Subgroups — C₃×C₅×#3045 #14276079

Order 1920.

1008 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#535	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#520	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#464	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#463	C₂	1	2	yes	maximal, complemented
7	C₃×C₅×#460	C₂	1	2	yes	maximal, complemented
8	C₃×C₅×#459	C₂	1	2	yes	maximal, complemented
9	C₃×C₅×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₅×#417	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×C₅×#102	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#3045	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
16	C₃×C₅×AGL(1,ℤ/8)	—	2	4	no	—
17	C₃×C₅×#135	—	2	4	no	—
18	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₃×C₅×SD₁₆	C₂²	1	2	yes	—
20	C₂×C₃×C₅×SD₁₆	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃×C₅×SD₁₆	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
23	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
25	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
26	C₃×C₅×#127	C₂²	1	2	yes	—
27	C₃×C₅×#124	C₂²	1	2	yes	—
28	C₃×C₅×#121	—	2	1	no	—
29	C₃×C₅×#121	—	2	1	no	—
30	C₃×C₅×#120	—	2	2	no	—
31	C₃×C₄×C₅×Q₈	C₂²	1	2	yes	complemented
32	C₃×C₄×C₅×D₄	—	2	2	no	—
33	C₃×C₄×C₅×D₄	C₂²	1	2	yes	complemented
34	C₃×C₅×#117	—	2	1	no	—
35	C₃×C₅×#117	C₂²	1	1	yes	characteristic
36	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
37	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
38	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
39	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
40	C₃×C₅×#103	C₂²	1	2	yes	complemented
41	C₃×C₅×#102	C₂²	1	2	yes	—
42	C₃×C₅×#102	C₂²	1	2	yes	—
43	C₃×C₅×#102	C₂²	1	2	yes	complemented
44	C₃×#3045	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
45	C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×C₅×#135	C₂×C₃	1	1	yes	characteristic, complemented
47	C₅×#535	C₂×C₃	1	1	yes	characteristic, complemented
48	C₅×#520	C₂×C₃	1	1	yes	characteristic, complemented
49	C₅×#464	C₂×C₃	1	2	yes	complemented
50	C₅×#463	C₂×C₃	1	2	yes	complemented
51	C₅×#460	C₂×C₃	1	2	yes	complemented
52	C₅×#459	C₂×C₃	1	2	yes	complemented
53	C₅×#424	C₂×C₃	1	1	yes	characteristic, complemented
54	C₅×#417	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×C₅×#102	C₂×C₃	1	1	yes	characteristic, complemented
56	C₂⁴×C₃×C₅	—	2	1	no	—
57	C₃×C₅×#41	—	2	2	no	—
58	C₃×C₅×#41	—	2	2	no	—
59	C₃×C₅×#41	—	2	2	no	—
60	C₃×C₅×#41	D₄	1	4	yes	complemented
61	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₅×D₄	—	2	1	no	—
64	C₂×C₃×C₅×D₄	—	2	1	no	—
65	C₂×C₃×C₅×D₄	—	2	2	no	—
66	C₂×C₃×C₅×D₄	—	2	1	no	—
67	C₂×C₃×C₅×D₄	—	2	2	no	—
68	C₂×C₃×C₅×D₄	—	2	2	no	—
69	C₂×C₃×C₅×D₄	—	2	2	no	—
70	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
71	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
72	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
73	C₂²×C₃×C₄×C₅	—	2	1	no	—
74	C₂²×C₃×C₄×C₅	—	2	1	no	—
75	C₂²×C₃×C₄×C₅	—	2	1	no	—
76	C₂²×C₃×C₄×C₅	—	2	2	no	—
77	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
78	C₃×C₅×Q₁₆	—	2	2	no	—
79	C₃×C₅×SD₁₆	—	2	4	no	—
80	C₃×C₅×SD₁₆	—	2	2	no	—
81	C₃×C₅×SD₁₆	—	2	2	no	—
82	C₃×C₅×D₈	—	2	4	no	—
83	C₃×C₅×D₈	—	2	2	no	—
84	C₃×C₅×M₁₆	—	2	2	no	—
85	C₂×C₃×C₅×C₈	—	2	2	no	—
86	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
87	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
88	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
89	C₃×C₅×C₄⋊C₄	—	2	1	no	—
90	C₃×C₅×C₄⋊C₄	—	2	2	no	—
91	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
92	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
93	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
94	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
95	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
96	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
97	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
98	C₃×C₄²×C₅	—	2	2	no	—
99	C₃×C₄²×C₅	C₂³	1	2	yes	—
100	C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	1	1	yes	characteristic, complemented
101	C₂×C₃×#135	C₂×C₅	1	1	yes	characteristic, complemented
102	C₃×#535	C₂×C₅	1	1	yes	characteristic, complemented
103	C₃×#520	C₂×C₅	1	1	yes	characteristic, complemented
104	C₃×#464	C₂×C₅	1	2	yes	complemented
105	C₃×#463	C₂×C₅	1	2	yes	complemented
106	C₃×#460	C₂×C₅	1	2	yes	complemented
107	C₃×#459	C₂×C₅	1	2	yes	complemented
108	C₃×#424	C₂×C₅	1	1	yes	characteristic, complemented
109	C₃×#417	C₂×C₅	1	1	yes	characteristic, complemented
110	C₂×C₃×#102	C₂×C₅	1	1	yes	characteristic, complemented
111	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
112	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
113	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
114	C₅×AGL(1,ℤ/8)	—	2	4	no	—
115	C₅×#135	—	2	4	no	—
116	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
117	C₂×C₅×SD₁₆	C₂²×C₃	1	2	yes	—
118	C₂×C₅×SD₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
119	C₂×C₅×SD₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
120	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
121	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic, complemented
122	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
123	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
124	C₅×#127	C₂²×C₃	1	2	yes	—
125	C₅×#124	C₂²×C₃	1	2	yes	—
126	C₅×#121	—	2	1	no	—
127	C₅×#121	—	2	1	no	—
128	C₅×#120	—	2	2	no	—
129	C₄×C₅×Q₈	C₂²×C₃	1	2	yes	complemented
130	C₄×C₅×D₄	—	2	2	no	—
131	C₄×C₅×D₄	C₂²×C₃	1	2	yes	complemented
132	C₅×#117	—	2	1	no	—
133	C₅×#117	C₂²×C₃	1	1	yes	characteristic
134	C₂×C₅×C₄⋊C₄	—	2	1	no	—
135	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
136	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
137	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
138	C₅×#103	C₂²×C₃	1	2	yes	complemented
139	C₅×#102	C₂²×C₃	1	2	yes	—
140	C₅×#102	C₂²×C₃	1	2	yes	—
141	C₅×#102	C₂²×C₃	1	2	yes	complemented
142	#3045	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
143	C₂³×C₃×C₅	—	2	1	no	—
144	C₂³×C₃×C₅	—	2	2	no	—
145	C₂³×C₃×C₅	—	2	1	no	—
146	C₂³×C₃×C₅	—	4	2	no	—
147	C₂³×C₃×C₅	—	8	1	no	—
148	C₂³×C₃×C₅	—	8	1	no	—
149	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
150	C₃×C₅×Q₈	—	2	1	no	—
151	C₃×C₅×Q₈	—	2	1	no	—
152	C₃×C₅×Q₈	—	2	1	no	—
153	C₃×C₅×Q₈	C₂×D₄	1	2	yes	—
154	C₃×C₅×D₄	—	2	4	no	—
155	C₃×C₅×D₄	—	2	2	no	—
156	C₃×C₅×D₄	—	2	1	no	—
157	C₃×C₅×D₄	—	2	1	no	—
158	C₃×C₅×D₄	—	2	2	no	—
159	C₃×C₅×D₄	—	2	1	no	—
160	C₃×C₅×D₄	—	4	2	no	—
161	C₃×C₅×D₄	—	4	2	no	—
162	C₃×C₅×D₄	—	8	2	no	—
163	C₃×C₅×D₄	—	8	2	no	—
164	C₃×C₅×D₄	C₂×D₄	1	2	yes	—
165	C₂×C₃×C₄×C₅	—	2	1	no	—
166	C₂×C₃×C₄×C₅	—	2	4	no	—
167	C₂×C₃×C₄×C₅	—	2	2	no	—
168	C₂×C₃×C₄×C₅	—	2	1	no	—
169	C₂×C₃×C₄×C₅	—	4	2	no	—
170	C₂×C₃×C₄×C₅	—	8	2	no	—
171	C₂×C₃×C₄×C₅	—	8	1	no	—
172	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
173	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
174	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
176	C₂×C₃×C₄×C₅	#41	1	2	yes	—
177	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
178	C₃×C₅×C₈	—	2	1	no	—
179	C₃×C₅×C₈	—	2	1	no	—
180	C₃×C₅×C₈	—	2	2	no	—
181	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
182	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
183	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
184	C₃×AGL(1,ℤ/8)	—	2	4	no	—
185	C₃×#135	—	2	4	no	—
186	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
187	C₂×C₃×SD₁₆	C₂²×C₅	1	2	yes	—
188	C₂×C₃×SD₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
189	C₂×C₃×SD₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
190	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
191	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic, complemented
192	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
193	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
194	C₃×#127	C₂²×C₅	1	2	yes	—
195	C₃×#124	C₂²×C₅	1	2	yes	—
196	C₃×#121	—	2	1	no	—
197	C₃×#121	—	2	1	no	—
198	C₃×#120	—	2	2	no	—
199	C₃×C₄×Q₈	C₂²×C₅	1	2	yes	complemented
200	C₃×C₄×D₄	—	2	2	no	—
201	C₃×C₄×D₄	C₂²×C₅	1	2	yes	complemented
202	C₃×#117	—	2	1	no	—
203	C₃×#117	C₂²×C₅	1	1	yes	characteristic
204	C₂×C₃×C₄⋊C₄	—	2	1	no	—
205	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
206	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
207	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
208	C₃×#103	C₂²×C₅	1	2	yes	complemented
209	C₃×#102	C₂²×C₅	1	2	yes	—
210	C₃×#102	C₂²×C₅	1	2	yes	—
211	C₃×#102	C₂²×C₅	1	2	yes	complemented
212	C₂⁴×C₅	—	2	1	no	—
213	C₅×#41	—	2	2	no	—
214	C₅×#41	—	2	2	no	—
215	C₅×#41	—	2	2	no	—
216	C₅×#41	C₃×D₄	1	4	yes	complemented
217	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
218	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
219	C₂×C₅×D₄	—	2	1	no	—
220	C₂×C₅×D₄	—	2	1	no	—
221	C₂×C₅×D₄	—	2	2	no	—
222	C₂×C₅×D₄	—	2	1	no	—
223	C₂×C₅×D₄	—	2	2	no	—
224	C₂×C₅×D₄	—	2	2	no	—
225	C₂×C₅×D₄	—	2	2	no	—
226	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
227	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
228	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
229	C₂²×C₄×C₅	—	2	1	no	—
230	C₂²×C₄×C₅	—	2	1	no	—
231	C₂²×C₄×C₅	—	2	1	no	—
232	C₂²×C₄×C₅	—	2	2	no	—
233	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
234	C₅×Q₁₆	—	2	2	no	—
235	C₅×SD₁₆	—	2	4	no	—
236	C₅×SD₁₆	—	2	2	no	—
237	C₅×SD₁₆	—	2	2	no	—
238	C₅×D₈	—	2	4	no	—
239	C₅×D₈	—	2	2	no	—
240	C₅×M₁₆	—	2	2	no	—
241	C₂×C₅×C₈	—	2	2	no	—
242	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
243	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
244	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
245	C₅×C₄⋊C₄	—	2	1	no	—
246	C₅×C₄⋊C₄	—	2	2	no	—
247	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
248	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
249	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
250	C₅×C₂²⋊C₄	—	2	2	no	—
251	C₅×C₂²⋊C₄	—	2	2	no	—
252	C₅×C₂²⋊C₄	—	2	2	no	—
253	C₅×C₂²⋊C₄	C₃×D₄	1	2	yes	—
254	C₄²×C₅	—	2	2	no	—
255	C₄²×C₅	C₂³×C₃	1	2	yes	—
256	C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	1	1	yes	characteristic, complemented
257	C₂×#135	C₂×C₃×C₅	1	1	yes	characteristic, complemented
258	#535	C₂×C₃×C₅	1	1	yes	characteristic, complemented
259	#520	C₂×C₃×C₅	1	1	yes	characteristic, complemented
260	#464	C₂×C₃×C₅	1	2	yes	complemented
261	#463	C₂×C₃×C₅	1	2	yes	complemented
262	#460	C₂×C₃×C₅	1	2	yes	complemented
263	#459	C₂×C₃×C₅	1	2	yes	complemented
264	#424	C₂×C₃×C₅	1	1	yes	characteristic, complemented
265	#417	C₂×C₃×C₅	1	1	yes	characteristic, complemented
266	C₂×#102	C₂×C₃×C₅	1	1	yes	characteristic, complemented
267	C₂²×C₃×C₅	—	2	1	no	—
268	C₂²×C₃×C₅	—	2	1	no	—
269	C₂²×C₃×C₅	—	2	2	no	—
270	C₂²×C₃×C₅	—	4	1	no	—
271	C₂²×C₃×C₅	—	4	1	no	—
272	C₂²×C₃×C₅	—	4	1	no	—
273	C₂²×C₃×C₅	—	4	2	no	—
274	C₂²×C₃×C₅	—	4	1	no	—
275	C₂²×C₃×C₅	—	4	2	no	—
276	C₂²×C₃×C₅	—	4	1	no	—
277	C₂²×C₃×C₅	—	4	2	no	—
278	C₂²×C₃×C₅	—	8	4	no	—
279	C₂²×C₃×C₅	#121	1	2	yes	—
280	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
281	C₃×C₄×C₅	—	2	2	no	—
282	C₃×C₄×C₅	—	2	2	no	—
283	C₃×C₄×C₅	—	2	2	no	—
284	C₃×C₄×C₅	—	4	1	no	—
285	C₃×C₄×C₅	—	4	2	no	—
286	C₃×C₄×C₅	#121	1	2	yes	—
287	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
288	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
289	C₂⁴×C₃	—	2	1	no	—
290	C₃×#41	—	2	2	no	—
291	C₃×#41	—	2	2	no	—
292	C₃×#41	—	2	2	no	—
293	C₃×#41	C₅×D₄	1	4	yes	complemented
294	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
295	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
296	C₂×C₃×D₄	—	2	1	no	—
297	C₂×C₃×D₄	—	2	1	no	—
298	C₂×C₃×D₄	—	2	2	no	—
299	C₂×C₃×D₄	—	2	1	no	—
300	C₂×C₃×D₄	—	2	2	no	—
301	C₂×C₃×D₄	—	2	2	no	—
302	C₂×C₃×D₄	—	2	2	no	—
303	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
304	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
305	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
306	C₂²×C₃×C₄	—	2	1	no	—
307	C₂²×C₃×C₄	—	2	1	no	—
308	C₂²×C₃×C₄	—	2	1	no	—
309	C₂²×C₃×C₄	—	2	2	no	—
310	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
311	C₃×Q₁₆	—	2	2	no	—
312	C₃×SD₁₆	—	2	4	no	—
313	C₃×SD₁₆	—	2	2	no	—
314	C₃×SD₁₆	—	2	2	no	—
315	C₃×D₈	—	2	4	no	—
316	C₃×D₈	—	2	2	no	—
317	C₃×M₁₆	—	2	2	no	—
318	C₂×C₃×C₈	—	2	2	no	—
319	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
320	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
321	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
322	C₃×C₄⋊C₄	—	2	1	no	—
323	C₃×C₄⋊C₄	—	2	2	no	—
324	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
325	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
326	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
327	C₃×C₂²⋊C₄	—	2	2	no	—
328	C₃×C₂²⋊C₄	—	2	2	no	—
329	C₃×C₂²⋊C₄	—	2	2	no	—
330	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
331	C₃×C₄²	—	2	2	no	—
332	C₃×C₄²	C₂³×C₅	1	2	yes	—
333	C₂³×C₅	—	2	1	no	—
334	C₂³×C₅	—	2	2	no	—
335	C₂³×C₅	—	2	1	no	—
336	C₂³×C₅	—	4	2	no	—
337	C₂³×C₅	—	8	1	no	—
338	C₂³×C₅	—	8	1	no	—
339	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
340	C₅×Q₈	—	2	1	no	—
341	C₅×Q₈	—	2	1	no	—
342	C₅×Q₈	—	2	1	no	—
343	C₅×Q₈	C₂×C₃×D₄	1	2	yes	—
344	C₅×D₄	—	2	4	no	—
345	C₅×D₄	—	2	2	no	—
346	C₅×D₄	—	2	1	no	—
347	C₅×D₄	—	2	1	no	—
348	C₅×D₄	—	2	2	no	—
349	C₅×D₄	—	2	1	no	—
350	C₅×D₄	—	4	2	no	—
351	C₅×D₄	—	4	2	no	—
352	C₅×D₄	—	8	2	no	—
353	C₅×D₄	—	8	2	no	—
354	C₅×D₄	C₂×C₃×D₄	1	2	yes	—
355	C₂×C₄×C₅	—	2	1	no	—
356	C₂×C₄×C₅	—	2	4	no	—
357	C₂×C₄×C₅	—	2	2	no	—
358	C₂×C₄×C₅	—	2	1	no	—
359	C₂×C₄×C₅	—	4	2	no	—
360	C₂×C₄×C₅	—	8	2	no	—
361	C₂×C₄×C₅	—	8	1	no	—
362	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
363	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
364	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
365	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
366	C₂×C₄×C₅	C₃×#41	1	2	yes	—
367	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
368	C₅×C₈	—	2	1	no	—
369	C₅×C₈	—	2	1	no	—
370	C₅×C₈	—	2	2	no	—
371	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
372	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
373	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
374	AGL(1,ℤ/8)	—	2	4	no	—
375	#135	—	2	4	no	—
376	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
377	C₂×SD₁₆	C₂²×C₃×C₅	1	2	yes	—
378	C₂×SD₁₆	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
379	C₂×SD₁₆	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
380	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
381	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
382	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
383	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
384	#127	C₂²×C₃×C₅	1	2	yes	—
385	#124	C₂²×C₃×C₅	1	2	yes	—
386	#121	—	2	1	no	—
387	#121	—	2	1	no	—
388	#120	—	2	2	no	—
389	C₄×Q₈	C₂²×C₃×C₅	1	2	yes	complemented
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	C₂²×C₃×C₅	1	2	yes	complemented
392	#117	—	2	1	no	—
393	#117	C₂²×C₃×C₅	1	1	yes	characteristic
394	C₂×C₄⋊C₄	—	2	1	no	—
395	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
396	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	complemented
397	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	complemented
398	#103	C₂²×C₃×C₅	1	2	yes	complemented
399	#102	C₂²×C₃×C₅	1	2	yes	—
400	#102	C₂²×C₃×C₅	1	2	yes	—
401	#102	C₂²×C₃×C₅	1	2	yes	complemented
402	C₂×C₃×C₅	—	2	2	no	—
403	C₂×C₃×C₅	—	4	2	no	—
404	C₂×C₃×C₅	—	8	1	no	—
405	C₂×C₃×C₅	—	8	2	no	—
406	C₂×C₃×C₅	C₂×#121	1	1	yes	characteristic
407	C₂×C₃×C₅	#576	1	1	yes	characteristic
408	C₂×C₃×C₅	#577	1	1	yes	characteristic
409	C₂³×C₃	—	2	1	no	—
410	C₂³×C₃	—	2	2	no	—
411	C₂³×C₃	—	2	1	no	—
412	C₂³×C₃	—	4	2	no	—
413	C₂³×C₃	—	8	1	no	—
414	C₂³×C₃	—	8	1	no	—
415	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
416	C₃×Q₈	—	2	1	no	—
417	C₃×Q₈	—	2	1	no	—
418	C₃×Q₈	—	2	1	no	—
419	C₃×Q₈	C₂×C₅×D₄	1	2	yes	—
420	C₃×D₄	—	2	4	no	—
421	C₃×D₄	—	2	2	no	—
422	C₃×D₄	—	2	1	no	—
423	C₃×D₄	—	2	1	no	—
424	C₃×D₄	—	2	2	no	—
425	C₃×D₄	—	2	1	no	—
426	C₃×D₄	—	4	2	no	—
427	C₃×D₄	—	4	2	no	—
428	C₃×D₄	—	8	2	no	—
429	C₃×D₄	—	8	2	no	—
430	C₃×D₄	C₂×C₅×D₄	1	2	yes	—
431	C₂×C₃×C₄	—	2	1	no	—
432	C₂×C₃×C₄	—	2	4	no	—
433	C₂×C₃×C₄	—	2	2	no	—
434	C₂×C₃×C₄	—	2	1	no	—
435	C₂×C₃×C₄	—	4	2	no	—
436	C₂×C₃×C₄	—	8	2	no	—
437	C₂×C₃×C₄	—	8	1	no	—
438	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
439	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
440	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
441	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
442	C₂×C₃×C₄	C₅×#41	1	2	yes	—
443	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
444	C₃×C₈	—	2	1	no	—
445	C₃×C₈	—	2	1	no	—
446	C₃×C₈	—	2	2	no	—
447	C₂²×C₅	—	2	1	no	—
448	C₂²×C₅	—	2	1	no	—
449	C₂²×C₅	—	2	2	no	—
450	C₂²×C₅	—	4	1	no	—
451	C₂²×C₅	—	4	1	no	—
452	C₂²×C₅	—	4	1	no	—
453	C₂²×C₅	—	4	2	no	—
454	C₂²×C₅	—	4	1	no	—
455	C₂²×C₅	—	4	2	no	—
456	C₂²×C₅	—	4	1	no	—
457	C₂²×C₅	—	4	2	no	—
458	C₂²×C₅	—	8	4	no	—
459	C₂²×C₅	C₃×#121	1	2	yes	—
460	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
461	C₄×C₅	—	2	2	no	—
462	C₄×C₅	—	2	2	no	—
463	C₄×C₅	—	2	2	no	—
464	C₄×C₅	—	4	2	no	—
465	C₄×C₅	—	4	1	no	—
466	C₄×C₅	C₃×#121	1	2	yes	—
467	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
468	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
469	C₂⁴	—	2	1	no	—
470	#41	—	2	2	no	—
471	#41	—	2	2	no	—
472	#41	—	2	2	no	—
473	#41	C₃×C₅×D₄	1	4	yes	complemented
474	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
475	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
476	C₂×D₄	—	2	1	no	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	—	2	2	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	2	no	—
482	C₂×D₄	—	2	2	no	—
483	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
484	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
485	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
491	Q₁₆	—	2	2	no	—
492	SD₁₆	—	2	4	no	—
493	SD₁₆	—	2	2	no	—
494	SD₁₆	—	2	2	no	—
495	D₈	—	2	4	no	—
496	D₈	—	2	2	no	—
497	M₁₆	—	2	2	no	—
498	C₂×C₈	—	2	2	no	—
499	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
500	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
501	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
505	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
506	C₄⋊C₄	C₂³×C₃×C₅	1	2	yes	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	C₃×C₅×D₄	1	2	yes	—
511	C₄²	—	2	2	no	—
512	C₄²	C₂³×C₃×C₅	1	2	yes	—
513	C₃×C₅	#3045	1	1	yes	characteristic, direct product factor, Hall
514	C₂²×C₃	—	2	1	no	—
515	C₂²×C₃	—	2	1	no	—
516	C₂²×C₃	—	2	2	no	—
517	C₂²×C₃	—	4	1	no	—
518	C₂²×C₃	—	4	1	no	—
519	C₂²×C₃	—	4	1	no	—
520	C₂²×C₃	—	4	2	no	—
521	C₂²×C₃	—	4	1	no	—
522	C₂²×C₃	—	4	2	no	—
523	C₂²×C₃	—	4	1	no	—
524	C₂²×C₃	—	4	2	no	—
525	C₂²×C₃	—	8	4	no	—
526	C₂²×C₃	C₅×#121	1	2	yes	—
527	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
528	C₃×C₄	—	2	2	no	—
529	C₃×C₄	—	2	2	no	—
530	C₃×C₄	—	2	2	no	—
531	C₃×C₄	—	4	1	no	—
532	C₃×C₄	—	4	2	no	—
533	C₃×C₄	C₅×#121	1	2	yes	—
534	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
535	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
536	C₂×C₅	—	2	2	no	—
537	C₂×C₅	—	4	2	no	—
538	C₂×C₅	—	8	2	no	—
539	C₂×C₅	—	8	1	no	—
540	C₂×C₅	C₂×C₃×#121	1	1	yes	characteristic
541	C₂×C₅	C₃×#576	1	1	yes	characteristic
542	C₂×C₅	C₃×#577	1	1	yes	characteristic
543	C₂³	—	2	1	no	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	1	no	—
546	C₂³	—	4	2	no	—
547	C₂³	—	8	1	no	—
548	C₂³	—	8	1	no	—
549	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
550	Q₈	—	2	1	no	—
551	Q₈	—	2	1	no	—
552	Q₈	—	2	1	no	—
553	Q₈	C₂×C₃×C₅×D₄	1	2	yes	—
554	D₄	—	2	4	no	—
555	D₄	—	2	2	no	—
556	D₄	—	2	1	no	—
557	D₄	—	2	1	no	—
558	D₄	—	2	2	no	—
559	D₄	—	2	1	no	—
560	D₄	—	4	2	no	—
561	D₄	—	4	2	no	—
562	D₄	—	8	2	no	—
563	D₄	—	8	2	no	—
564	D₄	C₂×C₃×C₅×D₄	1	2	yes	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	4	2	no	—
570	C₂×C₄	—	8	2	no	—
571	C₂×C₄	—	8	1	no	—
572	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
573	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
574	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
575	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
576	C₂×C₄	C₃×C₅×#41	1	2	yes	—
577	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
578	C₈	—	2	1	no	—
579	C₈	—	2	1	no	—
580	C₈	—	2	2	no	—
581	C₂×C₃	—	2	2	no	—
582	C₂×C₃	—	4	2	no	—
583	C₂×C₃	—	8	2	no	—
584	C₂×C₃	—	8	1	no	—
585	C₂×C₃	C₂×C₅×#121	1	1	yes	characteristic
586	C₂×C₃	C₅×#576	1	1	yes	characteristic
587	C₂×C₃	C₅×#577	1	1	yes	characteristic
588	C₅	C₃×#3045	1	1	yes	characteristic, direct product factor, Sylow
589	C₂²	—	2	1	no	—
590	C₂²	—	2	1	no	—
591	C₂²	—	2	2	no	—
592	C₂²	—	4	1	no	—
593	C₂²	—	4	1	no	—
594	C₂²	—	4	1	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	4	1	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	4	1	no	—
599	C₂²	—	4	2	no	—
600	C₂²	—	8	4	no	—
601	C₂²	C₃×C₅×#121	1	2	yes	—
602	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
603	C₄	—	2	2	no	—
604	C₄	—	2	2	no	—
605	C₄	—	2	2	no	—
606	C₄	—	4	1	no	—
607	C₄	—	4	2	no	—
608	C₄	C₃×C₅×#121	1	2	yes	—
609	C₄	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
610	C₄	C₂×C₃×C₅×#41	1	1	yes	characteristic
611	C₃	C₅×#3045	1	1	yes	characteristic, direct product factor, Sylow
612	C₂	—	2	2	no	—
613	C₂	—	4	2	no	—
614	C₂	—	8	2	no	—
615	C₂	—	8	1	no	—
616	C₂	C₂×C₃×C₅×#121	1	1	yes	characteristic
617	C₂	C₃×C₅×#576	1	1	yes	characteristic
618	C₂	C₃×C₅×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	semidirect
2. C₂×C₃×C₅×#135	C₂	semidirect
3. C₃×C₅×#535	C₂	semidirect
4. C₃×C₅×#520	C₂	semidirect
5. C₃×C₅×#464	C₂	semidirect
6. C₃×C₅×#463	C₂	semidirect
7. C₃×C₅×#460	C₂	semidirect
8. C₃×C₅×#459	C₂	semidirect
9. C₃×C₅×#424	C₂	semidirect
10. C₃×C₅×#417	C₂	semidirect
11. C₂×C₃×C₅×#102	C₂	semidirect
12. C₅×#3045	C₃	direct (N × H)
18. C₂×C₃×C₅×Q₁₆	C₂²	semidirect
20. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
21. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
23. C₂×C₃×C₅×D₈	C₂²	semidirect
31. C₃×C₄×C₅×Q₈	C₂²	semidirect
33. C₃×C₄×C₅×D₄	C₂²	semidirect
38. C₃×C₅×C₄⋊C₈	C₂²	semidirect
39. C₃×C₅×C₄⋊C₈	C₂²	semidirect
40. C₃×C₅×#103	C₂²	semidirect
43. C₃×C₅×#102	C₂²	semidirect
44. C₃×#3045	C₅	direct (N × H)
45. C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	semidirect
46. C₂×C₅×#135	C₂×C₃	semidirect
47. C₅×#535	C₂×C₃	semidirect
48. C₅×#520	C₂×C₃	semidirect
49. C₅×#464	C₂×C₃	semidirect
50. C₅×#463	C₂×C₃	semidirect
51. C₅×#460	C₂×C₃	semidirect
52. C₅×#459	C₂×C₃	semidirect
53. C₅×#424	C₂×C₃	semidirect
54. C₅×#417	C₂×C₃	semidirect
55. C₂×C₅×#102	C₂×C₃	semidirect
60. C₃×C₅×#41	D₄	semidirect
100. C₂×C₃×AGL(1,ℤ/8)	C₂×C₅	semidirect
101. C₂×C₃×#135	C₂×C₅	semidirect
102. C₃×#535	C₂×C₅	semidirect
103. C₃×#520	C₂×C₅	semidirect
104. C₃×#464	C₂×C₅	semidirect
105. C₃×#463	C₂×C₅	semidirect
106. C₃×#460	C₂×C₅	semidirect
107. C₃×#459	C₂×C₅	semidirect
108. C₃×#424	C₂×C₅	semidirect
109. C₃×#417	C₂×C₅	semidirect
110. C₂×C₃×#102	C₂×C₅	semidirect
116. C₂×C₅×Q₁₆	C₂²×C₃	semidirect
118. C₂×C₅×SD₁₆	C₂²×C₃	semidirect
119. C₂×C₅×SD₁₆	C₂²×C₃	semidirect
121. C₂×C₅×D₈	C₂²×C₃	semidirect
129. C₄×C₅×Q₈	C₂²×C₃	semidirect
131. C₄×C₅×D₄	C₂²×C₃	semidirect
136. C₅×C₄⋊C₈	C₂²×C₃	semidirect
137. C₅×C₄⋊C₈	C₂²×C₃	semidirect
138. C₅×#103	C₂²×C₃	semidirect
141. C₅×#102	C₂²×C₃	semidirect
142. #3045	C₃×C₅	direct (N × H)
186. C₂×C₃×Q₁₆	C₂²×C₅	semidirect
188. C₂×C₃×SD₁₆	C₂²×C₅	semidirect
189. C₂×C₃×SD₁₆	C₂²×C₅	semidirect
191. C₂×C₃×D₈	C₂²×C₅	semidirect
199. C₃×C₄×Q₈	C₂²×C₅	semidirect
201. C₃×C₄×D₄	C₂²×C₅	semidirect
206. C₃×C₄⋊C₈	C₂²×C₅	semidirect
207. C₃×C₄⋊C₈	C₂²×C₅	semidirect
208. C₃×#103	C₂²×C₅	semidirect
211. C₃×#102	C₂²×C₅	semidirect
216. C₅×#41	C₃×D₄	semidirect
256. C₂×AGL(1,ℤ/8)	C₂×C₃×C₅	semidirect
257. C₂×#135	C₂×C₃×C₅	semidirect
258. #535	C₂×C₃×C₅	semidirect
259. #520	C₂×C₃×C₅	semidirect
260. #464	C₂×C₃×C₅	semidirect
261. #463	C₂×C₃×C₅	semidirect
262. #460	C₂×C₃×C₅	semidirect
263. #459	C₂×C₃×C₅	semidirect
264. #424	C₂×C₃×C₅	semidirect
265. #417	C₂×C₃×C₅	semidirect
266. C₂×#102	C₂×C₃×C₅	semidirect
293. C₃×#41	C₅×D₄	semidirect
376. C₂×Q₁₆	C₂²×C₃×C₅	semidirect
378. C₂×SD₁₆	C₂²×C₃×C₅	semidirect
379. C₂×SD₁₆	C₂²×C₃×C₅	semidirect
381. C₂×D₈	C₂²×C₃×C₅	semidirect
389. C₄×Q₈	C₂²×C₃×C₅	semidirect
391. C₄×D₄	C₂²×C₃×C₅	semidirect
396. C₄⋊C₈	C₂²×C₃×C₅	semidirect
397. C₄⋊C₈	C₂²×C₃×C₅	semidirect
398. #103	C₂²×C₃×C₅	semidirect
401. #102	C₂²×C₃×C₅	semidirect
473. #41	C₃×C₅×D₄	semidirect
513. C₃×C₅	#3045	direct (N × H)
588. C₅	C₃×#3045	direct (N × H)
611. C₃	C₅×#3045	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×AGL(1,ℤ/8)	2
C₂×C₃×C₅×#135	2
C₃×C₅×#535	2
C₃×C₅×#520	2
C₃×C₅×#464	2
C₃×C₅×#463	2
C₃×C₅×#460	2
C₃×C₅×#459	2
C₃×C₅×#424	2
C₃×C₅×#417	2
C₂×C₃×C₅×#102	2
C₅×#3045	3
C₃×#3045	5

Subgroups — C3×C5×#3045 #14276079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3045 #14276079