Subgroups — C₂×C₃×#66138 #14480294

Order 1920.

3968 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#4099	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#66138	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₃×#4085	C₂	1	4	yes	maximal, complemented
7	C₂×C₃×#4083	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#4081	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×#4075	C₂	1	2	yes	maximal, complemented
10	C₂×#66138	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
12	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic, complemented
13	C₂⁴×C₃×D₅	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₅×D₄	—	2	8	no	—
15	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
16	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	complemented
17	C₂²×C₃×C₅×D₄	C₂²	1	8	yes	complemented
18	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₃×C₂²⋊D₅	—	2	8	no	—
20	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
21	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	complemented
22	C₂²×C₃×C₂²⋊D₅	C₂²	1	4	yes	complemented
23	C₂²×C₃×C₂²⋊D₅	C₂²	1	8	yes	complemented
24	C₂³×C₃×Dic₅	C₂²	1	1	yes	characteristic, complemented
25	C₂³×C₃×Dic₅	C₂²	1	2	yes	complemented
26	C₃×#4099	C₂²	1	16	yes	complemented
27	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	complemented
28	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	complemented
29	C₂×C₃×C₂²⋊Dic₅	C₂²	1	4	yes	complemented
30	C₂×C₃×C₂²⋊Dic₅	C₂²	1	8	yes	complemented
31	C₃×#4085	C₂²	1	32	yes	complemented
32	C₃×#4083	C₂²	1	16	yes	complemented
33	C₃×#4081	C₂²	1	16	yes	complemented
34	C₃×#4075	C₂²	1	16	yes	complemented
35	C₂×C₃×#685	C₂²	1	4	yes	complemented
36	C₂×C₃×#683	C₂²	1	4	yes	complemented
37	C₂×C₃×#534	—	5	1	no	maximal, Hall
38	C₂³×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂³×C₂²⋊D₅	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×#4099	C₂×C₃	1	2	yes	complemented
41	C₂²×C₂²⋊Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
42	#66138	C₂×C₃	1	16	yes	direct product factor
43	C₂×#4085	C₂×C₃	1	4	yes	complemented
44	C₂×#4083	C₂×C₃	1	2	yes	complemented
45	C₂×#4081	C₂×C₃	1	2	yes	complemented
46	C₂×#4075	C₂×C₃	1	2	yes	complemented
47	C₂⁴×C₃×C₅	—	2	4	no	—
48	C₂⁴×C₃×C₅	—	4	4	no	—
49	C₂⁴×C₃×C₅	—	4	4	no	—
50	C₂⁴×C₃×C₅	D₄	1	8	yes	complemented
51	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
52	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
53	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
54	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
55	C₂⁴×C₃×C₅	C₂³	1	4	yes	—
56	C₂⁴×C₃×C₅	C₂³	1	4	yes	complemented
57	C₂³×C₃×D₅	—	2	4	no	—
58	C₂³×C₃×D₅	—	4	4	no	—
59	C₂³×C₃×D₅	C₂³	1	2	yes	—
60	C₂³×C₃×D₅	C₂³	1	4	yes	complemented
61	C₂×C₃×C₅×D₄	—	2	8	no	—
62	C₂×C₃×C₅×D₄	—	2	8	no	—
63	C₂×C₃×C₅×D₄	—	2	32	no	—
64	C₂×C₃×C₅×D₄	C₂³	1	8	yes	—
65	C₂×C₃×C₅×D₄	C₂³	1	8	yes	complemented
66	C₂²×C₃×C₄×C₅	—	2	4	no	—
67	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
68	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	complemented
69	C₂×C₃×C₂²⋊D₅	—	2	8	no	—
70	C₂×C₃×C₂²⋊D₅	—	2	8	no	—
71	C₂×C₃×C₂²⋊D₅	—	2	32	no	—
72	C₂×C₃×C₂²⋊D₅	—	4	8	no	—
73	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	—
74	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	complemented
75	C₂×C₃×C₂²⋊D₅	C₂³	1	16	yes	complemented
76	C₂²×C₃×Dic₅	—	2	4	no	—
77	C₂²×C₃×Dic₅	—	4	4	no	—
78	C₂²×C₃×Dic₅	C₂³	1	4	yes	—
79	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
80	C₂²×C₃×Dic₅	C₂³	1	4	yes	complemented
81	C₂²×C₃×Dic₅	C₂³	1	8	yes	complemented
82	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	complemented
83	C₃×C₂²⋊Dic₅	C₂³	1	16	yes	complemented
84	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	complemented
85	C₃×#685	C₂³	1	16	yes	complemented
86	C₃×#683	C₂³	1	16	yes	complemented
87	C₂³×C₃×D₄	—	5	1	no	—
88	C₂³×C₃×D₄	D₅	1	1	yes	characteristic, complemented
89	C₃×#534	—	5	16	no	—
90	C₂×C₃×#123	—	5	2	no	—
91	C₂×C₃×#123	—	5	2	no	—
92	C₂×C₃×#121	—	5	4	no	—
93	C₂×C₃×#120	—	5	2	no	—
94	C₂×C₃×#120	—	5	2	no	—
95	C₂²×C₃×C₂²⋊C₄	—	5	1	no	—
96	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
97	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
98	C₂⁴×D₅	C₂²×C₃	1	1	yes	characteristic, complemented
99	C₂²×C₅×D₄	—	2	8	no	—
100	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
101	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	complemented
102	C₂²×C₅×D₄	C₂²×C₃	1	8	yes	complemented
103	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
104	C₂²×C₂²⋊D₅	—	2	8	no	—
105	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
106	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	complemented
107	C₂²×C₂²⋊D₅	C₂²×C₃	1	4	yes	complemented
108	C₂²×C₂²⋊D₅	C₂²×C₃	1	8	yes	complemented
109	C₂³×Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
110	C₂³×Dic₅	C₂²×C₃	1	2	yes	complemented
111	#4099	C₂²×C₃	1	16	yes	complemented
112	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	complemented
113	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	complemented
114	C₂×C₂²⋊Dic₅	C₂²×C₃	1	4	yes	complemented
115	C₂×C₂²⋊Dic₅	C₂²×C₃	1	8	yes	complemented
116	#4085	C₂²×C₃	1	32	yes	complemented
117	#4083	C₂²×C₃	1	16	yes	complemented
118	#4081	C₂²×C₃	1	16	yes	complemented
119	#4075	C₂²×C₃	1	16	yes	complemented
120	C₂×#685	C₂²×C₃	1	4	yes	complemented
121	C₂×#683	C₂²×C₃	1	4	yes	complemented
122	C₂×#534	—	5	1	no	Sylow
123	C₂³×C₃×C₅	—	2	4	no	—
124	C₂³×C₃×C₅	—	2	4	no	—
125	C₂³×C₃×C₅	—	2	4	no	—
126	C₂³×C₃×C₅	—	2	4	no	—
127	C₂³×C₃×C₅	—	2	8	no	—
128	C₂³×C₃×C₅	—	2	8	no	—
129	C₂³×C₃×C₅	—	4	16	no	—
130	C₂³×C₃×C₅	—	4	8	no	—
131	C₂³×C₃×C₅	—	4	8	no	—
132	C₂³×C₃×C₅	—	4	16	no	—
133	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
134	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
135	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
136	C₂³×C₃×C₅	C₂×D₄	1	16	yes	complemented
137	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
138	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
139	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
140	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
141	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
142	C₂²×C₃×D₅	—	2	4	no	—
143	C₂²×C₃×D₅	—	2	8	no	—
144	C₂²×C₃×D₅	—	2	8	no	—
145	C₂²×C₃×D₅	—	4	8	no	—
146	C₂²×C₃×D₅	—	4	16	no	—
147	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
148	C₃×C₅×D₄	—	2	16	no	—
149	C₃×C₅×D₄	—	2	16	no	—
150	C₂×C₃×C₄×C₅	—	2	4	no	—
151	C₂×C₃×C₄×C₅	—	2	8	no	—
152	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
153	C₃×C₂²⋊D₅	—	2	16	no	—
154	C₃×C₂²⋊D₅	—	2	16	no	—
155	C₃×C₂²⋊D₅	—	4	16	no	—
156	C₂×C₃×Dic₅	—	2	4	no	—
157	C₂×C₃×Dic₅	—	2	8	no	—
158	C₂×C₃×Dic₅	—	2	8	no	—
159	C₂×C₃×Dic₅	—	4	8	no	—
160	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
161	C₂×C₃×Dic₅	C₂⁴	1	8	yes	—
162	C₂⁵×C₃	—	5	1	no	—
163	C₂⁵×C₃	C₂×D₅	1	1	yes	characteristic
164	C₂⁵×C₃	C₂×D₅	1	1	yes	characteristic, complemented
165	C₂²×C₃×D₄	—	2	8	no	—
166	C₂²×C₃×D₄	—	5	2	no	—
167	C₂²×C₃×D₄	—	5	2	no	—
168	C₂²×C₃×D₄	—	5	8	no	—
169	C₂²×C₃×D₄	—	5	4	no	—
170	C₂²×C₃×D₄	—	10	8	no	—
171	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
172	C₂²×C₃×D₄	C₂×D₅	1	2	yes	complemented
173	C₂²×C₃×D₄	C₂×D₅	1	8	yes	complemented
174	C₂³×C₃×C₄	—	5	1	no	—
175	C₂³×C₃×C₄	—	5	2	no	—
176	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic, complemented
177	C₃×#123	—	5	16	no	—
178	C₃×#123	—	5	16	no	—
179	C₃×#121	—	5	32	no	—
180	C₃×#120	—	5	16	no	—
181	C₃×#120	—	5	16	no	—
182	C₂×C₃×C₄⋊C₄	—	5	4	no	—
183	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
184	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
185	C₂×C₃×C₂²⋊C₄	—	5	8	no	—
186	C₂×C₃×C₂²⋊C₄	—	5	4	no	—
187	C₂×C₃×C₂²⋊C₄	—	5	4	no	—
188	C₂⁴×C₅	—	2	4	no	—
189	C₂⁴×C₅	—	4	4	no	—
190	C₂⁴×C₅	—	4	4	no	—
191	C₂⁴×C₅	C₃×D₄	1	8	yes	complemented
192	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
193	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
194	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
195	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
196	C₂⁴×C₅	C₂³×C₃	1	4	yes	—
197	C₂⁴×C₅	C₂³×C₃	1	4	yes	complemented
198	C₂³×D₅	—	2	4	no	—
199	C₂³×D₅	—	4	4	no	—
200	C₂³×D₅	C₂³×C₃	1	2	yes	—
201	C₂³×D₅	C₂³×C₃	1	4	yes	complemented
202	C₂×C₅×D₄	—	2	8	no	—
203	C₂×C₅×D₄	—	2	8	no	—
204	C₂×C₅×D₄	—	2	32	no	—
205	C₂×C₅×D₄	C₂³×C₃	1	8	yes	—
206	C₂×C₅×D₄	C₂³×C₃	1	8	yes	complemented
207	C₂²×C₄×C₅	—	2	4	no	—
208	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
209	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	complemented
210	C₂×C₂²⋊D₅	—	2	8	no	—
211	C₂×C₂²⋊D₅	—	2	8	no	—
212	C₂×C₂²⋊D₅	—	2	32	no	—
213	C₂×C₂²⋊D₅	—	4	8	no	—
214	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	—
215	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	complemented
216	C₂×C₂²⋊D₅	C₂³×C₃	1	16	yes	complemented
217	C₂²×Dic₅	—	2	4	no	—
218	C₂²×Dic₅	—	4	4	no	—
219	C₂²×Dic₅	C₂³×C₃	1	4	yes	—
220	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
221	C₂²×Dic₅	C₂³×C₃	1	4	yes	complemented
222	C₂²×Dic₅	C₂³×C₃	1	8	yes	complemented
223	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	complemented
224	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	complemented
225	C₂²⋊Dic₅	C₂³×C₃	1	16	yes	complemented
226	#685	C₂³×C₃	1	16	yes	complemented
227	#683	C₂³×C₃	1	16	yes	complemented
228	C₂³×D₄	—	5	1	no	—
229	C₂³×D₄	C₃×D₅	1	1	yes	characteristic, complemented
230	#534	—	5	16	no	—
231	C₂×#123	—	5	2	no	—
232	C₂×#123	—	5	2	no	—
233	C₂×#121	—	5	4	no	—
234	C₂×#120	—	5	2	no	—
235	C₂×#120	—	5	2	no	—
236	C₂²×C₂²⋊C₄	—	5	1	no	—
237	C₂²×C₃×C₅	—	2	8	no	—
238	C₂²×C₃×C₅	—	2	8	no	—
239	C₂²×C₃×C₅	—	2	8	no	—
240	C₂²×C₃×C₅	—	2	4	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	—	2	8	no	—
245	C₂²×C₃×C₅	—	4	16	no	—
246	C₂²×C₃×C₅	—	4	16	no	—
247	C₂²×C₃×C₅	—	4	8	no	—
248	C₂²×C₃×C₅	C₂²×D₄	1	4	yes	—
249	C₂²×C₃×C₅	C₂²×D₄	1	4	yes	—
250	C₂²×C₃×C₅	C₂²×D₄	1	4	yes	—
251	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
252	C₂²×C₃×C₅	H(5,𝔽₂)	1	4	yes	—
253	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
254	C₂×C₃×D₅	—	2	4	no	—
255	C₂×C₃×D₅	—	2	8	no	—
256	C₂×C₃×D₅	—	4	16	no	—
257	C₂×C₃×D₅	—	4	8	no	—
258	C₃×C₄×C₅	—	2	4	no	—
259	C₃×Dic₅	—	2	4	no	—
260	C₃×Dic₅	—	2	8	no	—
261	C₂⁴×C₃	—	2	4	no	—
262	C₂⁴×C₃	—	4	4	no	—
263	C₂⁴×C₃	—	4	4	no	—
264	C₂⁴×C₃	—	5	2	no	—
265	C₂⁴×C₃	—	5	4	no	—
266	C₂⁴×C₃	—	10	4	no	—
267	C₂⁴×C₃	—	20	4	no	—
268	C₂⁴×C₃	C₂²⋊D₅	1	8	yes	complemented
269	C₂⁴×C₃	C₂²×D₅	1	2	yes	—
270	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
271	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
272	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
273	C₂⁴×C₃	C₂²×D₅	1	4	yes	—
274	C₂⁴×C₃	C₂²×D₅	1	4	yes	complemented
275	C₂×C₃×D₄	—	2	8	no	—
276	C₂×C₃×D₄	—	2	8	no	—
277	C₂×C₃×D₄	—	2	32	no	—
278	C₂×C₃×D₄	—	5	8	no	—
279	C₂×C₃×D₄	—	5	8	no	—
280	C₂×C₃×D₄	—	5	16	no	—
281	C₂×C₃×D₄	—	10	8	no	—
282	C₂×C₃×D₄	—	10	8	no	—
283	C₂×C₃×D₄	—	10	32	no	—
284	C₂×C₃×D₄	—	20	8	no	—
285	C₂×C₃×D₄	C₂²×D₅	1	8	yes	—
286	C₂×C₃×D₄	C₂²×D₅	1	8	yes	complemented
287	C₂²×C₃×C₄	—	2	4	no	—
288	C₂²×C₃×C₄	—	5	4	no	—
289	C₂²×C₃×C₄	—	5	4	no	—
290	C₂²×C₃×C₄	—	5	2	no	—
291	C₂²×C₃×C₄	—	5	8	no	—
292	C₂²×C₃×C₄	—	10	4	no	—
293	C₂²×C₃×C₄	—	20	4	no	—
294	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
295	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	complemented
296	C₃×C₄⋊C₄	—	5	16	no	—
297	C₃×C₂²⋊C₄	—	5	8	no	—
298	C₃×C₂²⋊C₄	—	5	8	no	—
299	C₃×C₂²⋊C₄	—	5	16	no	—
300	C₃×C₂²⋊C₄	—	5	16	no	—
301	C₂³×C₅	—	2	4	no	—
302	C₂³×C₅	—	2	4	no	—
303	C₂³×C₅	—	2	4	no	—
304	C₂³×C₅	—	2	4	no	—
305	C₂³×C₅	—	2	8	no	—
306	C₂³×C₅	—	2	8	no	—
307	C₂³×C₅	—	4	16	no	—
308	C₂³×C₅	—	4	8	no	—
309	C₂³×C₅	—	4	8	no	—
310	C₂³×C₅	—	4	16	no	—
311	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
312	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
313	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
314	C₂³×C₅	C₂×C₃×D₄	1	16	yes	complemented
315	C₂³×C₅	C₂⁴×C₃	1	4	yes	—
316	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
317	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
318	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
319	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
320	C₂²×D₅	—	2	4	no	—
321	C₂²×D₅	—	2	8	no	—
322	C₂²×D₅	—	2	8	no	—
323	C₂²×D₅	—	4	8	no	—
324	C₂²×D₅	—	4	16	no	—
325	C₂²×D₅	C₂⁴×C₃	1	4	yes	—
326	C₅×D₄	—	2	16	no	—
327	C₅×D₄	—	2	16	no	—
328	C₂×C₄×C₅	—	2	4	no	—
329	C₂×C₄×C₅	—	2	8	no	—
330	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
331	C₂²⋊D₅	—	2	16	no	—
332	C₂²⋊D₅	—	2	16	no	—
333	C₂²⋊D₅	—	4	16	no	—
334	C₂×Dic₅	—	2	4	no	—
335	C₂×Dic₅	—	2	8	no	—
336	C₂×Dic₅	—	2	8	no	—
337	C₂×Dic₅	—	4	8	no	—
338	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
339	C₂×Dic₅	C₂⁴×C₃	1	8	yes	—
340	C₂⁵	—	5	1	no	—
341	C₂⁵	C₂×C₃×D₅	1	1	yes	characteristic
342	C₂⁵	C₂×C₃×D₅	1	1	yes	characteristic, complemented
343	C₂²×D₄	—	2	8	no	—
344	C₂²×D₄	—	5	2	no	—
345	C₂²×D₄	—	5	2	no	—
346	C₂²×D₄	—	5	8	no	—
347	C₂²×D₄	—	5	4	no	—
348	C₂²×D₄	—	10	8	no	—
349	C₂²×D₄	C₂×C₃×D₅	1	2	yes	—
350	C₂²×D₄	C₂×C₃×D₅	1	2	yes	complemented
351	C₂²×D₄	C₂×C₃×D₅	1	8	yes	complemented
352	C₂³×C₄	—	5	1	no	—
353	C₂³×C₄	—	5	2	no	—
354	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic, complemented
355	#123	—	5	16	no	—
356	#123	—	5	16	no	—
357	#121	—	5	32	no	—
358	#120	—	5	16	no	—
359	#120	—	5	16	no	—
360	C₂×C₄⋊C₄	—	5	4	no	—
361	C₂×C₂²⋊C₄	—	5	2	no	—
362	C₂×C₂²⋊C₄	—	5	2	no	—
363	C₂×C₂²⋊C₄	—	5	8	no	—
364	C₂×C₂²⋊C₄	—	5	4	no	—
365	C₂×C₂²⋊C₄	—	5	4	no	—
366	C₂×C₃×C₅	—	2	4	no	—
367	C₂×C₃×C₅	—	2	4	no	—
368	C₂×C₃×C₅	—	2	4	no	—
369	C₂×C₃×C₅	—	4	4	no	—
370	C₂×C₃×C₅	#534	1	4	yes	complemented
371	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
372	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
373	C₃×D₅	—	4	4	no	—
374	C₂³×C₃	—	2	4	no	—
375	C₂³×C₃	—	2	4	no	—
376	C₂³×C₃	—	2	4	no	—
377	C₂³×C₃	—	2	8	no	—
378	C₂³×C₃	—	2	4	no	—
379	C₂³×C₃	—	2	8	no	—
380	C₂³×C₃	—	4	16	no	—
381	C₂³×C₃	—	4	8	no	—
382	C₂³×C₃	—	4	8	no	—
383	C₂³×C₃	—	4	16	no	—
384	C₂³×C₃	—	5	4	no	—
385	C₂³×C₃	—	10	8	no	—
386	C₂³×C₃	—	10	8	no	—
387	C₂³×C₃	—	10	4	no	—
388	C₂³×C₃	—	20	8	no	—
389	C₂³×C₃	—	20	16	no	—
390	C₂³×C₃	C₂×C₂²⋊D₅	1	4	yes	—
391	C₂³×C₃	C₂×C₂²⋊D₅	1	4	yes	—
392	C₂³×C₃	C₂×C₂²⋊D₅	1	4	yes	—
393	C₂³×C₃	C₂×C₂²⋊D₅	1	16	yes	complemented
394	C₂³×C₃	C₂³×D₅	1	4	yes	—
395	C₂³×C₃	C₂³×D₅	1	2	yes	—
396	C₂³×C₃	C₂³×D₅	1	2	yes	—
397	C₂³×C₃	C₂³×D₅	1	2	yes	—
398	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
399	C₃×D₄	—	2	16	no	—
400	C₃×D₄	—	2	16	no	—
401	C₃×D₄	—	10	16	no	—
402	C₃×D₄	—	10	16	no	—
403	C₃×D₄	—	20	16	no	—
404	C₂×C₃×C₄	—	2	4	no	—
405	C₂×C₃×C₄	—	2	8	no	—
406	C₂×C₃×C₄	—	5	8	no	—
407	C₂×C₃×C₄	—	5	4	no	—
408	C₂×C₃×C₄	—	10	8	no	—
409	C₂×C₃×C₄	—	10	8	no	—
410	C₂×C₃×C₄	—	10	4	no	—
411	C₂×C₃×C₄	—	20	8	no	—
412	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
413	C₂²×C₅	—	2	4	no	—
414	C₂²×C₅	—	2	4	no	—
415	C₂²×C₅	—	2	4	no	—
416	C₂²×C₅	—	2	8	no	—
417	C₂²×C₅	—	2	8	no	—
418	C₂²×C₅	—	2	8	no	—
419	C₂²×C₅	—	2	4	no	—
420	C₂²×C₅	—	2	8	no	—
421	C₂²×C₅	—	4	16	no	—
422	C₂²×C₅	—	4	16	no	—
423	C₂²×C₅	—	4	8	no	—
424	C₂²×C₅	C₂²×C₃×D₄	1	4	yes	—
425	C₂²×C₅	C₂²×C₃×D₄	1	4	yes	—
426	C₂²×C₅	C₂²×C₃×D₄	1	4	yes	—
427	C₂²×C₅	C₂²×C₃×D₄	1	2	yes	—
428	C₂²×C₅	C₃×H(5,𝔽₂)	1	4	yes	—
429	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
430	C₂×D₅	—	2	4	no	—
431	C₂×D₅	—	2	8	no	—
432	C₂×D₅	—	4	16	no	—
433	C₂×D₅	—	4	8	no	—
434	C₄×C₅	—	2	4	no	—
435	Dic₅	—	2	4	no	—
436	Dic₅	—	2	8	no	—
437	C₂⁴	—	2	4	no	—
438	C₂⁴	—	4	4	no	—
439	C₂⁴	—	4	4	no	—
440	C₂⁴	—	5	2	no	—
441	C₂⁴	—	5	4	no	—
442	C₂⁴	—	10	4	no	—
443	C₂⁴	—	20	4	no	—
444	C₂⁴	C₃×C₂²⋊D₅	1	8	yes	complemented
445	C₂⁴	C₂²×C₃×D₅	1	2	yes	—
446	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
447	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
448	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
449	C₂⁴	C₂²×C₃×D₅	1	4	yes	—
450	C₂⁴	C₂²×C₃×D₅	1	4	yes	complemented
451	C₂×D₄	—	2	8	no	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	2	32	no	—
454	C₂×D₄	—	5	8	no	—
455	C₂×D₄	—	5	8	no	—
456	C₂×D₄	—	5	16	no	—
457	C₂×D₄	—	10	8	no	—
458	C₂×D₄	—	10	8	no	—
459	C₂×D₄	—	10	32	no	—
460	C₂×D₄	—	20	8	no	—
461	C₂×D₄	C₂²×C₃×D₅	1	8	yes	—
462	C₂×D₄	C₂²×C₃×D₅	1	8	yes	complemented
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	5	4	no	—
465	C₂²×C₄	—	5	2	no	—
466	C₂²×C₄	—	5	4	no	—
467	C₂²×C₄	—	5	8	no	—
468	C₂²×C₄	—	10	4	no	—
469	C₂²×C₄	—	20	4	no	—
470	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
471	C₂²×C₄	C₂²×C₃×D₅	1	4	yes	complemented
472	C₄⋊C₄	—	5	16	no	—
473	C₂²⋊C₄	—	5	8	no	—
474	C₂²⋊C₄	—	5	8	no	—
475	C₂²⋊C₄	—	5	16	no	—
476	C₂²⋊C₄	—	5	16	no	—
477	C₃×C₅	C₂×#534	1	1	yes	characteristic, complemented, Hall
478	C₂²×C₃	—	2	8	no	—
479	C₂²×C₃	—	2	8	no	—
480	C₂²×C₃	—	2	8	no	—
481	C₂²×C₃	—	2	4	no	—
482	C₂²×C₃	—	2	4	no	—
483	C₂²×C₃	—	2	4	no	—
484	C₂²×C₃	—	2	8	no	—
485	C₂²×C₃	—	2	4	no	—
486	C₂²×C₃	—	4	16	no	—
487	C₂²×C₃	—	4	16	no	—
488	C₂²×C₃	—	4	8	no	—
489	C₂²×C₃	—	10	8	no	—
490	C₂²×C₃	—	10	4	no	—
491	C₂²×C₃	—	20	16	no	—
492	C₂²×C₃	—	20	8	no	—
493	C₂²×C₃	#4144	1	4	yes	—
494	C₂²×C₃	C₂²×C₂²⋊D₅	1	4	yes	—
495	C₂²×C₃	C₂²×C₂²⋊D₅	1	4	yes	—
496	C₂²×C₃	C₂²×C₂²⋊D₅	1	4	yes	—
497	C₂²×C₃	C₂²×C₂²⋊D₅	1	2	yes	—
498	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
499	C₃×C₄	—	2	4	no	—
500	C₃×C₄	—	10	8	no	—
501	C₃×C₄	—	10	4	no	—
502	C₂×C₅	—	2	4	no	—
503	C₂×C₅	—	2	4	no	—
504	C₂×C₅	—	2	4	no	—
505	C₂×C₅	—	4	4	no	—
506	C₂×C₅	C₃×#534	1	4	yes	complemented
507	C₂×C₅	C₂³×C₃×D₄	1	1	yes	characteristic
508	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	2	yes	—
509	D₅	—	4	4	no	—
510	C₂³	—	2	4	no	—
511	C₂³	—	2	4	no	—
512	C₂³	—	2	4	no	—
513	C₂³	—	2	8	no	—
514	C₂³	—	2	4	no	—
515	C₂³	—	2	8	no	—
516	C₂³	—	4	16	no	—
517	C₂³	—	4	8	no	—
518	C₂³	—	4	8	no	—
519	C₂³	—	4	16	no	—
520	C₂³	—	5	4	no	—
521	C₂³	—	10	8	no	—
522	C₂³	—	10	8	no	—
523	C₂³	—	10	4	no	—
524	C₂³	—	20	8	no	—
525	C₂³	—	20	16	no	—
526	C₂³	C₂×C₃×C₂²⋊D₅	1	4	yes	—
527	C₂³	C₂×C₃×C₂²⋊D₅	1	4	yes	—
528	C₂³	C₂×C₃×C₂²⋊D₅	1	4	yes	—
529	C₂³	C₂×C₃×C₂²⋊D₅	1	16	yes	complemented
530	C₂³	C₂³×C₃×D₅	1	4	yes	—
531	C₂³	C₂³×C₃×D₅	1	2	yes	—
532	C₂³	C₂³×C₃×D₅	1	2	yes	—
533	C₂³	C₂³×C₃×D₅	1	2	yes	—
534	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
535	D₄	—	2	16	no	—
536	D₄	—	2	16	no	—
537	D₄	—	10	16	no	—
538	D₄	—	10	16	no	—
539	D₄	—	20	16	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	5	8	no	—
543	C₂×C₄	—	5	4	no	—
544	C₂×C₄	—	10	8	no	—
545	C₂×C₄	—	10	8	no	—
546	C₂×C₄	—	10	4	no	—
547	C₂×C₄	—	20	8	no	—
548	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
549	C₂×C₃	—	2	4	no	—
550	C₂×C₃	—	2	4	no	—
551	C₂×C₃	—	2	4	no	—
552	C₂×C₃	—	4	4	no	—
553	C₂×C₃	—	20	4	no	—
554	C₂×C₃	#66138	1	4	yes	direct product factor
555	C₂×C₃	C₂×#4144	1	2	yes	—
556	C₂×C₃	C₂³×C₂²⋊D₅	1	1	yes	characteristic
557	C₅	C₂×C₃×#534	1	1	yes	characteristic, complemented, Sylow
558	C₂²	—	2	8	no	—
559	C₂²	—	2	8	no	—
560	C₂²	—	2	8	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	2	8	no	—
565	C₂²	—	2	4	no	—
566	C₂²	—	4	16	no	—
567	C₂²	—	4	16	no	—
568	C₂²	—	4	8	no	—
569	C₂²	—	10	8	no	—
570	C₂²	—	10	4	no	—
571	C₂²	—	20	16	no	—
572	C₂²	—	20	8	no	—
573	C₂²	C₃×#4144	1	4	yes	—
574	C₂²	C₂²×C₃×C₂²⋊D₅	1	4	yes	—
575	C₂²	C₂²×C₃×C₂²⋊D₅	1	4	yes	—
576	C₂²	C₂²×C₃×C₂²⋊D₅	1	4	yes	—
577	C₂²	C₂²×C₃×C₂²⋊D₅	1	2	yes	—
578	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
579	C₄	—	2	4	no	—
580	C₄	—	10	8	no	—
581	C₄	—	10	4	no	—
582	C₃	C₂×#66138	1	1	yes	characteristic, direct product factor, Sylow
583	C₂	—	2	4	no	—
584	C₂	—	2	4	no	—
585	C₂	—	2	4	no	—
586	C₂	—	4	4	no	—
587	C₂	—	20	4	no	—
588	C₂	C₃×#66138	1	4	yes	direct product factor
589	C₂	C₂×C₃×#4144	1	2	yes	—
590	C₂	C₂³×C₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₅
Fitting	C₂³×C₃×C₅×D₄	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂³×C₃×D₄	D₅
hypocenter	C₅	C₂×C₃×#534
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅×D₄	C₂	semidirect
2. C₂³×C₃×C₂²⋊D₅	C₂	semidirect
3. C₂×C₃×#4099	C₂	semidirect
4. C₂²×C₃×C₂²⋊Dic₅	C₂	semidirect
5. C₃×#66138	C₂	direct (N × H)
6. C₂×C₃×#4085	C₂	semidirect
7. C₂×C₃×#4083	C₂	semidirect
8. C₂×C₃×#4081	C₂	semidirect
9. C₂×C₃×#4075	C₂	semidirect
10. C₂×#66138	C₃	direct (N × H)
12. C₂⁵×C₃×C₅	C₂²	semidirect
13. C₂⁴×C₃×D₅	C₂²	semidirect
16. C₂²×C₃×C₅×D₄	C₂²	semidirect
17. C₂²×C₃×C₅×D₄	C₂²	semidirect
18. C₂³×C₃×C₄×C₅	C₂²	semidirect
21. C₂²×C₃×C₂²⋊D₅	C₂²	semidirect
22. C₂²×C₃×C₂²⋊D₅	C₂²	semidirect
23. C₂²×C₃×C₂²⋊D₅	C₂²	semidirect
24. C₂³×C₃×Dic₅	C₂²	semidirect
25. C₂³×C₃×Dic₅	C₂²	semidirect
26. C₃×#4099	C₂²	semidirect
27. C₂×C₃×C₂²⋊Dic₅	C₂²	semidirect
28. C₂×C₃×C₂²⋊Dic₅	C₂²	semidirect
29. C₂×C₃×C₂²⋊Dic₅	C₂²	semidirect
30. C₂×C₃×C₂²⋊Dic₅	C₂²	semidirect
31. C₃×#4085	C₂²	semidirect
32. C₃×#4083	C₂²	semidirect
33. C₃×#4081	C₂²	semidirect
34. C₃×#4075	C₂²	semidirect
35. C₂×C₃×#685	C₂²	semidirect
36. C₂×C₃×#683	C₂²	semidirect
38. C₂³×C₅×D₄	C₂×C₃	semidirect
39. C₂³×C₂²⋊D₅	C₂×C₃	semidirect
40. C₂×#4099	C₂×C₃	semidirect
41. C₂²×C₂²⋊Dic₅	C₂×C₃	semidirect
42. #66138	C₂×C₃	direct (N × H)
43. C₂×#4085	C₂×C₃	semidirect
44. C₂×#4083	C₂×C₃	semidirect
45. C₂×#4081	C₂×C₃	semidirect
46. C₂×#4075	C₂×C₃	semidirect
50. C₂⁴×C₃×C₅	D₄	semidirect
56. C₂⁴×C₃×C₅	C₂³	semidirect
60. C₂³×C₃×D₅	C₂³	semidirect
65. C₂×C₃×C₅×D₄	C₂³	semidirect
68. C₂²×C₃×C₄×C₅	C₂³	semidirect
74. C₂×C₃×C₂²⋊D₅	C₂³	semidirect
75. C₂×C₃×C₂²⋊D₅	C₂³	semidirect
80. C₂²×C₃×Dic₅	C₂³	semidirect
81. C₂²×C₃×Dic₅	C₂³	semidirect
82. C₃×C₂²⋊Dic₅	C₂³	semidirect
83. C₃×C₂²⋊Dic₅	C₂³	semidirect
84. C₃×C₂²⋊Dic₅	C₂³	semidirect
85. C₃×#685	C₂³	semidirect
86. C₃×#683	C₂³	semidirect
88. C₂³×C₃×D₄	D₅	semidirect
97. C₂⁵×C₅	C₂²×C₃	semidirect
98. C₂⁴×D₅	C₂²×C₃	semidirect
101. C₂²×C₅×D₄	C₂²×C₃	semidirect
102. C₂²×C₅×D₄	C₂²×C₃	semidirect
103. C₂³×C₄×C₅	C₂²×C₃	semidirect
106. C₂²×C₂²⋊D₅	C₂²×C₃	semidirect
107. C₂²×C₂²⋊D₅	C₂²×C₃	semidirect
108. C₂²×C₂²⋊D₅	C₂²×C₃	semidirect
109. C₂³×Dic₅	C₂²×C₃	semidirect
110. C₂³×Dic₅	C₂²×C₃	semidirect
111. #4099	C₂²×C₃	semidirect
112. C₂×C₂²⋊Dic₅	C₂²×C₃	semidirect
113. C₂×C₂²⋊Dic₅	C₂²×C₃	semidirect
114. C₂×C₂²⋊Dic₅	C₂²×C₃	semidirect
115. C₂×C₂²⋊Dic₅	C₂²×C₃	semidirect
116. #4085	C₂²×C₃	semidirect
117. #4083	C₂²×C₃	semidirect
118. #4081	C₂²×C₃	semidirect
119. #4075	C₂²×C₃	semidirect
120. C₂×#685	C₂²×C₃	semidirect
121. C₂×#683	C₂²×C₃	semidirect
136. C₂³×C₃×C₅	C₂×D₄	semidirect
164. C₂⁵×C₃	C₂×D₅	semidirect
172. C₂²×C₃×D₄	C₂×D₅	semidirect
173. C₂²×C₃×D₄	C₂×D₅	semidirect
176. C₂³×C₃×C₄	C₂×D₅	semidirect
191. C₂⁴×C₅	C₃×D₄	semidirect
197. C₂⁴×C₅	C₂³×C₃	semidirect
201. C₂³×D₅	C₂³×C₃	semidirect
206. C₂×C₅×D₄	C₂³×C₃	semidirect
209. C₂²×C₄×C₅	C₂³×C₃	semidirect
215. C₂×C₂²⋊D₅	C₂³×C₃	semidirect
216. C₂×C₂²⋊D₅	C₂³×C₃	semidirect
221. C₂²×Dic₅	C₂³×C₃	semidirect
222. C₂²×Dic₅	C₂³×C₃	semidirect
223. C₂²⋊Dic₅	C₂³×C₃	semidirect
224. C₂²⋊Dic₅	C₂³×C₃	semidirect
225. C₂²⋊Dic₅	C₂³×C₃	semidirect
226. #685	C₂³×C₃	semidirect
227. #683	C₂³×C₃	semidirect
229. C₂³×D₄	C₃×D₅	semidirect
268. C₂⁴×C₃	C₂²⋊D₅	semidirect
274. C₂⁴×C₃	C₂²×D₅	semidirect
286. C₂×C₃×D₄	C₂²×D₅	semidirect
295. C₂²×C₃×C₄	C₂²×D₅	semidirect
314. C₂³×C₅	C₂×C₃×D₄	semidirect
342. C₂⁵	C₂×C₃×D₅	semidirect
350. C₂²×D₄	C₂×C₃×D₅	semidirect
351. C₂²×D₄	C₂×C₃×D₅	semidirect
354. C₂³×C₄	C₂×C₃×D₅	semidirect
370. C₂×C₃×C₅	#534	semidirect
393. C₂³×C₃	C₂×C₂²⋊D₅	semidirect
444. C₂⁴	C₃×C₂²⋊D₅	semidirect
450. C₂⁴	C₂²×C₃×D₅	semidirect
462. C₂×D₄	C₂²×C₃×D₅	semidirect
471. C₂²×C₄	C₂²×C₃×D₅	semidirect
477. C₃×C₅	C₂×#534	semidirect
506. C₂×C₅	C₃×#534	semidirect
529. C₂³	C₂×C₃×C₂²⋊D₅	semidirect
554. C₂×C₃	#66138	direct (N × H)
557. C₅	C₂×C₃×#534	semidirect
582. C₃	C₂×#66138	direct (N × H)
588. C₂	C₃×#66138	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃×C₅×D₄	2
C₂³×C₃×C₂²⋊D₅	2
C₂×C₃×#4099	2
C₂²×C₃×C₂²⋊Dic₅	2
C₃×#66138	2
C₂×C₃×#4085	2
C₂×C₃×#4083	2
C₂×C₃×#4081	2
C₂×C₃×#4075	2
C₂×#66138	3
C₂×C₃×#534	5

Subgroups — C2×C3×#66138 #14480294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#66138 #14480294