Subgroups — C₂×C₃×#65921 #14483257

Order 1920.

2656 conjugacy classes of subgroups.

The subgroup lattice has 552 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
2	C₂²×C₃×C₂²⋊Dic₅	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#4082	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#4081	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#4081	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×D₄×Dic₅	C₂	1	2	yes	maximal, complemented
7	C₃×#65921	C₂	1	16	yes	maximal, direct product factor
8	C₂×C₃×#4025	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×#4024	C₂	1	2	yes	maximal, complemented
10	C₂×C₃×#4023	C₂	1	2	yes	maximal, complemented
11	C₂×#65921	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	2	yes	—
14	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₃×Dic₅	C₂²	1	2	yes	—
16	C₂³×C₃×Dic₅	C₂²	1	2	yes	—
17	C₂³×C₃×Dic₅	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
19	C₃×C₅×#120	C₂²	1	8	yes	—
20	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
24	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
25	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
26	C₂×C₃×C₂²⋊Dic₅	C₂²	1	16	yes	complemented
27	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	complemented
28	C₂×C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
29	C₃×#4082	C₂²	1	8	yes	complemented
30	C₃×#4081	C₂²	1	8	yes	complemented
31	C₃×#4081	C₂²	1	16	yes	complemented
32	C₃×D₄×Dic₅	C₂²	1	16	yes	complemented
33	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	complemented
34	C₂×C₃×#683	C₂²	1	2	yes	complemented
35	C₂×C₃×#683	C₂²	1	2	yes	complemented
36	C₂×C₃×#683	C₂²	1	2	yes	complemented
37	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	complemented
38	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic, complemented
39	C₃×#4025	C₂²	1	16	yes	complemented
40	C₃×#4024	C₂²	1	16	yes	complemented
41	C₃×#4023	C₂²	1	16	yes	complemented
42	C₂×C₃×#551	—	5	1	no	maximal, Hall
43	C₂×C₅×#120	C₂×C₃	1	1	yes	characteristic
44	C₂²×C₂²⋊Dic₅	C₂×C₃	1	2	yes	complemented
45	C₂×#4082	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×#4081	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂×#4081	C₂×C₃	1	2	yes	complemented
48	C₂×D₄×Dic₅	C₂×C₃	1	2	yes	complemented
49	#65921	C₂×C₃	1	16	yes	direct product factor
50	C₂×#4025	C₂×C₃	1	2	yes	complemented
51	C₂×#4024	C₂×C₃	1	2	yes	complemented
52	C₂×#4023	C₂×C₃	1	2	yes	complemented
53	C₂⁴×C₃×C₅	—	2	8	no	—
54	C₂⁴×C₃×C₅	—	4	2	no	—
55	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
56	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
57	C₂⁴×C₃×C₅	C₂³	1	4	yes	—
58	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
59	C₂×C₃×C₅×D₄	—	2	8	no	—
60	C₂×C₃×C₅×D₄	—	4	2	no	—
61	C₂×C₃×C₅×D₄	C₂³	1	8	yes	—
62	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
63	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
64	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
65	C₂²×C₃×Dic₅	—	2	8	no	—
66	C₂²×C₃×Dic₅	—	4	4	no	—
67	C₂²×C₃×Dic₅	—	4	2	no	—
68	C₂²×C₃×Dic₅	C₂³	1	8	yes	—
69	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
70	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
71	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
72	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
73	C₂²×C₃×Dic₅	C₂³	1	8	yes	—
74	C₂²×C₃×Dic₅	C₂³	1	4	yes	complemented
75	C₂×C₃×Dic₁₀	—	4	2	no	—
76	C₂×C₃×Dic₁₀	C₂³	1	4	yes	complemented
77	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
78	C₃×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
79	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	—
80	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	—
81	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	—
82	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	—
83	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	complemented
84	C₃×C₂²⋊Dic₅	C₂³	1	4	yes	complemented
85	C₃×C₄⋊Dic₅	C₂³	1	8	yes	complemented
86	C₃×#683	C₂³	1	8	yes	complemented
87	C₃×#683	C₂³	1	8	yes	complemented
88	C₃×#683	C₂³	1	8	yes	complemented
89	C₃×C₄×Dic₅	C₂³	1	8	yes	complemented
90	C₃×C₄×Dic₅	C₂³	1	4	yes	complemented
91	C₃×#551	—	5	16	no	—
92	C₂×C₃×#126	—	5	2	no	—
93	C₂×C₃×#124	—	5	1	no	—
94	C₂×C₃×#123	—	5	2	no	—
95	C₂×C₃×#123	—	5	1	no	—
96	C₂×C₃×C₂²⋊Q₈	—	5	2	no	—
97	C₂×C₃×#120	D₅	1	1	yes	characteristic
98	C₂×C₃×C₄×D₄	—	5	2	no	—
99	C₂×C₃×#117	—	5	2	no	—
100	C₂²×C₃×C₂²⋊C₄	—	5	2	no	—
101	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
102	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
103	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
104	C₂³×Dic₅	C₂²×C₃	1	2	yes	—
105	C₂³×Dic₅	C₂²×C₃	1	2	yes	—
106	C₂³×Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
107	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic, complemented
108	C₅×#120	C₂²×C₃	1	8	yes	—
109	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
110	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
111	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
112	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
113	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
114	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
115	C₂×C₂²⋊Dic₅	C₂²×C₃	1	16	yes	complemented
116	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	complemented
117	C₂×C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
118	#4082	C₂²×C₃	1	8	yes	complemented
119	#4081	C₂²×C₃	1	8	yes	complemented
120	#4081	C₂²×C₃	1	16	yes	complemented
121	D₄×Dic₅	C₂²×C₃	1	16	yes	complemented
122	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	complemented
123	C₂×#683	C₂²×C₃	1	2	yes	complemented
124	C₂×#683	C₂²×C₃	1	2	yes	complemented
125	C₂×#683	C₂²×C₃	1	2	yes	complemented
126	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
127	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	complemented
128	#4025	C₂²×C₃	1	16	yes	complemented
129	#4024	C₂²×C₃	1	16	yes	complemented
130	#4023	C₂²×C₃	1	16	yes	complemented
131	C₂×#551	—	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	—	2	16	no	—
136	C₂³×C₃×C₅	—	2	4	no	—
137	C₂³×C₃×C₅	—	2	2	no	—
138	C₂³×C₃×C₅	—	2	2	no	—
139	C₂³×C₃×C₅	—	4	4	no	—
140	C₂³×C₃×C₅	—	4	16	no	—
141	C₂³×C₃×C₅	#41	1	8	yes	—
142	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
143	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
144	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
145	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
146	C₃×C₅×D₄	—	2	16	no	—
147	C₃×C₅×D₄	—	4	4	no	—
148	C₂×C₃×C₄×C₅	—	2	4	no	—
149	C₂×C₃×C₄×C₅	—	2	2	no	—
150	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
151	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
152	C₂×C₃×Dic₅	—	2	16	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	—	2	4	no	—
157	C₂×C₃×Dic₅	—	4	8	no	—
158	C₂×C₃×Dic₅	—	4	4	no	—
159	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
160	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
161	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
162	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
163	C₃×Dic₁₀	—	4	4	no	—
164	C₂⁵×C₃	C₂×D₅	1	1	yes	characteristic
165	C₂²×C₃×Q₈	—	5	1	no	—
166	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
167	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
168	C₂³×C₃×C₄	—	5	2	no	—
169	C₂³×C₃×C₄	—	5	2	no	—
170	C₂³×C₃×C₄	—	5	1	no	—
171	C₃×#126	—	5	16	no	—
172	C₃×#124	—	5	8	no	—
173	C₃×#123	—	5	16	no	—
174	C₃×#123	—	5	8	no	—
175	C₃×C₂²⋊Q₈	—	5	16	no	—
176	C₃×#120	C₂×D₅	1	8	yes	—
177	C₃×C₄×D₄	—	5	16	no	—
178	C₃×#117	—	5	16	no	—
179	C₂×C₃×C₄⋊C₄	—	5	2	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×C₂²⋊C₄	—	5	2	no	—
184	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
185	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
186	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
187	C₂×C₃×C₂²⋊C₄	—	5	16	no	—
188	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
189	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
190	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
191	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
192	C₂×C₃×C₄²	—	5	1	no	—
193	C₂×C₃×C₄²	—	5	2	no	—
194	C₂⁴×C₅	—	2	8	no	—
195	C₂⁴×C₅	—	4	2	no	—
196	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
197	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
198	C₂⁴×C₅	C₂³×C₃	1	4	yes	—
199	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
200	C₂×C₅×D₄	—	2	8	no	—
201	C₂×C₅×D₄	—	4	2	no	—
202	C₂×C₅×D₄	C₂³×C₃	1	8	yes	—
203	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
204	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
205	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
206	C₂²×Dic₅	—	2	8	no	—
207	C₂²×Dic₅	—	4	4	no	—
208	C₂²×Dic₅	—	4	2	no	—
209	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
210	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
211	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
212	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
213	C₂²×Dic₅	C₂³×C₃	1	8	yes	—
214	C₂²×Dic₅	C₂³×C₃	1	8	yes	—
215	C₂²×Dic₅	C₂³×C₃	1	4	yes	complemented
216	C₂×Dic₁₀	—	4	2	no	—
217	C₂×Dic₁₀	C₂³×C₃	1	4	yes	complemented
218	C₅×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
219	C₅×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
220	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	—
221	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	—
222	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	—
223	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	—
224	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	complemented
225	C₂²⋊Dic₅	C₂³×C₃	1	4	yes	complemented
226	C₄⋊Dic₅	C₂³×C₃	1	8	yes	complemented
227	#683	C₂³×C₃	1	8	yes	complemented
228	#683	C₂³×C₃	1	8	yes	complemented
229	#683	C₂³×C₃	1	8	yes	complemented
230	C₄×Dic₅	C₂³×C₃	1	4	yes	complemented
231	C₄×Dic₅	C₂³×C₃	1	8	yes	complemented
232	#551	—	5	16	no	—
233	C₂×#126	—	5	2	no	—
234	C₂×#124	—	5	1	no	—
235	C₂×#123	—	5	2	no	—
236	C₂×#123	—	5	1	no	—
237	C₂×C₂²⋊Q₈	—	5	2	no	—
238	C₂×#120	C₃×D₅	1	1	yes	characteristic
239	C₂×C₄×D₄	—	5	2	no	—
240	C₂×#117	—	5	2	no	—
241	C₂²×C₂²⋊C₄	—	5	2	no	—
242	C₂²×C₃×C₅	—	2	16	no	—
243	C₂²×C₃×C₅	—	2	4	no	—
244	C₂²×C₃×C₅	—	2	4	no	—
245	C₂²×C₃×C₅	—	2	4	no	—
246	C₂²×C₃×C₅	—	2	4	no	—
247	C₂²×C₃×C₅	—	2	2	no	—
248	C₂²×C₃×C₅	—	2	2	no	—
249	C₂²×C₃×C₅	—	4	16	no	—
250	C₂²×C₃×C₅	—	4	4	no	—
251	C₂²×C₃×C₅	—	4	4	no	—
252	C₂²×C₃×C₅	C₂×#41	1	8	yes	—
253	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
254	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
255	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
256	C₃×C₄×C₅	—	2	4	no	—
257	C₃×C₄×C₅	—	2	2	no	—
258	C₃×Dic₅	—	2	4	no	—
259	C₃×Dic₅	—	2	4	no	—
260	C₃×Dic₅	—	2	4	no	—
261	C₃×Dic₅	—	2	4	no	—
262	C₂⁴×C₃	—	2	8	no	—
263	C₂⁴×C₃	—	4	2	no	—
264	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
265	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
266	C₂⁴×C₃	C₂²×D₅	1	4	yes	—
267	C₂⁴×C₃	C₂²×D₅	1	2	yes	—
268	C₂×C₃×Q₈	—	5	4	no	—
269	C₂×C₃×Q₈	—	20	2	no	—
270	C₂×C₃×D₄	—	2	8	no	—
271	C₂×C₃×D₄	—	4	2	no	—
272	C₂×C₃×D₄	C₂²×D₅	1	8	yes	—
273	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
274	C₂²×C₃×C₄	—	5	8	no	—
275	C₂²×C₃×C₄	—	5	2	no	—
276	C₂²×C₃×C₄	—	5	2	no	—
277	C₂²×C₃×C₄	—	5	2	no	—
278	C₂²×C₃×C₄	—	5	2	no	—
279	C₂²×C₃×C₄	—	5	8	no	—
280	C₂²×C₃×C₄	—	5	4	no	—
281	C₂²×C₃×C₄	—	10	8	no	—
282	C₂²×C₃×C₄	—	20	2	no	—
283	C₂²×C₃×C₄	—	20	4	no	—
284	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
285	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
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	8	no	—
290	C₃×C₂²⋊C₄	—	5	8	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	—	5	4	no	—
296	C₃×C₂²⋊C₄	C₂²×D₅	1	8	yes	—
297	C₃×C₂²⋊C₄	C₂²×D₅	1	4	yes	—
298	C₃×C₄²	—	5	4	no	—
299	C₃×C₄²	—	5	8	no	—
300	C₂³×C₅	—	2	4	no	—
301	C₂³×C₅	—	2	4	no	—
302	C₂³×C₅	—	2	4	no	—
303	C₂³×C₅	—	2	16	no	—
304	C₂³×C₅	—	2	4	no	—
305	C₂³×C₅	—	2	2	no	—
306	C₂³×C₅	—	2	2	no	—
307	C₂³×C₅	—	4	4	no	—
308	C₂³×C₅	—	4	16	no	—
309	C₂³×C₅	C₃×#41	1	8	yes	—
310	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
311	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
312	C₂³×C₅	C₂⁴×C₃	1	4	yes	—
313	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
314	C₅×D₄	—	2	16	no	—
315	C₅×D₄	—	4	4	no	—
316	C₂×C₄×C₅	—	2	4	no	—
317	C₂×C₄×C₅	—	2	2	no	—
318	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
319	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
320	C₂×Dic₅	—	2	16	no	—
321	C₂×Dic₅	—	2	4	no	—
322	C₂×Dic₅	—	2	4	no	—
323	C₂×Dic₅	—	2	4	no	—
324	C₂×Dic₅	—	2	4	no	—
325	C₂×Dic₅	—	4	8	no	—
326	C₂×Dic₅	—	4	4	no	—
327	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
328	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
329	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
330	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
331	Dic₁₀	—	4	4	no	—
332	C₂⁵	C₂×C₃×D₅	1	1	yes	characteristic
333	C₂²×Q₈	—	5	1	no	—
334	C₂²×D₄	C₂×C₃×D₅	1	2	yes	—
335	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
336	C₂³×C₄	—	5	2	no	—
337	C₂³×C₄	—	5	2	no	—
338	C₂³×C₄	—	5	1	no	—
339	#126	—	5	16	no	—
340	#124	—	5	8	no	—
341	#123	—	5	16	no	—
342	#123	—	5	8	no	—
343	C₂²⋊Q₈	—	5	16	no	—
344	#120	C₂×C₃×D₅	1	8	yes	—
345	C₄×D₄	—	5	16	no	—
346	#117	—	5	16	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	2	no	—
351	C₂×C₂²⋊C₄	—	5	2	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	16	no	—
356	C₂×C₂²⋊C₄	—	5	2	no	—
357	C₂×C₂²⋊C₄	—	5	1	no	—
358	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
359	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
360	C₂×C₄²	—	5	1	no	—
361	C₂×C₄²	—	5	2	no	—
362	C₂×C₃×C₅	—	2	8	no	—
363	C₂×C₃×C₅	—	4	2	no	—
364	C₂×C₃×C₅	—	4	2	no	—
365	C₂×C₃×C₅	#551	1	4	yes	complemented
366	C₂×C₃×C₅	C₂²×#41	1	2	yes	—
367	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
368	C₂³×C₃	—	2	4	no	—
369	C₂³×C₃	—	2	4	no	—
370	C₂³×C₃	—	2	4	no	—
371	C₂³×C₃	—	2	16	no	—
372	C₂³×C₃	—	2	4	no	—
373	C₂³×C₃	—	2	2	no	—
374	C₂³×C₃	—	2	2	no	—
375	C₂³×C₃	—	4	4	no	—
376	C₂³×C₃	—	4	16	no	—
377	C₂³×C₃	#711	1	8	yes	—
378	C₂³×C₃	C₂³×D₅	1	2	yes	—
379	C₂³×C₃	C₂³×D₅	1	2	yes	—
380	C₂³×C₃	C₂³×D₅	1	4	yes	—
381	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
382	C₃×Q₈	—	20	4	no	—
383	C₃×D₄	—	2	16	no	—
384	C₃×D₄	—	4	4	no	—
385	C₂×C₃×C₄	—	2	4	no	—
386	C₂×C₃×C₄	—	2	2	no	—
387	C₂×C₃×C₄	—	5	4	no	—
388	C₂×C₃×C₄	—	5	4	no	—
389	C₂×C₃×C₄	—	5	4	no	—
390	C₂×C₃×C₄	—	5	4	no	—
391	C₂×C₃×C₄	—	10	16	no	—
392	C₂×C₃×C₄	—	10	4	no	—
393	C₂×C₃×C₄	—	10	4	no	—
394	C₂×C₃×C₄	—	10	4	no	—
395	C₂×C₃×C₄	—	10	4	no	—
396	C₂×C₃×C₄	—	20	4	no	—
397	C₂×C₃×C₄	—	20	8	no	—
398	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
399	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
400	C₂²×C₅	—	2	4	no	—
401	C₂²×C₅	—	2	4	no	—
402	C₂²×C₅	—	2	16	no	—
403	C₂²×C₅	—	2	4	no	—
404	C₂²×C₅	—	2	4	no	—
405	C₂²×C₅	—	2	2	no	—
406	C₂²×C₅	—	2	2	no	—
407	C₂²×C₅	—	4	16	no	—
408	C₂²×C₅	—	4	4	no	—
409	C₂²×C₅	—	4	4	no	—
410	C₂²×C₅	C₂×C₃×#41	1	8	yes	—
411	C₂²×C₅	C₂×C₃×#41	1	4	yes	—
412	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
413	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
414	C₄×C₅	—	2	4	no	—
415	C₄×C₅	—	2	2	no	—
416	Dic₅	—	2	4	no	—
417	Dic₅	—	2	4	no	—
418	Dic₅	—	2	4	no	—
419	Dic₅	—	2	4	no	—
420	C₂⁴	—	2	8	no	—
421	C₂⁴	—	4	2	no	—
422	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
423	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
424	C₂⁴	C₂²×C₃×D₅	1	4	yes	—
425	C₂⁴	C₂²×C₃×D₅	1	2	yes	—
426	C₂×Q₈	—	5	4	no	—
427	C₂×Q₈	—	20	2	no	—
428	C₂×D₄	—	2	8	no	—
429	C₂×D₄	—	4	2	no	—
430	C₂×D₄	C₂²×C₃×D₅	1	8	yes	—
431	C₂×D₄	C₂²×C₃×D₅	1	4	yes	—
432	C₂²×C₄	—	5	2	no	—
433	C₂²×C₄	—	5	2	no	—
434	C₂²×C₄	—	5	2	no	—
435	C₂²×C₄	—	5	2	no	—
436	C₂²×C₄	—	5	8	no	—
437	C₂²×C₄	—	5	8	no	—
438	C₂²×C₄	—	5	4	no	—
439	C₂²×C₄	—	10	8	no	—
440	C₂²×C₄	—	20	2	no	—
441	C₂²×C₄	—	20	4	no	—
442	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
443	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
444	C₄⋊C₄	—	5	8	no	—
445	C₄⋊C₄	—	5	8	no	—
446	C₄⋊C₄	—	5	8	no	—
447	C₄⋊C₄	—	5	8	no	—
448	C₂²⋊C₄	—	5	8	no	—
449	C₂²⋊C₄	—	5	8	no	—
450	C₂²⋊C₄	—	5	8	no	—
451	C₂²⋊C₄	—	5	8	no	—
452	C₂²⋊C₄	—	5	8	no	—
453	C₂²⋊C₄	—	5	4	no	—
454	C₂²⋊C₄	C₂²×C₃×D₅	1	8	yes	—
455	C₂²⋊C₄	C₂²×C₃×D₅	1	4	yes	—
456	C₄²	—	5	4	no	—
457	C₄²	—	5	8	no	—
458	C₃×C₅	C₂×#551	1	1	yes	characteristic, complemented, Hall
459	C₂²×C₃	—	2	16	no	—
460	C₂²×C₃	—	2	4	no	—
461	C₂²×C₃	—	2	4	no	—
462	C₂²×C₃	—	2	4	no	—
463	C₂²×C₃	—	2	4	no	—
464	C₂²×C₃	—	2	2	no	—
465	C₂²×C₃	—	2	2	no	—
466	C₂²×C₃	—	4	16	no	—
467	C₂²×C₃	—	4	4	no	—
468	C₂²×C₃	—	4	4	no	—
469	C₂²×C₃	C₂×#711	1	8	yes	—
470	C₂²×C₃	C₂×#711	1	4	yes	—
471	C₂²×C₃	#4144	1	2	yes	—
472	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
473	C₃×C₄	—	2	4	no	—
474	C₃×C₄	—	2	2	no	—
475	C₃×C₄	—	10	4	no	—
476	C₃×C₄	—	10	4	no	—
477	C₃×C₄	—	10	4	no	—
478	C₃×C₄	—	10	4	no	—
479	C₂×C₅	—	2	8	no	—
480	C₂×C₅	—	4	2	no	—
481	C₂×C₅	—	4	2	no	—
482	C₂×C₅	C₃×#551	1	4	yes	complemented
483	C₂×C₅	C₂²×C₃×#41	1	2	yes	—
484	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
485	C₂³	—	2	4	no	—
486	C₂³	—	2	4	no	—
487	C₂³	—	2	4	no	—
488	C₂³	—	2	16	no	—
489	C₂³	—	2	4	no	—
490	C₂³	—	2	2	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	4	4	no	—
493	C₂³	—	4	16	no	—
494	C₂³	C₃×#711	1	8	yes	—
495	C₂³	C₂³×C₃×D₅	1	2	yes	—
496	C₂³	C₂³×C₃×D₅	1	2	yes	—
497	C₂³	C₂³×C₃×D₅	1	4	yes	—
498	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
499	Q₈	—	20	4	no	—
500	D₄	—	2	16	no	—
501	D₄	—	4	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	5	4	no	—
505	C₂×C₄	—	5	4	no	—
506	C₂×C₄	—	5	4	no	—
507	C₂×C₄	—	5	4	no	—
508	C₂×C₄	—	10	16	no	—
509	C₂×C₄	—	10	4	no	—
510	C₂×C₄	—	10	4	no	—
511	C₂×C₄	—	10	4	no	—
512	C₂×C₄	—	10	4	no	—
513	C₂×C₄	—	20	4	no	—
514	C₂×C₄	—	20	8	no	—
515	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
516	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
517	C₂×C₃	—	2	8	no	—
518	C₂×C₃	—	4	2	no	—
519	C₂×C₃	—	4	2	no	—
520	C₂×C₃	#65921	1	4	yes	direct product factor
521	C₂×C₃	C₂²×#711	1	2	yes	—
522	C₂×C₃	C₂×#4144	1	1	yes	characteristic
523	C₅	C₂×C₃×#551	1	1	yes	characteristic, complemented, Sylow
524	C₂²	—	2	4	no	—
525	C₂²	—	2	4	no	—
526	C₂²	—	2	16	no	—
527	C₂²	—	2	4	no	—
528	C₂²	—	2	4	no	—
529	C₂²	—	2	2	no	—
530	C₂²	—	2	2	no	—
531	C₂²	—	4	16	no	—
532	C₂²	—	4	4	no	—
533	C₂²	—	4	4	no	—
534	C₂²	C₂×C₃×#711	1	8	yes	—
535	C₂²	C₂×C₃×#711	1	4	yes	—
536	C₂²	C₃×#4144	1	2	yes	—
537	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
538	C₄	—	2	4	no	—
539	C₄	—	2	2	no	—
540	C₄	—	10	4	no	—
541	C₄	—	10	4	no	—
542	C₄	—	10	4	no	—
543	C₄	—	10	4	no	—
544	C₃	C₂×#65921	1	1	yes	characteristic, direct product factor, Sylow
545	C₂	—	2	8	no	—
546	C₂	—	4	2	no	—
547	C₂	—	4	2	no	—
548	C₂	C₃×#65921	1	4	yes	direct product factor
549	C₂	C₂²×C₃×#711	1	2	yes	—
550	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, prime radical	C₂×C₃×C₅×#120	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₃×#120	D₅
hypocenter	C₅	C₂×C₃×#551
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
2. C₂²×C₃×C₂²⋊Dic₅	C₂	semidirect
3. C₂×C₃×#4082	C₂	semidirect
4. C₂×C₃×#4081	C₂	semidirect
5. C₂×C₃×#4081	C₂	semidirect
6. C₂×C₃×D₄×Dic₅	C₂	semidirect
7. C₃×#65921	C₂	direct (N × H)
8. C₂×C₃×#4025	C₂	semidirect
9. C₂×C₃×#4024	C₂	semidirect
10. C₂×C₃×#4023	C₂	semidirect
11. C₂×#65921	C₃	direct (N × H)
17. C₂³×C₃×Dic₅	C₂²	semidirect
18. C₂²×C₃×Dic₁₀	C₂²	semidirect
26. C₂×C₃×C₂²⋊Dic₅	C₂²	semidirect
27. C₂×C₃×C₂²⋊Dic₅	C₂²	semidirect
28. C₂×C₃×C₂²⋊Dic₅	C₂²	semidirect
29. C₃×#4082	C₂²	semidirect
30. C₃×#4081	C₂²	semidirect
31. C₃×#4081	C₂²	semidirect
32. C₃×D₄×Dic₅	C₂²	semidirect
33. C₂×C₃×C₄⋊Dic₅	C₂²	semidirect
34. C₂×C₃×#683	C₂²	semidirect
35. C₂×C₃×#683	C₂²	semidirect
36. C₂×C₃×#683	C₂²	semidirect
37. C₂×C₃×C₄×Dic₅	C₂²	semidirect
38. C₂×C₃×C₄×Dic₅	C₂²	semidirect
39. C₃×#4025	C₂²	semidirect
40. C₃×#4024	C₂²	semidirect
41. C₃×#4023	C₂²	semidirect
44. C₂²×C₂²⋊Dic₅	C₂×C₃	semidirect
45. C₂×#4082	C₂×C₃	semidirect
46. C₂×#4081	C₂×C₃	semidirect
47. C₂×#4081	C₂×C₃	semidirect
48. C₂×D₄×Dic₅	C₂×C₃	semidirect
49. #65921	C₂×C₃	direct (N × H)
50. C₂×#4025	C₂×C₃	semidirect
51. C₂×#4024	C₂×C₃	semidirect
52. C₂×#4023	C₂×C₃	semidirect
74. C₂²×C₃×Dic₅	C₂³	semidirect
76. C₂×C₃×Dic₁₀	C₂³	semidirect
83. C₃×C₂²⋊Dic₅	C₂³	semidirect
84. C₃×C₂²⋊Dic₅	C₂³	semidirect
85. C₃×C₄⋊Dic₅	C₂³	semidirect
86. C₃×#683	C₂³	semidirect
87. C₃×#683	C₂³	semidirect
88. C₃×#683	C₂³	semidirect
89. C₃×C₄×Dic₅	C₂³	semidirect
90. C₃×C₄×Dic₅	C₂³	semidirect
106. C₂³×Dic₅	C₂²×C₃	semidirect
107. C₂²×Dic₁₀	C₂²×C₃	semidirect
115. C₂×C₂²⋊Dic₅	C₂²×C₃	semidirect
116. C₂×C₂²⋊Dic₅	C₂²×C₃	semidirect
117. C₂×C₂²⋊Dic₅	C₂²×C₃	semidirect
118. #4082	C₂²×C₃	semidirect
119. #4081	C₂²×C₃	semidirect
120. #4081	C₂²×C₃	semidirect
121. D₄×Dic₅	C₂²×C₃	semidirect
122. C₂×C₄⋊Dic₅	C₂²×C₃	semidirect
123. C₂×#683	C₂²×C₃	semidirect
124. C₂×#683	C₂²×C₃	semidirect
125. C₂×#683	C₂²×C₃	semidirect
126. C₂×C₄×Dic₅	C₂²×C₃	semidirect
127. C₂×C₄×Dic₅	C₂²×C₃	semidirect
128. #4025	C₂²×C₃	semidirect
129. #4024	C₂²×C₃	semidirect
130. #4023	C₂²×C₃	semidirect
215. C₂²×Dic₅	C₂³×C₃	semidirect
217. C₂×Dic₁₀	C₂³×C₃	semidirect
224. C₂²⋊Dic₅	C₂³×C₃	semidirect
225. C₂²⋊Dic₅	C₂³×C₃	semidirect
226. C₄⋊Dic₅	C₂³×C₃	semidirect
227. #683	C₂³×C₃	semidirect
228. #683	C₂³×C₃	semidirect
229. #683	C₂³×C₃	semidirect
230. C₄×Dic₅	C₂³×C₃	semidirect
231. C₄×Dic₅	C₂³×C₃	semidirect
365. C₂×C₃×C₅	#551	semidirect
458. C₃×C₅	C₂×#551	semidirect
482. C₂×C₅	C₃×#551	semidirect
520. C₂×C₃	#65921	direct (N × H)
523. C₅	C₂×C₃×#551	semidirect
544. C₃	C₂×#65921	direct (N × H)
548. C₂	C₃×#65921	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
C₂²×C₃×C₂²⋊Dic₅	2
C₂×C₃×#4082	2
C₂×C₃×#4081	2
C₂×C₃×D₄×Dic₅	2
C₃×#65921	2
C₂×C₃×#4025	2
C₂×C₃×#4024	2
C₂×C₃×#4023	2
C₂×#65921	3
C₂×C₃×#551	5

Subgroups — C2×C3×#65921 #14483257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65921 #14483257