Subgroups — C₂×C₃×C₅×#454 #14276050

Order 1920.

1456 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#454	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×C₅×#104	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×C₅×#101	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#454	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃×C₅×2₋¹⁺⁴	—	2	4	no	—
9	C₃×C₅×2₋¹⁺⁴	—	2	2	no	—
10	C₃×C₅×2₋¹⁺⁴	C₂²	1	4	yes	complemented
11	C₂×C₃×C₅×#41	—	2	2	no	—
12	C₂×C₃×C₅×#41	—	2	2	no	—
13	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
14	C₂²×C₃×C₅×Q₈	—	2	2	no	—
15	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
17	C₃×C₅×AGL(1,ℤ/8)	—	2	4	no	—
18	C₃×C₅×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
19	C₂×C₃×C₅×SD₁₆	—	2	2	no	—
20	C₂×C₃×C₅×D₈	—	2	2	no	—
21	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
22	C₃×C₅×#124	—	2	2	no	—
23	C₃×C₅×#124	C₂²	1	4	yes	complemented
24	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
25	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
26	C₃×C₅×#104	C₂²	1	8	yes	complemented
27	C₃×C₅×#101	C₂²	1	4	yes	complemented
28	C₂×C₃×#454	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₂×C₅×2₋¹⁺⁴	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	1	2	yes	complemented
31	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic, complemented
32	C₅×#454	C₂×C₃	1	8	yes	direct product factor
33	C₂×C₅×#104	C₂×C₃	1	2	yes	complemented
34	C₂×C₅×#101	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂⁴×C₃×C₅	—	2	1	no	—
36	C₃×C₅×#41	—	2	8	no	—
37	C₃×C₅×#41	—	2	8	no	—
38	C₃×C₅×#41	—	2	8	no	—
39	C₃×C₅×#41	—	2	8	no	—
40	C₃×C₅×#41	—	2	2	no	—
41	C₃×C₅×#41	—	2	2	no	—
42	C₃×C₅×#41	D₄	1	4	yes	—
43	C₃×C₅×#41	C₂³	1	4	yes	—
44	C₂×C₃×C₅×Q₈	—	2	2	no	—
45	C₂×C₃×C₅×Q₈	—	2	8	no	—
46	C₂×C₃×C₅×Q₈	—	2	8	no	—
47	C₂×C₃×C₅×Q₈	—	2	2	no	—
48	C₂×C₃×C₅×Q₈	—	2	2	no	—
49	C₂×C₃×C₅×Q₈	D₄	1	2	yes	—
50	C₂×C₃×C₅×Q₈	D₄	1	2	yes	—
51	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
52	C₂×C₃×C₅×D₄	—	2	2	no	—
53	C₂×C₃×C₅×D₄	—	2	2	no	—
54	C₂×C₃×C₅×D₄	—	2	2	no	—
55	C₂×C₃×C₅×D₄	—	2	2	no	—
56	C₂×C₃×C₅×D₄	—	2	1	no	—
57	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
58	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
59	C₂²×C₃×C₄×C₅	—	2	4	no	—
60	C₂²×C₃×C₄×C₅	—	2	2	no	—
61	C₂²×C₃×C₄×C₅	—	2	1	no	—
62	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
63	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
64	C₃×C₅×SD₁₆	—	2	8	no	—
65	C₃×C₅×D₈	—	2	8	no	—
66	C₃×C₅×M₁₆	—	2	2	no	—
67	C₃×C₅×M₁₆	C₂³	1	4	yes	—
68	C₂×C₃×C₅×C₈	—	2	2	no	—
69	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
70	C₃×C₄²×C₅	—	2	1	no	—
71	C₃×C₄²×C₅	C₂³	1	2	yes	—
72	C₂×C₃×2₋¹⁺⁴	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	1	2	yes	complemented
74	C₂×C₃×#124	C₂×C₅	1	1	yes	characteristic, complemented
75	C₃×#454	C₂×C₅	1	8	yes	direct product factor
76	C₂×C₃×#104	C₂×C₅	1	2	yes	complemented
77	C₂×C₃×#101	C₂×C₅	1	1	yes	characteristic, complemented
78	C₅×2₋¹⁺⁴	—	2	4	no	—
79	C₅×2₋¹⁺⁴	—	2	2	no	—
80	C₅×2₋¹⁺⁴	C₂²×C₃	1	4	yes	complemented
81	C₂×C₅×#41	—	2	2	no	—
82	C₂×C₅×#41	—	2	2	no	—
83	C₂×C₅×#41	C₂²×C₃	1	2	yes	—
84	C₂²×C₅×Q₈	—	2	2	no	—
85	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
86	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
87	C₅×AGL(1,ℤ/8)	—	2	4	no	—
88	C₅×AGL(1,ℤ/8)	C₂²×C₃	1	8	yes	complemented
89	C₂×C₅×SD₁₆	—	2	2	no	—
90	C₂×C₅×D₈	—	2	2	no	—
91	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
92	C₅×#124	—	2	2	no	—
93	C₅×#124	C₂²×C₃	1	4	yes	complemented
94	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
95	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
96	C₅×#104	C₂²×C₃	1	8	yes	complemented
97	C₅×#101	C₂²×C₃	1	4	yes	complemented
98	C₂×#454	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
99	C₂³×C₃×C₅	—	2	2	no	—
100	C₂³×C₃×C₅	—	2	2	no	—
101	C₂³×C₃×C₅	—	4	4	no	—
102	C₂³×C₃×C₅	—	4	1	no	—
103	C₂³×C₃×C₅	—	4	1	no	—
104	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
105	C₃×C₅×Q₈	—	2	8	no	—
106	C₃×C₅×Q₈	—	2	2	no	—
107	C₃×C₅×Q₈	—	2	4	no	—
108	C₃×C₅×Q₈	—	2	4	no	—
109	C₃×C₅×Q₈	C₂×D₄	1	4	yes	—
110	C₃×C₅×D₄	—	2	8	no	—
111	C₃×C₅×D₄	—	2	4	no	—
112	C₃×C₅×D₄	—	2	4	no	—
113	C₃×C₅×D₄	—	2	2	no	—
114	C₃×C₅×D₄	—	2	4	no	—
115	C₃×C₅×D₄	—	4	2	no	—
116	C₃×C₅×D₄	C₂×D₄	1	4	yes	—
117	C₂×C₃×C₄×C₅	—	2	16	no	—
118	C₂×C₃×C₄×C₅	—	2	2	no	—
119	C₂×C₃×C₄×C₅	—	2	2	no	—
120	C₂×C₃×C₄×C₅	—	2	2	no	—
121	C₂×C₃×C₄×C₅	—	2	4	no	—
122	C₂×C₃×C₄×C₅	—	2	4	no	—
123	C₂×C₃×C₄×C₅	—	2	2	no	—
124	C₂×C₃×C₄×C₅	—	4	2	no	—
125	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
126	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
127	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
128	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
129	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
130	C₃×C₅×C₈	—	2	4	no	—
131	C₃×2₋¹⁺⁴	—	2	4	no	—
132	C₃×2₋¹⁺⁴	—	2	2	no	—
133	C₃×2₋¹⁺⁴	C₂²×C₅	1	4	yes	complemented
134	C₂×C₃×#41	—	2	2	no	—
135	C₂×C₃×#41	—	2	2	no	—
136	C₂×C₃×#41	C₂²×C₅	1	2	yes	—
137	C₂²×C₃×Q₈	—	2	2	no	—
138	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
139	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
140	C₃×AGL(1,ℤ/8)	—	2	4	no	—
141	C₃×AGL(1,ℤ/8)	C₂²×C₅	1	8	yes	complemented
142	C₂×C₃×SD₁₆	—	2	2	no	—
143	C₂×C₃×D₈	—	2	2	no	—
144	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
145	C₃×#124	—	2	2	no	—
146	C₃×#124	C₂²×C₅	1	4	yes	complemented
147	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
148	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
149	C₃×#104	C₂²×C₅	1	8	yes	complemented
150	C₃×#101	C₂²×C₅	1	4	yes	complemented
151	C₂⁴×C₅	—	2	1	no	—
152	C₅×#41	—	2	8	no	—
153	C₅×#41	—	2	8	no	—
154	C₅×#41	—	2	8	no	—
155	C₅×#41	—	2	8	no	—
156	C₅×#41	—	2	2	no	—
157	C₅×#41	—	2	2	no	—
158	C₅×#41	C₃×D₄	1	4	yes	—
159	C₅×#41	C₂³×C₃	1	4	yes	—
160	C₂×C₅×Q₈	—	2	2	no	—
161	C₂×C₅×Q₈	—	2	8	no	—
162	C₂×C₅×Q₈	—	2	8	no	—
163	C₂×C₅×Q₈	—	2	2	no	—
164	C₂×C₅×Q₈	—	2	2	no	—
165	C₂×C₅×Q₈	C₃×D₄	1	2	yes	—
166	C₂×C₅×Q₈	C₃×D₄	1	2	yes	—
167	C₂×C₅×Q₈	C₂³×C₃	1	2	yes	—
168	C₂×C₅×D₄	—	2	2	no	—
169	C₂×C₅×D₄	—	2	2	no	—
170	C₂×C₅×D₄	—	2	2	no	—
171	C₂×C₅×D₄	—	2	2	no	—
172	C₂×C₅×D₄	—	2	1	no	—
173	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
174	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
175	C₂²×C₄×C₅	—	2	4	no	—
176	C₂²×C₄×C₅	—	2	2	no	—
177	C₂²×C₄×C₅	—	2	1	no	—
178	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
179	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
180	C₅×SD₁₆	—	2	8	no	—
181	C₅×D₈	—	2	8	no	—
182	C₅×M₁₆	—	2	2	no	—
183	C₅×M₁₆	C₂³×C₃	1	4	yes	—
184	C₂×C₅×C₈	—	2	2	no	—
185	C₅×C₂²⋊C₄	—	2	8	no	—
186	C₄²×C₅	—	2	1	no	—
187	C₄²×C₅	C₂³×C₃	1	2	yes	—
188	C₂×2₋¹⁺⁴	C₂×C₃×C₅	1	1	yes	characteristic, complemented
189	C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	1	2	yes	complemented
190	C₂×#124	C₂×C₃×C₅	1	1	yes	characteristic, complemented
191	#454	C₂×C₃×C₅	1	8	yes	direct product factor
192	C₂×#104	C₂×C₃×C₅	1	2	yes	complemented
193	C₂×#101	C₂×C₃×C₅	1	1	yes	characteristic, complemented
194	C₂²×C₃×C₅	—	2	2	no	—
195	C₂²×C₃×C₅	—	2	4	no	—
196	C₂²×C₃×C₅	—	4	4	no	—
197	C₂²×C₃×C₅	—	4	4	no	—
198	C₂²×C₃×C₅	—	4	2	no	—
199	C₂²×C₃×C₅	—	8	2	no	—
200	C₂²×C₃×C₅	—	8	2	no	—
201	C₂²×C₃×C₅	#120	1	1	yes	characteristic
202	C₂²×C₃×C₅	#120	1	1	yes	characteristic
203	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
204	C₃×C₄×C₅	—	2	4	no	—
205	C₃×C₄×C₅	—	2	4	no	—
206	C₃×C₄×C₅	—	4	2	no	—
207	C₃×C₄×C₅	#120	1	2	yes	—
208	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
209	C₂⁴×C₃	—	2	1	no	—
210	C₃×#41	—	2	8	no	—
211	C₃×#41	—	2	8	no	—
212	C₃×#41	—	2	8	no	—
213	C₃×#41	—	2	8	no	—
214	C₃×#41	—	2	2	no	—
215	C₃×#41	—	2	2	no	—
216	C₃×#41	C₅×D₄	1	4	yes	—
217	C₃×#41	C₂³×C₅	1	4	yes	—
218	C₂×C₃×Q₈	—	2	2	no	—
219	C₂×C₃×Q₈	—	2	8	no	—
220	C₂×C₃×Q₈	—	2	8	no	—
221	C₂×C₃×Q₈	—	2	2	no	—
222	C₂×C₃×Q₈	—	2	2	no	—
223	C₂×C₃×Q₈	C₅×D₄	1	2	yes	—
224	C₂×C₃×Q₈	C₅×D₄	1	2	yes	—
225	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
226	C₂×C₃×D₄	—	2	2	no	—
227	C₂×C₃×D₄	—	2	2	no	—
228	C₂×C₃×D₄	—	2	2	no	—
229	C₂×C₃×D₄	—	2	2	no	—
230	C₂×C₃×D₄	—	2	1	no	—
231	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
232	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
233	C₂²×C₃×C₄	—	2	4	no	—
234	C₂²×C₃×C₄	—	2	2	no	—
235	C₂²×C₃×C₄	—	2	1	no	—
236	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
237	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
238	C₃×SD₁₆	—	2	8	no	—
239	C₃×D₈	—	2	8	no	—
240	C₃×M₁₆	—	2	2	no	—
241	C₃×M₁₆	C₂³×C₅	1	4	yes	—
242	C₂×C₃×C₈	—	2	2	no	—
243	C₃×C₂²⋊C₄	—	2	8	no	—
244	C₃×C₄²	—	2	1	no	—
245	C₃×C₄²	C₂³×C₅	1	2	yes	—
246	C₂³×C₅	—	2	2	no	—
247	C₂³×C₅	—	2	2	no	—
248	C₂³×C₅	—	4	4	no	—
249	C₂³×C₅	—	4	1	no	—
250	C₂³×C₅	—	4	1	no	—
251	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
252	C₅×Q₈	—	2	8	no	—
253	C₅×Q₈	—	2	2	no	—
254	C₅×Q₈	—	2	4	no	—
255	C₅×Q₈	—	2	4	no	—
256	C₅×Q₈	C₂×C₃×D₄	1	4	yes	—
257	C₅×D₄	—	2	8	no	—
258	C₅×D₄	—	2	4	no	—
259	C₅×D₄	—	2	4	no	—
260	C₅×D₄	—	2	2	no	—
261	C₅×D₄	—	2	4	no	—
262	C₅×D₄	—	4	2	no	—
263	C₅×D₄	C₂×C₃×D₄	1	4	yes	—
264	C₂×C₄×C₅	—	2	16	no	—
265	C₂×C₄×C₅	—	2	2	no	—
266	C₂×C₄×C₅	—	2	2	no	—
267	C₂×C₄×C₅	—	2	2	no	—
268	C₂×C₄×C₅	—	2	4	no	—
269	C₂×C₄×C₅	—	2	4	no	—
270	C₂×C₄×C₅	—	2	2	no	—
271	C₂×C₄×C₅	—	4	2	no	—
272	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
273	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
274	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
275	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
276	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
277	C₅×C₈	—	2	4	no	—
278	2₋¹⁺⁴	—	2	4	no	—
279	2₋¹⁺⁴	—	2	2	no	—
280	2₋¹⁺⁴	C₂²×C₃×C₅	1	4	yes	complemented
281	C₂×#41	—	2	2	no	—
282	C₂×#41	—	2	2	no	—
283	C₂×#41	C₂²×C₃×C₅	1	2	yes	—
284	C₂²×Q₈	—	2	2	no	—
285	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
286	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
287	AGL(1,ℤ/8)	—	2	4	no	—
288	AGL(1,ℤ/8)	C₂²×C₃×C₅	1	8	yes	complemented
289	C₂×SD₁₆	—	2	2	no	—
290	C₂×D₈	—	2	2	no	—
291	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
292	#124	—	2	2	no	—
293	#124	C₂²×C₃×C₅	1	4	yes	complemented
294	C₂×C₂²⋊C₄	—	2	2	no	—
295	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
296	#104	C₂²×C₃×C₅	1	8	yes	complemented
297	#101	C₂²×C₃×C₅	1	4	yes	complemented
298	C₂×C₃×C₅	—	2	1	no	—
299	C₂×C₃×C₅	—	2	1	no	—
300	C₂×C₃×C₅	—	4	4	no	—
301	C₂×C₃×C₅	—	8	2	no	—
302	C₂×C₃×C₅	#454	1	2	yes	direct product factor
303	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
304	C₂³×C₃	—	2	2	no	—
305	C₂³×C₃	—	2	2	no	—
306	C₂³×C₃	—	4	4	no	—
307	C₂³×C₃	—	4	1	no	—
308	C₂³×C₃	—	4	1	no	—
309	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
310	C₃×Q₈	—	2	8	no	—
311	C₃×Q₈	—	2	2	no	—
312	C₃×Q₈	—	2	4	no	—
313	C₃×Q₈	—	2	4	no	—
314	C₃×Q₈	C₂×C₅×D₄	1	4	yes	—
315	C₃×D₄	—	2	8	no	—
316	C₃×D₄	—	2	4	no	—
317	C₃×D₄	—	2	4	no	—
318	C₃×D₄	—	2	2	no	—
319	C₃×D₄	—	2	4	no	—
320	C₃×D₄	—	4	2	no	—
321	C₃×D₄	C₂×C₅×D₄	1	4	yes	—
322	C₂×C₃×C₄	—	2	16	no	—
323	C₂×C₃×C₄	—	2	2	no	—
324	C₂×C₃×C₄	—	2	2	no	—
325	C₂×C₃×C₄	—	2	2	no	—
326	C₂×C₃×C₄	—	2	4	no	—
327	C₂×C₃×C₄	—	2	4	no	—
328	C₂×C₃×C₄	—	2	2	no	—
329	C₂×C₃×C₄	—	4	2	no	—
330	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
331	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
332	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
333	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
334	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
335	C₃×C₈	—	2	4	no	—
336	C₂²×C₅	—	2	2	no	—
337	C₂²×C₅	—	2	4	no	—
338	C₂²×C₅	—	4	4	no	—
339	C₂²×C₅	—	4	4	no	—
340	C₂²×C₅	—	4	2	no	—
341	C₂²×C₅	—	8	2	no	—
342	C₂²×C₅	—	8	2	no	—
343	C₂²×C₅	C₃×#120	1	1	yes	characteristic
344	C₂²×C₅	C₃×#120	1	1	yes	characteristic
345	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
346	C₄×C₅	—	2	4	no	—
347	C₄×C₅	—	2	4	no	—
348	C₄×C₅	—	4	2	no	—
349	C₄×C₅	C₃×#120	1	2	yes	—
350	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
351	C₂⁴	—	2	1	no	—
352	#41	—	2	8	no	—
353	#41	—	2	8	no	—
354	#41	—	2	8	no	—
355	#41	—	2	8	no	—
356	#41	—	2	2	no	—
357	#41	—	2	2	no	—
358	#41	C₃×C₅×D₄	1	4	yes	—
359	#41	C₂³×C₃×C₅	1	4	yes	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	8	no	—
362	C₂×Q₈	—	2	8	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	C₃×C₅×D₄	1	2	yes	—
366	C₂×Q₈	C₃×C₅×D₄	1	2	yes	—
367	C₂×Q₈	C₂³×C₃×C₅	1	2	yes	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	C₃×C₅×D₄	1	2	yes	—
374	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
379	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
380	SD₁₆	—	2	8	no	—
381	D₈	—	2	8	no	—
382	M₁₆	—	2	2	no	—
383	M₁₆	C₂³×C₃×C₅	1	4	yes	—
384	C₂×C₈	—	2	2	no	—
385	C₂²⋊C₄	—	2	8	no	—
386	C₄²	—	2	1	no	—
387	C₄²	C₂³×C₃×C₅	1	2	yes	—
388	C₃×C₅	C₂×#454	1	1	yes	characteristic, direct product factor, Hall
389	C₂²×C₃	—	2	2	no	—
390	C₂²×C₃	—	2	4	no	—
391	C₂²×C₃	—	4	4	no	—
392	C₂²×C₃	—	4	4	no	—
393	C₂²×C₃	—	4	2	no	—
394	C₂²×C₃	—	8	2	no	—
395	C₂²×C₃	—	8	2	no	—
396	C₂²×C₃	C₅×#120	1	1	yes	characteristic
397	C₂²×C₃	C₅×#120	1	1	yes	characteristic
398	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
399	C₃×C₄	—	2	4	no	—
400	C₃×C₄	—	2	4	no	—
401	C₃×C₄	—	4	2	no	—
402	C₃×C₄	C₅×#120	1	2	yes	—
403	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
404	C₂×C₅	—	2	1	no	—
405	C₂×C₅	—	2	1	no	—
406	C₂×C₅	—	4	4	no	—
407	C₂×C₅	—	8	2	no	—
408	C₂×C₅	C₃×#454	1	2	yes	direct product factor
409	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
410	C₂³	—	2	2	no	—
411	C₂³	—	2	2	no	—
412	C₂³	—	4	4	no	—
413	C₂³	—	4	1	no	—
414	C₂³	—	4	1	no	—
415	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
416	Q₈	—	2	8	no	—
417	Q₈	—	2	2	no	—
418	Q₈	—	2	4	no	—
419	Q₈	—	2	4	no	—
420	Q₈	C₂×C₃×C₅×D₄	1	4	yes	—
421	D₄	—	2	8	no	—
422	D₄	—	2	4	no	—
423	D₄	—	2	4	no	—
424	D₄	—	2	2	no	—
425	D₄	—	2	4	no	—
426	D₄	—	4	2	no	—
427	D₄	C₂×C₃×C₅×D₄	1	4	yes	—
428	C₂×C₄	—	2	16	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
437	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
438	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
439	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
440	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
441	C₈	—	2	4	no	—
442	C₂×C₃	—	2	1	no	—
443	C₂×C₃	—	2	1	no	—
444	C₂×C₃	—	4	4	no	—
445	C₂×C₃	—	8	2	no	—
446	C₂×C₃	C₅×#454	1	2	yes	direct product factor
447	C₂×C₃	C₂×C₅×#120	1	1	yes	characteristic
448	C₅	C₂×C₃×#454	1	1	yes	characteristic, direct product factor, Sylow
449	C₂²	—	2	2	no	—
450	C₂²	—	2	4	no	—
451	C₂²	—	4	4	no	—
452	C₂²	—	4	4	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	8	2	no	—
455	C₂²	—	8	2	no	—
456	C₂²	C₃×C₅×#120	1	1	yes	characteristic
457	C₂²	C₃×C₅×#120	1	1	yes	characteristic
458	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
459	C₄	—	2	4	no	—
460	C₄	—	2	4	no	—
461	C₄	—	4	2	no	—
462	C₄	C₃×C₅×#120	1	2	yes	—
463	C₄	C₂²×C₃×C₅×D₄	1	2	yes	—
464	C₃	C₂×C₅×#454	1	1	yes	characteristic, direct product factor, Sylow
465	C₂	—	2	1	no	—
466	C₂	—	2	1	no	—
467	C₂	—	4	4	no	—
468	C₂	—	8	2	no	—
469	C₂	C₃×C₅×#454	1	2	yes	direct product factor
470	C₂	C₂×C₃×C₅×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×2₋¹⁺⁴	C₂	semidirect
2. C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	semidirect
3. C₂×C₃×C₅×#124	C₂	semidirect
4. C₃×C₅×#454	C₂	direct (N × H)
5. C₂×C₃×C₅×#104	C₂	semidirect
6. C₂×C₃×C₅×#101	C₂	semidirect
7. C₂×C₅×#454	C₃	direct (N × H)
10. C₃×C₅×2₋¹⁺⁴	C₂²	semidirect
18. C₃×C₅×AGL(1,ℤ/8)	C₂²	semidirect
23. C₃×C₅×#124	C₂²	semidirect
26. C₃×C₅×#104	C₂²	semidirect
27. C₃×C₅×#101	C₂²	semidirect
28. C₂×C₃×#454	C₅	direct (N × H)
29. C₂×C₅×2₋¹⁺⁴	C₂×C₃	semidirect
30. C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	semidirect
31. C₂×C₅×#124	C₂×C₃	semidirect
32. C₅×#454	C₂×C₃	direct (N × H)
33. C₂×C₅×#104	C₂×C₃	semidirect
34. C₂×C₅×#101	C₂×C₃	semidirect
72. C₂×C₃×2₋¹⁺⁴	C₂×C₅	semidirect
73. C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	semidirect
74. C₂×C₃×#124	C₂×C₅	semidirect
75. C₃×#454	C₂×C₅	direct (N × H)
76. C₂×C₃×#104	C₂×C₅	semidirect
77. C₂×C₃×#101	C₂×C₅	semidirect
80. C₅×2₋¹⁺⁴	C₂²×C₃	semidirect
88. C₅×AGL(1,ℤ/8)	C₂²×C₃	semidirect
93. C₅×#124	C₂²×C₃	semidirect
96. C₅×#104	C₂²×C₃	semidirect
97. C₅×#101	C₂²×C₃	semidirect
98. C₂×#454	C₃×C₅	direct (N × H)
133. C₃×2₋¹⁺⁴	C₂²×C₅	semidirect
141. C₃×AGL(1,ℤ/8)	C₂²×C₅	semidirect
146. C₃×#124	C₂²×C₅	semidirect
149. C₃×#104	C₂²×C₅	semidirect
150. C₃×#101	C₂²×C₅	semidirect
188. C₂×2₋¹⁺⁴	C₂×C₃×C₅	semidirect
189. C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	semidirect
190. C₂×#124	C₂×C₃×C₅	semidirect
191. #454	C₂×C₃×C₅	direct (N × H)
192. C₂×#104	C₂×C₃×C₅	semidirect
193. C₂×#101	C₂×C₃×C₅	semidirect
280. 2₋¹⁺⁴	C₂²×C₃×C₅	semidirect
288. AGL(1,ℤ/8)	C₂²×C₃×C₅	semidirect
293. #124	C₂²×C₃×C₅	semidirect
296. #104	C₂²×C₃×C₅	semidirect
297. #101	C₂²×C₃×C₅	semidirect
302. C₂×C₃×C₅	#454	direct (N × H)
388. C₃×C₅	C₂×#454	direct (N × H)
408. C₂×C₅	C₃×#454	direct (N × H)
446. C₂×C₃	C₅×#454	direct (N × H)
448. C₅	C₂×C₃×#454	direct (N × H)
464. C₃	C₂×C₅×#454	direct (N × H)
469. C₂	C₃×C₅×#454	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×C₅×2₋¹⁺⁴	2
C₂×C₃×C₅×AGL(1,ℤ/8)	2
C₂×C₃×C₅×#124	2
C₃×C₅×#454	2
C₂×C₃×C₅×#104	2
C₂×C₃×C₅×#101	2
C₂×C₅×#454	3
C₂×C₃×#454	5

Subgroups — C2×C3×C5×#454 #14276050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#454 #14276050