Subgroups — C₂×C₃×#65902 #14484001

Order 1920.

2224 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4093	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#4090	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65902	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃×#4044	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#4042	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#4039	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃×#4033	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#4022	C₂	1	2	yes	maximal, complemented
10	C₂×C₃×#4021	C₂	1	1	yes	maximal, characteristic
11	C₂×C₃×#4018	C₂	1	2	yes	maximal, complemented
12	C₂×C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#65902	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	—
17	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
19	C₃×C₄×C₅×Q₈	C₂²	1	8	yes	complemented
20	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
24	C₃×#4093	C₂²	1	8	yes	—
25	C₃×#4090	C₂²	1	8	yes	complemented
26	C₂×C₃×#685	C₂²	1	2	yes	—
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×#685	C₂²	1	2	yes	—
31	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
32	C₂×C₃×#683	C₂²	1	2	yes	—
33	C₂×C₃×#683	C₂²	1	2	yes	—
34	C₂×C₃×#683	C₂²	1	1	yes	characteristic
35	C₂×C₃×#683	C₂²	1	1	yes	characteristic
36	C₂×C₃×#683	C₂²	1	2	yes	—
37	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
38	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
39	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
40	C₃×#4044	C₂²	1	16	yes	complemented
41	C₃×#4042	C₂²	1	16	yes	complemented
42	C₃×#4039	C₂²	1	8	yes	—
43	C₃×#4033	C₂²	1	8	yes	complemented
44	C₃×#4022	C₂²	1	16	yes	complemented
45	C₃×#4021	C₂²	1	8	yes	—
46	C₃×#4018	C₂²	1	16	yes	complemented
47	C₃×C₄×Dic₁₀	C₂²	1	8	yes	complemented
48	C₂×C₃×#556	—	5	1	no	maximal, Hall
49	C₂×C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂×#4093	C₂×C₃	1	1	yes	characteristic
51	C₂×#4090	C₂×C₃	1	1	yes	characteristic, complemented
52	#65902	C₂×C₃	1	16	yes	direct product factor
53	C₂×#4044	C₂×C₃	1	2	yes	complemented
54	C₂×#4042	C₂×C₃	1	2	yes	complemented
55	C₂×#4039	C₂×C₃	1	1	yes	characteristic
56	C₂×#4033	C₂×C₃	1	1	yes	characteristic, complemented
57	C₂×#4022	C₂×C₃	1	2	yes	complemented
58	C₂×#4021	C₂×C₃	1	1	yes	characteristic
59	C₂×#4018	C₂×C₃	1	2	yes	complemented
60	C₂×C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
61	C₂³×C₃×D₅	C₂³	1	2	yes	—
62	C₂×C₃×C₅×Q₈	—	2	4	no	—
63	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
64	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
66	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
68	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
69	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
70	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
71	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
72	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
73	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
74	C₂×C₃×D₂₀	—	2	4	no	—
75	C₂×C₃×D₂₀	C₂³	1	4	yes	—
76	C₂×C₃×C₄×D₅	—	2	8	no	—
77	C₂×C₃×C₄×D₅	C₂³	1	8	yes	—
78	C₂×C₃×Dic₁₀	—	2	4	no	—
79	C₂×C₃×Dic₁₀	—	2	4	no	—
80	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
81	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
82	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
83	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
84	C₃×C₄²×C₅	C₂³	1	4	yes	—
85	C₃×C₄²×C₅	C₂³	1	8	yes	—
86	C₃×#685	C₂³	1	8	yes	—
87	C₃×#685	C₂³	1	8	yes	—
88	C₃×#685	C₂³	1	8	yes	—
89	C₃×#685	C₂³	1	8	yes	—
90	C₃×#685	C₂³	1	8	yes	—
91	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
92	C₃×#683	C₂³	1	8	yes	—
93	C₃×#683	C₂³	1	8	yes	—
94	C₃×#683	C₂³	1	4	yes	—
95	C₃×#683	C₂³	1	4	yes	—
96	C₃×#683	C₂³	1	8	yes	—
97	C₃×C₄×Dic₅	C₂³	1	4	yes	—
98	C₃×C₄×Dic₅	C₂³	1	4	yes	—
99	C₃×C₄×Dic₅	C₂³	1	8	yes	—
100	C₃×#556	—	5	16	no	—
101	C₂×C₃×C₄⋊Q₈	—	5	1	no	—
102	C₂×C₃×#126	—	5	2	no	—
103	C₂×C₃×#126	—	5	2	no	—
104	C₂×C₃×#124	—	5	1	no	—
105	C₂×C₃×#124	—	5	1	no	—
106	C₂×C₃×C₂²⋊Q₈	—	5	2	no	—
107	C₂×C₃×C₄×Q₈	—	5	1	no	—
108	C₂×C₃×C₄×Q₈	—	5	1	no	—
109	C₂×C₃×C₄×Q₈	D₅	1	1	yes	characteristic, complemented
110	C₂×C₃×C₄×D₄	—	5	1	no	—
111	C₂×C₃×#117	—	5	2	no	—
112	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
113	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
114	C₂²×C₄×D₅	C₂²×C₃	1	2	yes	—
115	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
116	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
117	C₄×C₅×Q₈	C₂²×C₃	1	8	yes	complemented
118	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
119	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
120	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
121	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
122	#4093	C₂²×C₃	1	8	yes	—
123	#4090	C₂²×C₃	1	8	yes	complemented
124	C₂×#685	C₂²×C₃	1	2	yes	—
125	C₂×#685	C₂²×C₃	1	2	yes	—
126	C₂×#685	C₂²×C₃	1	2	yes	—
127	C₂×#685	C₂²×C₃	1	2	yes	—
128	C₂×#685	C₂²×C₃	1	2	yes	—
129	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
130	C₂×#683	C₂²×C₃	1	1	yes	characteristic
131	C₂×#683	C₂²×C₃	1	1	yes	characteristic
132	C₂×#683	C₂²×C₃	1	2	yes	—
133	C₂×#683	C₂²×C₃	1	2	yes	—
134	C₂×#683	C₂²×C₃	1	2	yes	—
135	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
136	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
137	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	—
138	#4044	C₂²×C₃	1	16	yes	complemented
139	#4042	C₂²×C₃	1	16	yes	complemented
140	#4039	C₂²×C₃	1	8	yes	—
141	#4033	C₂²×C₃	1	8	yes	complemented
142	#4022	C₂²×C₃	1	16	yes	complemented
143	#4021	C₂²×C₃	1	8	yes	—
144	#4018	C₂²×C₃	1	16	yes	complemented
145	C₄×Dic₁₀	C₂²×C₃	1	8	yes	complemented
146	C₂×#556	—	5	1	no	Sylow
147	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
148	C₂²×C₃×D₅	—	2	4	no	—
149	C₂²×C₃×D₅	—	2	4	no	—
150	C₂²×C₃×D₅	—	2	4	no	—
151	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
152	C₃×C₅×Q₈	—	2	8	no	—
153	C₂×C₃×C₄×C₅	—	2	2	no	—
154	C₂×C₃×C₄×C₅	—	2	4	no	—
155	C₂×C₃×C₄×C₅	—	2	4	no	—
156	C₂×C₃×C₄×C₅	—	2	2	no	—
157	C₂×C₃×C₄×C₅	#41	1	4	yes	—
158	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
159	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
160	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
161	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
162	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
163	C₂×C₃×Dic₅	—	2	4	no	—
164	C₂×C₃×Dic₅	—	2	2	no	—
165	C₂×C₃×Dic₅	—	2	2	no	—
166	C₂×C₃×Dic₅	—	2	2	no	—
167	C₂×C₃×Dic₅	#41	1	4	yes	—
168	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
169	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
170	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
171	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
172	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
173	C₃×D₂₀	—	2	8	no	—
174	C₃×C₄×D₅	—	2	16	no	—
175	C₃×Dic₁₀	—	2	8	no	—
176	C₃×Dic₁₀	—	2	8	no	—
177	C₂²×C₃×Q₈	—	5	1	no	—
178	C₂²×C₃×Q₈	—	5	1	no	—
179	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
180	C₂²×C₃×D₄	—	5	1	no	—
181	C₂³×C₃×C₄	—	5	2	no	—
182	C₃×C₄⋊Q₈	—	5	8	no	—
183	C₃×#126	—	5	16	no	—
184	C₃×#126	—	5	16	no	—
185	C₃×#124	—	5	8	no	—
186	C₃×#124	—	5	8	no	—
187	C₃×C₂²⋊Q₈	—	5	16	no	—
188	C₃×C₄×Q₈	—	5	8	no	—
189	C₃×C₄×Q₈	—	5	8	no	—
190	C₃×C₄×Q₈	C₂×D₅	1	8	yes	complemented
191	C₃×C₄×D₄	—	5	8	no	—
192	C₃×#117	—	5	16	no	—
193	C₂×C₃×C₄⋊C₄	—	5	1	no	—
194	C₂×C₃×C₄⋊C₄	—	5	2	no	—
195	C₂×C₃×C₄⋊C₄	—	5	2	no	—
196	C₂×C₃×C₄⋊C₄	—	5	1	no	—
197	C₂×C₃×C₄⋊C₄	—	5	1	no	—
198	C₂×C₃×C₄⋊C₄	—	5	2	no	—
199	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
200	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
201	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
202	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
203	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
204	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
205	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
206	C₂×C₃×C₄²	—	5	2	no	—
207	C₂×C₃×C₄²	—	5	1	no	—
208	C₂×C₃×C₄²	—	5	1	no	—
209	C₂×C₃×C₄²	C₂×D₅	1	2	yes	—
210	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
211	C₂³×D₅	C₂³×C₃	1	2	yes	—
212	C₂×C₅×Q₈	—	2	4	no	—
213	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
214	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
215	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
216	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
217	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
218	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
219	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
220	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
221	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
222	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
223	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
224	C₂×D₂₀	—	2	4	no	—
225	C₂×D₂₀	C₂³×C₃	1	4	yes	—
226	C₂×C₄×D₅	—	2	8	no	—
227	C₂×C₄×D₅	C₂³×C₃	1	8	yes	—
228	C₂×Dic₁₀	—	2	4	no	—
229	C₂×Dic₁₀	—	2	4	no	—
230	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
231	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
232	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
233	C₅×C₄⋊C₄	C₂³×C₃	1	8	yes	—
234	C₄²×C₅	C₂³×C₃	1	4	yes	—
235	C₄²×C₅	C₂³×C₃	1	8	yes	—
236	#685	C₂³×C₃	1	8	yes	—
237	#685	C₂³×C₃	1	8	yes	—
238	#685	C₂³×C₃	1	8	yes	—
239	#685	C₂³×C₃	1	8	yes	—
240	#685	C₂³×C₃	1	8	yes	—
241	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
242	#683	C₂³×C₃	1	4	yes	—
243	#683	C₂³×C₃	1	4	yes	—
244	#683	C₂³×C₃	1	8	yes	—
245	#683	C₂³×C₃	1	8	yes	—
246	#683	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	#556	—	5	16	no	—
251	C₂×C₄⋊Q₈	—	5	1	no	—
252	C₂×#126	—	5	2	no	—
253	C₂×#126	—	5	2	no	—
254	C₂×#124	—	5	1	no	—
255	C₂×#124	—	5	1	no	—
256	C₂×C₂²⋊Q₈	—	5	2	no	—
257	C₂×C₄×Q₈	—	5	1	no	—
258	C₂×C₄×Q₈	—	5	1	no	—
259	C₂×C₄×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
260	C₂×C₄×D₄	—	5	1	no	—
261	C₂×#117	—	5	2	no	—
262	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
263	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
264	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
265	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
266	C₂×C₃×D₅	—	2	4	no	—
267	C₂×C₃×D₅	—	2	4	no	—
268	C₂×C₃×D₅	—	2	4	no	—
269	C₂×C₃×D₅	—	4	8	no	—
270	C₃×C₄×C₅	—	2	2	no	—
271	C₃×C₄×C₅	—	2	4	no	—
272	C₃×C₄×C₅	—	2	4	no	—
273	C₃×C₄×C₅	—	2	2	no	—
274	C₃×C₄×C₅	C₂×#41	1	4	yes	—
275	C₃×Dic₅	—	2	4	no	—
276	C₃×Dic₅	—	2	2	no	—
277	C₃×Dic₅	—	2	2	no	—
278	C₃×Dic₅	—	2	2	no	—
279	C₃×Dic₅	C₂×#41	1	4	yes	—
280	C₂⁴×C₃	—	5	2	no	—
281	C₂×C₃×Q₈	—	2	4	no	—
282	C₂×C₃×Q₈	—	5	4	no	—
283	C₂×C₃×Q₈	—	5	4	no	—
284	C₂×C₃×Q₈	—	10	4	no	—
285	C₂×C₃×Q₈	—	10	4	no	—
286	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	—
287	C₂×C₃×D₄	—	5	4	no	—
288	C₂×C₃×D₄	—	10	4	no	—
289	C₂²×C₃×C₄	—	5	1	no	—
290	C₂²×C₃×C₄	—	5	1	no	—
291	C₂²×C₃×C₄	—	5	1	no	—
292	C₂²×C₃×C₄	—	5	2	no	—
293	C₂²×C₃×C₄	—	5	1	no	—
294	C₂²×C₃×C₄	—	5	8	no	—
295	C₂²×C₃×C₄	—	10	8	no	—
296	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
297	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
298	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
299	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
300	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
301	C₃×C₄⋊C₄	—	5	4	no	—
302	C₃×C₄⋊C₄	—	5	8	no	—
303	C₃×C₄⋊C₄	—	5	8	no	—
304	C₃×C₄⋊C₄	—	5	4	no	—
305	C₃×C₄⋊C₄	—	5	8	no	—
306	C₃×C₄⋊C₄	—	5	4	no	—
307	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
308	C₃×C₄⋊C₄	C₂²×D₅	1	8	yes	—
309	C₃×C₂²⋊C₄	—	5	8	no	—
310	C₃×C₂²⋊C₄	—	5	8	no	—
311	C₃×C₂²⋊C₄	—	5	8	no	—
312	C₃×C₂²⋊C₄	—	5	8	no	—
313	C₃×C₂²⋊C₄	—	5	8	no	—
314	C₃×C₄²	—	5	8	no	—
315	C₃×C₄²	—	5	4	no	—
316	C₃×C₄²	—	5	4	no	—
317	C₃×C₄²	C₂²×D₅	1	8	yes	—
318	C₃×C₄²	C₂²×D₅	1	4	yes	—
319	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
320	C₂²×D₅	—	2	4	no	—
321	C₂²×D₅	—	2	4	no	—
322	C₂²×D₅	—	2	4	no	—
323	C₂²×D₅	C₂⁴×C₃	1	4	yes	—
324	C₅×Q₈	—	2	8	no	—
325	C₂×C₄×C₅	—	2	2	no	—
326	C₂×C₄×C₅	—	2	4	no	—
327	C₂×C₄×C₅	—	2	4	no	—
328	C₂×C₄×C₅	—	2	2	no	—
329	C₂×C₄×C₅	C₃×#41	1	4	yes	—
330	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
331	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
332	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
333	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
334	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
335	C₂×Dic₅	—	2	4	no	—
336	C₂×Dic₅	—	2	2	no	—
337	C₂×Dic₅	—	2	2	no	—
338	C₂×Dic₅	—	2	2	no	—
339	C₂×Dic₅	C₃×#41	1	4	yes	—
340	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
341	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
342	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
343	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
344	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
345	D₂₀	—	2	8	no	—
346	C₄×D₅	—	2	16	no	—
347	Dic₁₀	—	2	8	no	—
348	Dic₁₀	—	2	8	no	—
349	C₂²×Q₈	—	5	1	no	—
350	C₂²×Q₈	—	5	1	no	—
351	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
352	C₂²×D₄	—	5	1	no	—
353	C₂³×C₄	—	5	2	no	—
354	C₄⋊Q₈	—	5	8	no	—
355	#126	—	5	16	no	—
356	#126	—	5	16	no	—
357	#124	—	5	8	no	—
358	#124	—	5	8	no	—
359	C₂²⋊Q₈	—	5	16	no	—
360	C₄×Q₈	—	5	8	no	—
361	C₄×Q₈	—	5	8	no	—
362	C₄×Q₈	C₂×C₃×D₅	1	8	yes	complemented
363	C₄×D₄	—	5	8	no	—
364	#117	—	5	16	no	—
365	C₂×C₄⋊C₄	—	5	1	no	—
366	C₂×C₄⋊C₄	—	5	2	no	—
367	C₂×C₄⋊C₄	—	5	2	no	—
368	C₂×C₄⋊C₄	—	5	1	no	—
369	C₂×C₄⋊C₄	—	5	1	no	—
370	C₂×C₄⋊C₄	—	5	2	no	—
371	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
372	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
373	C₂×C₂²⋊C₄	—	5	2	no	—
374	C₂×C₂²⋊C₄	—	5	2	no	—
375	C₂×C₂²⋊C₄	—	5	2	no	—
376	C₂×C₂²⋊C₄	—	5	2	no	—
377	C₂×C₂²⋊C₄	—	5	2	no	—
378	C₂×C₄²	—	5	2	no	—
379	C₂×C₄²	—	5	1	no	—
380	C₂×C₄²	—	5	1	no	—
381	C₂×C₄²	C₂×C₃×D₅	1	2	yes	—
382	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
383	C₂×C₃×C₅	#556	1	4	yes	complemented
384	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
385	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
386	C₂×C₃×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
387	C₃×D₅	—	4	4	no	—
388	C₂³×C₃	—	5	4	no	—
389	C₂³×C₃	—	10	4	no	—
390	C₂³×C₃	—	10	4	no	—
391	C₂³×C₃	—	10	4	no	—
392	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
393	C₃×Q₈	—	2	8	no	—
394	C₃×Q₈	—	10	8	no	—
395	C₃×Q₈	—	10	8	no	—
396	C₃×D₄	—	10	8	no	—
397	C₂×C₃×C₄	—	2	4	no	—
398	C₂×C₃×C₄	—	2	2	no	—
399	C₂×C₃×C₄	—	2	4	no	—
400	C₂×C₃×C₄	—	2	2	no	—
401	C₂×C₃×C₄	—	5	4	no	—
402	C₂×C₃×C₄	—	5	2	no	—
403	C₂×C₃×C₄	—	5	2	no	—
404	C₂×C₃×C₄	—	5	2	no	—
405	C₂×C₃×C₄	—	5	4	no	—
406	C₂×C₃×C₄	—	5	2	no	—
407	C₂×C₃×C₄	—	10	16	no	—
408	C₂×C₃×C₄	—	10	4	no	—
409	C₂×C₃×C₄	—	10	2	no	—
410	C₂×C₃×C₄	—	10	2	no	—
411	C₂×C₃×C₄	—	10	2	no	—
412	C₂×C₃×C₄	#709	1	4	yes	—
413	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
414	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
415	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
416	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
417	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
418	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
419	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
420	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
421	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
422	C₂×D₅	—	2	4	no	—
423	C₂×D₅	—	2	4	no	—
424	C₂×D₅	—	2	4	no	—
425	C₂×D₅	—	4	8	no	—
426	C₄×C₅	—	2	2	no	—
427	C₄×C₅	—	2	4	no	—
428	C₄×C₅	—	2	4	no	—
429	C₄×C₅	—	2	2	no	—
430	C₄×C₅	C₂×C₃×#41	1	4	yes	—
431	Dic₅	—	2	4	no	—
432	Dic₅	—	2	2	no	—
433	Dic₅	—	2	2	no	—
434	Dic₅	—	2	2	no	—
435	Dic₅	C₂×C₃×#41	1	4	yes	—
436	C₂⁴	—	5	2	no	—
437	C₂×Q₈	—	2	4	no	—
438	C₂×Q₈	—	5	4	no	—
439	C₂×Q₈	—	5	4	no	—
440	C₂×Q₈	—	10	4	no	—
441	C₂×Q₈	—	10	4	no	—
442	C₂×Q₈	C₂²×C₃×D₅	1	4	yes	—
443	C₂×D₄	—	5	4	no	—
444	C₂×D₄	—	10	4	no	—
445	C₂²×C₄	—	5	1	no	—
446	C₂²×C₄	—	5	1	no	—
447	C₂²×C₄	—	5	1	no	—
448	C₂²×C₄	—	5	2	no	—
449	C₂²×C₄	—	5	1	no	—
450	C₂²×C₄	—	5	8	no	—
451	C₂²×C₄	—	10	8	no	—
452	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
453	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
454	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
455	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
456	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
457	C₄⋊C₄	—	5	4	no	—
458	C₄⋊C₄	—	5	8	no	—
459	C₄⋊C₄	—	5	8	no	—
460	C₄⋊C₄	—	5	4	no	—
461	C₄⋊C₄	—	5	4	no	—
462	C₄⋊C₄	—	5	8	no	—
463	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
464	C₄⋊C₄	C₂²×C₃×D₅	1	8	yes	—
465	C₂²⋊C₄	—	5	8	no	—
466	C₂²⋊C₄	—	5	8	no	—
467	C₂²⋊C₄	—	5	8	no	—
468	C₂²⋊C₄	—	5	8	no	—
469	C₂²⋊C₄	—	5	8	no	—
470	C₄²	—	5	8	no	—
471	C₄²	—	5	4	no	—
472	C₄²	—	5	4	no	—
473	C₄²	C₂²×C₃×D₅	1	8	yes	—
474	C₄²	C₂²×C₃×D₅	1	4	yes	—
475	C₃×C₅	C₂×#556	1	1	yes	characteristic, complemented, Hall
476	C₂²×C₃	—	10	4	no	—
477	C₂²×C₃	—	10	4	no	—
478	C₂²×C₃	—	10	4	no	—
479	C₂²×C₃	—	20	8	no	—
480	C₂²×C₃	C₂×#709	1	2	yes	—
481	C₂²×C₃	#4147	1	2	yes	—
482	C₂²×C₃	D₅×#41	1	2	yes	—
483	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
484	C₃×C₄	—	2	4	no	—
485	C₃×C₄	—	2	2	no	—
486	C₃×C₄	—	2	4	no	—
487	C₃×C₄	—	2	2	no	—
488	C₃×C₄	—	5	4	no	—
489	C₃×C₄	—	10	4	no	—
490	C₃×C₄	—	10	2	no	—
491	C₃×C₄	—	10	2	no	—
492	C₃×C₄	—	10	2	no	—
493	C₃×C₄	C₂×#709	1	4	yes	—
494	C₂×C₅	C₃×#556	1	4	yes	complemented
495	C₂×C₅	C₂²×C₃×#41	1	1	yes	characteristic
496	C₂×C₅	C₂²×C₃×#41	1	1	yes	characteristic
497	C₂×C₅	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic
498	D₅	—	4	4	no	—
499	C₂³	—	5	4	no	—
500	C₂³	—	10	4	no	—
501	C₂³	—	10	4	no	—
502	C₂³	—	10	4	no	—
503	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
504	Q₈	—	2	8	no	—
505	Q₈	—	10	8	no	—
506	Q₈	—	10	8	no	—
507	D₄	—	10	8	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	5	4	no	—
513	C₂×C₄	—	5	2	no	—
514	C₂×C₄	—	5	2	no	—
515	C₂×C₄	—	5	2	no	—
516	C₂×C₄	—	5	4	no	—
517	C₂×C₄	—	5	2	no	—
518	C₂×C₄	—	10	16	no	—
519	C₂×C₄	—	10	4	no	—
520	C₂×C₄	—	10	2	no	—
521	C₂×C₄	—	10	2	no	—
522	C₂×C₄	—	10	2	no	—
523	C₂×C₄	C₃×#709	1	4	yes	—
524	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
525	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
526	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
527	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
528	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
529	C₂×C₃	—	20	4	no	—
530	C₂×C₃	#65902	1	4	yes	direct product factor
531	C₂×C₃	C₂²×#709	1	1	yes	characteristic
532	C₂×C₃	C₂×#4147	1	1	yes	characteristic
533	C₂×C₃	C₂×D₅×#41	1	1	yes	characteristic
534	C₅	C₂×C₃×#556	1	1	yes	characteristic, complemented, Sylow
535	C₂²	—	10	4	no	—
536	C₂²	—	10	4	no	—
537	C₂²	—	10	4	no	—
538	C₂²	—	20	8	no	—
539	C₂²	C₂×C₃×#709	1	2	yes	—
540	C₂²	C₃×#4147	1	2	yes	—
541	C₂²	C₃×D₅×#41	1	2	yes	—
542	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
543	C₄	—	2	4	no	—
544	C₄	—	2	2	no	—
545	C₄	—	2	4	no	—
546	C₄	—	2	2	no	—
547	C₄	—	5	4	no	—
548	C₄	—	10	4	no	—
549	C₄	—	10	2	no	—
550	C₄	—	10	2	no	—
551	C₄	—	10	2	no	—
552	C₄	C₂×C₃×#709	1	4	yes	—
553	C₃	C₂×#65902	1	1	yes	characteristic, direct product factor, Sylow
554	C₂	—	20	4	no	—
555	C₂	C₃×#65902	1	4	yes	direct product factor
556	C₂	C₂²×C₃×#709	1	1	yes	characteristic
557	C₂	C₂×C₃×#4147	1	1	yes	characteristic
558	C₂	C₂×C₃×D₅×#41	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×C₄×C₅×Q₈	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂²×C₃×D₂₀	C₂²
hypercenter	C₂×C₃×C₄×Q₈	D₅
hypocenter	C₅	C₂×C₃×#556
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₅×Q₈	C₂	semidirect
3. C₂×C₃×#4090	C₂	semidirect
4. C₃×#65902	C₂	direct (N × H)
5. C₂×C₃×#4044	C₂	semidirect
6. C₂×C₃×#4042	C₂	semidirect
8. C₂×C₃×#4033	C₂	semidirect
9. C₂×C₃×#4022	C₂	semidirect
11. C₂×C₃×#4018	C₂	semidirect
12. C₂×C₃×C₄×Dic₁₀	C₂	semidirect
13. C₂×#65902	C₃	direct (N × H)
19. C₃×C₄×C₅×Q₈	C₂²	semidirect
25. C₃×#4090	C₂²	semidirect
40. C₃×#4044	C₂²	semidirect
41. C₃×#4042	C₂²	semidirect
43. C₃×#4033	C₂²	semidirect
44. C₃×#4022	C₂²	semidirect
46. C₃×#4018	C₂²	semidirect
47. C₃×C₄×Dic₁₀	C₂²	semidirect
49. C₂×C₄×C₅×Q₈	C₂×C₃	semidirect
51. C₂×#4090	C₂×C₃	semidirect
52. #65902	C₂×C₃	direct (N × H)
53. C₂×#4044	C₂×C₃	semidirect
54. C₂×#4042	C₂×C₃	semidirect
56. C₂×#4033	C₂×C₃	semidirect
57. C₂×#4022	C₂×C₃	semidirect
59. C₂×#4018	C₂×C₃	semidirect
60. C₂×C₄×Dic₁₀	C₂×C₃	semidirect
109. C₂×C₃×C₄×Q₈	D₅	semidirect
117. C₄×C₅×Q₈	C₂²×C₃	semidirect
123. #4090	C₂²×C₃	semidirect
138. #4044	C₂²×C₃	semidirect
139. #4042	C₂²×C₃	semidirect
141. #4033	C₂²×C₃	semidirect
142. #4022	C₂²×C₃	semidirect
144. #4018	C₂²×C₃	semidirect
145. C₄×Dic₁₀	C₂²×C₃	semidirect
190. C₃×C₄×Q₈	C₂×D₅	semidirect
259. C₂×C₄×Q₈	C₃×D₅	semidirect
362. C₄×Q₈	C₂×C₃×D₅	semidirect
383. C₂×C₃×C₅	#556	semidirect
475. C₃×C₅	C₂×#556	semidirect
494. C₂×C₅	C₃×#556	semidirect
530. C₂×C₃	#65902	direct (N × H)
534. C₅	C₂×C₃×#556	semidirect
553. C₃	C₂×#65902	direct (N × H)
555. C₂	C₃×#65902	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×C₄×C₅×Q₈	2
C₂×C₃×#4093	2
C₂×C₃×#4090	2
C₃×#65902	2
C₂×C₃×#4044	2
C₂×C₃×#4042	2
C₂×C₃×#4039	2
C₂×C₃×#4033	2
C₂×C₃×#4022	2
C₂×C₃×#4021	2
C₂×C₃×#4018	2
C₂×C₃×C₄×Dic₁₀	2
C₂×#65902	3
C₂×C₃×#556	5

Subgroups — C2×C3×#65902 #14484001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65902 #14484001