Subgroups — C₂×C₃×#65929 #14495185

Order 1920.

2688 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#120	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4099	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#4086	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#4085	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#4084	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#4081	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#65929	C₂	1	16	yes	maximal, direct product factor
8	C₂×C₃×#4030	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×#4029	C₂	1	2	yes	maximal, complemented
10	C₂×C₃×#4025	C₂	1	2	yes	maximal, complemented
11	C₂×#65929	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
18	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	complemented
19	C₂³×C₃×Dic₅	C₂²	1	1	yes	characteristic, complemented
20	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic, complemented
21	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	complemented
22	C₃×C₅×#120	C₂²	1	8	yes	complemented
23	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₃×#4099	C₂²	1	16	yes	complemented
26	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
27	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
28	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
29	C₂×C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
30	C₃×#4086	C₂²	1	8	yes	complemented
31	C₃×#4085	C₂²	1	16	yes	complemented
32	C₃×#4084	C₂²	1	16	yes	complemented
33	C₃×#4081	C₂²	1	8	yes	complemented
34	C₂×C₃×#685	C₂²	1	2	yes	complemented
35	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	complemented
36	C₂×C₃×#683	C₂²	1	2	yes	complemented
37	C₂×C₃×#683	C₂²	1	2	yes	complemented
38	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic, complemented
39	C₃×#4030	C₂²	1	16	yes	complemented
40	C₃×#4029	C₂²	1	16	yes	complemented
41	C₃×#4025	C₂²	1	16	yes	complemented
42	C₂×C₃×#560	—	5	1	no	maximal, Hall
43	C₂×C₅×#120	C₂×C₃	1	1	yes	characteristic, complemented
44	C₂×#4099	C₂×C₃	1	2	yes	complemented
45	C₂×#4086	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×#4085	C₂×C₃	1	2	yes	complemented
47	C₂×#4084	C₂×C₃	1	2	yes	complemented
48	C₂×#4081	C₂×C₃	1	1	yes	characteristic, complemented
49	#65929	C₂×C₃	1	16	yes	direct product factor
50	C₂×#4030	C₂×C₃	1	2	yes	complemented
51	C₂×#4029	C₂×C₃	1	2	yes	complemented
52	C₂×#4025	C₂×C₃	1	2	yes	complemented
53	C₂⁴×C₃×C₅	—	4	4	no	—
54	C₂⁴×C₃×C₅	—	4	2	no	—
55	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
56	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
57	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
58	C₂⁴×C₃×C₅	C₂³	1	4	yes	—
59	C₂³×C₃×D₅	C₂³	1	2	yes	—
60	C₂×C₃×C₅×D₄	—	4	4	no	—
61	C₂×C₃×C₅×D₄	—	4	2	no	—
62	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
63	C₂×C₃×C₅×D₄	C₂³	1	8	yes	—
64	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
65	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
66	C₂×C₃×C₂²⋊D₅	—	4	4	no	—
67	C₂×C₃×C₂²⋊D₅	—	4	4	no	—
68	C₂×C₃×C₂²⋊D₅	—	4	4	no	—
69	C₂×C₃×C₂²⋊D₅	—	4	4	no	—
70	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	—
71	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	—
72	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	—
73	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	complemented
74	C₂²×C₃×Dic₅	—	4	2	no	—
75	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
76	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
77	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
78	C₂²×C₃×Dic₅	C₂³	1	4	yes	complemented
79	C₂×C₃×D₂₀	—	4	2	no	—
80	C₂×C₃×D₂₀	C₂³	1	4	yes	complemented
81	C₂×C₃×C₄×D₅	—	4	4	no	—
82	C₂×C₃×C₄×D₅	C₂³	1	8	yes	complemented
83	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
84	C₃×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
85	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	—
86	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	—
87	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	—
88	C₃×C₂²⋊Dic₅	C₂³	1	4	yes	complemented
89	C₃×#685	C₂³	1	8	yes	complemented
90	C₃×C₄⋊Dic₅	C₂³	1	8	yes	complemented
91	C₃×#683	C₂³	1	8	yes	complemented
92	C₃×#683	C₂³	1	8	yes	complemented
93	C₃×C₄×Dic₅	C₂³	1	4	yes	complemented
94	C₃×#560	—	5	16	no	—
95	C₂×C₃×#127	—	5	1	no	—
96	C₂×C₃×#126	—	5	2	no	—
97	C₂×C₃×#123	—	5	1	no	—
98	C₂×C₃×#123	—	5	2	no	—
99	C₂×C₃×#121	—	5	2	no	—
100	C₂×C₃×#121	—	5	2	no	—
101	C₂×C₃×#121	—	5	2	no	—
102	C₂×C₃×#120	—	5	2	no	—
103	C₂×C₃×#120	D₅	1	1	yes	characteristic, complemented
104	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
105	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
106	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
107	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
108	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
109	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
110	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	complemented
111	C₂³×Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
112	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic, complemented
113	C₂²×C₄×D₅	C₂²×C₃	1	2	yes	complemented
114	C₅×#120	C₂²×C₃	1	8	yes	complemented
115	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
116	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
117	#4099	C₂²×C₃	1	16	yes	complemented
118	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
119	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
120	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
121	C₂×C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
122	#4086	C₂²×C₃	1	8	yes	complemented
123	#4085	C₂²×C₃	1	16	yes	complemented
124	#4084	C₂²×C₃	1	16	yes	complemented
125	#4081	C₂²×C₃	1	8	yes	complemented
126	C₂×#685	C₂²×C₃	1	2	yes	complemented
127	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	complemented
128	C₂×#683	C₂²×C₃	1	2	yes	complemented
129	C₂×#683	C₂²×C₃	1	2	yes	complemented
130	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
131	#4030	C₂²×C₃	1	16	yes	complemented
132	#4029	C₂²×C₃	1	16	yes	complemented
133	#4025	C₂²×C₃	1	16	yes	complemented
134	C₂×#560	—	5	1	no	Sylow
135	C₂³×C₃×C₅	—	2	2	no	—
136	C₂³×C₃×C₅	—	2	4	no	—
137	C₂³×C₃×C₅	—	2	4	no	—
138	C₂³×C₃×C₅	—	2	4	no	—
139	C₂³×C₃×C₅	—	2	4	no	—
140	C₂³×C₃×C₅	—	2	4	no	—
141	C₂³×C₃×C₅	—	2	2	no	—
142	C₂³×C₃×C₅	—	4	8	no	—
143	C₂³×C₃×C₅	—	4	4	no	—
144	C₂³×C₃×C₅	—	4	16	no	—
145	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
146	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
147	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
148	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
149	C₂²×C₃×D₅	—	2	4	no	—
150	C₂²×C₃×D₅	—	2	4	no	—
151	C₂²×C₃×D₅	—	2	4	no	—
152	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
153	C₃×C₅×D₄	—	4	8	no	—
154	C₃×C₅×D₄	—	4	4	no	—
155	C₂×C₃×C₄×C₅	—	2	4	no	—
156	C₂×C₃×C₄×C₅	—	2	2	no	—
157	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
158	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
159	C₃×C₂²⋊D₅	—	4	8	no	—
160	C₃×C₂²⋊D₅	—	4	8	no	—
161	C₃×C₂²⋊D₅	—	4	8	no	—
162	C₃×C₂²⋊D₅	—	4	8	no	—
163	C₂×C₃×Dic₅	—	2	4	no	—
164	C₂×C₃×Dic₅	—	2	4	no	—
165	C₂×C₃×Dic₅	—	2	4	no	—
166	C₂×C₃×Dic₅	—	4	4	no	—
167	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
168	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
169	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
170	C₃×D₂₀	—	4	4	no	—
171	C₃×C₄×D₅	—	4	8	no	—
172	C₂⁵×C₃	C₂×D₅	1	1	yes	characteristic
173	C₂²×C₃×D₄	—	5	2	no	—
174	C₂²×C₃×D₄	—	5	2	no	—
175	C₂²×C₃×D₄	—	5	2	no	—
176	C₂²×C₃×D₄	—	5	2	no	—
177	C₂²×C₃×D₄	—	5	1	no	—
178	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
179	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
180	C₂³×C₃×C₄	—	5	1	no	—
181	C₂³×C₃×C₄	—	5	2	no	—
182	C₃×#127	—	5	8	no	—
183	C₃×#126	—	5	16	no	—
184	C₃×#123	—	5	8	no	—
185	C₃×#123	—	5	16	no	—
186	C₃×#121	—	5	16	no	—
187	C₃×#121	—	5	16	no	—
188	C₃×#121	—	5	16	no	—
189	C₃×#120	—	5	16	no	—
190	C₃×#120	C₂×D₅	1	8	yes	complemented
191	C₂×C₃×C₄⋊C₄	—	5	2	no	—
192	C₂×C₃×C₄⋊C₄	—	5	2	no	—
193	C₂×C₃×C₄⋊C₄	—	5	2	no	—
194	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
195	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
196	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
197	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
198	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
199	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
200	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
201	C₂×C₃×C₄²	—	5	1	no	—
202	C₂⁴×C₅	—	4	4	no	—
203	C₂⁴×C₅	—	4	2	no	—
204	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
205	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
206	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
207	C₂⁴×C₅	C₂³×C₃	1	4	yes	—
208	C₂³×D₅	C₂³×C₃	1	2	yes	—
209	C₂×C₅×D₄	—	4	4	no	—
210	C₂×C₅×D₄	—	4	2	no	—
211	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
212	C₂×C₅×D₄	C₂³×C₃	1	8	yes	—
213	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
214	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
215	C₂×C₂²⋊D₅	—	4	4	no	—
216	C₂×C₂²⋊D₅	—	4	4	no	—
217	C₂×C₂²⋊D₅	—	4	4	no	—
218	C₂×C₂²⋊D₅	—	4	4	no	—
219	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	—
220	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	—
221	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	—
222	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	complemented
223	C₂²×Dic₅	—	4	2	no	—
224	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
225	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
226	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
227	C₂²×Dic₅	C₂³×C₃	1	4	yes	complemented
228	C₂×D₂₀	—	4	2	no	—
229	C₂×D₂₀	C₂³×C₃	1	4	yes	complemented
230	C₂×C₄×D₅	—	4	4	no	—
231	C₂×C₄×D₅	C₂³×C₃	1	8	yes	complemented
232	C₅×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
233	C₅×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
234	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	—
235	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	—
236	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	—
237	C₂²⋊Dic₅	C₂³×C₃	1	4	yes	complemented
238	#685	C₂³×C₃	1	8	yes	complemented
239	C₄⋊Dic₅	C₂³×C₃	1	8	yes	complemented
240	#683	C₂³×C₃	1	8	yes	complemented
241	#683	C₂³×C₃	1	8	yes	complemented
242	C₄×Dic₅	C₂³×C₃	1	4	yes	complemented
243	#560	—	5	16	no	—
244	C₂×#127	—	5	1	no	—
245	C₂×#126	—	5	2	no	—
246	C₂×#123	—	5	1	no	—
247	C₂×#123	—	5	2	no	—
248	C₂×#121	—	5	2	no	—
249	C₂×#121	—	5	2	no	—
250	C₂×#121	—	5	2	no	—
251	C₂×#120	—	5	2	no	—
252	C₂×#120	C₃×D₅	1	1	yes	characteristic, complemented
253	C₂²×C₃×C₅	—	2	2	no	—
254	C₂²×C₃×C₅	—	2	4	no	—
255	C₂²×C₃×C₅	—	2	4	no	—
256	C₂²×C₃×C₅	—	2	4	no	—
257	C₂²×C₃×C₅	—	2	4	no	—
258	C₂²×C₃×C₅	—	2	4	no	—
259	C₂²×C₃×C₅	—	2	2	no	—
260	C₂²×C₃×C₅	—	4	16	no	—
261	C₂²×C₃×C₅	—	4	4	no	—
262	C₂²×C₃×C₅	—	4	8	no	—
263	C₂²×C₃×C₅	—	4	4	no	—
264	C₂²×C₃×C₅	H(5,𝔽₂)	1	4	yes	—
265	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
266	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
267	C₂×C₃×D₅	—	2	4	no	—
268	C₂×C₃×D₅	—	2	4	no	—
269	C₂×C₃×D₅	—	2	4	no	—
270	C₂×C₃×D₅	—	4	8	no	—
271	C₃×C₄×C₅	—	2	4	no	—
272	C₃×C₄×C₅	—	2	2	no	—
273	C₃×Dic₅	—	2	4	no	—
274	C₃×Dic₅	—	2	4	no	—
275	C₃×Dic₅	—	2	4	no	—
276	C₂⁴×C₃	—	4	4	no	—
277	C₂⁴×C₃	—	4	2	no	—
278	C₂⁴×C₃	—	5	2	no	—
279	C₂⁴×C₃	C₂²×D₅	1	2	yes	—
280	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
281	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
282	C₂⁴×C₃	C₂²×D₅	1	4	yes	—
283	C₂×C₃×D₄	—	4	4	no	—
284	C₂×C₃×D₄	—	4	2	no	—
285	C₂×C₃×D₄	—	5	8	no	—
286	C₂×C₃×D₄	—	5	8	no	—
287	C₂×C₃×D₄	—	5	8	no	—
288	C₂×C₃×D₄	—	5	8	no	—
289	C₂×C₃×D₄	—	5	4	no	—
290	C₂×C₃×D₄	—	20	2	no	—
291	C₂×C₃×D₄	—	20	4	no	—
292	C₂×C₃×D₄	—	20	4	no	—
293	C₂×C₃×D₄	—	20	4	no	—
294	C₂×C₃×D₄	—	20	4	no	—
295	C₂×C₃×D₄	C₂²×D₅	1	8	yes	—
296	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
297	C₂²×C₃×C₄	—	5	2	no	—
298	C₂²×C₃×C₄	—	5	2	no	—
299	C₂²×C₃×C₄	—	5	2	no	—
300	C₂²×C₃×C₄	—	5	4	no	—
301	C₂²×C₃×C₄	—	5	8	no	—
302	C₂²×C₃×C₄	—	20	4	no	—
303	C₂²×C₃×C₄	—	20	2	no	—
304	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
305	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
306	C₃×C₄⋊C₄	—	5	8	no	—
307	C₃×C₄⋊C₄	—	5	8	no	—
308	C₃×C₄⋊C₄	—	5	8	no	—
309	C₃×C₂²⋊C₄	—	5	8	no	—
310	C₃×C₂²⋊C₄	—	5	8	no	—
311	C₃×C₂²⋊C₄	—	5	8	no	—
312	C₃×C₂²⋊C₄	—	5	4	no	—
313	C₃×C₂²⋊C₄	—	5	8	no	—
314	C₃×C₂²⋊C₄	C₂²×D₅	1	8	yes	—
315	C₃×C₂²⋊C₄	C₂²×D₅	1	4	yes	—
316	C₃×C₄²	—	5	4	no	—
317	C₂³×C₅	—	2	2	no	—
318	C₂³×C₅	—	2	4	no	—
319	C₂³×C₅	—	2	4	no	—
320	C₂³×C₅	—	2	4	no	—
321	C₂³×C₅	—	2	4	no	—
322	C₂³×C₅	—	2	4	no	—
323	C₂³×C₅	—	2	2	no	—
324	C₂³×C₅	—	4	8	no	—
325	C₂³×C₅	—	4	4	no	—
326	C₂³×C₅	—	4	16	no	—
327	C₂³×C₅	C₂⁴×C₃	1	4	yes	—
328	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
329	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
330	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
331	C₂²×D₅	—	2	4	no	—
332	C₂²×D₅	—	2	4	no	—
333	C₂²×D₅	—	2	4	no	—
334	C₂²×D₅	C₂⁴×C₃	1	4	yes	—
335	C₅×D₄	—	4	8	no	—
336	C₅×D₄	—	4	4	no	—
337	C₂×C₄×C₅	—	2	4	no	—
338	C₂×C₄×C₅	—	2	2	no	—
339	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
340	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
341	C₂²⋊D₅	—	4	8	no	—
342	C₂²⋊D₅	—	4	8	no	—
343	C₂²⋊D₅	—	4	8	no	—
344	C₂²⋊D₅	—	4	8	no	—
345	C₂×Dic₅	—	2	4	no	—
346	C₂×Dic₅	—	2	4	no	—
347	C₂×Dic₅	—	2	4	no	—
348	C₂×Dic₅	—	4	4	no	—
349	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
350	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
351	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
352	D₂₀	—	4	4	no	—
353	C₄×D₅	—	4	8	no	—
354	C₂⁵	C₂×C₃×D₅	1	1	yes	characteristic
355	C₂²×D₄	—	5	2	no	—
356	C₂²×D₄	—	5	2	no	—
357	C₂²×D₄	—	5	2	no	—
358	C₂²×D₄	—	5	2	no	—
359	C₂²×D₄	—	5	1	no	—
360	C₂²×D₄	C₂×C₃×D₅	1	2	yes	—
361	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
362	C₂³×C₄	—	5	1	no	—
363	C₂³×C₄	—	5	2	no	—
364	#127	—	5	8	no	—
365	#126	—	5	16	no	—
366	#123	—	5	8	no	—
367	#123	—	5	16	no	—
368	#121	—	5	16	no	—
369	#121	—	5	16	no	—
370	#121	—	5	16	no	—
371	#120	—	5	16	no	—
372	#120	C₂×C₃×D₅	1	8	yes	complemented
373	C₂×C₄⋊C₄	—	5	2	no	—
374	C₂×C₄⋊C₄	—	5	2	no	—
375	C₂×C₄⋊C₄	—	5	2	no	—
376	C₂×C₂²⋊C₄	—	5	2	no	—
377	C₂×C₂²⋊C₄	—	5	2	no	—
378	C₂×C₂²⋊C₄	—	5	2	no	—
379	C₂×C₂²⋊C₄	—	5	1	no	—
380	C₂×C₂²⋊C₄	—	5	2	no	—
381	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
382	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
383	C₂×C₄²	—	5	1	no	—
384	C₂×C₃×C₅	—	4	2	no	—
385	C₂×C₃×C₅	—	4	4	no	—
386	C₂×C₃×C₅	—	4	2	no	—
387	C₂×C₃×C₅	#560	1	4	yes	complemented
388	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
389	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
390	C₃×D₅	—	4	4	no	—
391	C₂³×C₃	—	2	2	no	—
392	C₂³×C₃	—	2	4	no	—
393	C₂³×C₃	—	2	4	no	—
394	C₂³×C₃	—	2	2	no	—
395	C₂³×C₃	—	2	4	no	—
396	C₂³×C₃	—	2	4	no	—
397	C₂³×C₃	—	2	4	no	—
398	C₂³×C₃	—	4	8	no	—
399	C₂³×C₃	—	4	4	no	—
400	C₂³×C₃	—	4	16	no	—
401	C₂³×C₃	—	5	4	no	—
402	C₂³×C₃	—	10	4	no	—
403	C₂³×C₃	—	10	4	no	—
404	C₂³×C₃	—	10	4	no	—
405	C₂³×C₃	C₂³×D₅	1	4	yes	—
406	C₂³×C₃	C₂³×D₅	1	2	yes	—
407	C₂³×C₃	C₂³×D₅	1	2	yes	—
408	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
409	C₃×D₄	—	4	8	no	—
410	C₃×D₄	—	4	4	no	—
411	C₃×D₄	—	20	4	no	—
412	C₃×D₄	—	20	8	no	—
413	C₃×D₄	—	20	8	no	—
414	C₃×D₄	—	20	8	no	—
415	C₃×D₄	—	20	8	no	—
416	C₂×C₃×C₄	—	2	4	no	—
417	C₂×C₃×C₄	—	2	2	no	—
418	C₂×C₃×C₄	—	5	4	no	—
419	C₂×C₃×C₄	—	5	4	no	—
420	C₂×C₃×C₄	—	5	4	no	—
421	C₂×C₃×C₄	—	10	4	no	—
422	C₂×C₃×C₄	—	10	4	no	—
423	C₂×C₃×C₄	—	10	4	no	—
424	C₂×C₃×C₄	—	20	8	no	—
425	C₂×C₃×C₄	—	20	4	no	—
426	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
427	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
428	C₂²×C₅	—	2	2	no	—
429	C₂²×C₅	—	2	4	no	—
430	C₂²×C₅	—	2	4	no	—
431	C₂²×C₅	—	2	4	no	—
432	C₂²×C₅	—	2	4	no	—
433	C₂²×C₅	—	2	4	no	—
434	C₂²×C₅	—	2	2	no	—
435	C₂²×C₅	—	4	16	no	—
436	C₂²×C₅	—	4	4	no	—
437	C₂²×C₅	—	4	8	no	—
438	C₂²×C₅	—	4	4	no	—
439	C₂²×C₅	C₃×H(5,𝔽₂)	1	4	yes	—
440	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
441	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
442	C₂×D₅	—	2	4	no	—
443	C₂×D₅	—	2	4	no	—
444	C₂×D₅	—	2	4	no	—
445	C₂×D₅	—	4	8	no	—
446	C₄×C₅	—	2	4	no	—
447	C₄×C₅	—	2	2	no	—
448	Dic₅	—	2	4	no	—
449	Dic₅	—	2	4	no	—
450	Dic₅	—	2	4	no	—
451	C₂⁴	—	4	4	no	—
452	C₂⁴	—	4	2	no	—
453	C₂⁴	—	5	2	no	—
454	C₂⁴	C₂²×C₃×D₅	1	2	yes	—
455	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
456	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
457	C₂⁴	C₂²×C₃×D₅	1	4	yes	—
458	C₂×D₄	—	4	4	no	—
459	C₂×D₄	—	4	2	no	—
460	C₂×D₄	—	5	8	no	—
461	C₂×D₄	—	5	8	no	—
462	C₂×D₄	—	5	8	no	—
463	C₂×D₄	—	5	8	no	—
464	C₂×D₄	—	5	4	no	—
465	C₂×D₄	—	20	2	no	—
466	C₂×D₄	—	20	4	no	—
467	C₂×D₄	—	20	4	no	—
468	C₂×D₄	—	20	4	no	—
469	C₂×D₄	—	20	4	no	—
470	C₂×D₄	C₂²×C₃×D₅	1	8	yes	—
471	C₂×D₄	C₂²×C₃×D₅	1	4	yes	—
472	C₂²×C₄	—	5	2	no	—
473	C₂²×C₄	—	5	2	no	—
474	C₂²×C₄	—	5	2	no	—
475	C₂²×C₄	—	5	4	no	—
476	C₂²×C₄	—	5	8	no	—
477	C₂²×C₄	—	20	4	no	—
478	C₂²×C₄	—	20	2	no	—
479	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
480	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
481	C₄⋊C₄	—	5	8	no	—
482	C₄⋊C₄	—	5	8	no	—
483	C₄⋊C₄	—	5	8	no	—
484	C₂²⋊C₄	—	5	8	no	—
485	C₂²⋊C₄	—	5	8	no	—
486	C₂²⋊C₄	—	5	8	no	—
487	C₂²⋊C₄	—	5	4	no	—
488	C₂²⋊C₄	—	5	8	no	—
489	C₂²⋊C₄	C₂²×C₃×D₅	1	8	yes	—
490	C₂²⋊C₄	C₂²×C₃×D₅	1	4	yes	—
491	C₄²	—	5	4	no	—
492	C₃×C₅	C₂×#560	1	1	yes	characteristic, complemented, Hall
493	C₂²×C₃	—	2	2	no	—
494	C₂²×C₃	—	2	4	no	—
495	C₂²×C₃	—	2	4	no	—
496	C₂²×C₃	—	2	2	no	—
497	C₂²×C₃	—	2	4	no	—
498	C₂²×C₃	—	2	4	no	—
499	C₂²×C₃	—	2	4	no	—
500	C₂²×C₃	—	4	16	no	—
501	C₂²×C₃	—	4	4	no	—
502	C₂²×C₃	—	4	8	no	—
503	C₂²×C₃	—	4	4	no	—
504	C₂²×C₃	—	10	4	no	—
505	C₂²×C₃	—	10	4	no	—
506	C₂²×C₃	—	10	4	no	—
507	C₂²×C₃	—	20	8	no	—
508	C₂²×C₃	#4144	1	4	yes	—
509	C₂²×C₃	#4144	1	2	yes	—
510	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
511	C₃×C₄	—	2	4	no	—
512	C₃×C₄	—	2	2	no	—
513	C₃×C₄	—	10	4	no	—
514	C₃×C₄	—	10	4	no	—
515	C₃×C₄	—	10	4	no	—
516	C₂×C₅	—	4	4	no	—
517	C₂×C₅	—	4	2	no	—
518	C₂×C₅	—	4	2	no	—
519	C₂×C₅	C₃×#560	1	4	yes	complemented
520	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	2	yes	—
521	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
522	D₅	—	4	4	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	4	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	4	no	—
528	C₂³	—	2	4	no	—
529	C₂³	—	2	4	no	—
530	C₂³	—	4	8	no	—
531	C₂³	—	4	4	no	—
532	C₂³	—	4	16	no	—
533	C₂³	—	5	4	no	—
534	C₂³	—	10	4	no	—
535	C₂³	—	10	4	no	—
536	C₂³	—	10	4	no	—
537	C₂³	C₂³×C₃×D₅	1	4	yes	—
538	C₂³	C₂³×C₃×D₅	1	2	yes	—
539	C₂³	C₂³×C₃×D₅	1	2	yes	—
540	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
541	D₄	—	4	8	no	—
542	D₄	—	4	4	no	—
543	D₄	—	20	4	no	—
544	D₄	—	20	8	no	—
545	D₄	—	20	8	no	—
546	D₄	—	20	8	no	—
547	D₄	—	20	8	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	5	4	no	—
551	C₂×C₄	—	5	4	no	—
552	C₂×C₄	—	5	4	no	—
553	C₂×C₄	—	10	4	no	—
554	C₂×C₄	—	10	4	no	—
555	C₂×C₄	—	10	4	no	—
556	C₂×C₄	—	20	8	no	—
557	C₂×C₄	—	20	4	no	—
558	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
559	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
560	C₂×C₃	—	4	4	no	—
561	C₂×C₃	—	4	2	no	—
562	C₂×C₃	—	4	2	no	—
563	C₂×C₃	—	20	4	no	—
564	C₂×C₃	#65929	1	4	yes	direct product factor
565	C₂×C₃	C₂×#4144	1	2	yes	—
566	C₂×C₃	C₂×#4144	1	1	yes	characteristic
567	C₅	C₂×C₃×#560	1	1	yes	characteristic, complemented, Sylow
568	C₂²	—	2	2	no	—
569	C₂²	—	2	4	no	—
570	C₂²	—	2	4	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	4	no	—
573	C₂²	—	2	4	no	—
574	C₂²	—	2	4	no	—
575	C₂²	—	4	16	no	—
576	C₂²	—	4	8	no	—
577	C₂²	—	4	4	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	10	4	no	—
580	C₂²	—	10	4	no	—
581	C₂²	—	10	4	no	—
582	C₂²	—	20	8	no	—
583	C₂²	C₃×#4144	1	4	yes	—
584	C₂²	C₃×#4144	1	2	yes	—
585	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
586	C₄	—	2	4	no	—
587	C₄	—	2	2	no	—
588	C₄	—	10	4	no	—
589	C₄	—	10	4	no	—
590	C₄	—	10	4	no	—
591	C₃	C₂×#65929	1	1	yes	characteristic, direct product factor, Sylow
592	C₂	—	4	4	no	—
593	C₂	—	4	2	no	—
594	C₂	—	4	2	no	—
595	C₂	—	20	4	no	—
596	C₂	C₃×#65929	1	4	yes	direct product factor
597	C₂	C₂×C₃×#4144	1	2	yes	—
598	C₂	C₂×C₃×#4144	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×C₅×#120	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₃×#120	D₅
hypocenter	C₅	C₂×C₃×#560
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×#120

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#120	C₂	semidirect
2. C₂×C₃×#4099	C₂	semidirect
3. C₂×C₃×#4086	C₂	semidirect
4. C₂×C₃×#4085	C₂	semidirect
5. C₂×C₃×#4084	C₂	semidirect
6. C₂×C₃×#4081	C₂	semidirect
7. C₃×#65929	C₂	direct (N × H)
8. C₂×C₃×#4030	C₂	semidirect
9. C₂×C₃×#4029	C₂	semidirect
10. C₂×C₃×#4025	C₂	semidirect
11. C₂×#65929	C₃	direct (N × H)
18. C₂²×C₃×C₂²⋊D₅	C₂²	semidirect
19. C₂³×C₃×Dic₅	C₂²	semidirect
20. C₂²×C₃×D₂₀	C₂²	semidirect
21. C₂²×C₃×C₄×D₅	C₂²	semidirect
22. C₃×C₅×#120	C₂²	semidirect
25. C₃×#4099	C₂²	semidirect
29. C₂×C₃×C₂²⋊Dic₅	C₂²	semidirect
30. C₃×#4086	C₂²	semidirect
31. C₃×#4085	C₂²	semidirect
32. C₃×#4084	C₂²	semidirect
33. C₃×#4081	C₂²	semidirect
34. C₂×C₃×#685	C₂²	semidirect
35. C₂×C₃×C₄⋊Dic₅	C₂²	semidirect
36. C₂×C₃×#683	C₂²	semidirect
37. C₂×C₃×#683	C₂²	semidirect
38. C₂×C₃×C₄×Dic₅	C₂²	semidirect
39. C₃×#4030	C₂²	semidirect
40. C₃×#4029	C₂²	semidirect
41. C₃×#4025	C₂²	semidirect
43. C₂×C₅×#120	C₂×C₃	semidirect
44. C₂×#4099	C₂×C₃	semidirect
45. C₂×#4086	C₂×C₃	semidirect
46. C₂×#4085	C₂×C₃	semidirect
47. C₂×#4084	C₂×C₃	semidirect
48. C₂×#4081	C₂×C₃	semidirect
49. #65929	C₂×C₃	direct (N × H)
50. C₂×#4030	C₂×C₃	semidirect
51. C₂×#4029	C₂×C₃	semidirect
52. C₂×#4025	C₂×C₃	semidirect
73. C₂×C₃×C₂²⋊D₅	C₂³	semidirect
78. C₂²×C₃×Dic₅	C₂³	semidirect
80. C₂×C₃×D₂₀	C₂³	semidirect
82. C₂×C₃×C₄×D₅	C₂³	semidirect
88. C₃×C₂²⋊Dic₅	C₂³	semidirect
89. C₃×#685	C₂³	semidirect
90. C₃×C₄⋊Dic₅	C₂³	semidirect
91. C₃×#683	C₂³	semidirect
92. C₃×#683	C₂³	semidirect
93. C₃×C₄×Dic₅	C₂³	semidirect
103. C₂×C₃×#120	D₅	semidirect
110. C₂²×C₂²⋊D₅	C₂²×C₃	semidirect
111. C₂³×Dic₅	C₂²×C₃	semidirect
112. C₂²×D₂₀	C₂²×C₃	semidirect
113. C₂²×C₄×D₅	C₂²×C₃	semidirect
114. C₅×#120	C₂²×C₃	semidirect
117. #4099	C₂²×C₃	semidirect
121. C₂×C₂²⋊Dic₅	C₂²×C₃	semidirect
122. #4086	C₂²×C₃	semidirect
123. #4085	C₂²×C₃	semidirect
124. #4084	C₂²×C₃	semidirect
125. #4081	C₂²×C₃	semidirect
126. C₂×#685	C₂²×C₃	semidirect
127. C₂×C₄⋊Dic₅	C₂²×C₃	semidirect
128. C₂×#683	C₂²×C₃	semidirect
129. C₂×#683	C₂²×C₃	semidirect
130. C₂×C₄×Dic₅	C₂²×C₃	semidirect
131. #4030	C₂²×C₃	semidirect
132. #4029	C₂²×C₃	semidirect
133. #4025	C₂²×C₃	semidirect
190. C₃×#120	C₂×D₅	semidirect
222. C₂×C₂²⋊D₅	C₂³×C₃	semidirect
227. C₂²×Dic₅	C₂³×C₃	semidirect
229. C₂×D₂₀	C₂³×C₃	semidirect
231. C₂×C₄×D₅	C₂³×C₃	semidirect
237. C₂²⋊Dic₅	C₂³×C₃	semidirect
238. #685	C₂³×C₃	semidirect
239. C₄⋊Dic₅	C₂³×C₃	semidirect
240. #683	C₂³×C₃	semidirect
241. #683	C₂³×C₃	semidirect
242. C₄×Dic₅	C₂³×C₃	semidirect
252. C₂×#120	C₃×D₅	semidirect
372. #120	C₂×C₃×D₅	semidirect
387. C₂×C₃×C₅	#560	semidirect
492. C₃×C₅	C₂×#560	semidirect
519. C₂×C₅	C₃×#560	semidirect
564. C₂×C₃	#65929	direct (N × H)
567. C₅	C₂×C₃×#560	semidirect
591. C₃	C₂×#65929	direct (N × H)
596. C₂	C₃×#65929	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×#120	2
C₂×C₃×#4099	2
C₂×C₃×#4086	2
C₂×C₃×#4085	2
C₂×C₃×#4084	2
C₂×C₃×#4081	2
C₃×#65929	2
C₂×C₃×#4030	2
C₂×C₃×#4029	2
C₂×C₃×#4025	2
C₂×#65929	3
C₂×C₃×#560	5

Subgroups — C2×C3×#65929 #14495185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65929 #14495185