Subgroups — C₃×C₅×#2728 #14275762

Order 1920.

1024 conjugacy classes of subgroups.

The subgroup lattice has 480 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×#400	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#398	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#397	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#394	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#388	C₂	1	2	yes	maximal, complemented
8	C₃×C₅×#386	C₂	1	2	yes	maximal
9	C₃×C₅×#385	C₂	1	2	yes	maximal, complemented
10	C₅×#2728	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×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₅×C₂²⋊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×C₄²×C₅	C₂²	1	1	yes	characteristic
26	C₃×C₅×#95	C₂²	1	2	yes	—
27	C₃×C₅×#95	C₂²	1	2	yes	—
28	C₃×C₅×#95	C₂²	1	2	yes	—
29	C₃×C₅×#95	C₂²	1	2	yes	—
30	C₃×C₅×#95	C₂²	1	2	yes	—
31	C₃×C₅×#95	C₂²	1	1	yes	characteristic
32	C₃×C₅×#95	C₂²	1	1	yes	characteristic
33	C₃×#2728	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
34	C₂×C₅×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
35	C₅×#400	C₂×C₃	1	2	yes	complemented
36	C₅×#398	C₂×C₃	1	1	yes	characteristic, complemented
37	C₅×#397	C₂×C₃	1	2	yes	complemented
38	C₅×#394	C₂×C₃	1	1	yes	characteristic
39	C₅×#391	C₂×C₃	1	1	yes	characteristic, complemented
40	C₅×#388	C₂×C₃	1	2	yes	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	—	2	4	no	—
48	C₂×C₃×C₅×D₄	—	4	1	no	—
49	C₂²×C₃×C₄×C₅	—	2	4	no	—
50	C₂²×C₃×C₄×C₅	—	4	2	no	—
51	C₂²×C₃×C₄×C₅	—	4	2	no	—
52	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
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	1	yes	characteristic
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	1	yes	characteristic
59	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
60	C₃×C₅×C₄⋊C₄	—	2	4	no	—
61	C₃×C₅×C₄⋊C₄	—	2	4	no	—
62	C₃×C₅×C₄⋊C₄	—	2	4	no	—
63	C₃×C₅×C₄⋊C₄	—	4	1	no	—
64	C₃×C₅×C₄⋊C₄	—	4	1	no	—
65	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
66	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
67	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
68	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
69	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
70	C₃×C₄²×C₅	—	4	1	no	—
71	C₂×C₃×C₂²⋊Q₈	C₂×C₅	1	2	yes	complemented
72	C₃×#400	C₂×C₅	1	2	yes	complemented
73	C₃×#398	C₂×C₅	1	1	yes	characteristic, complemented
74	C₃×#397	C₂×C₅	1	2	yes	complemented
75	C₃×#394	C₂×C₅	1	1	yes	characteristic
76	C₃×#391	C₂×C₅	1	1	yes	characteristic, complemented
77	C₃×#388	C₂×C₅	1	2	yes	complemented
78	C₃×#386	C₂×C₅	1	2	yes	—
79	C₃×#385	C₂×C₅	1	2	yes	complemented
80	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
81	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
82	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
83	C₅×C₂²⋊Q₈	—	2	8	no	—
84	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
85	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
86	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
89	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
90	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
91	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
92	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
93	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
94	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
95	C₅×#95	C₂²×C₃	1	2	yes	—
96	C₅×#95	C₂²×C₃	1	2	yes	—
97	C₅×#95	C₂²×C₃	1	2	yes	—
98	C₅×#95	C₂²×C₃	1	2	yes	—
99	C₅×#95	C₂²×C₃	1	2	yes	—
100	C₅×#95	C₂²×C₃	1	1	yes	characteristic
101	C₅×#95	C₂²×C₃	1	1	yes	characteristic
102	#2728	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
103	C₂³×C₃×C₅	—	2	2	no	—
104	C₂³×C₃×C₅	—	2	2	no	—
105	C₂³×C₃×C₅	—	2	2	no	—
106	C₂³×C₃×C₅	—	2	2	no	—
107	C₂³×C₃×C₅	—	2	2	no	—
108	C₂³×C₃×C₅	—	2	2	no	—
109	C₂³×C₃×C₅	#41	1	4	yes	—
110	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
111	C₃×C₅×Q₈	—	4	4	no	—
112	C₃×C₅×D₄	—	4	4	no	—
113	C₂×C₃×C₄×C₅	—	2	1	no	—
114	C₂×C₃×C₄×C₅	—	2	1	no	—
115	C₂×C₃×C₄×C₅	—	2	2	no	—
116	C₂×C₃×C₄×C₅	—	2	2	no	—
117	C₂×C₃×C₄×C₅	—	2	2	no	—
118	C₂×C₃×C₄×C₅	—	2	2	no	—
119	C₂×C₃×C₄×C₅	—	2	2	no	—
120	C₂×C₃×C₄×C₅	—	2	2	no	—
121	C₂×C₃×C₄×C₅	—	2	2	no	—
122	C₂×C₃×C₄×C₅	—	2	2	no	—
123	C₂×C₃×C₄×C₅	—	2	2	no	—
124	C₂×C₃×C₄×C₅	—	2	2	no	—
125	C₂×C₃×C₄×C₅	—	2	2	no	—
126	C₂×C₃×C₄×C₅	—	2	2	no	—
127	C₂×C₃×C₄×C₅	—	2	1	no	—
128	C₂×C₃×C₄×C₅	—	2	2	no	—
129	C₂×C₃×C₄×C₅	—	2	1	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	—	4	8	no	—
134	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
135	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
136	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
137	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
138	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
139	C₃×C₂²⋊Q₈	—	2	8	no	—
140	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
141	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
142	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
143	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
144	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
145	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
146	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
147	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
148	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
149	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
150	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
151	C₃×#95	C₂²×C₅	1	2	yes	—
152	C₃×#95	C₂²×C₅	1	2	yes	—
153	C₃×#95	C₂²×C₅	1	2	yes	—
154	C₃×#95	C₂²×C₅	1	2	yes	—
155	C₃×#95	C₂²×C₅	1	2	yes	—
156	C₃×#95	C₂²×C₅	1	1	yes	characteristic
157	C₃×#95	C₂²×C₅	1	1	yes	characteristic
158	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
159	C₂×C₅×Q₈	—	4	1	no	—
160	C₂×C₅×Q₈	—	4	1	no	—
161	C₂×C₅×Q₈	C₃×D₄	1	4	yes	complemented
162	C₂×C₅×D₄	—	2	4	no	—
163	C₂×C₅×D₄	—	4	1	no	—
164	C₂²×C₄×C₅	—	2	4	no	—
165	C₂²×C₄×C₅	—	4	2	no	—
166	C₂²×C₄×C₅	—	4	2	no	—
167	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
168	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
169	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
170	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
171	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
172	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
173	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
174	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
175	C₅×C₄⋊C₄	—	2	4	no	—
176	C₅×C₄⋊C₄	—	2	4	no	—
177	C₅×C₄⋊C₄	—	2	4	no	—
178	C₅×C₄⋊C₄	—	4	1	no	—
179	C₅×C₄⋊C₄	—	4	1	no	—
180	C₅×C₂²⋊C₄	—	2	4	no	—
181	C₅×C₂²⋊C₄	—	2	4	no	—
182	C₅×C₂²⋊C₄	—	2	4	no	—
183	C₅×C₂²⋊C₄	—	2	4	no	—
184	C₅×C₂²⋊C₄	—	4	2	no	—
185	C₄²×C₅	—	4	1	no	—
186	C₂×C₂²⋊Q₈	C₂×C₃×C₅	1	2	yes	complemented
187	#400	C₂×C₃×C₅	1	2	yes	complemented
188	#398	C₂×C₃×C₅	1	1	yes	characteristic, complemented
189	#397	C₂×C₃×C₅	1	2	yes	complemented
190	#394	C₂×C₃×C₅	1	1	yes	characteristic
191	#391	C₂×C₃×C₅	1	1	yes	characteristic, complemented
192	#388	C₂×C₃×C₅	1	2	yes	complemented
193	#386	C₂×C₃×C₅	1	2	yes	—
194	#385	C₂×C₃×C₅	1	2	yes	complemented
195	C₂²×C₃×C₅	—	2	4	no	—
196	C₂²×C₃×C₅	—	2	4	no	—
197	C₂²×C₃×C₅	—	2	4	no	—
198	C₂²×C₃×C₅	—	4	2	no	—
199	C₂²×C₃×C₅	—	4	2	no	—
200	C₂²×C₃×C₅	—	4	2	no	—
201	C₂²×C₃×C₅	—	4	2	no	—
202	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
203	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
204	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
205	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
206	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
207	C₃×C₄×C₅	—	2	4	no	—
208	C₃×C₄×C₅	—	2	2	no	—
209	C₃×C₄×C₅	—	4	2	no	—
210	C₃×C₄×C₅	—	4	2	no	—
211	C₃×C₄×C₅	—	4	2	no	—
212	C₃×C₄×C₅	—	4	2	no	—
213	C₃×C₄×C₅	—	4	1	no	—
214	C₃×C₄×C₅	—	4	1	no	—
215	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
216	C₂×C₃×Q₈	—	4	1	no	—
217	C₂×C₃×Q₈	—	4	1	no	—
218	C₂×C₃×Q₈	C₅×D₄	1	4	yes	complemented
219	C₂×C₃×D₄	—	2	4	no	—
220	C₂×C₃×D₄	—	4	1	no	—
221	C₂²×C₃×C₄	—	2	4	no	—
222	C₂²×C₃×C₄	—	4	2	no	—
223	C₂²×C₃×C₄	—	4	2	no	—
224	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
225	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
226	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
227	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
228	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
229	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
230	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
231	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
232	C₃×C₄⋊C₄	—	2	4	no	—
233	C₃×C₄⋊C₄	—	2	4	no	—
234	C₃×C₄⋊C₄	—	2	4	no	—
235	C₃×C₄⋊C₄	—	4	1	no	—
236	C₃×C₄⋊C₄	—	4	1	no	—
237	C₃×C₂²⋊C₄	—	2	4	no	—
238	C₃×C₂²⋊C₄	—	2	4	no	—
239	C₃×C₂²⋊C₄	—	2	4	no	—
240	C₃×C₂²⋊C₄	—	2	4	no	—
241	C₃×C₂²⋊C₄	—	4	2	no	—
242	C₃×C₄²	—	4	1	no	—
243	C₂³×C₅	—	2	2	no	—
244	C₂³×C₅	—	2	2	no	—
245	C₂³×C₅	—	2	2	no	—
246	C₂³×C₅	—	2	2	no	—
247	C₂³×C₅	—	2	2	no	—
248	C₂³×C₅	—	2	2	no	—
249	C₂³×C₅	C₃×#41	1	4	yes	—
250	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
251	C₅×Q₈	—	4	4	no	—
252	C₅×D₄	—	4	4	no	—
253	C₂×C₄×C₅	—	2	1	no	—
254	C₂×C₄×C₅	—	2	1	no	—
255	C₂×C₄×C₅	—	2	2	no	—
256	C₂×C₄×C₅	—	2	2	no	—
257	C₂×C₄×C₅	—	2	2	no	—
258	C₂×C₄×C₅	—	2	2	no	—
259	C₂×C₄×C₅	—	2	2	no	—
260	C₂×C₄×C₅	—	2	2	no	—
261	C₂×C₄×C₅	—	2	2	no	—
262	C₂×C₄×C₅	—	2	2	no	—
263	C₂×C₄×C₅	—	2	2	no	—
264	C₂×C₄×C₅	—	2	2	no	—
265	C₂×C₄×C₅	—	2	2	no	—
266	C₂×C₄×C₅	—	2	2	no	—
267	C₂×C₄×C₅	—	2	1	no	—
268	C₂×C₄×C₅	—	2	2	no	—
269	C₂×C₄×C₅	—	2	1	no	—
270	C₂×C₄×C₅	—	2	2	no	—
271	C₂×C₄×C₅	—	2	2	no	—
272	C₂×C₄×C₅	—	2	2	no	—
273	C₂×C₄×C₅	—	4	8	no	—
274	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
275	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
276	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
277	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
278	C₂³×C₄	C₂²×C₃×C₅	1	2	yes	—
279	C₂²⋊Q₈	—	2	8	no	—
280	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
281	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
282	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
283	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
284	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
285	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
286	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
287	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
288	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
289	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
290	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
291	#95	C₂²×C₃×C₅	1	2	yes	—
292	#95	C₂²×C₃×C₅	1	2	yes	—
293	#95	C₂²×C₃×C₅	1	2	yes	—
294	#95	C₂²×C₃×C₅	1	2	yes	—
295	#95	C₂²×C₃×C₅	1	2	yes	—
296	#95	C₂²×C₃×C₅	1	1	yes	characteristic
297	#95	C₂²×C₃×C₅	1	1	yes	characteristic
298	C₂×C₃×C₅	—	4	4	no	—
299	C₂×C₃×C₅	#534	1	1	yes	characteristic
300	C₂×C₃×C₅	#539	1	2	yes	—
301	C₂×C₃×C₅	#548	1	2	yes	—
302	C₂×C₃×C₅	#551	1	1	yes	characteristic
303	C₂×C₃×C₅	#563	1	1	yes	characteristic
304	C₂³×C₃	—	2	2	no	—
305	C₂³×C₃	—	2	2	no	—
306	C₂³×C₃	—	2	2	no	—
307	C₂³×C₃	—	2	2	no	—
308	C₂³×C₃	—	2	2	no	—
309	C₂³×C₃	—	2	2	no	—
310	C₂³×C₃	C₅×#41	1	4	yes	—
311	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
312	C₃×Q₈	—	4	4	no	—
313	C₃×D₄	—	4	4	no	—
314	C₂×C₃×C₄	—	2	1	no	—
315	C₂×C₃×C₄	—	2	1	no	—
316	C₂×C₃×C₄	—	2	2	no	—
317	C₂×C₃×C₄	—	2	2	no	—
318	C₂×C₃×C₄	—	2	2	no	—
319	C₂×C₃×C₄	—	2	2	no	—
320	C₂×C₃×C₄	—	2	2	no	—
321	C₂×C₃×C₄	—	2	2	no	—
322	C₂×C₃×C₄	—	2	2	no	—
323	C₂×C₃×C₄	—	2	2	no	—
324	C₂×C₃×C₄	—	2	2	no	—
325	C₂×C₃×C₄	—	2	2	no	—
326	C₂×C₃×C₄	—	2	2	no	—
327	C₂×C₃×C₄	—	2	2	no	—
328	C₂×C₃×C₄	—	2	1	no	—
329	C₂×C₃×C₄	—	2	2	no	—
330	C₂×C₃×C₄	—	2	1	no	—
331	C₂×C₃×C₄	—	2	2	no	—
332	C₂×C₃×C₄	—	2	2	no	—
333	C₂×C₃×C₄	—	2	2	no	—
334	C₂×C₃×C₄	—	4	8	no	—
335	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
336	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
337	C₂²×C₅	—	2	4	no	—
338	C₂²×C₅	—	2	4	no	—
339	C₂²×C₅	—	2	4	no	—
340	C₂²×C₅	—	4	2	no	—
341	C₂²×C₅	—	4	2	no	—
342	C₂²×C₅	—	4	2	no	—
343	C₂²×C₅	—	4	2	no	—
344	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
345	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
346	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
347	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
348	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
349	C₄×C₅	—	2	4	no	—
350	C₄×C₅	—	2	2	no	—
351	C₄×C₅	—	4	2	no	—
352	C₄×C₅	—	4	2	no	—
353	C₄×C₅	—	4	2	no	—
354	C₄×C₅	—	4	2	no	—
355	C₄×C₅	—	4	1	no	—
356	C₄×C₅	—	4	1	no	—
357	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
358	C₂×Q₈	—	4	1	no	—
359	C₂×Q₈	—	4	1	no	—
360	C₂×Q₈	C₃×C₅×D₄	1	4	yes	complemented
361	C₂×D₄	—	2	4	no	—
362	C₂×D₄	—	4	1	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	4	2	no	—
365	C₂²×C₄	—	4	2	no	—
366	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
367	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
368	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
369	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
370	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
371	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
372	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
373	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
374	C₄⋊C₄	—	2	4	no	—
375	C₄⋊C₄	—	2	4	no	—
376	C₄⋊C₄	—	2	4	no	—
377	C₄⋊C₄	—	4	1	no	—
378	C₄⋊C₄	—	4	1	no	—
379	C₂²⋊C₄	—	2	4	no	—
380	C₂²⋊C₄	—	2	4	no	—
381	C₂²⋊C₄	—	2	4	no	—
382	C₂²⋊C₄	—	2	4	no	—
383	C₂²⋊C₄	—	4	2	no	—
384	C₄²	—	4	1	no	—
385	C₃×C₅	#2728	1	1	yes	characteristic, direct product factor, Hall
386	C₂²×C₃	—	2	4	no	—
387	C₂²×C₃	—	2	4	no	—
388	C₂²×C₃	—	2	4	no	—
389	C₂²×C₃	—	4	2	no	—
390	C₂²×C₃	—	4	2	no	—
391	C₂²×C₃	—	4	2	no	—
392	C₂²×C₃	—	4	2	no	—
393	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
394	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
395	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
396	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
397	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
398	C₃×C₄	—	2	4	no	—
399	C₃×C₄	—	2	2	no	—
400	C₃×C₄	—	4	2	no	—
401	C₃×C₄	—	4	2	no	—
402	C₃×C₄	—	4	2	no	—
403	C₃×C₄	—	4	2	no	—
404	C₃×C₄	—	4	1	no	—
405	C₃×C₄	—	4	1	no	—
406	C₂×C₅	—	4	4	no	—
407	C₂×C₅	C₃×#534	1	1	yes	characteristic
408	C₂×C₅	C₃×#539	1	2	yes	—
409	C₂×C₅	C₃×#548	1	2	yes	—
410	C₂×C₅	C₃×#551	1	1	yes	characteristic
411	C₂×C₅	C₃×#563	1	1	yes	characteristic
412	C₂³	—	2	2	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	2	no	—
415	C₂³	—	2	2	no	—
416	C₂³	—	2	2	no	—
417	C₂³	—	2	2	no	—
418	C₂³	C₃×C₅×#41	1	4	yes	—
419	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
420	Q₈	—	4	4	no	—
421	D₄	—	4	4	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	4	8	no	—
443	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
444	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
445	C₂×C₃	—	4	4	no	—
446	C₂×C₃	C₅×#534	1	1	yes	characteristic
447	C₂×C₃	C₅×#539	1	2	yes	—
448	C₂×C₃	C₅×#548	1	2	yes	—
449	C₂×C₃	C₅×#551	1	1	yes	characteristic
450	C₂×C₃	C₅×#563	1	1	yes	characteristic
451	C₅	C₃×#2728	1	1	yes	characteristic, direct product factor, Sylow
452	C₂²	—	2	4	no	—
453	C₂²	—	2	4	no	—
454	C₂²	—	2	4	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	4	2	no	—
459	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
460	C₂²	C₂×C₃×C₅×#41	1	2	yes	—
461	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
462	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
463	C₂²	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
464	C₄	—	2	4	no	—
465	C₄	—	2	2	no	—
466	C₄	—	4	2	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	2	no	—
469	C₄	—	4	2	no	—
470	C₄	—	4	1	no	—
471	C₄	—	4	1	no	—
472	C₃	C₅×#2728	1	1	yes	characteristic, direct product factor, Sylow
473	C₂	—	4	4	no	—
474	C₂	C₃×C₅×#534	1	1	yes	characteristic
475	C₂	C₃×C₅×#539	1	2	yes	—
476	C₂	C₃×C₅×#548	1	2	yes	—
477	C₂	C₃×C₅×#551	1	1	yes	characteristic
478	C₂	C₃×C₅×#563	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×#400	C₂	semidirect
3. C₃×C₅×#398	C₂	semidirect
4. C₃×C₅×#397	C₂	semidirect
6. C₃×C₅×#391	C₂	semidirect
7. C₃×C₅×#388	C₂	semidirect
9. C₃×C₅×#385	C₂	semidirect
10. C₅×#2728	C₃	direct (N × H)
33. C₃×#2728	C₅	direct (N × H)
34. C₂×C₅×C₂²⋊Q₈	C₂×C₃	semidirect
35. C₅×#400	C₂×C₃	semidirect
36. C₅×#398	C₂×C₃	semidirect
37. C₅×#397	C₂×C₃	semidirect
39. C₅×#391	C₂×C₃	semidirect
40. C₅×#388	C₂×C₃	semidirect
42. C₅×#385	C₂×C₃	semidirect
46. C₂×C₃×C₅×Q₈	D₄	semidirect
71. C₂×C₃×C₂²⋊Q₈	C₂×C₅	semidirect
72. C₃×#400	C₂×C₅	semidirect
73. C₃×#398	C₂×C₅	semidirect
74. C₃×#397	C₂×C₅	semidirect
76. C₃×#391	C₂×C₅	semidirect
77. C₃×#388	C₂×C₅	semidirect
79. C₃×#385	C₂×C₅	semidirect
102. #2728	C₃×C₅	direct (N × H)
161. C₂×C₅×Q₈	C₃×D₄	semidirect
186. C₂×C₂²⋊Q₈	C₂×C₃×C₅	semidirect
187. #400	C₂×C₃×C₅	semidirect
188. #398	C₂×C₃×C₅	semidirect
189. #397	C₂×C₃×C₅	semidirect
191. #391	C₂×C₃×C₅	semidirect
192. #388	C₂×C₃×C₅	semidirect
194. #385	C₂×C₃×C₅	semidirect
218. C₂×C₃×Q₈	C₅×D₄	semidirect
360. C₂×Q₈	C₃×C₅×D₄	semidirect
385. C₃×C₅	#2728	direct (N × H)
451. C₅	C₃×#2728	direct (N × H)
472. C₃	C₅×#2728	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×#400	2
C₃×C₅×#398	2
C₃×C₅×#397	2
C₃×C₅×#394	2
C₃×C₅×#391	2
C₃×C₅×#388	2
C₃×C₅×#386	2
C₃×C₅×#385	2
C₅×#2728	3
C₃×#2728	5

Subgroups — C3×C5×#2728 #14275762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2728 #14275762