Subgroups — C₃×C₅×#3217 #14276251

Order 1920.

980 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#496	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#490	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#460	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#459	C₂	1	2	yes	maximal, complemented
7	C₃×C₅×#450	C₂	1	2	yes	maximal, complemented
8	C₃×C₅×#447	C₂	1	2	yes	maximal, complemented
9	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#3217	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₃×C₅×D₄	—	2	1	no	—
12	C₂²×C₃×C₅×D₄	—	2	1	no	—
13	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
14	C₂×C₃×C₅×SD₁₆	C₂²	1	4	yes	complemented
15	C₂×C₃×C₅×D₈	C₂²	1	4	yes	complemented
16	C₃×C₅×#127	—	2	1	no	—
17	C₃×C₅×#127	—	2	1	no	—
18	C₃×C₅×#127	C₂²	1	2	yes	—
19	C₃×C₅×#127	C₂²	1	2	yes	—
20	C₃×C₅×#126	—	2	1	no	—
21	C₃×C₅×#125	C₂²	1	1	yes	characteristic, complemented
22	C₃×C₅×#124	C₂²	1	1	yes	characteristic, complemented
23	C₃×C₅×#123	—	2	1	no	—
24	C₃×C₅×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₃×C₅×#121	C₂²	1	1	yes	characteristic
26	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
27	C₃×C₄×C₅×D₄	—	2	1	no	—
28	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
29	C₃×C₅×#117	—	2	1	no	—
30	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
32	C₃×C₅×#102	C₂²	1	4	yes	complemented
33	C₃×C₅×#102	C₂²	1	4	yes	complemented
34	C₃×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₃×C₅×#97	C₂²	1	2	yes	complemented
36	C₃×#3217	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic, complemented
38	C₅×#529	C₂×C₃	1	1	yes	characteristic, complemented
39	C₅×#496	C₂×C₃	1	2	yes	complemented
40	C₅×#490	C₂×C₃	1	2	yes	complemented
41	C₅×#460	C₂×C₃	1	2	yes	complemented
42	C₅×#459	C₂×C₃	1	2	yes	complemented
43	C₅×#450	C₂×C₃	1	2	yes	complemented
44	C₅×#447	C₂×C₃	1	2	yes	complemented
45	C₅×#432	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂⁴×C₃×C₅	—	2	2	no	—
47	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₅×D₄	—	2	1	no	—
49	C₂×C₃×C₅×D₄	—	2	2	no	—
50	C₂×C₃×C₅×D₄	—	2	2	no	—
51	C₂×C₃×C₅×D₄	—	2	2	no	—
52	C₂×C₃×C₅×D₄	—	2	2	no	—
53	C₂×C₃×C₅×D₄	—	2	2	no	—
54	C₂×C₃×C₅×D₄	—	4	4	no	—
55	C₂×C₃×C₅×D₄	—	4	4	no	—
56	C₂×C₃×C₅×D₄	—	4	4	no	—
57	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
59	C₂²×C₃×C₄×C₅	—	2	1	no	—
60	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
61	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
62	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
63	C₃×C₅×SD₁₆	—	4	4	no	—
64	C₃×C₅×D₈	—	4	4	no	—
65	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
66	C₃×C₅×C₄⋊C₄	—	2	1	no	—
67	C₃×C₅×C₄⋊C₄	—	2	1	no	—
68	C₃×C₅×C₄⋊C₄	—	2	1	no	—
69	C₃×C₅×C₄⋊C₄	—	2	1	no	—
70	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
71	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
73	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
74	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
75	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
76	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
77	C₃×C₄²×C₅	—	2	1	no	—
78	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
79	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
80	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
81	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
82	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic, complemented
83	C₃×#529	C₂×C₅	1	1	yes	characteristic, complemented
84	C₃×#496	C₂×C₅	1	2	yes	complemented
85	C₃×#490	C₂×C₅	1	2	yes	complemented
86	C₃×#460	C₂×C₅	1	2	yes	complemented
87	C₃×#459	C₂×C₅	1	2	yes	complemented
88	C₃×#450	C₂×C₅	1	2	yes	complemented
89	C₃×#447	C₂×C₅	1	2	yes	complemented
90	C₃×#432	C₂×C₅	1	1	yes	characteristic, complemented
91	C₂²×C₅×D₄	—	2	1	no	—
92	C₂²×C₅×D₄	—	2	1	no	—
93	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
94	C₂×C₅×SD₁₆	C₂²×C₃	1	4	yes	complemented
95	C₂×C₅×D₈	C₂²×C₃	1	4	yes	complemented
96	C₅×#127	—	2	1	no	—
97	C₅×#127	—	2	1	no	—
98	C₅×#127	C₂²×C₃	1	2	yes	—
99	C₅×#127	C₂²×C₃	1	2	yes	—
100	C₅×#126	—	2	1	no	—
101	C₅×#125	C₂²×C₃	1	1	yes	characteristic, complemented
102	C₅×#124	C₂²×C₃	1	1	yes	characteristic, complemented
103	C₅×#123	—	2	1	no	—
104	C₅×C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
105	C₅×#121	C₂²×C₃	1	1	yes	characteristic
106	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
107	C₄×C₅×D₄	—	2	1	no	—
108	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
109	C₅×#117	—	2	1	no	—
110	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
111	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
112	C₅×#102	C₂²×C₃	1	4	yes	complemented
113	C₅×#102	C₂²×C₃	1	4	yes	complemented
114	C₅×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
115	C₅×#97	C₂²×C₃	1	2	yes	complemented
116	#3217	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
117	C₂³×C₃×C₅	—	2	1	no	—
118	C₂³×C₃×C₅	—	2	4	no	—
119	C₂³×C₃×C₅	—	4	4	no	—
120	C₂³×C₃×C₅	—	4	4	no	—
121	C₂³×C₃×C₅	—	4	4	no	—
122	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
123	C₃×C₅×Q₈	—	2	2	no	—
124	C₃×C₅×D₄	—	2	2	no	—
125	C₃×C₅×D₄	—	2	4	no	—
126	C₃×C₅×D₄	—	2	4	no	—
127	C₃×C₅×D₄	—	4	4	no	—
128	C₃×C₅×D₄	—	4	4	no	—
129	C₃×C₅×D₄	—	4	4	no	—
130	C₃×C₅×D₄	—	4	4	no	—
131	C₃×C₅×D₄	—	4	2	no	—
132	C₃×C₅×D₄	—	4	4	no	—
133	C₂×C₃×C₄×C₅	—	2	2	no	—
134	C₂×C₃×C₄×C₅	—	2	2	no	—
135	C₂×C₃×C₄×C₅	—	2	2	no	—
136	C₂×C₃×C₄×C₅	—	2	1	no	—
137	C₂×C₃×C₄×C₅	—	2	1	no	—
138	C₂×C₃×C₄×C₅	—	2	1	no	—
139	C₂×C₃×C₄×C₅	—	4	2	no	—
140	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
143	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
144	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
145	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
146	C₃×C₅×C₈	—	2	4	no	—
147	C₂²×C₃×D₄	—	2	1	no	—
148	C₂²×C₃×D₄	—	2	1	no	—
149	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
150	C₂×C₃×SD₁₆	C₂²×C₅	1	4	yes	complemented
151	C₂×C₃×D₈	C₂²×C₅	1	4	yes	complemented
152	C₃×#127	—	2	1	no	—
153	C₃×#127	—	2	1	no	—
154	C₃×#127	C₂²×C₅	1	2	yes	—
155	C₃×#127	C₂²×C₅	1	2	yes	—
156	C₃×#126	—	2	1	no	—
157	C₃×#125	C₂²×C₅	1	1	yes	characteristic, complemented
158	C₃×#124	C₂²×C₅	1	1	yes	characteristic, complemented
159	C₃×#123	—	2	1	no	—
160	C₃×C₂²⋊Q₈	C₂²×C₅	1	1	yes	characteristic
161	C₃×#121	C₂²×C₅	1	1	yes	characteristic
162	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
163	C₃×C₄×D₄	—	2	1	no	—
164	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
165	C₃×#117	—	2	1	no	—
166	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
167	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
168	C₃×#102	C₂²×C₅	1	4	yes	complemented
169	C₃×#102	C₂²×C₅	1	4	yes	complemented
170	C₃×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
171	C₃×#97	C₂²×C₅	1	2	yes	complemented
172	C₂⁴×C₅	—	2	2	no	—
173	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
174	C₂×C₅×D₄	—	2	1	no	—
175	C₂×C₅×D₄	—	2	2	no	—
176	C₂×C₅×D₄	—	2	2	no	—
177	C₂×C₅×D₄	—	2	2	no	—
178	C₂×C₅×D₄	—	2	2	no	—
179	C₂×C₅×D₄	—	2	2	no	—
180	C₂×C₅×D₄	—	4	4	no	—
181	C₂×C₅×D₄	—	4	4	no	—
182	C₂×C₅×D₄	—	4	4	no	—
183	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
184	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
185	C₂²×C₄×C₅	—	2	1	no	—
186	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
189	C₅×SD₁₆	—	4	4	no	—
190	C₅×D₈	—	4	4	no	—
191	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
192	C₅×C₄⋊C₄	—	2	1	no	—
193	C₅×C₄⋊C₄	—	2	1	no	—
194	C₅×C₄⋊C₄	—	2	1	no	—
195	C₅×C₄⋊C₄	—	2	1	no	—
196	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
197	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
198	C₅×C₂²⋊C₄	—	2	1	no	—
199	C₅×C₂²⋊C₄	—	2	1	no	—
200	C₅×C₂²⋊C₄	—	2	1	no	—
201	C₅×C₂²⋊C₄	—	2	1	no	—
202	C₅×C₂²⋊C₄	—	2	1	no	—
203	C₄²×C₅	—	2	1	no	—
204	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
205	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
206	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
207	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
208	C₂×#127	C₂×C₃×C₅	1	1	yes	characteristic, complemented
209	#529	C₂×C₃×C₅	1	1	yes	characteristic, complemented
210	#496	C₂×C₃×C₅	1	2	yes	complemented
211	#490	C₂×C₃×C₅	1	2	yes	complemented
212	#460	C₂×C₃×C₅	1	2	yes	complemented
213	#459	C₂×C₃×C₅	1	2	yes	complemented
214	#450	C₂×C₃×C₅	1	2	yes	complemented
215	#447	C₂×C₃×C₅	1	2	yes	complemented
216	#432	C₂×C₃×C₅	1	1	yes	characteristic, complemented
217	C₂²×C₃×C₅	—	2	1	no	—
218	C₂²×C₃×C₅	—	2	1	no	—
219	C₂²×C₃×C₅	—	4	4	no	—
220	C₂²×C₃×C₅	—	4	1	no	—
221	C₂²×C₃×C₅	—	4	1	no	—
222	C₂²×C₃×C₅	—	4	4	no	—
223	C₂²×C₃×C₅	—	4	4	no	—
224	C₂²×C₃×C₅	—	4	1	no	—
225	C₂²×C₃×C₅	—	8	8	no	—
226	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
227	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
228	C₃×C₄×C₅	—	2	1	no	—
229	C₃×C₄×C₅	—	2	1	no	—
230	C₃×C₄×C₅	—	2	1	no	—
231	C₃×C₄×C₅	—	2	1	no	—
232	C₃×C₄×C₅	—	4	1	no	—
233	C₃×C₄×C₅	—	4	1	no	—
234	C₃×C₄×C₅	—	4	1	no	—
235	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
236	C₃×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
237	C₂⁴×C₃	—	2	2	no	—
238	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
239	C₂×C₃×D₄	—	2	1	no	—
240	C₂×C₃×D₄	—	2	2	no	—
241	C₂×C₃×D₄	—	2	2	no	—
242	C₂×C₃×D₄	—	2	2	no	—
243	C₂×C₃×D₄	—	2	2	no	—
244	C₂×C₃×D₄	—	2	2	no	—
245	C₂×C₃×D₄	—	4	4	no	—
246	C₂×C₃×D₄	—	4	4	no	—
247	C₂×C₃×D₄	—	4	4	no	—
248	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
249	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
250	C₂²×C₃×C₄	—	2	1	no	—
251	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
252	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
253	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
254	C₃×SD₁₆	—	4	4	no	—
255	C₃×D₈	—	4	4	no	—
256	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
257	C₃×C₄⋊C₄	—	2	1	no	—
258	C₃×C₄⋊C₄	—	2	1	no	—
259	C₃×C₄⋊C₄	—	2	1	no	—
260	C₃×C₄⋊C₄	—	2	1	no	—
261	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
262	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
263	C₃×C₂²⋊C₄	—	2	1	no	—
264	C₃×C₂²⋊C₄	—	2	1	no	—
265	C₃×C₂²⋊C₄	—	2	1	no	—
266	C₃×C₂²⋊C₄	—	2	1	no	—
267	C₃×C₂²⋊C₄	—	2	1	no	—
268	C₃×C₄²	—	2	1	no	—
269	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
270	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
271	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
272	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
273	C₂³×C₅	—	2	1	no	—
274	C₂³×C₅	—	2	4	no	—
275	C₂³×C₅	—	4	4	no	—
276	C₂³×C₅	—	4	4	no	—
277	C₂³×C₅	—	4	4	no	—
278	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
279	C₅×Q₈	—	2	2	no	—
280	C₅×D₄	—	2	2	no	—
281	C₅×D₄	—	2	4	no	—
282	C₅×D₄	—	2	4	no	—
283	C₅×D₄	—	4	4	no	—
284	C₅×D₄	—	4	4	no	—
285	C₅×D₄	—	4	4	no	—
286	C₅×D₄	—	4	4	no	—
287	C₅×D₄	—	4	2	no	—
288	C₅×D₄	—	4	4	no	—
289	C₂×C₄×C₅	—	2	2	no	—
290	C₂×C₄×C₅	—	2	2	no	—
291	C₂×C₄×C₅	—	2	2	no	—
292	C₂×C₄×C₅	—	2	1	no	—
293	C₂×C₄×C₅	—	2	1	no	—
294	C₂×C₄×C₅	—	2	1	no	—
295	C₂×C₄×C₅	—	4	2	no	—
296	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
297	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
298	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
299	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
300	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
301	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
302	C₅×C₈	—	2	4	no	—
303	C₂²×D₄	—	2	1	no	—
304	C₂²×D₄	—	2	1	no	—
305	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
306	C₂×SD₁₆	C₂²×C₃×C₅	1	4	yes	complemented
307	C₂×D₈	C₂²×C₃×C₅	1	4	yes	complemented
308	#127	—	2	1	no	—
309	#127	—	2	1	no	—
310	#127	C₂²×C₃×C₅	1	2	yes	—
311	#127	C₂²×C₃×C₅	1	2	yes	—
312	#126	—	2	1	no	—
313	#125	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
314	#124	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
315	#123	—	2	1	no	—
316	C₂²⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
317	#121	C₂²×C₃×C₅	1	1	yes	characteristic
318	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
319	C₄×D₄	—	2	1	no	—
320	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
321	#117	—	2	1	no	—
322	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
323	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	complemented
324	#102	C₂²×C₃×C₅	1	4	yes	complemented
325	#102	C₂²×C₃×C₅	1	4	yes	complemented
326	C₂²⋊C₈	C₂²×C₃×C₅	1	2	yes	—
327	#97	C₂²×C₃×C₅	1	2	yes	complemented
328	C₂×C₃×C₅	—	2	2	no	—
329	C₂×C₃×C₅	—	8	1	no	—
330	C₂×C₃×C₅	—	8	4	no	—
331	C₂×C₃×C₅	#535	1	1	yes	characteristic
332	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
333	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₂³×C₃	—	2	1	no	—
335	C₂³×C₃	—	2	4	no	—
336	C₂³×C₃	—	4	4	no	—
337	C₂³×C₃	—	4	4	no	—
338	C₂³×C₃	—	4	4	no	—
339	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
340	C₃×Q₈	—	2	2	no	—
341	C₃×D₄	—	2	2	no	—
342	C₃×D₄	—	2	4	no	—
343	C₃×D₄	—	2	4	no	—
344	C₃×D₄	—	4	4	no	—
345	C₃×D₄	—	4	4	no	—
346	C₃×D₄	—	4	4	no	—
347	C₃×D₄	—	4	4	no	—
348	C₃×D₄	—	4	2	no	—
349	C₃×D₄	—	4	4	no	—
350	C₂×C₃×C₄	—	2	2	no	—
351	C₂×C₃×C₄	—	2	2	no	—
352	C₂×C₃×C₄	—	2	2	no	—
353	C₂×C₃×C₄	—	2	1	no	—
354	C₂×C₃×C₄	—	2	1	no	—
355	C₂×C₃×C₄	—	2	1	no	—
356	C₂×C₃×C₄	—	4	2	no	—
357	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
358	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
359	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
360	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
361	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
362	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
363	C₃×C₈	—	2	4	no	—
364	C₂²×C₅	—	2	1	no	—
365	C₂²×C₅	—	2	1	no	—
366	C₂²×C₅	—	4	1	no	—
367	C₂²×C₅	—	4	4	no	—
368	C₂²×C₅	—	4	4	no	—
369	C₂²×C₅	—	4	1	no	—
370	C₂²×C₅	—	4	4	no	—
371	C₂²×C₅	—	4	1	no	—
372	C₂²×C₅	—	8	8	no	—
373	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
374	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
375	C₄×C₅	—	2	1	no	—
376	C₄×C₅	—	2	1	no	—
377	C₄×C₅	—	2	1	no	—
378	C₄×C₅	—	2	1	no	—
379	C₄×C₅	—	4	1	no	—
380	C₄×C₅	—	4	1	no	—
381	C₄×C₅	—	4	1	no	—
382	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
383	C₄×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
384	C₂⁴	—	2	2	no	—
385	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	4	4	no	—
393	C₂×D₄	—	4	4	no	—
394	C₂×D₄	—	4	4	no	—
395	C₂×D₄	C₂³×C₃×C₅	1	4	yes	—
396	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
399	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
400	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
401	SD₁₆	—	4	4	no	—
402	D₈	—	4	4	no	—
403	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
409	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
410	C₂²⋊C₄	—	2	1	no	—
411	C₂²⋊C₄	—	2	1	no	—
412	C₂²⋊C₄	—	2	1	no	—
413	C₂²⋊C₄	—	2	1	no	—
414	C₂²⋊C₄	—	2	1	no	—
415	C₄²	—	2	1	no	—
416	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
417	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
418	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
419	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
420	C₃×C₅	#3217	1	1	yes	characteristic, direct product factor, Hall
421	C₂²×C₃	—	2	1	no	—
422	C₂²×C₃	—	2	1	no	—
423	C₂²×C₃	—	4	4	no	—
424	C₂²×C₃	—	4	1	no	—
425	C₂²×C₃	—	4	1	no	—
426	C₂²×C₃	—	4	4	no	—
427	C₂²×C₃	—	4	4	no	—
428	C₂²×C₃	—	4	1	no	—
429	C₂²×C₃	—	8	8	no	—
430	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
431	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
432	C₃×C₄	—	2	1	no	—
433	C₃×C₄	—	2	1	no	—
434	C₃×C₄	—	2	1	no	—
435	C₃×C₄	—	2	1	no	—
436	C₃×C₄	—	4	1	no	—
437	C₃×C₄	—	4	1	no	—
438	C₃×C₄	—	4	1	no	—
439	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
440	C₃×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
441	C₂×C₅	—	2	2	no	—
442	C₂×C₅	—	8	4	no	—
443	C₂×C₅	—	8	1	no	—
444	C₂×C₅	C₃×#535	1	1	yes	characteristic
445	C₂×C₅	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
446	C₂×C₅	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
447	C₂³	—	2	1	no	—
448	C₂³	—	2	4	no	—
449	C₂³	—	4	4	no	—
450	C₂³	—	4	4	no	—
451	C₂³	—	4	4	no	—
452	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
453	Q₈	—	2	2	no	—
454	D₄	—	2	2	no	—
455	D₄	—	2	4	no	—
456	D₄	—	2	4	no	—
457	D₄	—	4	4	no	—
458	D₄	—	4	4	no	—
459	D₄	—	4	4	no	—
460	D₄	—	4	4	no	—
461	D₄	—	4	2	no	—
462	D₄	—	4	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
471	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
472	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
473	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
474	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
475	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
476	C₈	—	2	4	no	—
477	C₂×C₃	—	2	2	no	—
478	C₂×C₃	—	8	4	no	—
479	C₂×C₃	—	8	1	no	—
480	C₂×C₃	C₅×#535	1	1	yes	characteristic
481	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
482	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
483	C₅	C₃×#3217	1	1	yes	characteristic, direct product factor, Sylow
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	4	1	no	—
488	C₂²	—	4	1	no	—
489	C₂²	—	4	4	no	—
490	C₂²	—	4	4	no	—
491	C₂²	—	4	1	no	—
492	C₂²	—	8	8	no	—
493	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
494	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
495	C₄	—	2	1	no	—
496	C₄	—	2	1	no	—
497	C₄	—	2	1	no	—
498	C₄	—	2	1	no	—
499	C₄	—	4	1	no	—
500	C₄	—	4	1	no	—
501	C₄	—	4	1	no	—
502	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
503	C₄	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
504	C₃	C₅×#3217	1	1	yes	characteristic, direct product factor, Sylow
505	C₂	—	2	2	no	—
506	C₂	—	8	4	no	—
507	C₂	—	8	1	no	—
508	C₂	C₃×C₅×#535	1	1	yes	characteristic
509	C₂	C₂×C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
510	C₂	C₂×C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#127	C₂	semidirect
2. C₃×C₅×#529	C₂	semidirect
3. C₃×C₅×#496	C₂	semidirect
4. C₃×C₅×#490	C₂	semidirect
5. C₃×C₅×#460	C₂	semidirect
6. C₃×C₅×#459	C₂	semidirect
7. C₃×C₅×#450	C₂	semidirect
8. C₃×C₅×#447	C₂	semidirect
9. C₃×C₅×#432	C₂	semidirect
10. C₅×#3217	C₃	direct (N × H)
14. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
15. C₂×C₃×C₅×D₈	C₂²	semidirect
21. C₃×C₅×#125	C₂²	semidirect
22. C₃×C₅×#124	C₂²	semidirect
26. C₃×C₄×C₅×Q₈	C₂²	semidirect
28. C₃×C₄×C₅×D₄	C₂²	semidirect
31. C₃×C₅×C₄⋊C₈	C₂²	semidirect
32. C₃×C₅×#102	C₂²	semidirect
33. C₃×C₅×#102	C₂²	semidirect
35. C₃×C₅×#97	C₂²	semidirect
36. C₃×#3217	C₅	direct (N × H)
37. C₂×C₅×#127	C₂×C₃	semidirect
38. C₅×#529	C₂×C₃	semidirect
39. C₅×#496	C₂×C₃	semidirect
40. C₅×#490	C₂×C₃	semidirect
41. C₅×#460	C₂×C₃	semidirect
42. C₅×#459	C₂×C₃	semidirect
43. C₅×#450	C₂×C₃	semidirect
44. C₅×#447	C₂×C₃	semidirect
45. C₅×#432	C₂×C₃	semidirect
82. C₂×C₃×#127	C₂×C₅	semidirect
83. C₃×#529	C₂×C₅	semidirect
84. C₃×#496	C₂×C₅	semidirect
85. C₃×#490	C₂×C₅	semidirect
86. C₃×#460	C₂×C₅	semidirect
87. C₃×#459	C₂×C₅	semidirect
88. C₃×#450	C₂×C₅	semidirect
89. C₃×#447	C₂×C₅	semidirect
90. C₃×#432	C₂×C₅	semidirect
94. C₂×C₅×SD₁₆	C₂²×C₃	semidirect
95. C₂×C₅×D₈	C₂²×C₃	semidirect
101. C₅×#125	C₂²×C₃	semidirect
102. C₅×#124	C₂²×C₃	semidirect
106. C₄×C₅×Q₈	C₂²×C₃	semidirect
108. C₄×C₅×D₄	C₂²×C₃	semidirect
111. C₅×C₄⋊C₈	C₂²×C₃	semidirect
112. C₅×#102	C₂²×C₃	semidirect
113. C₅×#102	C₂²×C₃	semidirect
115. C₅×#97	C₂²×C₃	semidirect
116. #3217	C₃×C₅	direct (N × H)
150. C₂×C₃×SD₁₆	C₂²×C₅	semidirect
151. C₂×C₃×D₈	C₂²×C₅	semidirect
157. C₃×#125	C₂²×C₅	semidirect
158. C₃×#124	C₂²×C₅	semidirect
162. C₃×C₄×Q₈	C₂²×C₅	semidirect
164. C₃×C₄×D₄	C₂²×C₅	semidirect
167. C₃×C₄⋊C₈	C₂²×C₅	semidirect
168. C₃×#102	C₂²×C₅	semidirect
169. C₃×#102	C₂²×C₅	semidirect
171. C₃×#97	C₂²×C₅	semidirect
208. C₂×#127	C₂×C₃×C₅	semidirect
209. #529	C₂×C₃×C₅	semidirect
210. #496	C₂×C₃×C₅	semidirect
211. #490	C₂×C₃×C₅	semidirect
212. #460	C₂×C₃×C₅	semidirect
213. #459	C₂×C₃×C₅	semidirect
214. #450	C₂×C₃×C₅	semidirect
215. #447	C₂×C₃×C₅	semidirect
216. #432	C₂×C₃×C₅	semidirect
306. C₂×SD₁₆	C₂²×C₃×C₅	semidirect
307. C₂×D₈	C₂²×C₃×C₅	semidirect
313. #125	C₂²×C₃×C₅	semidirect
314. #124	C₂²×C₃×C₅	semidirect
318. C₄×Q₈	C₂²×C₃×C₅	semidirect
320. C₄×D₄	C₂²×C₃×C₅	semidirect
323. C₄⋊C₈	C₂²×C₃×C₅	semidirect
324. #102	C₂²×C₃×C₅	semidirect
325. #102	C₂²×C₃×C₅	semidirect
327. #97	C₂²×C₃×C₅	semidirect
420. C₃×C₅	#3217	direct (N × H)
483. C₅	C₃×#3217	direct (N × H)
504. C₃	C₅×#3217	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×#127	2
C₃×C₅×#529	2
C₃×C₅×#496	2
C₃×C₅×#490	2
C₃×C₅×#460	2
C₃×C₅×#459	2
C₃×C₅×#450	2
C₃×C₅×#447	2
C₃×C₅×#432	2
C₅×#3217	3
C₃×#3217	5

Subgroups — C3×C5×#3217 #14276251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3217 #14276251