Subgroups — C₂×C₃×#66018 #14495276

Order 1920.

2304 conjugacy classes of subgroups.

The subgroup lattice has 544 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, complemented
2	C₂×C₃×#4092	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#4085	C₂	1	2	yes	maximal, complemented
4	C₃×#66018	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃×#4032	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#4030	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#4029	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#4024	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×#4021	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃×#4016	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#66018	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
15	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂³×C₃×Dic₅	C₂²	1	2	yes	—
17	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
18	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	—
19	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
20	C₃×C₅×#124	C₂²	1	8	yes	complemented
21	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
25	C₃×#4092	C₂²	1	16	yes	complemented
26	C₃×#4085	C₂²	1	16	yes	complemented
27	C₂×C₃×#685	C₂²	1	2	yes	—
28	C₂×C₃×#685	C₂²	1	2	yes	—
29	C₂×C₃×#685	C₂²	1	2	yes	—
30	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
31	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
32	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
33	C₂×C₃×#683	C₂²	1	2	yes	—
34	C₂×C₃×#683	C₂²	1	2	yes	—
35	C₂×C₃×#683	C₂²	1	2	yes	—
36	C₃×#4032	C₂²	1	16	yes	complemented
37	C₃×#4030	C₂²	1	16	yes	complemented
38	C₃×#4029	C₂²	1	16	yes	complemented
39	C₃×#4024	C₂²	1	16	yes	complemented
40	C₃×#4021	C₂²	1	8	yes	complemented
41	C₃×#4016	C₂²	1	8	yes	complemented
42	C₂×C₃×#562	—	5	1	no	maximal, Hall
43	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic, complemented
44	C₂×#4092	C₂×C₃	1	2	yes	complemented
45	C₂×#4085	C₂×C₃	1	2	yes	complemented
46	#66018	C₂×C₃	1	16	yes	direct product factor
47	C₂×#4032	C₂×C₃	1	2	yes	complemented
48	C₂×#4030	C₂×C₃	1	2	yes	complemented
49	C₂×#4029	C₂×C₃	1	2	yes	complemented
50	C₂×#4024	C₂×C₃	1	2	yes	complemented
51	C₂×#4021	C₂×C₃	1	1	yes	characteristic, complemented
52	C₂×#4016	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
54	C₂³×C₃×D₅	C₂³	1	2	yes	—
55	C₂×C₃×C₅×Q₈	—	4	2	no	—
56	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
57	C₂×C₃×C₅×D₄	—	4	2	no	—
58	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
59	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
61	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
62	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₂²⋊D₅	—	4	4	no	—
64	C₂×C₃×C₂²⋊D₅	—	4	4	no	—
65	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	—
66	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	—
67	C₂²×C₃×Dic₅	—	4	4	no	—
68	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
69	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
70	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
71	C₂²×C₃×Dic₅	C₂³	1	8	yes	—
72	C₂×C₃×D₂₀	—	4	2	no	—
73	C₂×C₃×D₂₀	C₂³	1	4	yes	—
74	C₂×C₃×C₄×D₅	—	4	4	no	—
75	C₂×C₃×C₄×D₅	C₂³	1	8	yes	—
76	C₂×C₃×Dic₁₀	—	4	2	no	—
77	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
78	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
79	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
80	C₃×C₄²×C₅	C₂³	1	4	yes	—
81	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	—
82	C₃×#685	C₂³	1	8	yes	—
83	C₃×#685	C₂³	1	8	yes	—
84	C₃×#685	C₂³	1	8	yes	—
85	C₃×C₄⋊Dic₅	C₂³	1	8	yes	—
86	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
87	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
88	C₃×#683	C₂³	1	8	yes	—
89	C₃×#683	C₂³	1	8	yes	—
90	C₃×#683	C₂³	1	8	yes	—
91	C₃×#562	—	5	16	no	—
92	C₂×C₃×#125	—	5	1	no	—
93	C₂×C₃×#124	—	5	1	no	—
94	C₂×C₃×#124	D₅	1	1	yes	characteristic, complemented
95	C₂×C₃×#123	—	5	2	no	—
96	C₂×C₃×#123	—	5	2	no	—
97	C₂×C₃×C₂²⋊Q₈	—	5	2	no	—
98	C₂×C₃×C₂²⋊Q₈	—	5	2	no	—
99	C₂×C₃×#121	—	5	2	no	—
100	C₂×C₃×#121	—	5	2	no	—
101	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
102	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
103	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
104	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
105	C₂³×Dic₅	C₂²×C₃	1	2	yes	—
106	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
107	C₂²×C₄×D₅	C₂²×C₃	1	2	yes	—
108	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
109	C₅×#124	C₂²×C₃	1	8	yes	complemented
110	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
111	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
112	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
113	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
114	#4092	C₂²×C₃	1	16	yes	complemented
115	#4085	C₂²×C₃	1	16	yes	complemented
116	C₂×#685	C₂²×C₃	1	2	yes	—
117	C₂×#685	C₂²×C₃	1	2	yes	—
118	C₂×#685	C₂²×C₃	1	2	yes	—
119	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
120	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
121	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
122	C₂×#683	C₂²×C₃	1	2	yes	—
123	C₂×#683	C₂²×C₃	1	2	yes	—
124	C₂×#683	C₂²×C₃	1	2	yes	—
125	#4032	C₂²×C₃	1	16	yes	complemented
126	#4030	C₂²×C₃	1	16	yes	complemented
127	#4029	C₂²×C₃	1	16	yes	complemented
128	#4024	C₂²×C₃	1	16	yes	complemented
129	#4021	C₂²×C₃	1	8	yes	complemented
130	#4016	C₂²×C₃	1	8	yes	complemented
131	C₂×#562	—	5	1	no	Sylow
132	C₂³×C₃×C₅	—	2	4	no	—
133	C₂³×C₃×C₅	—	2	4	no	—
134	C₂³×C₃×C₅	—	2	4	no	—
135	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
136	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
137	C₂²×C₃×D₅	—	2	4	no	—
138	C₂²×C₃×D₅	—	2	4	no	—
139	C₂²×C₃×D₅	—	2	4	no	—
140	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
141	C₃×C₅×Q₈	—	4	4	no	—
142	C₃×C₅×D₄	—	4	4	no	—
143	C₂×C₃×C₄×C₅	—	2	2	no	—
144	C₂×C₃×C₄×C₅	—	2	4	no	—
145	C₂×C₃×C₄×C₅	—	2	2	no	—
146	C₂×C₃×C₄×C₅	—	2	2	no	—
147	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
148	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
149	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
150	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
151	C₃×C₂²⋊D₅	—	4	8	no	—
152	C₃×C₂²⋊D₅	—	4	8	no	—
153	C₂×C₃×Dic₅	—	2	4	no	—
154	C₂×C₃×Dic₅	—	2	4	no	—
155	C₂×C₃×Dic₅	—	2	4	no	—
156	C₂×C₃×Dic₅	—	4	8	no	—
157	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
158	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
159	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
160	C₃×D₂₀	—	4	4	no	—
161	C₃×C₄×D₅	—	4	8	no	—
162	C₃×Dic₁₀	—	4	4	no	—
163	C₂²×C₃×Q₈	—	5	1	no	—
164	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
165	C₂²×C₃×D₄	—	5	2	no	—
166	C₂²×C₃×D₄	—	5	2	no	—
167	C₂²×C₃×D₄	—	5	1	no	—
168	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
169	C₂³×C₃×C₄	—	5	2	no	—
170	C₂³×C₃×C₄	—	5	2	no	—
171	C₃×#125	—	5	8	no	—
172	C₃×#124	—	5	8	no	—
173	C₃×#124	C₂×D₅	1	8	yes	complemented
174	C₃×#123	—	5	16	no	—
175	C₃×#123	—	5	16	no	—
176	C₃×C₂²⋊Q₈	—	5	16	no	—
177	C₃×C₂²⋊Q₈	—	5	16	no	—
178	C₃×#121	—	5	16	no	—
179	C₃×#121	—	5	16	no	—
180	C₂×C₃×C₄⋊C₄	—	5	2	no	—
181	C₂×C₃×C₄⋊C₄	—	5	2	no	—
182	C₂×C₃×C₄⋊C₄	—	5	2	no	—
183	C₂×C₃×C₄⋊C₄	—	5	2	no	—
184	C₂×C₃×C₄⋊C₄	—	5	1	no	—
185	C₂×C₃×C₄⋊C₄	—	5	1	no	—
186	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
187	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
188	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
189	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
190	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
191	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
192	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
193	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
194	C₂³×D₅	C₂³×C₃	1	2	yes	—
195	C₂×C₅×Q₈	—	4	2	no	—
196	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
197	C₂×C₅×D₄	—	4	2	no	—
198	C₂×C₅×D₄	C₂³×C₃	1	4	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	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
203	C₂×C₂²⋊D₅	—	4	4	no	—
204	C₂×C₂²⋊D₅	—	4	4	no	—
205	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	—
206	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	—
207	C₂²×Dic₅	—	4	4	no	—
208	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
209	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
210	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
211	C₂²×Dic₅	C₂³×C₃	1	8	yes	—
212	C₂×D₂₀	—	4	2	no	—
213	C₂×D₂₀	C₂³×C₃	1	4	yes	—
214	C₂×C₄×D₅	—	4	4	no	—
215	C₂×C₄×D₅	C₂³×C₃	1	8	yes	—
216	C₂×Dic₁₀	—	4	2	no	—
217	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
218	C₅×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
219	C₅×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
220	C₄²×C₅	C₂³×C₃	1	4	yes	—
221	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	—
222	#685	C₂³×C₃	1	8	yes	—
223	#685	C₂³×C₃	1	8	yes	—
224	#685	C₂³×C₃	1	8	yes	—
225	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
226	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
227	C₄⋊Dic₅	C₂³×C₃	1	8	yes	—
228	#683	C₂³×C₃	1	8	yes	—
229	#683	C₂³×C₃	1	8	yes	—
230	#683	C₂³×C₃	1	8	yes	—
231	#562	—	5	16	no	—
232	C₂×#125	—	5	1	no	—
233	C₂×#124	—	5	1	no	—
234	C₂×#124	C₃×D₅	1	1	yes	characteristic, complemented
235	C₂×#123	—	5	2	no	—
236	C₂×#123	—	5	2	no	—
237	C₂×C₂²⋊Q₈	—	5	2	no	—
238	C₂×C₂²⋊Q₈	—	5	2	no	—
239	C₂×#121	—	5	2	no	—
240	C₂×#121	—	5	2	no	—
241	C₂²×C₃×C₅	—	2	4	no	—
242	C₂²×C₃×C₅	—	2	4	no	—
243	C₂²×C₃×C₅	—	2	4	no	—
244	C₂²×C₃×C₅	—	4	8	no	—
245	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
246	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
247	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
248	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
249	C₂×C₃×D₅	—	2	4	no	—
250	C₂×C₃×D₅	—	2	4	no	—
251	C₂×C₃×D₅	—	2	4	no	—
252	C₂×C₃×D₅	—	4	8	no	—
253	C₃×C₄×C₅	—	2	2	no	—
254	C₃×C₄×C₅	—	2	4	no	—
255	C₃×C₄×C₅	—	2	2	no	—
256	C₃×C₄×C₅	—	2	2	no	—
257	C₃×Dic₅	—	2	4	no	—
258	C₃×Dic₅	—	2	4	no	—
259	C₃×Dic₅	—	2	4	no	—
260	C₂⁴×C₃	—	5	2	no	—
261	C₂⁴×C₃	C₂²×D₅	1	2	yes	—
262	C₂×C₃×Q₈	—	4	2	no	—
263	C₂×C₃×Q₈	—	5	4	no	—
264	C₂×C₃×Q₈	—	20	2	no	—
265	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	—
266	C₂×C₃×D₄	—	4	2	no	—
267	C₂×C₃×D₄	—	5	8	no	—
268	C₂×C₃×D₄	—	5	8	no	—
269	C₂×C₃×D₄	—	5	4	no	—
270	C₂×C₃×D₄	—	20	2	no	—
271	C₂×C₃×D₄	—	20	4	no	—
272	C₂×C₃×D₄	—	20	4	no	—
273	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
274	C₂²×C₃×C₄	—	5	2	no	—
275	C₂²×C₃×C₄	—	5	2	no	—
276	C₂²×C₃×C₄	—	5	2	no	—
277	C₂²×C₃×C₄	—	5	8	no	—
278	C₂²×C₃×C₄	—	5	8	no	—
279	C₂²×C₃×C₄	—	20	4	no	—
280	C₂²×C₃×C₄	—	20	4	no	—
281	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
282	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
283	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
284	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
285	C₃×C₄⋊C₄	—	5	8	no	—
286	C₃×C₄⋊C₄	—	5	8	no	—
287	C₃×C₄⋊C₄	—	5	8	no	—
288	C₃×C₄⋊C₄	—	5	8	no	—
289	C₃×C₄⋊C₄	—	5	4	no	—
290	C₃×C₄⋊C₄	—	5	4	no	—
291	C₃×C₂²⋊C₄	—	5	8	no	—
292	C₃×C₂²⋊C₄	—	5	8	no	—
293	C₃×C₂²⋊C₄	—	5	8	no	—
294	C₃×C₂²⋊C₄	—	5	8	no	—
295	C₃×C₂²⋊C₄	C₂²×D₅	1	8	yes	—
296	C₃×C₂²⋊C₄	C₂²×D₅	1	8	yes	—
297	C₃×C₄²	C₂²×D₅	1	4	yes	—
298	C₂³×C₅	—	2	4	no	—
299	C₂³×C₅	—	2	4	no	—
300	C₂³×C₅	—	2	4	no	—
301	C₂³×C₅	C₂⁴×C₃	1	4	yes	—
302	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
303	C₂²×D₅	—	2	4	no	—
304	C₂²×D₅	—	2	4	no	—
305	C₂²×D₅	—	2	4	no	—
306	C₂²×D₅	C₂⁴×C₃	1	4	yes	—
307	C₅×Q₈	—	4	4	no	—
308	C₅×D₄	—	4	4	no	—
309	C₂×C₄×C₅	—	2	2	no	—
310	C₂×C₄×C₅	—	2	4	no	—
311	C₂×C₄×C₅	—	2	2	no	—
312	C₂×C₄×C₅	—	2	2	no	—
313	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
314	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
315	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
316	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
317	C₂²⋊D₅	—	4	8	no	—
318	C₂²⋊D₅	—	4	8	no	—
319	C₂×Dic₅	—	2	4	no	—
320	C₂×Dic₅	—	2	4	no	—
321	C₂×Dic₅	—	2	4	no	—
322	C₂×Dic₅	—	4	8	no	—
323	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
324	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
325	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
326	D₂₀	—	4	4	no	—
327	C₄×D₅	—	4	8	no	—
328	Dic₁₀	—	4	4	no	—
329	C₂²×Q₈	—	5	1	no	—
330	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
331	C₂²×D₄	—	5	2	no	—
332	C₂²×D₄	—	5	2	no	—
333	C₂²×D₄	—	5	1	no	—
334	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
335	C₂³×C₄	—	5	2	no	—
336	C₂³×C₄	—	5	2	no	—
337	#125	—	5	8	no	—
338	#124	—	5	8	no	—
339	#124	C₂×C₃×D₅	1	8	yes	complemented
340	#123	—	5	16	no	—
341	#123	—	5	16	no	—
342	C₂²⋊Q₈	—	5	16	no	—
343	C₂²⋊Q₈	—	5	16	no	—
344	#121	—	5	16	no	—
345	#121	—	5	16	no	—
346	C₂×C₄⋊C₄	—	5	2	no	—
347	C₂×C₄⋊C₄	—	5	2	no	—
348	C₂×C₄⋊C₄	—	5	2	no	—
349	C₂×C₄⋊C₄	—	5	2	no	—
350	C₂×C₄⋊C₄	—	5	1	no	—
351	C₂×C₄⋊C₄	—	5	1	no	—
352	C₂×C₂²⋊C₄	—	5	2	no	—
353	C₂×C₂²⋊C₄	—	5	2	no	—
354	C₂×C₂²⋊C₄	—	5	2	no	—
355	C₂×C₂²⋊C₄	—	5	2	no	—
356	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
357	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
358	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
359	C₂×C₃×C₅	—	4	4	no	—
360	C₂×C₃×C₅	#562	1	4	yes	complemented
361	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
362	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
363	C₂×C₃×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
364	C₃×D₅	—	4	4	no	—
365	C₂³×C₃	—	2	4	no	—
366	C₂³×C₃	—	2	4	no	—
367	C₂³×C₃	—	2	4	no	—
368	C₂³×C₃	—	5	4	no	—
369	C₂³×C₃	—	10	4	no	—
370	C₂³×C₃	—	10	4	no	—
371	C₂³×C₃	—	10	4	no	—
372	C₂³×C₃	C₂³×D₅	1	4	yes	—
373	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
374	C₃×Q₈	—	4	4	no	—
375	C₃×Q₈	—	20	4	no	—
376	C₃×D₄	—	4	4	no	—
377	C₃×D₄	—	20	4	no	—
378	C₃×D₄	—	20	8	no	—
379	C₃×D₄	—	20	8	no	—
380	C₂×C₃×C₄	—	2	4	no	—
381	C₂×C₃×C₄	—	2	2	no	—
382	C₂×C₃×C₄	—	2	2	no	—
383	C₂×C₃×C₄	—	2	2	no	—
384	C₂×C₃×C₄	—	5	4	no	—
385	C₂×C₃×C₄	—	5	4	no	—
386	C₂×C₃×C₄	—	5	4	no	—
387	C₂×C₃×C₄	—	10	4	no	—
388	C₂×C₃×C₄	—	10	4	no	—
389	C₂×C₃×C₄	—	10	4	no	—
390	C₂×C₃×C₄	—	20	8	no	—
391	C₂×C₃×C₄	—	20	8	no	—
392	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
393	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
394	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
395	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
396	C₂²×C₅	—	2	4	no	—
397	C₂²×C₅	—	2	4	no	—
398	C₂²×C₅	—	2	4	no	—
399	C₂²×C₅	—	4	8	no	—
400	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
401	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
402	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
403	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
404	C₂×D₅	—	2	4	no	—
405	C₂×D₅	—	2	4	no	—
406	C₂×D₅	—	2	4	no	—
407	C₂×D₅	—	4	8	no	—
408	C₄×C₅	—	2	2	no	—
409	C₄×C₅	—	2	4	no	—
410	C₄×C₅	—	2	2	no	—
411	C₄×C₅	—	2	2	no	—
412	Dic₅	—	2	4	no	—
413	Dic₅	—	2	4	no	—
414	Dic₅	—	2	4	no	—
415	C₂⁴	—	5	2	no	—
416	C₂⁴	C₂²×C₃×D₅	1	2	yes	—
417	C₂×Q₈	—	4	2	no	—
418	C₂×Q₈	—	5	4	no	—
419	C₂×Q₈	—	20	2	no	—
420	C₂×Q₈	C₂²×C₃×D₅	1	4	yes	—
421	C₂×D₄	—	4	2	no	—
422	C₂×D₄	—	5	8	no	—
423	C₂×D₄	—	5	8	no	—
424	C₂×D₄	—	5	4	no	—
425	C₂×D₄	—	20	2	no	—
426	C₂×D₄	—	20	4	no	—
427	C₂×D₄	—	20	4	no	—
428	C₂×D₄	C₂²×C₃×D₅	1	4	yes	—
429	C₂²×C₄	—	5	2	no	—
430	C₂²×C₄	—	5	2	no	—
431	C₂²×C₄	—	5	2	no	—
432	C₂²×C₄	—	5	8	no	—
433	C₂²×C₄	—	5	8	no	—
434	C₂²×C₄	—	20	4	no	—
435	C₂²×C₄	—	20	4	no	—
436	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
437	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
438	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
439	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
440	C₄⋊C₄	—	5	8	no	—
441	C₄⋊C₄	—	5	8	no	—
442	C₄⋊C₄	—	5	8	no	—
443	C₄⋊C₄	—	5	8	no	—
444	C₄⋊C₄	—	5	4	no	—
445	C₄⋊C₄	—	5	4	no	—
446	C₂²⋊C₄	—	5	8	no	—
447	C₂²⋊C₄	—	5	8	no	—
448	C₂²⋊C₄	—	5	8	no	—
449	C₂²⋊C₄	—	5	8	no	—
450	C₂²⋊C₄	C₂²×C₃×D₅	1	8	yes	—
451	C₂²⋊C₄	C₂²×C₃×D₅	1	8	yes	—
452	C₄²	C₂²×C₃×D₅	1	4	yes	—
453	C₃×C₅	C₂×#562	1	1	yes	characteristic, complemented, Hall
454	C₂²×C₃	—	2	4	no	—
455	C₂²×C₃	—	2	4	no	—
456	C₂²×C₃	—	2	4	no	—
457	C₂²×C₃	—	4	8	no	—
458	C₂²×C₃	—	10	4	no	—
459	C₂²×C₃	—	10	4	no	—
460	C₂²×C₃	—	10	4	no	—
461	C₂²×C₃	—	20	8	no	—
462	C₂²×C₃	#4144	1	2	yes	—
463	C₂²×C₃	#4149	1	2	yes	—
464	C₂²×C₃	#4150	1	2	yes	—
465	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
466	C₃×C₄	—	2	4	no	—
467	C₃×C₄	—	2	2	no	—
468	C₃×C₄	—	2	2	no	—
469	C₃×C₄	—	2	2	no	—
470	C₃×C₄	—	10	4	no	—
471	C₃×C₄	—	10	4	no	—
472	C₃×C₄	—	10	4	no	—
473	C₂×C₅	—	4	4	no	—
474	C₂×C₅	C₃×#562	1	4	yes	complemented
475	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
476	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
477	C₂×C₅	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic
478	D₅	—	4	4	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	2	4	no	—
481	C₂³	—	2	4	no	—
482	C₂³	—	5	4	no	—
483	C₂³	—	10	4	no	—
484	C₂³	—	10	4	no	—
485	C₂³	—	10	4	no	—
486	C₂³	C₂³×C₃×D₅	1	4	yes	—
487	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
488	Q₈	—	4	4	no	—
489	Q₈	—	20	4	no	—
490	D₄	—	4	4	no	—
491	D₄	—	20	4	no	—
492	D₄	—	20	8	no	—
493	D₄	—	20	8	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	5	4	no	—
499	C₂×C₄	—	5	4	no	—
500	C₂×C₄	—	5	4	no	—
501	C₂×C₄	—	10	4	no	—
502	C₂×C₄	—	10	4	no	—
503	C₂×C₄	—	10	4	no	—
504	C₂×C₄	—	20	8	no	—
505	C₂×C₄	—	20	8	no	—
506	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
507	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
508	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
509	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
510	C₂×C₃	—	4	4	no	—
511	C₂×C₃	—	20	4	no	—
512	C₂×C₃	#66018	1	4	yes	direct product factor
513	C₂×C₃	C₂×#4144	1	1	yes	characteristic
514	C₂×C₃	C₂×#4149	1	1	yes	characteristic
515	C₂×C₃	C₂×#4150	1	1	yes	characteristic
516	C₅	C₂×C₃×#562	1	1	yes	characteristic, complemented, Sylow
517	C₂²	—	2	4	no	—
518	C₂²	—	2	4	no	—
519	C₂²	—	2	4	no	—
520	C₂²	—	4	8	no	—
521	C₂²	—	10	4	no	—
522	C₂²	—	10	4	no	—
523	C₂²	—	10	4	no	—
524	C₂²	—	20	8	no	—
525	C₂²	C₃×#4144	1	2	yes	—
526	C₂²	C₃×#4149	1	2	yes	—
527	C₂²	C₃×#4150	1	2	yes	—
528	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
529	C₄	—	2	4	no	—
530	C₄	—	2	2	no	—
531	C₄	—	2	2	no	—
532	C₄	—	2	2	no	—
533	C₄	—	10	4	no	—
534	C₄	—	10	4	no	—
535	C₄	—	10	4	no	—
536	C₃	C₂×#66018	1	1	yes	characteristic, direct product factor, Sylow
537	C₂	—	4	4	no	—
538	C₂	—	20	4	no	—
539	C₂	C₃×#66018	1	4	yes	direct product factor
540	C₂	C₂×C₃×#4144	1	1	yes	characteristic
541	C₂	C₂×C₃×#4149	1	1	yes	characteristic
542	C₂	C₂×C₃×#4150	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×#124	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₃×#124	D₅
hypocenter	C₅	C₂×C₃×#562
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×#124

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#124	C₂	semidirect
2. C₂×C₃×#4092	C₂	semidirect
3. C₂×C₃×#4085	C₂	semidirect
4. C₃×#66018	C₂	direct (N × H)
5. C₂×C₃×#4032	C₂	semidirect
6. C₂×C₃×#4030	C₂	semidirect
7. C₂×C₃×#4029	C₂	semidirect
8. C₂×C₃×#4024	C₂	semidirect
9. C₂×C₃×#4021	C₂	semidirect
10. C₂×C₃×#4016	C₂	semidirect
11. C₂×#66018	C₃	direct (N × H)
20. C₃×C₅×#124	C₂²	semidirect
25. C₃×#4092	C₂²	semidirect
26. C₃×#4085	C₂²	semidirect
36. C₃×#4032	C₂²	semidirect
37. C₃×#4030	C₂²	semidirect
38. C₃×#4029	C₂²	semidirect
39. C₃×#4024	C₂²	semidirect
40. C₃×#4021	C₂²	semidirect
41. C₃×#4016	C₂²	semidirect
43. C₂×C₅×#124	C₂×C₃	semidirect
44. C₂×#4092	C₂×C₃	semidirect
45. C₂×#4085	C₂×C₃	semidirect
46. #66018	C₂×C₃	direct (N × H)
47. C₂×#4032	C₂×C₃	semidirect
48. C₂×#4030	C₂×C₃	semidirect
49. C₂×#4029	C₂×C₃	semidirect
50. C₂×#4024	C₂×C₃	semidirect
51. C₂×#4021	C₂×C₃	semidirect
52. C₂×#4016	C₂×C₃	semidirect
94. C₂×C₃×#124	D₅	semidirect
109. C₅×#124	C₂²×C₃	semidirect
114. #4092	C₂²×C₃	semidirect
115. #4085	C₂²×C₃	semidirect
125. #4032	C₂²×C₃	semidirect
126. #4030	C₂²×C₃	semidirect
127. #4029	C₂²×C₃	semidirect
128. #4024	C₂²×C₃	semidirect
129. #4021	C₂²×C₃	semidirect
130. #4016	C₂²×C₃	semidirect
173. C₃×#124	C₂×D₅	semidirect
234. C₂×#124	C₃×D₅	semidirect
339. #124	C₂×C₃×D₅	semidirect
360. C₂×C₃×C₅	#562	semidirect
453. C₃×C₅	C₂×#562	semidirect
474. C₂×C₅	C₃×#562	semidirect
512. C₂×C₃	#66018	direct (N × H)
516. C₅	C₂×C₃×#562	semidirect
536. C₃	C₂×#66018	direct (N × H)
539. C₂	C₃×#66018	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×#4092	2
C₂×C₃×#4085	2
C₃×#66018	2
C₂×C₃×#4032	2
C₂×C₃×#4030	2
C₂×C₃×#4029	2
C₂×C₃×#4024	2
C₂×C₃×#4021	2
C₂×C₃×#4016	2
C₂×#66018	3
C₂×C₃×#562	5

Subgroups — C2×C3×#66018 #14495276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#66018 #14495276