Subgroups — C₂×C₃×#65899 #14483602

Order 1920.

2504 conjugacy classes of subgroups.

The subgroup lattice has 632 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, complemented
2	C₂×C₃×#4085	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#4076	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#4076	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#4072	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#65899	C₂	1	16	yes	maximal, direct product factor
8	C₂×C₃×#4044	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×#4029	C₂	1	2	yes	maximal, complemented
10	C₂×C₃×#4027	C₂	1	2	yes	maximal, complemented
11	C₂×C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃×#4016	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#65899	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
18	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
19	C₂³×C₃×Dic₅	C₂²	1	2	yes	—
20	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
21	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	—
22	C₃×C₄×C₅×D₄	C₂²	1	8	yes	complemented
23	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
27	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
28	C₃×#4085	C₂²	1	16	yes	complemented
29	C₃×#4076	C₂²	1	8	yes	complemented
30	C₃×#4076	C₂²	1	8	yes	complemented
31	C₂×C₃×#685	C₂²	1	2	yes	—
32	C₂×C₃×#685	C₂²	1	2	yes	—
33	C₂×C₃×#685	C₂²	1	2	yes	—
34	C₃×#4072	C₂²	1	8	yes	complemented
35	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
36	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
37	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
38	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
39	C₂×C₃×C₄⋊Dic₅	C₂²	1	8	yes	complemented
40	C₂×C₃×#683	C₂²	1	2	yes	—
41	C₂×C₃×#683	C₂²	1	2	yes	—
42	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
43	C₃×#4044	C₂²	1	16	yes	complemented
44	C₃×#4029	C₂²	1	16	yes	complemented
45	C₃×#4027	C₂²	1	16	yes	complemented
46	C₃×C₄×D₂₀	C₂²	1	8	yes	complemented
47	C₃×#4016	C₂²	1	8	yes	complemented
48	C₂×C₃×#553	—	5	1	no	maximal, Hall
49	C₂×C₄×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂×#4085	C₂×C₃	1	2	yes	complemented
51	C₂×#4076	C₂×C₃	1	1	yes	characteristic, complemented
52	C₂×#4076	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂×#4072	C₂×C₃	1	1	yes	characteristic, complemented
54	C₂²×C₄⋊Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
55	#65899	C₂×C₃	1	16	yes	direct product factor
56	C₂×#4044	C₂×C₃	1	2	yes	complemented
57	C₂×#4029	C₂×C₃	1	2	yes	complemented
58	C₂×#4027	C₂×C₃	1	2	yes	complemented
59	C₂×C₄×D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂×#4016	C₂×C₃	1	1	yes	characteristic, complemented
61	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
62	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
63	C₂³×C₃×D₅	C₂³	1	2	yes	—
64	C₂×C₃×C₅×D₄	—	2	4	no	—
65	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
66	C₂²×C₃×C₄×C₅	—	2	4	no	—
67	C₂²×C₃×C₄×C₅	—	2	4	no	—
68	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
71	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
72	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
73	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
74	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
75	C₂×C₃×C₂²⋊D₅	—	2	8	no	—
76	C₂×C₃×C₂²⋊D₅	—	4	4	no	—
77	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	—
78	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	—
79	C₂²×C₃×Dic₅	—	4	4	no	—
80	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
81	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
82	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
83	C₂²×C₃×Dic₅	C₂³	1	8	yes	—
84	C₂×C₃×D₂₀	—	2	4	no	—
85	C₂×C₃×D₂₀	C₂³	1	4	yes	—
86	C₂×C₃×C₄×D₅	—	4	4	no	—
87	C₂×C₃×C₄×D₅	C₂³	1	8	yes	—
88	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
89	C₃×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
90	C₃×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
91	C₃×C₄²×C₅	C₂³	1	4	yes	—
92	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	—
93	C₃×#685	C₂³	1	8	yes	—
94	C₃×#685	C₂³	1	8	yes	—
95	C₃×#685	C₂³	1	8	yes	—
96	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
97	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
98	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
99	C₃×C₄⋊Dic₅	C₂³	1	8	yes	—
100	C₃×#683	C₂³	1	8	yes	—
101	C₃×#683	C₂³	1	8	yes	—
102	C₃×C₄×Dic₅	C₂³	1	8	yes	—
103	C₃×#553	—	5	16	no	—
104	C₂×C₃×#126	—	5	2	no	—
105	C₂×C₃×#125	—	5	1	no	—
106	C₂×C₃×#123	—	5	2	no	—
107	C₂×C₃×#121	—	5	2	no	—
108	C₂×C₃×#121	—	5	1	no	—
109	C₂×C₃×#121	—	5	1	no	—
110	C₂×C₃×C₄×D₄	—	5	2	no	—
111	C₂×C₃×C₄×D₄	—	5	1	no	—
112	C₂×C₃×C₄×D₄	D₅	1	1	yes	characteristic, complemented
113	C₂×C₃×#117	—	5	1	no	—
114	C₂²×C₃×C₄⋊C₄	—	5	1	no	—
115	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
116	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
117	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
118	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
119	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
120	C₂³×Dic₅	C₂²×C₃	1	2	yes	—
121	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
122	C₂²×C₄×D₅	C₂²×C₃	1	2	yes	—
123	C₄×C₅×D₄	C₂²×C₃	1	8	yes	complemented
124	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
125	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
126	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
127	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
128	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
129	#4085	C₂²×C₃	1	16	yes	complemented
130	#4076	C₂²×C₃	1	8	yes	complemented
131	#4076	C₂²×C₃	1	8	yes	complemented
132	C₂×#685	C₂²×C₃	1	2	yes	—
133	C₂×#685	C₂²×C₃	1	2	yes	—
134	C₂×#685	C₂²×C₃	1	2	yes	—
135	#4072	C₂²×C₃	1	8	yes	complemented
136	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
137	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
138	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
139	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
140	C₂×C₄⋊Dic₅	C₂²×C₃	1	8	yes	complemented
141	C₂×#683	C₂²×C₃	1	2	yes	—
142	C₂×#683	C₂²×C₃	1	2	yes	—
143	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	—
144	#4044	C₂²×C₃	1	16	yes	complemented
145	#4029	C₂²×C₃	1	16	yes	complemented
146	#4027	C₂²×C₃	1	16	yes	complemented
147	C₄×D₂₀	C₂²×C₃	1	8	yes	complemented
148	#4016	C₂²×C₃	1	8	yes	complemented
149	C₂×#553	—	5	1	no	Sylow
150	C₂³×C₃×C₅	—	2	2	no	—
151	C₂³×C₃×C₅	—	2	2	no	—
152	C₂³×C₃×C₅	—	2	2	no	—
153	C₂³×C₃×C₅	—	2	2	no	—
154	C₂³×C₃×C₅	—	2	2	no	—
155	C₂³×C₃×C₅	#41	1	4	yes	—
156	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
157	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
158	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
159	C₂²×C₃×D₅	—	2	4	no	—
160	C₂²×C₃×D₅	—	2	4	no	—
161	C₂²×C₃×D₅	—	2	4	no	—
162	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
163	C₃×C₅×D₄	—	2	8	no	—
164	C₂×C₃×C₄×C₅	—	2	2	no	—
165	C₂×C₃×C₄×C₅	—	2	2	no	—
166	C₂×C₃×C₄×C₅	—	2	2	no	—
167	C₂×C₃×C₄×C₅	—	2	2	no	—
168	C₂×C₃×C₄×C₅	—	2	8	no	—
169	C₂×C₃×C₄×C₅	—	2	8	no	—
170	C₂×C₃×C₄×C₅	#41	1	4	yes	—
171	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
172	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
173	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
174	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
175	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
176	C₃×C₂²⋊D₅	—	2	16	no	—
177	C₃×C₂²⋊D₅	—	4	8	no	—
178	C₂×C₃×Dic₅	—	2	4	no	—
179	C₂×C₃×Dic₅	—	2	4	no	—
180	C₂×C₃×Dic₅	—	2	4	no	—
181	C₂×C₃×Dic₅	—	4	8	no	—
182	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
183	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
184	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
185	C₃×D₂₀	—	2	8	no	—
186	C₃×C₄×D₅	—	4	8	no	—
187	C₂²×C₃×D₄	—	5	2	no	—
188	C₂²×C₃×D₄	—	5	2	no	—
189	C₂²×C₃×D₄	—	5	1	no	—
190	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
191	C₂³×C₃×C₄	—	5	2	no	—
192	C₂³×C₃×C₄	—	5	2	no	—
193	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
194	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
195	C₃×#126	—	5	16	no	—
196	C₃×#125	—	5	8	no	—
197	C₃×#123	—	5	16	no	—
198	C₃×#121	—	5	16	no	—
199	C₃×#121	—	5	8	no	—
200	C₃×#121	—	5	8	no	—
201	C₃×C₄×D₄	—	5	16	no	—
202	C₃×C₄×D₄	—	5	8	no	—
203	C₃×C₄×D₄	C₂×D₅	1	8	yes	complemented
204	C₃×#117	—	5	8	no	—
205	C₂×C₃×C₄⋊C₄	—	5	1	no	—
206	C₂×C₃×C₄⋊C₄	—	5	1	no	—
207	C₂×C₃×C₄⋊C₄	—	5	2	no	—
208	C₂×C₃×C₄⋊C₄	—	5	8	no	—
209	C₂×C₃×C₄⋊C₄	—	5	1	no	—
210	C₂×C₃×C₄⋊C₄	—	5	2	no	—
211	C₂×C₃×C₄⋊C₄	—	5	2	no	—
212	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
213	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
214	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
215	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
216	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
217	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
218	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
219	C₂×C₃×C₄²	—	5	2	no	—
220	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
221	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
222	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
223	C₂³×D₅	C₂³×C₃	1	2	yes	—
224	C₂×C₅×D₄	—	2	4	no	—
225	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
226	C₂²×C₄×C₅	—	2	4	no	—
227	C₂²×C₄×C₅	—	2	4	no	—
228	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
229	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
230	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
231	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
232	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
233	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
234	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
235	C₂×C₂²⋊D₅	—	2	8	no	—
236	C₂×C₂²⋊D₅	—	4	4	no	—
237	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	—
238	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	—
239	C₂²×Dic₅	—	4	4	no	—
240	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
241	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
242	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
243	C₂²×Dic₅	C₂³×C₃	1	8	yes	—
244	C₂×D₂₀	—	2	4	no	—
245	C₂×D₂₀	C₂³×C₃	1	4	yes	—
246	C₂×C₄×D₅	—	4	4	no	—
247	C₂×C₄×D₅	C₂³×C₃	1	8	yes	—
248	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
249	C₅×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
250	C₅×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
251	C₄²×C₅	C₂³×C₃	1	4	yes	—
252	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	—
253	#685	C₂³×C₃	1	8	yes	—
254	#685	C₂³×C₃	1	8	yes	—
255	#685	C₂³×C₃	1	8	yes	—
256	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
257	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
258	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
259	C₄⋊Dic₅	C₂³×C₃	1	8	yes	—
260	#683	C₂³×C₃	1	8	yes	—
261	#683	C₂³×C₃	1	8	yes	—
262	C₄×Dic₅	C₂³×C₃	1	8	yes	—
263	#553	—	5	16	no	—
264	C₂×#126	—	5	2	no	—
265	C₂×#125	—	5	1	no	—
266	C₂×#123	—	5	2	no	—
267	C₂×#121	—	5	2	no	—
268	C₂×#121	—	5	1	no	—
269	C₂×#121	—	5	1	no	—
270	C₂×C₄×D₄	—	5	2	no	—
271	C₂×C₄×D₄	—	5	1	no	—
272	C₂×C₄×D₄	C₃×D₅	1	1	yes	characteristic, complemented
273	C₂×#117	—	5	1	no	—
274	C₂²×C₄⋊C₄	—	5	1	no	—
275	C₂²×C₃×C₅	—	2	8	no	—
276	C₂²×C₃×C₅	—	2	2	no	—
277	C₂²×C₃×C₅	—	2	2	no	—
278	C₂²×C₃×C₅	—	2	2	no	—
279	C₂²×C₃×C₅	—	2	2	no	—
280	C₂²×C₃×C₅	—	2	2	no	—
281	C₂²×C₃×C₅	—	4	4	no	—
282	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
283	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
284	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
285	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
286	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
287	C₂×C₃×D₅	—	2	4	no	—
288	C₂×C₃×D₅	—	2	4	no	—
289	C₂×C₃×D₅	—	2	4	no	—
290	C₂×C₃×D₅	—	4	8	no	—
291	C₃×C₄×C₅	—	2	2	no	—
292	C₃×C₄×C₅	—	2	2	no	—
293	C₃×C₄×C₅	—	2	2	no	—
294	C₃×C₄×C₅	—	2	2	no	—
295	C₃×C₄×C₅	C₂×#41	1	4	yes	—
296	C₃×Dic₅	—	2	4	no	—
297	C₃×Dic₅	—	2	4	no	—
298	C₃×Dic₅	—	2	4	no	—
299	C₂⁴×C₃	—	5	2	no	—
300	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
301	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
302	C₂×C₃×D₄	—	2	4	no	—
303	C₂×C₃×D₄	—	5	8	no	—
304	C₂×C₃×D₄	—	5	8	no	—
305	C₂×C₃×D₄	—	5	4	no	—
306	C₂×C₃×D₄	—	10	8	no	—
307	C₂×C₃×D₄	—	10	4	no	—
308	C₂×C₃×D₄	—	20	4	no	—
309	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
310	C₂²×C₃×C₄	—	2	4	no	—
311	C₂²×C₃×C₄	—	2	4	no	—
312	C₂²×C₃×C₄	—	5	2	no	—
313	C₂²×C₃×C₄	—	5	2	no	—
314	C₂²×C₃×C₄	—	5	2	no	—
315	C₂²×C₃×C₄	—	5	8	no	—
316	C₂²×C₃×C₄	—	5	8	no	—
317	C₂²×C₃×C₄	—	20	4	no	—
318	C₂²×C₃×C₄	—	20	4	no	—
319	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
320	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
321	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
322	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
323	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
324	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
325	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
326	C₃×C₄⋊C₄	—	5	8	no	—
327	C₃×C₄⋊C₄	—	5	8	no	—
328	C₃×C₄⋊C₄	—	5	4	no	—
329	C₃×C₄⋊C₄	—	5	4	no	—
330	C₃×C₄⋊C₄	—	5	8	no	—
331	C₃×C₄⋊C₄	—	5	4	no	—
332	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
333	C₃×C₂²⋊C₄	—	5	8	no	—
334	C₃×C₂²⋊C₄	—	5	8	no	—
335	C₃×C₂²⋊C₄	—	5	8	no	—
336	C₃×C₂²⋊C₄	—	5	8	no	—
337	C₃×C₂²⋊C₄	C₂²×D₅	1	4	yes	—
338	C₃×C₂²⋊C₄	C₂²×D₅	1	4	yes	—
339	C₃×C₄²	—	5	8	no	—
340	C₃×C₄²	C₂²×D₅	1	4	yes	—
341	C₂³×C₅	—	2	2	no	—
342	C₂³×C₅	—	2	2	no	—
343	C₂³×C₅	—	2	2	no	—
344	C₂³×C₅	—	2	2	no	—
345	C₂³×C₅	—	2	2	no	—
346	C₂³×C₅	C₃×#41	1	4	yes	—
347	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
348	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
349	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
350	C₂²×D₅	—	2	4	no	—
351	C₂²×D₅	—	2	4	no	—
352	C₂²×D₅	—	2	4	no	—
353	C₂²×D₅	C₂⁴×C₃	1	4	yes	—
354	C₅×D₄	—	2	8	no	—
355	C₂×C₄×C₅	—	2	2	no	—
356	C₂×C₄×C₅	—	2	2	no	—
357	C₂×C₄×C₅	—	2	2	no	—
358	C₂×C₄×C₅	—	2	2	no	—
359	C₂×C₄×C₅	—	2	8	no	—
360	C₂×C₄×C₅	—	2	8	no	—
361	C₂×C₄×C₅	C₃×#41	1	4	yes	—
362	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
363	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
364	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
365	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
366	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
367	C₂²⋊D₅	—	2	16	no	—
368	C₂²⋊D₅	—	4	8	no	—
369	C₂×Dic₅	—	2	4	no	—
370	C₂×Dic₅	—	2	4	no	—
371	C₂×Dic₅	—	2	4	no	—
372	C₂×Dic₅	—	4	8	no	—
373	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
374	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
375	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
376	D₂₀	—	2	8	no	—
377	C₄×D₅	—	4	8	no	—
378	C₂²×D₄	—	5	2	no	—
379	C₂²×D₄	—	5	2	no	—
380	C₂²×D₄	—	5	1	no	—
381	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
382	C₂³×C₄	—	5	2	no	—
383	C₂³×C₄	—	5	2	no	—
384	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
385	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
386	#126	—	5	16	no	—
387	#125	—	5	8	no	—
388	#123	—	5	16	no	—
389	#121	—	5	16	no	—
390	#121	—	5	8	no	—
391	#121	—	5	8	no	—
392	C₄×D₄	—	5	16	no	—
393	C₄×D₄	—	5	8	no	—
394	C₄×D₄	C₂×C₃×D₅	1	8	yes	complemented
395	#117	—	5	8	no	—
396	C₂×C₄⋊C₄	—	5	1	no	—
397	C₂×C₄⋊C₄	—	5	1	no	—
398	C₂×C₄⋊C₄	—	5	2	no	—
399	C₂×C₄⋊C₄	—	5	8	no	—
400	C₂×C₄⋊C₄	—	5	1	no	—
401	C₂×C₄⋊C₄	—	5	2	no	—
402	C₂×C₄⋊C₄	—	5	2	no	—
403	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
404	C₂×C₂²⋊C₄	—	5	2	no	—
405	C₂×C₂²⋊C₄	—	5	2	no	—
406	C₂×C₂²⋊C₄	—	5	2	no	—
407	C₂×C₂²⋊C₄	—	5	2	no	—
408	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
409	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
410	C₂×C₄²	—	5	2	no	—
411	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
412	C₂×C₃×C₅	—	2	4	no	—
413	C₂×C₃×C₅	—	4	2	no	—
414	C₂×C₃×C₅	#553	1	4	yes	complemented
415	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
416	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
417	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
418	C₃×D₅	—	4	4	no	—
419	C₂³×C₃	—	2	2	no	—
420	C₂³×C₃	—	2	2	no	—
421	C₂³×C₃	—	2	2	no	—
422	C₂³×C₃	—	2	2	no	—
423	C₂³×C₃	—	2	2	no	—
424	C₂³×C₃	—	5	4	no	—
425	C₂³×C₃	—	10	4	no	—
426	C₂³×C₃	—	10	4	no	—
427	C₂³×C₃	—	10	4	no	—
428	C₂³×C₃	#709	1	4	yes	—
429	C₂³×C₃	C₂³×D₅	1	2	yes	—
430	C₂³×C₃	C₂³×D₅	1	2	yes	—
431	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
432	C₃×D₄	—	2	8	no	—
433	C₃×D₄	—	10	16	no	—
434	C₃×D₄	—	10	8	no	—
435	C₃×D₄	—	20	8	no	—
436	C₂×C₃×C₄	—	2	2	no	—
437	C₂×C₃×C₄	—	2	2	no	—
438	C₂×C₃×C₄	—	2	2	no	—
439	C₂×C₃×C₄	—	2	2	no	—
440	C₂×C₃×C₄	—	2	8	no	—
441	C₂×C₃×C₄	—	2	8	no	—
442	C₂×C₃×C₄	—	5	4	no	—
443	C₂×C₃×C₄	—	5	4	no	—
444	C₂×C₃×C₄	—	5	4	no	—
445	C₂×C₃×C₄	—	10	4	no	—
446	C₂×C₃×C₄	—	10	4	no	—
447	C₂×C₃×C₄	—	10	4	no	—
448	C₂×C₃×C₄	—	20	8	no	—
449	C₂×C₃×C₄	—	20	8	no	—
450	C₂×C₃×C₄	#711	1	4	yes	—
451	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
452	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
453	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
454	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
455	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
456	C₂²×C₅	—	2	2	no	—
457	C₂²×C₅	—	2	2	no	—
458	C₂²×C₅	—	2	8	no	—
459	C₂²×C₅	—	2	2	no	—
460	C₂²×C₅	—	2	2	no	—
461	C₂²×C₅	—	2	2	no	—
462	C₂²×C₅	—	4	4	no	—
463	C₂²×C₅	C₂×C₃×#41	1	4	yes	—
464	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
465	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
466	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
467	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
468	C₂×D₅	—	2	4	no	—
469	C₂×D₅	—	2	4	no	—
470	C₂×D₅	—	2	4	no	—
471	C₂×D₅	—	4	8	no	—
472	C₄×C₅	—	2	2	no	—
473	C₄×C₅	—	2	2	no	—
474	C₄×C₅	—	2	2	no	—
475	C₄×C₅	—	2	2	no	—
476	C₄×C₅	C₂×C₃×#41	1	4	yes	—
477	Dic₅	—	2	4	no	—
478	Dic₅	—	2	4	no	—
479	Dic₅	—	2	4	no	—
480	C₂⁴	—	5	2	no	—
481	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
482	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	5	8	no	—
485	C₂×D₄	—	5	8	no	—
486	C₂×D₄	—	5	4	no	—
487	C₂×D₄	—	10	8	no	—
488	C₂×D₄	—	10	4	no	—
489	C₂×D₄	—	20	4	no	—
490	C₂×D₄	C₂²×C₃×D₅	1	4	yes	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	5	2	no	—
494	C₂²×C₄	—	5	2	no	—
495	C₂²×C₄	—	5	2	no	—
496	C₂²×C₄	—	5	8	no	—
497	C₂²×C₄	—	5	8	no	—
498	C₂²×C₄	—	20	4	no	—
499	C₂²×C₄	—	20	4	no	—
500	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
501	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
502	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
503	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
504	C₂²×C₄	C₂²×C₃×D₅	1	4	yes	—
505	C₂²×C₄	C₂²×C₃×D₅	1	4	yes	—
506	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
507	C₄⋊C₄	—	5	4	no	—
508	C₄⋊C₄	—	5	4	no	—
509	C₄⋊C₄	—	5	8	no	—
510	C₄⋊C₄	—	5	4	no	—
511	C₄⋊C₄	—	5	8	no	—
512	C₄⋊C₄	—	5	8	no	—
513	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
514	C₂²⋊C₄	—	5	8	no	—
515	C₂²⋊C₄	—	5	8	no	—
516	C₂²⋊C₄	—	5	8	no	—
517	C₂²⋊C₄	—	5	8	no	—
518	C₂²⋊C₄	C₂²×C₃×D₅	1	4	yes	—
519	C₂²⋊C₄	C₂²×C₃×D₅	1	4	yes	—
520	C₄²	—	5	8	no	—
521	C₄²	C₂²×C₃×D₅	1	4	yes	—
522	C₃×C₅	C₂×#553	1	1	yes	characteristic, complemented, Hall
523	C₂²×C₃	—	2	8	no	—
524	C₂²×C₃	—	2	2	no	—
525	C₂²×C₃	—	2	2	no	—
526	C₂²×C₃	—	2	2	no	—
527	C₂²×C₃	—	2	2	no	—
528	C₂²×C₃	—	2	2	no	—
529	C₂²×C₃	—	4	4	no	—
530	C₂²×C₃	—	10	4	no	—
531	C₂²×C₃	—	10	4	no	—
532	C₂²×C₃	—	10	4	no	—
533	C₂²×C₃	—	20	8	no	—
534	C₂²×C₃	C₂×#709	1	4	yes	—
535	C₂²×C₃	C₂×#709	1	2	yes	—
536	C₂²×C₃	C₂×#711	1	2	yes	—
537	C₂²×C₃	#4149	1	2	yes	—
538	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
539	C₃×C₄	—	2	2	no	—
540	C₃×C₄	—	2	2	no	—
541	C₃×C₄	—	2	2	no	—
542	C₃×C₄	—	2	2	no	—
543	C₃×C₄	—	10	4	no	—
544	C₃×C₄	—	10	4	no	—
545	C₃×C₄	—	10	4	no	—
546	C₃×C₄	C₂×#711	1	4	yes	—
547	C₂×C₅	—	2	4	no	—
548	C₂×C₅	—	4	2	no	—
549	C₂×C₅	C₃×#553	1	4	yes	complemented
550	C₂×C₅	C₂²×C₃×#41	1	1	yes	characteristic
551	C₂×C₅	C₂²×C₃×#41	1	1	yes	characteristic
552	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
553	D₅	—	4	4	no	—
554	C₂³	—	2	2	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	2	2	no	—
558	C₂³	—	2	2	no	—
559	C₂³	—	5	4	no	—
560	C₂³	—	10	4	no	—
561	C₂³	—	10	4	no	—
562	C₂³	—	10	4	no	—
563	C₂³	C₃×#709	1	4	yes	—
564	C₂³	C₂³×C₃×D₅	1	2	yes	—
565	C₂³	C₂³×C₃×D₅	1	2	yes	—
566	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
567	D₄	—	2	8	no	—
568	D₄	—	10	16	no	—
569	D₄	—	10	8	no	—
570	D₄	—	20	8	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	8	no	—
576	C₂×C₄	—	2	8	no	—
577	C₂×C₄	—	5	4	no	—
578	C₂×C₄	—	5	4	no	—
579	C₂×C₄	—	5	4	no	—
580	C₂×C₄	—	10	4	no	—
581	C₂×C₄	—	10	4	no	—
582	C₂×C₄	—	10	4	no	—
583	C₂×C₄	—	20	8	no	—
584	C₂×C₄	—	20	8	no	—
585	C₂×C₄	C₃×#711	1	4	yes	—
586	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
587	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
588	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
589	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
590	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
591	C₂×C₃	—	2	4	no	—
592	C₂×C₃	—	4	2	no	—
593	C₂×C₃	—	20	4	no	—
594	C₂×C₃	#65899	1	4	yes	direct product factor
595	C₂×C₃	C₂²×#709	1	1	yes	characteristic
596	C₂×C₃	C₂²×#711	1	1	yes	characteristic
597	C₂×C₃	C₂×#4149	1	1	yes	characteristic
598	C₅	C₂×C₃×#553	1	1	yes	characteristic, complemented, Sylow
599	C₂²	—	2	2	no	—
600	C₂²	—	2	2	no	—
601	C₂²	—	2	8	no	—
602	C₂²	—	2	2	no	—
603	C₂²	—	2	2	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	4	4	no	—
606	C₂²	—	10	4	no	—
607	C₂²	—	10	4	no	—
608	C₂²	—	10	4	no	—
609	C₂²	—	20	8	no	—
610	C₂²	C₂×C₃×#709	1	4	yes	—
611	C₂²	C₂×C₃×#709	1	2	yes	—
612	C₂²	C₂×C₃×#711	1	2	yes	—
613	C₂²	C₃×#4149	1	2	yes	—
614	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
615	C₄	—	2	2	no	—
616	C₄	—	2	2	no	—
617	C₄	—	2	2	no	—
618	C₄	—	2	2	no	—
619	C₄	—	10	4	no	—
620	C₄	—	10	4	no	—
621	C₄	—	10	4	no	—
622	C₄	C₂×C₃×#711	1	4	yes	—
623	C₃	C₂×#65899	1	1	yes	characteristic, direct product factor, Sylow
624	C₂	—	2	4	no	—
625	C₂	—	4	2	no	—
626	C₂	—	20	4	no	—
627	C₂	C₃×#65899	1	4	yes	direct product factor
628	C₂	C₂²×C₃×#709	1	1	yes	characteristic
629	C₂	C₂²×C₃×#711	1	1	yes	characteristic
630	C₂	C₂×C₃×#4149	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×D₄	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₃×C₄×D₄	D₅
hypocenter	C₅	C₂×C₃×#553
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₅×D₄	C₂	semidirect
2. C₂×C₃×#4085	C₂	semidirect
3. C₂×C₃×#4076	C₂	semidirect
4. C₂×C₃×#4076	C₂	semidirect
5. C₂×C₃×#4072	C₂	semidirect
6. C₂²×C₃×C₄⋊Dic₅	C₂	semidirect
7. C₃×#65899	C₂	direct (N × H)
8. C₂×C₃×#4044	C₂	semidirect
9. C₂×C₃×#4029	C₂	semidirect
10. C₂×C₃×#4027	C₂	semidirect
11. C₂×C₃×C₄×D₂₀	C₂	semidirect
12. C₂×C₃×#4016	C₂	semidirect
13. C₂×#65899	C₃	direct (N × H)
22. C₃×C₄×C₅×D₄	C₂²	semidirect
28. C₃×#4085	C₂²	semidirect
29. C₃×#4076	C₂²	semidirect
30. C₃×#4076	C₂²	semidirect
34. C₃×#4072	C₂²	semidirect
39. C₂×C₃×C₄⋊Dic₅	C₂²	semidirect
43. C₃×#4044	C₂²	semidirect
44. C₃×#4029	C₂²	semidirect
45. C₃×#4027	C₂²	semidirect
46. C₃×C₄×D₂₀	C₂²	semidirect
47. C₃×#4016	C₂²	semidirect
49. C₂×C₄×C₅×D₄	C₂×C₃	semidirect
50. C₂×#4085	C₂×C₃	semidirect
51. C₂×#4076	C₂×C₃	semidirect
52. C₂×#4076	C₂×C₃	semidirect
53. C₂×#4072	C₂×C₃	semidirect
54. C₂²×C₄⋊Dic₅	C₂×C₃	semidirect
55. #65899	C₂×C₃	direct (N × H)
56. C₂×#4044	C₂×C₃	semidirect
57. C₂×#4029	C₂×C₃	semidirect
58. C₂×#4027	C₂×C₃	semidirect
59. C₂×C₄×D₂₀	C₂×C₃	semidirect
60. C₂×#4016	C₂×C₃	semidirect
112. C₂×C₃×C₄×D₄	D₅	semidirect
123. C₄×C₅×D₄	C₂²×C₃	semidirect
129. #4085	C₂²×C₃	semidirect
130. #4076	C₂²×C₃	semidirect
131. #4076	C₂²×C₃	semidirect
135. #4072	C₂²×C₃	semidirect
140. C₂×C₄⋊Dic₅	C₂²×C₃	semidirect
144. #4044	C₂²×C₃	semidirect
145. #4029	C₂²×C₃	semidirect
146. #4027	C₂²×C₃	semidirect
147. C₄×D₂₀	C₂²×C₃	semidirect
148. #4016	C₂²×C₃	semidirect
203. C₃×C₄×D₄	C₂×D₅	semidirect
272. C₂×C₄×D₄	C₃×D₅	semidirect
394. C₄×D₄	C₂×C₃×D₅	semidirect
414. C₂×C₃×C₅	#553	semidirect
522. C₃×C₅	C₂×#553	semidirect
549. C₂×C₅	C₃×#553	semidirect
594. C₂×C₃	#65899	direct (N × H)
598. C₅	C₂×C₃×#553	semidirect
623. C₃	C₂×#65899	direct (N × H)
627. C₂	C₃×#65899	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
C₂×C₃×#4085	2
C₂×C₃×#4076	2
C₂×C₃×#4072	2
C₂²×C₃×C₄⋊Dic₅	2
C₃×#65899	2
C₂×C₃×#4044	2
C₂×C₃×#4029	2
C₂×C₃×#4027	2
C₂×C₃×C₄×D₂₀	2
C₂×C₃×#4016	2
C₂×#65899	3
C₂×C₃×#553	5

Subgroups — C2×C3×#65899 #14483602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65899 #14483602