Subgroups — C₂×C₃×#66016 #14481317

Order 1920.

2264 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	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4092	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#4084	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×D₄×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#66016	C₂	1	16	yes	maximal, direct product factor
7	C₂×C₃×#4032	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#4031	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×#4025	C₂	1	2	yes	maximal, complemented
10	C₂×C₃×#4024	C₂	1	2	yes	maximal, complemented
11	C₂×C₃×#4018	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃×#4015	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#66016	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂³×C₃×Dic₅	C₂²	1	2	yes	—
18	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
19	C₂²×C₃×Dic₁₀	C₂²	1	2	yes	—
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	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
26	C₃×#4092	C₂²	1	8	yes	complemented
27	C₃×Q₈×Dic₅	C₂²	1	8	yes	complemented
28	C₃×#4084	C₂²	1	8	yes	—
29	C₃×D₄×Dic₅	C₂²	1	8	yes	complemented
30	C₂×C₃×#685	C₂²	1	2	yes	—
31	C₂×C₃×#685	C₂²	1	1	yes	characteristic
32	C₂×C₃×#685	C₂²	1	1	yes	characteristic
33	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
35	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
36	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
37	C₂×C₃×#683	C₂²	1	2	yes	—
38	C₂×C₃×#683	C₂²	1	1	yes	characteristic
39	C₂×C₃×#683	C₂²	1	1	yes	characteristic
40	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
41	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
42	C₃×#4032	C₂²	1	16	yes	complemented
43	C₃×#4031	C₂²	1	16	yes	complemented
44	C₃×#4025	C₂²	1	16	yes	complemented
45	C₃×#4024	C₂²	1	16	yes	complemented
46	C₃×#4018	C₂²	1	8	yes	complemented
47	C₃×#4015	C₂²	1	8	yes	complemented
48	C₂×C₃×#542	—	5	1	no	maximal, Hall
49	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂×#4092	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂×Q₈×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
52	C₂×#4084	C₂×C₃	1	1	yes	characteristic
53	C₂×D₄×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
54	#66016	C₂×C₃	1	16	yes	direct product factor
55	C₂×#4032	C₂×C₃	1	2	yes	complemented
56	C₂×#4031	C₂×C₃	1	2	yes	complemented
57	C₂×#4025	C₂×C₃	1	2	yes	complemented
58	C₂×#4024	C₂×C₃	1	2	yes	complemented
59	C₂×#4018	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂×#4015	C₂×C₃	1	1	yes	characteristic, complemented
61	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
62	C₂³×C₃×D₅	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₅×Q₈	—	2	4	no	—
64	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
65	C₂×C₃×C₅×D₄	—	2	4	no	—
66	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
67	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
70	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
71	C₂×C₃×C₂²⋊D₅	—	4	4	no	—
72	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	—
73	C₂²×C₃×Dic₅	—	4	4	no	—
74	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
75	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
76	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
77	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
78	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
79	C₂²×C₃×Dic₅	C₂³	1	8	yes	—
80	C₂×C₃×C₄×D₅	—	2	4	no	—
81	C₂×C₃×C₄×D₅	C₂³	1	4	yes	—
82	C₂×C₃×Dic₁₀	—	4	4	no	—
83	C₂×C₃×Dic₁₀	C₂³	1	8	yes	—
84	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
85	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
86	C₃×C₄²×C₅	C₂³	1	4	yes	—
87	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	—
88	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	—
89	C₃×#685	C₂³	1	8	yes	—
90	C₃×#685	C₂³	1	4	yes	—
91	C₃×#685	C₂³	1	4	yes	—
92	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
93	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
94	C₃×C₄⋊Dic₅	C₂³	1	8	yes	—
95	C₃×C₄⋊Dic₅	C₂³	1	8	yes	—
96	C₃×#683	C₂³	1	8	yes	—
97	C₃×#683	C₂³	1	4	yes	—
98	C₃×#683	C₂³	1	4	yes	—
99	C₃×C₄×Dic₅	C₂³	1	4	yes	—
100	C₃×C₄×Dic₅	C₂³	1	8	yes	—
101	C₃×#542	—	5	16	no	—
102	C₂×C₃×C₄⋊Q₈	—	5	1	no	—
103	C₂×C₃×#126	—	5	2	no	—
104	C₂×C₃×#124	—	5	2	no	—
105	C₂×C₃×#124	D₅	1	1	yes	characteristic, complemented
106	C₂×C₃×#123	—	5	2	no	—
107	C₂×C₃×C₂²⋊Q₈	—	5	1	no	—
108	C₂×C₃×C₂²⋊Q₈	—	5	2	no	—
109	C₂×C₃×#121	—	5	1	no	—
110	C₂×C₃×C₄×Q₈	—	5	1	no	—
111	C₂×C₃×C₄×D₄	—	5	1	no	—
112	C₂×C₃×#117	—	5	1	no	—
113	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
114	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
115	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
116	C₂³×Dic₅	C₂²×C₃	1	2	yes	—
117	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
118	C₂²×Dic₁₀	C₂²×C₃	1	2	yes	—
119	C₅×#124	C₂²×C₃	1	8	yes	complemented
120	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
121	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
122	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
123	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
124	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
125	#4092	C₂²×C₃	1	8	yes	complemented
126	Q₈×Dic₅	C₂²×C₃	1	8	yes	complemented
127	#4084	C₂²×C₃	1	8	yes	—
128	D₄×Dic₅	C₂²×C₃	1	8	yes	complemented
129	C₂×#685	C₂²×C₃	1	1	yes	characteristic
130	C₂×#685	C₂²×C₃	1	1	yes	characteristic
131	C₂×#685	C₂²×C₃	1	2	yes	—
132	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
133	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
134	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
135	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
136	C₂×#683	C₂²×C₃	1	1	yes	characteristic
137	C₂×#683	C₂²×C₃	1	1	yes	characteristic
138	C₂×#683	C₂²×C₃	1	2	yes	—
139	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
140	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	—
141	#4032	C₂²×C₃	1	16	yes	complemented
142	#4031	C₂²×C₃	1	16	yes	complemented
143	#4025	C₂²×C₃	1	16	yes	complemented
144	#4024	C₂²×C₃	1	16	yes	complemented
145	#4018	C₂²×C₃	1	8	yes	complemented
146	#4015	C₂²×C₃	1	8	yes	complemented
147	C₂×#542	—	5	1	no	Sylow
148	C₂³×C₃×C₅	—	2	4	no	—
149	C₂³×C₃×C₅	—	2	4	no	—
150	C₂³×C₃×C₅	—	2	4	no	—
151	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
152	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
153	C₂²×C₃×D₅	—	2	2	no	—
154	C₂²×C₃×D₅	—	2	2	no	—
155	C₂²×C₃×D₅	—	2	2	no	—
156	C₂²×C₃×D₅	C₂⁴	1	2	yes	—
157	C₃×C₅×Q₈	—	2	8	no	—
158	C₃×C₅×D₄	—	2	8	no	—
159	C₂×C₃×C₄×C₅	—	2	4	no	—
160	C₂×C₃×C₄×C₅	—	2	2	no	—
161	C₂×C₃×C₄×C₅	—	2	2	no	—
162	C₂×C₃×C₄×C₅	#41	1	4	yes	—
163	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
164	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
165	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
166	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
167	C₃×C₂²⋊D₅	—	4	8	no	—
168	C₂×C₃×Dic₅	—	2	2	no	—
169	C₂×C₃×Dic₅	—	2	4	no	—
170	C₂×C₃×Dic₅	—	2	4	no	—
171	C₂×C₃×Dic₅	—	2	2	no	—
172	C₂×C₃×Dic₅	—	2	2	no	—
173	C₂×C₃×Dic₅	—	4	8	no	—
174	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
175	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
176	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
177	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
178	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
179	C₃×C₄×D₅	—	2	8	no	—
180	C₃×Dic₁₀	—	4	8	no	—
181	C₂²×C₃×Q₈	—	5	2	no	—
182	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
183	C₂²×C₃×D₄	—	5	2	no	—
184	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
185	C₂³×C₃×C₄	—	5	2	no	—
186	C₂³×C₃×C₄	—	5	1	no	—
187	C₃×C₄⋊Q₈	—	5	8	no	—
188	C₃×#126	—	5	16	no	—
189	C₃×#124	—	5	16	no	—
190	C₃×#124	C₂×D₅	1	8	yes	complemented
191	C₃×#123	—	5	16	no	—
192	C₃×C₂²⋊Q₈	—	5	8	no	—
193	C₃×C₂²⋊Q₈	—	5	16	no	—
194	C₃×#121	—	5	8	no	—
195	C₃×C₄×Q₈	—	5	8	no	—
196	C₃×C₄×D₄	—	5	8	no	—
197	C₃×#117	—	5	8	no	—
198	C₂×C₃×C₄⋊C₄	—	5	1	no	—
199	C₂×C₃×C₄⋊C₄	—	5	1	no	—
200	C₂×C₃×C₄⋊C₄	—	5	1	no	—
201	C₂×C₃×C₄⋊C₄	—	5	1	no	—
202	C₂×C₃×C₄⋊C₄	—	5	2	no	—
203	C₂×C₃×C₄⋊C₄	—	5	2	no	—
204	C₂×C₃×C₄⋊C₄	—	5	2	no	—
205	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
206	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
207	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
208	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
209	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
210	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
211	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
212	C₂×C₃×C₄²	—	5	1	no	—
213	C₂×C₃×C₄²	—	5	2	no	—
214	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
215	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
216	C₂³×D₅	C₂³×C₃	1	1	yes	characteristic
217	C₂×C₅×Q₈	—	2	4	no	—
218	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
219	C₂×C₅×D₄	—	2	4	no	—
220	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
221	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
222	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
223	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
224	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
225	C₂×C₂²⋊D₅	—	4	4	no	—
226	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	—
227	C₂²×Dic₅	—	4	4	no	—
228	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
229	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
230	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
231	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
232	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
233	C₂²×Dic₅	C₂³×C₃	1	8	yes	—
234	C₂×C₄×D₅	—	2	4	no	—
235	C₂×C₄×D₅	C₂³×C₃	1	4	yes	—
236	C₂×Dic₁₀	—	4	4	no	—
237	C₂×Dic₁₀	C₂³×C₃	1	8	yes	—
238	C₅×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
239	C₅×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
240	C₄²×C₅	C₂³×C₃	1	4	yes	—
241	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	—
242	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	—
243	#685	C₂³×C₃	1	4	yes	—
244	#685	C₂³×C₃	1	4	yes	—
245	#685	C₂³×C₃	1	8	yes	—
246	C₄⋊Dic₅	C₂³×C₃	1	8	yes	—
247	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
248	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
249	C₄⋊Dic₅	C₂³×C₃	1	8	yes	—
250	#683	C₂³×C₃	1	4	yes	—
251	#683	C₂³×C₃	1	4	yes	—
252	#683	C₂³×C₃	1	8	yes	—
253	C₄×Dic₅	C₂³×C₃	1	4	yes	—
254	C₄×Dic₅	C₂³×C₃	1	8	yes	—
255	#542	—	5	16	no	—
256	C₂×C₄⋊Q₈	—	5	1	no	—
257	C₂×#126	—	5	2	no	—
258	C₂×#124	—	5	2	no	—
259	C₂×#124	C₃×D₅	1	1	yes	characteristic, complemented
260	C₂×#123	—	5	2	no	—
261	C₂×C₂²⋊Q₈	—	5	1	no	—
262	C₂×C₂²⋊Q₈	—	5	2	no	—
263	C₂×#121	—	5	1	no	—
264	C₂×C₄×Q₈	—	5	1	no	—
265	C₂×C₄×D₄	—	5	1	no	—
266	C₂×#117	—	5	1	no	—
267	C₂²×C₃×C₅	—	2	4	no	—
268	C₂²×C₃×C₅	—	2	4	no	—
269	C₂²×C₃×C₅	—	2	4	no	—
270	C₂²×C₃×C₅	—	4	8	no	—
271	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
272	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
273	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
274	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
275	C₂×C₃×D₅	—	2	2	no	—
276	C₂×C₃×D₅	—	2	2	no	—
277	C₂×C₃×D₅	—	2	2	no	—
278	C₂×C₃×D₅	—	4	4	no	—
279	C₃×C₄×C₅	—	2	4	no	—
280	C₃×C₄×C₅	—	2	2	no	—
281	C₃×C₄×C₅	—	2	2	no	—
282	C₃×C₄×C₅	C₂×#41	1	4	yes	—
283	C₃×Dic₅	—	2	2	no	—
284	C₃×Dic₅	—	2	4	no	—
285	C₃×Dic₅	—	2	4	no	—
286	C₃×Dic₅	—	2	2	no	—
287	C₃×Dic₅	—	2	2	no	—
288	C₂⁴×C₃	—	5	1	no	—
289	C₂⁴×C₃	C₂²×D₅	1	2	yes	—
290	C₂×C₃×Q₈	—	2	4	no	—
291	C₂×C₃×Q₈	—	5	8	no	—
292	C₂×C₃×Q₈	—	20	4	no	—
293	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	—
294	C₂×C₃×D₄	—	2	4	no	—
295	C₂×C₃×D₄	—	5	8	no	—
296	C₂×C₃×D₄	—	20	4	no	—
297	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
298	C₂²×C₃×C₄	—	5	1	no	—
299	C₂²×C₃×C₄	—	5	1	no	—
300	C₂²×C₃×C₄	—	5	2	no	—
301	C₂²×C₃×C₄	—	5	1	no	—
302	C₂²×C₃×C₄	—	5	2	no	—
303	C₂²×C₃×C₄	—	5	8	no	—
304	C₂²×C₃×C₄	—	5	4	no	—
305	C₂²×C₃×C₄	—	10	4	no	—
306	C₂²×C₃×C₄	—	20	4	no	—
307	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
308	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
309	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
310	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
311	C₃×C₄⋊C₄	—	5	4	no	—
312	C₃×C₄⋊C₄	—	5	4	no	—
313	C₃×C₄⋊C₄	—	5	8	no	—
314	C₃×C₄⋊C₄	—	5	4	no	—
315	C₃×C₄⋊C₄	—	5	4	no	—
316	C₃×C₄⋊C₄	—	5	8	no	—
317	C₃×C₄⋊C₄	—	5	8	no	—
318	C₃×C₂²⋊C₄	—	5	8	no	—
319	C₃×C₂²⋊C₄	—	5	8	no	—
320	C₃×C₂²⋊C₄	—	5	4	no	—
321	C₃×C₂²⋊C₄	—	5	4	no	—
322	C₃×C₂²⋊C₄	—	5	8	no	—
323	C₃×C₂²⋊C₄	C₂²×D₅	1	8	yes	—
324	C₃×C₂²⋊C₄	C₂²×D₅	1	8	yes	—
325	C₃×C₄²	—	5	4	no	—
326	C₃×C₄²	—	5	8	no	—
327	C₃×C₄²	C₂²×D₅	1	4	yes	—
328	C₂³×C₅	—	2	4	no	—
329	C₂³×C₅	—	2	4	no	—
330	C₂³×C₅	—	2	4	no	—
331	C₂³×C₅	C₂⁴×C₃	1	4	yes	—
332	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
333	C₂²×D₅	—	2	2	no	—
334	C₂²×D₅	—	2	2	no	—
335	C₂²×D₅	—	2	2	no	—
336	C₂²×D₅	C₂⁴×C₃	1	2	yes	—
337	C₅×Q₈	—	2	8	no	—
338	C₅×D₄	—	2	8	no	—
339	C₂×C₄×C₅	—	2	4	no	—
340	C₂×C₄×C₅	—	2	2	no	—
341	C₂×C₄×C₅	—	2	2	no	—
342	C₂×C₄×C₅	C₃×#41	1	4	yes	—
343	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
344	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
345	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
346	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
347	C₂²⋊D₅	—	4	8	no	—
348	C₂×Dic₅	—	2	2	no	—
349	C₂×Dic₅	—	2	4	no	—
350	C₂×Dic₅	—	2	4	no	—
351	C₂×Dic₅	—	2	2	no	—
352	C₂×Dic₅	—	2	2	no	—
353	C₂×Dic₅	—	4	8	no	—
354	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
355	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
356	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
357	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
358	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
359	C₄×D₅	—	2	8	no	—
360	Dic₁₀	—	4	8	no	—
361	C₂²×Q₈	—	5	2	no	—
362	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
363	C₂²×D₄	—	5	2	no	—
364	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
365	C₂³×C₄	—	5	2	no	—
366	C₂³×C₄	—	5	1	no	—
367	C₄⋊Q₈	—	5	8	no	—
368	#126	—	5	16	no	—
369	#124	—	5	16	no	—
370	#124	C₂×C₃×D₅	1	8	yes	complemented
371	#123	—	5	16	no	—
372	C₂²⋊Q₈	—	5	8	no	—
373	C₂²⋊Q₈	—	5	16	no	—
374	#121	—	5	8	no	—
375	C₄×Q₈	—	5	8	no	—
376	C₄×D₄	—	5	8	no	—
377	#117	—	5	8	no	—
378	C₂×C₄⋊C₄	—	5	1	no	—
379	C₂×C₄⋊C₄	—	5	1	no	—
380	C₂×C₄⋊C₄	—	5	1	no	—
381	C₂×C₄⋊C₄	—	5	1	no	—
382	C₂×C₄⋊C₄	—	5	2	no	—
383	C₂×C₄⋊C₄	—	5	2	no	—
384	C₂×C₄⋊C₄	—	5	2	no	—
385	C₂×C₂²⋊C₄	—	5	2	no	—
386	C₂×C₂²⋊C₄	—	5	2	no	—
387	C₂×C₂²⋊C₄	—	5	1	no	—
388	C₂×C₂²⋊C₄	—	5	1	no	—
389	C₂×C₂²⋊C₄	—	5	2	no	—
390	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
391	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
392	C₂×C₄²	—	5	1	no	—
393	C₂×C₄²	—	5	2	no	—
394	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
395	C₂×C₃×C₅	—	4	4	no	—
396	C₂×C₃×C₅	#542	1	4	yes	complemented
397	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
398	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
399	C₂×C₃×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
400	C₃×D₅	—	4	2	no	—
401	C₂³×C₃	—	2	4	no	—
402	C₂³×C₃	—	2	4	no	—
403	C₂³×C₃	—	2	4	no	—
404	C₂³×C₃	—	5	2	no	—
405	C₂³×C₃	—	10	2	no	—
406	C₂³×C₃	—	10	2	no	—
407	C₂³×C₃	—	10	2	no	—
408	C₂³×C₃	C₂³×D₅	1	4	yes	—
409	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
410	C₃×Q₈	—	2	8	no	—
411	C₃×Q₈	—	20	8	no	—
412	C₃×D₄	—	2	8	no	—
413	C₃×D₄	—	20	8	no	—
414	C₂×C₃×C₄	—	2	4	no	—
415	C₂×C₃×C₄	—	2	2	no	—
416	C₂×C₃×C₄	—	2	2	no	—
417	C₂×C₃×C₄	—	5	2	no	—
418	C₂×C₃×C₄	—	5	2	no	—
419	C₂×C₃×C₄	—	5	4	no	—
420	C₂×C₃×C₄	—	5	2	no	—
421	C₂×C₃×C₄	—	5	4	no	—
422	C₂×C₃×C₄	—	10	2	no	—
423	C₂×C₃×C₄	—	10	2	no	—
424	C₂×C₃×C₄	—	10	4	no	—
425	C₂×C₃×C₄	—	10	4	no	—
426	C₂×C₃×C₄	—	10	2	no	—
427	C₂×C₃×C₄	—	10	8	no	—
428	C₂×C₃×C₄	—	20	8	no	—
429	C₂×C₃×C₄	#711	1	4	yes	—
430	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
431	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
432	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
433	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
434	C₂²×C₅	—	2	4	no	—
435	C₂²×C₅	—	2	4	no	—
436	C₂²×C₅	—	2	4	no	—
437	C₂²×C₅	—	4	8	no	—
438	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
439	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
440	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
441	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
442	C₂×D₅	—	2	2	no	—
443	C₂×D₅	—	2	2	no	—
444	C₂×D₅	—	2	2	no	—
445	C₂×D₅	—	4	4	no	—
446	C₄×C₅	—	2	4	no	—
447	C₄×C₅	—	2	2	no	—
448	C₄×C₅	—	2	2	no	—
449	C₄×C₅	C₂×C₃×#41	1	4	yes	—
450	Dic₅	—	2	2	no	—
451	Dic₅	—	2	4	no	—
452	Dic₅	—	2	4	no	—
453	Dic₅	—	2	2	no	—
454	Dic₅	—	2	2	no	—
455	C₂⁴	—	5	1	no	—
456	C₂⁴	C₂²×C₃×D₅	1	2	yes	—
457	C₂×Q₈	—	2	4	no	—
458	C₂×Q₈	—	5	8	no	—
459	C₂×Q₈	—	20	4	no	—
460	C₂×Q₈	C₂²×C₃×D₅	1	4	yes	—
461	C₂×D₄	—	2	4	no	—
462	C₂×D₄	—	5	8	no	—
463	C₂×D₄	—	20	4	no	—
464	C₂×D₄	C₂²×C₃×D₅	1	4	yes	—
465	C₂²×C₄	—	5	1	no	—
466	C₂²×C₄	—	5	1	no	—
467	C₂²×C₄	—	5	2	no	—
468	C₂²×C₄	—	5	1	no	—
469	C₂²×C₄	—	5	2	no	—
470	C₂²×C₄	—	5	8	no	—
471	C₂²×C₄	—	5	4	no	—
472	C₂²×C₄	—	10	4	no	—
473	C₂²×C₄	—	20	4	no	—
474	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
475	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
476	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
477	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
478	C₄⋊C₄	—	5	4	no	—
479	C₄⋊C₄	—	5	4	no	—
480	C₄⋊C₄	—	5	4	no	—
481	C₄⋊C₄	—	5	4	no	—
482	C₄⋊C₄	—	5	8	no	—
483	C₄⋊C₄	—	5	8	no	—
484	C₄⋊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	4	no	—
489	C₂²⋊C₄	—	5	8	no	—
490	C₂²⋊C₄	C₂²×C₃×D₅	1	8	yes	—
491	C₂²⋊C₄	C₂²×C₃×D₅	1	8	yes	—
492	C₄²	—	5	4	no	—
493	C₄²	—	5	8	no	—
494	C₄²	C₂²×C₃×D₅	1	4	yes	—
495	C₃×C₅	C₂×#542	1	1	yes	characteristic, complemented, Hall
496	C₂²×C₃	—	2	4	no	—
497	C₂²×C₃	—	2	4	no	—
498	C₂²×C₃	—	2	4	no	—
499	C₂²×C₃	—	4	8	no	—
500	C₂²×C₃	—	10	2	no	—
501	C₂²×C₃	—	10	2	no	—
502	C₂²×C₃	—	10	2	no	—
503	C₂²×C₃	—	20	4	no	—
504	C₂²×C₃	C₂×#711	1	2	yes	—
505	C₂²×C₃	#4149	1	2	yes	—
506	C₂²×C₃	#4150	1	2	yes	—
507	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
508	C₃×C₄	—	2	4	no	—
509	C₃×C₄	—	2	2	no	—
510	C₃×C₄	—	2	2	no	—
511	C₃×C₄	—	10	2	no	—
512	C₃×C₄	—	10	2	no	—
513	C₃×C₄	—	10	4	no	—
514	C₃×C₄	—	10	4	no	—
515	C₃×C₄	—	10	2	no	—
516	C₃×C₄	C₂×#711	1	4	yes	—
517	C₂×C₅	—	4	4	no	—
518	C₂×C₅	C₃×#542	1	4	yes	complemented
519	C₂×C₅	C₂²×C₃×#41	1	1	yes	characteristic
520	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
521	C₂×C₅	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic
522	D₅	—	4	2	no	—
523	C₂³	—	2	4	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	4	no	—
526	C₂³	—	5	2	no	—
527	C₂³	—	10	2	no	—
528	C₂³	—	10	2	no	—
529	C₂³	—	10	2	no	—
530	C₂³	C₂³×C₃×D₅	1	4	yes	—
531	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
532	Q₈	—	2	8	no	—
533	Q₈	—	20	8	no	—
534	D₄	—	2	8	no	—
535	D₄	—	20	8	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	5	2	no	—
540	C₂×C₄	—	5	2	no	—
541	C₂×C₄	—	5	4	no	—
542	C₂×C₄	—	5	2	no	—
543	C₂×C₄	—	5	4	no	—
544	C₂×C₄	—	10	2	no	—
545	C₂×C₄	—	10	2	no	—
546	C₂×C₄	—	10	4	no	—
547	C₂×C₄	—	10	4	no	—
548	C₂×C₄	—	10	2	no	—
549	C₂×C₄	—	10	8	no	—
550	C₂×C₄	—	20	8	no	—
551	C₂×C₄	C₃×#711	1	4	yes	—
552	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
553	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
554	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
555	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
556	C₂×C₃	—	4	4	no	—
557	C₂×C₃	—	20	2	no	—
558	C₂×C₃	#66016	1	4	yes	direct product factor
559	C₂×C₃	C₂²×#711	1	1	yes	characteristic
560	C₂×C₃	C₂×#4149	1	1	yes	characteristic
561	C₂×C₃	C₂×#4150	1	1	yes	characteristic
562	C₅	C₂×C₃×#542	1	1	yes	characteristic, complemented, Sylow
563	C₂²	—	2	4	no	—
564	C₂²	—	2	4	no	—
565	C₂²	—	2	4	no	—
566	C₂²	—	4	8	no	—
567	C₂²	—	10	2	no	—
568	C₂²	—	10	2	no	—
569	C₂²	—	10	2	no	—
570	C₂²	—	20	4	no	—
571	C₂²	C₂×C₃×#711	1	2	yes	—
572	C₂²	C₃×#4149	1	2	yes	—
573	C₂²	C₃×#4150	1	2	yes	—
574	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
575	C₄	—	2	4	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	2	no	—
578	C₄	—	10	2	no	—
579	C₄	—	10	2	no	—
580	C₄	—	10	4	no	—
581	C₄	—	10	4	no	—
582	C₄	—	10	2	no	—
583	C₄	C₂×C₃×#711	1	4	yes	—
584	C₃	C₂×#66016	1	1	yes	characteristic, direct product factor, Sylow
585	C₂	—	4	4	no	—
586	C₂	—	20	2	no	—
587	C₂	C₃×#66016	1	4	yes	direct product factor
588	C₂	C₂²×C₃×#711	1	1	yes	characteristic
589	C₂	C₂×C₃×#4149	1	1	yes	characteristic
590	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
prime radical	C₂×C₃×#4084	C₂
hypercenter	C₂×C₃×#124	D₅
hypocenter	C₅	C₂×C₃×#542
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×Q₈×Dic₅	C₂	semidirect
5. C₂×C₃×D₄×Dic₅	C₂	semidirect
6. C₃×#66016	C₂	direct (N × H)
7. C₂×C₃×#4032	C₂	semidirect
8. C₂×C₃×#4031	C₂	semidirect
9. C₂×C₃×#4025	C₂	semidirect
10. C₂×C₃×#4024	C₂	semidirect
11. C₂×C₃×#4018	C₂	semidirect
12. C₂×C₃×#4015	C₂	semidirect
13. C₂×#66016	C₃	direct (N × H)
20. C₃×C₅×#124	C₂²	semidirect
26. C₃×#4092	C₂²	semidirect
27. C₃×Q₈×Dic₅	C₂²	semidirect
29. C₃×D₄×Dic₅	C₂²	semidirect
42. C₃×#4032	C₂²	semidirect
43. C₃×#4031	C₂²	semidirect
44. C₃×#4025	C₂²	semidirect
45. C₃×#4024	C₂²	semidirect
46. C₃×#4018	C₂²	semidirect
47. C₃×#4015	C₂²	semidirect
49. C₂×C₅×#124	C₂×C₃	semidirect
50. C₂×#4092	C₂×C₃	semidirect
51. C₂×Q₈×Dic₅	C₂×C₃	semidirect
53. C₂×D₄×Dic₅	C₂×C₃	semidirect
54. #66016	C₂×C₃	direct (N × H)
55. C₂×#4032	C₂×C₃	semidirect
56. C₂×#4031	C₂×C₃	semidirect
57. C₂×#4025	C₂×C₃	semidirect
58. C₂×#4024	C₂×C₃	semidirect
59. C₂×#4018	C₂×C₃	semidirect
60. C₂×#4015	C₂×C₃	semidirect
105. C₂×C₃×#124	D₅	semidirect
119. C₅×#124	C₂²×C₃	semidirect
125. #4092	C₂²×C₃	semidirect
126. Q₈×Dic₅	C₂²×C₃	semidirect
128. D₄×Dic₅	C₂²×C₃	semidirect
141. #4032	C₂²×C₃	semidirect
142. #4031	C₂²×C₃	semidirect
143. #4025	C₂²×C₃	semidirect
144. #4024	C₂²×C₃	semidirect
145. #4018	C₂²×C₃	semidirect
146. #4015	C₂²×C₃	semidirect
190. C₃×#124	C₂×D₅	semidirect
259. C₂×#124	C₃×D₅	semidirect
370. #124	C₂×C₃×D₅	semidirect
396. C₂×C₃×C₅	#542	semidirect
495. C₃×C₅	C₂×#542	semidirect
518. C₂×C₅	C₃×#542	semidirect
558. C₂×C₃	#66016	direct (N × H)
562. C₅	C₂×C₃×#542	semidirect
584. C₃	C₂×#66016	direct (N × H)
587. C₂	C₃×#66016	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×Q₈×Dic₅	2
C₂×C₃×#4084	2
C₂×C₃×D₄×Dic₅	2
C₃×#66016	2
C₂×C₃×#4032	2
C₂×C₃×#4031	2
C₂×C₃×#4025	2
C₂×C₃×#4024	2
C₂×C₃×#4018	2
C₂×C₃×#4015	2
C₂×#66016	3
C₂×C₃×#542	5

Subgroups — C2×C3×#66016 #14481317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#66016 #14481317