Subgroups — C₂×C₃×#66167 #14480564

Order 1920.

3280 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#66167	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#4092	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#4085	C₂	1	4	yes	maximal, complemented
6	C₂×C₃×#4084	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#4076	C₂	1	2	yes	maximal, complemented
8	C₂²×C₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#4070	C₂	1	2	yes	maximal, complemented
10	C₂×#66167	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅×#41	—	2	8	no	—
12	C₂×C₃×C₅×#41	C₂²	1	8	yes	complemented
13	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
16	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
17	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
18	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
19	C₂²×C₃×C₂²⋊D₅	C₂²	1	4	yes	—
20	C₂³×C₃×Dic₅	C₂²	1	2	yes	—
21	C₂×C₃×#709	—	2	8	no	—
22	C₂×C₃×#709	C₂²	1	8	yes	complemented
23	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
24	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	—
25	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₂²⋊Dic₅	C₂²	1	4	yes	—
27	C₃×#4092	C₂²	1	16	yes	complemented
28	C₃×#4085	C₂²	1	32	yes	complemented
29	C₃×#4084	C₂²	1	16	yes	complemented
30	C₃×#4076	C₂²	1	16	yes	complemented
31	C₂×C₃×#685	C₂²	1	4	yes	—
32	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
33	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
34	C₂×C₃×C₄⋊Dic₅	C₂²	1	8	yes	complemented
35	C₃×#4070	C₂²	1	16	yes	complemented
36	C₂×C₃×#683	C₂²	1	4	yes	—
37	C₂×C₃×#537	—	5	1	no	maximal, Hall
38	C₂²×C₅×#41	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂²×#709	C₂×C₃	1	1	yes	characteristic, complemented
40	#66167	C₂×C₃	1	16	yes	direct product factor
41	C₂×#4092	C₂×C₃	1	2	yes	complemented
42	C₂×#4085	C₂×C₃	1	4	yes	complemented
43	C₂×#4084	C₂×C₃	1	2	yes	complemented
44	C₂×#4076	C₂×C₃	1	2	yes	complemented
45	C₂²×C₄⋊Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×#4070	C₂×C₃	1	2	yes	complemented
47	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
48	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₂³×C₃×D₅	C₂³	1	2	yes	—
50	C₃×C₅×#41	—	2	32	no	—
51	C₂×C₃×C₅×Q₈	—	2	4	no	—
52	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
53	C₂×C₃×C₅×D₄	—	2	8	no	—
54	C₂×C₃×C₅×D₄	—	2	4	no	—
55	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
56	C₂×C₃×C₅×D₄	C₂³	1	8	yes	—
57	C₂²×C₃×C₄×C₅	—	2	8	no	—
58	C₂²×C₃×C₄×C₅	D₄	1	8	yes	complemented
59	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
60	C₂²×C₃×C₄×C₅	C₂³	1	8	yes	—
61	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
62	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
64	C₂×C₃×C₂²⋊D₅	—	2	8	no	—
65	C₂×C₃×C₂²⋊D₅	—	4	8	no	—
66	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	—
67	C₂×C₃×C₂²⋊D₅	C₂³	1	16	yes	—
68	C₂²×C₃×Dic₅	—	4	4	no	—
69	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
70	C₂²×C₃×Dic₅	C₂³	1	4	yes	—
71	C₂²×C₃×Dic₅	C₂³	1	8	yes	—
72	C₃×#709	—	2	32	no	—
73	C₂×C₃×D₂₀	—	2	4	no	—
74	C₂×C₃×D₂₀	C₂³	1	4	yes	—
75	C₂×C₃×C₄×D₅	—	2	8	no	—
76	C₂×C₃×C₄×D₅	C₂³	1	8	yes	—
77	C₂×C₃×Dic₁₀	—	2	4	no	—
78	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
79	C₃×C₂²⋊Dic₅	C₂³	1	16	yes	—
80	C₃×#685	C₂³	1	16	yes	—
81	C₃×C₄⋊Dic₅	C₂³	1	8	yes	—
82	C₃×C₄⋊Dic₅	C₂³	1	8	yes	—
83	C₃×#683	C₂³	1	16	yes	—
84	C₂²×C₃×#41	—	5	1	no	—
85	C₂²×C₃×#41	D₅	1	1	yes	characteristic, complemented
86	C₃×#537	—	5	16	no	—
87	C₂×C₃×C₂²⋊Q₈	—	5	2	no	—
88	C₂×C₃×C₂²⋊Q₈	—	5	2	no	—
89	C₂×C₃×#121	—	5	2	no	—
90	C₂×C₃×#121	—	5	4	no	—
91	C₂×C₃×#121	—	5	2	no	—
92	C₂²×C₃×C₄⋊C₄	—	5	1	no	—
93	C₂×C₅×#41	—	2	8	no	—
94	C₂×C₅×#41	C₂²×C₃	1	8	yes	complemented
95	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
96	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
97	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
98	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
99	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
100	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
101	C₂²×C₂²⋊D₅	C₂²×C₃	1	4	yes	—
102	C₂³×Dic₅	C₂²×C₃	1	2	yes	—
103	C₂×#709	—	2	8	no	—
104	C₂×#709	C₂²×C₃	1	8	yes	complemented
105	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
106	C₂²×C₄×D₅	C₂²×C₃	1	2	yes	—
107	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
108	C₂×C₂²⋊Dic₅	C₂²×C₃	1	4	yes	—
109	#4092	C₂²×C₃	1	16	yes	complemented
110	#4085	C₂²×C₃	1	32	yes	complemented
111	#4084	C₂²×C₃	1	16	yes	complemented
112	#4076	C₂²×C₃	1	16	yes	complemented
113	C₂×#685	C₂²×C₃	1	4	yes	—
114	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
115	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
116	C₂×C₄⋊Dic₅	C₂²×C₃	1	8	yes	complemented
117	#4070	C₂²×C₃	1	16	yes	complemented
118	C₂×#683	C₂²×C₃	1	4	yes	—
119	C₂×#537	—	5	1	no	Sylow
120	C₂³×C₃×C₅	—	2	2	no	—
121	C₂³×C₃×C₅	—	2	2	no	—
122	C₂³×C₃×C₅	—	2	4	no	—
123	C₂³×C₃×C₅	—	2	4	no	—
124	C₂³×C₃×C₅	—	2	4	no	—
125	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
126	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
127	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
128	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
129	C₂²×C₃×D₅	—	2	4	no	—
130	C₂²×C₃×D₅	—	2	4	no	—
131	C₂²×C₃×D₅	—	2	4	no	—
132	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
133	C₃×C₅×Q₈	—	2	8	no	—
134	C₃×C₅×D₄	—	2	16	no	—
135	C₃×C₅×D₄	—	2	8	no	—
136	C₂×C₃×C₄×C₅	—	2	4	no	—
137	C₂×C₃×C₄×C₅	—	2	16	no	—
138	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
139	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
140	C₂×C₃×C₄×C₅	C₂×D₄	1	16	yes	complemented
141	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
142	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
143	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
144	C₃×C₂²⋊D₅	—	2	16	no	—
145	C₃×C₂²⋊D₅	—	4	16	no	—
146	C₂×C₃×Dic₅	—	2	4	no	—
147	C₂×C₃×Dic₅	—	2	8	no	—
148	C₂×C₃×Dic₅	—	4	8	no	—
149	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
150	C₂×C₃×Dic₅	C₂⁴	1	8	yes	—
151	C₃×D₂₀	—	2	8	no	—
152	C₃×C₄×D₅	—	2	16	no	—
153	C₃×Dic₁₀	—	2	8	no	—
154	C₂×C₃×#41	—	2	8	no	—
155	C₂×C₃×#41	—	5	8	no	—
156	C₂×C₃×#41	—	10	8	no	—
157	C₂×C₃×#41	C₂×D₅	1	8	yes	complemented
158	C₂²×C₃×Q₈	—	5	1	no	—
159	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
160	C₂²×C₃×D₄	—	5	2	no	—
161	C₂²×C₃×D₄	—	5	4	no	—
162	C₂²×C₃×D₄	—	5	1	no	—
163	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
164	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
165	C₂³×C₃×C₄	—	5	2	no	—
166	C₂³×C₃×C₄	—	5	2	no	—
167	C₂³×C₃×C₄	C₂×D₅	1	2	yes	—
168	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
169	C₃×C₂²⋊Q₈	—	5	16	no	—
170	C₃×C₂²⋊Q₈	—	5	16	no	—
171	C₃×#121	—	5	16	no	—
172	C₃×#121	—	5	32	no	—
173	C₃×#121	—	5	16	no	—
174	C₂×C₃×C₄⋊C₄	—	5	2	no	—
175	C₂×C₃×C₄⋊C₄	—	5	2	no	—
176	C₂×C₃×C₄⋊C₄	—	5	8	no	—
177	C₂×C₃×C₄⋊C₄	—	5	4	no	—
178	C₂×C₃×C₂²⋊C₄	—	5	4	no	—
179	C₂×C₃×C₂²⋊C₄	—	5	4	no	—
180	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
181	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
182	C₂³×D₅	C₂³×C₃	1	2	yes	—
183	C₅×#41	—	2	32	no	—
184	C₂×C₅×Q₈	—	2	4	no	—
185	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
186	C₂×C₅×D₄	—	2	8	no	—
187	C₂×C₅×D₄	—	2	4	no	—
188	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
189	C₂×C₅×D₄	C₂³×C₃	1	8	yes	—
190	C₂²×C₄×C₅	—	2	8	no	—
191	C₂²×C₄×C₅	C₃×D₄	1	8	yes	complemented
192	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
193	C₂²×C₄×C₅	C₂³×C₃	1	8	yes	—
194	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
195	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
196	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
197	C₂×C₂²⋊D₅	—	2	8	no	—
198	C₂×C₂²⋊D₅	—	4	8	no	—
199	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	—
200	C₂×C₂²⋊D₅	C₂³×C₃	1	16	yes	—
201	C₂²×Dic₅	—	4	4	no	—
202	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
203	C₂²×Dic₅	C₂³×C₃	1	4	yes	—
204	C₂²×Dic₅	C₂³×C₃	1	8	yes	—
205	#709	—	2	32	no	—
206	C₂×D₂₀	—	2	4	no	—
207	C₂×D₂₀	C₂³×C₃	1	4	yes	—
208	C₂×C₄×D₅	—	2	8	no	—
209	C₂×C₄×D₅	C₂³×C₃	1	8	yes	—
210	C₂×Dic₁₀	—	2	4	no	—
211	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
212	C₂²⋊Dic₅	C₂³×C₃	1	16	yes	—
213	#685	C₂³×C₃	1	16	yes	—
214	C₄⋊Dic₅	C₂³×C₃	1	8	yes	—
215	C₄⋊Dic₅	C₂³×C₃	1	8	yes	—
216	#683	C₂³×C₃	1	16	yes	—
217	C₂²×#41	—	5	1	no	—
218	C₂²×#41	C₃×D₅	1	1	yes	characteristic, complemented
219	#537	—	5	16	no	—
220	C₂×C₂²⋊Q₈	—	5	2	no	—
221	C₂×C₂²⋊Q₈	—	5	2	no	—
222	C₂×#121	—	5	2	no	—
223	C₂×#121	—	5	4	no	—
224	C₂×#121	—	5	2	no	—
225	C₂²×C₄⋊C₄	—	5	1	no	—
226	C₂²×C₃×C₅	—	2	8	no	—
227	C₂²×C₃×C₅	—	2	2	no	—
228	C₂²×C₃×C₅	—	2	2	no	—
229	C₂²×C₃×C₅	—	2	4	no	—
230	C₂²×C₃×C₅	—	2	4	no	—
231	C₂²×C₃×C₅	—	2	4	no	—
232	C₂²×C₃×C₅	—	4	8	no	—
233	C₂²×C₃×C₅	C₂²×D₄	1	4	yes	—
234	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
235	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
236	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
237	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
238	C₂×C₃×D₅	—	2	4	no	—
239	C₂×C₃×D₅	—	2	4	no	—
240	C₂×C₃×D₅	—	2	4	no	—
241	C₂×C₃×D₅	—	4	8	no	—
242	C₃×C₄×C₅	—	2	4	no	—
243	C₃×C₄×C₅	C₂²×D₄	1	4	yes	—
244	C₃×C₄×C₅	C₂²×D₄	1	4	yes	—
245	C₃×Dic₅	—	2	4	no	—
246	C₃×Dic₅	—	2	8	no	—
247	C₂⁴×C₃	—	5	2	no	—
248	C₂⁴×C₃	C₂²×D₅	1	2	yes	—
249	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
250	C₃×#41	—	2	32	no	—
251	C₃×#41	—	10	32	no	—
252	C₂×C₃×Q₈	—	2	4	no	—
253	C₂×C₃×Q₈	—	5	4	no	—
254	C₂×C₃×Q₈	—	10	4	no	—
255	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	—
256	C₂×C₃×D₄	—	2	8	no	—
257	C₂×C₃×D₄	—	2	4	no	—
258	C₂×C₃×D₄	—	5	8	no	—
259	C₂×C₃×D₄	—	5	16	no	—
260	C₂×C₃×D₄	—	5	4	no	—
261	C₂×C₃×D₄	—	10	8	no	—
262	C₂×C₃×D₄	—	10	4	no	—
263	C₂×C₃×D₄	—	20	8	no	—
264	C₂×C₃×D₄	C₂²×D₅	1	8	yes	—
265	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
266	C₂²×C₃×C₄	—	2	8	no	—
267	C₂²×C₃×C₄	—	5	4	no	—
268	C₂²×C₃×C₄	—	5	2	no	—
269	C₂²×C₃×C₄	—	5	8	no	—
270	C₂²×C₃×C₄	—	5	8	no	—
271	C₂²×C₃×C₄	—	10	8	no	—
272	C₂²×C₃×C₄	—	20	4	no	—
273	C₂²×C₃×C₄	C₂²⋊D₅	1	8	yes	complemented
274	C₂²×C₃×C₄	C₂²×D₅	1	8	yes	—
275	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
276	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
277	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
278	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
279	C₃×C₄⋊C₄	—	5	16	no	—
280	C₃×C₄⋊C₄	—	5	8	no	—
281	C₃×C₄⋊C₄	—	5	8	no	—
282	C₃×C₂²⋊C₄	—	5	16	no	—
283	C₃×C₂²⋊C₄	—	5	16	no	—
284	C₂³×C₅	—	2	2	no	—
285	C₂³×C₅	—	2	2	no	—
286	C₂³×C₅	—	2	4	no	—
287	C₂³×C₅	—	2	4	no	—
288	C₂³×C₅	—	2	4	no	—
289	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
290	C₂³×C₅	C₂⁴×C₃	1	4	yes	—
291	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
292	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
293	C₂²×D₅	—	2	4	no	—
294	C₂²×D₅	—	2	4	no	—
295	C₂²×D₅	—	2	4	no	—
296	C₂²×D₅	C₂⁴×C₃	1	4	yes	—
297	C₅×Q₈	—	2	8	no	—
298	C₅×D₄	—	2	16	no	—
299	C₅×D₄	—	2	8	no	—
300	C₂×C₄×C₅	—	2	4	no	—
301	C₂×C₄×C₅	—	2	16	no	—
302	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
303	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
304	C₂×C₄×C₅	C₂×C₃×D₄	1	16	yes	complemented
305	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
306	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
307	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
308	C₂²⋊D₅	—	2	16	no	—
309	C₂²⋊D₅	—	4	16	no	—
310	C₂×Dic₅	—	2	4	no	—
311	C₂×Dic₅	—	2	8	no	—
312	C₂×Dic₅	—	4	8	no	—
313	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
314	C₂×Dic₅	C₂⁴×C₃	1	8	yes	—
315	D₂₀	—	2	8	no	—
316	C₄×D₅	—	2	16	no	—
317	Dic₁₀	—	2	8	no	—
318	C₂×#41	—	2	8	no	—
319	C₂×#41	—	5	8	no	—
320	C₂×#41	—	10	8	no	—
321	C₂×#41	C₂×C₃×D₅	1	8	yes	complemented
322	C₂²×Q₈	—	5	1	no	—
323	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
324	C₂²×D₄	—	5	2	no	—
325	C₂²×D₄	—	5	4	no	—
326	C₂²×D₄	—	5	1	no	—
327	C₂²×D₄	C₂×C₃×D₅	1	2	yes	—
328	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
329	C₂³×C₄	—	5	2	no	—
330	C₂³×C₄	—	5	2	no	—
331	C₂³×C₄	C₂×C₃×D₅	1	2	yes	—
332	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
333	C₂²⋊Q₈	—	5	16	no	—
334	C₂²⋊Q₈	—	5	16	no	—
335	#121	—	5	16	no	—
336	#121	—	5	32	no	—
337	#121	—	5	16	no	—
338	C₂×C₄⋊C₄	—	5	2	no	—
339	C₂×C₄⋊C₄	—	5	2	no	—
340	C₂×C₄⋊C₄	—	5	8	no	—
341	C₂×C₄⋊C₄	—	5	4	no	—
342	C₂×C₂²⋊C₄	—	5	4	no	—
343	C₂×C₂²⋊C₄	—	5	4	no	—
344	C₂×C₃×C₅	—	2	4	no	—
345	C₂×C₃×C₅	—	4	4	no	—
346	C₂×C₃×C₅	#537	1	4	yes	complemented
347	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
348	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
349	C₂×C₃×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
350	C₃×D₅	—	4	4	no	—
351	C₂³×C₃	—	2	2	no	—
352	C₂³×C₃	—	2	2	no	—
353	C₂³×C₃	—	2	4	no	—
354	C₂³×C₃	—	2	4	no	—
355	C₂³×C₃	—	2	4	no	—
356	C₂³×C₃	—	5	4	no	—
357	C₂³×C₃	—	10	4	no	—
358	C₂³×C₃	—	10	4	no	—
359	C₂³×C₃	—	10	4	no	—
360	C₂³×C₃	C₂×C₂²⋊D₅	1	4	yes	—
361	C₂³×C₃	C₂³×D₅	1	4	yes	—
362	C₂³×C₃	C₂³×D₅	1	2	yes	—
363	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
364	C₃×Q₈	—	2	8	no	—
365	C₃×Q₈	—	10	8	no	—
366	C₃×D₄	—	2	16	no	—
367	C₃×D₄	—	2	8	no	—
368	C₃×D₄	—	10	16	no	—
369	C₃×D₄	—	10	8	no	—
370	C₃×D₄	—	20	16	no	—
371	C₂×C₃×C₄	—	2	4	no	—
372	C₂×C₃×C₄	—	2	16	no	—
373	C₂×C₃×C₄	—	5	8	no	—
374	C₂×C₃×C₄	—	5	4	no	—
375	C₂×C₃×C₄	—	10	8	no	—
376	C₂×C₃×C₄	—	10	4	no	—
377	C₂×C₃×C₄	—	10	16	no	—
378	C₂×C₃×C₄	—	20	8	no	—
379	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
380	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
381	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	16	yes	complemented
382	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
383	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
384	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
385	C₂²×C₅	—	2	2	no	—
386	C₂²×C₅	—	2	2	no	—
387	C₂²×C₅	—	2	8	no	—
388	C₂²×C₅	—	2	4	no	—
389	C₂²×C₅	—	2	4	no	—
390	C₂²×C₅	—	2	4	no	—
391	C₂²×C₅	—	4	8	no	—
392	C₂²×C₅	C₂²×C₃×D₄	1	4	yes	—
393	C₂²×C₅	C₂²×C₃×D₄	1	2	yes	—
394	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
395	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
396	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
397	C₂×D₅	—	2	4	no	—
398	C₂×D₅	—	2	4	no	—
399	C₂×D₅	—	2	4	no	—
400	C₂×D₅	—	4	8	no	—
401	C₄×C₅	—	2	4	no	—
402	C₄×C₅	C₂²×C₃×D₄	1	4	yes	—
403	C₄×C₅	C₂²×C₃×D₄	1	4	yes	—
404	Dic₅	—	2	4	no	—
405	Dic₅	—	2	8	no	—
406	C₂⁴	—	5	2	no	—
407	C₂⁴	C₂²×C₃×D₅	1	2	yes	—
408	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
409	#41	—	2	32	no	—
410	#41	—	10	32	no	—
411	C₂×Q₈	—	2	4	no	—
412	C₂×Q₈	—	5	4	no	—
413	C₂×Q₈	—	10	4	no	—
414	C₂×Q₈	C₂²×C₃×D₅	1	4	yes	—
415	C₂×D₄	—	2	8	no	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	5	8	no	—
418	C₂×D₄	—	5	16	no	—
419	C₂×D₄	—	5	4	no	—
420	C₂×D₄	—	10	8	no	—
421	C₂×D₄	—	10	4	no	—
422	C₂×D₄	—	20	8	no	—
423	C₂×D₄	C₂²×C₃×D₅	1	8	yes	—
424	C₂×D₄	C₂²×C₃×D₅	1	4	yes	—
425	C₂²×C₄	—	2	8	no	—
426	C₂²×C₄	—	5	4	no	—
427	C₂²×C₄	—	5	2	no	—
428	C₂²×C₄	—	5	8	no	—
429	C₂²×C₄	—	5	8	no	—
430	C₂²×C₄	—	10	8	no	—
431	C₂²×C₄	—	20	4	no	—
432	C₂²×C₄	C₃×C₂²⋊D₅	1	8	yes	complemented
433	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
434	C₂²×C₄	C₂²×C₃×D₅	1	8	yes	—
435	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
436	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
437	C₂²×C₄	C₂²×C₃×D₅	1	4	yes	—
438	C₄⋊C₄	—	5	8	no	—
439	C₄⋊C₄	—	5	8	no	—
440	C₄⋊C₄	—	5	16	no	—
441	C₂²⋊C₄	—	5	16	no	—
442	C₂²⋊C₄	—	5	16	no	—
443	C₃×C₅	C₂×#537	1	1	yes	characteristic, complemented, Hall
444	C₂²×C₃	—	2	8	no	—
445	C₂²×C₃	—	2	2	no	—
446	C₂²×C₃	—	2	2	no	—
447	C₂²×C₃	—	2	4	no	—
448	C₂²×C₃	—	2	4	no	—
449	C₂²×C₃	—	2	4	no	—
450	C₂²×C₃	—	4	8	no	—
451	C₂²×C₃	—	10	4	no	—
452	C₂²×C₃	—	10	4	no	—
453	C₂²×C₃	—	10	4	no	—
454	C₂²×C₃	—	20	8	no	—
455	C₂²×C₃	#4149	1	2	yes	—
456	C₂²×C₃	#4150	1	2	yes	—
457	C₂²×C₃	C₂²×C₂²⋊D₅	1	4	yes	—
458	C₂²×C₃	C₂²×C₂²⋊D₅	1	2	yes	—
459	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
460	C₃×C₄	—	2	4	no	—
461	C₃×C₄	—	10	8	no	—
462	C₃×C₄	—	10	4	no	—
463	C₃×C₄	C₂²×C₂²⋊D₅	1	4	yes	—
464	C₃×C₄	C₂²×C₂²⋊D₅	1	4	yes	—
465	C₂×C₅	—	2	4	no	—
466	C₂×C₅	—	4	4	no	—
467	C₂×C₅	C₃×#537	1	4	yes	complemented
468	C₂×C₅	C₂³×C₃×D₄	1	1	yes	characteristic
469	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
470	C₂×C₅	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic
471	D₅	—	4	4	no	—
472	C₂³	—	2	2	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	4	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	5	4	no	—
478	C₂³	—	10	4	no	—
479	C₂³	—	10	4	no	—
480	C₂³	—	10	4	no	—
481	C₂³	C₂×C₃×C₂²⋊D₅	1	4	yes	—
482	C₂³	C₂³×C₃×D₅	1	4	yes	—
483	C₂³	C₂³×C₃×D₅	1	2	yes	—
484	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
485	Q₈	—	2	8	no	—
486	Q₈	—	10	8	no	—
487	D₄	—	2	16	no	—
488	D₄	—	2	8	no	—
489	D₄	—	10	16	no	—
490	D₄	—	10	8	no	—
491	D₄	—	20	16	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	16	no	—
494	C₂×C₄	—	5	8	no	—
495	C₂×C₄	—	5	4	no	—
496	C₂×C₄	—	10	8	no	—
497	C₂×C₄	—	10	4	no	—
498	C₂×C₄	—	10	16	no	—
499	C₂×C₄	—	20	8	no	—
500	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	4	yes	—
501	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	4	yes	—
502	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	16	yes	complemented
503	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
504	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
505	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
506	C₂×C₃	—	2	4	no	—
507	C₂×C₃	—	4	4	no	—
508	C₂×C₃	—	20	4	no	—
509	C₂×C₃	#66167	1	4	yes	direct product factor
510	C₂×C₃	C₂×#4149	1	1	yes	characteristic
511	C₂×C₃	C₂×#4150	1	1	yes	characteristic
512	C₂×C₃	C₂³×C₂²⋊D₅	1	1	yes	characteristic
513	C₅	C₂×C₃×#537	1	1	yes	characteristic, complemented, Sylow
514	C₂²	—	2	2	no	—
515	C₂²	—	2	2	no	—
516	C₂²	—	2	8	no	—
517	C₂²	—	2	4	no	—
518	C₂²	—	2	4	no	—
519	C₂²	—	2	4	no	—
520	C₂²	—	4	8	no	—
521	C₂²	—	10	4	no	—
522	C₂²	—	10	4	no	—
523	C₂²	—	10	4	no	—
524	C₂²	—	20	8	no	—
525	C₂²	C₃×#4149	1	2	yes	—
526	C₂²	C₃×#4150	1	2	yes	—
527	C₂²	C₂²×C₃×C₂²⋊D₅	1	4	yes	—
528	C₂²	C₂²×C₃×C₂²⋊D₅	1	2	yes	—
529	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
530	C₄	—	2	4	no	—
531	C₄	—	10	8	no	—
532	C₄	—	10	4	no	—
533	C₄	C₂²×C₃×C₂²⋊D₅	1	4	yes	—
534	C₄	C₂²×C₃×C₂²⋊D₅	1	4	yes	—
535	C₃	C₂×#66167	1	1	yes	characteristic, direct product factor, Sylow
536	C₂	—	2	4	no	—
537	C₂	—	4	4	no	—
538	C₂	—	20	4	no	—
539	C₂	C₃×#66167	1	4	yes	direct product factor
540	C₂	C₂×C₃×#4149	1	1	yes	characteristic
541	C₂	C₂×C₃×#4150	1	1	yes	characteristic
542	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	C₂²×C₃×C₅×#41	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂²×C₃×#41	D₅
hypocenter	C₅	C₂×C₃×#537
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×#41	C₂	semidirect
2. C₂²×C₃×#709	C₂	semidirect
3. C₃×#66167	C₂	direct (N × H)
4. C₂×C₃×#4092	C₂	semidirect
5. C₂×C₃×#4085	C₂	semidirect
6. C₂×C₃×#4084	C₂	semidirect
7. C₂×C₃×#4076	C₂	semidirect
8. C₂²×C₃×C₄⋊Dic₅	C₂	semidirect
9. C₂×C₃×#4070	C₂	semidirect
10. C₂×#66167	C₃	direct (N × H)
12. C₂×C₃×C₅×#41	C₂²	semidirect
22. C₂×C₃×#709	C₂²	semidirect
27. C₃×#4092	C₂²	semidirect
28. C₃×#4085	C₂²	semidirect
29. C₃×#4084	C₂²	semidirect
30. C₃×#4076	C₂²	semidirect
34. C₂×C₃×C₄⋊Dic₅	C₂²	semidirect
35. C₃×#4070	C₂²	semidirect
38. C₂²×C₅×#41	C₂×C₃	semidirect
39. C₂²×#709	C₂×C₃	semidirect
40. #66167	C₂×C₃	direct (N × H)
41. C₂×#4092	C₂×C₃	semidirect
42. C₂×#4085	C₂×C₃	semidirect
43. C₂×#4084	C₂×C₃	semidirect
44. C₂×#4076	C₂×C₃	semidirect
45. C₂²×C₄⋊Dic₅	C₂×C₃	semidirect
46. C₂×#4070	C₂×C₃	semidirect
58. C₂²×C₃×C₄×C₅	D₄	semidirect
85. C₂²×C₃×#41	D₅	semidirect
94. C₂×C₅×#41	C₂²×C₃	semidirect
104. C₂×#709	C₂²×C₃	semidirect
109. #4092	C₂²×C₃	semidirect
110. #4085	C₂²×C₃	semidirect
111. #4084	C₂²×C₃	semidirect
112. #4076	C₂²×C₃	semidirect
116. C₂×C₄⋊Dic₅	C₂²×C₃	semidirect
117. #4070	C₂²×C₃	semidirect
140. C₂×C₃×C₄×C₅	C₂×D₄	semidirect
157. C₂×C₃×#41	C₂×D₅	semidirect
191. C₂²×C₄×C₅	C₃×D₄	semidirect
218. C₂²×#41	C₃×D₅	semidirect
273. C₂²×C₃×C₄	C₂²⋊D₅	semidirect
304. C₂×C₄×C₅	C₂×C₃×D₄	semidirect
321. C₂×#41	C₂×C₃×D₅	semidirect
346. C₂×C₃×C₅	#537	semidirect
381. C₂×C₃×C₄	C₂×C₂²⋊D₅	semidirect
432. C₂²×C₄	C₃×C₂²⋊D₅	semidirect
443. C₃×C₅	C₂×#537	semidirect
467. C₂×C₅	C₃×#537	semidirect
502. C₂×C₄	C₂×C₃×C₂²⋊D₅	semidirect
509. C₂×C₃	#66167	direct (N × H)
513. C₅	C₂×C₃×#537	semidirect
535. C₃	C₂×#66167	direct (N × H)
539. C₂	C₃×#66167	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×#41	2
C₂²×C₃×#709	2
C₃×#66167	2
C₂×C₃×#4092	2
C₂×C₃×#4085	2
C₂×C₃×#4084	2
C₂×C₃×#4076	2
C₂²×C₃×C₄⋊Dic₅	2
C₂×C₃×#4070	2
C₂×#66167	3
C₂×C₃×#537	5

Subgroups — C2×C3×#66167 #14480564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#66167 #14480564