Subgroups — C₃×C₅×#2311 #14275345

Order 1920.

1760 conjugacy classes of subgroups.

The subgroup lattice has 464 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
2	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#391	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#390	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#386	C₂	1	4	yes	maximal, complemented
7	C₃×C₅×#380	C₂	1	4	yes	maximal, complemented
8	C₅×#2311	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×#41	—	2	4	no	—
10	C₂×C₃×C₅×#41	—	2	4	no	—
11	C₂×C₃×C₅×#41	C₄	1	8	yes	complemented
12	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
14	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
16	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
17	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
18	C₃×C₅×#117	—	2	4	no	—
19	C₂×C₃×C₅×C₄⋊C₄	—	2	4	no	—
20	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	8	yes	—
25	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
26	C₃×C₅×#95	C₂²	1	8	yes	—
27	C₃×#2311	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂²×C₅×#41	C₂×C₃	1	1	yes	characteristic
29	C₂×C₅×#117	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂²×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
31	C₅×#391	C₂×C₃	1	2	yes	complemented
32	C₅×#390	C₂×C₃	1	2	yes	complemented
33	C₅×#386	C₂×C₃	1	4	yes	complemented
34	C₅×#380	C₂×C₃	1	4	yes	complemented
35	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
36	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
37	C₃×C₅×#41	—	2	32	no	—
38	C₂×C₃×C₅×Q₈	—	2	2	no	—
39	C₂×C₃×C₅×Q₈	—	2	2	no	—
40	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	complemented
41	C₂×C₃×C₅×D₄	—	2	4	no	—
42	C₂×C₃×C₅×D₄	—	2	4	no	—
43	C₂×C₃×C₅×D₄	—	2	2	no	—
44	C₂×C₃×C₅×D₄	—	2	2	no	—
45	C₂×C₃×C₅×D₄	C₂×C₄	1	8	yes	—
46	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	complemented
47	C₂²×C₃×C₄×C₅	—	2	4	no	—
48	C₂²×C₃×C₄×C₅	—	2	4	no	—
49	C₂²×C₃×C₄×C₅	—	2	4	no	—
50	C₂²×C₃×C₄×C₅	—	4	4	no	—
51	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
52	C₂²×C₃×C₄×C₅	C₂×C₄	1	8	yes	complemented
53	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
54	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
55	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
56	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
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	C₃×C₅×C₄⋊C₄	—	2	4	no	—
61	C₃×C₅×C₄⋊C₄	—	2	4	no	—
62	C₃×C₅×C₄⋊C₄	—	2	4	no	—
63	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
64	C₃×C₅×C₂²⋊C₄	—	4	8	no	—
65	C₃×C₄²×C₅	—	2	4	no	—
66	C₂²×C₃×#41	C₂×C₅	1	1	yes	characteristic
67	C₂×C₃×#117	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂²×C₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
69	C₃×#391	C₂×C₅	1	2	yes	complemented
70	C₃×#390	C₂×C₅	1	2	yes	complemented
71	C₃×#386	C₂×C₅	1	4	yes	complemented
72	C₃×#380	C₂×C₅	1	4	yes	complemented
73	C₂×C₅×#41	—	2	4	no	—
74	C₂×C₅×#41	—	2	4	no	—
75	C₂×C₅×#41	C₃×C₄	1	8	yes	complemented
76	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
77	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
78	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
79	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
80	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
81	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
82	C₅×#117	—	2	4	no	—
83	C₂×C₅×C₄⋊C₄	—	2	4	no	—
84	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
85	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
86	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
89	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
90	C₅×#95	C₂²×C₃	1	8	yes	—
91	#2311	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
92	C₂³×C₃×C₅	—	2	2	no	—
93	C₂³×C₃×C₅	—	2	2	no	—
94	C₂³×C₃×C₅	—	2	2	no	—
95	C₂³×C₃×C₅	—	2	2	no	—
96	C₂³×C₃×C₅	—	2	4	no	—
97	C₂³×C₃×C₅	—	2	4	no	—
98	C₂³×C₃×C₅	C₂²×C₄	1	4	yes	—
99	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
100	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
101	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
102	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
103	C₃×C₅×Q₈	—	2	8	no	—
104	C₃×C₅×D₄	—	2	16	no	—
105	C₃×C₅×D₄	—	2	8	no	—
106	C₂×C₃×C₄×C₅	—	2	2	no	—
107	C₂×C₃×C₄×C₅	—	2	2	no	—
108	C₂×C₃×C₄×C₅	—	2	16	no	—
109	C₂×C₃×C₄×C₅	—	2	4	no	—
110	C₂×C₃×C₄×C₅	—	2	4	no	—
111	C₂×C₃×C₄×C₅	—	2	4	no	—
112	C₂×C₃×C₄×C₅	—	2	8	no	—
113	C₂×C₃×C₄×C₅	—	2	4	no	—
114	C₂×C₃×C₄×C₅	—	4	8	no	—
115	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	16	yes	complemented
116	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
117	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
118	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
119	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
120	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
121	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
122	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
123	C₂×C₃×#41	—	2	4	no	—
124	C₂×C₃×#41	—	2	4	no	—
125	C₂×C₃×#41	C₄×C₅	1	8	yes	complemented
126	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
127	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
128	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
129	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
130	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
131	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
132	C₃×#117	—	2	4	no	—
133	C₂×C₃×C₄⋊C₄	—	2	4	no	—
134	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
135	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
136	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
137	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
138	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	8	yes	—
139	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
140	C₃×#95	C₂²×C₅	1	8	yes	—
141	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
142	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
143	C₅×#41	—	2	32	no	—
144	C₂×C₅×Q₈	—	2	2	no	—
145	C₂×C₅×Q₈	—	2	2	no	—
146	C₂×C₅×Q₈	C₂×C₃×C₄	1	4	yes	complemented
147	C₂×C₅×D₄	—	2	4	no	—
148	C₂×C₅×D₄	—	2	4	no	—
149	C₂×C₅×D₄	—	2	2	no	—
150	C₂×C₅×D₄	—	2	2	no	—
151	C₂×C₅×D₄	C₂×C₃×C₄	1	8	yes	—
152	C₂×C₅×D₄	C₂×C₃×C₄	1	4	yes	complemented
153	C₂²×C₄×C₅	—	2	4	no	—
154	C₂²×C₄×C₅	—	2	4	no	—
155	C₂²×C₄×C₅	—	2	4	no	—
156	C₂²×C₄×C₅	—	4	4	no	—
157	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
158	C₂²×C₄×C₅	C₂×C₃×C₄	1	8	yes	complemented
159	C₂²×C₄×C₅	C₃×D₄	1	4	yes	—
160	C₂²×C₄×C₅	C₃×D₄	1	4	yes	—
161	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
162	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
163	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
164	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
165	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
166	C₅×C₄⋊C₄	—	2	4	no	—
167	C₅×C₄⋊C₄	—	2	4	no	—
168	C₅×C₄⋊C₄	—	2	4	no	—
169	C₅×C₂²⋊C₄	—	2	4	no	—
170	C₅×C₂²⋊C₄	—	4	8	no	—
171	C₄²×C₅	—	2	4	no	—
172	C₂²×#41	C₂×C₃×C₅	1	1	yes	characteristic
173	C₂×#117	C₂×C₃×C₅	1	1	yes	characteristic, complemented
174	C₂²×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
175	#391	C₂×C₃×C₅	1	2	yes	complemented
176	#390	C₂×C₃×C₅	1	2	yes	complemented
177	#386	C₂×C₃×C₅	1	4	yes	complemented
178	#380	C₂×C₃×C₅	1	4	yes	complemented
179	C₂²×C₃×C₅	—	2	4	no	—
180	C₂²×C₃×C₅	—	2	2	no	—
181	C₂²×C₃×C₅	—	2	2	no	—
182	C₂²×C₃×C₅	—	2	4	no	—
183	C₂²×C₃×C₅	—	2	4	no	—
184	C₂²×C₃×C₅	—	2	8	no	—
185	C₂²×C₃×C₅	—	4	2	no	—
186	C₂²×C₃×C₅	—	4	2	no	—
187	C₂²×C₃×C₅	—	4	4	no	—
188	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	4	yes	—
189	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
190	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
191	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
192	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
193	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
194	C₃×C₄×C₅	—	2	4	no	—
195	C₃×C₄×C₅	—	4	4	no	—
196	C₃×C₄×C₅	—	4	4	no	—
197	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
198	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
199	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
200	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
201	C₃×#41	—	2	32	no	—
202	C₂×C₃×Q₈	—	2	2	no	—
203	C₂×C₃×Q₈	—	2	2	no	—
204	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	complemented
205	C₂×C₃×D₄	—	2	4	no	—
206	C₂×C₃×D₄	—	2	4	no	—
207	C₂×C₃×D₄	—	2	2	no	—
208	C₂×C₃×D₄	—	2	2	no	—
209	C₂×C₃×D₄	C₂×C₄×C₅	1	8	yes	—
210	C₂×C₃×D₄	C₂×C₄×C₅	1	4	yes	complemented
211	C₂²×C₃×C₄	—	2	4	no	—
212	C₂²×C₃×C₄	—	2	4	no	—
213	C₂²×C₃×C₄	—	2	4	no	—
214	C₂²×C₃×C₄	—	4	4	no	—
215	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
216	C₂²×C₃×C₄	C₂×C₄×C₅	1	8	yes	complemented
217	C₂²×C₃×C₄	C₅×D₄	1	4	yes	—
218	C₂²×C₃×C₄	C₅×D₄	1	4	yes	—
219	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
220	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
221	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
222	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
223	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
224	C₃×C₄⋊C₄	—	2	4	no	—
225	C₃×C₄⋊C₄	—	2	4	no	—
226	C₃×C₄⋊C₄	—	2	4	no	—
227	C₃×C₂²⋊C₄	—	2	4	no	—
228	C₃×C₂²⋊C₄	—	4	8	no	—
229	C₃×C₄²	—	2	4	no	—
230	C₂³×C₅	—	2	2	no	—
231	C₂³×C₅	—	2	2	no	—
232	C₂³×C₅	—	2	2	no	—
233	C₂³×C₅	—	2	2	no	—
234	C₂³×C₅	—	2	4	no	—
235	C₂³×C₅	—	2	4	no	—
236	C₂³×C₅	C₂²×C₃×C₄	1	4	yes	—
237	C₂³×C₅	C₂²×C₃×C₄	1	2	yes	—
238	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
239	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
240	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
241	C₅×Q₈	—	2	8	no	—
242	C₅×D₄	—	2	16	no	—
243	C₅×D₄	—	2	8	no	—
244	C₂×C₄×C₅	—	2	2	no	—
245	C₂×C₄×C₅	—	2	2	no	—
246	C₂×C₄×C₅	—	2	16	no	—
247	C₂×C₄×C₅	—	2	4	no	—
248	C₂×C₄×C₅	—	2	4	no	—
249	C₂×C₄×C₅	—	2	4	no	—
250	C₂×C₄×C₅	—	2	8	no	—
251	C₂×C₄×C₅	—	2	4	no	—
252	C₂×C₄×C₅	—	4	8	no	—
253	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	16	yes	complemented
254	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
255	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
256	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
257	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
258	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
259	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
260	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
261	C₂×#41	—	2	4	no	—
262	C₂×#41	—	2	4	no	—
263	C₂×#41	C₃×C₄×C₅	1	8	yes	complemented
264	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
265	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
266	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
267	C₂³×C₄	C₂²×C₃×C₅	1	2	yes	—
268	C₂³×C₄	C₂²×C₃×C₅	1	2	yes	—
269	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
270	#117	—	2	4	no	—
271	C₂×C₄⋊C₄	—	2	4	no	—
272	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
273	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
274	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
275	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
276	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	8	yes	—
277	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
278	#95	C₂²×C₃×C₅	1	8	yes	—
279	C₂×C₃×C₅	—	2	4	no	—
280	C₂×C₃×C₅	—	4	4	no	—
281	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
282	C₂×C₃×C₅	#520	1	2	yes	—
283	C₂×C₃×C₅	#535	1	1	yes	characteristic
284	C₂×C₃×C₅	#536	1	1	yes	characteristic
285	C₂×C₃×C₅	#537	1	2	yes	—
286	C₂³×C₃	—	2	2	no	—
287	C₂³×C₃	—	2	2	no	—
288	C₂³×C₃	—	2	2	no	—
289	C₂³×C₃	—	2	2	no	—
290	C₂³×C₃	—	2	4	no	—
291	C₂³×C₃	—	2	4	no	—
292	C₂³×C₃	C₂²×C₄×C₅	1	4	yes	—
293	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
294	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
295	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
296	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
297	C₃×Q₈	—	2	8	no	—
298	C₃×D₄	—	2	16	no	—
299	C₃×D₄	—	2	8	no	—
300	C₂×C₃×C₄	—	2	2	no	—
301	C₂×C₃×C₄	—	2	2	no	—
302	C₂×C₃×C₄	—	2	16	no	—
303	C₂×C₃×C₄	—	2	4	no	—
304	C₂×C₃×C₄	—	2	4	no	—
305	C₂×C₃×C₄	—	2	4	no	—
306	C₂×C₃×C₄	—	2	8	no	—
307	C₂×C₃×C₄	—	2	4	no	—
308	C₂×C₃×C₄	—	4	8	no	—
309	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	16	yes	complemented
310	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
311	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
312	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
313	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
314	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
315	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
316	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
317	C₂²×C₅	—	2	4	no	—
318	C₂²×C₅	—	2	2	no	—
319	C₂²×C₅	—	2	2	no	—
320	C₂²×C₅	—	2	4	no	—
321	C₂²×C₅	—	2	4	no	—
322	C₂²×C₅	—	2	8	no	—
323	C₂²×C₅	—	4	2	no	—
324	C₂²×C₅	—	4	2	no	—
325	C₂²×C₅	—	4	4	no	—
326	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	4	yes	—
327	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
328	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
329	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
330	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
331	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
332	C₄×C₅	—	2	4	no	—
333	C₄×C₅	—	4	4	no	—
334	C₄×C₅	—	4	4	no	—
335	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	4	yes	—
336	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	4	yes	—
337	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
338	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
339	#41	—	2	32	no	—
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	C₂×C₃×C₄×C₅	1	4	yes	complemented
343	C₂×D₄	—	2	4	no	—
344	C₂×D₄	—	2	4	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	C₂×C₃×C₄×C₅	1	8	yes	—
348	C₂×D₄	C₂×C₃×C₄×C₅	1	4	yes	complemented
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	4	no	—
352	C₂²×C₄	—	4	4	no	—
353	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
354	C₂²×C₄	C₂×C₃×C₄×C₅	1	8	yes	complemented
355	C₂²×C₄	C₃×C₅×D₄	1	4	yes	—
356	C₂²×C₄	C₃×C₅×D₄	1	4	yes	—
357	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
358	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
359	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
360	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
361	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
362	C₄⋊C₄	—	2	4	no	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	4	no	—
365	C₂²⋊C₄	—	2	4	no	—
366	C₂²⋊C₄	—	4	8	no	—
367	C₄²	—	2	4	no	—
368	C₃×C₅	#2311	1	1	yes	characteristic, direct product factor, Hall
369	C₂²×C₃	—	2	4	no	—
370	C₂²×C₃	—	2	2	no	—
371	C₂²×C₃	—	2	2	no	—
372	C₂²×C₃	—	2	4	no	—
373	C₂²×C₃	—	2	4	no	—
374	C₂²×C₃	—	2	8	no	—
375	C₂²×C₃	—	4	2	no	—
376	C₂²×C₃	—	4	2	no	—
377	C₂²×C₃	—	4	4	no	—
378	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	4	yes	—
379	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
380	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
381	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
382	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
383	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
384	C₃×C₄	—	2	4	no	—
385	C₃×C₄	—	4	4	no	—
386	C₃×C₄	—	4	4	no	—
387	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
388	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
389	C₂×C₅	—	2	4	no	—
390	C₂×C₅	—	4	4	no	—
391	C₂×C₅	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
392	C₂×C₅	C₃×#520	1	2	yes	—
393	C₂×C₅	C₃×#535	1	1	yes	characteristic
394	C₂×C₅	C₃×#536	1	1	yes	characteristic
395	C₂×C₅	C₃×#537	1	2	yes	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	2	no	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	4	no	—
401	C₂³	—	2	4	no	—
402	C₂³	C₂²×C₃×C₄×C₅	1	4	yes	—
403	C₂³	C₂²×C₃×C₄×C₅	1	2	yes	—
404	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
405	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
406	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
407	Q₈	—	2	8	no	—
408	D₄	—	2	16	no	—
409	D₄	—	2	8	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	16	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	8	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	4	8	no	—
419	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	16	yes	complemented
420	C₂×C₄	C₂²×C₃×C₄×C₅	1	4	yes	—
421	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
422	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
423	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
424	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
425	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
426	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
427	C₂×C₃	—	2	4	no	—
428	C₂×C₃	—	4	4	no	—
429	C₂×C₃	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
430	C₂×C₃	C₅×#520	1	2	yes	—
431	C₂×C₃	C₅×#535	1	1	yes	characteristic
432	C₂×C₃	C₅×#536	1	1	yes	characteristic
433	C₂×C₃	C₅×#537	1	2	yes	—
434	C₅	C₃×#2311	1	1	yes	characteristic, direct product factor, Sylow
435	C₂²	—	2	4	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	2	4	no	—
439	C₂²	—	2	4	no	—
440	C₂²	—	2	8	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	4	4	no	—
444	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	4	yes	—
445	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
446	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
447	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
448	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
449	C₂²	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
450	C₄	—	2	4	no	—
451	C₄	—	4	4	no	—
452	C₄	—	4	4	no	—
453	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	4	yes	—
454	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	4	yes	—
455	C₃	C₅×#2311	1	1	yes	characteristic, direct product factor, Sylow
456	C₂	—	2	4	no	—
457	C₂	—	4	4	no	—
458	C₂	C₂²×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
459	C₂	C₃×C₅×#520	1	2	yes	—
460	C₂	C₃×C₅×#535	1	1	yes	characteristic
461	C₂	C₃×C₅×#536	1	1	yes	characteristic
462	C₂	C₃×C₅×#537	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator	C₂²	C₂³×C₃×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂²×C₃×C₅×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#117	C₂	semidirect
3. C₂²×C₃×C₅×C₄⋊C₄	C₂	semidirect
4. C₃×C₅×#391	C₂	semidirect
5. C₃×C₅×#390	C₂	semidirect
6. C₃×C₅×#386	C₂	semidirect
7. C₃×C₅×#380	C₂	semidirect
8. C₅×#2311	C₃	direct (N × H)
11. C₂×C₃×C₅×#41	C₄	semidirect
27. C₃×#2311	C₅	direct (N × H)
29. C₂×C₅×#117	C₂×C₃	semidirect
30. C₂²×C₅×C₄⋊C₄	C₂×C₃	semidirect
31. C₅×#391	C₂×C₃	semidirect
32. C₅×#390	C₂×C₃	semidirect
33. C₅×#386	C₂×C₃	semidirect
34. C₅×#380	C₂×C₃	semidirect
40. C₂×C₃×C₅×Q₈	C₂×C₄	semidirect
46. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
52. C₂²×C₃×C₄×C₅	C₂×C₄	semidirect
67. C₂×C₃×#117	C₂×C₅	semidirect
68. C₂²×C₃×C₄⋊C₄	C₂×C₅	semidirect
69. C₃×#391	C₂×C₅	semidirect
70. C₃×#390	C₂×C₅	semidirect
71. C₃×#386	C₂×C₅	semidirect
72. C₃×#380	C₂×C₅	semidirect
75. C₂×C₅×#41	C₃×C₄	semidirect
91. #2311	C₃×C₅	direct (N × H)
115. C₂×C₃×C₄×C₅	C₂²⋊C₄	semidirect
125. C₂×C₃×#41	C₄×C₅	semidirect
146. C₂×C₅×Q₈	C₂×C₃×C₄	semidirect
152. C₂×C₅×D₄	C₂×C₃×C₄	semidirect
158. C₂²×C₄×C₅	C₂×C₃×C₄	semidirect
173. C₂×#117	C₂×C₃×C₅	semidirect
174. C₂²×C₄⋊C₄	C₂×C₃×C₅	semidirect
175. #391	C₂×C₃×C₅	semidirect
176. #390	C₂×C₃×C₅	semidirect
177. #386	C₂×C₃×C₅	semidirect
178. #380	C₂×C₃×C₅	semidirect
204. C₂×C₃×Q₈	C₂×C₄×C₅	semidirect
210. C₂×C₃×D₄	C₂×C₄×C₅	semidirect
216. C₂²×C₃×C₄	C₂×C₄×C₅	semidirect
253. C₂×C₄×C₅	C₃×C₂²⋊C₄	semidirect
263. C₂×#41	C₃×C₄×C₅	semidirect
309. C₂×C₃×C₄	C₅×C₂²⋊C₄	semidirect
342. C₂×Q₈	C₂×C₃×C₄×C₅	semidirect
348. C₂×D₄	C₂×C₃×C₄×C₅	semidirect
354. C₂²×C₄	C₂×C₃×C₄×C₅	semidirect
368. C₃×C₅	#2311	direct (N × H)
419. C₂×C₄	C₃×C₅×C₂²⋊C₄	semidirect
434. C₅	C₃×#2311	direct (N × H)
455. C₃	C₅×#2311	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×#117	2
C₂²×C₃×C₅×C₄⋊C₄	2
C₃×C₅×#391	2
C₃×C₅×#390	2
C₃×C₅×#386	2
C₃×C₅×#380	2
C₅×#2311	3
C₃×#2311	5

Subgroups — C3×C5×#2311 #14275345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2311 #14275345