Subgroups — C₃×C₅×#2404 #14275438

Order 1920.

2112 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×#120	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#393	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#392	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#386	C₂	1	4	yes	maximal, complemented
7	C₃×C₅×#386	C₂	1	2	yes	maximal, complemented
8	C₃×C₅×#381	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2404	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×#41	—	2	8	no	—
12	C₂×C₃×C₅×#41	—	2	4	no	—
13	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₅×D₄	C₂²	1	4	yes	—
15	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
16	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
18	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
19	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	complemented
20	C₃×C₅×#120	—	2	8	no	—
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	2	yes	—
25	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
27	C₃×C₅×#95	C₂²	1	2	yes	complemented
28	C₃×C₅×#95	C₂²	1	2	yes	complemented
29	C₃×C₅×#95	C₂²	1	4	yes	complemented
30	C₃×#2404	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₂²×C₅×#41	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×C₅×#120	C₂×C₃	1	2	yes	complemented
33	C₂²×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
34	C₅×#393	C₂×C₃	1	2	yes	complemented
35	C₅×#392	C₂×C₃	1	2	yes	complemented
36	C₅×#386	C₂×C₃	1	4	yes	complemented
37	C₅×#386	C₂×C₃	1	2	yes	complemented
38	C₅×#381	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	C₃×C₅×#41	—	2	32	no	—
47	C₂×C₃×C₅×Q₈	—	2	4	no	—
48	C₂×C₃×C₅×Q₈	—	2	2	no	—
49	C₂×C₃×C₅×D₄	—	2	4	no	—
50	C₂×C₃×C₅×D₄	—	2	4	no	—
51	C₂×C₃×C₅×D₄	—	2	4	no	—
52	C₂×C₃×C₅×D₄	—	2	8	no	—
53	C₂×C₃×C₅×D₄	—	2	8	no	—
54	C₂×C₃×C₅×D₄	—	2	2	no	—
55	C₂×C₃×C₅×D₄	—	4	4	no	—
56	C₂×C₃×C₅×D₄	D₄	1	8	yes	complemented
57	C₂²×C₃×C₄×C₅	—	2	4	no	—
58	C₂²×C₃×C₄×C₅	—	2	4	no	—
59	C₂²×C₃×C₄×C₅	—	2	4	no	—
60	C₂²×C₃×C₄×C₅	—	2	4	no	—
61	C₂²×C₃×C₄×C₅	—	2	2	no	—
62	C₂²×C₃×C₄×C₅	—	4	2	no	—
63	C₂²×C₃×C₄×C₅	D₄	1	4	yes	complemented
64	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
65	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
66	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
69	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
70	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
71	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
72	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
73	C₃×C₅×C₂²⋊C₄	—	4	4	no	—
74	C₂²×C₃×#41	C₂×C₅	1	1	yes	characteristic, complemented
75	C₂×C₃×#120	C₂×C₅	1	2	yes	complemented
76	C₂²×C₃×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
77	C₃×#393	C₂×C₅	1	2	yes	complemented
78	C₃×#392	C₂×C₅	1	2	yes	complemented
79	C₃×#386	C₂×C₅	1	4	yes	complemented
80	C₃×#386	C₂×C₅	1	2	yes	complemented
81	C₃×#381	C₂×C₅	1	1	yes	characteristic, complemented
82	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₅×#41	—	2	8	no	—
84	C₂×C₅×#41	—	2	4	no	—
85	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
86	C₂²×C₅×D₄	C₂²×C₃	1	4	yes	—
87	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
88	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
89	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
90	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
91	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	complemented
92	C₅×#120	—	2	8	no	—
93	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
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	4	yes	—
97	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
98	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
99	C₅×#95	C₂²×C₃	1	2	yes	complemented
100	C₅×#95	C₂²×C₃	1	2	yes	complemented
101	C₅×#95	C₂²×C₃	1	4	yes	complemented
102	#2404	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
103	C₂³×C₃×C₅	—	2	2	no	—
104	C₂³×C₃×C₅	—	2	2	no	—
105	C₂³×C₃×C₅	—	2	8	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	4	no	—
111	C₂³×C₃×C₅	—	2	2	no	—
112	C₂³×C₃×C₅	—	2	4	no	—
113	C₂³×C₃×C₅	—	2	4	no	—
114	C₂³×C₃×C₅	—	4	16	no	—
115	C₂³×C₃×C₅	—	4	4	no	—
116	C₂³×C₃×C₅	—	4	4	no	—
117	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
118	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
119	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
120	C₂³×C₃×C₅	#41	1	4	yes	complemented
121	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
122	C₃×C₅×Q₈	—	2	8	no	—
123	C₃×C₅×D₄	—	2	8	no	—
124	C₃×C₅×D₄	—	2	16	no	—
125	C₃×C₅×D₄	—	4	16	no	—
126	C₂×C₃×C₄×C₅	—	2	16	no	—
127	C₂×C₃×C₄×C₅	—	2	8	no	—
128	C₂×C₃×C₄×C₅	—	2	2	no	—
129	C₂×C₃×C₄×C₅	—	2	2	no	—
130	C₂×C₃×C₄×C₅	—	2	2	no	—
131	C₂×C₃×C₄×C₅	—	2	2	no	—
132	C₂×C₃×C₄×C₅	—	2	2	no	—
133	C₂×C₃×C₄×C₅	—	2	4	no	—
134	C₂×C₃×C₄×C₅	—	2	4	no	—
135	C₂×C₃×C₄×C₅	—	2	4	no	—
136	C₂×C₃×C₄×C₅	—	2	4	no	—
137	C₂×C₃×C₄×C₅	—	4	4	no	—
138	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
139	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
140	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
141	C₂⁵×C₃	C₂²×C₅	1	1	yes	characteristic
142	C₂×C₃×#41	—	2	8	no	—
143	C₂×C₃×#41	—	2	4	no	—
144	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
145	C₂²×C₃×D₄	C₂²×C₅	1	4	yes	—
146	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
147	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
148	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
149	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
150	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	complemented
151	C₃×#120	—	2	8	no	—
152	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
153	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
154	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
155	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
156	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
157	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
158	C₃×#95	C₂²×C₅	1	2	yes	complemented
159	C₃×#95	C₂²×C₅	1	2	yes	complemented
160	C₃×#95	C₂²×C₅	1	4	yes	complemented
161	C₂⁴×C₅	—	2	2	no	—
162	C₂⁴×C₅	—	2	4	no	—
163	C₂⁴×C₅	—	2	4	no	—
164	C₂⁴×C₅	—	4	2	no	—
165	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
166	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
167	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
168	C₅×#41	—	2	32	no	—
169	C₂×C₅×Q₈	—	2	4	no	—
170	C₂×C₅×Q₈	—	2	2	no	—
171	C₂×C₅×D₄	—	2	4	no	—
172	C₂×C₅×D₄	—	2	4	no	—
173	C₂×C₅×D₄	—	2	4	no	—
174	C₂×C₅×D₄	—	2	8	no	—
175	C₂×C₅×D₄	—	2	8	no	—
176	C₂×C₅×D₄	—	2	2	no	—
177	C₂×C₅×D₄	—	4	4	no	—
178	C₂×C₅×D₄	C₃×D₄	1	8	yes	complemented
179	C₂²×C₄×C₅	—	2	4	no	—
180	C₂²×C₄×C₅	—	2	4	no	—
181	C₂²×C₄×C₅	—	2	4	no	—
182	C₂²×C₄×C₅	—	2	4	no	—
183	C₂²×C₄×C₅	—	2	2	no	—
184	C₂²×C₄×C₅	—	4	2	no	—
185	C₂²×C₄×C₅	C₃×D₄	1	4	yes	complemented
186	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
187	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
188	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
189	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
190	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
191	C₅×C₂²⋊C₄	—	2	4	no	—
192	C₅×C₂²⋊C₄	—	2	4	no	—
193	C₅×C₂²⋊C₄	—	2	4	no	—
194	C₅×C₂²⋊C₄	—	2	8	no	—
195	C₅×C₂²⋊C₄	—	4	4	no	—
196	C₂²×#41	C₂×C₃×C₅	1	1	yes	characteristic, complemented
197	C₂×#120	C₂×C₃×C₅	1	2	yes	complemented
198	C₂²×C₂²⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
199	#393	C₂×C₃×C₅	1	2	yes	complemented
200	#392	C₂×C₃×C₅	1	2	yes	complemented
201	#386	C₂×C₃×C₅	1	4	yes	complemented
202	#386	C₂×C₃×C₅	1	2	yes	complemented
203	#381	C₂×C₃×C₅	1	1	yes	characteristic, complemented
204	C₂²×C₃×C₅	—	2	2	no	—
205	C₂²×C₃×C₅	—	2	2	no	—
206	C₂²×C₃×C₅	—	2	8	no	—
207	C₂²×C₃×C₅	—	2	4	no	—
208	C₂²×C₃×C₅	—	2	4	no	—
209	C₂²×C₃×C₅	—	2	8	no	—
210	C₂²×C₃×C₅	—	2	8	no	—
211	C₂²×C₃×C₅	—	4	16	no	—
212	C₂²×C₃×C₅	—	4	2	no	—
213	C₂²×C₃×C₅	—	4	2	no	—
214	C₂²×C₃×C₅	—	4	2	no	—
215	C₂²×C₃×C₅	—	4	2	no	—
216	C₂²×C₃×C₅	—	4	4	no	—
217	C₂²×C₃×C₅	—	4	4	no	—
218	C₂²×C₃×C₅	#120	1	4	yes	—
219	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
220	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
221	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
222	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
223	C₃×C₄×C₅	—	2	2	no	—
224	C₃×C₄×C₅	—	2	4	no	—
225	C₃×C₄×C₅	—	2	2	no	—
226	C₃×C₄×C₅	—	4	2	no	—
227	C₃×C₄×C₅	—	4	4	no	—
228	C₂⁴×C₃	—	2	2	no	—
229	C₂⁴×C₃	—	2	4	no	—
230	C₂⁴×C₃	—	2	4	no	—
231	C₂⁴×C₃	—	4	2	no	—
232	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
233	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
234	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
235	C₃×#41	—	2	32	no	—
236	C₂×C₃×Q₈	—	2	4	no	—
237	C₂×C₃×Q₈	—	2	2	no	—
238	C₂×C₃×D₄	—	2	4	no	—
239	C₂×C₃×D₄	—	2	4	no	—
240	C₂×C₃×D₄	—	2	4	no	—
241	C₂×C₃×D₄	—	2	8	no	—
242	C₂×C₃×D₄	—	2	8	no	—
243	C₂×C₃×D₄	—	2	2	no	—
244	C₂×C₃×D₄	—	4	4	no	—
245	C₂×C₃×D₄	C₅×D₄	1	8	yes	complemented
246	C₂²×C₃×C₄	—	2	4	no	—
247	C₂²×C₃×C₄	—	2	4	no	—
248	C₂²×C₃×C₄	—	2	4	no	—
249	C₂²×C₃×C₄	—	2	4	no	—
250	C₂²×C₃×C₄	—	2	2	no	—
251	C₂²×C₃×C₄	—	4	2	no	—
252	C₂²×C₃×C₄	C₅×D₄	1	4	yes	complemented
253	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
254	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
255	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
256	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
257	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
258	C₃×C₂²⋊C₄	—	2	4	no	—
259	C₃×C₂²⋊C₄	—	2	4	no	—
260	C₃×C₂²⋊C₄	—	2	4	no	—
261	C₃×C₂²⋊C₄	—	2	8	no	—
262	C₃×C₂²⋊C₄	—	4	4	no	—
263	C₂³×C₅	—	2	2	no	—
264	C₂³×C₅	—	2	2	no	—
265	C₂³×C₅	—	2	8	no	—
266	C₂³×C₅	—	2	2	no	—
267	C₂³×C₅	—	2	4	no	—
268	C₂³×C₅	—	2	2	no	—
269	C₂³×C₅	—	2	2	no	—
270	C₂³×C₅	—	2	4	no	—
271	C₂³×C₅	—	2	2	no	—
272	C₂³×C₅	—	2	4	no	—
273	C₂³×C₅	—	2	4	no	—
274	C₂³×C₅	—	4	16	no	—
275	C₂³×C₅	—	4	4	no	—
276	C₂³×C₅	—	4	4	no	—
277	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
278	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
279	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
280	C₂³×C₅	C₃×#41	1	4	yes	complemented
281	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
282	C₅×Q₈	—	2	8	no	—
283	C₅×D₄	—	2	8	no	—
284	C₅×D₄	—	2	16	no	—
285	C₅×D₄	—	4	16	no	—
286	C₂×C₄×C₅	—	2	16	no	—
287	C₂×C₄×C₅	—	2	8	no	—
288	C₂×C₄×C₅	—	2	2	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	2	no	—
293	C₂×C₄×C₅	—	2	4	no	—
294	C₂×C₄×C₅	—	2	4	no	—
295	C₂×C₄×C₅	—	2	4	no	—
296	C₂×C₄×C₅	—	2	4	no	—
297	C₂×C₄×C₅	—	4	4	no	—
298	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
299	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
300	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
301	C₂⁵	C₂²×C₃×C₅	1	1	yes	characteristic
302	C₂×#41	—	2	8	no	—
303	C₂×#41	—	2	4	no	—
304	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
305	C₂²×D₄	C₂²×C₃×C₅	1	4	yes	—
306	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
307	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
308	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
309	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
310	C₂³×C₄	C₂²×C₃×C₅	1	2	yes	complemented
311	#120	—	2	8	no	—
312	C₂×C₂²⋊C₄	—	2	4	no	—
313	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
314	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
315	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
316	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
317	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	complemented
318	#95	C₂²×C₃×C₅	1	2	yes	complemented
319	#95	C₂²×C₃×C₅	1	2	yes	complemented
320	#95	C₂²×C₃×C₅	1	4	yes	complemented
321	C₂×C₃×C₅	—	2	4	no	—
322	C₂×C₃×C₅	—	4	4	no	—
323	C₂×C₃×C₅	—	4	4	no	—
324	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
325	C₂×C₃×C₅	#525	1	1	yes	characteristic
326	C₂×C₃×C₅	#535	1	1	yes	characteristic
327	C₂×C₃×C₅	#546	1	4	yes	—
328	C₂³×C₃	—	2	2	no	—
329	C₂³×C₃	—	2	2	no	—
330	C₂³×C₃	—	2	8	no	—
331	C₂³×C₃	—	2	2	no	—
332	C₂³×C₃	—	2	4	no	—
333	C₂³×C₃	—	2	2	no	—
334	C₂³×C₃	—	2	2	no	—
335	C₂³×C₃	—	2	4	no	—
336	C₂³×C₃	—	2	2	no	—
337	C₂³×C₃	—	2	4	no	—
338	C₂³×C₃	—	2	4	no	—
339	C₂³×C₃	—	4	16	no	—
340	C₂³×C₃	—	4	4	no	—
341	C₂³×C₃	—	4	4	no	—
342	C₂³×C₃	C₂×C₅×D₄	1	4	yes	—
343	C₂³×C₃	C₂×C₅×D₄	1	4	yes	—
344	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
345	C₂³×C₃	C₅×#41	1	4	yes	complemented
346	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
347	C₃×Q₈	—	2	8	no	—
348	C₃×D₄	—	2	8	no	—
349	C₃×D₄	—	2	16	no	—
350	C₃×D₄	—	4	16	no	—
351	C₂×C₃×C₄	—	2	16	no	—
352	C₂×C₃×C₄	—	2	8	no	—
353	C₂×C₃×C₄	—	2	2	no	—
354	C₂×C₃×C₄	—	2	2	no	—
355	C₂×C₃×C₄	—	2	2	no	—
356	C₂×C₃×C₄	—	2	2	no	—
357	C₂×C₃×C₄	—	2	2	no	—
358	C₂×C₃×C₄	—	2	4	no	—
359	C₂×C₃×C₄	—	2	4	no	—
360	C₂×C₃×C₄	—	2	4	no	—
361	C₂×C₃×C₄	—	2	4	no	—
362	C₂×C₃×C₄	—	4	4	no	—
363	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
364	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
365	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
366	C₂²×C₅	—	2	2	no	—
367	C₂²×C₅	—	2	2	no	—
368	C₂²×C₅	—	2	8	no	—
369	C₂²×C₅	—	2	4	no	—
370	C₂²×C₅	—	2	4	no	—
371	C₂²×C₅	—	2	8	no	—
372	C₂²×C₅	—	2	8	no	—
373	C₂²×C₅	—	4	16	no	—
374	C₂²×C₅	—	4	2	no	—
375	C₂²×C₅	—	4	2	no	—
376	C₂²×C₅	—	4	2	no	—
377	C₂²×C₅	—	4	2	no	—
378	C₂²×C₅	—	4	4	no	—
379	C₂²×C₅	—	4	4	no	—
380	C₂²×C₅	C₃×#120	1	4	yes	—
381	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
382	C₂²×C₅	C₂²×C₃×D₄	1	2	yes	—
383	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
384	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
385	C₄×C₅	—	2	2	no	—
386	C₄×C₅	—	2	4	no	—
387	C₄×C₅	—	2	2	no	—
388	C₄×C₅	—	4	2	no	—
389	C₄×C₅	—	4	4	no	—
390	C₂⁴	—	2	2	no	—
391	C₂⁴	—	2	4	no	—
392	C₂⁴	—	2	4	no	—
393	C₂⁴	—	4	2	no	—
394	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
395	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
396	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
397	#41	—	2	32	no	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	8	no	—
404	C₂×D₄	—	2	8	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	4	4	no	—
407	C₂×D₄	C₃×C₅×D₄	1	8	yes	complemented
408	C₂²×C₄	—	2	4	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	4	2	no	—
414	C₂²×C₄	C₃×C₅×D₄	1	4	yes	complemented
415	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
416	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
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	—	2	4	no	—
421	C₂²⋊C₄	—	2	4	no	—
422	C₂²⋊C₄	—	2	4	no	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₂²⋊C₄	—	4	4	no	—
425	C₃×C₅	#2404	1	1	yes	characteristic, direct product factor, Hall
426	C₂²×C₃	—	2	2	no	—
427	C₂²×C₃	—	2	2	no	—
428	C₂²×C₃	—	2	8	no	—
429	C₂²×C₃	—	2	4	no	—
430	C₂²×C₃	—	2	4	no	—
431	C₂²×C₃	—	2	8	no	—
432	C₂²×C₃	—	2	8	no	—
433	C₂²×C₃	—	4	16	no	—
434	C₂²×C₃	—	4	2	no	—
435	C₂²×C₃	—	4	2	no	—
436	C₂²×C₃	—	4	2	no	—
437	C₂²×C₃	—	4	2	no	—
438	C₂²×C₃	—	4	4	no	—
439	C₂²×C₃	—	4	4	no	—
440	C₂²×C₃	C₅×#120	1	4	yes	—
441	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
442	C₂²×C₃	C₂²×C₅×D₄	1	2	yes	—
443	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
444	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
445	C₃×C₄	—	2	2	no	—
446	C₃×C₄	—	2	4	no	—
447	C₃×C₄	—	2	2	no	—
448	C₃×C₄	—	4	2	no	—
449	C₃×C₄	—	4	4	no	—
450	C₂×C₅	—	2	4	no	—
451	C₂×C₅	—	4	4	no	—
452	C₂×C₅	—	4	4	no	—
453	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
454	C₂×C₅	C₃×#525	1	1	yes	characteristic
455	C₂×C₅	C₃×#535	1	1	yes	characteristic
456	C₂×C₅	C₃×#546	1	4	yes	—
457	C₂³	—	2	2	no	—
458	C₂³	—	2	2	no	—
459	C₂³	—	2	8	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	2	no	—
464	C₂³	—	2	4	no	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	4	no	—
467	C₂³	—	2	4	no	—
468	C₂³	—	4	16	no	—
469	C₂³	—	4	4	no	—
470	C₂³	—	4	4	no	—
471	C₂³	C₂×C₃×C₅×D₄	1	4	yes	—
472	C₂³	C₂×C₃×C₅×D₄	1	4	yes	—
473	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
474	C₂³	C₃×C₅×#41	1	4	yes	complemented
475	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
476	Q₈	—	2	8	no	—
477	D₄	—	2	8	no	—
478	D₄	—	2	16	no	—
479	D₄	—	4	16	no	—
480	C₂×C₄	—	2	16	no	—
481	C₂×C₄	—	2	8	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	4	4	no	—
492	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
493	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
494	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
495	C₂×C₃	—	2	4	no	—
496	C₂×C₃	—	4	4	no	—
497	C₂×C₃	—	4	4	no	—
498	C₂×C₃	C₂×C₅×#120	1	1	yes	characteristic
499	C₂×C₃	C₅×#525	1	1	yes	characteristic
500	C₂×C₃	C₅×#535	1	1	yes	characteristic
501	C₂×C₃	C₅×#546	1	4	yes	—
502	C₅	C₃×#2404	1	1	yes	characteristic, direct product factor, Sylow
503	C₂²	—	2	2	no	—
504	C₂²	—	2	2	no	—
505	C₂²	—	2	8	no	—
506	C₂²	—	2	4	no	—
507	C₂²	—	2	4	no	—
508	C₂²	—	2	8	no	—
509	C₂²	—	2	8	no	—
510	C₂²	—	4	16	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	4	2	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	4	4	no	—
517	C₂²	C₃×C₅×#120	1	4	yes	—
518	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
519	C₂²	C₂²×C₃×C₅×D₄	1	2	yes	—
520	C₂²	C₂×C₃×C₅×#41	1	2	yes	—
521	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
522	C₄	—	2	2	no	—
523	C₄	—	2	4	no	—
524	C₄	—	2	2	no	—
525	C₄	—	4	2	no	—
526	C₄	—	4	4	no	—
527	C₃	C₅×#2404	1	1	yes	characteristic, direct product factor, Sylow
528	C₂	—	2	4	no	—
529	C₂	—	4	4	no	—
530	C₂	—	4	4	no	—
531	C₂	C₂×C₃×C₅×#120	1	1	yes	characteristic
532	C₂	C₃×C₅×#525	1	1	yes	characteristic
533	C₂	C₃×C₅×#535	1	1	yes	characteristic
534	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. C₂²×C₃×C₅×#41	C₂	semidirect
2. C₂×C₃×C₅×#120	C₂	semidirect
3. C₂²×C₃×C₅×C₂²⋊C₄	C₂	semidirect
4. C₃×C₅×#393	C₂	semidirect
5. C₃×C₅×#392	C₂	semidirect
6. C₃×C₅×#386	C₂	semidirect
7. C₃×C₅×#386	C₂	semidirect
8. C₃×C₅×#381	C₂	semidirect
9. C₅×#2404	C₃	direct (N × H)
13. C₂²×C₃×C₅×Q₈	C₂²	semidirect
16. C₂²×C₃×C₅×D₄	C₂²	semidirect
19. C₂³×C₃×C₄×C₅	C₂²	semidirect
26. C₂×C₃×C₅×C₂²⋊C₄	C₂²	semidirect
27. C₃×C₅×#95	C₂²	semidirect
28. C₃×C₅×#95	C₂²	semidirect
29. C₃×C₅×#95	C₂²	semidirect
30. C₃×#2404	C₅	direct (N × H)
31. C₂²×C₅×#41	C₂×C₃	semidirect
32. C₂×C₅×#120	C₂×C₃	semidirect
33. C₂²×C₅×C₂²⋊C₄	C₂×C₃	semidirect
34. C₅×#393	C₂×C₃	semidirect
35. C₅×#392	C₂×C₃	semidirect
36. C₅×#386	C₂×C₃	semidirect
37. C₅×#386	C₂×C₃	semidirect
38. C₅×#381	C₂×C₃	semidirect
56. C₂×C₃×C₅×D₄	D₄	semidirect
63. C₂²×C₃×C₄×C₅	D₄	semidirect
74. C₂²×C₃×#41	C₂×C₅	semidirect
75. C₂×C₃×#120	C₂×C₅	semidirect
76. C₂²×C₃×C₂²⋊C₄	C₂×C₅	semidirect
77. C₃×#393	C₂×C₅	semidirect
78. C₃×#392	C₂×C₅	semidirect
79. C₃×#386	C₂×C₅	semidirect
80. C₃×#386	C₂×C₅	semidirect
81. C₃×#381	C₂×C₅	semidirect
85. C₂²×C₅×Q₈	C₂²×C₃	semidirect
88. C₂²×C₅×D₄	C₂²×C₃	semidirect
91. C₂³×C₄×C₅	C₂²×C₃	semidirect
98. C₂×C₅×C₂²⋊C₄	C₂²×C₃	semidirect
99. C₅×#95	C₂²×C₃	semidirect
100. C₅×#95	C₂²×C₃	semidirect
101. C₅×#95	C₂²×C₃	semidirect
102. #2404	C₃×C₅	direct (N × H)
120. C₂³×C₃×C₅	#41	semidirect
144. C₂²×C₃×Q₈	C₂²×C₅	semidirect
147. C₂²×C₃×D₄	C₂²×C₅	semidirect
150. C₂³×C₃×C₄	C₂²×C₅	semidirect
157. C₂×C₃×C₂²⋊C₄	C₂²×C₅	semidirect
158. C₃×#95	C₂²×C₅	semidirect
159. C₃×#95	C₂²×C₅	semidirect
160. C₃×#95	C₂²×C₅	semidirect
178. C₂×C₅×D₄	C₃×D₄	semidirect
185. C₂²×C₄×C₅	C₃×D₄	semidirect
196. C₂²×#41	C₂×C₃×C₅	semidirect
197. C₂×#120	C₂×C₃×C₅	semidirect
198. C₂²×C₂²⋊C₄	C₂×C₃×C₅	semidirect
199. #393	C₂×C₃×C₅	semidirect
200. #392	C₂×C₃×C₅	semidirect
201. #386	C₂×C₃×C₅	semidirect
202. #386	C₂×C₃×C₅	semidirect
203. #381	C₂×C₃×C₅	semidirect
245. C₂×C₃×D₄	C₅×D₄	semidirect
252. C₂²×C₃×C₄	C₅×D₄	semidirect
280. C₂³×C₅	C₃×#41	semidirect
304. C₂²×Q₈	C₂²×C₃×C₅	semidirect
307. C₂²×D₄	C₂²×C₃×C₅	semidirect
310. C₂³×C₄	C₂²×C₃×C₅	semidirect
317. C₂×C₂²⋊C₄	C₂²×C₃×C₅	semidirect
318. #95	C₂²×C₃×C₅	semidirect
319. #95	C₂²×C₃×C₅	semidirect
320. #95	C₂²×C₃×C₅	semidirect
345. C₂³×C₃	C₅×#41	semidirect
407. C₂×D₄	C₃×C₅×D₄	semidirect
414. C₂²×C₄	C₃×C₅×D₄	semidirect
425. C₃×C₅	#2404	direct (N × H)
474. C₂³	C₃×C₅×#41	semidirect
502. C₅	C₃×#2404	direct (N × H)
527. C₃	C₅×#2404	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×#41	2
C₂×C₃×C₅×#120	2
C₂²×C₃×C₅×C₂²⋊C₄	2
C₃×C₅×#393	2
C₃×C₅×#392	2
C₃×C₅×#386	2
C₃×C₅×#381	2
C₅×#2404	3
C₃×#2404	5

Subgroups — C3×C5×#2404 #14275438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2404 #14275438