Subgroups — C₂×C₃×#65848 #14483608

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	C₂²×C₃×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4076	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#4076	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×C₄×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
6	C₃×#65848	C₂	1	16	yes	maximal, direct product factor
7	C₂×C₃×#4044	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#4041	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#4040	C₂	1	2	yes	maximal, complemented
10	C₂×C₃×#4039	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#4038	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃×#4036	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#65848	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
16	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂²×C₃×C₂²⋊D₅	C₂²	1	1	yes	characteristic
18	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
19	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
20	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
21	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	—
22	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	8	yes	complemented
26	C₂×C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
27	C₃×#4076	C₂²	1	8	yes	complemented
28	C₃×#4076	C₂²	1	16	yes	complemented
29	C₃×C₄×C₂²⋊D₅	C₂²	1	8	yes	complemented
30	C₃×C₄×C₂²⋊D₅	C₂²	1	16	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	C₂×C₃×#685	C₂²	1	1	yes	characteristic
35	C₂×C₃×#685	C₂²	1	2	yes	—
36	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
37	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
38	C₂×C₃×#683	C₂²	1	1	yes	characteristic
39	C₂×C₃×#683	C₂²	1	2	yes	—
40	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
41	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
42	C₃×#4044	C₂²	1	16	yes	complemented
43	C₃×#4041	C₂²	1	8	yes	complemented
44	C₃×#4040	C₂²	1	16	yes	complemented
45	C₃×#4039	C₂²	1	8	yes	complemented
46	C₃×#4038	C₂²	1	8	yes	complemented
47	C₃×#4036	C₂²	1	8	yes	complemented
48	C₂×C₃×#553	—	5	1	no	maximal, Hall
49	C₂²×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂×#4076	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂×#4076	C₂×C₃	1	2	yes	complemented
52	C₂×C₄×C₂²⋊D₅	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂×C₄×C₂²⋊D₅	C₂×C₃	1	2	yes	complemented
54	#65848	C₂×C₃	1	16	yes	direct product factor
55	C₂×#4044	C₂×C₃	1	2	yes	complemented
56	C₂×#4041	C₂×C₃	1	1	yes	characteristic, complemented
57	C₂×#4040	C₂×C₃	1	2	yes	complemented
58	C₂×#4039	C₂×C₃	1	1	yes	characteristic, complemented
59	C₂×#4038	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂×#4036	C₂×C₃	1	1	yes	characteristic, complemented
61	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
62	C₂³×C₃×D₅	C₂³	1	1	yes	characteristic
63	C₂³×C₃×D₅	C₂³	1	2	yes	—
64	C₂²×C₃×C₄×C₅	—	2	4	no	—
65	C₂²×C₃×C₄×C₅	—	4	4	no	—
66	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
67	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
68	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
70	C₂²×C₃×C₄×C₅	C₂³	1	8	yes	—
71	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
72	C₂×C₃×C₂²⋊D₅	—	2	8	no	—
73	C₂×C₃×C₂²⋊D₅	—	2	4	no	—
74	C₂×C₃×C₂²⋊D₅	C₂³	1	4	yes	—
75	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	—
76	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
77	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
78	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
79	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
80	C₂×C₃×D₂₀	—	2	4	no	—
81	C₂×C₃×D₂₀	—	4	4	no	—
82	C₂×C₃×D₂₀	C₂³	1	4	yes	—
83	C₂×C₃×D₂₀	C₂³	1	8	yes	—
84	C₂×C₃×C₄×D₅	—	2	4	no	—
85	C₂×C₃×C₄×D₅	—	4	4	no	—
86	C₂×C₃×C₄×D₅	C₂³	1	4	yes	—
87	C₂×C₃×C₄×D₅	C₂³	1	8	yes	—
88	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
89	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
90	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
91	C₃×C₂²⋊Dic₅	C₂³	1	4	yes	—
92	C₃×#685	C₂³	1	8	yes	—
93	C₃×#685	C₂³	1	8	yes	—
94	C₃×#685	C₂³	1	4	yes	—
95	C₃×#685	C₂³	1	8	yes	—
96	C₃×#685	C₂³	1	8	yes	—
97	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
98	C₃×C₄⋊Dic₅	C₂³	1	8	yes	—
99	C₃×#683	C₂³	1	8	yes	—
100	C₃×#683	C₂³	1	4	yes	—
101	C₃×C₄×Dic₅	C₂³	1	4	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	—	5	1	no	—
113	C₂×C₃×#117	—	5	1	no	—
114	C₂²×C₃×C₄⋊C₄	D₅	1	1	yes	characteristic, complemented
115	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
116	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
117	C₂²×C₂²⋊D₅	C₂²×C₃	1	1	yes	characteristic
118	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
119	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
120	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
121	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
122	C₂²×C₄×D₅	C₂²×C₃	1	2	yes	—
123	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
124	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
125	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
126	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
127	C₂×C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
128	#4076	C₂²×C₃	1	8	yes	complemented
129	#4076	C₂²×C₃	1	16	yes	complemented
130	C₄×C₂²⋊D₅	C₂²×C₃	1	8	yes	complemented
131	C₄×C₂²⋊D₅	C₂²×C₃	1	16	yes	complemented
132	C₂×#685	C₂²×C₃	1	2	yes	—
133	C₂×#685	C₂²×C₃	1	2	yes	—
134	C₂×#685	C₂²×C₃	1	1	yes	characteristic
135	C₂×#685	C₂²×C₃	1	2	yes	—
136	C₂×#685	C₂²×C₃	1	2	yes	—
137	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
138	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
139	C₂×#683	C₂²×C₃	1	2	yes	—
140	C₂×#683	C₂²×C₃	1	1	yes	characteristic
141	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
142	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	—
143	#4044	C₂²×C₃	1	16	yes	complemented
144	#4041	C₂²×C₃	1	8	yes	complemented
145	#4040	C₂²×C₃	1	16	yes	complemented
146	#4039	C₂²×C₃	1	8	yes	complemented
147	#4038	C₂²×C₃	1	8	yes	complemented
148	#4036	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	#41	1	4	yes	—
153	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
154	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
155	C₂²×C₃×D₅	—	2	2	no	—
156	C₂²×C₃×D₅	—	2	4	no	—
157	C₂²×C₃×D₅	—	2	2	no	—
158	C₂²×C₃×D₅	—	2	4	no	—
159	C₂²×C₃×D₅	—	2	4	no	—
160	C₂²×C₃×D₅	—	2	2	no	—
161	C₂²×C₃×D₅	C₂⁴	1	2	yes	—
162	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
163	C₂×C₃×C₄×C₅	—	2	2	no	—
164	C₂×C₃×C₄×C₅	—	2	4	no	—
165	C₂×C₃×C₄×C₅	—	2	4	no	—
166	C₂×C₃×C₄×C₅	—	2	8	no	—
167	C₂×C₃×C₄×C₅	—	4	8	no	—
168	C₂×C₃×C₄×C₅	#41	1	4	yes	—
169	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
170	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
171	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
172	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
173	C₃×C₂²⋊D₅	—	2	8	no	—
174	C₃×C₂²⋊D₅	—	2	16	no	—
175	C₂×C₃×Dic₅	—	2	2	no	—
176	C₂×C₃×Dic₅	—	2	4	no	—
177	C₂×C₃×Dic₅	—	2	2	no	—
178	C₂×C₃×Dic₅	—	2	2	no	—
179	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
180	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
181	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
182	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
183	C₃×D₂₀	—	2	8	no	—
184	C₃×D₂₀	—	4	8	no	—
185	C₃×C₄×D₅	—	2	8	no	—
186	C₃×C₄×D₅	—	4	8	no	—
187	C₂²×C₃×D₄	—	5	2	no	—
188	C₂²×C₃×D₄	—	5	1	no	—
189	C₂²×C₃×D₄	—	5	1	no	—
190	C₂²×C₃×D₄	—	5	2	no	—
191	C₂³×C₃×C₄	—	5	1	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	2	yes	—
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	—	5	8	no	—
204	C₃×#117	—	5	8	no	—
205	C₂×C₃×C₄⋊C₄	—	5	1	no	—
206	C₂×C₃×C₄⋊C₄	—	5	2	no	—
207	C₂×C₃×C₄⋊C₄	—	5	1	no	—
208	C₂×C₃×C₄⋊C₄	—	5	2	no	—
209	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
210	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
211	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
212	C₂×C₃×C₄⋊C₄	C₂×D₅	1	8	yes	complemented
213	C₂×C₃×C₂²⋊C₄	—	5	1	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	—	5	1	no	—
218	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
219	C₂×C₃×C₄²	—	5	2	no	—
220	C₂×C₃×C₄²	—	5	1	no	—
221	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
222	C₂³×D₅	C₂³×C₃	1	1	yes	characteristic
223	C₂³×D₅	C₂³×C₃	1	2	yes	—
224	C₂²×C₄×C₅	—	2	4	no	—
225	C₂²×C₄×C₅	—	4	4	no	—
226	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
227	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
228	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
229	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
230	C₂²×C₄×C₅	C₂³×C₃	1	8	yes	—
231	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
232	C₂×C₂²⋊D₅	—	2	4	no	—
233	C₂×C₂²⋊D₅	—	2	8	no	—
234	C₂×C₂²⋊D₅	C₂³×C₃	1	4	yes	—
235	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	—
236	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
237	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
238	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
239	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
240	C₂×D₂₀	—	2	4	no	—
241	C₂×D₂₀	—	4	4	no	—
242	C₂×D₂₀	C₂³×C₃	1	4	yes	—
243	C₂×D₂₀	C₂³×C₃	1	8	yes	—
244	C₂×C₄×D₅	—	2	4	no	—
245	C₂×C₄×D₅	—	4	4	no	—
246	C₂×C₄×D₅	C₂³×C₃	1	4	yes	—
247	C₂×C₄×D₅	C₂³×C₃	1	8	yes	—
248	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
249	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
250	C₅×C₄⋊C₄	C₂³×C₃	1	8	yes	—
251	C₂²⋊Dic₅	C₂³×C₃	1	4	yes	—
252	#685	C₂³×C₃	1	8	yes	—
253	#685	C₂³×C₃	1	8	yes	—
254	#685	C₂³×C₃	1	4	yes	—
255	#685	C₂³×C₃	1	8	yes	—
256	#685	C₂³×C₃	1	8	yes	—
257	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
258	C₄⋊Dic₅	C₂³×C₃	1	8	yes	—
259	#683	C₂³×C₃	1	8	yes	—
260	#683	C₂³×C₃	1	4	yes	—
261	C₄×Dic₅	C₂³×C₃	1	4	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	—	5	1	no	—
273	C₂×#117	—	5	1	no	—
274	C₂²×C₄⋊C₄	C₃×D₅	1	1	yes	characteristic, complemented
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	C₂×#41	1	2	yes	—
279	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
280	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
281	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
282	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
283	C₂×C₃×D₅	—	2	2	no	—
284	C₂×C₃×D₅	—	2	4	no	—
285	C₂×C₃×D₅	—	2	2	no	—
286	C₂×C₃×D₅	—	2	4	no	—
287	C₂×C₃×D₅	—	2	4	no	—
288	C₂×C₃×D₅	—	2	2	no	—
289	C₂×C₃×D₅	—	4	4	no	—
290	C₂×C₃×D₅	—	4	8	no	—
291	C₃×C₄×C₅	—	2	2	no	—
292	C₃×C₄×C₅	—	2	4	no	—
293	C₃×C₄×C₅	—	2	4	no	—
294	C₃×C₄×C₅	C₂×#41	1	4	yes	—
295	C₃×Dic₅	—	2	2	no	—
296	C₃×Dic₅	—	2	2	no	—
297	C₃×Dic₅	—	2	4	no	—
298	C₃×Dic₅	—	2	2	no	—
299	C₂⁴×C₃	—	5	1	no	—
300	C₂⁴×C₃	—	5	2	no	—
301	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
302	C₂×C₃×D₄	—	5	8	no	—
303	C₂×C₃×D₄	—	5	4	no	—
304	C₂×C₃×D₄	—	5	4	no	—
305	C₂×C₃×D₄	—	5	8	no	—
306	C₂×C₃×D₄	—	10	8	no	—
307	C₂×C₃×D₄	—	10	4	no	—
308	C₂×C₃×D₄	—	10	4	no	—
309	C₂×C₃×D₄	—	20	4	no	—
310	C₂²×C₃×C₄	—	2	4	no	—
311	C₂²×C₃×C₄	—	4	4	no	—
312	C₂²×C₃×C₄	—	5	1	no	—
313	C₂²×C₃×C₄	—	5	2	no	—
314	C₂²×C₃×C₄	—	5	1	no	—
315	C₂²×C₃×C₄	—	5	1	no	—
316	C₂²×C₃×C₄	—	5	4	no	—
317	C₂²×C₃×C₄	—	5	8	no	—
318	C₂²×C₃×C₄	—	10	4	no	—
319	C₂²×C₃×C₄	—	20	4	no	—
320	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
321	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
322	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
323	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
324	C₂²×C₃×C₄	C₂²×D₅	1	8	yes	—
325	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
326	C₃×C₄⋊C₄	—	5	8	no	—
327	C₃×C₄⋊C₄	—	5	4	no	—
328	C₃×C₄⋊C₄	—	5	4	no	—
329	C₃×C₄⋊C₄	—	5	8	no	—
330	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
331	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
332	C₃×C₄⋊C₄	C₂²×D₅	1	8	yes	—
333	C₃×C₂²⋊C₄	—	5	4	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	—	5	4	no	—
338	C₃×C₂²⋊C₄	—	5	8	no	—
339	C₃×C₄²	—	5	8	no	—
340	C₃×C₄²	—	5	4	no	—
341	C₂³×C₅	—	2	2	no	—
342	C₂³×C₅	—	2	2	no	—
343	C₂³×C₅	C₃×#41	1	4	yes	—
344	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
345	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
346	C₂²×D₅	—	2	2	no	—
347	C₂²×D₅	—	2	4	no	—
348	C₂²×D₅	—	2	2	no	—
349	C₂²×D₅	—	2	4	no	—
350	C₂²×D₅	—	2	4	no	—
351	C₂²×D₅	—	2	2	no	—
352	C₂²×D₅	C₂⁴×C₃	1	2	yes	—
353	C₂²×D₅	C₂⁴×C₃	1	4	yes	—
354	C₂×C₄×C₅	—	2	2	no	—
355	C₂×C₄×C₅	—	2	4	no	—
356	C₂×C₄×C₅	—	2	4	no	—
357	C₂×C₄×C₅	—	2	8	no	—
358	C₂×C₄×C₅	—	4	8	no	—
359	C₂×C₄×C₅	C₃×#41	1	4	yes	—
360	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
361	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
362	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
363	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
364	C₂²⋊D₅	—	2	8	no	—
365	C₂²⋊D₅	—	2	16	no	—
366	C₂×Dic₅	—	2	2	no	—
367	C₂×Dic₅	—	2	2	no	—
368	C₂×Dic₅	—	2	4	no	—
369	C₂×Dic₅	—	2	2	no	—
370	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
371	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
372	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
373	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
374	D₂₀	—	2	8	no	—
375	D₂₀	—	4	8	no	—
376	C₄×D₅	—	2	8	no	—
377	C₄×D₅	—	4	8	no	—
378	C₂²×D₄	—	5	2	no	—
379	C₂²×D₄	—	5	1	no	—
380	C₂²×D₄	—	5	1	no	—
381	C₂²×D₄	—	5	2	no	—
382	C₂³×C₄	—	5	1	no	—
383	C₂³×C₄	—	5	2	no	—
384	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
385	C₂³×C₄	C₂×C₃×D₅	1	2	yes	—
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	—	5	8	no	—
395	#117	—	5	8	no	—
396	C₂×C₄⋊C₄	—	5	1	no	—
397	C₂×C₄⋊C₄	—	5	2	no	—
398	C₂×C₄⋊C₄	—	5	1	no	—
399	C₂×C₄⋊C₄	—	5	2	no	—
400	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
403	C₂×C₄⋊C₄	C₂×C₃×D₅	1	8	yes	complemented
404	C₂×C₂²⋊C₄	—	5	1	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	—	5	1	no	—
409	C₂×C₂²⋊C₄	—	5	2	no	—
410	C₂×C₄²	—	5	2	no	—
411	C₂×C₄²	—	5	1	no	—
412	C₂×C₃×C₅	—	2	4	no	—
413	C₂×C₃×C₅	#553	1	4	yes	complemented
414	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
415	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
416	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
417	C₃×D₅	—	4	2	no	—
418	C₃×D₅	—	4	4	no	—
419	C₂³×C₃	—	2	2	no	—
420	C₂³×C₃	—	2	2	no	—
421	C₂³×C₃	—	5	2	no	—
422	C₂³×C₃	—	5	4	no	—
423	C₂³×C₃	—	10	2	no	—
424	C₂³×C₃	—	10	4	no	—
425	C₂³×C₃	—	10	4	no	—
426	C₂³×C₃	—	10	2	no	—
427	C₂³×C₃	—	10	2	no	—
428	C₂³×C₃	—	10	4	no	—
429	C₂³×C₃	#713	1	4	yes	—
430	C₂³×C₃	C₂³×D₅	1	2	yes	—
431	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
432	C₃×D₄	—	10	16	no	—
433	C₃×D₄	—	10	8	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	4	no	—
438	C₂×C₃×C₄	—	2	4	no	—
439	C₂×C₃×C₄	—	2	8	no	—
440	C₂×C₃×C₄	—	4	8	no	—
441	C₂×C₃×C₄	—	5	2	no	—
442	C₂×C₃×C₄	—	5	4	no	—
443	C₂×C₃×C₄	—	5	2	no	—
444	C₂×C₃×C₄	—	5	2	no	—
445	C₂×C₃×C₄	—	10	2	no	—
446	C₂×C₃×C₄	—	10	2	no	—
447	C₂×C₃×C₄	—	10	4	no	—
448	C₂×C₃×C₄	—	10	2	no	—
449	C₂×C₃×C₄	—	10	8	no	—
450	C₂×C₃×C₄	—	20	8	no	—
451	C₂×C₃×C₄	#709	1	4	yes	—
452	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
453	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
454	C₂×C₃×C₄	C₂³×D₅	1	4	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	C₂×C₃×#41	1	2	yes	—
460	C₂²×C₅	C₂×C₃×#41	1	4	yes	—
461	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
462	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
463	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
464	C₂×D₅	—	2	2	no	—
465	C₂×D₅	—	2	4	no	—
466	C₂×D₅	—	2	2	no	—
467	C₂×D₅	—	2	4	no	—
468	C₂×D₅	—	2	4	no	—
469	C₂×D₅	—	2	2	no	—
470	C₂×D₅	—	4	4	no	—
471	C₂×D₅	—	4	8	no	—
472	C₄×C₅	—	2	2	no	—
473	C₄×C₅	—	2	4	no	—
474	C₄×C₅	—	2	4	no	—
475	C₄×C₅	C₂×C₃×#41	1	4	yes	—
476	Dic₅	—	2	2	no	—
477	Dic₅	—	2	2	no	—
478	Dic₅	—	2	4	no	—
479	Dic₅	—	2	2	no	—
480	C₂⁴	—	5	1	no	—
481	C₂⁴	—	5	2	no	—
482	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
483	C₂×D₄	—	5	8	no	—
484	C₂×D₄	—	5	4	no	—
485	C₂×D₄	—	5	4	no	—
486	C₂×D₄	—	5	8	no	—
487	C₂×D₄	—	10	8	no	—
488	C₂×D₄	—	10	4	no	—
489	C₂×D₄	—	10	4	no	—
490	C₂×D₄	—	20	4	no	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	—	4	4	no	—
493	C₂²×C₄	—	5	1	no	—
494	C₂²×C₄	—	5	2	no	—
495	C₂²×C₄	—	5	1	no	—
496	C₂²×C₄	—	5	1	no	—
497	C₂²×C₄	—	5	4	no	—
498	C₂²×C₄	—	5	8	no	—
499	C₂²×C₄	—	10	4	no	—
500	C₂²×C₄	—	20	4	no	—
501	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
502	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
503	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
504	C₂²×C₄	C₂²×C₃×D₅	1	4	yes	—
505	C₂²×C₄	C₂²×C₃×D₅	1	8	yes	—
506	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
507	C₄⋊C₄	—	5	4	no	—
508	C₄⋊C₄	—	5	8	no	—
509	C₄⋊C₄	—	5	4	no	—
510	C₄⋊C₄	—	5	8	no	—
511	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
512	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
513	C₄⋊C₄	C₂²×C₃×D₅	1	8	yes	—
514	C₂²⋊C₄	—	5	4	no	—
515	C₂²⋊C₄	—	5	8	no	—
516	C₂²⋊C₄	—	5	8	no	—
517	C₂²⋊C₄	—	5	8	no	—
518	C₂²⋊C₄	—	5	4	no	—
519	C₂²⋊C₄	—	5	8	no	—
520	C₄²	—	5	8	no	—
521	C₄²	—	5	4	no	—
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	—	10	2	no	—
527	C₂²×C₃	—	10	4	no	—
528	C₂²×C₃	—	10	4	no	—
529	C₂²×C₃	—	10	2	no	—
530	C₂²×C₃	—	10	2	no	—
531	C₂²×C₃	—	10	4	no	—
532	C₂²×C₃	—	20	4	no	—
533	C₂²×C₃	—	20	8	no	—
534	C₂²×C₃	C₂×#709	1	2	yes	—
535	C₂²×C₃	#4144	1	2	yes	—
536	C₂²×C₃	C₂×#713	1	4	yes	—
537	C₂²×C₃	C₂×#713	1	2	yes	—
538	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
539	C₃×C₄	—	2	2	no	—
540	C₃×C₄	—	2	4	no	—
541	C₃×C₄	—	2	4	no	—
542	C₃×C₄	—	10	2	no	—
543	C₃×C₄	—	10	2	no	—
544	C₃×C₄	—	10	4	no	—
545	C₃×C₄	—	10	2	no	—
546	C₃×C₄	C₂×#709	1	4	yes	—
547	C₂×C₅	—	2	4	no	—
548	C₂×C₅	C₃×#553	1	4	yes	complemented
549	C₂×C₅	C₂²×C₃×#41	1	1	yes	characteristic
550	C₂×C₅	C₂²×C₃×#41	1	1	yes	characteristic
551	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
552	D₅	—	4	2	no	—
553	D₅	—	4	4	no	—
554	C₂³	—	2	2	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	5	2	no	—
557	C₂³	—	5	4	no	—
558	C₂³	—	10	2	no	—
559	C₂³	—	10	4	no	—
560	C₂³	—	10	4	no	—
561	C₂³	—	10	2	no	—
562	C₂³	—	10	2	no	—
563	C₂³	—	10	4	no	—
564	C₂³	C₃×#713	1	4	yes	—
565	C₂³	C₂³×C₃×D₅	1	2	yes	—
566	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
567	D₄	—	10	16	no	—
568	D₄	—	10	8	no	—
569	D₄	—	10	8	no	—
570	D₄	—	20	8	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	8	no	—
575	C₂×C₄	—	4	8	no	—
576	C₂×C₄	—	5	2	no	—
577	C₂×C₄	—	5	4	no	—
578	C₂×C₄	—	5	2	no	—
579	C₂×C₄	—	5	2	no	—
580	C₂×C₄	—	10	2	no	—
581	C₂×C₄	—	10	2	no	—
582	C₂×C₄	—	10	4	no	—
583	C₂×C₄	—	10	2	no	—
584	C₂×C₄	—	10	8	no	—
585	C₂×C₄	—	20	8	no	—
586	C₂×C₄	C₃×#709	1	4	yes	—
587	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
588	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
589	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
590	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
591	C₂×C₃	—	2	4	no	—
592	C₂×C₃	—	20	2	no	—
593	C₂×C₃	—	20	4	no	—
594	C₂×C₃	#65848	1	4	yes	direct product factor
595	C₂×C₃	C₂²×#709	1	1	yes	characteristic
596	C₂×C₃	C₂×#4144	1	1	yes	characteristic
597	C₂×C₃	C₂²×#713	1	1	yes	characteristic
598	C₅	C₂×C₃×#553	1	1	yes	characteristic, complemented, Sylow
599	C₂²	—	2	8	no	—
600	C₂²	—	2	2	no	—
601	C₂²	—	2	2	no	—
602	C₂²	—	10	2	no	—
603	C₂²	—	10	4	no	—
604	C₂²	—	10	4	no	—
605	C₂²	—	10	2	no	—
606	C₂²	—	10	2	no	—
607	C₂²	—	10	4	no	—
608	C₂²	—	20	4	no	—
609	C₂²	—	20	8	no	—
610	C₂²	C₂×C₃×#709	1	2	yes	—
611	C₂²	C₃×#4144	1	2	yes	—
612	C₂²	C₂×C₃×#713	1	4	yes	—
613	C₂²	C₂×C₃×#713	1	2	yes	—
614	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
615	C₄	—	2	2	no	—
616	C₄	—	2	4	no	—
617	C₄	—	2	4	no	—
618	C₄	—	10	2	no	—
619	C₄	—	10	2	no	—
620	C₄	—	10	4	no	—
621	C₄	—	10	2	no	—
622	C₄	C₂×C₃×#709	1	4	yes	—
623	C₃	C₂×#65848	1	1	yes	characteristic, direct product factor, Sylow
624	C₂	—	2	4	no	—
625	C₂	—	20	2	no	—
626	C₂	—	20	4	no	—
627	C₂	C₃×#65848	1	4	yes	direct product factor
628	C₂	C₂²×C₃×#709	1	1	yes	characteristic
629	C₂	C₂×C₃×#4144	1	1	yes	characteristic
630	C₂	C₂²×C₃×#713	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×C₄⋊C₄	C₂	semidirect
2. C₂×C₃×#4076	C₂	semidirect
3. C₂×C₃×#4076	C₂	semidirect
4. C₂×C₃×C₄×C₂²⋊D₅	C₂	semidirect
5. C₂×C₃×C₄×C₂²⋊D₅	C₂	semidirect
6. C₃×#65848	C₂	direct (N × H)
7. C₂×C₃×#4044	C₂	semidirect
8. C₂×C₃×#4041	C₂	semidirect
9. C₂×C₃×#4040	C₂	semidirect
10. C₂×C₃×#4039	C₂	semidirect
11. C₂×C₃×#4038	C₂	semidirect
12. C₂×C₃×#4036	C₂	semidirect
13. C₂×#65848	C₃	direct (N × H)
25. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
27. C₃×#4076	C₂²	semidirect
28. C₃×#4076	C₂²	semidirect
29. C₃×C₄×C₂²⋊D₅	C₂²	semidirect
30. C₃×C₄×C₂²⋊D₅	C₂²	semidirect
42. C₃×#4044	C₂²	semidirect
43. C₃×#4041	C₂²	semidirect
44. C₃×#4040	C₂²	semidirect
45. C₃×#4039	C₂²	semidirect
46. C₃×#4038	C₂²	semidirect
47. C₃×#4036	C₂²	semidirect
49. C₂²×C₅×C₄⋊C₄	C₂×C₃	semidirect
50. C₂×#4076	C₂×C₃	semidirect
51. C₂×#4076	C₂×C₃	semidirect
52. C₂×C₄×C₂²⋊D₅	C₂×C₃	semidirect
53. C₂×C₄×C₂²⋊D₅	C₂×C₃	semidirect
54. #65848	C₂×C₃	direct (N × H)
55. C₂×#4044	C₂×C₃	semidirect
56. C₂×#4041	C₂×C₃	semidirect
57. C₂×#4040	C₂×C₃	semidirect
58. C₂×#4039	C₂×C₃	semidirect
59. C₂×#4038	C₂×C₃	semidirect
60. C₂×#4036	C₂×C₃	semidirect
114. C₂²×C₃×C₄⋊C₄	D₅	semidirect
126. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
128. #4076	C₂²×C₃	semidirect
129. #4076	C₂²×C₃	semidirect
130. C₄×C₂²⋊D₅	C₂²×C₃	semidirect
131. C₄×C₂²⋊D₅	C₂²×C₃	semidirect
143. #4044	C₂²×C₃	semidirect
144. #4041	C₂²×C₃	semidirect
145. #4040	C₂²×C₃	semidirect
146. #4039	C₂²×C₃	semidirect
147. #4038	C₂²×C₃	semidirect
148. #4036	C₂²×C₃	semidirect
212. C₂×C₃×C₄⋊C₄	C₂×D₅	semidirect
274. C₂²×C₄⋊C₄	C₃×D₅	semidirect
403. C₂×C₄⋊C₄	C₂×C₃×D₅	semidirect
413. C₂×C₃×C₅	#553	semidirect
522. C₃×C₅	C₂×#553	semidirect
548. C₂×C₅	C₃×#553	semidirect
594. C₂×C₃	#65848	direct (N × H)
598. C₅	C₂×C₃×#553	semidirect
623. C₃	C₂×#65848	direct (N × H)
627. C₂	C₃×#65848	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×C₄⋊C₄	2
C₂×C₃×#4076	2
C₂×C₃×C₄×C₂²⋊D₅	2
C₃×#65848	2
C₂×C₃×#4044	2
C₂×C₃×#4041	2
C₂×C₃×#4040	2
C₂×C₃×#4039	2
C₂×C₃×#4038	2
C₂×C₃×#4036	2
C₂×#65848	3
C₂×C₃×#553	5

Subgroups — C2×C3×#65848 #14483608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65848 #14483608