Subgroups — C₃×C₅×#2431 #14275465

Order 1920.

1904 conjugacy classes of subgroups.

The subgroup lattice has 596 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×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₅×#121	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#392	C₂	1	4	yes	maximal, complemented
6	C₃×C₅×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#386	C₂	1	4	yes	maximal, complemented
9	C₃×C₄×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#2431	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅×#41	—	2	4	no	—
12	C₂×C₃×C₅×#41	—	2	4	no	—
13	C₂×C₃×C₅×#41	—	2	4	no	—
14	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
17	C₂²×C₃×C₅×D₄	C₂²	1	4	yes	—
18	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
19	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
20	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
21	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
22	C₃×C₅×#127	—	2	4	no	—
23	C₃×C₅×#124	—	2	4	no	—
24	C₃×C₅×#121	—	2	4	no	—
25	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
28	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
30	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
31	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
32	C₃×C₅×#95	C₂²	1	2	yes	—
33	C₃×C₅×#95	C₂²	1	4	yes	—
34	C₃×#2431	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
35	C₂²×C₅×#41	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×C₅×#121	C₂×C₃	1	1	yes	characteristic, complemented
39	C₅×#392	C₂×C₃	1	4	yes	complemented
40	C₅×#391	C₂×C₃	1	1	yes	characteristic, complemented
41	C₅×#390	C₂×C₃	1	1	yes	characteristic, complemented
42	C₅×#386	C₂×C₃	1	4	yes	complemented
43	C₄×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
44	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
45	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
46	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
47	C₃×C₅×#41	—	2	32	no	—
48	C₂×C₃×C₅×Q₈	—	2	2	no	—
49	C₂×C₃×C₅×Q₈	—	2	2	no	—
50	C₂×C₃×C₅×Q₈	D₄	1	4	yes	complemented
51	C₂×C₃×C₅×D₄	—	2	4	no	—
52	C₂×C₃×C₅×D₄	—	2	4	no	—
53	C₂×C₃×C₅×D₄	—	2	4	no	—
54	C₂×C₃×C₅×D₄	—	2	8	no	—
55	C₂×C₃×C₅×D₄	—	2	4	no	—
56	C₂×C₃×C₅×D₄	—	2	2	no	—
57	C₂×C₃×C₅×D₄	—	2	2	no	—
58	C₂×C₃×C₅×D₄	—	2	2	no	—
59	C₂×C₃×C₅×D₄	—	2	2	no	—
60	C₂×C₃×C₅×D₄	—	2	2	no	—
61	C₂×C₃×C₅×D₄	—	2	8	no	—
62	C₂×C₃×C₅×D₄	—	4	4	no	—
63	C₂×C₃×C₅×D₄	—	4	4	no	—
64	C₂×C₃×C₅×D₄	D₄	1	4	yes	complemented
65	C₂²×C₃×C₄×C₅	—	2	4	no	—
66	C₂²×C₃×C₄×C₅	—	2	4	no	—
67	C₂²×C₃×C₄×C₅	—	2	4	no	—
68	C₂²×C₃×C₄×C₅	—	2	2	no	—
69	C₂²×C₃×C₄×C₅	—	2	2	no	—
70	C₂²×C₃×C₄×C₅	D₄	1	4	yes	complemented
71	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
72	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
73	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
74	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
75	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
76	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
77	C₃×C₅×C₄⋊C₄	—	2	2	no	—
78	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
79	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
80	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
81	C₃×C₅×C₂²⋊C₄	—	4	4	no	—
82	C₃×C₄²×C₅	—	2	2	no	—
83	C₃×C₄²×C₅	—	2	2	no	—
84	C₂²×C₃×#41	C₂×C₅	1	1	yes	characteristic, complemented
85	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic, complemented
86	C₂×C₃×#124	C₂×C₅	1	1	yes	characteristic, complemented
87	C₂×C₃×#121	C₂×C₅	1	1	yes	characteristic, complemented
88	C₃×#392	C₂×C₅	1	4	yes	complemented
89	C₃×#391	C₂×C₅	1	1	yes	characteristic, complemented
90	C₃×#390	C₂×C₅	1	1	yes	characteristic, complemented
91	C₃×#386	C₂×C₅	1	4	yes	complemented
92	C₃×C₄×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
93	C₂×C₅×#41	—	2	4	no	—
94	C₂×C₅×#41	—	2	4	no	—
95	C₂×C₅×#41	—	2	4	no	—
96	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
97	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
98	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
99	C₂²×C₅×D₄	C₂²×C₃	1	4	yes	—
100	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
101	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
102	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
103	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
104	C₅×#127	—	2	4	no	—
105	C₅×#124	—	2	4	no	—
106	C₅×#121	—	2	4	no	—
107	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
108	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
109	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
110	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
111	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
112	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
113	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
114	C₅×#95	C₂²×C₃	1	2	yes	—
115	C₅×#95	C₂²×C₃	1	4	yes	—
116	#2431	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
117	C₂³×C₃×C₅	—	2	2	no	—
118	C₂³×C₃×C₅	—	2	1	no	—
119	C₂³×C₃×C₅	—	2	2	no	—
120	C₂³×C₃×C₅	—	2	1	no	—
121	C₂³×C₃×C₅	—	2	2	no	—
122	C₂³×C₃×C₅	—	2	4	no	—
123	C₂³×C₃×C₅	—	2	4	no	—
124	C₂³×C₃×C₅	—	2	2	no	—
125	C₂³×C₃×C₅	—	2	2	no	—
126	C₂³×C₃×C₅	—	2	2	no	—
127	C₂³×C₃×C₅	—	2	2	no	—
128	C₂³×C₃×C₅	—	2	4	no	—
129	C₂³×C₃×C₅	—	2	4	no	—
130	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
131	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
132	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
133	C₃×C₅×Q₈	—	2	8	no	—
134	C₃×C₅×D₄	—	2	16	no	—
135	C₃×C₅×D₄	—	2	8	no	—
136	C₃×C₅×D₄	—	4	16	no	—
137	C₃×C₅×D₄	—	4	8	no	—
138	C₃×C₅×D₄	—	4	4	no	—
139	C₂×C₃×C₄×C₅	—	2	16	no	—
140	C₂×C₃×C₄×C₅	—	2	2	no	—
141	C₂×C₃×C₄×C₅	—	2	1	no	—
142	C₂×C₃×C₄×C₅	—	2	1	no	—
143	C₂×C₃×C₄×C₅	—	2	2	no	—
144	C₂×C₃×C₄×C₅	—	2	2	no	—
145	C₂×C₃×C₄×C₅	—	2	2	no	—
146	C₂×C₃×C₄×C₅	—	2	8	no	—
147	C₂×C₃×C₄×C₅	—	2	2	no	—
148	C₂×C₃×C₄×C₅	—	2	2	no	—
149	C₂×C₃×C₄×C₅	—	2	2	no	—
150	C₂×C₃×C₄×C₅	—	2	2	no	—
151	C₂×C₃×C₄×C₅	—	2	4	no	—
152	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
153	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
154	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
155	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
156	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
157	C₂×C₃×C₄×C₅	#41	1	4	yes	complemented
158	C₂×C₃×#41	—	2	4	no	—
159	C₂×C₃×#41	—	2	4	no	—
160	C₂×C₃×#41	—	2	4	no	—
161	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
162	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
163	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
164	C₂²×C₃×D₄	C₂²×C₅	1	4	yes	—
165	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
166	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
167	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
168	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
169	C₃×#127	—	2	4	no	—
170	C₃×#124	—	2	4	no	—
171	C₃×#121	—	2	4	no	—
172	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
173	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
174	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
175	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
176	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
177	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
178	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
179	C₃×#95	C₂²×C₅	1	2	yes	—
180	C₃×#95	C₂²×C₅	1	4	yes	—
181	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
182	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
183	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
184	C₅×#41	—	2	32	no	—
185	C₂×C₅×Q₈	—	2	2	no	—
186	C₂×C₅×Q₈	—	2	2	no	—
187	C₂×C₅×Q₈	C₃×D₄	1	4	yes	complemented
188	C₂×C₅×D₄	—	2	4	no	—
189	C₂×C₅×D₄	—	2	4	no	—
190	C₂×C₅×D₄	—	2	4	no	—
191	C₂×C₅×D₄	—	2	8	no	—
192	C₂×C₅×D₄	—	2	4	no	—
193	C₂×C₅×D₄	—	2	2	no	—
194	C₂×C₅×D₄	—	2	2	no	—
195	C₂×C₅×D₄	—	2	2	no	—
196	C₂×C₅×D₄	—	2	2	no	—
197	C₂×C₅×D₄	—	2	2	no	—
198	C₂×C₅×D₄	—	2	8	no	—
199	C₂×C₅×D₄	—	4	4	no	—
200	C₂×C₅×D₄	—	4	4	no	—
201	C₂×C₅×D₄	C₃×D₄	1	4	yes	complemented
202	C₂²×C₄×C₅	—	2	4	no	—
203	C₂²×C₄×C₅	—	2	4	no	—
204	C₂²×C₄×C₅	—	2	4	no	—
205	C₂²×C₄×C₅	—	2	2	no	—
206	C₂²×C₄×C₅	—	2	2	no	—
207	C₂²×C₄×C₅	C₃×D₄	1	4	yes	complemented
208	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
209	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
210	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
211	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
212	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
213	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
214	C₅×C₄⋊C₄	—	2	2	no	—
215	C₅×C₂²⋊C₄	—	2	4	no	—
216	C₅×C₂²⋊C₄	—	2	8	no	—
217	C₅×C₂²⋊C₄	—	2	4	no	—
218	C₅×C₂²⋊C₄	—	4	4	no	—
219	C₄²×C₅	—	2	2	no	—
220	C₄²×C₅	—	2	2	no	—
221	C₂²×#41	C₂×C₃×C₅	1	1	yes	characteristic, complemented
222	C₂×#127	C₂×C₃×C₅	1	1	yes	characteristic, complemented
223	C₂×#124	C₂×C₃×C₅	1	1	yes	characteristic, complemented
224	C₂×#121	C₂×C₃×C₅	1	1	yes	characteristic, complemented
225	#392	C₂×C₃×C₅	1	4	yes	complemented
226	#391	C₂×C₃×C₅	1	1	yes	characteristic, complemented
227	#390	C₂×C₃×C₅	1	1	yes	characteristic, complemented
228	#386	C₂×C₃×C₅	1	4	yes	complemented
229	C₄×C₂²⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
230	C₂²×C₃×C₅	—	2	4	no	—
231	C₂²×C₃×C₅	—	2	2	no	—
232	C₂²×C₃×C₅	—	2	2	no	—
233	C₂²×C₃×C₅	—	2	2	no	—
234	C₂²×C₃×C₅	—	2	8	no	—
235	C₂²×C₃×C₅	—	4	2	no	—
236	C₂²×C₃×C₅	—	4	2	no	—
237	C₂²×C₃×C₅	—	4	2	no	—
238	C₂²×C₃×C₅	—	4	2	no	—
239	C₂²×C₃×C₅	—	4	4	no	—
240	C₂²×C₃×C₅	—	4	2	no	—
241	C₂²×C₃×C₅	—	4	2	no	—
242	C₂²×C₃×C₅	—	4	2	no	—
243	C₂²×C₃×C₅	—	4	4	no	—
244	C₂²×C₃×C₅	—	4	4	no	—
245	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
246	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
247	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
248	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
249	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
250	C₃×C₄×C₅	—	2	2	no	—
251	C₃×C₄×C₅	—	2	4	no	—
252	C₃×C₄×C₅	—	2	4	no	—
253	C₃×C₄×C₅	—	4	2	no	—
254	C₃×C₄×C₅	—	4	2	no	—
255	C₃×C₄×C₅	#120	1	4	yes	—
256	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
257	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
258	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
259	C₃×#41	—	2	32	no	—
260	C₂×C₃×Q₈	—	2	2	no	—
261	C₂×C₃×Q₈	—	2	2	no	—
262	C₂×C₃×Q₈	C₅×D₄	1	4	yes	complemented
263	C₂×C₃×D₄	—	2	4	no	—
264	C₂×C₃×D₄	—	2	4	no	—
265	C₂×C₃×D₄	—	2	4	no	—
266	C₂×C₃×D₄	—	2	8	no	—
267	C₂×C₃×D₄	—	2	4	no	—
268	C₂×C₃×D₄	—	2	2	no	—
269	C₂×C₃×D₄	—	2	2	no	—
270	C₂×C₃×D₄	—	2	2	no	—
271	C₂×C₃×D₄	—	2	2	no	—
272	C₂×C₃×D₄	—	2	2	no	—
273	C₂×C₃×D₄	—	2	8	no	—
274	C₂×C₃×D₄	—	4	4	no	—
275	C₂×C₃×D₄	—	4	4	no	—
276	C₂×C₃×D₄	C₅×D₄	1	4	yes	complemented
277	C₂²×C₃×C₄	—	2	4	no	—
278	C₂²×C₃×C₄	—	2	4	no	—
279	C₂²×C₃×C₄	—	2	4	no	—
280	C₂²×C₃×C₄	—	2	2	no	—
281	C₂²×C₃×C₄	—	2	2	no	—
282	C₂²×C₃×C₄	C₅×D₄	1	4	yes	complemented
283	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
284	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
285	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
286	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
287	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
288	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
289	C₃×C₄⋊C₄	—	2	2	no	—
290	C₃×C₂²⋊C₄	—	2	4	no	—
291	C₃×C₂²⋊C₄	—	2	8	no	—
292	C₃×C₂²⋊C₄	—	2	4	no	—
293	C₃×C₂²⋊C₄	—	4	4	no	—
294	C₃×C₄²	—	2	2	no	—
295	C₃×C₄²	—	2	2	no	—
296	C₂³×C₅	—	2	2	no	—
297	C₂³×C₅	—	2	1	no	—
298	C₂³×C₅	—	2	2	no	—
299	C₂³×C₅	—	2	1	no	—
300	C₂³×C₅	—	2	2	no	—
301	C₂³×C₅	—	2	4	no	—
302	C₂³×C₅	—	2	4	no	—
303	C₂³×C₅	—	2	2	no	—
304	C₂³×C₅	—	2	2	no	—
305	C₂³×C₅	—	2	2	no	—
306	C₂³×C₅	—	2	2	no	—
307	C₂³×C₅	—	2	4	no	—
308	C₂³×C₅	—	2	4	no	—
309	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
310	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
311	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
312	C₅×Q₈	—	2	8	no	—
313	C₅×D₄	—	2	16	no	—
314	C₅×D₄	—	2	8	no	—
315	C₅×D₄	—	4	16	no	—
316	C₅×D₄	—	4	8	no	—
317	C₅×D₄	—	4	4	no	—
318	C₂×C₄×C₅	—	2	16	no	—
319	C₂×C₄×C₅	—	2	2	no	—
320	C₂×C₄×C₅	—	2	1	no	—
321	C₂×C₄×C₅	—	2	1	no	—
322	C₂×C₄×C₅	—	2	2	no	—
323	C₂×C₄×C₅	—	2	2	no	—
324	C₂×C₄×C₅	—	2	2	no	—
325	C₂×C₄×C₅	—	2	8	no	—
326	C₂×C₄×C₅	—	2	2	no	—
327	C₂×C₄×C₅	—	2	2	no	—
328	C₂×C₄×C₅	—	2	2	no	—
329	C₂×C₄×C₅	—	2	2	no	—
330	C₂×C₄×C₅	—	2	4	no	—
331	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
332	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
333	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
334	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
335	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
336	C₂×C₄×C₅	C₃×#41	1	4	yes	complemented
337	C₂×#41	—	2	4	no	—
338	C₂×#41	—	2	4	no	—
339	C₂×#41	—	2	4	no	—
340	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
341	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
342	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
343	C₂²×D₄	C₂²×C₃×C₅	1	4	yes	—
344	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
345	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
346	C₂³×C₄	C₂²×C₃×C₅	1	2	yes	—
347	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
348	#127	—	2	4	no	—
349	#124	—	2	4	no	—
350	#121	—	2	4	no	—
351	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
352	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
353	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
354	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
355	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
356	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
357	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
358	#95	C₂²×C₃×C₅	1	2	yes	—
359	#95	C₂²×C₃×C₅	1	4	yes	—
360	C₂×C₃×C₅	—	4	2	no	—
361	C₂×C₃×C₅	—	4	4	no	—
362	C₂×C₃×C₅	—	8	2	no	—
363	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
364	C₂×C₃×C₅	#532	1	1	yes	characteristic
365	C₂×C₃×C₅	#535	1	1	yes	characteristic
366	C₂×C₃×C₅	#546	1	2	yes	—
367	C₂×C₃×C₅	#550	1	2	yes	—
368	C₂³×C₃	—	2	2	no	—
369	C₂³×C₃	—	2	1	no	—
370	C₂³×C₃	—	2	2	no	—
371	C₂³×C₃	—	2	1	no	—
372	C₂³×C₃	—	2	2	no	—
373	C₂³×C₃	—	2	4	no	—
374	C₂³×C₃	—	2	4	no	—
375	C₂³×C₃	—	2	2	no	—
376	C₂³×C₃	—	2	2	no	—
377	C₂³×C₃	—	2	2	no	—
378	C₂³×C₃	—	2	2	no	—
379	C₂³×C₃	—	2	4	no	—
380	C₂³×C₃	—	2	4	no	—
381	C₂³×C₃	C₂×C₅×D₄	1	4	yes	—
382	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
383	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
384	C₃×Q₈	—	2	8	no	—
385	C₃×D₄	—	2	16	no	—
386	C₃×D₄	—	2	8	no	—
387	C₃×D₄	—	4	16	no	—
388	C₃×D₄	—	4	8	no	—
389	C₃×D₄	—	4	4	no	—
390	C₂×C₃×C₄	—	2	16	no	—
391	C₂×C₃×C₄	—	2	2	no	—
392	C₂×C₃×C₄	—	2	1	no	—
393	C₂×C₃×C₄	—	2	1	no	—
394	C₂×C₃×C₄	—	2	2	no	—
395	C₂×C₃×C₄	—	2	2	no	—
396	C₂×C₃×C₄	—	2	2	no	—
397	C₂×C₃×C₄	—	2	8	no	—
398	C₂×C₃×C₄	—	2	2	no	—
399	C₂×C₃×C₄	—	2	2	no	—
400	C₂×C₃×C₄	—	2	2	no	—
401	C₂×C₃×C₄	—	2	2	no	—
402	C₂×C₃×C₄	—	2	4	no	—
403	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
404	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
405	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
406	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
407	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
408	C₂×C₃×C₄	C₅×#41	1	4	yes	complemented
409	C₂²×C₅	—	2	4	no	—
410	C₂²×C₅	—	2	2	no	—
411	C₂²×C₅	—	2	2	no	—
412	C₂²×C₅	—	2	2	no	—
413	C₂²×C₅	—	2	8	no	—
414	C₂²×C₅	—	4	2	no	—
415	C₂²×C₅	—	4	2	no	—
416	C₂²×C₅	—	4	2	no	—
417	C₂²×C₅	—	4	2	no	—
418	C₂²×C₅	—	4	4	no	—
419	C₂²×C₅	—	4	2	no	—
420	C₂²×C₅	—	4	2	no	—
421	C₂²×C₅	—	4	2	no	—
422	C₂²×C₅	—	4	4	no	—
423	C₂²×C₅	—	4	4	no	—
424	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
425	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
426	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
427	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
428	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
429	C₄×C₅	—	2	2	no	—
430	C₄×C₅	—	2	4	no	—
431	C₄×C₅	—	2	4	no	—
432	C₄×C₅	—	4	2	no	—
433	C₄×C₅	—	4	2	no	—
434	C₄×C₅	C₃×#120	1	4	yes	—
435	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
436	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
437	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
438	#41	—	2	32	no	—
439	C₂×Q₈	—	2	2	no	—
440	C₂×Q₈	—	2	2	no	—
441	C₂×Q₈	C₃×C₅×D₄	1	4	yes	complemented
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	2	8	no	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	4	4	no	—
454	C₂×D₄	—	4	4	no	—
455	C₂×D₄	C₃×C₅×D₄	1	4	yes	complemented
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	C₃×C₅×D₄	1	4	yes	complemented
462	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
463	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
464	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
465	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
466	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
467	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
468	C₄⋊C₄	—	2	2	no	—
469	C₂²⋊C₄	—	2	4	no	—
470	C₂²⋊C₄	—	2	8	no	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	—	4	4	no	—
473	C₄²	—	2	2	no	—
474	C₄²	—	2	2	no	—
475	C₃×C₅	#2431	1	1	yes	characteristic, direct product factor, Hall
476	C₂²×C₃	—	2	4	no	—
477	C₂²×C₃	—	2	2	no	—
478	C₂²×C₃	—	2	2	no	—
479	C₂²×C₃	—	2	2	no	—
480	C₂²×C₃	—	2	8	no	—
481	C₂²×C₃	—	4	2	no	—
482	C₂²×C₃	—	4	2	no	—
483	C₂²×C₃	—	4	2	no	—
484	C₂²×C₃	—	4	2	no	—
485	C₂²×C₃	—	4	4	no	—
486	C₂²×C₃	—	4	2	no	—
487	C₂²×C₃	—	4	2	no	—
488	C₂²×C₃	—	4	2	no	—
489	C₂²×C₃	—	4	4	no	—
490	C₂²×C₃	—	4	4	no	—
491	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
492	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
493	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
494	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
495	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
496	C₃×C₄	—	2	2	no	—
497	C₃×C₄	—	2	4	no	—
498	C₃×C₄	—	2	4	no	—
499	C₃×C₄	—	4	2	no	—
500	C₃×C₄	—	4	2	no	—
501	C₃×C₄	C₅×#120	1	4	yes	—
502	C₂×C₅	—	4	2	no	—
503	C₂×C₅	—	4	4	no	—
504	C₂×C₅	—	8	2	no	—
505	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
506	C₂×C₅	C₃×#532	1	1	yes	characteristic
507	C₂×C₅	C₃×#535	1	1	yes	characteristic
508	C₂×C₅	C₃×#546	1	2	yes	—
509	C₂×C₅	C₃×#550	1	2	yes	—
510	C₂³	—	2	2	no	—
511	C₂³	—	2	1	no	—
512	C₂³	—	2	2	no	—
513	C₂³	—	2	1	no	—
514	C₂³	—	2	2	no	—
515	C₂³	—	2	4	no	—
516	C₂³	—	2	4	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	2	no	—
519	C₂³	—	2	2	no	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	4	no	—
522	C₂³	—	2	4	no	—
523	C₂³	C₂×C₃×C₅×D₄	1	4	yes	—
524	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
525	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
526	Q₈	—	2	8	no	—
527	D₄	—	2	16	no	—
528	D₄	—	2	8	no	—
529	D₄	—	4	16	no	—
530	D₄	—	4	8	no	—
531	D₄	—	4	4	no	—
532	C₂×C₄	—	2	16	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	8	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
546	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
547	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
548	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
549	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
550	C₂×C₄	C₃×C₅×#41	1	4	yes	complemented
551	C₂×C₃	—	4	2	no	—
552	C₂×C₃	—	4	4	no	—
553	C₂×C₃	—	8	2	no	—
554	C₂×C₃	C₂×C₅×#120	1	1	yes	characteristic
555	C₂×C₃	C₅×#532	1	1	yes	characteristic
556	C₂×C₃	C₅×#535	1	1	yes	characteristic
557	C₂×C₃	C₅×#546	1	2	yes	—
558	C₂×C₃	C₅×#550	1	2	yes	—
559	C₅	C₃×#2431	1	1	yes	characteristic, direct product factor, Sylow
560	C₂²	—	2	4	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	2	8	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	2	no	—
571	C₂²	—	4	2	no	—
572	C₂²	—	4	2	no	—
573	C₂²	—	4	4	no	—
574	C₂²	—	4	4	no	—
575	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
576	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
577	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
578	C₂²	C₂×C₃×C₅×#41	1	2	yes	—
579	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
580	C₄	—	2	2	no	—
581	C₄	—	2	4	no	—
582	C₄	—	2	4	no	—
583	C₄	—	4	2	no	—
584	C₄	—	4	2	no	—
585	C₄	C₃×C₅×#120	1	4	yes	—
586	C₃	C₅×#2431	1	1	yes	characteristic, direct product factor, Sylow
587	C₂	—	4	2	no	—
588	C₂	—	4	4	no	—
589	C₂	—	8	2	no	—
590	C₂	C₂×C₃×C₅×#120	1	1	yes	characteristic
591	C₂	C₃×C₅×#532	1	1	yes	characteristic
592	C₂	C₃×C₅×#535	1	1	yes	characteristic
593	C₂	C₃×C₅×#546	1	2	yes	—
594	C₂	C₃×C₅×#550	1	2	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×#127	C₂	semidirect
3. C₂×C₃×C₅×#124	C₂	semidirect
4. C₂×C₃×C₅×#121	C₂	semidirect
5. C₃×C₅×#392	C₂	semidirect
6. C₃×C₅×#391	C₂	semidirect
7. C₃×C₅×#390	C₂	semidirect
8. C₃×C₅×#386	C₂	semidirect
9. C₃×C₄×C₅×C₂²⋊C₄	C₂	semidirect
10. C₅×#2431	C₃	direct (N × H)
34. C₃×#2431	C₅	direct (N × H)
35. C₂²×C₅×#41	C₂×C₃	semidirect
36. C₂×C₅×#127	C₂×C₃	semidirect
37. C₂×C₅×#124	C₂×C₃	semidirect
38. C₂×C₅×#121	C₂×C₃	semidirect
39. C₅×#392	C₂×C₃	semidirect
40. C₅×#391	C₂×C₃	semidirect
41. C₅×#390	C₂×C₃	semidirect
42. C₅×#386	C₂×C₃	semidirect
43. C₄×C₅×C₂²⋊C₄	C₂×C₃	semidirect
50. C₂×C₃×C₅×Q₈	D₄	semidirect
64. C₂×C₃×C₅×D₄	D₄	semidirect
70. C₂²×C₃×C₄×C₅	D₄	semidirect
84. C₂²×C₃×#41	C₂×C₅	semidirect
85. C₂×C₃×#127	C₂×C₅	semidirect
86. C₂×C₃×#124	C₂×C₅	semidirect
87. C₂×C₃×#121	C₂×C₅	semidirect
88. C₃×#392	C₂×C₅	semidirect
89. C₃×#391	C₂×C₅	semidirect
90. C₃×#390	C₂×C₅	semidirect
91. C₃×#386	C₂×C₅	semidirect
92. C₃×C₄×C₂²⋊C₄	C₂×C₅	semidirect
116. #2431	C₃×C₅	direct (N × H)
157. C₂×C₃×C₄×C₅	#41	semidirect
187. C₂×C₅×Q₈	C₃×D₄	semidirect
201. C₂×C₅×D₄	C₃×D₄	semidirect
207. C₂²×C₄×C₅	C₃×D₄	semidirect
221. C₂²×#41	C₂×C₃×C₅	semidirect
222. C₂×#127	C₂×C₃×C₅	semidirect
223. C₂×#124	C₂×C₃×C₅	semidirect
224. C₂×#121	C₂×C₃×C₅	semidirect
225. #392	C₂×C₃×C₅	semidirect
226. #391	C₂×C₃×C₅	semidirect
227. #390	C₂×C₃×C₅	semidirect
228. #386	C₂×C₃×C₅	semidirect
229. C₄×C₂²⋊C₄	C₂×C₃×C₅	semidirect
262. C₂×C₃×Q₈	C₅×D₄	semidirect
276. C₂×C₃×D₄	C₅×D₄	semidirect
282. C₂²×C₃×C₄	C₅×D₄	semidirect
336. C₂×C₄×C₅	C₃×#41	semidirect
408. C₂×C₃×C₄	C₅×#41	semidirect
441. C₂×Q₈	C₃×C₅×D₄	semidirect
455. C₂×D₄	C₃×C₅×D₄	semidirect
461. C₂²×C₄	C₃×C₅×D₄	semidirect
475. C₃×C₅	#2431	direct (N × H)
550. C₂×C₄	C₃×C₅×#41	semidirect
559. C₅	C₃×#2431	direct (N × H)
586. C₃	C₅×#2431	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×#127	2
C₂×C₃×C₅×#124	2
C₂×C₃×C₅×#121	2
C₃×C₅×#392	2
C₃×C₅×#391	2
C₃×C₅×#390	2
C₃×C₅×#386	2
C₃×C₄×C₅×C₂²⋊C₄	2
C₅×#2431	3
C₃×#2431	5

Subgroups — C3×C5×#2431 #14275465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2431 #14275465