Subgroups — C₃×C₅×#2448 #14275482

Order 1920.

1024 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#399	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#397	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#390	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#390	C₂	1	1	yes	maximal, characteristic
8	C₃×C₅×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₅×#388	C₂	1	2	yes	maximal, complemented
10	C₃×C₅×#388	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#2448	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₃×C₅×#124	—	2	4	no	—
16	C₃×C₄×C₅×Q₈	—	2	4	no	—
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
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×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	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	1	yes	characteristic
30	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
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	1	yes	characteristic
35	C₃×C₅×#95	C₂²	1	2	yes	—
36	C₃×C₅×#95	C₂²	1	2	yes	—
37	C₃×C₅×#95	C₂²	1	2	yes	—
38	C₃×#2448	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
39	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic
40	C₂×C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
41	C₅×#399	C₂×C₃	1	2	yes	complemented
42	C₅×#397	C₂×C₃	1	2	yes	complemented
43	C₅×#391	C₂×C₃	1	1	yes	characteristic, complemented
44	C₅×#390	C₂×C₃	1	1	yes	characteristic
45	C₅×#390	C₂×C₃	1	1	yes	characteristic
46	C₅×#389	C₂×C₃	1	1	yes	characteristic, complemented
47	C₅×#388	C₂×C₃	1	2	yes	complemented
48	C₅×#388	C₂×C₃	1	2	yes	complemented
49	C₄×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
51	C₂×C₃×C₅×Q₈	—	2	2	no	—
52	C₂×C₃×C₅×Q₈	—	2	2	no	—
53	C₂×C₃×C₅×Q₈	D₄	1	4	yes	—
54	C₂×C₃×C₅×D₄	—	2	2	no	—
55	C₂×C₃×C₅×D₄	—	2	2	no	—
56	C₂×C₃×C₅×D₄	—	2	2	no	—
57	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
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	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	C₃×C₅×C₄⋊C₄	—	2	4	no	—
68	C₃×C₅×C₄⋊C₄	—	2	2	no	—
69	C₃×C₅×C₄⋊C₄	—	2	2	no	—
70	C₃×C₅×C₄⋊C₄	—	2	2	no	—
71	C₃×C₅×C₄⋊C₄	—	2	2	no	—
72	C₃×C₅×C₄⋊C₄	—	4	2	no	—
73	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
74	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
75	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
76	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
77	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
78	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
79	C₃×C₄²×C₅	—	2	4	no	—
80	C₃×C₄²×C₅	—	2	2	no	—
81	C₃×C₄²×C₅	—	2	2	no	—
82	C₃×C₄²×C₅	—	2	2	no	—
83	C₂×C₃×#124	C₂×C₅	1	1	yes	characteristic
84	C₂×C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
85	C₃×#399	C₂×C₅	1	2	yes	complemented
86	C₃×#397	C₂×C₅	1	2	yes	complemented
87	C₃×#391	C₂×C₅	1	1	yes	characteristic, complemented
88	C₃×#390	C₂×C₅	1	1	yes	characteristic
89	C₃×#390	C₂×C₅	1	1	yes	characteristic
90	C₃×#389	C₂×C₅	1	1	yes	characteristic, complemented
91	C₃×#388	C₂×C₅	1	2	yes	complemented
92	C₃×#388	C₂×C₅	1	2	yes	complemented
93	C₃×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
94	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
95	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
96	C₅×#124	—	2	4	no	—
97	C₄×C₅×Q₈	—	2	4	no	—
98	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
99	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
100	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
101	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
102	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
103	C₂×C₅×C₄⋊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×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
109	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
110	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
111	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
112	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
113	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
114	C₅×#95	C₂²×C₃	1	1	yes	characteristic
115	C₅×#95	C₂²×C₃	1	1	yes	characteristic
116	C₅×#95	C₂²×C₃	1	2	yes	—
117	C₅×#95	C₂²×C₃	1	2	yes	—
118	C₅×#95	C₂²×C₃	1	2	yes	—
119	#2448	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
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	—	2	2	no	—
125	C₂³×C₃×C₅	—	2	2	no	—
126	C₂³×C₃×C₅	—	2	2	no	—
127	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
128	C₃×C₅×Q₈	—	2	8	no	—
129	C₃×C₅×D₄	—	4	4	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	2	no	—
134	C₂×C₃×C₄×C₅	—	2	2	no	—
135	C₂×C₃×C₄×C₅	—	2	1	no	—
136	C₂×C₃×C₄×C₅	—	2	1	no	—
137	C₂×C₃×C₄×C₅	—	2	1	no	—
138	C₂×C₃×C₄×C₅	—	2	2	no	—
139	C₂×C₃×C₄×C₅	—	2	1	no	—
140	C₂×C₃×C₄×C₅	—	2	1	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	1	no	—
148	C₂×C₃×C₄×C₅	—	2	1	no	—
149	C₂×C₃×C₄×C₅	—	2	2	no	—
150	C₂×C₃×C₄×C₅	—	2	2	no	—
151	C₂×C₃×C₄×C₅	—	2	1	no	—
152	C₂×C₃×C₄×C₅	—	2	1	no	—
153	C₂×C₃×C₄×C₅	—	2	1	no	—
154	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
155	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
156	C₂×C₃×C₄×C₅	#41	1	2	yes	—
157	C₂×C₃×C₄×C₅	#41	1	2	yes	—
158	C₂×C₃×C₄×C₅	#41	1	2	yes	—
159	C₂×C₃×C₄×C₅	#41	1	2	yes	—
160	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
161	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
162	C₃×#124	—	2	4	no	—
163	C₃×C₄×Q₈	—	2	4	no	—
164	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
165	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
166	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
167	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
168	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
169	C₂×C₃×C₄⋊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×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
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	2	yes	—
176	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
177	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
178	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
179	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
180	C₃×#95	C₂²×C₅	1	1	yes	characteristic
181	C₃×#95	C₂²×C₅	1	1	yes	characteristic
182	C₃×#95	C₂²×C₅	1	2	yes	—
183	C₃×#95	C₂²×C₅	1	2	yes	—
184	C₃×#95	C₂²×C₅	1	2	yes	—
185	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
186	C₂×C₅×Q₈	—	2	2	no	—
187	C₂×C₅×Q₈	—	2	2	no	—
188	C₂×C₅×Q₈	C₃×D₄	1	4	yes	—
189	C₂×C₅×D₄	—	2	2	no	—
190	C₂×C₅×D₄	—	2	2	no	—
191	C₂×C₅×D₄	—	2	2	no	—
192	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
193	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
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	2	yes	—
198	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
199	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
200	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
201	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
202	C₅×C₄⋊C₄	—	2	4	no	—
203	C₅×C₄⋊C₄	—	2	2	no	—
204	C₅×C₄⋊C₄	—	2	2	no	—
205	C₅×C₄⋊C₄	—	2	2	no	—
206	C₅×C₄⋊C₄	—	2	2	no	—
207	C₅×C₄⋊C₄	—	4	2	no	—
208	C₅×C₂²⋊C₄	—	2	4	no	—
209	C₅×C₂²⋊C₄	—	2	4	no	—
210	C₅×C₂²⋊C₄	—	2	4	no	—
211	C₅×C₂²⋊C₄	—	2	4	no	—
212	C₅×C₂²⋊C₄	—	4	2	no	—
213	C₅×C₂²⋊C₄	—	4	2	no	—
214	C₄²×C₅	—	2	4	no	—
215	C₄²×C₅	—	2	2	no	—
216	C₄²×C₅	—	2	2	no	—
217	C₄²×C₅	—	2	2	no	—
218	C₂×#124	C₂×C₃×C₅	1	1	yes	characteristic
219	C₂×C₄×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
220	#399	C₂×C₃×C₅	1	2	yes	complemented
221	#397	C₂×C₃×C₅	1	2	yes	complemented
222	#391	C₂×C₃×C₅	1	1	yes	characteristic, complemented
223	#390	C₂×C₃×C₅	1	1	yes	characteristic
224	#390	C₂×C₃×C₅	1	1	yes	characteristic
225	#389	C₂×C₃×C₅	1	1	yes	characteristic, complemented
226	#388	C₂×C₃×C₅	1	2	yes	complemented
227	#388	C₂×C₃×C₅	1	2	yes	complemented
228	C₄×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
229	C₂²×C₃×C₅	—	4	2	no	—
230	C₂²×C₃×C₅	—	4	2	no	—
231	C₂²×C₃×C₅	—	4	2	no	—
232	C₂²×C₃×C₅	—	4	2	no	—
233	C₂²×C₃×C₅	—	4	2	no	—
234	C₂²×C₃×C₅	—	4	2	no	—
235	C₂²×C₃×C₅	—	4	2	no	—
236	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
237	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
238	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
239	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
240	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
241	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
242	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
243	C₃×C₄×C₅	—	2	4	no	—
244	C₃×C₄×C₅	—	2	2	no	—
245	C₃×C₄×C₅	—	2	2	no	—
246	C₃×C₄×C₅	—	4	1	no	—
247	C₃×C₄×C₅	—	4	1	no	—
248	C₃×C₄×C₅	—	4	1	no	—
249	C₃×C₄×C₅	—	4	2	no	—
250	C₃×C₄×C₅	—	4	1	no	—
251	C₃×C₄×C₅	—	4	2	no	—
252	C₃×C₄×C₅	#123	1	4	yes	—
253	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
254	C₂×C₃×Q₈	—	2	2	no	—
255	C₂×C₃×Q₈	—	2	2	no	—
256	C₂×C₃×Q₈	C₅×D₄	1	4	yes	—
257	C₂×C₃×D₄	—	2	2	no	—
258	C₂×C₃×D₄	—	2	2	no	—
259	C₂×C₃×D₄	—	2	2	no	—
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	2	yes	—
263	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
264	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
265	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
266	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
267	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
268	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
269	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
270	C₃×C₄⋊C₄	—	2	4	no	—
271	C₃×C₄⋊C₄	—	2	2	no	—
272	C₃×C₄⋊C₄	—	2	2	no	—
273	C₃×C₄⋊C₄	—	2	2	no	—
274	C₃×C₄⋊C₄	—	2	2	no	—
275	C₃×C₄⋊C₄	—	4	2	no	—
276	C₃×C₂²⋊C₄	—	2	4	no	—
277	C₃×C₂²⋊C₄	—	2	4	no	—
278	C₃×C₂²⋊C₄	—	2	4	no	—
279	C₃×C₂²⋊C₄	—	2	4	no	—
280	C₃×C₂²⋊C₄	—	4	2	no	—
281	C₃×C₂²⋊C₄	—	4	2	no	—
282	C₃×C₄²	—	2	4	no	—
283	C₃×C₄²	—	2	2	no	—
284	C₃×C₄²	—	2	2	no	—
285	C₃×C₄²	—	2	2	no	—
286	C₂³×C₅	—	2	2	no	—
287	C₂³×C₅	—	2	2	no	—
288	C₂³×C₅	—	2	2	no	—
289	C₂³×C₅	—	2	2	no	—
290	C₂³×C₅	—	2	2	no	—
291	C₂³×C₅	—	2	2	no	—
292	C₂³×C₅	—	2	2	no	—
293	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
294	C₅×Q₈	—	2	8	no	—
295	C₅×D₄	—	4	4	no	—
296	C₂×C₄×C₅	—	2	1	no	—
297	C₂×C₄×C₅	—	2	1	no	—
298	C₂×C₄×C₅	—	2	1	no	—
299	C₂×C₄×C₅	—	2	2	no	—
300	C₂×C₄×C₅	—	2	2	no	—
301	C₂×C₄×C₅	—	2	1	no	—
302	C₂×C₄×C₅	—	2	1	no	—
303	C₂×C₄×C₅	—	2	1	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	2	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	—	2	1	no	—
312	C₂×C₄×C₅	—	2	1	no	—
313	C₂×C₄×C₅	—	2	1	no	—
314	C₂×C₄×C₅	—	2	1	no	—
315	C₂×C₄×C₅	—	2	2	no	—
316	C₂×C₄×C₅	—	2	2	no	—
317	C₂×C₄×C₅	—	2	1	no	—
318	C₂×C₄×C₅	—	2	1	no	—
319	C₂×C₄×C₅	—	2	1	no	—
320	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
321	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
322	C₂×C₄×C₅	C₃×#41	1	2	yes	—
323	C₂×C₄×C₅	C₃×#41	1	2	yes	—
324	C₂×C₄×C₅	C₃×#41	1	2	yes	—
325	C₂×C₄×C₅	C₃×#41	1	2	yes	—
326	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
327	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
328	#124	—	2	4	no	—
329	C₄×Q₈	—	2	4	no	—
330	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
331	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
332	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
333	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
334	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
336	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
337	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
338	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
339	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
340	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
341	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
342	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
343	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
344	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
345	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
346	#95	C₂²×C₃×C₅	1	1	yes	characteristic
347	#95	C₂²×C₃×C₅	1	1	yes	characteristic
348	#95	C₂²×C₃×C₅	1	2	yes	—
349	#95	C₂²×C₃×C₅	1	2	yes	—
350	#95	C₂²×C₃×C₅	1	2	yes	—
351	C₂×C₃×C₅	—	8	2	no	—
352	C₂×C₃×C₅	C₂×#123	1	1	yes	characteristic
353	C₂×C₃×C₅	#529	1	1	yes	characteristic
354	C₂×C₃×C₅	#542	1	1	yes	characteristic
355	C₂×C₃×C₅	#548	1	1	yes	characteristic
356	C₂×C₃×C₅	#550	1	1	yes	characteristic
357	C₂×C₃×C₅	#556	1	1	yes	characteristic
358	C₂×C₃×C₅	#559	1	1	yes	characteristic
359	C₂³×C₃	—	2	2	no	—
360	C₂³×C₃	—	2	2	no	—
361	C₂³×C₃	—	2	2	no	—
362	C₂³×C₃	—	2	2	no	—
363	C₂³×C₃	—	2	2	no	—
364	C₂³×C₃	—	2	2	no	—
365	C₂³×C₃	—	2	2	no	—
366	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
367	C₃×Q₈	—	2	8	no	—
368	C₃×D₄	—	4	4	no	—
369	C₂×C₃×C₄	—	2	1	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	1	no	—
375	C₂×C₃×C₄	—	2	1	no	—
376	C₂×C₃×C₄	—	2	1	no	—
377	C₂×C₃×C₄	—	2	2	no	—
378	C₂×C₃×C₄	—	2	1	no	—
379	C₂×C₃×C₄	—	2	1	no	—
380	C₂×C₃×C₄	—	2	2	no	—
381	C₂×C₃×C₄	—	2	2	no	—
382	C₂×C₃×C₄	—	2	1	no	—
383	C₂×C₃×C₄	—	2	1	no	—
384	C₂×C₃×C₄	—	2	1	no	—
385	C₂×C₃×C₄	—	2	2	no	—
386	C₂×C₃×C₄	—	2	1	no	—
387	C₂×C₃×C₄	—	2	1	no	—
388	C₂×C₃×C₄	—	2	2	no	—
389	C₂×C₃×C₄	—	2	2	no	—
390	C₂×C₃×C₄	—	2	1	no	—
391	C₂×C₃×C₄	—	2	1	no	—
392	C₂×C₃×C₄	—	2	1	no	—
393	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
394	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
395	C₂×C₃×C₄	C₅×#41	1	2	yes	—
396	C₂×C₃×C₄	C₅×#41	1	2	yes	—
397	C₂×C₃×C₄	C₅×#41	1	2	yes	—
398	C₂×C₃×C₄	C₅×#41	1	2	yes	—
399	C₂²×C₅	—	4	2	no	—
400	C₂²×C₅	—	4	2	no	—
401	C₂²×C₅	—	4	2	no	—
402	C₂²×C₅	—	4	2	no	—
403	C₂²×C₅	—	4	2	no	—
404	C₂²×C₅	—	4	2	no	—
405	C₂²×C₅	—	4	2	no	—
406	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
407	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
408	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
409	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
410	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
411	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
412	C₂²×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
413	C₄×C₅	—	2	4	no	—
414	C₄×C₅	—	2	2	no	—
415	C₄×C₅	—	2	2	no	—
416	C₄×C₅	—	4	2	no	—
417	C₄×C₅	—	4	1	no	—
418	C₄×C₅	—	4	1	no	—
419	C₄×C₅	—	4	1	no	—
420	C₄×C₅	—	4	2	no	—
421	C₄×C₅	—	4	1	no	—
422	C₄×C₅	C₃×#123	1	4	yes	—
423	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
424	C₂×Q₈	—	2	2	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	C₃×C₅×D₄	1	4	yes	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
431	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
432	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
433	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
434	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
435	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
436	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
437	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
438	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
439	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
440	C₄⋊C₄	—	2	4	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	4	2	no	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	4	2	no	—
451	C₂²⋊C₄	—	4	2	no	—
452	C₄²	—	2	4	no	—
453	C₄²	—	2	2	no	—
454	C₄²	—	2	2	no	—
455	C₄²	—	2	2	no	—
456	C₃×C₅	#2448	1	1	yes	characteristic, direct product factor, Hall
457	C₂²×C₃	—	4	2	no	—
458	C₂²×C₃	—	4	2	no	—
459	C₂²×C₃	—	4	2	no	—
460	C₂²×C₃	—	4	2	no	—
461	C₂²×C₃	—	4	2	no	—
462	C₂²×C₃	—	4	2	no	—
463	C₂²×C₃	—	4	2	no	—
464	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
465	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
466	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
467	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
468	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
469	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
470	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
471	C₃×C₄	—	2	4	no	—
472	C₃×C₄	—	2	2	no	—
473	C₃×C₄	—	2	2	no	—
474	C₃×C₄	—	4	1	no	—
475	C₃×C₄	—	4	1	no	—
476	C₃×C₄	—	4	1	no	—
477	C₃×C₄	—	4	2	no	—
478	C₃×C₄	—	4	1	no	—
479	C₃×C₄	—	4	2	no	—
480	C₃×C₄	C₅×#123	1	4	yes	—
481	C₂×C₅	—	8	2	no	—
482	C₂×C₅	C₂×C₃×#123	1	1	yes	characteristic
483	C₂×C₅	C₃×#529	1	1	yes	characteristic
484	C₂×C₅	C₃×#542	1	1	yes	characteristic
485	C₂×C₅	C₃×#548	1	1	yes	characteristic
486	C₂×C₅	C₃×#550	1	1	yes	characteristic
487	C₂×C₅	C₃×#556	1	1	yes	characteristic
488	C₂×C₅	C₃×#559	1	1	yes	characteristic
489	C₂³	—	2	2	no	—
490	C₂³	—	2	2	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
497	Q₈	—	2	8	no	—
498	D₄	—	4	4	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	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	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
524	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
525	C₂×C₄	C₃×C₅×#41	1	2	yes	—
526	C₂×C₄	C₃×C₅×#41	1	2	yes	—
527	C₂×C₄	C₃×C₅×#41	1	2	yes	—
528	C₂×C₄	C₃×C₅×#41	1	2	yes	—
529	C₂×C₃	—	8	2	no	—
530	C₂×C₃	C₂×C₅×#123	1	1	yes	characteristic
531	C₂×C₃	C₅×#529	1	1	yes	characteristic
532	C₂×C₃	C₅×#542	1	1	yes	characteristic
533	C₂×C₃	C₅×#548	1	1	yes	characteristic
534	C₂×C₃	C₅×#550	1	1	yes	characteristic
535	C₂×C₃	C₅×#556	1	1	yes	characteristic
536	C₂×C₃	C₅×#559	1	1	yes	characteristic
537	C₅	C₃×#2448	1	1	yes	characteristic, direct product factor, Sylow
538	C₂²	—	4	2	no	—
539	C₂²	—	4	2	no	—
540	C₂²	—	4	2	no	—
541	C₂²	—	4	2	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	4	2	no	—
544	C₂²	—	4	2	no	—
545	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
546	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
547	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
548	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
549	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
550	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
551	C₂²	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
552	C₄	—	2	4	no	—
553	C₄	—	2	2	no	—
554	C₄	—	2	2	no	—
555	C₄	—	4	1	no	—
556	C₄	—	4	1	no	—
557	C₄	—	4	1	no	—
558	C₄	—	4	2	no	—
559	C₄	—	4	1	no	—
560	C₄	—	4	2	no	—
561	C₄	C₃×C₅×#123	1	4	yes	—
562	C₃	C₅×#2448	1	1	yes	characteristic, direct product factor, Sylow
563	C₂	—	8	2	no	—
564	C₂	C₂×C₃×C₅×#123	1	1	yes	characteristic
565	C₂	C₃×C₅×#529	1	1	yes	characteristic
566	C₂	C₃×C₅×#542	1	1	yes	characteristic
567	C₂	C₃×C₅×#548	1	1	yes	characteristic
568	C₂	C₃×C₅×#550	1	1	yes	characteristic
569	C₂	C₃×C₅×#556	1	1	yes	characteristic
570	C₂	C₃×C₅×#559	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
2. C₂×C₃×C₄×C₅×Q₈	C₂	semidirect
3. C₃×C₅×#399	C₂	semidirect
4. C₃×C₅×#397	C₂	semidirect
5. C₃×C₅×#391	C₂	semidirect
8. C₃×C₅×#389	C₂	semidirect
9. C₃×C₅×#388	C₂	semidirect
10. C₃×C₅×#388	C₂	semidirect
11. C₃×C₄×C₅×C₄⋊C₄	C₂	semidirect
12. C₅×#2448	C₃	direct (N × H)
38. C₃×#2448	C₅	direct (N × H)
40. C₂×C₄×C₅×Q₈	C₂×C₃	semidirect
41. C₅×#399	C₂×C₃	semidirect
42. C₅×#397	C₂×C₃	semidirect
43. C₅×#391	C₂×C₃	semidirect
46. C₅×#389	C₂×C₃	semidirect
47. C₅×#388	C₂×C₃	semidirect
48. C₅×#388	C₂×C₃	semidirect
49. C₄×C₅×C₄⋊C₄	C₂×C₃	semidirect
84. C₂×C₃×C₄×Q₈	C₂×C₅	semidirect
85. C₃×#399	C₂×C₅	semidirect
86. C₃×#397	C₂×C₅	semidirect
87. C₃×#391	C₂×C₅	semidirect
90. C₃×#389	C₂×C₅	semidirect
91. C₃×#388	C₂×C₅	semidirect
92. C₃×#388	C₂×C₅	semidirect
93. C₃×C₄×C₄⋊C₄	C₂×C₅	semidirect
119. #2448	C₃×C₅	direct (N × H)
219. C₂×C₄×Q₈	C₂×C₃×C₅	semidirect
220. #399	C₂×C₃×C₅	semidirect
221. #397	C₂×C₃×C₅	semidirect
222. #391	C₂×C₃×C₅	semidirect
225. #389	C₂×C₃×C₅	semidirect
226. #388	C₂×C₃×C₅	semidirect
227. #388	C₂×C₃×C₅	semidirect
228. C₄×C₄⋊C₄	C₂×C₃×C₅	semidirect
456. C₃×C₅	#2448	direct (N × H)
537. C₅	C₃×#2448	direct (N × H)
562. C₃	C₅×#2448	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×#124	2
C₂×C₃×C₄×C₅×Q₈	2
C₃×C₅×#399	2
C₃×C₅×#397	2
C₃×C₅×#391	2
C₃×C₅×#390	2
C₃×C₅×#389	2
C₃×C₅×#388	2
C₃×C₄×C₅×C₄⋊C₄	2
C₅×#2448	3
C₃×#2448	5

Subgroups — C3×C5×#2448 #14275482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2448 #14275482