Subgroups — C₃×C₅×#2512 #14275546

Order 1920.

992 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#126	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#397	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#396	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#390	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#390	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#390	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#388	C₂	1	2	yes	maximal, complemented
9	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#2512	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₃×C₅×#126	—	2	8	no	—
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
30	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
31	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
32	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
33	C₃×C₅×#95	C₂²	1	1	yes	characteristic
34	C₃×C₅×#95	C₂²	1	2	yes	—
35	C₃×C₅×#95	C₂²	1	1	yes	characteristic
36	C₃×C₅×#95	C₂²	1	2	yes	—
37	C₃×#2512	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
38	C₂×C₅×#126	C₂×C₃	1	2	yes	complemented
39	C₅×#397	C₂×C₃	1	2	yes	complemented
40	C₅×#396	C₂×C₃	1	2	yes	complemented
41	C₅×#390	C₂×C₃	1	1	yes	characteristic
42	C₅×#390	C₂×C₃	1	1	yes	characteristic
43	C₅×#390	C₂×C₃	1	1	yes	characteristic
44	C₅×#389	C₂×C₃	1	1	yes	characteristic, complemented
45	C₅×#388	C₂×C₃	1	2	yes	complemented
46	C₅×#384	C₂×C₃	1	1	yes	characteristic, complemented
47	C₄×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
48	C₄×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
49	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
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	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
55	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
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×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
62	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
63	C₃×C₅×C₄⋊C₄	—	2	4	no	—
64	C₃×C₅×C₄⋊C₄	—	2	4	no	—
65	C₃×C₅×C₄⋊C₄	—	2	2	no	—
66	C₃×C₅×C₄⋊C₄	—	2	2	no	—
67	C₃×C₅×C₄⋊C₄	—	2	2	no	—
68	C₃×C₅×C₄⋊C₄	—	2	2	no	—
69	C₃×C₅×C₄⋊C₄	—	4	2	no	—
70	C₃×C₅×C₄⋊C₄	D₄	1	4	yes	complemented
71	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
72	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
73	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
74	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
75	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
76	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
77	C₃×C₄²×C₅	—	2	4	no	—
78	C₃×C₄²×C₅	—	2	2	no	—
79	C₃×C₄²×C₅	—	2	2	no	—
80	C₃×C₄²×C₅	—	2	2	no	—
81	C₂×C₃×#126	C₂×C₅	1	2	yes	complemented
82	C₃×#397	C₂×C₅	1	2	yes	complemented
83	C₃×#396	C₂×C₅	1	2	yes	complemented
84	C₃×#390	C₂×C₅	1	1	yes	characteristic
85	C₃×#390	C₂×C₅	1	1	yes	characteristic
86	C₃×#390	C₂×C₅	1	1	yes	characteristic
87	C₃×#389	C₂×C₅	1	1	yes	characteristic, complemented
88	C₃×#388	C₂×C₅	1	2	yes	complemented
89	C₃×#384	C₂×C₅	1	1	yes	characteristic, complemented
90	C₃×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
91	C₃×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
92	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
93	C₅×#126	—	2	8	no	—
94	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
95	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
97	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
98	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
99	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
100	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
101	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
102	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
103	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
104	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
105	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
106	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
107	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
108	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
109	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
110	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
111	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
112	C₅×#95	C₂²×C₃	1	1	yes	characteristic
113	C₅×#95	C₂²×C₃	1	2	yes	—
114	C₅×#95	C₂²×C₃	1	1	yes	characteristic
115	C₅×#95	C₂²×C₃	1	2	yes	—
116	#2512	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
117	C₂³×C₃×C₅	—	2	2	no	—
118	C₂³×C₃×C₅	—	2	2	no	—
119	C₂³×C₃×C₅	—	2	2	no	—
120	C₂³×C₃×C₅	—	2	2	no	—
121	C₂³×C₃×C₅	—	2	2	no	—
122	C₂³×C₃×C₅	—	2	2	no	—
123	C₂³×C₃×C₅	—	2	2	no	—
124	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
125	C₃×C₅×D₄	—	4	4	no	—
126	C₂×C₃×C₄×C₅	—	2	1	no	—
127	C₂×C₃×C₄×C₅	—	2	1	no	—
128	C₂×C₃×C₄×C₅	—	2	1	no	—
129	C₂×C₃×C₄×C₅	—	2	1	no	—
130	C₂×C₃×C₄×C₅	—	2	1	no	—
131	C₂×C₃×C₄×C₅	—	2	1	no	—
132	C₂×C₃×C₄×C₅	—	2	1	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	1	no	—
138	C₂×C₃×C₄×C₅	—	2	1	no	—
139	C₂×C₃×C₄×C₅	—	2	2	no	—
140	C₂×C₃×C₄×C₅	—	2	2	no	—
141	C₂×C₃×C₄×C₅	—	2	2	no	—
142	C₂×C₃×C₄×C₅	—	2	2	no	—
143	C₂×C₃×C₄×C₅	—	2	1	no	—
144	C₂×C₃×C₄×C₅	—	2	1	no	—
145	C₂×C₃×C₄×C₅	—	2	1	no	—
146	C₂×C₃×C₄×C₅	—	2	2	no	—
147	C₂×C₃×C₄×C₅	—	2	2	no	—
148	C₂×C₃×C₄×C₅	—	2	1	no	—
149	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
150	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
151	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
152	C₂×C₃×C₄×C₅	#41	1	2	yes	—
153	C₂×C₃×C₄×C₅	#41	1	2	yes	—
154	C₂×C₃×C₄×C₅	#41	1	2	yes	—
155	C₂×C₃×C₄×C₅	#41	1	2	yes	—
156	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
157	C₃×#126	—	2	8	no	—
158	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
159	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
160	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
161	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
162	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
163	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
164	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
165	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
166	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
167	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
168	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
169	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
170	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
171	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
172	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
173	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
174	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
175	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
176	C₃×#95	C₂²×C₅	1	1	yes	characteristic
177	C₃×#95	C₂²×C₅	1	2	yes	—
178	C₃×#95	C₂²×C₅	1	1	yes	characteristic
179	C₃×#95	C₂²×C₅	1	2	yes	—
180	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
181	C₂×C₅×D₄	—	2	2	no	—
182	C₂×C₅×D₄	—	2	2	no	—
183	C₂×C₅×D₄	—	2	2	no	—
184	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
185	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
186	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
187	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
188	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
189	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
190	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
191	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
192	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
193	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
194	C₅×C₄⋊C₄	—	2	4	no	—
195	C₅×C₄⋊C₄	—	2	4	no	—
196	C₅×C₄⋊C₄	—	2	2	no	—
197	C₅×C₄⋊C₄	—	2	2	no	—
198	C₅×C₄⋊C₄	—	2	2	no	—
199	C₅×C₄⋊C₄	—	2	2	no	—
200	C₅×C₄⋊C₄	—	4	2	no	—
201	C₅×C₄⋊C₄	C₃×D₄	1	4	yes	complemented
202	C₅×C₂²⋊C₄	—	2	4	no	—
203	C₅×C₂²⋊C₄	—	2	4	no	—
204	C₅×C₂²⋊C₄	—	2	4	no	—
205	C₅×C₂²⋊C₄	—	2	4	no	—
206	C₅×C₂²⋊C₄	—	4	2	no	—
207	C₅×C₂²⋊C₄	—	4	2	no	—
208	C₄²×C₅	—	2	4	no	—
209	C₄²×C₅	—	2	2	no	—
210	C₄²×C₅	—	2	2	no	—
211	C₄²×C₅	—	2	2	no	—
212	C₂×#126	C₂×C₃×C₅	1	2	yes	complemented
213	#397	C₂×C₃×C₅	1	2	yes	complemented
214	#396	C₂×C₃×C₅	1	2	yes	complemented
215	#390	C₂×C₃×C₅	1	1	yes	characteristic
216	#390	C₂×C₃×C₅	1	1	yes	characteristic
217	#390	C₂×C₃×C₅	1	1	yes	characteristic
218	#389	C₂×C₃×C₅	1	1	yes	characteristic, complemented
219	#388	C₂×C₃×C₅	1	2	yes	complemented
220	#384	C₂×C₃×C₅	1	1	yes	characteristic, complemented
221	C₄×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
222	C₄×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
223	C₂²×C₃×C₅	—	4	2	no	—
224	C₂²×C₃×C₅	—	4	2	no	—
225	C₂²×C₃×C₅	—	4	2	no	—
226	C₂²×C₃×C₅	—	4	2	no	—
227	C₂²×C₃×C₅	—	4	2	no	—
228	C₂²×C₃×C₅	—	4	2	no	—
229	C₂²×C₃×C₅	—	4	2	no	—
230	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
231	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
232	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
233	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
234	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
235	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
236	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
237	C₃×C₄×C₅	—	2	2	no	—
238	C₃×C₄×C₅	—	2	2	no	—
239	C₃×C₄×C₅	—	2	2	no	—
240	C₃×C₄×C₅	—	2	2	no	—
241	C₃×C₄×C₅	—	4	2	no	—
242	C₃×C₄×C₅	—	4	1	no	—
243	C₃×C₄×C₅	—	4	2	no	—
244	C₃×C₄×C₅	—	4	2	no	—
245	C₃×C₄×C₅	—	4	1	no	—
246	C₃×C₄×C₅	#126	1	4	yes	complemented
247	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
248	C₂×C₃×D₄	—	2	2	no	—
249	C₂×C₃×D₄	—	2	2	no	—
250	C₂×C₃×D₄	—	2	2	no	—
251	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
252	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
253	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
254	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
255	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
256	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
257	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
258	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
259	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
260	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
261	C₃×C₄⋊C₄	—	2	4	no	—
262	C₃×C₄⋊C₄	—	2	4	no	—
263	C₃×C₄⋊C₄	—	2	2	no	—
264	C₃×C₄⋊C₄	—	2	2	no	—
265	C₃×C₄⋊C₄	—	2	2	no	—
266	C₃×C₄⋊C₄	—	2	2	no	—
267	C₃×C₄⋊C₄	—	4	2	no	—
268	C₃×C₄⋊C₄	C₅×D₄	1	4	yes	complemented
269	C₃×C₂²⋊C₄	—	2	4	no	—
270	C₃×C₂²⋊C₄	—	2	4	no	—
271	C₃×C₂²⋊C₄	—	2	4	no	—
272	C₃×C₂²⋊C₄	—	2	4	no	—
273	C₃×C₂²⋊C₄	—	4	2	no	—
274	C₃×C₂²⋊C₄	—	4	2	no	—
275	C₃×C₄²	—	2	4	no	—
276	C₃×C₄²	—	2	2	no	—
277	C₃×C₄²	—	2	2	no	—
278	C₃×C₄²	—	2	2	no	—
279	C₂³×C₅	—	2	2	no	—
280	C₂³×C₅	—	2	2	no	—
281	C₂³×C₅	—	2	2	no	—
282	C₂³×C₅	—	2	2	no	—
283	C₂³×C₅	—	2	2	no	—
284	C₂³×C₅	—	2	2	no	—
285	C₂³×C₅	—	2	2	no	—
286	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
287	C₅×D₄	—	4	4	no	—
288	C₂×C₄×C₅	—	2	1	no	—
289	C₂×C₄×C₅	—	2	1	no	—
290	C₂×C₄×C₅	—	2	1	no	—
291	C₂×C₄×C₅	—	2	1	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	—	2	1	no	—
296	C₂×C₄×C₅	—	2	2	no	—
297	C₂×C₄×C₅	—	2	2	no	—
298	C₂×C₄×C₅	—	2	2	no	—
299	C₂×C₄×C₅	—	2	1	no	—
300	C₂×C₄×C₅	—	2	1	no	—
301	C₂×C₄×C₅	—	2	2	no	—
302	C₂×C₄×C₅	—	2	2	no	—
303	C₂×C₄×C₅	—	2	2	no	—
304	C₂×C₄×C₅	—	2	2	no	—
305	C₂×C₄×C₅	—	2	1	no	—
306	C₂×C₄×C₅	—	2	1	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	C₂×C₃×D₄	1	2	yes	—
312	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
313	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
314	C₂×C₄×C₅	C₃×#41	1	2	yes	—
315	C₂×C₄×C₅	C₃×#41	1	2	yes	—
316	C₂×C₄×C₅	C₃×#41	1	2	yes	—
317	C₂×C₄×C₅	C₃×#41	1	2	yes	—
318	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
319	#126	—	2	8	no	—
320	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
321	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
322	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
325	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
326	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
327	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
328	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
329	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
330	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
331	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
332	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
333	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
334	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
335	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
336	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
337	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
338	#95	C₂²×C₃×C₅	1	1	yes	characteristic
339	#95	C₂²×C₃×C₅	1	2	yes	—
340	#95	C₂²×C₃×C₅	1	1	yes	characteristic
341	#95	C₂²×C₃×C₅	1	2	yes	—
342	C₂×C₃×C₅	—	8	2	no	—
343	C₂×C₃×C₅	C₂×#126	1	1	yes	characteristic
344	C₂×C₃×C₅	#532	1	1	yes	characteristic
345	C₂×C₃×C₅	#542	1	1	yes	characteristic
346	C₂×C₃×C₅	#547	1	1	yes	characteristic
347	C₂×C₃×C₅	#550	1	1	yes	characteristic
348	C₂×C₃×C₅	#553	1	1	yes	characteristic
349	C₂×C₃×C₅	#556	1	1	yes	characteristic
350	C₂³×C₃	—	2	2	no	—
351	C₂³×C₃	—	2	2	no	—
352	C₂³×C₃	—	2	2	no	—
353	C₂³×C₃	—	2	2	no	—
354	C₂³×C₃	—	2	2	no	—
355	C₂³×C₃	—	2	2	no	—
356	C₂³×C₃	—	2	2	no	—
357	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
358	C₃×D₄	—	4	4	no	—
359	C₂×C₃×C₄	—	2	1	no	—
360	C₂×C₃×C₄	—	2	1	no	—
361	C₂×C₃×C₄	—	2	1	no	—
362	C₂×C₃×C₄	—	2	1	no	—
363	C₂×C₃×C₄	—	2	1	no	—
364	C₂×C₃×C₄	—	2	1	no	—
365	C₂×C₃×C₄	—	2	1	no	—
366	C₂×C₃×C₄	—	2	1	no	—
367	C₂×C₃×C₄	—	2	2	no	—
368	C₂×C₃×C₄	—	2	2	no	—
369	C₂×C₃×C₄	—	2	2	no	—
370	C₂×C₃×C₄	—	2	1	no	—
371	C₂×C₃×C₄	—	2	1	no	—
372	C₂×C₃×C₄	—	2	2	no	—
373	C₂×C₃×C₄	—	2	2	no	—
374	C₂×C₃×C₄	—	2	2	no	—
375	C₂×C₃×C₄	—	2	2	no	—
376	C₂×C₃×C₄	—	2	1	no	—
377	C₂×C₃×C₄	—	2	1	no	—
378	C₂×C₃×C₄	—	2	1	no	—
379	C₂×C₃×C₄	—	2	2	no	—
380	C₂×C₃×C₄	—	2	2	no	—
381	C₂×C₃×C₄	—	2	1	no	—
382	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
383	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
384	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
385	C₂×C₃×C₄	C₅×#41	1	2	yes	—
386	C₂×C₃×C₄	C₅×#41	1	2	yes	—
387	C₂×C₃×C₄	C₅×#41	1	2	yes	—
388	C₂×C₃×C₄	C₅×#41	1	2	yes	—
389	C₂²×C₅	—	4	2	no	—
390	C₂²×C₅	—	4	2	no	—
391	C₂²×C₅	—	4	2	no	—
392	C₂²×C₅	—	4	2	no	—
393	C₂²×C₅	—	4	2	no	—
394	C₂²×C₅	—	4	2	no	—
395	C₂²×C₅	—	4	2	no	—
396	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
397	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
398	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
399	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
400	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
401	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
402	C₂²×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
403	C₄×C₅	—	2	2	no	—
404	C₄×C₅	—	2	2	no	—
405	C₄×C₅	—	2	2	no	—
406	C₄×C₅	—	2	2	no	—
407	C₄×C₅	—	4	2	no	—
408	C₄×C₅	—	4	1	no	—
409	C₄×C₅	—	4	2	no	—
410	C₄×C₅	—	4	2	no	—
411	C₄×C₅	—	4	1	no	—
412	C₄×C₅	C₃×#126	1	4	yes	complemented
413	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
418	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
419	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
420	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
421	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
422	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
423	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
424	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
425	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
426	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	4	2	no	—
434	C₄⋊C₄	C₃×C₅×D₄	1	4	yes	complemented
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	2	4	no	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₂²⋊C₄	—	2	4	no	—
439	C₂²⋊C₄	—	4	2	no	—
440	C₂²⋊C₄	—	4	2	no	—
441	C₄²	—	2	4	no	—
442	C₄²	—	2	2	no	—
443	C₄²	—	2	2	no	—
444	C₄²	—	2	2	no	—
445	C₃×C₅	#2512	1	1	yes	characteristic, direct product factor, Hall
446	C₂²×C₃	—	4	2	no	—
447	C₂²×C₃	—	4	2	no	—
448	C₂²×C₃	—	4	2	no	—
449	C₂²×C₃	—	4	2	no	—
450	C₂²×C₃	—	4	2	no	—
451	C₂²×C₃	—	4	2	no	—
452	C₂²×C₃	—	4	2	no	—
453	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
454	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
455	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
456	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
457	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
458	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
459	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
460	C₃×C₄	—	2	2	no	—
461	C₃×C₄	—	2	2	no	—
462	C₃×C₄	—	2	2	no	—
463	C₃×C₄	—	2	2	no	—
464	C₃×C₄	—	4	2	no	—
465	C₃×C₄	—	4	1	no	—
466	C₃×C₄	—	4	2	no	—
467	C₃×C₄	—	4	2	no	—
468	C₃×C₄	—	4	1	no	—
469	C₃×C₄	C₅×#126	1	4	yes	complemented
470	C₂×C₅	—	8	2	no	—
471	C₂×C₅	C₂×C₃×#126	1	1	yes	characteristic
472	C₂×C₅	C₃×#532	1	1	yes	characteristic
473	C₂×C₅	C₃×#542	1	1	yes	characteristic
474	C₂×C₅	C₃×#547	1	1	yes	characteristic
475	C₂×C₅	C₃×#550	1	1	yes	characteristic
476	C₂×C₅	C₃×#553	1	1	yes	characteristic
477	C₂×C₅	C₃×#556	1	1	yes	characteristic
478	C₂³	—	2	2	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	2	no	—
484	C₂³	—	2	2	no	—
485	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
486	D₄	—	4	4	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
511	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
512	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
513	C₂×C₄	C₃×C₅×#41	1	2	yes	—
514	C₂×C₄	C₃×C₅×#41	1	2	yes	—
515	C₂×C₄	C₃×C₅×#41	1	2	yes	—
516	C₂×C₄	C₃×C₅×#41	1	2	yes	—
517	C₂×C₃	—	8	2	no	—
518	C₂×C₃	C₂×C₅×#126	1	1	yes	characteristic
519	C₂×C₃	C₅×#532	1	1	yes	characteristic
520	C₂×C₃	C₅×#542	1	1	yes	characteristic
521	C₂×C₃	C₅×#547	1	1	yes	characteristic
522	C₂×C₃	C₅×#550	1	1	yes	characteristic
523	C₂×C₃	C₅×#553	1	1	yes	characteristic
524	C₂×C₃	C₅×#556	1	1	yes	characteristic
525	C₅	C₃×#2512	1	1	yes	characteristic, direct product factor, Sylow
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
534	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
535	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
536	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
537	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
538	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
539	C₂²	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
540	C₄	—	2	2	no	—
541	C₄	—	2	2	no	—
542	C₄	—	2	2	no	—
543	C₄	—	2	2	no	—
544	C₄	—	4	2	no	—
545	C₄	—	4	1	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	1	no	—
549	C₄	C₃×C₅×#126	1	4	yes	complemented
550	C₃	C₅×#2512	1	1	yes	characteristic, direct product factor, Sylow
551	C₂	—	8	2	no	—
552	C₂	C₂×C₃×C₅×#126	1	1	yes	characteristic
553	C₂	C₃×C₅×#532	1	1	yes	characteristic
554	C₂	C₃×C₅×#542	1	1	yes	characteristic
555	C₂	C₃×C₅×#547	1	1	yes	characteristic
556	C₂	C₃×C₅×#550	1	1	yes	characteristic
557	C₂	C₃×C₅×#553	1	1	yes	characteristic
558	C₂	C₃×C₅×#556	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂²×C₃×C₅×D₄	C₂²
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×#126	C₂	semidirect
2. C₃×C₅×#397	C₂	semidirect
3. C₃×C₅×#396	C₂	semidirect
7. C₃×C₅×#389	C₂	semidirect
8. C₃×C₅×#388	C₂	semidirect
9. C₃×C₅×#384	C₂	semidirect
10. C₃×C₄×C₅×C₄⋊C₄	C₂	semidirect
11. C₃×C₄×C₅×C₄⋊C₄	C₂	semidirect
12. C₅×#2512	C₃	direct (N × H)
37. C₃×#2512	C₅	direct (N × H)
38. C₂×C₅×#126	C₂×C₃	semidirect
39. C₅×#397	C₂×C₃	semidirect
40. C₅×#396	C₂×C₃	semidirect
44. C₅×#389	C₂×C₃	semidirect
45. C₅×#388	C₂×C₃	semidirect
46. C₅×#384	C₂×C₃	semidirect
47. C₄×C₅×C₄⋊C₄	C₂×C₃	semidirect
48. C₄×C₅×C₄⋊C₄	C₂×C₃	semidirect
70. C₃×C₅×C₄⋊C₄	D₄	semidirect
81. C₂×C₃×#126	C₂×C₅	semidirect
82. C₃×#397	C₂×C₅	semidirect
83. C₃×#396	C₂×C₅	semidirect
87. C₃×#389	C₂×C₅	semidirect
88. C₃×#388	C₂×C₅	semidirect
89. C₃×#384	C₂×C₅	semidirect
90. C₃×C₄×C₄⋊C₄	C₂×C₅	semidirect
91. C₃×C₄×C₄⋊C₄	C₂×C₅	semidirect
116. #2512	C₃×C₅	direct (N × H)
201. C₅×C₄⋊C₄	C₃×D₄	semidirect
212. C₂×#126	C₂×C₃×C₅	semidirect
213. #397	C₂×C₃×C₅	semidirect
214. #396	C₂×C₃×C₅	semidirect
218. #389	C₂×C₃×C₅	semidirect
219. #388	C₂×C₃×C₅	semidirect
220. #384	C₂×C₃×C₅	semidirect
221. C₄×C₄⋊C₄	C₂×C₃×C₅	semidirect
222. C₄×C₄⋊C₄	C₂×C₃×C₅	semidirect
246. C₃×C₄×C₅	#126	semidirect
268. C₃×C₄⋊C₄	C₅×D₄	semidirect
412. C₄×C₅	C₃×#126	semidirect
434. C₄⋊C₄	C₃×C₅×D₄	semidirect
445. C₃×C₅	#2512	direct (N × H)
469. C₃×C₄	C₅×#126	semidirect
525. C₅	C₃×#2512	direct (N × H)
549. C₄	C₃×C₅×#126	semidirect
550. C₃	C₅×#2512	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×#126	2
C₃×C₅×#397	2
C₃×C₅×#396	2
C₃×C₅×#390	2
C₃×C₅×#389	2
C₃×C₅×#388	2
C₃×C₅×#384	2
C₃×C₄×C₅×C₄⋊C₄	2
C₅×#2512	3
C₃×#2512	5

Subgroups — C3×C5×#2512 #14275546

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2512 #14275546