Subgroups — C₃×C₅×#3238 #14276272

Order 1920.

724 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#529	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#501	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#489	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#483	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#480	C₂	1	2	yes	maximal
7	C₃×C₅×#477	C₂	1	2	yes	maximal, complemented
8	C₃×C₅×#476	C₂	1	2	yes	maximal, complemented
9	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#3238	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₃×C₅×Q₈	—	2	1	no	—
12	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
13	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
14	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₃×C₅×#126	—	2	1	no	—
17	C₃×C₅×#125	C₂²	1	1	yes	characteristic
18	C₃×C₅×#124	C₂²	1	1	yes	characteristic
19	C₃×C₅×#123	—	2	1	no	—
20	C₃×C₅×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₃×C₅×#121	C₂²	1	1	yes	characteristic
22	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
23	C₃×C₄×C₅×D₄	—	2	1	no	—
24	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
25	C₃×C₅×#117	—	2	1	no	—
26	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
27	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
29	C₃×C₅×#107	C₂²	1	4	yes	—
30	C₃×C₅×#106	C₂²	1	4	yes	—
31	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
32	C₃×C₅×#103	C₂²	1	4	yes	—
33	C₃×C₅×#102	C₂²	1	4	yes	—
34	C₃×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₃×C₅×#97	C₂²	1	2	yes	—
36	C₃×#3238	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
38	C₅×#529	C₂×C₃	1	1	yes	characteristic
39	C₅×#501	C₂×C₃	1	2	yes	complemented
40	C₅×#489	C₂×C₃	1	2	yes	complemented
41	C₅×#483	C₂×C₃	1	2	yes	complemented
42	C₅×#480	C₂×C₃	1	2	yes	—
43	C₅×#477	C₂×C₃	1	2	yes	complemented
44	C₅×#476	C₂×C₃	1	2	yes	complemented
45	C₅×#432	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×C₃×C₅×Q₈	—	2	2	no	—
47	C₂×C₃×C₅×Q₈	—	2	2	no	—
48	C₂×C₃×C₅×Q₈	—	4	4	no	—
49	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₅×D₄	—	2	1	no	—
51	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₄×C₅	—	2	2	no	—
53	C₂²×C₃×C₄×C₅	—	2	1	no	—
54	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
55	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
56	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
58	C₃×C₅×C₄⋊C₄	—	2	2	no	—
59	C₃×C₅×C₄⋊C₄	—	2	2	no	—
60	C₃×C₅×C₄⋊C₄	—	2	2	no	—
61	C₃×C₅×C₄⋊C₄	—	2	1	no	—
62	C₃×C₅×C₄⋊C₄	—	2	1	no	—
63	C₃×C₅×C₄⋊C₄	—	2	1	no	—
64	C₃×C₅×C₄⋊C₄	—	2	1	no	—
65	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
66	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
69	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
70	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
71	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
72	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
73	C₃×C₄²×C₅	—	2	1	no	—
74	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
75	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
76	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
77	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
78	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
79	C₃×#529	C₂×C₅	1	1	yes	characteristic
80	C₃×#501	C₂×C₅	1	2	yes	complemented
81	C₃×#489	C₂×C₅	1	2	yes	complemented
82	C₃×#483	C₂×C₅	1	2	yes	complemented
83	C₃×#480	C₂×C₅	1	2	yes	—
84	C₃×#477	C₂×C₅	1	2	yes	complemented
85	C₃×#476	C₂×C₅	1	2	yes	complemented
86	C₃×#432	C₂×C₅	1	1	yes	characteristic, complemented
87	C₂²×C₅×Q₈	—	2	1	no	—
88	C₅×C₄⋊Q₈	—	2	1	no	—
89	C₅×C₄⋊Q₈	—	2	1	no	—
90	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
91	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
92	C₅×#126	—	2	1	no	—
93	C₅×#125	C₂²×C₃	1	1	yes	characteristic
94	C₅×#124	C₂²×C₃	1	1	yes	characteristic
95	C₅×#123	—	2	1	no	—
96	C₅×C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
97	C₅×#121	C₂²×C₃	1	1	yes	characteristic
98	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
99	C₄×C₅×D₄	—	2	1	no	—
100	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
101	C₅×#117	—	2	1	no	—
102	C₂×C₅×C₄⋊C₄	—	2	1	no	—
103	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
104	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
105	C₅×#107	C₂²×C₃	1	4	yes	—
106	C₅×#106	C₂²×C₃	1	4	yes	—
107	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
108	C₅×#103	C₂²×C₃	1	4	yes	—
109	C₅×#102	C₂²×C₃	1	4	yes	—
110	C₅×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
111	C₅×#97	C₂²×C₃	1	2	yes	—
112	#3238	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
113	C₂³×C₃×C₅	—	2	1	no	—
114	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
115	C₃×C₅×Q₈	—	2	2	no	—
116	C₃×C₅×Q₈	—	4	4	no	—
117	C₃×C₅×Q₈	—	4	4	no	—
118	C₃×C₅×D₄	—	2	2	no	—
119	C₃×C₅×D₄	—	4	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	1	no	—
124	C₂×C₃×C₄×C₅	—	2	1	no	—
125	C₂×C₃×C₄×C₅	—	2	1	no	—
126	C₂×C₃×C₄×C₅	—	2	4	no	—
127	C₂×C₃×C₄×C₅	—	4	4	no	—
128	C₂×C₃×C₄×C₅	—	4	2	no	—
129	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
130	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
131	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
132	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
135	C₃×C₅×C₈	—	2	4	no	—
136	C₂²×C₃×Q₈	—	2	1	no	—
137	C₃×C₄⋊Q₈	—	2	1	no	—
138	C₃×C₄⋊Q₈	—	2	1	no	—
139	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
140	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
141	C₃×#126	—	2	1	no	—
142	C₃×#125	C₂²×C₅	1	1	yes	characteristic
143	C₃×#124	C₂²×C₅	1	1	yes	characteristic
144	C₃×#123	—	2	1	no	—
145	C₃×C₂²⋊Q₈	C₂²×C₅	1	1	yes	characteristic
146	C₃×#121	C₂²×C₅	1	1	yes	characteristic
147	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
148	C₃×C₄×D₄	—	2	1	no	—
149	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
150	C₃×#117	—	2	1	no	—
151	C₂×C₃×C₄⋊C₄	—	2	1	no	—
152	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
153	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
154	C₃×#107	C₂²×C₅	1	4	yes	—
155	C₃×#106	C₂²×C₅	1	4	yes	—
156	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
157	C₃×#103	C₂²×C₅	1	4	yes	—
158	C₃×#102	C₂²×C₅	1	4	yes	—
159	C₃×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
160	C₃×#97	C₂²×C₅	1	2	yes	—
161	C₂×C₅×Q₈	—	2	2	no	—
162	C₂×C₅×Q₈	—	2	2	no	—
163	C₂×C₅×Q₈	—	4	4	no	—
164	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
165	C₂×C₅×D₄	—	2	1	no	—
166	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
167	C₂²×C₄×C₅	—	2	2	no	—
168	C₂²×C₄×C₅	—	2	1	no	—
169	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
172	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
173	C₅×C₄⋊C₄	—	2	2	no	—
174	C₅×C₄⋊C₄	—	2	2	no	—
175	C₅×C₄⋊C₄	—	2	2	no	—
176	C₅×C₄⋊C₄	—	2	1	no	—
177	C₅×C₄⋊C₄	—	2	1	no	—
178	C₅×C₄⋊C₄	—	2	1	no	—
179	C₅×C₄⋊C₄	—	2	1	no	—
180	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
181	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
182	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
183	C₅×C₂²⋊C₄	—	2	1	no	—
184	C₅×C₂²⋊C₄	—	2	1	no	—
185	C₅×C₂²⋊C₄	—	2	1	no	—
186	C₅×C₂²⋊C₄	—	2	1	no	—
187	C₅×C₂²⋊C₄	—	2	1	no	—
188	C₄²×C₅	—	2	1	no	—
189	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
190	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
191	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
192	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
193	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
194	#529	C₂×C₃×C₅	1	1	yes	characteristic
195	#501	C₂×C₃×C₅	1	2	yes	complemented
196	#489	C₂×C₃×C₅	1	2	yes	complemented
197	#483	C₂×C₃×C₅	1	2	yes	complemented
198	#480	C₂×C₃×C₅	1	2	yes	—
199	#477	C₂×C₃×C₅	1	2	yes	complemented
200	#476	C₂×C₃×C₅	1	2	yes	complemented
201	#432	C₂×C₃×C₅	1	1	yes	characteristic, complemented
202	C₂²×C₃×C₅	—	2	1	no	—
203	C₂²×C₃×C₅	—	2	1	no	—
204	C₂²×C₃×C₅	—	4	1	no	—
205	C₂²×C₃×C₅	—	4	1	no	—
206	C₂²×C₃×C₅	—	4	1	no	—
207	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
208	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
209	C₃×C₄×C₅	—	2	1	no	—
210	C₃×C₄×C₅	—	2	1	no	—
211	C₃×C₄×C₅	—	2	1	no	—
212	C₃×C₄×C₅	—	2	1	no	—
213	C₃×C₄×C₅	—	4	1	no	—
214	C₃×C₄×C₅	—	4	4	no	—
215	C₃×C₄×C₅	—	4	1	no	—
216	C₃×C₄×C₅	—	4	1	no	—
217	C₃×C₄×C₅	#137	1	2	yes	—
218	C₃×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
219	C₂×C₃×Q₈	—	2	2	no	—
220	C₂×C₃×Q₈	—	2	2	no	—
221	C₂×C₃×Q₈	—	4	4	no	—
222	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
223	C₂×C₃×D₄	—	2	1	no	—
224	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
225	C₂²×C₃×C₄	—	2	2	no	—
226	C₂²×C₃×C₄	—	2	1	no	—
227	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
228	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
229	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
230	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
231	C₃×C₄⋊C₄	—	2	2	no	—
232	C₃×C₄⋊C₄	—	2	2	no	—
233	C₃×C₄⋊C₄	—	2	2	no	—
234	C₃×C₄⋊C₄	—	2	1	no	—
235	C₃×C₄⋊C₄	—	2	1	no	—
236	C₃×C₄⋊C₄	—	2	1	no	—
237	C₃×C₄⋊C₄	—	2	1	no	—
238	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
239	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
240	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
241	C₃×C₂²⋊C₄	—	2	1	no	—
242	C₃×C₂²⋊C₄	—	2	1	no	—
243	C₃×C₂²⋊C₄	—	2	1	no	—
244	C₃×C₂²⋊C₄	—	2	1	no	—
245	C₃×C₂²⋊C₄	—	2	1	no	—
246	C₃×C₄²	—	2	1	no	—
247	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
248	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
249	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
250	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
251	C₂³×C₅	—	2	1	no	—
252	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
253	C₅×Q₈	—	2	2	no	—
254	C₅×Q₈	—	4	4	no	—
255	C₅×Q₈	—	4	4	no	—
256	C₅×D₄	—	2	2	no	—
257	C₅×D₄	—	4	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	1	no	—
262	C₂×C₄×C₅	—	2	1	no	—
263	C₂×C₄×C₅	—	2	1	no	—
264	C₂×C₄×C₅	—	2	4	no	—
265	C₂×C₄×C₅	—	4	4	no	—
266	C₂×C₄×C₅	—	4	2	no	—
267	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
268	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
269	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
270	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
271	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
272	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
273	C₅×C₈	—	2	4	no	—
274	C₂²×Q₈	—	2	1	no	—
275	C₄⋊Q₈	—	2	1	no	—
276	C₄⋊Q₈	—	2	1	no	—
277	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
278	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
279	#126	—	2	1	no	—
280	#125	C₂²×C₃×C₅	1	1	yes	characteristic
281	#124	C₂²×C₃×C₅	1	1	yes	characteristic
282	#123	—	2	1	no	—
283	C₂²⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
284	#121	C₂²×C₃×C₅	1	1	yes	characteristic
285	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
286	C₄×D₄	—	2	1	no	—
287	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
288	#117	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
291	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
292	#107	C₂²×C₃×C₅	1	4	yes	—
293	#106	C₂²×C₃×C₅	1	4	yes	—
294	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
295	#103	C₂²×C₃×C₅	1	4	yes	—
296	#102	C₂²×C₃×C₅	1	4	yes	—
297	C₂²⋊C₈	C₂²×C₃×C₅	1	2	yes	—
298	#97	C₂²×C₃×C₅	1	2	yes	—
299	C₂×C₃×C₅	—	2	2	no	—
300	C₂×C₃×C₅	—	8	1	no	—
301	C₂×C₃×C₅	#536	1	1	yes	characteristic
302	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
303	C₂×C₃×C₅	C₂×#137	1	1	yes	characteristic
304	C₂³×C₃	—	2	1	no	—
305	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
306	C₃×Q₈	—	2	2	no	—
307	C₃×Q₈	—	4	4	no	—
308	C₃×Q₈	—	4	4	no	—
309	C₃×D₄	—	2	2	no	—
310	C₃×D₄	—	4	2	no	—
311	C₂×C₃×C₄	—	2	2	no	—
312	C₂×C₃×C₄	—	2	2	no	—
313	C₂×C₃×C₄	—	2	2	no	—
314	C₂×C₃×C₄	—	2	1	no	—
315	C₂×C₃×C₄	—	2	1	no	—
316	C₂×C₃×C₄	—	2	1	no	—
317	C₂×C₃×C₄	—	2	4	no	—
318	C₂×C₃×C₄	—	4	4	no	—
319	C₂×C₃×C₄	—	4	2	no	—
320	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
321	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
322	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
323	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
324	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
325	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
326	C₃×C₈	—	2	4	no	—
327	C₂²×C₅	—	2	1	no	—
328	C₂²×C₅	—	2	1	no	—
329	C₂²×C₅	—	4	1	no	—
330	C₂²×C₅	—	4	1	no	—
331	C₂²×C₅	—	4	1	no	—
332	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
333	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
334	C₄×C₅	—	2	1	no	—
335	C₄×C₅	—	2	1	no	—
336	C₄×C₅	—	2	1	no	—
337	C₄×C₅	—	2	1	no	—
338	C₄×C₅	—	4	4	no	—
339	C₄×C₅	—	4	1	no	—
340	C₄×C₅	—	4	1	no	—
341	C₄×C₅	—	4	1	no	—
342	C₄×C₅	C₃×#137	1	2	yes	—
343	C₄×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	4	4	no	—
347	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
353	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
354	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
355	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	1	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
364	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
365	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
366	C₂²⋊C₄	—	2	1	no	—
367	C₂²⋊C₄	—	2	1	no	—
368	C₂²⋊C₄	—	2	1	no	—
369	C₂²⋊C₄	—	2	1	no	—
370	C₂²⋊C₄	—	2	1	no	—
371	C₄²	—	2	1	no	—
372	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
373	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
374	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
375	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
376	C₃×C₅	#3238	1	1	yes	characteristic, direct product factor, Hall
377	C₂²×C₃	—	2	1	no	—
378	C₂²×C₃	—	2	1	no	—
379	C₂²×C₃	—	4	1	no	—
380	C₂²×C₃	—	4	1	no	—
381	C₂²×C₃	—	4	1	no	—
382	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
383	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
384	C₃×C₄	—	2	1	no	—
385	C₃×C₄	—	2	1	no	—
386	C₃×C₄	—	2	1	no	—
387	C₃×C₄	—	2	1	no	—
388	C₃×C₄	—	4	1	no	—
389	C₃×C₄	—	4	4	no	—
390	C₃×C₄	—	4	1	no	—
391	C₃×C₄	—	4	1	no	—
392	C₃×C₄	C₅×#137	1	2	yes	—
393	C₃×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
394	C₂×C₅	—	2	2	no	—
395	C₂×C₅	—	8	1	no	—
396	C₂×C₅	C₃×#536	1	1	yes	characteristic
397	C₂×C₅	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
398	C₂×C₅	C₂×C₃×#137	1	1	yes	characteristic
399	C₂³	—	2	1	no	—
400	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
401	Q₈	—	2	2	no	—
402	Q₈	—	4	4	no	—
403	Q₈	—	4	4	no	—
404	D₄	—	2	2	no	—
405	D₄	—	4	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	4	4	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
416	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
417	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
418	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
419	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
420	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
421	C₈	—	2	4	no	—
422	C₂×C₃	—	2	2	no	—
423	C₂×C₃	—	8	1	no	—
424	C₂×C₃	C₅×#536	1	1	yes	characteristic
425	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
426	C₂×C₃	C₂×C₅×#137	1	1	yes	characteristic
427	C₅	C₃×#3238	1	1	yes	characteristic, direct product factor, Sylow
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	4	1	no	—
431	C₂²	—	4	1	no	—
432	C₂²	—	4	1	no	—
433	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
434	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
435	C₄	—	2	1	no	—
436	C₄	—	2	1	no	—
437	C₄	—	2	1	no	—
438	C₄	—	2	1	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	4	no	—
441	C₄	—	4	1	no	—
442	C₄	—	4	1	no	—
443	C₄	C₃×C₅×#137	1	2	yes	—
444	C₄	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
445	C₃	C₅×#3238	1	1	yes	characteristic, direct product factor, Sylow
446	C₂	—	2	2	no	—
447	C₂	—	8	1	no	—
448	C₂	C₃×C₅×#536	1	1	yes	characteristic
449	C₂	C₂×C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
450	C₂	C₂×C₃×C₅×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
prime radical	C₃×C₅×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₄⋊Q₈	C₂	semidirect
3. C₃×C₅×#501	C₂	semidirect
4. C₃×C₅×#489	C₂	semidirect
5. C₃×C₅×#483	C₂	semidirect
7. C₃×C₅×#477	C₂	semidirect
8. C₃×C₅×#476	C₂	semidirect
9. C₃×C₅×#432	C₂	semidirect
10. C₅×#3238	C₃	direct (N × H)
36. C₃×#3238	C₅	direct (N × H)
37. C₂×C₅×C₄⋊Q₈	C₂×C₃	semidirect
39. C₅×#501	C₂×C₃	semidirect
40. C₅×#489	C₂×C₃	semidirect
41. C₅×#483	C₂×C₃	semidirect
43. C₅×#477	C₂×C₃	semidirect
44. C₅×#476	C₂×C₃	semidirect
45. C₅×#432	C₂×C₃	semidirect
78. C₂×C₃×C₄⋊Q₈	C₂×C₅	semidirect
80. C₃×#501	C₂×C₅	semidirect
81. C₃×#489	C₂×C₅	semidirect
82. C₃×#483	C₂×C₅	semidirect
84. C₃×#477	C₂×C₅	semidirect
85. C₃×#476	C₂×C₅	semidirect
86. C₃×#432	C₂×C₅	semidirect
112. #3238	C₃×C₅	direct (N × H)
193. C₂×C₄⋊Q₈	C₂×C₃×C₅	semidirect
195. #501	C₂×C₃×C₅	semidirect
196. #489	C₂×C₃×C₅	semidirect
197. #483	C₂×C₃×C₅	semidirect
199. #477	C₂×C₃×C₅	semidirect
200. #476	C₂×C₃×C₅	semidirect
201. #432	C₂×C₃×C₅	semidirect
376. C₃×C₅	#3238	direct (N × H)
427. C₅	C₃×#3238	direct (N × H)
445. C₃	C₅×#3238	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×C₄⋊Q₈	2
C₃×C₅×#529	2
C₃×C₅×#501	2
C₃×C₅×#489	2
C₃×C₅×#483	2
C₃×C₅×#480	2
C₃×C₅×#477	2
C₃×C₅×#476	2
C₃×C₅×#432	2
C₅×#3238	3
C₃×#3238	5

Subgroups — C3×C5×#3238 #14276272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3238 #14276272