Subgroups — C₃×C₅×#3278 #14276312

Order 1920.

964 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#137	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#550	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#468	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#452	C₂	1	2	yes	maximal, complemented
9	C₃×C₅×#448	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₅×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#3278	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
14	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
15	C₃×C₅×#137	—	2	8	no	—
16	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
17	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₅×SD₁₆	C₂²	1	2	yes	—
19	C₂×C₃×C₅×SD₁₆	C₂²	1	2	yes	—
20	C₂×C₃×C₅×SD₁₆	C₂²	1	2	yes	—
21	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
22	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
23	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₃×C₅×#127	—	2	1	no	—
25	C₃×C₅×#127	C₂²	1	1	yes	characteristic
26	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
27	C₃×C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₃×C₅×#121	—	2	2	no	—
29	C₃×C₅×#121	C₂²	1	2	yes	—
30	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
31	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
32	C₃×C₄×C₅×D₄	—	2	1	no	—
33	C₃×C₄×C₅×D₄	—	2	1	no	—
34	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
35	C₃×C₅×#107	C₂²	1	1	yes	characteristic
36	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃×C₅×#103	C₂²	1	1	yes	characteristic
38	C₃×C₅×#103	C₂²	1	2	yes	—
39	C₃×C₅×#103	C₂²	1	1	yes	characteristic
40	C₃×C₅×#102	C₂²	1	2	yes	—
41	C₃×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
42	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
43	C₃×#3278	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
44	C₂×C₅×#137	C₂×C₃	1	2	yes	complemented
45	C₅×#550	C₂×C₃	1	1	yes	characteristic
46	C₅×D₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
47	C₅×#492	C₂×C₃	1	1	yes	characteristic, complemented
48	C₅×#468	C₂×C₃	1	2	yes	complemented
49	C₅×#462	C₂×C₃	1	1	yes	characteristic, complemented
50	C₅×#460	C₂×C₃	1	1	yes	characteristic, complemented
51	C₅×#452	C₂×C₃	1	2	yes	complemented
52	C₅×#448	C₂×C₃	1	2	yes	complemented
53	C₄×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
54	C₅×#436	C₂×C₃	1	1	yes	characteristic, complemented
55	C₃×C₅×#41	—	2	4	no	—
56	C₃×C₅×#41	—	4	2	no	—
57	C₂×C₃×C₅×Q₈	—	2	2	no	—
58	C₂×C₃×C₅×Q₈	—	2	4	no	—
59	C₂×C₃×C₅×Q₈	—	4	2	no	—
60	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
61	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₅×D₄	—	2	2	no	—
64	C₂×C₃×C₅×D₄	—	2	2	no	—
65	C₂×C₃×C₅×D₄	—	2	1	no	—
66	C₂×C₃×C₅×D₄	—	2	1	no	—
67	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
68	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
69	C₂²×C₃×C₄×C₅	—	2	2	no	—
70	C₂²×C₃×C₄×C₅	—	2	2	no	—
71	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
72	C₃×C₅×Q₁₆	—	2	4	no	—
73	C₃×C₅×Q₁₆	D₄	1	4	yes	complemented
74	C₃×C₅×SD₁₆	—	2	4	no	—
75	C₃×C₅×SD₁₆	—	2	4	no	—
76	C₃×C₅×SD₁₆	—	4	2	no	—
77	C₃×C₅×M₁₆	—	2	4	no	—
78	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
79	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
80	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
81	C₃×C₅×C₄⋊C₄	—	2	1	no	—
82	C₃×C₅×C₄⋊C₄	—	2	1	no	—
83	C₃×C₅×C₄⋊C₄	—	2	1	no	—
84	C₃×C₅×C₄⋊C₄	—	2	1	no	—
85	C₃×C₅×C₄⋊C₄	—	2	2	no	—
86	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
87	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
90	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
91	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
92	C₃×C₄²×C₅	—	2	1	no	—
93	C₃×C₄²×C₅	—	2	1	no	—
94	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
95	C₂×C₃×#137	C₂×C₅	1	2	yes	complemented
96	C₃×#550	C₂×C₅	1	1	yes	characteristic
97	C₃×D₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
98	C₃×#492	C₂×C₅	1	1	yes	characteristic, complemented
99	C₃×#468	C₂×C₅	1	2	yes	complemented
100	C₃×#462	C₂×C₅	1	1	yes	characteristic, complemented
101	C₃×#460	C₂×C₅	1	1	yes	characteristic, complemented
102	C₃×#452	C₂×C₅	1	2	yes	complemented
103	C₃×#448	C₂×C₅	1	2	yes	complemented
104	C₃×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
105	C₃×#436	C₂×C₅	1	1	yes	characteristic, complemented
106	C₂×C₅×#41	C₂²×C₃	1	2	yes	—
107	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
108	C₅×#137	—	2	8	no	—
109	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
110	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
111	C₂×C₅×SD₁₆	C₂²×C₃	1	2	yes	—
112	C₂×C₅×SD₁₆	C₂²×C₃	1	2	yes	—
113	C₂×C₅×SD₁₆	C₂²×C₃	1	2	yes	—
114	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
115	C₅×C₄⋊Q₈	—	2	1	no	—
116	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
117	C₅×#127	—	2	1	no	—
118	C₅×#127	C₂²×C₃	1	1	yes	characteristic
119	C₅×C₂²⋊Q₈	—	2	2	no	—
120	C₅×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
121	C₅×#121	—	2	2	no	—
122	C₅×#121	C₂²×C₃	1	2	yes	—
123	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
124	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
125	C₄×C₅×D₄	—	2	1	no	—
126	C₄×C₅×D₄	—	2	1	no	—
127	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
128	C₅×#107	C₂²×C₃	1	1	yes	characteristic
129	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
130	C₅×#103	C₂²×C₃	1	1	yes	characteristic
131	C₅×#103	C₂²×C₃	1	1	yes	characteristic
132	C₅×#103	C₂²×C₃	1	2	yes	—
133	C₅×#102	C₂²×C₃	1	2	yes	—
134	C₅×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
135	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
136	#3278	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
137	C₂³×C₃×C₅	—	2	2	no	—
138	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
139	C₃×C₅×Q₈	—	2	1	no	—
140	C₃×C₅×Q₈	—	2	1	no	—
141	C₃×C₅×Q₈	—	2	2	no	—
142	C₃×C₅×Q₈	—	2	2	no	—
143	C₃×C₅×Q₈	—	4	4	no	—
144	C₃×C₅×Q₈	C₂×D₄	1	2	yes	—
145	C₃×C₅×Q₈	C₂×D₄	1	2	yes	—
146	C₃×C₅×D₄	—	2	2	no	—
147	C₃×C₅×D₄	—	2	2	no	—
148	C₃×C₅×D₄	—	2	2	no	—
149	C₃×C₅×D₄	—	4	4	no	—
150	C₃×C₅×D₄	—	4	2	no	—
151	C₃×C₅×D₄	—	8	1	no	—
152	C₃×C₅×D₄	—	8	2	no	—
153	C₂×C₃×C₄×C₅	—	2	1	no	—
154	C₂×C₃×C₄×C₅	—	2	1	no	—
155	C₂×C₃×C₄×C₅	—	2	2	no	—
156	C₂×C₃×C₄×C₅	—	2	2	no	—
157	C₂×C₃×C₄×C₅	—	2	1	no	—
158	C₂×C₃×C₄×C₅	—	2	1	no	—
159	C₂×C₃×C₄×C₅	—	2	2	no	—
160	C₂×C₃×C₄×C₅	—	4	4	no	—
161	C₂×C₃×C₄×C₅	—	4	4	no	—
162	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
164	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
166	C₃×C₅×C₈	—	2	2	no	—
167	C₃×C₅×C₈	—	2	1	no	—
168	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
169	C₂×C₃×#41	C₂²×C₅	1	2	yes	—
170	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
171	C₃×#137	—	2	8	no	—
172	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
173	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
174	C₂×C₃×SD₁₆	C₂²×C₅	1	2	yes	—
175	C₂×C₃×SD₁₆	C₂²×C₅	1	2	yes	—
176	C₂×C₃×SD₁₆	C₂²×C₅	1	2	yes	—
177	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
178	C₃×C₄⋊Q₈	—	2	1	no	—
179	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
180	C₃×#127	—	2	1	no	—
181	C₃×#127	C₂²×C₅	1	1	yes	characteristic
182	C₃×C₂²⋊Q₈	—	2	2	no	—
183	C₃×C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
184	C₃×#121	—	2	2	no	—
185	C₃×#121	C₂²×C₅	1	2	yes	—
186	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
187	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
188	C₃×C₄×D₄	—	2	1	no	—
189	C₃×C₄×D₄	—	2	1	no	—
190	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
191	C₃×#107	C₂²×C₅	1	1	yes	characteristic
192	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
193	C₃×#103	C₂²×C₅	1	1	yes	characteristic
194	C₃×#103	C₂²×C₅	1	1	yes	characteristic
195	C₃×#103	C₂²×C₅	1	2	yes	—
196	C₃×#102	C₂²×C₅	1	2	yes	—
197	C₃×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
198	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
199	C₅×#41	—	2	4	no	—
200	C₅×#41	—	4	2	no	—
201	C₂×C₅×Q₈	—	2	2	no	—
202	C₂×C₅×Q₈	—	2	4	no	—
203	C₂×C₅×Q₈	—	4	2	no	—
204	C₂×C₅×Q₈	C₂³×C₃	1	2	yes	—
205	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
206	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
207	C₂×C₅×D₄	—	2	2	no	—
208	C₂×C₅×D₄	—	2	2	no	—
209	C₂×C₅×D₄	—	2	1	no	—
210	C₂×C₅×D₄	—	2	1	no	—
211	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
212	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
213	C₂²×C₄×C₅	—	2	2	no	—
214	C₂²×C₄×C₅	—	2	2	no	—
215	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
216	C₅×Q₁₆	—	2	4	no	—
217	C₅×Q₁₆	C₃×D₄	1	4	yes	complemented
218	C₅×SD₁₆	—	2	4	no	—
219	C₅×SD₁₆	—	2	4	no	—
220	C₅×SD₁₆	—	4	2	no	—
221	C₅×M₁₆	—	2	4	no	—
222	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
223	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
224	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
225	C₅×C₄⋊C₄	—	2	1	no	—
226	C₅×C₄⋊C₄	—	2	1	no	—
227	C₅×C₄⋊C₄	—	2	1	no	—
228	C₅×C₄⋊C₄	—	2	1	no	—
229	C₅×C₄⋊C₄	—	2	2	no	—
230	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
231	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
232	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
233	C₅×C₂²⋊C₄	—	2	2	no	—
234	C₅×C₂²⋊C₄	—	2	2	no	—
235	C₅×C₂²⋊C₄	C₃×D₄	1	2	yes	—
236	C₄²×C₅	—	2	1	no	—
237	C₄²×C₅	—	2	1	no	—
238	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
239	C₂×#137	C₂×C₃×C₅	1	2	yes	complemented
240	#550	C₂×C₃×C₅	1	1	yes	characteristic
241	D₄×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
242	#492	C₂×C₃×C₅	1	1	yes	characteristic, complemented
243	#468	C₂×C₃×C₅	1	2	yes	complemented
244	#462	C₂×C₃×C₅	1	1	yes	characteristic, complemented
245	#460	C₂×C₃×C₅	1	1	yes	characteristic, complemented
246	#452	C₂×C₃×C₅	1	2	yes	complemented
247	#448	C₂×C₃×C₅	1	2	yes	complemented
248	C₄×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
249	#436	C₂×C₃×C₅	1	1	yes	characteristic, complemented
250	C₂²×C₃×C₅	—	2	2	no	—
251	C₂²×C₃×C₅	—	2	2	no	—
252	C₂²×C₃×C₅	—	2	2	no	—
253	C₂²×C₃×C₅	—	4	2	no	—
254	C₂²×C₃×C₅	—	4	2	no	—
255	C₂²×C₃×C₅	—	4	2	no	—
256	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
257	C₃×C₄×C₅	—	2	2	no	—
258	C₃×C₄×C₅	—	2	2	no	—
259	C₃×C₄×C₅	—	2	2	no	—
260	C₃×C₄×C₅	—	2	1	no	—
261	C₃×C₄×C₅	—	4	2	no	—
262	C₃×C₄×C₅	—	4	1	no	—
263	C₃×C₄×C₅	—	4	1	no	—
264	C₃×C₄×C₅	#137	1	2	yes	complemented
265	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
266	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
267	C₃×#41	—	2	4	no	—
268	C₃×#41	—	4	2	no	—
269	C₂×C₃×Q₈	—	2	2	no	—
270	C₂×C₃×Q₈	—	2	4	no	—
271	C₂×C₃×Q₈	—	4	2	no	—
272	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
273	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
274	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
275	C₂×C₃×D₄	—	2	2	no	—
276	C₂×C₃×D₄	—	2	2	no	—
277	C₂×C₃×D₄	—	2	1	no	—
278	C₂×C₃×D₄	—	2	1	no	—
279	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
280	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
281	C₂²×C₃×C₄	—	2	2	no	—
282	C₂²×C₃×C₄	—	2	2	no	—
283	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
284	C₃×Q₁₆	—	2	4	no	—
285	C₃×Q₁₆	C₅×D₄	1	4	yes	complemented
286	C₃×SD₁₆	—	2	4	no	—
287	C₃×SD₁₆	—	2	4	no	—
288	C₃×SD₁₆	—	4	2	no	—
289	C₃×M₁₆	—	2	4	no	—
290	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
291	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
292	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
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	—	2	1	no	—
297	C₃×C₄⋊C₄	—	2	2	no	—
298	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
299	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
300	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
301	C₃×C₂²⋊C₄	—	2	2	no	—
302	C₃×C₂²⋊C₄	—	2	2	no	—
303	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
304	C₃×C₄²	—	2	1	no	—
305	C₃×C₄²	—	2	1	no	—
306	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
307	C₂³×C₅	—	2	2	no	—
308	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
309	C₅×Q₈	—	2	1	no	—
310	C₅×Q₈	—	2	1	no	—
311	C₅×Q₈	—	2	2	no	—
312	C₅×Q₈	—	2	2	no	—
313	C₅×Q₈	—	4	4	no	—
314	C₅×Q₈	C₂×C₃×D₄	1	2	yes	—
315	C₅×Q₈	C₂×C₃×D₄	1	2	yes	—
316	C₅×D₄	—	2	2	no	—
317	C₅×D₄	—	2	2	no	—
318	C₅×D₄	—	2	2	no	—
319	C₅×D₄	—	4	4	no	—
320	C₅×D₄	—	4	2	no	—
321	C₅×D₄	—	8	1	no	—
322	C₅×D₄	—	8	2	no	—
323	C₂×C₄×C₅	—	2	1	no	—
324	C₂×C₄×C₅	—	2	1	no	—
325	C₂×C₄×C₅	—	2	2	no	—
326	C₂×C₄×C₅	—	2	2	no	—
327	C₂×C₄×C₅	—	2	1	no	—
328	C₂×C₄×C₅	—	2	1	no	—
329	C₂×C₄×C₅	—	2	2	no	—
330	C₂×C₄×C₅	—	4	4	no	—
331	C₂×C₄×C₅	—	4	4	no	—
332	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
333	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
334	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
335	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
336	C₅×C₈	—	2	2	no	—
337	C₅×C₈	—	2	1	no	—
338	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
339	C₂×#41	C₂²×C₃×C₅	1	2	yes	—
340	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
341	#137	—	2	8	no	—
342	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
343	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
344	C₂×SD₁₆	C₂²×C₃×C₅	1	2	yes	—
345	C₂×SD₁₆	C₂²×C₃×C₅	1	2	yes	—
346	C₂×SD₁₆	C₂²×C₃×C₅	1	2	yes	—
347	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
348	C₄⋊Q₈	—	2	1	no	—
349	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
350	#127	—	2	1	no	—
351	#127	C₂²×C₃×C₅	1	1	yes	characteristic
352	C₂²⋊Q₈	—	2	2	no	—
353	C₂²⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
354	#121	—	2	2	no	—
355	#121	C₂²×C₃×C₅	1	2	yes	—
356	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
357	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
358	C₄×D₄	—	2	1	no	—
359	C₄×D₄	—	2	1	no	—
360	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
361	#107	C₂²×C₃×C₅	1	1	yes	characteristic
362	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
363	#103	C₂²×C₃×C₅	1	1	yes	characteristic
364	#103	C₂²×C₃×C₅	1	1	yes	characteristic
365	#103	C₂²×C₃×C₅	1	2	yes	—
366	#102	C₂²×C₃×C₅	1	2	yes	—
367	C₂²⋊C₈	C₂²×C₃×C₅	1	2	yes	—
368	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
369	C₂×C₃×C₅	—	4	2	no	—
370	C₂×C₃×C₅	—	8	2	no	—
371	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
372	C₂×C₃×C₅	C₂×#137	1	1	yes	characteristic
373	C₂×C₃×C₅	#577	1	1	yes	characteristic
374	C₂³×C₃	—	2	2	no	—
375	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
376	C₃×Q₈	—	2	1	no	—
377	C₃×Q₈	—	2	1	no	—
378	C₃×Q₈	—	2	2	no	—
379	C₃×Q₈	—	2	2	no	—
380	C₃×Q₈	—	4	4	no	—
381	C₃×Q₈	C₂×C₅×D₄	1	2	yes	—
382	C₃×Q₈	C₂×C₅×D₄	1	2	yes	—
383	C₃×D₄	—	2	2	no	—
384	C₃×D₄	—	2	2	no	—
385	C₃×D₄	—	2	2	no	—
386	C₃×D₄	—	4	4	no	—
387	C₃×D₄	—	4	2	no	—
388	C₃×D₄	—	8	1	no	—
389	C₃×D₄	—	8	2	no	—
390	C₂×C₃×C₄	—	2	1	no	—
391	C₂×C₃×C₄	—	2	1	no	—
392	C₂×C₃×C₄	—	2	2	no	—
393	C₂×C₃×C₄	—	2	2	no	—
394	C₂×C₃×C₄	—	2	1	no	—
395	C₂×C₃×C₄	—	2	1	no	—
396	C₂×C₃×C₄	—	2	2	no	—
397	C₂×C₃×C₄	—	4	4	no	—
398	C₂×C₃×C₄	—	4	4	no	—
399	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
400	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
401	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
402	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
403	C₃×C₈	—	2	2	no	—
404	C₃×C₈	—	2	1	no	—
405	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
406	C₂²×C₅	—	2	2	no	—
407	C₂²×C₅	—	2	2	no	—
408	C₂²×C₅	—	2	2	no	—
409	C₂²×C₅	—	4	2	no	—
410	C₂²×C₅	—	4	2	no	—
411	C₂²×C₅	—	4	2	no	—
412	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
413	C₄×C₅	—	2	1	no	—
414	C₄×C₅	—	2	2	no	—
415	C₄×C₅	—	2	2	no	—
416	C₄×C₅	—	2	2	no	—
417	C₄×C₅	—	4	1	no	—
418	C₄×C₅	—	4	1	no	—
419	C₄×C₅	—	4	2	no	—
420	C₄×C₅	C₃×#137	1	2	yes	complemented
421	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
422	C₄×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
423	#41	—	2	4	no	—
424	#41	—	4	2	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	4	no	—
427	C₂×Q₈	—	4	2	no	—
428	C₂×Q₈	C₂³×C₃×C₅	1	2	yes	—
429	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
430	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
436	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
440	Q₁₆	—	2	4	no	—
441	Q₁₆	C₃×C₅×D₄	1	4	yes	complemented
442	SD₁₆	—	2	4	no	—
443	SD₁₆	—	2	4	no	—
444	SD₁₆	—	4	2	no	—
445	M₁₆	—	2	4	no	—
446	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
447	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
448	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
455	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
456	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
457	C₂²⋊C₄	—	2	2	no	—
458	C₂²⋊C₄	—	2	2	no	—
459	C₂²⋊C₄	C₃×C₅×D₄	1	2	yes	—
460	C₄²	—	2	1	no	—
461	C₄²	—	2	1	no	—
462	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
463	C₃×C₅	#3278	1	1	yes	characteristic, direct product factor, Hall
464	C₂²×C₃	—	2	2	no	—
465	C₂²×C₃	—	2	2	no	—
466	C₂²×C₃	—	2	2	no	—
467	C₂²×C₃	—	4	2	no	—
468	C₂²×C₃	—	4	2	no	—
469	C₂²×C₃	—	4	2	no	—
470	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
471	C₃×C₄	—	2	2	no	—
472	C₃×C₄	—	2	2	no	—
473	C₃×C₄	—	2	2	no	—
474	C₃×C₄	—	2	1	no	—
475	C₃×C₄	—	4	2	no	—
476	C₃×C₄	—	4	1	no	—
477	C₃×C₄	—	4	1	no	—
478	C₃×C₄	C₅×#137	1	2	yes	complemented
479	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
480	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
481	C₂×C₅	—	4	2	no	—
482	C₂×C₅	—	8	2	no	—
483	C₂×C₅	C₃×D₄²	1	1	yes	characteristic
484	C₂×C₅	C₂×C₃×#137	1	1	yes	characteristic
485	C₂×C₅	C₃×#577	1	1	yes	characteristic
486	C₂³	—	2	2	no	—
487	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
488	Q₈	—	2	1	no	—
489	Q₈	—	2	1	no	—
490	Q₈	—	2	2	no	—
491	Q₈	—	2	2	no	—
492	Q₈	—	4	4	no	—
493	Q₈	C₂×C₃×C₅×D₄	1	2	yes	—
494	Q₈	C₂×C₃×C₅×D₄	1	2	yes	—
495	D₄	—	2	2	no	—
496	D₄	—	2	2	no	—
497	D₄	—	2	2	no	—
498	D₄	—	4	4	no	—
499	D₄	—	4	2	no	—
500	D₄	—	8	1	no	—
501	D₄	—	8	2	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	4	4	no	—
510	C₂×C₄	—	4	4	no	—
511	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
512	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
513	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
514	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
515	C₈	—	2	2	no	—
516	C₈	—	2	1	no	—
517	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
518	C₂×C₃	—	4	2	no	—
519	C₂×C₃	—	8	2	no	—
520	C₂×C₃	C₅×D₄²	1	1	yes	characteristic
521	C₂×C₃	C₂×C₅×#137	1	1	yes	characteristic
522	C₂×C₃	C₅×#577	1	1	yes	characteristic
523	C₅	C₃×#3278	1	1	yes	characteristic, direct product factor, Sylow
524	C₂²	—	2	2	no	—
525	C₂²	—	2	2	no	—
526	C₂²	—	2	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
531	C₄	—	2	2	no	—
532	C₄	—	2	2	no	—
533	C₄	—	2	2	no	—
534	C₄	—	2	1	no	—
535	C₄	—	4	2	no	—
536	C₄	—	4	1	no	—
537	C₄	—	4	1	no	—
538	C₄	C₃×C₅×#137	1	2	yes	complemented
539	C₄	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
540	C₄	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
541	C₃	C₅×#3278	1	1	yes	characteristic, direct product factor, Sylow
542	C₂	—	4	2	no	—
543	C₂	—	8	2	no	—
544	C₂	C₃×C₅×D₄²	1	1	yes	characteristic
545	C₂	C₂×C₃×C₅×#137	1	1	yes	characteristic
546	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	C₃×C₅×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄×C₅×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#137	C₂	semidirect
3. C₃×C₅×D₄×Q₈	C₂	semidirect
4. C₃×C₅×#492	C₂	semidirect
5. C₃×C₅×#468	C₂	semidirect
6. C₃×C₅×#462	C₂	semidirect
7. C₃×C₅×#460	C₂	semidirect
8. C₃×C₅×#452	C₂	semidirect
9. C₃×C₅×#448	C₂	semidirect
10. C₃×C₄×C₅×Q₁₆	C₂	semidirect
11. C₃×C₅×#436	C₂	semidirect
12. C₅×#3278	C₃	direct (N × H)
43. C₃×#3278	C₅	direct (N × H)
44. C₂×C₅×#137	C₂×C₃	semidirect
46. C₅×D₄×Q₈	C₂×C₃	semidirect
47. C₅×#492	C₂×C₃	semidirect
48. C₅×#468	C₂×C₃	semidirect
49. C₅×#462	C₂×C₃	semidirect
50. C₅×#460	C₂×C₃	semidirect
51. C₅×#452	C₂×C₃	semidirect
52. C₅×#448	C₂×C₃	semidirect
53. C₄×C₅×Q₁₆	C₂×C₃	semidirect
54. C₅×#436	C₂×C₃	semidirect
73. C₃×C₅×Q₁₆	D₄	semidirect
95. C₂×C₃×#137	C₂×C₅	semidirect
97. C₃×D₄×Q₈	C₂×C₅	semidirect
98. C₃×#492	C₂×C₅	semidirect
99. C₃×#468	C₂×C₅	semidirect
100. C₃×#462	C₂×C₅	semidirect
101. C₃×#460	C₂×C₅	semidirect
102. C₃×#452	C₂×C₅	semidirect
103. C₃×#448	C₂×C₅	semidirect
104. C₃×C₄×Q₁₆	C₂×C₅	semidirect
105. C₃×#436	C₂×C₅	semidirect
136. #3278	C₃×C₅	direct (N × H)
217. C₅×Q₁₆	C₃×D₄	semidirect
239. C₂×#137	C₂×C₃×C₅	semidirect
241. D₄×Q₈	C₂×C₃×C₅	semidirect
242. #492	C₂×C₃×C₅	semidirect
243. #468	C₂×C₃×C₅	semidirect
244. #462	C₂×C₃×C₅	semidirect
245. #460	C₂×C₃×C₅	semidirect
246. #452	C₂×C₃×C₅	semidirect
247. #448	C₂×C₃×C₅	semidirect
248. C₄×Q₁₆	C₂×C₃×C₅	semidirect
249. #436	C₂×C₃×C₅	semidirect
264. C₃×C₄×C₅	#137	semidirect
285. C₃×Q₁₆	C₅×D₄	semidirect
420. C₄×C₅	C₃×#137	semidirect
441. Q₁₆	C₃×C₅×D₄	semidirect
463. C₃×C₅	#3278	direct (N × H)
478. C₃×C₄	C₅×#137	semidirect
523. C₅	C₃×#3278	direct (N × H)
538. C₄	C₃×C₅×#137	semidirect
541. C₃	C₅×#3278	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×#137	2
C₃×C₅×#550	2
C₃×C₅×D₄×Q₈	2
C₃×C₅×#492	2
C₃×C₅×#468	2
C₃×C₅×#462	2
C₃×C₅×#460	2
C₃×C₅×#452	2
C₃×C₅×#448	2
C₃×C₄×C₅×Q₁₆	2
C₃×C₅×#436	2
C₅×#3278	3
C₃×#3278	5

Subgroups — C3×C5×#3278 #14276312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3278 #14276312