Subgroups — C₃×C₅×#2539 #14275573

Order 1920.

1800 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#123	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×#120	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#399	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#392	C₂	1	2	yes	maximal, complemented
7	C₃×C₅×#390	C₂	1	4	yes	maximal, complemented
8	C₃×C₄×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2539	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
12	C₂²×C₃×C₅×D₄	C₂²	1	4	yes	—
13	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
14	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
16	C₃×C₅×#123	—	2	8	no	—
17	C₃×C₅×#121	—	2	4	no	—
18	C₃×C₅×#120	—	2	8	no	—
19	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	complemented
21	C₂×C₃×C₅×C₂²⋊C₄	—	2	4	no	—
22	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
27	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
28	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	complemented
29	C₃×C₅×#95	C₂²	1	2	yes	complemented
30	C₃×#2539	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₂×C₅×#123	C₂×C₃	1	2	yes	complemented
32	C₂×C₅×#121	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×C₅×#120	C₂×C₃	1	2	yes	complemented
34	C₂²×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
35	C₅×#399	C₂×C₃	1	2	yes	complemented
36	C₅×#392	C₂×C₃	1	2	yes	complemented
37	C₅×#390	C₂×C₃	1	4	yes	complemented
38	C₄×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂⁴×C₃×C₅	—	2	2	no	—
40	C₂⁴×C₃×C₅	—	2	4	no	—
41	C₂⁴×C₃×C₅	—	2	4	no	—
42	C₂⁴×C₃×C₅	—	4	2	no	—
43	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
44	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
45	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₅×D₄	—	2	4	no	—
47	C₂×C₃×C₅×D₄	—	2	4	no	—
48	C₂×C₃×C₅×D₄	—	2	4	no	—
49	C₂×C₃×C₅×D₄	—	2	8	no	—
50	C₂×C₃×C₅×D₄	—	2	8	no	—
51	C₂×C₃×C₅×D₄	—	2	2	no	—
52	C₂×C₃×C₅×D₄	—	4	4	no	—
53	C₂×C₃×C₅×D₄	—	4	2	no	—
54	C₂²×C₃×C₄×C₅	—	2	2	no	—
55	C₂²×C₃×C₄×C₅	—	2	4	no	—
56	C₂²×C₃×C₄×C₅	—	4	2	no	—
57	C₂²×C₃×C₄×C₅	D₄	1	4	yes	complemented
58	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
59	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
60	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
62	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
63	C₃×C₅×C₄⋊C₄	—	2	2	no	—
64	C₃×C₅×C₄⋊C₄	—	2	8	no	—
65	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
66	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
67	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
68	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
69	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
70	C₃×C₅×C₂²⋊C₄	D₄	1	8	yes	complemented
71	C₃×C₄²×C₅	—	4	2	no	—
72	C₂×C₃×#123	C₂×C₅	1	2	yes	complemented
73	C₂×C₃×#121	C₂×C₅	1	1	yes	characteristic, complemented
74	C₂×C₃×#120	C₂×C₅	1	2	yes	complemented
75	C₂²×C₃×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
76	C₃×#399	C₂×C₅	1	2	yes	complemented
77	C₃×#392	C₂×C₅	1	2	yes	complemented
78	C₃×#390	C₂×C₅	1	4	yes	complemented
79	C₃×C₄×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
80	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
81	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
82	C₂²×C₅×D₄	C₂²×C₃	1	4	yes	—
83	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
84	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
85	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
86	C₅×#123	—	2	8	no	—
87	C₅×#121	—	2	4	no	—
88	C₅×#120	—	2	8	no	—
89	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
90	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
91	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
92	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
93	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
94	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
95	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
96	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
97	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
98	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	complemented
99	C₅×#95	C₂²×C₃	1	2	yes	complemented
100	#2539	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
101	C₂³×C₃×C₅	—	2	2	no	—
102	C₂³×C₃×C₅	—	2	2	no	—
103	C₂³×C₃×C₅	—	2	8	no	—
104	C₂³×C₃×C₅	—	2	2	no	—
105	C₂³×C₃×C₅	—	2	4	no	—
106	C₂³×C₃×C₅	—	2	2	no	—
107	C₂³×C₃×C₅	—	2	4	no	—
108	C₂³×C₃×C₅	—	2	2	no	—
109	C₂³×C₃×C₅	—	2	2	no	—
110	C₂³×C₃×C₅	—	2	2	no	—
111	C₂³×C₃×C₅	—	2	4	no	—
112	C₂³×C₃×C₅	—	2	4	no	—
113	C₂³×C₃×C₅	—	4	16	no	—
114	C₂³×C₃×C₅	—	4	4	no	—
115	C₂³×C₃×C₅	—	4	4	no	—
116	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
117	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
118	C₂³×C₃×C₅	#41	1	4	yes	—
119	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
120	C₃×C₅×D₄	—	4	16	no	—
121	C₃×C₅×D₄	—	4	4	no	—
122	C₃×C₅×D₄	—	4	8	no	—
123	C₂×C₃×C₄×C₅	—	2	4	no	—
124	C₂×C₃×C₄×C₅	—	2	2	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	2	no	—
129	C₂×C₃×C₄×C₅	—	2	4	no	—
130	C₂×C₃×C₄×C₅	—	2	2	no	—
131	C₂×C₃×C₄×C₅	—	2	2	no	—
132	C₂×C₃×C₄×C₅	—	4	4	no	—
133	C₂×C₃×C₄×C₅	—	4	4	no	—
134	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
135	C₂×C₃×C₄×C₅	C₂×D₄	1	8	yes	—
136	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
137	C₂⁵×C₃	C₂²×C₅	1	1	yes	characteristic
138	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
139	C₂²×C₃×D₄	C₂²×C₅	1	4	yes	—
140	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
141	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
142	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
143	C₃×#123	—	2	8	no	—
144	C₃×#121	—	2	4	no	—
145	C₃×#120	—	2	8	no	—
146	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	complemented
147	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
148	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
149	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
150	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
151	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
152	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
153	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	complemented
154	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
155	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	complemented
156	C₃×#95	C₂²×C₅	1	2	yes	complemented
157	C₂⁴×C₅	—	2	2	no	—
158	C₂⁴×C₅	—	2	4	no	—
159	C₂⁴×C₅	—	2	4	no	—
160	C₂⁴×C₅	—	4	2	no	—
161	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
162	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
163	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
164	C₂×C₅×D₄	—	2	4	no	—
165	C₂×C₅×D₄	—	2	4	no	—
166	C₂×C₅×D₄	—	2	4	no	—
167	C₂×C₅×D₄	—	2	8	no	—
168	C₂×C₅×D₄	—	2	8	no	—
169	C₂×C₅×D₄	—	2	2	no	—
170	C₂×C₅×D₄	—	4	4	no	—
171	C₂×C₅×D₄	—	4	2	no	—
172	C₂²×C₄×C₅	—	2	2	no	—
173	C₂²×C₄×C₅	—	2	4	no	—
174	C₂²×C₄×C₅	—	4	2	no	—
175	C₂²×C₄×C₅	C₃×D₄	1	4	yes	complemented
176	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
177	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
178	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
179	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
180	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
181	C₅×C₄⋊C₄	—	2	2	no	—
182	C₅×C₄⋊C₄	—	2	8	no	—
183	C₅×C₂²⋊C₄	—	2	4	no	—
184	C₅×C₂²⋊C₄	—	2	4	no	—
185	C₅×C₂²⋊C₄	—	2	8	no	—
186	C₅×C₂²⋊C₄	—	2	4	no	—
187	C₅×C₂²⋊C₄	—	2	8	no	—
188	C₅×C₂²⋊C₄	C₃×D₄	1	8	yes	complemented
189	C₄²×C₅	—	4	2	no	—
190	C₂×#123	C₂×C₃×C₅	1	2	yes	complemented
191	C₂×#121	C₂×C₃×C₅	1	1	yes	characteristic, complemented
192	C₂×#120	C₂×C₃×C₅	1	2	yes	complemented
193	C₂²×C₂²⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
194	#399	C₂×C₃×C₅	1	2	yes	complemented
195	#392	C₂×C₃×C₅	1	2	yes	complemented
196	#390	C₂×C₃×C₅	1	4	yes	complemented
197	C₄×C₂²⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
198	C₂²×C₃×C₅	—	2	2	no	—
199	C₂²×C₃×C₅	—	2	2	no	—
200	C₂²×C₃×C₅	—	2	8	no	—
201	C₂²×C₃×C₅	—	2	4	no	—
202	C₂²×C₃×C₅	—	2	8	no	—
203	C₂²×C₃×C₅	—	4	16	no	—
204	C₂²×C₃×C₅	—	4	2	no	—
205	C₂²×C₃×C₅	—	4	2	no	—
206	C₂²×C₃×C₅	—	4	4	no	—
207	C₂²×C₃×C₅	—	4	2	no	—
208	C₂²×C₃×C₅	—	4	2	no	—
209	C₂²×C₃×C₅	—	4	2	no	—
210	C₂²×C₃×C₅	—	4	4	no	—
211	C₂²×C₃×C₅	—	4	4	no	—
212	C₂²×C₃×C₅	#127	1	4	yes	—
213	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
214	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
215	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
216	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
217	C₃×C₄×C₅	—	2	2	no	—
218	C₃×C₄×C₅	—	2	2	no	—
219	C₃×C₄×C₅	—	2	8	no	—
220	C₃×C₄×C₅	—	4	2	no	—
221	C₃×C₄×C₅	—	4	2	no	—
222	C₂⁴×C₃	—	2	2	no	—
223	C₂⁴×C₃	—	2	4	no	—
224	C₂⁴×C₃	—	2	4	no	—
225	C₂⁴×C₃	—	4	2	no	—
226	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
227	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
228	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
229	C₂×C₃×D₄	—	2	4	no	—
230	C₂×C₃×D₄	—	2	4	no	—
231	C₂×C₃×D₄	—	2	4	no	—
232	C₂×C₃×D₄	—	2	8	no	—
233	C₂×C₃×D₄	—	2	8	no	—
234	C₂×C₃×D₄	—	2	2	no	—
235	C₂×C₃×D₄	—	4	4	no	—
236	C₂×C₃×D₄	—	4	2	no	—
237	C₂²×C₃×C₄	—	2	2	no	—
238	C₂²×C₃×C₄	—	2	4	no	—
239	C₂²×C₃×C₄	—	4	2	no	—
240	C₂²×C₃×C₄	C₅×D₄	1	4	yes	complemented
241	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
242	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
243	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
244	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
245	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
246	C₃×C₄⋊C₄	—	2	2	no	—
247	C₃×C₄⋊C₄	—	2	8	no	—
248	C₃×C₂²⋊C₄	—	2	4	no	—
249	C₃×C₂²⋊C₄	—	2	4	no	—
250	C₃×C₂²⋊C₄	—	2	8	no	—
251	C₃×C₂²⋊C₄	—	2	4	no	—
252	C₃×C₂²⋊C₄	—	2	8	no	—
253	C₃×C₂²⋊C₄	C₅×D₄	1	8	yes	complemented
254	C₃×C₄²	—	4	2	no	—
255	C₂³×C₅	—	2	2	no	—
256	C₂³×C₅	—	2	2	no	—
257	C₂³×C₅	—	2	8	no	—
258	C₂³×C₅	—	2	2	no	—
259	C₂³×C₅	—	2	4	no	—
260	C₂³×C₅	—	2	2	no	—
261	C₂³×C₅	—	2	4	no	—
262	C₂³×C₅	—	2	2	no	—
263	C₂³×C₅	—	2	2	no	—
264	C₂³×C₅	—	2	2	no	—
265	C₂³×C₅	—	2	4	no	—
266	C₂³×C₅	—	2	4	no	—
267	C₂³×C₅	—	4	16	no	—
268	C₂³×C₅	—	4	4	no	—
269	C₂³×C₅	—	4	4	no	—
270	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
271	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
272	C₂³×C₅	C₃×#41	1	4	yes	—
273	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
274	C₅×D₄	—	4	16	no	—
275	C₅×D₄	—	4	4	no	—
276	C₅×D₄	—	4	8	no	—
277	C₂×C₄×C₅	—	2	4	no	—
278	C₂×C₄×C₅	—	2	2	no	—
279	C₂×C₄×C₅	—	2	4	no	—
280	C₂×C₄×C₅	—	2	2	no	—
281	C₂×C₄×C₅	—	2	2	no	—
282	C₂×C₄×C₅	—	2	2	no	—
283	C₂×C₄×C₅	—	2	4	no	—
284	C₂×C₄×C₅	—	2	2	no	—
285	C₂×C₄×C₅	—	2	2	no	—
286	C₂×C₄×C₅	—	4	4	no	—
287	C₂×C₄×C₅	—	4	4	no	—
288	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
289	C₂×C₄×C₅	C₂×C₃×D₄	1	8	yes	—
290	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
291	C₂⁵	C₂²×C₃×C₅	1	1	yes	characteristic
292	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
293	C₂²×D₄	C₂²×C₃×C₅	1	4	yes	—
294	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
295	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
296	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
297	#123	—	2	8	no	—
298	#121	—	2	4	no	—
299	#120	—	2	8	no	—
300	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
301	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	complemented
302	C₂×C₂²⋊C₄	—	2	4	no	—
303	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
304	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
305	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
306	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
307	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	complemented
308	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	complemented
309	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	complemented
310	#95	C₂²×C₃×C₅	1	2	yes	complemented
311	C₂×C₃×C₅	—	2	4	no	—
312	C₂×C₃×C₅	—	4	4	no	—
313	C₂×C₃×C₅	—	8	2	no	—
314	C₂×C₃×C₅	#525	1	1	yes	characteristic
315	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
316	C₂×C₃×C₅	#534	1	1	yes	characteristic
317	C₂×C₃×C₅	#546	1	4	yes	—
318	C₂³×C₃	—	2	2	no	—
319	C₂³×C₃	—	2	2	no	—
320	C₂³×C₃	—	2	8	no	—
321	C₂³×C₃	—	2	2	no	—
322	C₂³×C₃	—	2	4	no	—
323	C₂³×C₃	—	2	2	no	—
324	C₂³×C₃	—	2	4	no	—
325	C₂³×C₃	—	2	2	no	—
326	C₂³×C₃	—	2	2	no	—
327	C₂³×C₃	—	2	2	no	—
328	C₂³×C₃	—	2	4	no	—
329	C₂³×C₃	—	2	4	no	—
330	C₂³×C₃	—	4	16	no	—
331	C₂³×C₃	—	4	4	no	—
332	C₂³×C₃	—	4	4	no	—
333	C₂³×C₃	C₂×C₅×D₄	1	4	yes	—
334	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
335	C₂³×C₃	C₅×#41	1	4	yes	—
336	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
337	C₃×D₄	—	4	16	no	—
338	C₃×D₄	—	4	4	no	—
339	C₃×D₄	—	4	8	no	—
340	C₂×C₃×C₄	—	2	4	no	—
341	C₂×C₃×C₄	—	2	2	no	—
342	C₂×C₃×C₄	—	2	4	no	—
343	C₂×C₃×C₄	—	2	2	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	2	no	—
348	C₂×C₃×C₄	—	2	2	no	—
349	C₂×C₃×C₄	—	4	4	no	—
350	C₂×C₃×C₄	—	4	4	no	—
351	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
352	C₂×C₃×C₄	C₂×C₅×D₄	1	8	yes	—
353	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
354	C₂²×C₅	—	2	2	no	—
355	C₂²×C₅	—	2	2	no	—
356	C₂²×C₅	—	2	8	no	—
357	C₂²×C₅	—	2	4	no	—
358	C₂²×C₅	—	2	8	no	—
359	C₂²×C₅	—	4	16	no	—
360	C₂²×C₅	—	4	2	no	—
361	C₂²×C₅	—	4	2	no	—
362	C₂²×C₅	—	4	4	no	—
363	C₂²×C₅	—	4	2	no	—
364	C₂²×C₅	—	4	2	no	—
365	C₂²×C₅	—	4	2	no	—
366	C₂²×C₅	—	4	4	no	—
367	C₂²×C₅	—	4	4	no	—
368	C₂²×C₅	C₃×#127	1	4	yes	—
369	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
370	C₂²×C₅	C₂²×C₃×D₄	1	2	yes	—
371	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
372	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
373	C₄×C₅	—	2	2	no	—
374	C₄×C₅	—	2	2	no	—
375	C₄×C₅	—	2	8	no	—
376	C₄×C₅	—	4	2	no	—
377	C₄×C₅	—	4	2	no	—
378	C₂⁴	—	2	2	no	—
379	C₂⁴	—	2	4	no	—
380	C₂⁴	—	2	4	no	—
381	C₂⁴	—	4	2	no	—
382	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
383	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
384	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	2	4	no	—
388	C₂×D₄	—	2	8	no	—
389	C₂×D₄	—	2	8	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	4	4	no	—
392	C₂×D₄	—	4	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	C₃×C₅×D₄	1	4	yes	complemented
397	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
398	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
399	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
400	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
401	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	8	no	—
404	C₂²⋊C₄	—	2	4	no	—
405	C₂²⋊C₄	—	2	4	no	—
406	C₂²⋊C₄	—	2	8	no	—
407	C₂²⋊C₄	—	2	4	no	—
408	C₂²⋊C₄	—	2	8	no	—
409	C₂²⋊C₄	C₃×C₅×D₄	1	8	yes	complemented
410	C₄²	—	4	2	no	—
411	C₃×C₅	#2539	1	1	yes	characteristic, direct product factor, Hall
412	C₂²×C₃	—	2	2	no	—
413	C₂²×C₃	—	2	2	no	—
414	C₂²×C₃	—	2	8	no	—
415	C₂²×C₃	—	2	4	no	—
416	C₂²×C₃	—	2	8	no	—
417	C₂²×C₃	—	4	16	no	—
418	C₂²×C₃	—	4	2	no	—
419	C₂²×C₃	—	4	2	no	—
420	C₂²×C₃	—	4	4	no	—
421	C₂²×C₃	—	4	2	no	—
422	C₂²×C₃	—	4	2	no	—
423	C₂²×C₃	—	4	2	no	—
424	C₂²×C₃	—	4	4	no	—
425	C₂²×C₃	—	4	4	no	—
426	C₂²×C₃	C₅×#127	1	4	yes	—
427	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
428	C₂²×C₃	C₂²×C₅×D₄	1	2	yes	—
429	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
430	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
431	C₃×C₄	—	2	2	no	—
432	C₃×C₄	—	2	2	no	—
433	C₃×C₄	—	2	8	no	—
434	C₃×C₄	—	4	2	no	—
435	C₃×C₄	—	4	2	no	—
436	C₂×C₅	—	2	4	no	—
437	C₂×C₅	—	4	4	no	—
438	C₂×C₅	—	8	2	no	—
439	C₂×C₅	C₃×#525	1	1	yes	characteristic
440	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
441	C₂×C₅	C₃×#534	1	1	yes	characteristic
442	C₂×C₅	C₃×#546	1	4	yes	—
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	8	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	4	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	2	2	no	—
453	C₂³	—	2	4	no	—
454	C₂³	—	2	4	no	—
455	C₂³	—	4	16	no	—
456	C₂³	—	4	4	no	—
457	C₂³	—	4	4	no	—
458	C₂³	C₂×C₃×C₅×D₄	1	4	yes	—
459	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
460	C₂³	C₃×C₅×#41	1	4	yes	—
461	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
462	D₄	—	4	16	no	—
463	D₄	—	4	4	no	—
464	D₄	—	4	8	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	4	4	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
477	C₂×C₄	C₂×C₃×C₅×D₄	1	8	yes	—
478	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
479	C₂×C₃	—	2	4	no	—
480	C₂×C₃	—	4	4	no	—
481	C₂×C₃	—	8	2	no	—
482	C₂×C₃	C₅×#525	1	1	yes	characteristic
483	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
484	C₂×C₃	C₅×#534	1	1	yes	characteristic
485	C₂×C₃	C₅×#546	1	4	yes	—
486	C₅	C₃×#2539	1	1	yes	characteristic, direct product factor, Sylow
487	C₂²	—	2	2	no	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	8	no	—
490	C₂²	—	2	4	no	—
491	C₂²	—	2	8	no	—
492	C₂²	—	4	16	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	4	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	4	2	no	—
499	C₂²	—	4	4	no	—
500	C₂²	—	4	4	no	—
501	C₂²	C₃×C₅×#127	1	4	yes	—
502	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
503	C₂²	C₂²×C₃×C₅×D₄	1	2	yes	—
504	C₂²	C₂×C₃×C₅×#41	1	2	yes	—
505	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
506	C₄	—	2	2	no	—
507	C₄	—	2	2	no	—
508	C₄	—	2	8	no	—
509	C₄	—	4	2	no	—
510	C₄	—	4	2	no	—
511	C₃	C₅×#2539	1	1	yes	characteristic, direct product factor, Sylow
512	C₂	—	2	4	no	—
513	C₂	—	4	4	no	—
514	C₂	—	8	2	no	—
515	C₂	C₃×C₅×#525	1	1	yes	characteristic
516	C₂	C₂×C₃×C₅×#127	1	1	yes	characteristic
517	C₂	C₃×C₅×#534	1	1	yes	characteristic
518	C₂	C₃×C₅×#546	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#123	C₂	semidirect
2. C₂×C₃×C₅×#121	C₂	semidirect
3. C₂×C₃×C₅×#120	C₂	semidirect
4. C₂²×C₃×C₅×C₂²⋊C₄	C₂	semidirect
5. C₃×C₅×#399	C₂	semidirect
6. C₃×C₅×#392	C₂	semidirect
7. C₃×C₅×#390	C₂	semidirect
8. C₃×C₄×C₅×C₂²⋊C₄	C₂	semidirect
9. C₅×#2539	C₃	direct (N × H)
13. C₂²×C₃×C₅×D₄	C₂²	semidirect
19. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
20. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
26. C₂×C₃×C₅×C₂²⋊C₄	C₂²	semidirect
27. C₂×C₃×C₅×C₂²⋊C₄	C₂²	semidirect
28. C₂×C₃×C₄²×C₅	C₂²	semidirect
29. C₃×C₅×#95	C₂²	semidirect
30. C₃×#2539	C₅	direct (N × H)
31. C₂×C₅×#123	C₂×C₃	semidirect
32. C₂×C₅×#121	C₂×C₃	semidirect
33. C₂×C₅×#120	C₂×C₃	semidirect
34. C₂²×C₅×C₂²⋊C₄	C₂×C₃	semidirect
35. C₅×#399	C₂×C₃	semidirect
36. C₅×#392	C₂×C₃	semidirect
37. C₅×#390	C₂×C₃	semidirect
38. C₄×C₅×C₂²⋊C₄	C₂×C₃	semidirect
57. C₂²×C₃×C₄×C₅	D₄	semidirect
70. C₃×C₅×C₂²⋊C₄	D₄	semidirect
72. C₂×C₃×#123	C₂×C₅	semidirect
73. C₂×C₃×#121	C₂×C₅	semidirect
74. C₂×C₃×#120	C₂×C₅	semidirect
75. C₂²×C₃×C₂²⋊C₄	C₂×C₅	semidirect
76. C₃×#399	C₂×C₅	semidirect
77. C₃×#392	C₂×C₅	semidirect
78. C₃×#390	C₂×C₅	semidirect
79. C₃×C₄×C₂²⋊C₄	C₂×C₅	semidirect
83. C₂²×C₅×D₄	C₂²×C₃	semidirect
89. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
90. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
96. C₂×C₅×C₂²⋊C₄	C₂²×C₃	semidirect
97. C₂×C₅×C₂²⋊C₄	C₂²×C₃	semidirect
98. C₂×C₄²×C₅	C₂²×C₃	semidirect
99. C₅×#95	C₂²×C₃	semidirect
100. #2539	C₃×C₅	direct (N × H)
140. C₂²×C₃×D₄	C₂²×C₅	semidirect
146. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
147. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
153. C₂×C₃×C₂²⋊C₄	C₂²×C₅	semidirect
154. C₂×C₃×C₂²⋊C₄	C₂²×C₅	semidirect
155. C₂×C₃×C₄²	C₂²×C₅	semidirect
156. C₃×#95	C₂²×C₅	semidirect
175. C₂²×C₄×C₅	C₃×D₄	semidirect
188. C₅×C₂²⋊C₄	C₃×D₄	semidirect
190. C₂×#123	C₂×C₃×C₅	semidirect
191. C₂×#121	C₂×C₃×C₅	semidirect
192. C₂×#120	C₂×C₃×C₅	semidirect
193. C₂²×C₂²⋊C₄	C₂×C₃×C₅	semidirect
194. #399	C₂×C₃×C₅	semidirect
195. #392	C₂×C₃×C₅	semidirect
196. #390	C₂×C₃×C₅	semidirect
197. C₄×C₂²⋊C₄	C₂×C₃×C₅	semidirect
240. C₂²×C₃×C₄	C₅×D₄	semidirect
253. C₃×C₂²⋊C₄	C₅×D₄	semidirect
294. C₂²×D₄	C₂²×C₃×C₅	semidirect
300. C₂×C₄⋊C₄	C₂²×C₃×C₅	semidirect
301. C₂×C₄⋊C₄	C₂²×C₃×C₅	semidirect
307. C₂×C₂²⋊C₄	C₂²×C₃×C₅	semidirect
308. C₂×C₂²⋊C₄	C₂²×C₃×C₅	semidirect
309. C₂×C₄²	C₂²×C₃×C₅	semidirect
310. #95	C₂²×C₃×C₅	semidirect
396. C₂²×C₄	C₃×C₅×D₄	semidirect
409. C₂²⋊C₄	C₃×C₅×D₄	semidirect
411. C₃×C₅	#2539	direct (N × H)
486. C₅	C₃×#2539	direct (N × H)
511. C₃	C₅×#2539	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×#123	2
C₂×C₃×C₅×#121	2
C₂×C₃×C₅×#120	2
C₂²×C₃×C₅×C₂²⋊C₄	2
C₃×C₅×#399	2
C₃×C₅×#392	2
C₃×C₅×#390	2
C₃×C₄×C₅×C₂²⋊C₄	2
C₅×#2539	3
C₃×#2539	5

Subgroups — C3×C5×#2539 #14275573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2539 #14275573