅	1	2	yes	complemented
256	C₂³×C₅	—	2	8	no	—
257	C₂³×C₅	—	2	2	no	—
258	C₂³×C₅	—	2	2	no	—
259	C₂³×C₅	—	2	2	no	—
260	C₂³×C₅	—	2	2	no	—
261	C₂³×C₅	—	4	4	no	—
262	C₂³×C₅	—	4	4	no	—
263	C₂³×C₅	—	4	1	no	—
264	C₂³×C₅	—	4	1	no	—
265	C₂³×C₅	—	8	4	no	—
266	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
267	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
268	C₅×Q₈	—	2	2	no	—
269	C₅×Q₈	C₂×C₃×D₄	1	4	yes	complemented
270	C₅×D₄	—	2	16	no	—
271	C₅×D₄	—	2	4	no	—
272	C₅×D₄	—	2	4	no	—
273	C₅×D₄	—	2	4	no	—
274	C₅×D₄	—	2	2	no	—
275	C₅×D₄	—	2	4	no	—
276	C₅×D₄	—	2	4	no	—
277	C₅×D₄	—	4	16	no	—
278	C₅×D₄	—	4	2	no	—
279	C₅×D₄	C₂×C₃×D₄	1	4	yes	complemented
280	C₂×C₄×C₅	—	2	8	no	—
281	C₂×C₄×C₅	—	2	4	no	—
282	C₂×C₄×C₅	—	2	2	no	—
283	C₂×C₄×C₅	—	2	2	no	—
284	C₂×C₄×C₅	—	2	4	no	—
285	C₂×C₄×C₅	—	2	4	no	—
286	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
287	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
288	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
289	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
290	C₅×C₈	—	2	4	no	—
291	H(5,𝔽₂)	—	2	4	no	—
292	H(5,𝔽₂)	—	2	2	no	—
293	H(5,𝔽₂)	C₂²×C₃×C₅	1	4	yes	complemented
294	C₂×#41	—	2	2	no	—
295	C₂×#41	C₂²×C₃×C₅	1	2	yes	—
296	C₂²×D₄	—	2	2	no	—
297	C₂²×D₄	—	2	2	no	—
298	C₂²×D₄	—	2	2	no	—
299	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
300	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
301	AGL(1,ℤ/8)	—	2	4	no	—
302	AGL(1,ℤ/8)	C₂²×C₃×C₅	1	8	yes	complemented
303	C₂×SD₁₆	—	2	2	no	—
304	C₂×D₈	—	2	2	no	—
305	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	complemented
306	#127	—	2	2	no	—
307	#127	C₂²×C₃×C₅	1	4	yes	complemented
308	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
309	#104	C₂²×C₃×C₅	1	8	yes	complemented
310	#100	C₂²×C₃×C₅	1	4	yes	complemented
311	C₂×C₃×C₅	—	2	1	no	—
312	C₂×C₃×C₅	—	2	1	no	—
313	C₂×C₃×C₅	—	4	4	no	—
314	C₂×C₃×C₅	—	4	4	no	—
315	C₂×C₃×C₅	—	8	2	no	—
316	C₂×C₃×C₅	#453	1	2	yes	direct product factor
317	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
318	C₂³×C₃	—	2	8	no	—
319	C₂³×C₃	—	2	2	no	—
320	C₂³×C₃	—	2	2	no	—
321	C₂³×C₃	—	2	2	no	—
322	C₂³×C₃	—	2	2	no	—
323	C₂³×C₃	—	4	4	no	—
324	C₂³×C₃	—	4	4	no	—
325	C₂³×C₃	—	4	1	no	—
326	C₂³×C₃	—	4	1	no	—
327	C₂³×C₃	—	8	4	no	—
328	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
329	C₂³×C₃	C₂×C₅×D₄	1	4	yes	—
330	C₃×Q₈	—	2	2	no	—
331	C₃×Q₈	C₂×C₅×D₄	1	4	yes	complemented
332	C₃×D₄	—	2	16	no	—
333	C₃×D₄	—	2	4	no	—
334	C₃×D₄	—	2	4	no	—
335	C₃×D₄	—	2	4	no	—
336	C₃×D₄	—	2	2	no	—
337	C₃×D₄	—	2	4	no	—
338	C₃×D₄	—	2	4	no	—
339	C₃×D₄	—	4	16	no	—
340	C₃×D₄	—	4	2	no	—
341	C₃×D₄	C₂×C₅×D₄	1	4	yes	complemented
342	C₂×C₃×C₄	—	2	8	no	—
343	C₂×C₃×C₄	—	2	4	no	—
344	C₂×C₃×C₄	—	2	2	no	—
345	C₂×C₃×C₄	—	2	2	no	—
346	C₂×C₃×C₄	—	2	4	no	—
347	C₂×C₃×C₄	—	2	4	no	—
348	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
349	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
350	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
351	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
352	C₃×C₈	—	2	4	no	—
353	C₂²×C₅	—	2	2	no	—
354	C₂²×C₅	—	2	4	no	—
355	C₂²×C₅	—	2	4	no	—
356	C₂²×C₅	—	4	4	no	—
357	C₂²×C₅	—	4	4	no	—
358	C₂²×C₅	—	4	4	no	—
359	C₂²×C₅	—	4	4	no	—
360	C₂²×C₅	—	4	2	no	—
361	C₂²×C₅	—	8	2	no	—
362	C₂²×C₅	—	8	2	no	—
363	C₂²×C₅	—	8	8	no	—
364	C₂²×C₅	—	8	2	no	—
365	C₂²×C₅	C₃×#120	1	1	yes	characteristic
366	C₂²×C₅	C₃×#120	1	1	yes	characteristic
367	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
368	C₄×C₅	—	2	4	no	—
369	C₄×C₅	—	2	4	no	—
370	C₄×C₅	C₃×#120	1	2	yes	—
371	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
372	C₂⁴	—	2	2	no	—
373	C₂⁴	—	2	1	no	—
374	C₂⁴	C₃×C₅×D₄	1	2	yes	—
375	#41	—	2	8	no	—
376	#41	—	2	8	no	—
377	#41	—	2	2	no	—
378	#41	—	2	2	no	—
379	#41	C₃×C₅×D₄	1	4	yes	complemented
380	#41	C₂³×C₃×C₅	1	4	yes	—
381	C₂×Q₈	C₃×C₅×D₄	1	2	yes	complemented
382	C₂×D₄	—	2	4	no	—
383	C₂×D₄	—	2	8	no	—
384	C₂×D₄	—	2	8	no	—
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	8	no	—
387	C₂×D₄	—	2	8	no	—
388	C₂×D₄	—	2	8	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	4	8	no	—
395	C₂×D₄	C₃×C₅×D₄	1	2	yes	—
396	C₂×D₄	C₃×C₅×D₄	1	2	yes	complemented
397	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
398	C₂×D₄	C₂³×C₃×C₅	1	2	yes	complemented
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
403	SD₁₆	—	2	8	no	—
404	D₈	—	2	8	no	—
405	M₁₆	—	2	2	no	—
406	M₁₆	C₂³×C₃×C₅	1	4	yes	complemented
407	C₂×C₈	—	2	2	no	—
408	C₄²	—	2	1	no	—
409	C₄²	C₂³×C₃×C₅	1	2	yes	complemented
410	C₃×C₅	C₂×#453	1	1	yes	characteristic, direct product factor, Hall
411	C₂²×C₃	—	2	2	no	—
412	C₂²×C₃	—	2	4	no	—
413	C₂²×C₃	—	2	4	no	—
414	C₂²×C₃	—	4	4	no	—
415	C₂²×C₃	—	4	4	no	—
416	C₂²×C₃	—	4	4	no	—
417	C₂²×C₃	—	4	4	no	—
418	C₂²×C₃	—	4	2	no	—
419	C₂²×C₃	—	8	2	no	—
420	C₂²×C₃	—	8	2	no	—
421	C₂²×C₃	—	8	8	no	—
422	C₂²×C₃	—	8	2	no	—
423	C₂²×C₃	C₅×#120	1	1	yes	characteristic
424	C₂²×C₃	C₅×#120	1	1	yes	characteristic
425	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
426	C₃×C₄	—	2	4	no	—
427	C₃×C₄	—	2	4	no	—
428	C₃×C₄	C₅×#120	1	2	yes	—
429	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
430	C₂×C₅	—	2	1	no	—
431	C₂×C₅	—	2	1	no	—
432	C₂×C₅	—	4	4	no	—
433	C₂×C₅	—	4	4	no	—
434	C₂×C₅	—	8	2	no	—
435	C₂×C₅	C₃×#453	1	2	yes	direct product factor
436	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
437	C₂³	—	2	8	no	—
438	C₂³	—	2	2	no	—
439	C₂³	—	2	2	no	—
440	C₂³	—	2	2	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	4	4	no	—
443	C₂³	—	4	4	no	—
444	C₂³	—	4	1	no	—
445	C₂³	—	4	1	no	—
446	C₂³	—	8	4	no	—
447	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
448	C₂³	C₂×C₃×C₅×D₄	1	4	yes	—
449	Q₈	—	2	2	no	—
450	Q₈	C₂×C₃×C₅×D₄	1	4	yes	complemented
451	D₄	—	2	16	no	—
452	D₄	—	2	4	no	—
453	D₄	—	2	4	no	—
454	D₄	—	2	4	no	—
455	D₄	—	2	2	no	—
456	D₄	—	2	4	no	—
457	D₄	—	2	4	no	—
458	D₄	—	4	16	no	—
459	D₄	—	4	2	no	—
460	D₄	C₂×C₃×C₅×D₄	1	4	yes	complemented
461	C₂×C₄	—	2	8	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
468	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
469	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
470	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
471	C₈	—	2	4	no	—
472	C₂×C₃	—	2	1	no	—
473	C₂×C₃	—	2	1	no	—
474	C₂×C₃	—	4	4	no	—
475	C₂×C₃	—	4	4	no	—
476	C₂×C₃	—	8	2	no	—
477	C₂×C₃	C₅×#453	1	2	yes	direct product factor
478	C₂×C₃	C₂×C₅×#120	1	1	yes	characteristic
479	C₅	C₂×C₃×#453	1	1	yes	characteristic, direct product factor, Sylow
480	C₂²	—	2	2	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	2	4	no	—
483	C₂²	—	4	4	no	—
484	C₂²	—	4	4	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	8	2	no	—
489	C₂²	—	8	2	no	—
490	C₂²	—	8	8	no	—
491	C₂²	—	8	2	no	—
492	C₂²	C₃×C₅×#120	1	1	yes	characteristic
493	C₂²	C₃×C₅×#120	1	1	yes	characteristic
494	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
495	C₄	—	2	4	no	—
496	C₄	—	2	4	no	—
497	C₄	C₃×C₅×#120	1	2	yes	—
498	C₄	C₂²×C₃×C₅×D₄	1	2	yes	—
499	C₃	C₂×C₅×#453	1	1	yes	characteristic, direct product factor, Sylow
500	C₂	—	2	1	no	—
501	C₂	—	2	1	no	—
502	C₂	—	4	4	no	—
503	C₂	—	4	4	no	—
504	C₂	—	8	2	no	—
505	C₂	C₃×C₅×#453	1	2	yes	direct product factor
506	C₂	C₂×C₃×C₅×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×H(5,𝔽₂)	C₂	semidirect
2. C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	semidirect
3. C₂×C₃×C₅×#127	C₂	semidirect
4. C₃×C₅×#453	C₂	direct (N × H)
5. C₂×C₃×C₅×#104	C₂	semidirect
6. C₂×C₃×C₅×#100	C₂	semidirect
7. C₂×C₅×#453	C₃	direct (N × H)
10. C₃×C₅×H(5,𝔽₂)	C₂²	semidirect
17. C₂²×C₃×C₅×D₄	C₂²	semidirect
19. C₃×C₅×AGL(1,ℤ/8)	C₂²	semidirect
22. C₂×C₃×C₅×M₁₆	C₂²	semidirect
24. C₃×C₅×#127	C₂²	semidirect
25. C₂×C₃×C₄²×C₅	C₂²	semidirect
26. C₃×C₅×#104	C₂²	semidirect
27. C₃×C₅×#100	C₂²	semidirect
28. C₂×C₃×#453	C₅	direct (N × H)
29. C₂×C₅×H(5,𝔽₂)	C₂×C₃	semidirect
30. C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	semidirect
31. C₂×C₅×#127	C₂×C₃	semidirect
32. C₅×#453	C₂×C₃	direct (N × H)
33. C₂×C₅×#104	C₂×C₃	semidirect
34. C₂×C₅×#100	C₂×C₃	semidirect
42. C₃×C₅×#41	D₄	semidirect
44. C₂×C₃×C₅×Q₈	D₄	semidirect
59. C₂×C₃×C₅×D₄	D₄	semidirect
61. C₂×C₃×C₅×D₄	C₂³	semidirect
69. C₃×C₅×M₁₆	C₂³	semidirect
72. C₃×C₄²×C₅	C₂³	semidirect
73. C₂×C₃×H(5,𝔽₂)	C₂×C₅	semidirect
74. C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	semidirect
75. C₂×C₃×#127	C₂×C₅	semidirect
76. C₃×#453	C₂×C₅	direct (N × H)
77. C₂×C₃×#104	C₂×C₅	semidirect
78. C₂×C₃×#100	C₂×C₅	semidirect
81. C₅×H(5,𝔽₂)	C₂²×C₃	semidirect
88. C₂²×C₅×D₄	C₂²×C₃	semidirect
90. C₅×AGL(1,ℤ/8)	C₂²×C₃	semidirect
93. C₂×C₅×M₁₆	C₂²×C₃	semidirect
95. C₅×#127	C₂²×C₃	semidirect
96. C₂×C₄²×C₅	C₂²×C₃	semidirect
97. C₅×#104	C₂²×C₃	semidirect
98. C₅×#100	C₂²×C₃	semidirect
99. C₂×#453	C₃×C₅	direct (N × H)
113. C₃×C₅×Q₈	C₂×D₄	semidirect
123. C₃×C₅×D₄	C₂×D₄	semidirect
137. C₃×H(5,𝔽₂)	C₂²×C₅	semidirect
144. C₂²×C₃×D₄	C₂²×C₅	semidirect
146. C₃×AGL(1,ℤ/8)	C₂²×C₅	semidirect
149. C₂×C₃×M₁₆	C₂²×C₅	semidirect
151. C₃×#127	C₂²×C₅	semidirect
152. C₂×C₃×C₄²	C₂²×C₅	semidirect
153. C₃×#104	C₂²×C₅	semidirect
154. C₃×#100	C₂²×C₅	semidirect
162. C₅×#41	C₃×D₄	semidirect
164. C₂×C₅×Q₈	C₃×D₄	semidirect
179. C₂×C₅×D₄	C₃×D₄	semidirect
181. C₂×C₅×D₄	C₂³×C₃	semidirect
189. C₅×M₁₆	C₂³×C₃	semidirect
192. C₄²×C₅	C₂³×C₃	semidirect
193. C₂×H(5,𝔽₂)	C₂×C₃×C₅	semidirect
194. C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	semidirect
195. C₂×#127	C₂×C₃×C₅	semidirect
196. #453	C₂×C₃×C₅	direct (N × H)
197. C₂×#104	C₂×C₃×C₅	semidirect
198. C₂×#100	C₂×C₃×C₅	semidirect
225. C₃×#41	C₅×D₄	semidirect
227. C₂×C₃×Q₈	C₅×D₄	semidirect
242. C₂×C₃×D₄	C₅×D₄	semidirect
244. C₂×C₃×D₄	C₂³×C₅	semidirect
252. C₃×M₁₆	C₂³×C₅	semidirect
255. C₃×C₄²	C₂³×C₅	semidirect
269. C₅×Q₈	C₂×C₃×D₄	semidirect
279. C₅×D₄	C₂×C₃×D₄	semidirect
293. H(5,𝔽₂)	C₂²×C₃×C₅	semidirect
300. C₂²×D₄	C₂²×C₃×C₅	semidirect
302. AGL(1,ℤ/8)	C₂²×C₃×C₅	semidirect
305. C₂×M₁₆	C₂²×C₃×C₅	semidirect
307. #127	C₂²×C₃×C₅	semidirect
308. C₂×C₄²	C₂²×C₃×C₅	semidirect
309. #104	C₂²×C₃×C₅	semidirect
310. #100	C₂²×C₃×C₅	semidirect
316. C₂×C₃×C₅	#453	direct (N × H)
331. C₃×Q₈	C₂×C₅×D₄	semidirect
341. C₃×D₄	C₂×C₅×D₄	semidirect
379. #41	C₃×C₅×D₄	semidirect
381. C₂×Q₈	C₃×C₅×D₄	semidirect
396. C₂×D₄	C₃×C₅×D₄	semidirect
398. C₂×D₄	C₂³×C₃×C₅	semidirect
406. M₁₆	C₂³×C₃×C₅	semidirect
409. C₄²	C₂³×C₃×C₅	semidirect
410. C₃×C₅	C₂×#453	direct (N × H)
435. C₂×C₅	C₃×#453	direct (N × H)
450. Q₈	C₂×C₃×C₅×D₄	semidirect
460. D₄	C₂×C₃×C₅×D₄	semidirect
477. C₂×C₃	C₅×#453	direct (N × H)
479. C₅	C₂×C₃×#453	direct (N × H)
499. C₃	C₂×C₅×#453	direct (N × H)
505. C₂	C₃×C₅×#453	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×C₅×H(5,𝔽₂)	2
C₂×C₃×C₅×AGL(1,ℤ/8)	2
C₂×C₃×C₅×#127	2
C₃×C₅×#453	2
C₂×C₃×C₅×#104	2
C₂×C₃×C₅×#100	2
C₂×C₅×#453	3
C₂×C₃×#453	5

Subgroups — C2×C3×C5×#453 #14276048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#453 #14276048