Subgroups — C₃×C₅×#2732 #14275766

Order 1920.

1056 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#399	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#398	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#396	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#390	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#386	C₂	1	2	yes	maximal
9	C₃×C₅×#385	C₂	1	2	yes	maximal, complemented
10	C₅×#2732	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
14	C₃×C₅×C₂²⋊Q₈	—	2	8	no	—
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
29	C₃×C₅×#95	C₂²	1	2	yes	—
30	C₃×C₅×#95	C₂²	1	2	yes	—
31	C₃×C₅×#95	C₂²	1	2	yes	—
32	C₃×C₅×#95	C₂²	1	2	yes	—
33	C₃×#2732	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
34	C₂×C₅×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
35	C₅×#399	C₂×C₃	1	2	yes	complemented
36	C₅×#398	C₂×C₃	1	2	yes	complemented
37	C₅×#396	C₂×C₃	1	2	yes	complemented
38	C₅×#391	C₂×C₃	1	1	yes	characteristic, complemented
39	C₅×#390	C₂×C₃	1	1	yes	characteristic
40	C₅×#389	C₂×C₃	1	1	yes	characteristic, complemented
41	C₅×#386	C₂×C₃	1	2	yes	—
42	C₅×#385	C₂×C₃	1	2	yes	complemented
43	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
44	C₂×C₃×C₅×Q₈	—	4	1	no	—
45	C₂×C₃×C₅×Q₈	—	4	1	no	—
46	C₂×C₃×C₅×Q₈	D₄	1	4	yes	complemented
47	C₂×C₃×C₅×D₄	—	4	1	no	—
48	C₂×C₃×C₅×D₄	—	4	1	no	—
49	C₂×C₃×C₅×D₄	Q₈	1	4	yes	complemented
50	C₂²×C₃×C₄×C₅	—	2	4	no	—
51	C₂²×C₃×C₄×C₅	—	4	2	no	—
52	C₂²×C₃×C₄×C₅	—	4	2	no	—
53	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
55	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
56	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
57	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
59	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
61	C₃×C₅×C₄⋊C₄	—	2	4	no	—
62	C₃×C₅×C₄⋊C₄	—	2	4	no	—
63	C₃×C₅×C₄⋊C₄	—	2	4	no	—
64	C₃×C₅×C₄⋊C₄	—	2	2	no	—
65	C₃×C₅×C₄⋊C₄	—	4	1	no	—
66	C₃×C₅×C₄⋊C₄	—	4	1	no	—
67	C₃×C₅×C₄⋊C₄	—	4	1	no	—
68	C₃×C₅×C₄⋊C₄	—	4	2	no	—
69	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
70	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
71	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
72	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
73	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
74	C₃×C₄²×C₅	—	4	1	no	—
75	C₂×C₃×C₂²⋊Q₈	C₂×C₅	1	2	yes	complemented
76	C₃×#399	C₂×C₅	1	2	yes	complemented
77	C₃×#398	C₂×C₅	1	2	yes	complemented
78	C₃×#396	C₂×C₅	1	2	yes	complemented
79	C₃×#391	C₂×C₅	1	1	yes	characteristic, complemented
80	C₃×#390	C₂×C₅	1	1	yes	characteristic
81	C₃×#389	C₂×C₅	1	1	yes	characteristic, complemented
82	C₃×#386	C₂×C₅	1	2	yes	—
83	C₃×#385	C₂×C₅	1	2	yes	complemented
84	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
85	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
86	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
87	C₅×C₂²⋊Q₈	—	2	8	no	—
88	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
89	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
90	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
91	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
92	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
93	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
94	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
95	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
96	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
97	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
98	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
99	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
100	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
101	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
102	C₅×#95	C₂²×C₃	1	2	yes	—
103	C₅×#95	C₂²×C₃	1	2	yes	—
104	C₅×#95	C₂²×C₃	1	2	yes	—
105	C₅×#95	C₂²×C₃	1	2	yes	—
106	#2732	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
107	C₂³×C₃×C₅	—	2	2	no	—
108	C₂³×C₃×C₅	—	2	2	no	—
109	C₂³×C₃×C₅	—	2	2	no	—
110	C₂³×C₃×C₅	—	2	2	no	—
111	C₂³×C₃×C₅	—	2	2	no	—
112	C₂³×C₃×C₅	—	2	2	no	—
113	C₂³×C₃×C₅	C₂×Q₈	1	4	yes	—
114	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
115	C₃×C₅×Q₈	—	4	4	no	—
116	C₃×C₅×D₄	—	4	4	no	—
117	C₂×C₃×C₄×C₅	—	2	1	no	—
118	C₂×C₃×C₄×C₅	—	2	1	no	—
119	C₂×C₃×C₄×C₅	—	2	1	no	—
120	C₂×C₃×C₄×C₅	—	2	1	no	—
121	C₂×C₃×C₄×C₅	—	2	1	no	—
122	C₂×C₃×C₄×C₅	—	2	2	no	—
123	C₂×C₃×C₄×C₅	—	2	2	no	—
124	C₂×C₃×C₄×C₅	—	2	1	no	—
125	C₂×C₃×C₄×C₅	—	2	2	no	—
126	C₂×C₃×C₄×C₅	—	2	2	no	—
127	C₂×C₃×C₄×C₅	—	2	2	no	—
128	C₂×C₃×C₄×C₅	—	2	2	no	—
129	C₂×C₃×C₄×C₅	—	2	2	no	—
130	C₂×C₃×C₄×C₅	—	2	2	no	—
131	C₂×C₃×C₄×C₅	—	2	2	no	—
132	C₂×C₃×C₄×C₅	—	2	2	no	—
133	C₂×C₃×C₄×C₅	—	2	2	no	—
134	C₂×C₃×C₄×C₅	—	2	2	no	—
135	C₂×C₃×C₄×C₅	—	2	2	no	—
136	C₂×C₃×C₄×C₅	—	2	2	no	—
137	C₂×C₃×C₄×C₅	—	4	8	no	—
138	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
139	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
140	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
141	C₂×C₃×C₄×C₅	#41	1	2	yes	—
142	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
143	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
144	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
145	C₃×C₂²⋊Q₈	—	2	8	no	—
146	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
147	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
148	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
149	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
150	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
151	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
152	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
153	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
154	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
155	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
156	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
157	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
158	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
159	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
160	C₃×#95	C₂²×C₅	1	2	yes	—
161	C₃×#95	C₂²×C₅	1	2	yes	—
162	C₃×#95	C₂²×C₅	1	2	yes	—
163	C₃×#95	C₂²×C₅	1	2	yes	—
164	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
165	C₂×C₅×Q₈	—	4	1	no	—
166	C₂×C₅×Q₈	—	4	1	no	—
167	C₂×C₅×Q₈	C₃×D₄	1	4	yes	complemented
168	C₂×C₅×D₄	—	4	1	no	—
169	C₂×C₅×D₄	—	4	1	no	—
170	C₂×C₅×D₄	C₃×Q₈	1	4	yes	complemented
171	C₂²×C₄×C₅	—	2	4	no	—
172	C₂²×C₄×C₅	—	4	2	no	—
173	C₂²×C₄×C₅	—	4	2	no	—
174	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
175	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
176	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
177	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
178	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
179	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
180	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
181	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
182	C₅×C₄⋊C₄	—	2	4	no	—
183	C₅×C₄⋊C₄	—	2	4	no	—
184	C₅×C₄⋊C₄	—	2	4	no	—
185	C₅×C₄⋊C₄	—	2	2	no	—
186	C₅×C₄⋊C₄	—	4	1	no	—
187	C₅×C₄⋊C₄	—	4	1	no	—
188	C₅×C₄⋊C₄	—	4	1	no	—
189	C₅×C₄⋊C₄	—	4	2	no	—
190	C₅×C₂²⋊C₄	—	2	4	no	—
191	C₅×C₂²⋊C₄	—	2	4	no	—
192	C₅×C₂²⋊C₄	—	2	4	no	—
193	C₅×C₂²⋊C₄	—	4	2	no	—
194	C₅×C₂²⋊C₄	—	4	2	no	—
195	C₄²×C₅	—	4	1	no	—
196	C₂×C₂²⋊Q₈	C₂×C₃×C₅	1	2	yes	complemented
197	#399	C₂×C₃×C₅	1	2	yes	complemented
198	#398	C₂×C₃×C₅	1	2	yes	complemented
199	#396	C₂×C₃×C₅	1	2	yes	complemented
200	#391	C₂×C₃×C₅	1	1	yes	characteristic, complemented
201	#390	C₂×C₃×C₅	1	1	yes	characteristic
202	#389	C₂×C₃×C₅	1	1	yes	characteristic, complemented
203	#386	C₂×C₃×C₅	1	2	yes	—
204	#385	C₂×C₃×C₅	1	2	yes	complemented
205	C₂²×C₃×C₅	—	2	4	no	—
206	C₂²×C₃×C₅	—	2	4	no	—
207	C₂²×C₃×C₅	—	2	4	no	—
208	C₂²×C₃×C₅	—	4	2	no	—
209	C₂²×C₃×C₅	—	4	2	no	—
210	C₂²×C₃×C₅	—	4	2	no	—
211	C₂²×C₃×C₅	—	4	2	no	—
212	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
213	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
214	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
215	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
216	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
217	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
218	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
219	C₃×C₄×C₅	—	2	4	no	—
220	C₃×C₄×C₅	—	2	2	no	—
221	C₃×C₄×C₅	—	2	2	no	—
222	C₃×C₄×C₅	—	2	2	no	—
223	C₃×C₄×C₅	—	4	2	no	—
224	C₃×C₄×C₅	—	4	2	no	—
225	C₃×C₄×C₅	—	4	2	no	—
226	C₃×C₄×C₅	—	4	2	no	—
227	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
228	C₂×C₃×Q₈	—	4	1	no	—
229	C₂×C₃×Q₈	—	4	1	no	—
230	C₂×C₃×Q₈	C₅×D₄	1	4	yes	complemented
231	C₂×C₃×D₄	—	4	1	no	—
232	C₂×C₃×D₄	—	4	1	no	—
233	C₂×C₃×D₄	C₅×Q₈	1	4	yes	complemented
234	C₂²×C₃×C₄	—	2	4	no	—
235	C₂²×C₃×C₄	—	4	2	no	—
236	C₂²×C₃×C₄	—	4	2	no	—
237	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
238	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
239	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
240	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
241	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
242	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
243	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
244	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
245	C₃×C₄⋊C₄	—	2	4	no	—
246	C₃×C₄⋊C₄	—	2	4	no	—
247	C₃×C₄⋊C₄	—	2	4	no	—
248	C₃×C₄⋊C₄	—	2	2	no	—
249	C₃×C₄⋊C₄	—	4	1	no	—
250	C₃×C₄⋊C₄	—	4	1	no	—
251	C₃×C₄⋊C₄	—	4	1	no	—
252	C₃×C₄⋊C₄	—	4	2	no	—
253	C₃×C₂²⋊C₄	—	2	4	no	—
254	C₃×C₂²⋊C₄	—	2	4	no	—
255	C₃×C₂²⋊C₄	—	2	4	no	—
256	C₃×C₂²⋊C₄	—	4	2	no	—
257	C₃×C₂²⋊C₄	—	4	2	no	—
258	C₃×C₄²	—	4	1	no	—
259	C₂³×C₅	—	2	2	no	—
260	C₂³×C₅	—	2	2	no	—
261	C₂³×C₅	—	2	2	no	—
262	C₂³×C₅	—	2	2	no	—
263	C₂³×C₅	—	2	2	no	—
264	C₂³×C₅	—	2	2	no	—
265	C₂³×C₅	C₂×C₃×Q₈	1	4	yes	—
266	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
267	C₅×Q₈	—	4	4	no	—
268	C₅×D₄	—	4	4	no	—
269	C₂×C₄×C₅	—	2	1	no	—
270	C₂×C₄×C₅	—	2	1	no	—
271	C₂×C₄×C₅	—	2	1	no	—
272	C₂×C₄×C₅	—	2	1	no	—
273	C₂×C₄×C₅	—	2	1	no	—
274	C₂×C₄×C₅	—	2	2	no	—
275	C₂×C₄×C₅	—	2	2	no	—
276	C₂×C₄×C₅	—	2	1	no	—
277	C₂×C₄×C₅	—	2	2	no	—
278	C₂×C₄×C₅	—	2	2	no	—
279	C₂×C₄×C₅	—	2	2	no	—
280	C₂×C₄×C₅	—	2	2	no	—
281	C₂×C₄×C₅	—	2	2	no	—
282	C₂×C₄×C₅	—	2	2	no	—
283	C₂×C₄×C₅	—	2	2	no	—
284	C₂×C₄×C₅	—	2	2	no	—
285	C₂×C₄×C₅	—	2	2	no	—
286	C₂×C₄×C₅	—	2	2	no	—
287	C₂×C₄×C₅	—	2	2	no	—
288	C₂×C₄×C₅	—	2	2	no	—
289	C₂×C₄×C₅	—	4	8	no	—
290	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
291	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
292	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
293	C₂×C₄×C₅	C₃×#41	1	2	yes	—
294	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
295	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
296	C₂³×C₄	C₂²×C₃×C₅	1	2	yes	—
297	C₂²⋊Q₈	—	2	8	no	—
298	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
299	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
300	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
301	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
306	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
307	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
308	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
309	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
310	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
311	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
312	#95	C₂²×C₃×C₅	1	2	yes	—
313	#95	C₂²×C₃×C₅	1	2	yes	—
314	#95	C₂²×C₃×C₅	1	2	yes	—
315	#95	C₂²×C₃×C₅	1	2	yes	—
316	C₂×C₃×C₅	—	4	4	no	—
317	C₂×C₃×C₅	#534	1	1	yes	characteristic
318	C₂×C₃×C₅	#539	1	1	yes	characteristic
319	C₂×C₃×C₅	#543	1	1	yes	characteristic
320	C₂×C₃×C₅	D₄×Q₈	1	1	yes	characteristic
321	C₂×C₃×C₅	#550	1	1	yes	characteristic
322	C₂×C₃×C₅	#554	1	1	yes	characteristic
323	C₂×C₃×C₅	#562	1	1	yes	characteristic
324	C₂³×C₃	—	2	2	no	—
325	C₂³×C₃	—	2	2	no	—
326	C₂³×C₃	—	2	2	no	—
327	C₂³×C₃	—	2	2	no	—
328	C₂³×C₃	—	2	2	no	—
329	C₂³×C₃	—	2	2	no	—
330	C₂³×C₃	C₂×C₅×Q₈	1	4	yes	—
331	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
332	C₃×Q₈	—	4	4	no	—
333	C₃×D₄	—	4	4	no	—
334	C₂×C₃×C₄	—	2	1	no	—
335	C₂×C₃×C₄	—	2	1	no	—
336	C₂×C₃×C₄	—	2	1	no	—
337	C₂×C₃×C₄	—	2	1	no	—
338	C₂×C₃×C₄	—	2	1	no	—
339	C₂×C₃×C₄	—	2	2	no	—
340	C₂×C₃×C₄	—	2	2	no	—
341	C₂×C₃×C₄	—	2	1	no	—
342	C₂×C₃×C₄	—	2	2	no	—
343	C₂×C₃×C₄	—	2	2	no	—
344	C₂×C₃×C₄	—	2	2	no	—
345	C₂×C₃×C₄	—	2	2	no	—
346	C₂×C₃×C₄	—	2	2	no	—
347	C₂×C₃×C₄	—	2	2	no	—
348	C₂×C₃×C₄	—	2	2	no	—
349	C₂×C₃×C₄	—	2	2	no	—
350	C₂×C₃×C₄	—	2	2	no	—
351	C₂×C₃×C₄	—	2	2	no	—
352	C₂×C₃×C₄	—	2	2	no	—
353	C₂×C₃×C₄	—	2	2	no	—
354	C₂×C₃×C₄	—	4	8	no	—
355	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
356	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
357	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
358	C₂×C₃×C₄	C₅×#41	1	2	yes	—
359	C₂²×C₅	—	2	4	no	—
360	C₂²×C₅	—	2	4	no	—
361	C₂²×C₅	—	2	4	no	—
362	C₂²×C₅	—	4	2	no	—
363	C₂²×C₅	—	4	2	no	—
364	C₂²×C₅	—	4	2	no	—
365	C₂²×C₅	—	4	2	no	—
366	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
367	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
368	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
369	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
370	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
371	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
372	C₂²×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
373	C₄×C₅	—	2	4	no	—
374	C₄×C₅	—	2	2	no	—
375	C₄×C₅	—	2	2	no	—
376	C₄×C₅	—	2	2	no	—
377	C₄×C₅	—	4	2	no	—
378	C₄×C₅	—	4	2	no	—
379	C₄×C₅	—	4	2	no	—
380	C₄×C₅	—	4	2	no	—
381	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
382	C₂×Q₈	—	4	1	no	—
383	C₂×Q₈	—	4	1	no	—
384	C₂×Q₈	C₃×C₅×D₄	1	4	yes	complemented
385	C₂×D₄	—	4	1	no	—
386	C₂×D₄	—	4	1	no	—
387	C₂×D₄	C₃×C₅×Q₈	1	4	yes	complemented
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	4	2	no	—
390	C₂²×C₄	—	4	2	no	—
391	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
392	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
393	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
394	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
395	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
396	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
397	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
398	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	4	1	no	—
404	C₄⋊C₄	—	4	1	no	—
405	C₄⋊C₄	—	4	1	no	—
406	C₄⋊C₄	—	4	2	no	—
407	C₂²⋊C₄	—	2	4	no	—
408	C₂²⋊C₄	—	2	4	no	—
409	C₂²⋊C₄	—	2	4	no	—
410	C₂²⋊C₄	—	4	2	no	—
411	C₂²⋊C₄	—	4	2	no	—
412	C₄²	—	4	1	no	—
413	C₃×C₅	#2732	1	1	yes	characteristic, direct product factor, Hall
414	C₂²×C₃	—	2	4	no	—
415	C₂²×C₃	—	2	4	no	—
416	C₂²×C₃	—	2	4	no	—
417	C₂²×C₃	—	4	2	no	—
418	C₂²×C₃	—	4	2	no	—
419	C₂²×C₃	—	4	2	no	—
420	C₂²×C₃	—	4	2	no	—
421	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
422	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
423	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
424	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
425	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
426	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
427	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
428	C₃×C₄	—	2	4	no	—
429	C₃×C₄	—	2	2	no	—
430	C₃×C₄	—	2	2	no	—
431	C₃×C₄	—	2	2	no	—
432	C₃×C₄	—	4	2	no	—
433	C₃×C₄	—	4	2	no	—
434	C₃×C₄	—	4	2	no	—
435	C₃×C₄	—	4	2	no	—
436	C₂×C₅	—	4	4	no	—
437	C₂×C₅	C₃×#534	1	1	yes	characteristic
438	C₂×C₅	C₃×#539	1	1	yes	characteristic
439	C₂×C₅	C₃×#543	1	1	yes	characteristic
440	C₂×C₅	C₃×D₄×Q₈	1	1	yes	characteristic
441	C₂×C₅	C₃×#550	1	1	yes	characteristic
442	C₂×C₅	C₃×#554	1	1	yes	characteristic
443	C₂×C₅	C₃×#562	1	1	yes	characteristic
444	C₂³	—	2	2	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	2	no	—
450	C₂³	C₂×C₃×C₅×Q₈	1	4	yes	—
451	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
452	Q₈	—	4	4	no	—
453	D₄	—	4	4	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	4	8	no	—
475	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
476	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
477	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
478	C₂×C₄	C₃×C₅×#41	1	2	yes	—
479	C₂×C₃	—	4	4	no	—
480	C₂×C₃	C₅×#534	1	1	yes	characteristic
481	C₂×C₃	C₅×#539	1	1	yes	characteristic
482	C₂×C₃	C₅×#543	1	1	yes	characteristic
483	C₂×C₃	C₅×D₄×Q₈	1	1	yes	characteristic
484	C₂×C₃	C₅×#550	1	1	yes	characteristic
485	C₂×C₃	C₅×#554	1	1	yes	characteristic
486	C₂×C₃	C₅×#562	1	1	yes	characteristic
487	C₅	C₃×#2732	1	1	yes	characteristic, direct product factor, Sylow
488	C₂²	—	2	4	no	—
489	C₂²	—	2	4	no	—
490	C₂²	—	2	4	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
496	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
497	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
498	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
499	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
500	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
501	C₂²	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
502	C₄	—	2	4	no	—
503	C₄	—	2	2	no	—
504	C₄	—	2	2	no	—
505	C₄	—	2	2	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	2	no	—
508	C₄	—	4	2	no	—
509	C₄	—	4	2	no	—
510	C₃	C₅×#2732	1	1	yes	characteristic, direct product factor, Sylow
511	C₂	—	4	4	no	—
512	C₂	C₃×C₅×#534	1	1	yes	characteristic
513	C₂	C₃×C₅×#539	1	1	yes	characteristic
514	C₂	C₃×C₅×#543	1	1	yes	characteristic
515	C₂	C₃×C₅×D₄×Q₈	1	1	yes	characteristic
516	C₂	C₃×C₅×#550	1	1	yes	characteristic
517	C₂	C₃×C₅×#554	1	1	yes	characteristic
518	C₂	C₃×C₅×#562	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₂²⋊Q₈	C₂	semidirect
2. C₃×C₅×#399	C₂	semidirect
3. C₃×C₅×#398	C₂	semidirect
4. C₃×C₅×#396	C₂	semidirect
5. C₃×C₅×#391	C₂	semidirect
7. C₃×C₅×#389	C₂	semidirect
9. C₃×C₅×#385	C₂	semidirect
10. C₅×#2732	C₃	direct (N × H)
33. C₃×#2732	C₅	direct (N × H)
34. C₂×C₅×C₂²⋊Q₈	C₂×C₃	semidirect
35. C₅×#399	C₂×C₃	semidirect
36. C₅×#398	C₂×C₃	semidirect
37. C₅×#396	C₂×C₃	semidirect
38. C₅×#391	C₂×C₃	semidirect
40. C₅×#389	C₂×C₃	semidirect
42. C₅×#385	C₂×C₃	semidirect
46. C₂×C₃×C₅×Q₈	D₄	semidirect
49. C₂×C₃×C₅×D₄	Q₈	semidirect
75. C₂×C₃×C₂²⋊Q₈	C₂×C₅	semidirect
76. C₃×#399	C₂×C₅	semidirect
77. C₃×#398	C₂×C₅	semidirect
78. C₃×#396	C₂×C₅	semidirect
79. C₃×#391	C₂×C₅	semidirect
81. C₃×#389	C₂×C₅	semidirect
83. C₃×#385	C₂×C₅	semidirect
106. #2732	C₃×C₅	direct (N × H)
167. C₂×C₅×Q₈	C₃×D₄	semidirect
170. C₂×C₅×D₄	C₃×Q₈	semidirect
196. C₂×C₂²⋊Q₈	C₂×C₃×C₅	semidirect
197. #399	C₂×C₃×C₅	semidirect
198. #398	C₂×C₃×C₅	semidirect
199. #396	C₂×C₃×C₅	semidirect
200. #391	C₂×C₃×C₅	semidirect
202. #389	C₂×C₃×C₅	semidirect
204. #385	C₂×C₃×C₅	semidirect
230. C₂×C₃×Q₈	C₅×D₄	semidirect
233. C₂×C₃×D₄	C₅×Q₈	semidirect
384. C₂×Q₈	C₃×C₅×D₄	semidirect
387. C₂×D₄	C₃×C₅×Q₈	semidirect
413. C₃×C₅	#2732	direct (N × H)
487. C₅	C₃×#2732	direct (N × H)
510. C₃	C₅×#2732	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×C₂²⋊Q₈	2
C₃×C₅×#399	2
C₃×C₅×#398	2
C₃×C₅×#396	2
C₃×C₅×#391	2
C₃×C₅×#390	2
C₃×C₅×#389	2
C₃×C₅×#386	2
C₃×C₅×#385	2
C₅×#2732	3
C₃×#2732	5

Subgroups — C3×C5×#2732 #14275766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2732 #14275766