Subgroups — C₃×C₅×#3081 #14276115

Order 1920.

796 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#518	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#478	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#478	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#476	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#474	C₂	1	2	yes	maximal, complemented
7	C₃×C₅×#428	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₅×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₅×#418	C₂	1	1	yes	maximal, characteristic
11	C₂×C₃×C₅×#102	C₂	1	1	yes	maximal, characteristic
12	C₅×#3081	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
17	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
18	C₃×C₅×#125	—	2	1	no	—
19	C₃×C₅×#125	C₂²	1	2	yes	complemented
20	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
21	C₃×C₄×C₅×D₄	—	2	1	no	—
22	C₃×C₄×C₅×D₄	—	2	1	no	—
23	C₃×C₄×C₅×D₄	C₂²	1	2	yes	—
24	C₃×C₄×C₅×D₄	C₂²	1	2	yes	—
25	C₃×C₅×#117	C₂²	1	1	yes	characteristic
26	C₃×C₅×#117	C₂²	1	1	yes	characteristic
27	C₃×C₅×#117	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
30	C₃×C₅×#107	C₂²	1	1	yes	characteristic, complemented
31	C₃×C₅×#107	C₂²	1	2	yes	complemented
32	C₃×C₅×#107	C₂²	1	1	yes	characteristic, complemented
33	C₃×C₅×#106	C₂²	1	1	yes	characteristic, complemented
34	C₃×C₅×#106	C₂²	1	2	yes	complemented
35	C₃×C₅×#106	C₂²	1	1	yes	characteristic, complemented
36	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
37	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
38	C₃×C₅×#102	C₂²	1	2	yes	—
39	C₃×C₅×#102	C₂²	1	2	yes	—
40	C₃×C₅×#102	C₂²	1	2	yes	—
41	C₃×C₅×#102	C₂²	1	2	yes	—
42	C₃×#3081	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×#544	C₂×C₃	1	1	yes	characteristic, complemented
44	C₅×#518	C₂×C₃	1	1	yes	characteristic
45	C₅×#478	C₂×C₃	1	2	yes	complemented
46	C₅×#478	C₂×C₃	1	2	yes	complemented
47	C₅×#476	C₂×C₃	1	2	yes	complemented
48	C₅×#474	C₂×C₃	1	2	yes	complemented
49	C₅×#428	C₂×C₃	1	1	yes	characteristic, complemented
50	C₅×#427	C₂×C₃	1	1	yes	characteristic, complemented
51	C₅×#424	C₂×C₃	1	1	yes	characteristic, complemented
52	C₅×#418	C₂×C₃	1	1	yes	characteristic
53	C₂×C₅×#102	C₂×C₃	1	1	yes	characteristic
54	C₂⁴×C₃×C₅	—	2	1	no	—
55	C₂×C₃×C₅×Q₈	—	2	2	no	—
56	C₂×C₃×C₅×D₄	—	2	1	no	—
57	C₂×C₃×C₅×D₄	—	2	2	no	—
58	C₂×C₃×C₅×D₄	Q₈	1	4	yes	complemented
59	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
60	C₂²×C₃×C₄×C₅	—	2	1	no	—
61	C₂²×C₃×C₄×C₅	—	2	2	no	—
62	C₂²×C₃×C₄×C₅	—	2	2	no	—
63	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
64	C₃×C₅×M₁₆	—	2	2	no	—
65	C₂×C₃×C₅×C₈	—	2	2	no	—
66	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
67	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
69	C₃×C₅×C₄⋊C₄	—	2	1	no	—
70	C₃×C₅×C₄⋊C₄	—	2	2	no	—
71	C₃×C₅×C₄⋊C₄	—	2	2	no	—
72	C₃×C₅×C₄⋊C₄	—	2	2	no	—
73	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
74	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
75	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
76	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
81	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
82	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
83	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
84	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
85	C₃×C₄²×C₅	—	2	1	no	—
86	C₃×C₄²×C₅	—	2	1	no	—
87	C₃×C₄²×C₅	C₂³	1	2	yes	—
88	C₃×#544	C₂×C₅	1	1	yes	characteristic, complemented
89	C₃×#518	C₂×C₅	1	1	yes	characteristic
90	C₃×#478	C₂×C₅	1	2	yes	complemented
91	C₃×#478	C₂×C₅	1	2	yes	complemented
92	C₃×#476	C₂×C₅	1	2	yes	complemented
93	C₃×#474	C₂×C₅	1	2	yes	complemented
94	C₃×#428	C₂×C₅	1	1	yes	characteristic, complemented
95	C₃×#427	C₂×C₅	1	1	yes	characteristic, complemented
96	C₃×#424	C₂×C₅	1	1	yes	characteristic, complemented
97	C₃×#418	C₂×C₅	1	1	yes	characteristic
98	C₂×C₃×#102	C₂×C₅	1	1	yes	characteristic
99	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
100	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
101	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
102	C₅×C₄⋊Q₈	—	2	1	no	—
103	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
104	C₅×#125	—	2	1	no	—
105	C₅×#125	C₂²×C₃	1	2	yes	complemented
106	C₅×C₂²⋊Q₈	—	2	2	no	—
107	C₄×C₅×D₄	—	2	1	no	—
108	C₄×C₅×D₄	—	2	1	no	—
109	C₄×C₅×D₄	C₂²×C₃	1	2	yes	—
110	C₄×C₅×D₄	C₂²×C₃	1	2	yes	—
111	C₅×#117	C₂²×C₃	1	1	yes	characteristic
112	C₅×#117	C₂²×C₃	1	1	yes	characteristic
113	C₅×#117	C₂²×C₃	1	1	yes	characteristic
114	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
115	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
116	C₅×#107	C₂²×C₃	1	1	yes	characteristic, complemented
117	C₅×#107	C₂²×C₃	1	1	yes	characteristic, complemented
118	C₅×#107	C₂²×C₃	1	2	yes	complemented
119	C₅×#106	C₂²×C₃	1	1	yes	characteristic, complemented
120	C₅×#106	C₂²×C₃	1	1	yes	characteristic, complemented
121	C₅×#106	C₂²×C₃	1	2	yes	complemented
122	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
123	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
124	C₅×#102	C₂²×C₃	1	2	yes	—
125	C₅×#102	C₂²×C₃	1	2	yes	—
126	C₅×#102	C₂²×C₃	1	2	yes	—
127	C₅×#102	C₂²×C₃	1	2	yes	—
128	#3081	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
129	C₂³×C₃×C₅	—	2	2	no	—
130	C₂³×C₃×C₅	—	2	4	no	—
131	C₂³×C₃×C₅	—	8	1	no	—
132	C₂³×C₃×C₅	—	8	1	no	—
133	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
134	C₃×C₅×Q₈	—	8	2	no	—
135	C₃×C₅×D₄	—	2	4	no	—
136	C₃×C₅×D₄	—	2	2	no	—
137	C₃×C₅×D₄	C₂×Q₈	1	4	yes	—
138	C₂×C₃×C₄×C₅	—	2	1	no	—
139	C₂×C₃×C₄×C₅	—	2	1	no	—
140	C₂×C₃×C₄×C₅	—	2	1	no	—
141	C₂×C₃×C₄×C₅	—	2	1	no	—
142	C₂×C₃×C₄×C₅	—	2	2	no	—
143	C₂×C₃×C₄×C₅	—	8	2	no	—
144	C₂×C₃×C₄×C₅	—	8	2	no	—
145	C₂×C₃×C₄×C₅	—	8	1	no	—
146	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
147	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
148	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
150	C₂×C₃×C₄×C₅	#41	1	2	yes	—
151	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
152	C₃×C₅×C₈	—	2	1	no	—
153	C₃×C₅×C₈	—	2	1	no	—
154	C₃×C₅×C₈	—	2	2	no	—
155	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
156	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
157	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
158	C₃×C₄⋊Q₈	—	2	1	no	—
159	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
160	C₃×#125	—	2	1	no	—
161	C₃×#125	C₂²×C₅	1	2	yes	complemented
162	C₃×C₂²⋊Q₈	—	2	2	no	—
163	C₃×C₄×D₄	—	2	1	no	—
164	C₃×C₄×D₄	—	2	1	no	—
165	C₃×C₄×D₄	C₂²×C₅	1	2	yes	—
166	C₃×C₄×D₄	C₂²×C₅	1	2	yes	—
167	C₃×#117	C₂²×C₅	1	1	yes	characteristic
168	C₃×#117	C₂²×C₅	1	1	yes	characteristic
169	C₃×#117	C₂²×C₅	1	1	yes	characteristic
170	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
171	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
172	C₃×#107	C₂²×C₅	1	1	yes	characteristic, complemented
173	C₃×#107	C₂²×C₅	1	1	yes	characteristic, complemented
174	C₃×#107	C₂²×C₅	1	2	yes	complemented
175	C₃×#106	C₂²×C₅	1	1	yes	characteristic, complemented
176	C₃×#106	C₂²×C₅	1	1	yes	characteristic, complemented
177	C₃×#106	C₂²×C₅	1	2	yes	complemented
178	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
179	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
180	C₃×#102	C₂²×C₅	1	2	yes	—
181	C₃×#102	C₂²×C₅	1	2	yes	—
182	C₃×#102	C₂²×C₅	1	2	yes	—
183	C₃×#102	C₂²×C₅	1	2	yes	—
184	C₂⁴×C₅	—	2	1	no	—
185	C₂×C₅×Q₈	—	2	2	no	—
186	C₂×C₅×D₄	—	2	1	no	—
187	C₂×C₅×D₄	—	2	2	no	—
188	C₂×C₅×D₄	C₃×Q₈	1	4	yes	complemented
189	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
190	C₂²×C₄×C₅	—	2	1	no	—
191	C₂²×C₄×C₅	—	2	2	no	—
192	C₂²×C₄×C₅	—	2	2	no	—
193	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
194	C₅×M₁₆	—	2	2	no	—
195	C₂×C₅×C₈	—	2	2	no	—
196	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
197	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
198	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
199	C₅×C₄⋊C₄	—	2	1	no	—
200	C₅×C₄⋊C₄	—	2	2	no	—
201	C₅×C₄⋊C₄	—	2	2	no	—
202	C₅×C₄⋊C₄	—	2	2	no	—
203	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
204	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
205	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
206	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
207	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
208	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
209	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
210	C₅×C₂²⋊C₄	—	2	1	no	—
211	C₅×C₂²⋊C₄	—	2	1	no	—
212	C₅×C₂²⋊C₄	—	2	2	no	—
213	C₅×C₂²⋊C₄	—	2	2	no	—
214	C₅×C₂²⋊C₄	C₃×D₄	1	2	yes	—
215	C₄²×C₅	—	2	1	no	—
216	C₄²×C₅	—	2	1	no	—
217	C₄²×C₅	C₂³×C₃	1	2	yes	—
218	#544	C₂×C₃×C₅	1	1	yes	characteristic, complemented
219	#518	C₂×C₃×C₅	1	1	yes	characteristic
220	#478	C₂×C₃×C₅	1	2	yes	complemented
221	#478	C₂×C₃×C₅	1	2	yes	complemented
222	#476	C₂×C₃×C₅	1	2	yes	complemented
223	#474	C₂×C₃×C₅	1	2	yes	complemented
224	#428	C₂×C₃×C₅	1	1	yes	characteristic, complemented
225	#427	C₂×C₃×C₅	1	1	yes	characteristic, complemented
226	#424	C₂×C₃×C₅	1	1	yes	characteristic, complemented
227	#418	C₂×C₃×C₅	1	1	yes	characteristic
228	C₂×#102	C₂×C₃×C₅	1	1	yes	characteristic
229	C₂²×C₃×C₅	—	2	1	no	—
230	C₂²×C₃×C₅	—	2	1	no	—
231	C₂²×C₃×C₅	—	2	4	no	—
232	C₂²×C₃×C₅	—	4	2	no	—
233	C₂²×C₃×C₅	—	4	2	no	—
234	C₂²×C₃×C₅	—	8	4	no	—
235	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
236	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
237	C₃×C₄×C₅	—	2	2	no	—
238	C₃×C₄×C₅	—	2	2	no	—
239	C₃×C₄×C₅	—	4	1	no	—
240	C₃×C₄×C₅	—	4	2	no	—
241	C₃×C₄×C₅	—	4	1	no	—
242	C₃×C₄×C₅	—	4	1	no	—
243	C₃×C₄×C₅	—	4	1	no	—
244	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
245	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
246	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
247	C₂⁴×C₃	—	2	1	no	—
248	C₂×C₃×Q₈	—	2	2	no	—
249	C₂×C₃×D₄	—	2	1	no	—
250	C₂×C₃×D₄	—	2	2	no	—
251	C₂×C₃×D₄	C₅×Q₈	1	4	yes	complemented
252	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
253	C₂²×C₃×C₄	—	2	1	no	—
254	C₂²×C₃×C₄	—	2	2	no	—
255	C₂²×C₃×C₄	—	2	2	no	—
256	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
257	C₃×M₁₆	—	2	2	no	—
258	C₂×C₃×C₈	—	2	2	no	—
259	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
260	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
261	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
262	C₃×C₄⋊C₄	—	2	1	no	—
263	C₃×C₄⋊C₄	—	2	2	no	—
264	C₃×C₄⋊C₄	—	2	2	no	—
265	C₃×C₄⋊C₄	—	2	2	no	—
266	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
267	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
268	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
269	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
270	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
271	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
272	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
273	C₃×C₂²⋊C₄	—	2	1	no	—
274	C₃×C₂²⋊C₄	—	2	1	no	—
275	C₃×C₂²⋊C₄	—	2	2	no	—
276	C₃×C₂²⋊C₄	—	2	2	no	—
277	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
278	C₃×C₄²	—	2	1	no	—
279	C₃×C₄²	—	2	1	no	—
280	C₃×C₄²	C₂³×C₅	1	2	yes	—
281	C₂³×C₅	—	2	2	no	—
282	C₂³×C₅	—	2	4	no	—
283	C₂³×C₅	—	8	1	no	—
284	C₂³×C₅	—	8	1	no	—
285	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
286	C₅×Q₈	—	8	2	no	—
287	C₅×D₄	—	2	4	no	—
288	C₅×D₄	—	2	2	no	—
289	C₅×D₄	C₂×C₃×Q₈	1	4	yes	—
290	C₂×C₄×C₅	—	2	1	no	—
291	C₂×C₄×C₅	—	2	1	no	—
292	C₂×C₄×C₅	—	2	1	no	—
293	C₂×C₄×C₅	—	2	1	no	—
294	C₂×C₄×C₅	—	2	2	no	—
295	C₂×C₄×C₅	—	8	2	no	—
296	C₂×C₄×C₅	—	8	2	no	—
297	C₂×C₄×C₅	—	8	1	no	—
298	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
299	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
300	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
301	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
302	C₂×C₄×C₅	C₃×#41	1	2	yes	—
303	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
304	C₅×C₈	—	2	1	no	—
305	C₅×C₈	—	2	1	no	—
306	C₅×C₈	—	2	2	no	—
307	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
308	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
309	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
310	C₄⋊Q₈	—	2	1	no	—
311	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	complemented
312	#125	—	2	1	no	—
313	#125	C₂²×C₃×C₅	1	2	yes	complemented
314	C₂²⋊Q₈	—	2	2	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	C₂²×C₃×C₅	1	2	yes	—
318	C₄×D₄	C₂²×C₃×C₅	1	2	yes	—
319	#117	C₂²×C₃×C₅	1	1	yes	characteristic
320	#117	C₂²×C₃×C₅	1	1	yes	characteristic
321	#117	C₂²×C₃×C₅	1	1	yes	characteristic
322	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
323	C₂×C₂²⋊C₄	—	2	2	no	—
324	#107	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
325	#107	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
326	#107	C₂²×C₃×C₅	1	2	yes	complemented
327	#106	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
328	#106	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
329	#106	C₂²×C₃×C₅	1	2	yes	complemented
330	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	complemented
331	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	complemented
332	#102	C₂²×C₃×C₅	1	2	yes	—
333	#102	C₂²×C₃×C₅	1	2	yes	—
334	#102	C₂²×C₃×C₅	1	2	yes	—
335	#102	C₂²×C₃×C₅	1	2	yes	—
336	C₂×C₃×C₅	—	2	2	no	—
337	C₂×C₃×C₅	—	4	4	no	—
338	C₂×C₃×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
339	C₂×C₃×C₅	#576	1	1	yes	characteristic
340	C₂×C₃×C₅	#577	1	1	yes	characteristic
341	C₂³×C₃	—	2	2	no	—
342	C₂³×C₃	—	2	4	no	—
343	C₂³×C₃	—	8	1	no	—
344	C₂³×C₃	—	8	1	no	—
345	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
346	C₃×Q₈	—	8	2	no	—
347	C₃×D₄	—	2	4	no	—
348	C₃×D₄	—	2	2	no	—
349	C₃×D₄	C₂×C₅×Q₈	1	4	yes	—
350	C₂×C₃×C₄	—	2	1	no	—
351	C₂×C₃×C₄	—	2	1	no	—
352	C₂×C₃×C₄	—	2	1	no	—
353	C₂×C₃×C₄	—	2	1	no	—
354	C₂×C₃×C₄	—	2	2	no	—
355	C₂×C₃×C₄	—	8	2	no	—
356	C₂×C₃×C₄	—	8	2	no	—
357	C₂×C₃×C₄	—	8	1	no	—
358	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
359	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
360	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
361	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
362	C₂×C₃×C₄	C₅×#41	1	2	yes	—
363	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
364	C₃×C₈	—	2	1	no	—
365	C₃×C₈	—	2	1	no	—
366	C₃×C₈	—	2	2	no	—
367	C₂²×C₅	—	2	1	no	—
368	C₂²×C₅	—	2	1	no	—
369	C₂²×C₅	—	2	4	no	—
370	C₂²×C₅	—	4	2	no	—
371	C₂²×C₅	—	4	2	no	—
372	C₂²×C₅	—	8	4	no	—
373	C₂²×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
374	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
375	C₄×C₅	—	2	2	no	—
376	C₄×C₅	—	2	2	no	—
377	C₄×C₅	—	4	1	no	—
378	C₄×C₅	—	4	2	no	—
379	C₄×C₅	—	4	1	no	—
380	C₄×C₅	—	4	1	no	—
381	C₄×C₅	—	4	1	no	—
382	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
383	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
384	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
385	C₂⁴	—	2	1	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	C₃×C₅×Q₈	1	4	yes	complemented
390	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
395	M₁₆	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
398	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
399	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
405	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
406	C₄⋊C₄	C₂³×C₃×C₅	1	2	yes	—
407	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
408	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
409	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
410	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
411	C₂²⋊C₄	—	2	1	no	—
412	C₂²⋊C₄	—	2	1	no	—
413	C₂²⋊C₄	—	2	2	no	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	C₃×C₅×D₄	1	2	yes	—
416	C₄²	—	2	1	no	—
417	C₄²	—	2	1	no	—
418	C₄²	C₂³×C₃×C₅	1	2	yes	—
419	C₃×C₅	#3081	1	1	yes	characteristic, direct product factor, Hall
420	C₂²×C₃	—	2	1	no	—
421	C₂²×C₃	—	2	1	no	—
422	C₂²×C₃	—	2	4	no	—
423	C₂²×C₃	—	4	2	no	—
424	C₂²×C₃	—	4	2	no	—
425	C₂²×C₃	—	8	4	no	—
426	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
427	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
428	C₃×C₄	—	2	2	no	—
429	C₃×C₄	—	2	2	no	—
430	C₃×C₄	—	4	1	no	—
431	C₃×C₄	—	4	2	no	—
432	C₃×C₄	—	4	1	no	—
433	C₃×C₄	—	4	1	no	—
434	C₃×C₄	—	4	1	no	—
435	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
436	C₃×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
437	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
438	C₂×C₅	—	2	2	no	—
439	C₂×C₅	—	4	4	no	—
440	C₂×C₅	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
441	C₂×C₅	C₃×#576	1	1	yes	characteristic
442	C₂×C₅	C₃×#577	1	1	yes	characteristic
443	C₂³	—	2	2	no	—
444	C₂³	—	2	4	no	—
445	C₂³	—	8	1	no	—
446	C₂³	—	8	1	no	—
447	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
448	Q₈	—	8	2	no	—
449	D₄	—	2	4	no	—
450	D₄	—	2	2	no	—
451	D₄	C₂×C₃×C₅×Q₈	1	4	yes	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	8	2	no	—
458	C₂×C₄	—	8	2	no	—
459	C₂×C₄	—	8	1	no	—
460	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
461	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
462	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
463	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
464	C₂×C₄	C₃×C₅×#41	1	2	yes	—
465	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
466	C₈	—	2	1	no	—
467	C₈	—	2	1	no	—
468	C₈	—	2	2	no	—
469	C₂×C₃	—	2	2	no	—
470	C₂×C₃	—	4	4	no	—
471	C₂×C₃	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
472	C₂×C₃	C₅×#576	1	1	yes	characteristic
473	C₂×C₃	C₅×#577	1	1	yes	characteristic
474	C₅	C₃×#3081	1	1	yes	characteristic, direct product factor, Sylow
475	C₂²	—	2	1	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	8	4	no	—
481	C₂²	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
482	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
483	C₄	—	2	2	no	—
484	C₄	—	2	2	no	—
485	C₄	—	4	1	no	—
486	C₄	—	4	2	no	—
487	C₄	—	4	1	no	—
488	C₄	—	4	1	no	—
489	C₄	—	4	1	no	—
490	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
491	C₄	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
492	C₄	C₂×C₃×C₅×#41	1	1	yes	characteristic
493	C₃	C₅×#3081	1	1	yes	characteristic, direct product factor, Sylow
494	C₂	—	2	2	no	—
495	C₂	—	4	4	no	—
496	C₂	C₂×C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
497	C₂	C₃×C₅×#576	1	1	yes	characteristic
498	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
prime radical	C₂²×C₃×C₅×D₄	C₂²
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. C₃×C₅×#544	C₂	semidirect
3. C₃×C₅×#478	C₂	semidirect
4. C₃×C₅×#478	C₂	semidirect
5. C₃×C₅×#476	C₂	semidirect
6. C₃×C₅×#474	C₂	semidirect
7. C₃×C₅×#428	C₂	semidirect
8. C₃×C₅×#427	C₂	semidirect
9. C₃×C₅×#424	C₂	semidirect
12. C₅×#3081	C₃	direct (N × H)
17. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
19. C₃×C₅×#125	C₂²	semidirect
30. C₃×C₅×#107	C₂²	semidirect
31. C₃×C₅×#107	C₂²	semidirect
32. C₃×C₅×#107	C₂²	semidirect
33. C₃×C₅×#106	C₂²	semidirect
34. C₃×C₅×#106	C₂²	semidirect
35. C₃×C₅×#106	C₂²	semidirect
36. C₃×C₅×C₄⋊C₈	C₂²	semidirect
37. C₃×C₅×C₄⋊C₈	C₂²	semidirect
42. C₃×#3081	C₅	direct (N × H)
43. C₅×#544	C₂×C₃	semidirect
45. C₅×#478	C₂×C₃	semidirect
46. C₅×#478	C₂×C₃	semidirect
47. C₅×#476	C₂×C₃	semidirect
48. C₅×#474	C₂×C₃	semidirect
49. C₅×#428	C₂×C₃	semidirect
50. C₅×#427	C₂×C₃	semidirect
51. C₅×#424	C₂×C₃	semidirect
58. C₂×C₃×C₅×D₄	Q₈	semidirect
88. C₃×#544	C₂×C₅	semidirect
90. C₃×#478	C₂×C₅	semidirect
91. C₃×#478	C₂×C₅	semidirect
92. C₃×#476	C₂×C₅	semidirect
93. C₃×#474	C₂×C₅	semidirect
94. C₃×#428	C₂×C₅	semidirect
95. C₃×#427	C₂×C₅	semidirect
96. C₃×#424	C₂×C₅	semidirect
103. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
105. C₅×#125	C₂²×C₃	semidirect
116. C₅×#107	C₂²×C₃	semidirect
117. C₅×#107	C₂²×C₃	semidirect
118. C₅×#107	C₂²×C₃	semidirect
119. C₅×#106	C₂²×C₃	semidirect
120. C₅×#106	C₂²×C₃	semidirect
121. C₅×#106	C₂²×C₃	semidirect
122. C₅×C₄⋊C₈	C₂²×C₃	semidirect
123. C₅×C₄⋊C₈	C₂²×C₃	semidirect
128. #3081	C₃×C₅	direct (N × H)
159. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
161. C₃×#125	C₂²×C₅	semidirect
172. C₃×#107	C₂²×C₅	semidirect
173. C₃×#107	C₂²×C₅	semidirect
174. C₃×#107	C₂²×C₅	semidirect
175. C₃×#106	C₂²×C₅	semidirect
176. C₃×#106	C₂²×C₅	semidirect
177. C₃×#106	C₂²×C₅	semidirect
178. C₃×C₄⋊C₈	C₂²×C₅	semidirect
179. C₃×C₄⋊C₈	C₂²×C₅	semidirect
188. C₂×C₅×D₄	C₃×Q₈	semidirect
218. #544	C₂×C₃×C₅	semidirect
220. #478	C₂×C₃×C₅	semidirect
221. #478	C₂×C₃×C₅	semidirect
222. #476	C₂×C₃×C₅	semidirect
223. #474	C₂×C₃×C₅	semidirect
224. #428	C₂×C₃×C₅	semidirect
225. #427	C₂×C₃×C₅	semidirect
226. #424	C₂×C₃×C₅	semidirect
251. C₂×C₃×D₄	C₅×Q₈	semidirect
311. C₄⋊Q₈	C₂²×C₃×C₅	semidirect
313. #125	C₂²×C₃×C₅	semidirect
324. #107	C₂²×C₃×C₅	semidirect
325. #107	C₂²×C₃×C₅	semidirect
326. #107	C₂²×C₃×C₅	semidirect
327. #106	C₂²×C₃×C₅	semidirect
328. #106	C₂²×C₃×C₅	semidirect
329. #106	C₂²×C₃×C₅	semidirect
330. C₄⋊C₈	C₂²×C₃×C₅	semidirect
331. C₄⋊C₈	C₂²×C₃×C₅	semidirect
389. C₂×D₄	C₃×C₅×Q₈	semidirect
419. C₃×C₅	#3081	direct (N × H)
474. C₅	C₃×#3081	direct (N × H)
493. C₃	C₅×#3081	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×#544	2
C₃×C₅×#518	2
C₃×C₅×#478	2
C₃×C₅×#476	2
C₃×C₅×#474	2
C₃×C₅×#428	2
C₃×C₅×#427	2
C₃×C₅×#424	2
C₃×C₅×#418	2
C₂×C₃×C₅×#102	2
C₅×#3081	3
C₃×#3081	5

Subgroups — C3×C5×#3081 #14276115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3081 #14276115