Subgroups — C₃×C₅×#3068 #14276102

Order 1920.

932 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	C₃×C₅×#575	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#472	C₂	1	4	yes	maximal, complemented
6	C₃×C₅×#471	C₂	1	4	yes	maximal, complemented
7	C₃×C₅×#430	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#413	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×C₅×#100	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#3068	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₃×C₅×#137	—	2	1	no	—
15	C₃×C₅×#137	C₂²	1	2	yes	complemented
16	C₃×C₅×AGL(1,ℤ/8)	—	2	2	no	—
17	C₃×C₅×AGL(1,ℤ/8)	—	2	1	no	—
18	C₃×C₅×AGL(1,ℤ/8)	—	2	2	no	—
19	C₃×C₅×AGL(1,ℤ/8)	—	2	2	no	—
20	C₃×C₅×AGL(1,ℤ/8)	C₂²	1	4	yes	—
21	C₃×C₅×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
22	C₃×C₅×#135	—	2	2	no	—
23	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
24	C₂×C₃×C₅×SD₁₆	C₂²	1	2	yes	complemented
25	C₂×C₃×C₅×D₈	—	2	1	no	—
26	C₂×C₃×C₅×D₈	C₂²	1	2	yes	complemented
27	C₃×C₅×#131	—	2	2	no	—
28	C₃×C₅×#131	C₂²	1	4	yes	complemented
29	C₂×C₃×C₅×M₁₆	—	2	2	no	—
30	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
31	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
32	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
33	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
34	C₃×C₅×#108	C₂²	1	4	yes	complemented
35	C₃×C₅×#101	C₂²	1	2	yes	complemented
36	C₃×C₅×#100	C₂²	1	4	yes	complemented
37	C₃×C₅×#100	C₂²	1	2	yes	complemented
38	C₃×#3068	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
39	C₅×#575	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic, complemented
42	C₅×#568	C₂×C₃	1	1	yes	characteristic, complemented
43	C₅×#472	C₂×C₃	1	4	yes	complemented
44	C₅×#471	C₂×C₃	1	4	yes	complemented
45	C₅×#430	C₂×C₃	1	1	yes	characteristic, complemented
46	C₅×#413	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂×C₅×#100	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂⁴×C₃×C₅	—	2	1	no	—
49	C₃×C₅×#41	—	2	1	no	—
50	C₃×C₅×#41	—	2	1	no	—
51	C₃×C₅×#41	—	2	1	no	—
52	C₃×C₅×#41	—	2	1	no	—
53	C₃×C₅×#41	—	2	1	no	—
54	C₃×C₅×#41	—	2	1	no	—
55	C₃×C₅×#41	D₄	1	2	yes	—
56	C₃×C₅×#41	C₂³	1	2	yes	—
57	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
58	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₅×D₄	—	2	1	no	—
60	C₂×C₃×C₅×D₄	—	2	1	no	—
61	C₂×C₃×C₅×D₄	—	2	1	no	—
62	C₂×C₃×C₅×D₄	—	2	1	no	—
63	C₂×C₃×C₅×D₄	—	2	1	no	—
64	C₂×C₃×C₅×D₄	—	2	1	no	—
65	C₂×C₃×C₅×D₄	—	2	1	no	—
66	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
67	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
68	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄×C₅	—	2	1	no	—
70	C₂²×C₃×C₄×C₅	—	2	1	no	—
71	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
72	C₃×C₅×Q₁₆	—	2	2	no	—
73	C₃×C₅×SD₁₆	—	2	4	no	—
74	C₃×C₅×SD₁₆	—	2	2	no	—
75	C₃×C₅×SD₁₆	—	2	2	no	—
76	C₃×C₅×SD₁₆	—	2	4	no	—
77	C₃×C₅×D₈	—	2	4	no	—
78	C₃×C₅×D₈	—	2	2	no	—
79	C₃×C₅×D₈	—	2	4	no	—
80	C₃×C₅×M₁₆	—	2	1	no	—
81	C₃×C₅×M₁₆	—	2	1	no	—
82	C₃×C₅×M₁₆	—	2	1	no	—
83	C₃×C₅×M₁₆	—	2	4	no	—
84	C₃×C₅×M₁₆	D₄	1	2	yes	—
85	C₃×C₅×M₁₆	D₄	1	2	yes	—
86	C₃×C₅×M₁₆	C₂³	1	2	yes	—
87	C₃×C₅×M₁₆	C₂³	1	2	yes	—
88	C₃×C₅×M₁₆	C₂³	1	2	yes	—
89	C₂×C₃×C₅×C₈	—	2	1	no	—
90	C₂×C₃×C₅×C₈	—	2	1	no	—
91	C₂×C₃×C₅×C₈	—	2	1	no	—
92	C₂×C₃×C₅×C₈	—	2	4	no	—
93	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
94	C₃×#575	C₂×C₅	1	1	yes	characteristic, complemented
95	C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	1	1	yes	characteristic, complemented
96	C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	1	1	yes	characteristic, complemented
97	C₃×#568	C₂×C₅	1	1	yes	characteristic, complemented
98	C₃×#472	C₂×C₅	1	4	yes	complemented
99	C₃×#471	C₂×C₅	1	4	yes	complemented
100	C₃×#430	C₂×C₅	1	1	yes	characteristic, complemented
101	C₃×#413	C₂×C₅	1	1	yes	characteristic, complemented
102	C₂×C₃×#100	C₂×C₅	1	1	yes	characteristic, complemented
103	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
104	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
105	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
106	C₅×#137	—	2	1	no	—
107	C₅×#137	C₂²×C₃	1	2	yes	complemented
108	C₅×AGL(1,ℤ/8)	—	2	2	no	—
109	C₅×AGL(1,ℤ/8)	—	2	1	no	—
110	C₅×AGL(1,ℤ/8)	—	2	2	no	—
111	C₅×AGL(1,ℤ/8)	—	2	2	no	—
112	C₅×AGL(1,ℤ/8)	C₂²×C₃	1	4	yes	—
113	C₅×AGL(1,ℤ/8)	C₂²×C₃	1	2	yes	complemented
114	C₅×#135	—	2	2	no	—
115	C₂×C₅×SD₁₆	—	2	1	no	—
116	C₂×C₅×SD₁₆	C₂²×C₃	1	2	yes	complemented
117	C₂×C₅×D₈	—	2	1	no	—
118	C₂×C₅×D₈	C₂²×C₃	1	2	yes	complemented
119	C₅×#131	—	2	2	no	—
120	C₅×#131	C₂²×C₃	1	4	yes	complemented
121	C₂×C₅×M₁₆	—	2	2	no	—
122	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
123	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
124	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
125	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
126	C₅×#108	C₂²×C₃	1	4	yes	complemented
127	C₅×#101	C₂²×C₃	1	2	yes	complemented
128	C₅×#100	C₂²×C₃	1	4	yes	complemented
129	C₅×#100	C₂²×C₃	1	2	yes	complemented
130	#3068	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³×C₃×C₅	—	2	1	no	—
132	C₂³×C₃×C₅	—	2	1	no	—
133	C₂³×C₃×C₅	—	2	2	no	—
134	C₂³×C₃×C₅	—	4	1	no	—
135	C₂³×C₃×C₅	—	4	1	no	—
136	C₂³×C₃×C₅	—	8	2	no	—
137	C₂³×C₃×C₅	#41	1	1	yes	characteristic
138	C₃×C₅×Q₈	—	2	1	no	—
139	C₃×C₅×Q₈	—	2	1	no	—
140	C₃×C₅×Q₈	—	2	1	no	—
141	C₃×C₅×Q₈	C₂×D₄	1	2	yes	—
142	C₃×C₅×D₄	—	2	2	no	—
143	C₃×C₅×D₄	—	2	2	no	—
144	C₃×C₅×D₄	—	2	1	no	—
145	C₃×C₅×D₄	—	2	1	no	—
146	C₃×C₅×D₄	—	2	2	no	—
147	C₃×C₅×D₄	—	2	2	no	—
148	C₃×C₅×D₄	—	2	1	no	—
149	C₃×C₅×D₄	—	4	1	no	—
150	C₃×C₅×D₄	—	4	1	no	—
151	C₃×C₅×D₄	—	4	1	no	—
152	C₃×C₅×D₄	—	4	1	no	—
153	C₃×C₅×D₄	C₂×D₄	1	2	yes	—
154	C₂×C₃×C₄×C₅	—	2	1	no	—
155	C₂×C₃×C₄×C₅	—	2	2	no	—
156	C₂×C₃×C₄×C₅	—	2	2	no	—
157	C₂×C₃×C₄×C₅	—	2	1	no	—
158	C₂×C₃×C₄×C₅	—	4	1	no	—
159	C₂×C₃×C₄×C₅	—	4	1	no	—
160	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
161	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
162	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
165	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
166	C₃×C₅×C₈	—	2	2	no	—
167	C₃×C₅×C₈	—	2	2	no	—
168	C₃×C₅×C₈	—	2	2	no	—
169	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
170	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
171	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
172	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
173	C₃×#137	—	2	1	no	—
174	C₃×#137	C₂²×C₅	1	2	yes	complemented
175	C₃×AGL(1,ℤ/8)	—	2	2	no	—
176	C₃×AGL(1,ℤ/8)	—	2	1	no	—
177	C₃×AGL(1,ℤ/8)	—	2	2	no	—
178	C₃×AGL(1,ℤ/8)	—	2	2	no	—
179	C₃×AGL(1,ℤ/8)	C₂²×C₅	1	4	yes	—
180	C₃×AGL(1,ℤ/8)	C₂²×C₅	1	2	yes	complemented
181	C₃×#135	—	2	2	no	—
182	C₂×C₃×SD₁₆	—	2	1	no	—
183	C₂×C₃×SD₁₆	C₂²×C₅	1	2	yes	complemented
184	C₂×C₃×D₈	—	2	1	no	—
185	C₂×C₃×D₈	C₂²×C₅	1	2	yes	complemented
186	C₃×#131	—	2	2	no	—
187	C₃×#131	C₂²×C₅	1	4	yes	complemented
188	C₂×C₃×M₁₆	—	2	2	no	—
189	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
190	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
191	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
192	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
193	C₃×#108	C₂²×C₅	1	4	yes	complemented
194	C₃×#101	C₂²×C₅	1	2	yes	complemented
195	C₃×#100	C₂²×C₅	1	4	yes	complemented
196	C₃×#100	C₂²×C₅	1	2	yes	complemented
197	C₂⁴×C₅	—	2	1	no	—
198	C₅×#41	—	2	1	no	—
199	C₅×#41	—	2	1	no	—
200	C₅×#41	—	2	1	no	—
201	C₅×#41	—	2	1	no	—
202	C₅×#41	—	2	1	no	—
203	C₅×#41	—	2	1	no	—
204	C₅×#41	C₃×D₄	1	2	yes	—
205	C₅×#41	C₂³×C₃	1	2	yes	—
206	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
207	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
208	C₂×C₅×D₄	—	2	1	no	—
209	C₂×C₅×D₄	—	2	1	no	—
210	C₂×C₅×D₄	—	2	1	no	—
211	C₂×C₅×D₄	—	2	1	no	—
212	C₂×C₅×D₄	—	2	1	no	—
213	C₂×C₅×D₄	—	2	1	no	—
214	C₂×C₅×D₄	—	2	1	no	—
215	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
216	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
217	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
218	C₂²×C₄×C₅	—	2	1	no	—
219	C₂²×C₄×C₅	—	2	1	no	—
220	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
221	C₅×Q₁₆	—	2	2	no	—
222	C₅×SD₁₆	—	2	4	no	—
223	C₅×SD₁₆	—	2	2	no	—
224	C₅×SD₁₆	—	2	2	no	—
225	C₅×SD₁₆	—	2	4	no	—
226	C₅×D₈	—	2	4	no	—
227	C₅×D₈	—	2	2	no	—
228	C₅×D₈	—	2	4	no	—
229	C₅×M₁₆	—	2	1	no	—
230	C₅×M₁₆	—	2	1	no	—
231	C₅×M₁₆	—	2	1	no	—
232	C₅×M₁₆	—	2	4	no	—
233	C₅×M₁₆	C₃×D₄	1	2	yes	—
234	C₅×M₁₆	C₃×D₄	1	2	yes	—
235	C₅×M₁₆	C₂³×C₃	1	2	yes	—
236	C₅×M₁₆	C₂³×C₃	1	2	yes	—
237	C₅×M₁₆	C₂³×C₃	1	2	yes	—
238	C₂×C₅×C₈	—	2	1	no	—
239	C₂×C₅×C₈	—	2	1	no	—
240	C₂×C₅×C₈	—	2	1	no	—
241	C₂×C₅×C₈	—	2	4	no	—
242	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
243	#575	C₂×C₃×C₅	1	1	yes	characteristic, complemented
244	C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	1	1	yes	characteristic, complemented
245	C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	1	1	yes	characteristic, complemented
246	#568	C₂×C₃×C₅	1	1	yes	characteristic, complemented
247	#472	C₂×C₃×C₅	1	4	yes	complemented
248	#471	C₂×C₃×C₅	1	4	yes	complemented
249	#430	C₂×C₃×C₅	1	1	yes	characteristic, complemented
250	#413	C₂×C₃×C₅	1	1	yes	characteristic, complemented
251	C₂×#100	C₂×C₃×C₅	1	1	yes	characteristic, complemented
252	C₂²×C₃×C₅	—	2	2	no	—
253	C₂²×C₃×C₅	—	4	1	no	—
254	C₂²×C₃×C₅	—	4	1	no	—
255	C₂²×C₃×C₅	—	4	2	no	—
256	C₂²×C₃×C₅	—	8	1	no	—
257	C₂²×C₃×C₅	—	8	1	no	—
258	C₂²×C₃×C₅	—	8	2	no	—
259	C₂²×C₃×C₅	—	8	2	no	—
260	C₂²×C₃×C₅	—	8	2	no	—
261	C₂²×C₃×C₅	#121	1	1	yes	characteristic
262	C₂²×C₃×C₅	#121	1	1	yes	characteristic
263	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
264	C₃×C₄×C₅	—	2	2	no	—
265	C₃×C₄×C₅	—	4	1	no	—
266	C₃×C₄×C₅	#121	1	1	yes	characteristic
267	C₃×C₄×C₅	#121	1	1	yes	characteristic
268	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
269	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
270	C₂⁴×C₃	—	2	1	no	—
271	C₃×#41	—	2	1	no	—
272	C₃×#41	—	2	1	no	—
273	C₃×#41	—	2	1	no	—
274	C₃×#41	—	2	1	no	—
275	C₃×#41	—	2	1	no	—
276	C₃×#41	—	2	1	no	—
277	C₃×#41	C₅×D₄	1	2	yes	—
278	C₃×#41	C₂³×C₅	1	2	yes	—
279	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
280	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
281	C₂×C₃×D₄	—	2	1	no	—
282	C₂×C₃×D₄	—	2	1	no	—
283	C₂×C₃×D₄	—	2	1	no	—
284	C₂×C₃×D₄	—	2	1	no	—
285	C₂×C₃×D₄	—	2	1	no	—
286	C₂×C₃×D₄	—	2	1	no	—
287	C₂×C₃×D₄	—	2	1	no	—
288	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
289	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
290	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
291	C₂²×C₃×C₄	—	2	1	no	—
292	C₂²×C₃×C₄	—	2	1	no	—
293	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
294	C₃×Q₁₆	—	2	2	no	—
295	C₃×SD₁₆	—	2	4	no	—
296	C₃×SD₁₆	—	2	2	no	—
297	C₃×SD₁₆	—	2	2	no	—
298	C₃×SD₁₆	—	2	4	no	—
299	C₃×D₈	—	2	4	no	—
300	C₃×D₈	—	2	2	no	—
301	C₃×D₈	—	2	4	no	—
302	C₃×M₁₆	—	2	1	no	—
303	C₃×M₁₆	—	2	1	no	—
304	C₃×M₁₆	—	2	1	no	—
305	C₃×M₁₆	—	2	4	no	—
306	C₃×M₁₆	C₅×D₄	1	2	yes	—
307	C₃×M₁₆	C₅×D₄	1	2	yes	—
308	C₃×M₁₆	C₂³×C₅	1	2	yes	—
309	C₃×M₁₆	C₂³×C₅	1	2	yes	—
310	C₃×M₁₆	C₂³×C₅	1	2	yes	—
311	C₂×C₃×C₈	—	2	1	no	—
312	C₂×C₃×C₈	—	2	1	no	—
313	C₂×C₃×C₈	—	2	1	no	—
314	C₂×C₃×C₈	—	2	4	no	—
315	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
316	C₂³×C₅	—	2	1	no	—
317	C₂³×C₅	—	2	1	no	—
318	C₂³×C₅	—	2	2	no	—
319	C₂³×C₅	—	4	1	no	—
320	C₂³×C₅	—	4	1	no	—
321	C₂³×C₅	—	8	2	no	—
322	C₂³×C₅	C₃×#41	1	1	yes	characteristic
323	C₅×Q₈	—	2	1	no	—
324	C₅×Q₈	—	2	1	no	—
325	C₅×Q₈	—	2	1	no	—
326	C₅×Q₈	C₂×C₃×D₄	1	2	yes	—
327	C₅×D₄	—	2	2	no	—
328	C₅×D₄	—	2	2	no	—
329	C₅×D₄	—	2	1	no	—
330	C₅×D₄	—	2	1	no	—
331	C₅×D₄	—	2	2	no	—
332	C₅×D₄	—	2	2	no	—
333	C₅×D₄	—	2	1	no	—
334	C₅×D₄	—	4	1	no	—
335	C₅×D₄	—	4	1	no	—
336	C₅×D₄	—	4	1	no	—
337	C₅×D₄	—	4	1	no	—
338	C₅×D₄	C₂×C₃×D₄	1	2	yes	—
339	C₂×C₄×C₅	—	2	1	no	—
340	C₂×C₄×C₅	—	2	2	no	—
341	C₂×C₄×C₅	—	2	2	no	—
342	C₂×C₄×C₅	—	2	1	no	—
343	C₂×C₄×C₅	—	4	1	no	—
344	C₂×C₄×C₅	—	4	1	no	—
345	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
346	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
347	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
348	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
349	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
350	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
351	C₅×C₈	—	2	2	no	—
352	C₅×C₈	—	2	2	no	—
353	C₅×C₈	—	2	2	no	—
354	C₅×C₈	C₂×C₃×D₄	1	4	yes	—
355	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
356	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
357	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
358	#137	—	2	1	no	—
359	#137	C₂²×C₃×C₅	1	2	yes	complemented
360	AGL(1,ℤ/8)	—	2	2	no	—
361	AGL(1,ℤ/8)	—	2	1	no	—
362	AGL(1,ℤ/8)	—	2	2	no	—
363	AGL(1,ℤ/8)	—	2	2	no	—
364	AGL(1,ℤ/8)	C₂²×C₃×C₅	1	4	yes	—
365	AGL(1,ℤ/8)	C₂²×C₃×C₅	1	2	yes	complemented
366	#135	—	2	2	no	—
367	C₂×SD₁₆	—	2	1	no	—
368	C₂×SD₁₆	C₂²×C₃×C₅	1	2	yes	complemented
369	C₂×D₈	—	2	1	no	—
370	C₂×D₈	C₂²×C₃×C₅	1	2	yes	complemented
371	#131	—	2	2	no	—
372	#131	C₂²×C₃×C₅	1	4	yes	complemented
373	C₂×M₁₆	—	2	2	no	—
374	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
375	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
376	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
377	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
378	#108	C₂²×C₃×C₅	1	4	yes	complemented
379	#101	C₂²×C₃×C₅	1	2	yes	complemented
380	#100	C₂²×C₃×C₅	1	4	yes	complemented
381	#100	C₂²×C₃×C₅	1	2	yes	complemented
382	C₂×C₃×C₅	—	2	1	no	—
383	C₂×C₃×C₅	—	2	1	no	—
384	C₂×C₃×C₅	—	2	1	no	—
385	C₂×C₃×C₅	—	4	2	no	—
386	C₂×C₃×C₅	—	8	1	no	—
387	C₂×C₃×C₅	—	8	2	no	—
388	C₂×C₃×C₅	C₂×#121	1	1	yes	characteristic
389	C₂³×C₃	—	2	1	no	—
390	C₂³×C₃	—	2	1	no	—
391	C₂³×C₃	—	2	2	no	—
392	C₂³×C₃	—	4	1	no	—
393	C₂³×C₃	—	4	1	no	—
394	C₂³×C₃	—	8	2	no	—
395	C₂³×C₃	C₅×#41	1	1	yes	characteristic
396	C₃×Q₈	—	2	1	no	—
397	C₃×Q₈	—	2	1	no	—
398	C₃×Q₈	—	2	1	no	—
399	C₃×Q₈	C₂×C₅×D₄	1	2	yes	—
400	C₃×D₄	—	2	2	no	—
401	C₃×D₄	—	2	2	no	—
402	C₃×D₄	—	2	1	no	—
403	C₃×D₄	—	2	1	no	—
404	C₃×D₄	—	2	2	no	—
405	C₃×D₄	—	2	2	no	—
406	C₃×D₄	—	2	1	no	—
407	C₃×D₄	—	4	1	no	—
408	C₃×D₄	—	4	1	no	—
409	C₃×D₄	—	4	1	no	—
410	C₃×D₄	—	4	1	no	—
411	C₃×D₄	C₂×C₅×D₄	1	2	yes	—
412	C₂×C₃×C₄	—	2	1	no	—
413	C₂×C₃×C₄	—	2	2	no	—
414	C₂×C₃×C₄	—	2	2	no	—
415	C₂×C₃×C₄	—	2	1	no	—
416	C₂×C₃×C₄	—	4	1	no	—
417	C₂×C₃×C₄	—	4	1	no	—
418	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
419	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
420	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
421	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
422	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
423	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
424	C₃×C₈	—	2	2	no	—
425	C₃×C₈	—	2	2	no	—
426	C₃×C₈	—	2	2	no	—
427	C₃×C₈	C₂×C₅×D₄	1	4	yes	—
428	C₂²×C₅	—	2	2	no	—
429	C₂²×C₅	—	4	1	no	—
430	C₂²×C₅	—	4	1	no	—
431	C₂²×C₅	—	4	2	no	—
432	C₂²×C₅	—	8	1	no	—
433	C₂²×C₅	—	8	1	no	—
434	C₂²×C₅	—	8	2	no	—
435	C₂²×C₅	—	8	2	no	—
436	C₂²×C₅	—	8	2	no	—
437	C₂²×C₅	C₃×#121	1	1	yes	characteristic
438	C₂²×C₅	C₃×#121	1	1	yes	characteristic
439	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
440	C₄×C₅	—	2	2	no	—
441	C₄×C₅	—	4	1	no	—
442	C₄×C₅	C₃×#121	1	1	yes	characteristic
443	C₄×C₅	C₃×#121	1	1	yes	characteristic
444	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
445	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
446	C₂⁴	—	2	1	no	—
447	#41	—	2	1	no	—
448	#41	—	2	1	no	—
449	#41	—	2	1	no	—
450	#41	—	2	1	no	—
451	#41	—	2	1	no	—
452	#41	—	2	1	no	—
453	#41	C₃×C₅×D₄	1	2	yes	—
454	#41	C₂³×C₃×C₅	1	2	yes	—
455	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
456	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	1	no	—
463	C₂×D₄	—	2	1	no	—
464	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
465	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
466	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
470	Q₁₆	—	2	2	no	—
471	SD₁₆	—	2	4	no	—
472	SD₁₆	—	2	2	no	—
473	SD₁₆	—	2	2	no	—
474	SD₁₆	—	2	4	no	—
475	D₈	—	2	4	no	—
476	D₈	—	2	2	no	—
477	D₈	—	2	4	no	—
478	M₁₆	—	2	1	no	—
479	M₁₆	—	2	1	no	—
480	M₁₆	—	2	1	no	—
481	M₁₆	—	2	4	no	—
482	M₁₆	C₃×C₅×D₄	1	2	yes	—
483	M₁₆	C₃×C₅×D₄	1	2	yes	—
484	M₁₆	C₂³×C₃×C₅	1	2	yes	—
485	M₁₆	C₂³×C₃×C₅	1	2	yes	—
486	M₁₆	C₂³×C₃×C₅	1	2	yes	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	4	no	—
491	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
492	C₃×C₅	#3068	1	1	yes	characteristic, direct product factor, Hall
493	C₂²×C₃	—	2	2	no	—
494	C₂²×C₃	—	4	1	no	—
495	C₂²×C₃	—	4	1	no	—
496	C₂²×C₃	—	4	2	no	—
497	C₂²×C₃	—	8	1	no	—
498	C₂²×C₃	—	8	1	no	—
499	C₂²×C₃	—	8	2	no	—
500	C₂²×C₃	—	8	2	no	—
501	C₂²×C₃	—	8	2	no	—
502	C₂²×C₃	C₅×#121	1	1	yes	characteristic
503	C₂²×C₃	C₅×#121	1	1	yes	characteristic
504	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
505	C₃×C₄	—	2	2	no	—
506	C₃×C₄	—	4	1	no	—
507	C₃×C₄	C₅×#121	1	1	yes	characteristic
508	C₃×C₄	C₅×#121	1	1	yes	characteristic
509	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
510	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
511	C₂×C₅	—	2	1	no	—
512	C₂×C₅	—	2	1	no	—
513	C₂×C₅	—	2	1	no	—
514	C₂×C₅	—	4	2	no	—
515	C₂×C₅	—	8	2	no	—
516	C₂×C₅	—	8	1	no	—
517	C₂×C₅	C₂×C₃×#121	1	1	yes	characteristic
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	2	2	no	—
521	C₂³	—	4	1	no	—
522	C₂³	—	4	1	no	—
523	C₂³	—	8	2	no	—
524	C₂³	C₃×C₅×#41	1	1	yes	characteristic
525	Q₈	—	2	1	no	—
526	Q₈	—	2	1	no	—
527	Q₈	—	2	1	no	—
528	Q₈	C₂×C₃×C₅×D₄	1	2	yes	—
529	D₄	—	2	2	no	—
530	D₄	—	2	2	no	—
531	D₄	—	2	1	no	—
532	D₄	—	2	2	no	—
533	D₄	—	2	2	no	—
534	D₄	—	2	1	no	—
535	D₄	—	2	1	no	—
536	D₄	—	4	1	no	—
537	D₄	—	4	1	no	—
538	D₄	—	4	1	no	—
539	D₄	—	4	1	no	—
540	D₄	C₂×C₃×C₅×D₄	1	2	yes	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
548	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
549	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
550	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
551	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
552	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
553	C₈	—	2	2	no	—
554	C₈	—	2	2	no	—
555	C₈	—	2	2	no	—
556	C₈	C₂×C₃×C₅×D₄	1	4	yes	—
557	C₂×C₃	—	2	1	no	—
558	C₂×C₃	—	2	1	no	—
559	C₂×C₃	—	2	1	no	—
560	C₂×C₃	—	4	2	no	—
561	C₂×C₃	—	8	2	no	—
562	C₂×C₃	—	8	1	no	—
563	C₂×C₃	C₂×C₅×#121	1	1	yes	characteristic
564	C₅	C₃×#3068	1	1	yes	characteristic, direct product factor, Sylow
565	C₂²	—	2	2	no	—
566	C₂²	—	4	1	no	—
567	C₂²	—	4	1	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	8	1	no	—
570	C₂²	—	8	2	no	—
571	C₂²	—	8	1	no	—
572	C₂²	—	8	2	no	—
573	C₂²	—	8	2	no	—
574	C₂²	C₃×C₅×#121	1	1	yes	characteristic
575	C₂²	C₃×C₅×#121	1	1	yes	characteristic
576	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
577	C₄	—	2	2	no	—
578	C₄	—	4	1	no	—
579	C₄	C₃×C₅×#121	1	1	yes	characteristic
580	C₄	C₃×C₅×#121	1	1	yes	characteristic
581	C₄	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
582	C₄	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
583	C₃	C₅×#3068	1	1	yes	characteristic, direct product factor, Sylow
584	C₂	—	2	1	no	—
585	C₂	—	2	1	no	—
586	C₂	—	2	1	no	—
587	C₂	—	4	2	no	—
588	C₂	—	8	2	no	—
589	C₂	—	8	1	no	—
590	C₂	C₂×C₃×C₅×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	C₂×#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#575	C₂	semidirect
2. C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	semidirect
3. C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	semidirect
4. C₃×C₅×#568	C₂	semidirect
5. C₃×C₅×#472	C₂	semidirect
6. C₃×C₅×#471	C₂	semidirect
7. C₃×C₅×#430	C₂	semidirect
8. C₃×C₅×#413	C₂	semidirect
9. C₂×C₃×C₅×#100	C₂	semidirect
10. C₅×#3068	C₃	direct (N × H)
15. C₃×C₅×#137	C₂²	semidirect
21. C₃×C₅×AGL(1,ℤ/8)	C₂²	semidirect
24. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
26. C₂×C₃×C₅×D₈	C₂²	semidirect
28. C₃×C₅×#131	C₂²	semidirect
34. C₃×C₅×#108	C₂²	semidirect
35. C₃×C₅×#101	C₂²	semidirect
36. C₃×C₅×#100	C₂²	semidirect
37. C₃×C₅×#100	C₂²	semidirect
38. C₃×#3068	C₅	direct (N × H)
39. C₅×#575	C₂×C₃	semidirect
40. C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	semidirect
41. C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	semidirect
42. C₅×#568	C₂×C₃	semidirect
43. C₅×#472	C₂×C₃	semidirect
44. C₅×#471	C₂×C₃	semidirect
45. C₅×#430	C₂×C₃	semidirect
46. C₅×#413	C₂×C₃	semidirect
47. C₂×C₅×#100	C₂×C₃	semidirect
94. C₃×#575	C₂×C₅	semidirect
95. C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	semidirect
96. C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	semidirect
97. C₃×#568	C₂×C₅	semidirect
98. C₃×#472	C₂×C₅	semidirect
99. C₃×#471	C₂×C₅	semidirect
100. C₃×#430	C₂×C₅	semidirect
101. C₃×#413	C₂×C₅	semidirect
102. C₂×C₃×#100	C₂×C₅	semidirect
107. C₅×#137	C₂²×C₃	semidirect
113. C₅×AGL(1,ℤ/8)	C₂²×C₃	semidirect
116. C₂×C₅×SD₁₆	C₂²×C₃	semidirect
118. C₂×C₅×D₈	C₂²×C₃	semidirect
120. C₅×#131	C₂²×C₃	semidirect
126. C₅×#108	C₂²×C₃	semidirect
127. C₅×#101	C₂²×C₃	semidirect
128. C₅×#100	C₂²×C₃	semidirect
129. C₅×#100	C₂²×C₃	semidirect
130. #3068	C₃×C₅	direct (N × H)
174. C₃×#137	C₂²×C₅	semidirect
180. C₃×AGL(1,ℤ/8)	C₂²×C₅	semidirect
183. C₂×C₃×SD₁₆	C₂²×C₅	semidirect
185. C₂×C₃×D₈	C₂²×C₅	semidirect
187. C₃×#131	C₂²×C₅	semidirect
193. C₃×#108	C₂²×C₅	semidirect
194. C₃×#101	C₂²×C₅	semidirect
195. C₃×#100	C₂²×C₅	semidirect
196. C₃×#100	C₂²×C₅	semidirect
243. #575	C₂×C₃×C₅	semidirect
244. C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	semidirect
245. C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	semidirect
246. #568	C₂×C₃×C₅	semidirect
247. #472	C₂×C₃×C₅	semidirect
248. #471	C₂×C₃×C₅	semidirect
249. #430	C₂×C₃×C₅	semidirect
250. #413	C₂×C₃×C₅	semidirect
251. C₂×#100	C₂×C₃×C₅	semidirect
359. #137	C₂²×C₃×C₅	semidirect
365. AGL(1,ℤ/8)	C₂²×C₃×C₅	semidirect
368. C₂×SD₁₆	C₂²×C₃×C₅	semidirect
370. C₂×D₈	C₂²×C₃×C₅	semidirect
372. #131	C₂²×C₃×C₅	semidirect
378. #108	C₂²×C₃×C₅	semidirect
379. #101	C₂²×C₃×C₅	semidirect
380. #100	C₂²×C₃×C₅	semidirect
381. #100	C₂²×C₃×C₅	semidirect
492. C₃×C₅	#3068	direct (N × H)
564. C₅	C₃×#3068	direct (N × H)
583. C₃	C₅×#3068	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#575	2
C₂×C₃×C₅×AGL(1,ℤ/8)	2
C₃×C₅×#568	2
C₃×C₅×#472	2
C₃×C₅×#471	2
C₃×C₅×#430	2
C₃×C₅×#413	2
C₂×C₃×C₅×#100	2
C₅×#3068	3
C₃×#3068	5

Subgroups — C3×C5×#3068 #14276102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3068 #14276102