Subgroups — C₃×C₅×#3091 #14276125

Order 1920.

840 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#525	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#485	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#484	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#483	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#482	C₂	1	2	yes	maximal, complemented
7	C₃×C₅×#481	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#480	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₅×#427	C₂	1	2	yes	maximal, complemented
10	C₃×C₅×#416	C₂	1	2	yes	maximal, complemented
11	C₂×C₃×C₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#3091	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
16	C₃×C₅×#123	—	2	1	no	—
17	C₃×C₅×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
19	C₃×C₅×#121	C₂²	1	1	yes	characteristic
20	C₃×C₅×#121	C₂²	1	1	yes	characteristic
21	C₃×C₅×#120	—	2	1	no	—
22	C₃×C₄×C₅×D₄	—	2	1	no	—
23	C₃×C₄×C₅×D₄	—	2	1	no	—
24	C₃×C₅×#117	—	2	1	no	—
25	C₃×C₅×#117	—	2	1	no	—
26	C₃×C₅×#117	C₂²	1	2	yes	—
27	C₃×C₅×#117	C₂²	1	2	yes	—
28	C₃×C₅×#117	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₃×C₅×C₂²⋊C₄	—	2	1	no	—
32	C₂×C₃×C₄²×C₅	—	2	1	no	—
33	C₃×C₅×#107	C₂²	1	2	yes	complemented
34	C₃×C₅×#107	C₂²	1	2	yes	complemented
35	C₃×C₅×#106	C₂²	1	2	yes	complemented
36	C₃×C₅×#106	C₂²	1	2	yes	complemented
37	C₃×C₅×#103	C₂²	1	2	yes	complemented
38	C₃×C₅×#103	C₂²	1	2	yes	complemented
39	C₃×C₅×#102	C₂²	1	2	yes	complemented
40	C₃×C₅×#102	C₂²	1	2	yes	complemented
41	C₃×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
44	C₃×#3091	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
45	C₅×#525	C₂×C₃	1	1	yes	characteristic
46	C₂×C₅×#117	C₂×C₃	1	1	yes	characteristic, complemented
47	C₅×#485	C₂×C₃	1	2	yes	complemented
48	C₅×#484	C₂×C₃	1	1	yes	characteristic, complemented
49	C₅×#483	C₂×C₃	1	1	yes	characteristic, complemented
50	C₅×#482	C₂×C₃	1	2	yes	complemented
51	C₅×#481	C₂×C₃	1	1	yes	characteristic, complemented
52	C₅×#480	C₂×C₃	1	1	yes	characteristic, complemented
53	C₅×#427	C₂×C₃	1	2	yes	complemented
54	C₅×#416	C₂×C₃	1	2	yes	complemented
55	C₂×C₅×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
56	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
57	C₃×C₅×#41	—	2	4	no	—
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	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄×C₅	—	2	4	no	—
64	C₂²×C₃×C₄×C₅	—	2	1	no	—
65	C₂²×C₃×C₄×C₅	—	2	1	no	—
66	C₂²×C₃×C₄×C₅	—	2	1	no	—
67	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
68	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
69	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
70	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
71	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
72	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
73	C₂×C₃×C₅×C₈	—	2	4	no	—
74	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
75	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
76	C₃×C₅×C₄⋊C₄	—	2	1	no	—
77	C₃×C₅×C₄⋊C₄	—	2	1	no	—
78	C₃×C₅×C₄⋊C₄	—	2	1	no	—
79	C₃×C₅×C₄⋊C₄	—	2	1	no	—
80	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
81	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
82	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
83	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
84	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
85	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
86	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
87	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
88	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
89	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
90	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
91	C₃×C₄²×C₅	—	2	2	no	—
92	C₃×C₄²×C₅	—	2	2	no	—
93	C₃×C₄²×C₅	—	2	1	no	—
94	C₃×#525	C₂×C₅	1	1	yes	characteristic
95	C₂×C₃×#117	C₂×C₅	1	1	yes	characteristic, complemented
96	C₃×#485	C₂×C₅	1	2	yes	complemented
97	C₃×#484	C₂×C₅	1	1	yes	characteristic, complemented
98	C₃×#483	C₂×C₅	1	1	yes	characteristic, complemented
99	C₃×#482	C₂×C₅	1	2	yes	complemented
100	C₃×#481	C₂×C₅	1	1	yes	characteristic, complemented
101	C₃×#480	C₂×C₅	1	1	yes	characteristic, complemented
102	C₃×#427	C₂×C₅	1	2	yes	complemented
103	C₃×#416	C₂×C₅	1	2	yes	complemented
104	C₂×C₃×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
105	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
106	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
107	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
108	C₅×#123	—	2	1	no	—
109	C₅×C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
110	C₅×C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
111	C₅×#121	C₂²×C₃	1	1	yes	characteristic
112	C₅×#121	C₂²×C₃	1	1	yes	characteristic
113	C₅×#120	—	2	1	no	—
114	C₄×C₅×D₄	—	2	1	no	—
115	C₄×C₅×D₄	—	2	1	no	—
116	C₅×#117	—	2	1	no	—
117	C₅×#117	—	2	1	no	—
118	C₅×#117	C₂²×C₃	1	2	yes	—
119	C₅×#117	C₂²×C₃	1	1	yes	characteristic
120	C₅×#117	C₂²×C₃	1	2	yes	—
121	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
122	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
123	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
124	C₂×C₄²×C₅	—	2	1	no	—
125	C₅×#107	C₂²×C₃	1	2	yes	complemented
126	C₅×#107	C₂²×C₃	1	2	yes	complemented
127	C₅×#106	C₂²×C₃	1	2	yes	complemented
128	C₅×#106	C₂²×C₃	1	2	yes	complemented
129	C₅×#103	C₂²×C₃	1	2	yes	complemented
130	C₅×#103	C₂²×C₃	1	2	yes	complemented
131	C₅×#102	C₂²×C₃	1	2	yes	complemented
132	C₅×#102	C₂²×C₃	1	2	yes	complemented
133	C₅×C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
134	C₅×C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
135	C₅×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
136	#3091	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
137	C₂³×C₃×C₅	—	2	2	no	—
138	C₂³×C₃×C₅	—	2	2	no	—
139	C₂³×C₃×C₅	—	2	1	no	—
140	C₂³×C₃×C₅	—	4	1	no	—
141	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
142	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
143	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
144	C₃×C₅×Q₈	—	2	2	no	—
145	C₃×C₅×D₄	—	2	2	no	—
146	C₃×C₅×D₄	—	4	2	no	—
147	C₃×C₅×D₄	—	4	2	no	—
148	C₃×C₅×D₄	—	4	2	no	—
149	C₂×C₃×C₄×C₅	—	2	2	no	—
150	C₂×C₃×C₄×C₅	—	2	2	no	—
151	C₂×C₃×C₄×C₅	—	2	2	no	—
152	C₂×C₃×C₄×C₅	—	2	2	no	—
153	C₂×C₃×C₄×C₅	—	2	2	no	—
154	C₂×C₃×C₄×C₅	—	2	2	no	—
155	C₂×C₃×C₄×C₅	—	2	2	no	—
156	C₂×C₃×C₄×C₅	—	2	1	no	—
157	C₂×C₃×C₄×C₅	—	2	1	no	—
158	C₂×C₃×C₄×C₅	—	2	1	no	—
159	C₂×C₃×C₄×C₅	—	4	2	no	—
160	C₂×C₃×C₄×C₅	—	4	2	no	—
161	C₂×C₃×C₄×C₅	—	4	2	no	—
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	C₂×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄×C₅	#41	1	4	yes	—
166	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
167	C₃×C₅×C₈	—	2	2	no	—
168	C₃×C₅×C₈	—	2	2	no	—
169	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
170	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
171	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
172	C₃×#123	—	2	1	no	—
173	C₃×C₂²⋊Q₈	C₂²×C₅	1	1	yes	characteristic
174	C₃×C₂²⋊Q₈	C₂²×C₅	1	1	yes	characteristic
175	C₃×#121	C₂²×C₅	1	1	yes	characteristic
176	C₃×#121	C₂²×C₅	1	1	yes	characteristic
177	C₃×#120	—	2	1	no	—
178	C₃×C₄×D₄	—	2	1	no	—
179	C₃×C₄×D₄	—	2	1	no	—
180	C₃×#117	—	2	1	no	—
181	C₃×#117	—	2	1	no	—
182	C₃×#117	C₂²×C₅	1	2	yes	—
183	C₃×#117	C₂²×C₅	1	1	yes	characteristic
184	C₃×#117	C₂²×C₅	1	2	yes	—
185	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
186	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
187	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
188	C₂×C₃×C₄²	—	2	1	no	—
189	C₃×#107	C₂²×C₅	1	2	yes	complemented
190	C₃×#107	C₂²×C₅	1	2	yes	complemented
191	C₃×#106	C₂²×C₅	1	2	yes	complemented
192	C₃×#106	C₂²×C₅	1	2	yes	complemented
193	C₃×#103	C₂²×C₅	1	2	yes	complemented
194	C₃×#103	C₂²×C₅	1	2	yes	complemented
195	C₃×#102	C₂²×C₅	1	2	yes	complemented
196	C₃×#102	C₂²×C₅	1	2	yes	complemented
197	C₃×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
198	C₃×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
199	C₃×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
200	C₂⁴×C₅	C₃×D₄	1	1	yes	characteristic
201	C₅×#41	—	2	4	no	—
202	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
203	C₂×C₅×D₄	—	2	1	no	—
204	C₂×C₅×D₄	—	2	1	no	—
205	C₂×C₅×D₄	—	2	1	no	—
206	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
207	C₂²×C₄×C₅	—	2	4	no	—
208	C₂²×C₄×C₅	—	2	1	no	—
209	C₂²×C₄×C₅	—	2	1	no	—
210	C₂²×C₄×C₅	—	2	1	no	—
211	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
212	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
213	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
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	1	yes	characteristic
217	C₂×C₅×C₈	—	2	4	no	—
218	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
219	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
220	C₅×C₄⋊C₄	—	2	1	no	—
221	C₅×C₄⋊C₄	—	2	1	no	—
222	C₅×C₄⋊C₄	—	2	1	no	—
223	C₅×C₄⋊C₄	—	2	1	no	—
224	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
225	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
226	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
227	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
228	C₅×C₂²⋊C₄	—	2	2	no	—
229	C₅×C₂²⋊C₄	—	2	2	no	—
230	C₅×C₂²⋊C₄	—	2	1	no	—
231	C₅×C₂²⋊C₄	—	2	1	no	—
232	C₅×C₂²⋊C₄	—	2	1	no	—
233	C₅×C₂²⋊C₄	—	2	1	no	—
234	C₅×C₂²⋊C₄	—	2	1	no	—
235	C₄²×C₅	—	2	2	no	—
236	C₄²×C₅	—	2	2	no	—
237	C₄²×C₅	—	2	1	no	—
238	#525	C₂×C₃×C₅	1	1	yes	characteristic
239	C₂×#117	C₂×C₃×C₅	1	1	yes	characteristic, complemented
240	#485	C₂×C₃×C₅	1	2	yes	complemented
241	#484	C₂×C₃×C₅	1	1	yes	characteristic, complemented
242	#483	C₂×C₃×C₅	1	1	yes	characteristic, complemented
243	#482	C₂×C₃×C₅	1	2	yes	complemented
244	#481	C₂×C₃×C₅	1	1	yes	characteristic, complemented
245	#480	C₂×C₃×C₅	1	1	yes	characteristic, complemented
246	#427	C₂×C₃×C₅	1	2	yes	complemented
247	#416	C₂×C₃×C₅	1	2	yes	complemented
248	C₂×C₂²⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
249	C₂²×C₃×C₅	—	2	1	no	—
250	C₂²×C₃×C₅	—	2	1	no	—
251	C₂²×C₃×C₅	—	2	1	no	—
252	C₂²×C₃×C₅	—	2	1	no	—
253	C₂²×C₃×C₅	—	2	1	no	—
254	C₂²×C₃×C₅	—	2	1	no	—
255	C₂²×C₃×C₅	—	2	1	no	—
256	C₂²×C₃×C₅	—	4	4	no	—
257	C₂²×C₃×C₅	—	4	1	no	—
258	C₂²×C₃×C₅	—	4	1	no	—
259	C₂²×C₃×C₅	—	4	1	no	—
260	C₂²×C₃×C₅	#123	1	2	yes	—
261	C₂²×C₃×C₅	#135	1	2	yes	—
262	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
263	C₃×C₄×C₅	—	2	1	no	—
264	C₃×C₄×C₅	—	2	1	no	—
265	C₃×C₄×C₅	—	4	1	no	—
266	C₃×C₄×C₅	—	4	2	no	—
267	C₃×C₄×C₅	—	4	1	no	—
268	C₃×C₄×C₅	—	4	2	no	—
269	C₃×C₄×C₅	—	4	1	no	—
270	C₃×C₄×C₅	#123	1	2	yes	—
271	C₃×C₄×C₅	C₂×#41	1	2	yes	—
272	C₂⁴×C₃	C₅×D₄	1	1	yes	characteristic
273	C₃×#41	—	2	4	no	—
274	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
275	C₂×C₃×D₄	—	2	1	no	—
276	C₂×C₃×D₄	—	2	1	no	—
277	C₂×C₃×D₄	—	2	1	no	—
278	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
279	C₂²×C₃×C₄	—	2	4	no	—
280	C₂²×C₃×C₄	—	2	1	no	—
281	C₂²×C₃×C₄	—	2	1	no	—
282	C₂²×C₃×C₄	—	2	1	no	—
283	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
284	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
285	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
286	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
287	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
288	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
289	C₂×C₃×C₈	—	2	4	no	—
290	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
291	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
292	C₃×C₄⋊C₄	—	2	1	no	—
293	C₃×C₄⋊C₄	—	2	1	no	—
294	C₃×C₄⋊C₄	—	2	1	no	—
295	C₃×C₄⋊C₄	—	2	1	no	—
296	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
297	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
298	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
299	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
300	C₃×C₂²⋊C₄	—	2	2	no	—
301	C₃×C₂²⋊C₄	—	2	2	no	—
302	C₃×C₂²⋊C₄	—	2	1	no	—
303	C₃×C₂²⋊C₄	—	2	1	no	—
304	C₃×C₂²⋊C₄	—	2	1	no	—
305	C₃×C₂²⋊C₄	—	2	1	no	—
306	C₃×C₂²⋊C₄	—	2	1	no	—
307	C₃×C₄²	—	2	2	no	—
308	C₃×C₄²	—	2	2	no	—
309	C₃×C₄²	—	2	1	no	—
310	C₂³×C₅	—	2	2	no	—
311	C₂³×C₅	—	2	2	no	—
312	C₂³×C₅	—	2	1	no	—
313	C₂³×C₅	—	4	1	no	—
314	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
315	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
316	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
317	C₅×Q₈	—	2	2	no	—
318	C₅×D₄	—	2	2	no	—
319	C₅×D₄	—	4	2	no	—
320	C₅×D₄	—	4	2	no	—
321	C₅×D₄	—	4	2	no	—
322	C₂×C₄×C₅	—	2	2	no	—
323	C₂×C₄×C₅	—	2	2	no	—
324	C₂×C₄×C₅	—	2	2	no	—
325	C₂×C₄×C₅	—	2	2	no	—
326	C₂×C₄×C₅	—	2	2	no	—
327	C₂×C₄×C₅	—	2	2	no	—
328	C₂×C₄×C₅	—	2	2	no	—
329	C₂×C₄×C₅	—	2	1	no	—
330	C₂×C₄×C₅	—	2	1	no	—
331	C₂×C₄×C₅	—	2	1	no	—
332	C₂×C₄×C₅	—	4	2	no	—
333	C₂×C₄×C₅	—	4	2	no	—
334	C₂×C₄×C₅	—	4	2	no	—
335	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
336	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
337	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
338	C₂×C₄×C₅	C₃×#41	1	4	yes	—
339	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
340	C₅×C₈	—	2	2	no	—
341	C₅×C₈	—	2	2	no	—
342	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
343	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
344	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
345	#123	—	2	1	no	—
346	C₂²⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
347	C₂²⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
348	#121	C₂²×C₃×C₅	1	1	yes	characteristic
349	#121	C₂²×C₃×C₅	1	1	yes	characteristic
350	#120	—	2	1	no	—
351	C₄×D₄	—	2	1	no	—
352	C₄×D₄	—	2	1	no	—
353	#117	—	2	1	no	—
354	#117	—	2	1	no	—
355	#117	C₂²×C₃×C₅	1	2	yes	—
356	#117	C₂²×C₃×C₅	1	2	yes	—
357	#117	C₂²×C₃×C₅	1	1	yes	characteristic
358	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
359	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
360	C₂×C₂²⋊C₄	—	2	1	no	—
361	C₂×C₄²	—	2	1	no	—
362	#107	C₂²×C₃×C₅	1	2	yes	complemented
363	#107	C₂²×C₃×C₅	1	2	yes	complemented
364	#106	C₂²×C₃×C₅	1	2	yes	complemented
365	#106	C₂²×C₃×C₅	1	2	yes	complemented
366	#103	C₂²×C₃×C₅	1	2	yes	complemented
367	#103	C₂²×C₃×C₅	1	2	yes	complemented
368	#102	C₂²×C₃×C₅	1	2	yes	complemented
369	#102	C₂²×C₃×C₅	1	2	yes	complemented
370	C₂²⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
371	C₂²⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
372	C₂²⋊C₈	C₂²×C₃×C₅	1	2	yes	—
373	C₂×C₃×C₅	—	2	2	no	—
374	C₂×C₃×C₅	—	2	2	no	—
375	C₂×C₃×C₅	—	4	1	no	—
376	C₂×C₃×C₅	—	8	1	no	—
377	C₂×C₃×C₅	C₂×#123	1	1	yes	characteristic
378	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
379	C₂×C₃×C₅	#575	1	1	yes	characteristic
380	C₂³×C₃	—	2	2	no	—
381	C₂³×C₃	—	2	2	no	—
382	C₂³×C₃	—	2	1	no	—
383	C₂³×C₃	—	4	1	no	—
384	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
385	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
386	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
387	C₃×Q₈	—	2	2	no	—
388	C₃×D₄	—	2	2	no	—
389	C₃×D₄	—	4	2	no	—
390	C₃×D₄	—	4	2	no	—
391	C₃×D₄	—	4	2	no	—
392	C₂×C₃×C₄	—	2	2	no	—
393	C₂×C₃×C₄	—	2	2	no	—
394	C₂×C₃×C₄	—	2	2	no	—
395	C₂×C₃×C₄	—	2	2	no	—
396	C₂×C₃×C₄	—	2	2	no	—
397	C₂×C₃×C₄	—	2	2	no	—
398	C₂×C₃×C₄	—	2	2	no	—
399	C₂×C₃×C₄	—	2	1	no	—
400	C₂×C₃×C₄	—	2	1	no	—
401	C₂×C₃×C₄	—	2	1	no	—
402	C₂×C₃×C₄	—	4	2	no	—
403	C₂×C₃×C₄	—	4	2	no	—
404	C₂×C₃×C₄	—	4	2	no	—
405	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
406	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
407	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
408	C₂×C₃×C₄	C₅×#41	1	4	yes	—
409	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
410	C₃×C₈	—	2	2	no	—
411	C₃×C₈	—	2	2	no	—
412	C₂²×C₅	—	2	1	no	—
413	C₂²×C₅	—	2	1	no	—
414	C₂²×C₅	—	2	1	no	—
415	C₂²×C₅	—	2	1	no	—
416	C₂²×C₅	—	2	1	no	—
417	C₂²×C₅	—	2	1	no	—
418	C₂²×C₅	—	2	1	no	—
419	C₂²×C₅	—	4	4	no	—
420	C₂²×C₅	—	4	1	no	—
421	C₂²×C₅	—	4	1	no	—
422	C₂²×C₅	—	4	1	no	—
423	C₂²×C₅	C₃×#123	1	2	yes	—
424	C₂²×C₅	C₃×#135	1	2	yes	—
425	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
426	C₄×C₅	—	2	1	no	—
427	C₄×C₅	—	2	1	no	—
428	C₄×C₅	—	4	2	no	—
429	C₄×C₅	—	4	1	no	—
430	C₄×C₅	—	4	2	no	—
431	C₄×C₅	—	4	1	no	—
432	C₄×C₅	—	4	1	no	—
433	C₄×C₅	C₃×#123	1	2	yes	—
434	C₄×C₅	C₂×C₃×#41	1	2	yes	—
435	C₂⁴	C₃×C₅×D₄	1	1	yes	characteristic
436	#41	—	2	4	no	—
437	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	2	1	no	—
441	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
447	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
448	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
449	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
450	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
451	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
452	C₂×C₈	—	2	4	no	—
453	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
454	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	C₂³×C₃×C₅	1	2	yes	—
460	C₄⋊C₄	C₂³×C₃×C₅	1	2	yes	—
461	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
462	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
463	C₂²⋊C₄	—	2	2	no	—
464	C₂²⋊C₄	—	2	2	no	—
465	C₂²⋊C₄	—	2	1	no	—
466	C₂²⋊C₄	—	2	1	no	—
467	C₂²⋊C₄	—	2	1	no	—
468	C₂²⋊C₄	—	2	1	no	—
469	C₂²⋊C₄	—	2	1	no	—
470	C₄²	—	2	2	no	—
471	C₄²	—	2	2	no	—
472	C₄²	—	2	1	no	—
473	C₃×C₅	#3091	1	1	yes	characteristic, direct product factor, Hall
474	C₂²×C₃	—	2	1	no	—
475	C₂²×C₃	—	2	1	no	—
476	C₂²×C₃	—	2	1	no	—
477	C₂²×C₃	—	2	1	no	—
478	C₂²×C₃	—	2	1	no	—
479	C₂²×C₃	—	2	1	no	—
480	C₂²×C₃	—	2	1	no	—
481	C₂²×C₃	—	4	4	no	—
482	C₂²×C₃	—	4	1	no	—
483	C₂²×C₃	—	4	1	no	—
484	C₂²×C₃	—	4	1	no	—
485	C₂²×C₃	C₅×#123	1	2	yes	—
486	C₂²×C₃	C₅×#135	1	2	yes	—
487	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
488	C₃×C₄	—	2	1	no	—
489	C₃×C₄	—	2	1	no	—
490	C₃×C₄	—	4	1	no	—
491	C₃×C₄	—	4	2	no	—
492	C₃×C₄	—	4	1	no	—
493	C₃×C₄	—	4	2	no	—
494	C₃×C₄	—	4	1	no	—
495	C₃×C₄	C₅×#123	1	2	yes	—
496	C₃×C₄	C₂×C₅×#41	1	2	yes	—
497	C₂×C₅	—	2	2	no	—
498	C₂×C₅	—	2	2	no	—
499	C₂×C₅	—	4	1	no	—
500	C₂×C₅	—	8	1	no	—
501	C₂×C₅	C₂×C₃×#123	1	1	yes	characteristic
502	C₂×C₅	C₂×C₃×#135	1	1	yes	characteristic
503	C₂×C₅	C₃×#575	1	1	yes	characteristic
504	C₂³	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	1	no	—
507	C₂³	—	4	1	no	—
508	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
509	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
510	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
511	Q₈	—	2	2	no	—
512	D₄	—	2	2	no	—
513	D₄	—	4	2	no	—
514	D₄	—	4	2	no	—
515	D₄	—	4	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
530	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
531	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
532	C₂×C₄	C₃×C₅×#41	1	4	yes	—
533	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
534	C₈	—	2	2	no	—
535	C₈	—	2	2	no	—
536	C₂×C₃	—	2	2	no	—
537	C₂×C₃	—	2	2	no	—
538	C₂×C₃	—	4	1	no	—
539	C₂×C₃	—	8	1	no	—
540	C₂×C₃	C₂×C₅×#123	1	1	yes	characteristic
541	C₂×C₃	C₂×C₅×#135	1	1	yes	characteristic
542	C₂×C₃	C₅×#575	1	1	yes	characteristic
543	C₅	C₃×#3091	1	1	yes	characteristic, direct product factor, Sylow
544	C₂²	—	2	1	no	—
545	C₂²	—	2	1	no	—
546	C₂²	—	2	1	no	—
547	C₂²	—	2	1	no	—
548	C₂²	—	2	1	no	—
549	C₂²	—	2	1	no	—
550	C₂²	—	2	1	no	—
551	C₂²	—	4	4	no	—
552	C₂²	—	4	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	1	no	—
555	C₂²	C₃×C₅×#123	1	2	yes	—
556	C₂²	C₃×C₅×#135	1	2	yes	—
557	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
558	C₄	—	2	1	no	—
559	C₄	—	2	1	no	—
560	C₄	—	4	1	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	1	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	1	no	—
565	C₄	C₃×C₅×#123	1	2	yes	—
566	C₄	C₂×C₃×C₅×#41	1	2	yes	—
567	C₃	C₅×#3091	1	1	yes	characteristic, direct product factor, Sylow
568	C₂	—	2	2	no	—
569	C₂	—	2	2	no	—
570	C₂	—	4	1	no	—
571	C₂	—	8	1	no	—
572	C₂	C₂×C₃×C₅×#123	1	1	yes	characteristic
573	C₂	C₂×C₃×C₅×#135	1	1	yes	characteristic
574	C₂	C₃×C₅×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₃×C₅×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂³×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#117	C₂	semidirect
3. C₃×C₅×#485	C₂	semidirect
4. C₃×C₅×#484	C₂	semidirect
5. C₃×C₅×#483	C₂	semidirect
6. C₃×C₅×#482	C₂	semidirect
7. C₃×C₅×#481	C₂	semidirect
8. C₃×C₅×#480	C₂	semidirect
9. C₃×C₅×#427	C₂	semidirect
10. C₃×C₅×#416	C₂	semidirect
11. C₂×C₃×C₅×C₂²⋊C₈	C₂	semidirect
12. C₅×#3091	C₃	direct (N × H)
33. C₃×C₅×#107	C₂²	semidirect
34. C₃×C₅×#107	C₂²	semidirect
35. C₃×C₅×#106	C₂²	semidirect
36. C₃×C₅×#106	C₂²	semidirect
37. C₃×C₅×#103	C₂²	semidirect
38. C₃×C₅×#103	C₂²	semidirect
39. C₃×C₅×#102	C₂²	semidirect
40. C₃×C₅×#102	C₂²	semidirect
44. C₃×#3091	C₅	direct (N × H)
46. C₂×C₅×#117	C₂×C₃	semidirect
47. C₅×#485	C₂×C₃	semidirect
48. C₅×#484	C₂×C₃	semidirect
49. C₅×#483	C₂×C₃	semidirect
50. C₅×#482	C₂×C₃	semidirect
51. C₅×#481	C₂×C₃	semidirect
52. C₅×#480	C₂×C₃	semidirect
53. C₅×#427	C₂×C₃	semidirect
54. C₅×#416	C₂×C₃	semidirect
55. C₂×C₅×C₂²⋊C₈	C₂×C₃	semidirect
95. C₂×C₃×#117	C₂×C₅	semidirect
96. C₃×#485	C₂×C₅	semidirect
97. C₃×#484	C₂×C₅	semidirect
98. C₃×#483	C₂×C₅	semidirect
99. C₃×#482	C₂×C₅	semidirect
100. C₃×#481	C₂×C₅	semidirect
101. C₃×#480	C₂×C₅	semidirect
102. C₃×#427	C₂×C₅	semidirect
103. C₃×#416	C₂×C₅	semidirect
104. C₂×C₃×C₂²⋊C₈	C₂×C₅	semidirect
125. C₅×#107	C₂²×C₃	semidirect
126. C₅×#107	C₂²×C₃	semidirect
127. C₅×#106	C₂²×C₃	semidirect
128. C₅×#106	C₂²×C₃	semidirect
129. C₅×#103	C₂²×C₃	semidirect
130. C₅×#103	C₂²×C₃	semidirect
131. C₅×#102	C₂²×C₃	semidirect
132. C₅×#102	C₂²×C₃	semidirect
136. #3091	C₃×C₅	direct (N × H)
189. C₃×#107	C₂²×C₅	semidirect
190. C₃×#107	C₂²×C₅	semidirect
191. C₃×#106	C₂²×C₅	semidirect
192. C₃×#106	C₂²×C₅	semidirect
193. C₃×#103	C₂²×C₅	semidirect
194. C₃×#103	C₂²×C₅	semidirect
195. C₃×#102	C₂²×C₅	semidirect
196. C₃×#102	C₂²×C₅	semidirect
239. C₂×#117	C₂×C₃×C₅	semidirect
240. #485	C₂×C₃×C₅	semidirect
241. #484	C₂×C₃×C₅	semidirect
242. #483	C₂×C₃×C₅	semidirect
243. #482	C₂×C₃×C₅	semidirect
244. #481	C₂×C₃×C₅	semidirect
245. #480	C₂×C₃×C₅	semidirect
246. #427	C₂×C₃×C₅	semidirect
247. #416	C₂×C₃×C₅	semidirect
248. C₂×C₂²⋊C₈	C₂×C₃×C₅	semidirect
362. #107	C₂²×C₃×C₅	semidirect
363. #107	C₂²×C₃×C₅	semidirect
364. #106	C₂²×C₃×C₅	semidirect
365. #106	C₂²×C₃×C₅	semidirect
366. #103	C₂²×C₃×C₅	semidirect
367. #103	C₂²×C₃×C₅	semidirect
368. #102	C₂²×C₃×C₅	semidirect
369. #102	C₂²×C₃×C₅	semidirect
473. C₃×C₅	#3091	direct (N × H)
543. C₅	C₃×#3091	direct (N × H)
567. C₃	C₅×#3091	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×#525	2
C₂×C₃×C₅×#117	2
C₃×C₅×#485	2
C₃×C₅×#484	2
C₃×C₅×#483	2
C₃×C₅×#482	2
C₃×C₅×#481	2
C₃×C₅×#480	2
C₃×C₅×#427	2
C₃×C₅×#416	2
C₂×C₃×C₅×C₂²⋊C₈	2
C₅×#3091	3
C₃×#3091	5

Subgroups — C3×C5×#3091 #14276125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3091 #14276125