Subgroups — C₃×C₅×#2353 #14275387

Order 1920.

1176 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#388	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#387	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#386	C₂	1	2	yes	maximal
5	C₃×C₅×#383	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#381	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×C₅×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×C₅×#95	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#2353	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
15	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₅×D₄	C₄	1	8	yes	complemented
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
26	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
27	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
28	C₃×C₅×#95	C₂²	1	1	yes	characteristic
29	C₃×C₅×#95	C₂²	1	1	yes	characteristic
30	C₃×C₅×#95	C₂²	1	2	yes	—
31	C₃×C₅×#95	C₂²	1	1	yes	characteristic
32	C₃×C₅×#95	C₂²	1	2	yes	—
33	C₃×C₅×#95	C₂²	1	1	yes	characteristic
34	C₃×C₅×#95	C₂²	1	2	yes	—
35	C₃×C₅×#95	C₂²	1	2	yes	—
36	C₃×C₅×#95	C₂²	1	2	yes	—
37	C₃×#2353	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
38	C₂×C₄×C₅×D₄	C₂×C₃	1	1	yes	characteristic
39	C₅×#388	C₂×C₃	1	2	yes	complemented
40	C₅×#387	C₂×C₃	1	2	yes	complemented
41	C₅×#386	C₂×C₃	1	2	yes	—
42	C₅×#383	C₂×C₃	1	1	yes	characteristic, complemented
43	C₅×#381	C₂×C₃	1	1	yes	characteristic, complemented
44	C₅×#380	C₂×C₃	1	1	yes	characteristic, complemented
45	C₄×C₅×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
46	C₄×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
47	C₅×#376	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂×C₅×#95	C₂×C₃	1	1	yes	characteristic, complemented
49	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
50	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₅×D₄	—	2	2	no	—
52	C₂×C₃×C₅×D₄	—	2	2	no	—
53	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	—
54	C₂²×C₃×C₄×C₅	—	2	4	no	—
55	C₂²×C₃×C₄×C₅	—	2	4	no	—
56	C₂²×C₃×C₄×C₅	—	4	2	no	—
57	C₂²×C₃×C₄×C₅	—	4	2	no	—
58	C₂²×C₃×C₄×C₅	—	4	2	no	—
59	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
60	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	complemented
61	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
62	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
63	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
64	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
69	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
70	C₃×C₅×C₄⋊C₄	—	4	2	no	—
71	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
72	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
73	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
74	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
75	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
76	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	complemented
77	C₃×C₄²×C₅	—	2	4	no	—
78	C₃×C₄²×C₅	—	4	2	no	—
79	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
80	C₂×C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic
81	C₃×#388	C₂×C₅	1	2	yes	complemented
82	C₃×#387	C₂×C₅	1	2	yes	complemented
83	C₃×#386	C₂×C₅	1	2	yes	—
84	C₃×#383	C₂×C₅	1	1	yes	characteristic, complemented
85	C₃×#381	C₂×C₅	1	1	yes	characteristic, complemented
86	C₃×#380	C₂×C₅	1	1	yes	characteristic, complemented
87	C₃×C₄×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
88	C₃×C₄×C₂²⋊C₄	C₂×C₅	1	2	yes	complemented
89	C₃×#376	C₂×C₅	1	1	yes	characteristic, complemented
90	C₂×C₃×#95	C₂×C₅	1	1	yes	characteristic, complemented
91	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
92	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
93	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
94	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
95	C₄×C₅×D₄	C₃×C₄	1	8	yes	complemented
96	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
97	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
98	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
99	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
100	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
101	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
102	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
103	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
104	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
105	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
106	C₅×#95	C₂²×C₃	1	1	yes	characteristic
107	C₅×#95	C₂²×C₃	1	1	yes	characteristic
108	C₅×#95	C₂²×C₃	1	2	yes	—
109	C₅×#95	C₂²×C₃	1	1	yes	characteristic
110	C₅×#95	C₂²×C₃	1	2	yes	—
111	C₅×#95	C₂²×C₃	1	1	yes	characteristic
112	C₅×#95	C₂²×C₃	1	2	yes	—
113	C₅×#95	C₂²×C₃	1	2	yes	—
114	C₅×#95	C₂²×C₃	1	2	yes	—
115	#2353	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
116	C₂³×C₃×C₅	—	2	2	no	—
117	C₂³×C₃×C₅	—	2	2	no	—
118	C₂³×C₃×C₅	—	2	2	no	—
119	C₂³×C₃×C₅	—	2	1	no	—
120	C₂³×C₃×C₅	—	2	1	no	—
121	C₂³×C₃×C₅	—	2	2	no	—
122	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
123	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
124	C₂³×C₃×C₅	#41	1	2	yes	—
125	C₂³×C₃×C₅	#41	1	2	yes	—
126	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
127	C₃×C₅×D₄	—	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	1	no	—
131	C₂×C₃×C₄×C₅	—	2	1	no	—
132	C₂×C₃×C₄×C₅	—	2	2	no	—
133	C₂×C₃×C₄×C₅	—	2	2	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	2	no	—
138	C₂×C₃×C₄×C₅	—	2	2	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	2	no	—
144	C₂×C₃×C₄×C₅	—	2	2	no	—
145	C₂×C₃×C₄×C₅	—	4	4	no	—
146	C₂×C₃×C₄×C₅	—	4	4	no	—
147	C₂×C₃×C₄×C₅	—	4	8	no	—
148	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
149	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
150	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
151	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
152	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
153	C₂×C₃×C₄×C₅	#41	1	4	yes	—
154	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
155	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
156	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
157	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
158	C₃×C₄×D₄	C₄×C₅	1	8	yes	complemented
159	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
160	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
161	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
162	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
163	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
164	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
165	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
166	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
167	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
168	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
169	C₃×#95	C₂²×C₅	1	1	yes	characteristic
170	C₃×#95	C₂²×C₅	1	1	yes	characteristic
171	C₃×#95	C₂²×C₅	1	2	yes	—
172	C₃×#95	C₂²×C₅	1	1	yes	characteristic
173	C₃×#95	C₂²×C₅	1	2	yes	—
174	C₃×#95	C₂²×C₅	1	1	yes	characteristic
175	C₃×#95	C₂²×C₅	1	2	yes	—
176	C₃×#95	C₂²×C₅	1	2	yes	—
177	C₃×#95	C₂²×C₅	1	2	yes	—
178	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
179	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
180	C₂×C₅×D₄	—	2	2	no	—
181	C₂×C₅×D₄	—	2	2	no	—
182	C₂×C₅×D₄	C₂×C₃×C₄	1	4	yes	—
183	C₂²×C₄×C₅	—	2	4	no	—
184	C₂²×C₄×C₅	—	2	4	no	—
185	C₂²×C₄×C₅	—	4	2	no	—
186	C₂²×C₄×C₅	—	4	2	no	—
187	C₂²×C₄×C₅	—	4	2	no	—
188	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
189	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	complemented
190	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
191	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
192	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
193	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
194	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
195	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
196	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
197	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
198	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
199	C₅×C₄⋊C₄	—	4	2	no	—
200	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
201	C₅×C₂²⋊C₄	—	2	4	no	—
202	C₅×C₂²⋊C₄	—	2	4	no	—
203	C₅×C₂²⋊C₄	—	4	2	no	—
204	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
205	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
206	C₄²×C₅	—	2	4	no	—
207	C₄²×C₅	—	4	2	no	—
208	C₄²×C₅	C₂×C₃×C₄	1	4	yes	complemented
209	C₂×C₄×D₄	C₂×C₃×C₅	1	1	yes	characteristic
210	#388	C₂×C₃×C₅	1	2	yes	complemented
211	#387	C₂×C₃×C₅	1	2	yes	complemented
212	#386	C₂×C₃×C₅	1	2	yes	—
213	#383	C₂×C₃×C₅	1	1	yes	characteristic, complemented
214	#381	C₂×C₃×C₅	1	1	yes	characteristic, complemented
215	#380	C₂×C₃×C₅	1	1	yes	characteristic, complemented
216	C₄×C₂²⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
217	C₄×C₂²⋊C₄	C₂×C₃×C₅	1	2	yes	complemented
218	#376	C₂×C₃×C₅	1	1	yes	characteristic, complemented
219	C₂×#95	C₂×C₃×C₅	1	1	yes	characteristic, complemented
220	C₂²×C₃×C₅	—	2	2	no	—
221	C₂²×C₃×C₅	—	2	2	no	—
222	C₂²×C₃×C₅	—	2	4	no	—
223	C₂²×C₃×C₅	—	2	4	no	—
224	C₂²×C₃×C₅	—	2	2	no	—
225	C₂²×C₃×C₅	—	2	2	no	—
226	C₂²×C₃×C₅	—	2	2	no	—
227	C₂²×C₃×C₅	—	4	2	no	—
228	C₂²×C₃×C₅	—	4	2	no	—
229	C₂²×C₃×C₅	#117	1	4	yes	—
230	C₂²×C₃×C₅	C₂³×C₄	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	2	yes	—
234	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
235	C₃×C₄×C₅	—	2	2	no	—
236	C₃×C₄×C₅	—	2	2	no	—
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	4	no	—
241	C₃×C₄×C₅	—	4	2	no	—
242	C₃×C₄×C₅	—	4	2	no	—
243	C₃×C₄×C₅	—	4	2	no	—
244	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
245	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
246	C₂×C₃×D₄	—	2	2	no	—
247	C₂×C₃×D₄	—	2	2	no	—
248	C₂×C₃×D₄	C₂×C₄×C₅	1	4	yes	—
249	C₂²×C₃×C₄	—	2	4	no	—
250	C₂²×C₃×C₄	—	2	4	no	—
251	C₂²×C₃×C₄	—	4	2	no	—
252	C₂²×C₃×C₄	—	4	2	no	—
253	C₂²×C₃×C₄	—	4	2	no	—
254	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
255	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	complemented
256	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
257	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
258	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
259	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
260	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
261	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
262	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
263	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
264	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
265	C₃×C₄⋊C₄	—	4	2	no	—
266	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	complemented
267	C₃×C₂²⋊C₄	—	2	4	no	—
268	C₃×C₂²⋊C₄	—	2	4	no	—
269	C₃×C₂²⋊C₄	—	4	2	no	—
270	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
271	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	complemented
272	C₃×C₄²	—	2	4	no	—
273	C₃×C₄²	—	4	2	no	—
274	C₃×C₄²	C₂×C₄×C₅	1	4	yes	complemented
275	C₂³×C₅	—	2	2	no	—
276	C₂³×C₅	—	2	2	no	—
277	C₂³×C₅	—	2	2	no	—
278	C₂³×C₅	—	2	1	no	—
279	C₂³×C₅	—	2	1	no	—
280	C₂³×C₅	—	2	2	no	—
281	C₂³×C₅	C₂²×C₃×C₄	1	2	yes	—
282	C₂³×C₅	C₂²×C₃×C₄	1	2	yes	—
283	C₂³×C₅	C₃×#41	1	2	yes	—
284	C₂³×C₅	C₃×#41	1	2	yes	—
285	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
286	C₅×D₄	—	2	8	no	—
287	C₂×C₄×C₅	—	2	2	no	—
288	C₂×C₄×C₅	—	2	2	no	—
289	C₂×C₄×C₅	—	2	1	no	—
290	C₂×C₄×C₅	—	2	1	no	—
291	C₂×C₄×C₅	—	2	2	no	—
292	C₂×C₄×C₅	—	2	2	no	—
293	C₂×C₄×C₅	—	2	2	no	—
294	C₂×C₄×C₅	—	2	2	no	—
295	C₂×C₄×C₅	—	2	2	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	2	no	—
300	C₂×C₄×C₅	—	2	2	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	—	4	4	no	—
305	C₂×C₄×C₅	—	4	4	no	—
306	C₂×C₄×C₅	—	4	8	no	—
307	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
308	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
309	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
310	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
311	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
312	C₂×C₄×C₅	C₃×#41	1	4	yes	—
313	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
314	C₂³×C₄	C₂²×C₃×C₅	1	2	yes	—
315	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
316	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
317	C₄×D₄	C₃×C₄×C₅	1	8	yes	complemented
318	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
319	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
320	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
321	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
322	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
323	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
324	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
325	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
326	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
327	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
328	#95	C₂²×C₃×C₅	1	1	yes	characteristic
329	#95	C₂²×C₃×C₅	1	1	yes	characteristic
330	#95	C₂²×C₃×C₅	1	2	yes	—
331	#95	C₂²×C₃×C₅	1	1	yes	characteristic
332	#95	C₂²×C₃×C₅	1	2	yes	—
333	#95	C₂²×C₃×C₅	1	1	yes	characteristic
334	#95	C₂²×C₃×C₅	1	2	yes	—
335	#95	C₂²×C₃×C₅	1	2	yes	—
336	#95	C₂²×C₃×C₅	1	2	yes	—
337	C₂×C₃×C₅	—	2	4	no	—
338	C₂×C₃×C₅	—	4	2	no	—
339	C₂×C₃×C₅	C₂×#117	1	1	yes	characteristic
340	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
341	C₂×C₃×C₅	#518	1	1	yes	characteristic
342	C₂×C₃×C₅	#551	1	2	yes	—
343	C₂×C₃×C₅	#553	1	2	yes	—
344	C₂³×C₃	—	2	2	no	—
345	C₂³×C₃	—	2	2	no	—
346	C₂³×C₃	—	2	2	no	—
347	C₂³×C₃	—	2	1	no	—
348	C₂³×C₃	—	2	1	no	—
349	C₂³×C₃	—	2	2	no	—
350	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
351	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
352	C₂³×C₃	C₅×#41	1	2	yes	—
353	C₂³×C₃	C₅×#41	1	2	yes	—
354	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
355	C₃×D₄	—	2	8	no	—
356	C₂×C₃×C₄	—	2	2	no	—
357	C₂×C₃×C₄	—	2	2	no	—
358	C₂×C₃×C₄	—	2	1	no	—
359	C₂×C₃×C₄	—	2	1	no	—
360	C₂×C₃×C₄	—	2	2	no	—
361	C₂×C₃×C₄	—	2	2	no	—
362	C₂×C₃×C₄	—	2	2	no	—
363	C₂×C₃×C₄	—	2	2	no	—
364	C₂×C₃×C₄	—	2	2	no	—
365	C₂×C₃×C₄	—	2	2	no	—
366	C₂×C₃×C₄	—	2	2	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	2	no	—
371	C₂×C₃×C₄	—	2	2	no	—
372	C₂×C₃×C₄	—	2	2	no	—
373	C₂×C₃×C₄	—	4	4	no	—
374	C₂×C₃×C₄	—	4	4	no	—
375	C₂×C₃×C₄	—	4	8	no	—
376	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
377	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
378	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
379	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
380	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
381	C₂×C₃×C₄	C₅×#41	1	4	yes	—
382	C₂²×C₅	—	2	2	no	—
383	C₂²×C₅	—	2	2	no	—
384	C₂²×C₅	—	2	4	no	—
385	C₂²×C₅	—	2	4	no	—
386	C₂²×C₅	—	2	2	no	—
387	C₂²×C₅	—	2	2	no	—
388	C₂²×C₅	—	2	2	no	—
389	C₂²×C₅	—	4	2	no	—
390	C₂²×C₅	—	4	2	no	—
391	C₂²×C₅	C₃×#117	1	4	yes	—
392	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
393	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
394	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
395	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
396	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
397	C₄×C₅	—	2	2	no	—
398	C₄×C₅	—	2	2	no	—
399	C₄×C₅	—	2	2	no	—
400	C₄×C₅	—	2	2	no	—
401	C₄×C₅	—	2	2	no	—
402	C₄×C₅	—	2	4	no	—
403	C₄×C₅	—	4	2	no	—
404	C₄×C₅	—	4	2	no	—
405	C₄×C₅	—	4	2	no	—
406	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
407	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	C₂×C₃×C₄×C₅	1	4	yes	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	4	2	no	—
414	C₂²×C₄	—	4	2	no	—
415	C₂²×C₄	—	4	2	no	—
416	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
417	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	complemented
418	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
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	1	yes	characteristic
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	2	yes	—
427	C₄⋊C₄	—	4	2	no	—
428	C₄⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	complemented
429	C₂²⋊C₄	—	2	4	no	—
430	C₂²⋊C₄	—	2	4	no	—
431	C₂²⋊C₄	—	4	2	no	—
432	C₂²⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
433	C₂²⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	complemented
434	C₄²	—	2	4	no	—
435	C₄²	—	4	2	no	—
436	C₄²	C₂×C₃×C₄×C₅	1	4	yes	complemented
437	C₃×C₅	#2353	1	1	yes	characteristic, direct product factor, Hall
438	C₂²×C₃	—	2	2	no	—
439	C₂²×C₃	—	2	2	no	—
440	C₂²×C₃	—	2	4	no	—
441	C₂²×C₃	—	2	4	no	—
442	C₂²×C₃	—	2	2	no	—
443	C₂²×C₃	—	2	2	no	—
444	C₂²×C₃	—	2	2	no	—
445	C₂²×C₃	—	4	2	no	—
446	C₂²×C₃	—	4	2	no	—
447	C₂²×C₃	C₅×#117	1	4	yes	—
448	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
449	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
450	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
451	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
452	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
453	C₃×C₄	—	2	2	no	—
454	C₃×C₄	—	2	2	no	—
455	C₃×C₄	—	2	2	no	—
456	C₃×C₄	—	2	2	no	—
457	C₃×C₄	—	2	2	no	—
458	C₃×C₄	—	2	4	no	—
459	C₃×C₄	—	4	2	no	—
460	C₃×C₄	—	4	2	no	—
461	C₃×C₄	—	4	2	no	—
462	C₂×C₅	—	2	4	no	—
463	C₂×C₅	—	4	2	no	—
464	C₂×C₅	C₂×C₃×#117	1	1	yes	characteristic
465	C₂×C₅	C₃×C₄×#41	1	1	yes	characteristic
466	C₂×C₅	C₃×#518	1	1	yes	characteristic
467	C₂×C₅	C₃×#551	1	2	yes	—
468	C₂×C₅	C₃×#553	1	2	yes	—
469	C₂³	—	2	2	no	—
470	C₂³	—	2	2	no	—
471	C₂³	—	2	2	no	—
472	C₂³	—	2	1	no	—
473	C₂³	—	2	1	no	—
474	C₂³	—	2	2	no	—
475	C₂³	C₂²×C₃×C₄×C₅	1	2	yes	—
476	C₂³	C₂²×C₃×C₄×C₅	1	2	yes	—
477	C₂³	C₃×C₅×#41	1	2	yes	—
478	C₂³	C₃×C₅×#41	1	2	yes	—
479	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
480	D₄	—	2	8	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	—	4	4	no	—
500	C₂×C₄	—	4	8	no	—
501	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
502	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
503	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
504	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
505	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
506	C₂×C₄	C₃×C₅×#41	1	4	yes	—
507	C₂×C₃	—	2	4	no	—
508	C₂×C₃	—	4	2	no	—
509	C₂×C₃	C₂×C₅×#117	1	1	yes	characteristic
510	C₂×C₃	C₄×C₅×#41	1	1	yes	characteristic
511	C₂×C₃	C₅×#518	1	1	yes	characteristic
512	C₂×C₃	C₅×#551	1	2	yes	—
513	C₂×C₃	C₅×#553	1	2	yes	—
514	C₅	C₃×#2353	1	1	yes	characteristic, direct product factor, Sylow
515	C₂²	—	2	2	no	—
516	C₂²	—	2	2	no	—
517	C₂²	—	2	4	no	—
518	C₂²	—	2	4	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	2	2	no	—
521	C₂²	—	2	2	no	—
522	C₂²	—	4	2	no	—
523	C₂²	—	4	2	no	—
524	C₂²	C₃×C₅×#117	1	4	yes	—
525	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
526	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
527	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
528	C₂²	C₂×C₃×C₅×#41	1	2	yes	—
529	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
530	C₄	—	2	2	no	—
531	C₄	—	2	2	no	—
532	C₄	—	2	2	no	—
533	C₄	—	2	2	no	—
534	C₄	—	2	2	no	—
535	C₄	—	2	4	no	—
536	C₄	—	4	2	no	—
537	C₄	—	4	2	no	—
538	C₄	—	4	2	no	—
539	C₃	C₅×#2353	1	1	yes	characteristic, direct product factor, Sylow
540	C₂	—	2	4	no	—
541	C₂	—	4	2	no	—
542	C₂	C₂×C₃×C₅×#117	1	1	yes	characteristic
543	C₂	C₃×C₄×C₅×#41	1	1	yes	characteristic
544	C₂	C₃×C₅×#518	1	1	yes	characteristic
545	C₂	C₃×C₅×#551	1	2	yes	—
546	C₂	C₃×C₅×#553	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×D₄	C₂²
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. C₃×C₅×#388	C₂	semidirect
3. C₃×C₅×#387	C₂	semidirect
5. C₃×C₅×#383	C₂	semidirect
6. C₃×C₅×#381	C₂	semidirect
7. C₃×C₅×#380	C₂	semidirect
8. C₃×C₄×C₅×C₂²⋊C₄	C₂	semidirect
9. C₃×C₄×C₅×C₂²⋊C₄	C₂	semidirect
10. C₃×C₅×#376	C₂	semidirect
11. C₂×C₃×C₅×#95	C₂	semidirect
12. C₅×#2353	C₃	direct (N × H)
17. C₃×C₄×C₅×D₄	C₄	semidirect
37. C₃×#2353	C₅	direct (N × H)
39. C₅×#388	C₂×C₃	semidirect
40. C₅×#387	C₂×C₃	semidirect
42. C₅×#383	C₂×C₃	semidirect
43. C₅×#381	C₂×C₃	semidirect
44. C₅×#380	C₂×C₃	semidirect
45. C₄×C₅×C₂²⋊C₄	C₂×C₃	semidirect
46. C₄×C₅×C₂²⋊C₄	C₂×C₃	semidirect
47. C₅×#376	C₂×C₃	semidirect
48. C₂×C₅×#95	C₂×C₃	semidirect
60. C₂²×C₃×C₄×C₅	C₂×C₄	semidirect
71. C₃×C₅×C₄⋊C₄	C₂×C₄	semidirect
76. C₃×C₅×C₂²⋊C₄	C₂×C₄	semidirect
79. C₃×C₄²×C₅	C₂×C₄	semidirect
81. C₃×#388	C₂×C₅	semidirect
82. C₃×#387	C₂×C₅	semidirect
84. C₃×#383	C₂×C₅	semidirect
85. C₃×#381	C₂×C₅	semidirect
86. C₃×#380	C₂×C₅	semidirect
87. C₃×C₄×C₂²⋊C₄	C₂×C₅	semidirect
88. C₃×C₄×C₂²⋊C₄	C₂×C₅	semidirect
89. C₃×#376	C₂×C₅	semidirect
90. C₂×C₃×#95	C₂×C₅	semidirect
95. C₄×C₅×D₄	C₃×C₄	semidirect
115. #2353	C₃×C₅	direct (N × H)
158. C₃×C₄×D₄	C₄×C₅	semidirect
189. C₂²×C₄×C₅	C₂×C₃×C₄	semidirect
200. C₅×C₄⋊C₄	C₂×C₃×C₄	semidirect
205. C₅×C₂²⋊C₄	C₂×C₃×C₄	semidirect
208. C₄²×C₅	C₂×C₃×C₄	semidirect
210. #388	C₂×C₃×C₅	semidirect
211. #387	C₂×C₃×C₅	semidirect
213. #383	C₂×C₃×C₅	semidirect
214. #381	C₂×C₃×C₅	semidirect
215. #380	C₂×C₃×C₅	semidirect
216. C₄×C₂²⋊C₄	C₂×C₃×C₅	semidirect
217. C₄×C₂²⋊C₄	C₂×C₃×C₅	semidirect
218. #376	C₂×C₃×C₅	semidirect
219. C₂×#95	C₂×C₃×C₅	semidirect
255. C₂²×C₃×C₄	C₂×C₄×C₅	semidirect
266. C₃×C₄⋊C₄	C₂×C₄×C₅	semidirect
271. C₃×C₂²⋊C₄	C₂×C₄×C₅	semidirect
274. C₃×C₄²	C₂×C₄×C₅	semidirect
317. C₄×D₄	C₃×C₄×C₅	semidirect
417. C₂²×C₄	C₂×C₃×C₄×C₅	semidirect
428. C₄⋊C₄	C₂×C₃×C₄×C₅	semidirect
433. C₂²⋊C₄	C₂×C₃×C₄×C₅	semidirect
436. C₄²	C₂×C₃×C₄×C₅	semidirect
437. C₃×C₅	#2353	direct (N × H)
514. C₅	C₃×#2353	direct (N × H)
539. C₃	C₅×#2353	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×C₄×C₅×D₄	2
C₃×C₅×#388	2
C₃×C₅×#387	2
C₃×C₅×#386	2
C₃×C₅×#383	2
C₃×C₅×#381	2
C₃×C₅×#380	2
C₃×C₄×C₅×C₂²⋊C₄	2
C₃×C₅×#376	2
C₂×C₃×C₅×#95	2
C₅×#2353	3
C₃×#2353	5

Subgroups — C3×C5×#2353 #14275387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2353 #14275387