Subgroups — C₃×C₅×#3324 #14276358

Order 1920.

896 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×D₄²	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#482	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#468	C₂	1	2	yes	maximal, complemented
7	C₃×C₅×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#450	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×C₅×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₅×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₅×#418	C₂	1	2	yes	maximal, complemented
12	C₅×#3324	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₃×C₅×D₄	—	2	1	no	—
14	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
15	C₂×C₃×C₅×SD₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₅×SD₁₆	C₂²	1	2	yes	complemented
17	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	complemented
18	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₃×C₅×#127	C₂²	1	1	yes	characteristic
20	C₃×C₅×#126	—	2	2	no	—
21	C₃×C₅×#124	—	2	1	no	—
22	C₃×C₅×C₂²⋊Q₈	C₂²	1	2	yes	complemented
23	C₃×C₅×#121	—	2	1	no	—
24	C₃×C₅×#121	C₂²	1	2	yes	—
25	C₃×C₅×#120	—	2	2	no	—
26	C₃×C₄×C₅×Q₈	—	2	1	no	—
27	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
28	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
29	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
30	C₃×C₅×#117	C₂²	1	2	yes	complemented
31	C₃×C₅×#107	C₂²	1	2	yes	complemented
32	C₃×C₅×#106	C₂²	1	1	yes	characteristic, complemented
33	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
34	C₃×C₅×#103	C₂²	1	1	yes	characteristic, complemented
35	C₃×C₅×#102	C₂²	1	2	yes	—
36	C₃×C₅×#102	C₂²	1	1	yes	characteristic
37	C₃×C₅×#102	C₂²	1	2	yes	complemented
38	C₃×C₅×#102	C₂²	1	2	yes	complemented
39	C₃×C₅×#102	C₂²	1	2	yes	complemented
40	C₃×C₅×C₂²⋊C₈	C₂²	1	2	yes	complemented
41	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
42	C₃×#3324	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×#556	C₂×C₃	1	1	yes	characteristic, complemented
44	C₅×D₄²	C₂×C₃	1	1	yes	characteristic
45	C₅×#486	C₂×C₃	1	1	yes	characteristic, complemented
46	C₅×#482	C₂×C₃	1	2	yes	complemented
47	C₅×#474	C₂×C₃	1	1	yes	characteristic, complemented
48	C₅×#468	C₂×C₃	1	2	yes	complemented
49	C₅×#460	C₂×C₃	1	1	yes	characteristic, complemented
50	C₅×#450	C₂×C₃	1	2	yes	complemented
51	C₄×C₅×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
52	C₅×#436	C₂×C₃	1	1	yes	characteristic, complemented
53	C₅×#418	C₂×C₃	1	2	yes	complemented
54	C₂⁴×C₃×C₅	—	2	2	no	—
55	C₂×C₃×C₅×Q₈	—	2	1	no	—
56	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₅×D₄	—	2	4	no	—
58	C₂×C₃×C₅×D₄	—	2	2	no	—
59	C₂×C₃×C₅×D₄	—	2	2	no	—
60	C₂×C₃×C₅×D₄	—	2	1	no	—
61	C₂×C₃×C₅×D₄	—	2	1	no	—
62	C₂×C₃×C₅×D₄	—	4	2	no	—
63	C₂×C₃×C₅×D₄	—	4	4	no	—
64	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
65	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
66	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄×C₅	—	2	1	no	—
68	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
69	C₃×C₅×SD₁₆	—	2	2	no	—
70	C₃×C₅×SD₁₆	—	4	2	no	—
71	C₃×C₅×M₁₆	—	4	2	no	—
72	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
74	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
75	C₃×C₅×C₄⋊C₄	—	2	2	no	—
76	C₃×C₅×C₄⋊C₄	—	2	1	no	—
77	C₃×C₅×C₄⋊C₄	—	2	1	no	—
78	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
79	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
81	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
82	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
83	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
84	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
85	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
86	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
87	C₃×C₄²×C₅	—	2	2	no	—
88	C₃×C₄²×C₅	—	2	1	no	—
89	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
90	C₃×#556	C₂×C₅	1	1	yes	characteristic, complemented
91	C₃×D₄²	C₂×C₅	1	1	yes	characteristic
92	C₃×#486	C₂×C₅	1	1	yes	characteristic, complemented
93	C₃×#482	C₂×C₅	1	2	yes	complemented
94	C₃×#474	C₂×C₅	1	1	yes	characteristic, complemented
95	C₃×#468	C₂×C₅	1	2	yes	complemented
96	C₃×#460	C₂×C₅	1	1	yes	characteristic, complemented
97	C₃×#450	C₂×C₅	1	2	yes	complemented
98	C₃×C₄×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
99	C₃×#436	C₂×C₅	1	1	yes	characteristic, complemented
100	C₃×#418	C₂×C₅	1	2	yes	complemented
101	C₂²×C₅×D₄	—	2	1	no	—
102	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
103	C₂×C₅×SD₁₆	C₂²×C₃	1	1	yes	characteristic
104	C₂×C₅×SD₁₆	C₂²×C₃	1	2	yes	complemented
105	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	complemented
106	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
107	C₅×#127	C₂²×C₃	1	1	yes	characteristic
108	C₅×#126	—	2	2	no	—
109	C₅×#124	—	2	1	no	—
110	C₅×C₂²⋊Q₈	C₂²×C₃	1	2	yes	complemented
111	C₅×#121	—	2	1	no	—
112	C₅×#121	C₂²×C₃	1	2	yes	—
113	C₅×#120	—	2	2	no	—
114	C₄×C₅×Q₈	—	2	1	no	—
115	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
116	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
117	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
118	C₅×#117	C₂²×C₃	1	2	yes	complemented
119	C₅×#107	C₂²×C₃	1	2	yes	complemented
120	C₅×#106	C₂²×C₃	1	1	yes	characteristic, complemented
121	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
122	C₅×#103	C₂²×C₃	1	1	yes	characteristic, complemented
123	C₅×#102	C₂²×C₃	1	1	yes	characteristic
124	C₅×#102	C₂²×C₃	1	2	yes	—
125	C₅×#102	C₂²×C₃	1	2	yes	complemented
126	C₅×#102	C₂²×C₃	1	2	yes	complemented
127	C₅×#102	C₂²×C₃	1	2	yes	complemented
128	C₅×C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
129	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
130	#3324	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³×C₃×C₅	—	2	1	no	—
132	C₂³×C₃×C₅	—	2	2	no	—
133	C₂³×C₃×C₅	—	4	4	no	—
134	C₂³×C₃×C₅	—	4	4	no	—
135	C₂³×C₃×C₅	—	8	2	no	—
136	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
137	C₃×C₅×Q₈	—	2	1	no	—
138	C₃×C₅×Q₈	—	2	1	no	—
139	C₃×C₅×Q₈	—	4	2	no	—
140	C₃×C₅×D₄	—	2	2	no	—
141	C₃×C₅×D₄	—	2	2	no	—
142	C₃×C₅×D₄	—	2	1	no	—
143	C₃×C₅×D₄	—	2	2	no	—
144	C₃×C₅×D₄	—	4	4	no	—
145	C₃×C₅×D₄	—	4	4	no	—
146	C₃×C₅×D₄	—	4	2	no	—
147	C₃×C₅×D₄	—	8	1	no	—
148	C₃×C₅×D₄	#41	1	2	yes	—
149	C₂×C₃×C₄×C₅	—	2	2	no	—
150	C₂×C₃×C₄×C₅	—	2	2	no	—
151	C₂×C₃×C₄×C₅	—	2	1	no	—
152	C₂×C₃×C₄×C₅	—	2	1	no	—
153	C₂×C₃×C₄×C₅	—	2	2	no	—
154	C₂×C₃×C₄×C₅	—	2	1	no	—
155	C₂×C₃×C₄×C₅	—	4	2	no	—
156	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
157	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
160	C₃×C₅×C₈	—	2	1	no	—
161	C₃×C₅×C₈	—	2	1	no	—
162	C₃×C₅×C₈	—	2	2	no	—
163	C₂²×C₃×D₄	—	2	1	no	—
164	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
165	C₂×C₃×SD₁₆	C₂²×C₅	1	1	yes	characteristic
166	C₂×C₃×SD₁₆	C₂²×C₅	1	2	yes	complemented
167	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	complemented
168	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
169	C₃×#127	C₂²×C₅	1	1	yes	characteristic
170	C₃×#126	—	2	2	no	—
171	C₃×#124	—	2	1	no	—
172	C₃×C₂²⋊Q₈	C₂²×C₅	1	2	yes	complemented
173	C₃×#121	—	2	1	no	—
174	C₃×#121	C₂²×C₅	1	2	yes	—
175	C₃×#120	—	2	2	no	—
176	C₃×C₄×Q₈	—	2	1	no	—
177	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
178	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
179	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
180	C₃×#117	C₂²×C₅	1	2	yes	complemented
181	C₃×#107	C₂²×C₅	1	2	yes	complemented
182	C₃×#106	C₂²×C₅	1	1	yes	characteristic, complemented
183	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
184	C₃×#103	C₂²×C₅	1	1	yes	characteristic, complemented
185	C₃×#102	C₂²×C₅	1	1	yes	characteristic
186	C₃×#102	C₂²×C₅	1	2	yes	—
187	C₃×#102	C₂²×C₅	1	2	yes	complemented
188	C₃×#102	C₂²×C₅	1	2	yes	complemented
189	C₃×#102	C₂²×C₅	1	2	yes	complemented
190	C₃×C₂²⋊C₈	C₂²×C₅	1	2	yes	complemented
191	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
192	C₂⁴×C₅	—	2	2	no	—
193	C₂×C₅×Q₈	—	2	1	no	—
194	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
195	C₂×C₅×D₄	—	2	4	no	—
196	C₂×C₅×D₄	—	2	2	no	—
197	C₂×C₅×D₄	—	2	2	no	—
198	C₂×C₅×D₄	—	2	1	no	—
199	C₂×C₅×D₄	—	2	1	no	—
200	C₂×C₅×D₄	—	4	2	no	—
201	C₂×C₅×D₄	—	4	4	no	—
202	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
203	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
204	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
205	C₂²×C₄×C₅	—	2	1	no	—
206	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
207	C₅×SD₁₆	—	2	2	no	—
208	C₅×SD₁₆	—	4	2	no	—
209	C₅×M₁₆	—	4	2	no	—
210	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
211	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
212	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
213	C₅×C₄⋊C₄	—	2	2	no	—
214	C₅×C₄⋊C₄	—	2	1	no	—
215	C₅×C₄⋊C₄	—	2	1	no	—
216	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
217	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
218	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
219	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
220	C₅×C₂²⋊C₄	—	2	2	no	—
221	C₅×C₂²⋊C₄	—	2	2	no	—
222	C₅×C₂²⋊C₄	—	2	1	no	—
223	C₅×C₂²⋊C₄	—	2	2	no	—
224	C₅×C₂²⋊C₄	C₃×D₄	1	2	yes	—
225	C₄²×C₅	—	2	2	no	—
226	C₄²×C₅	—	2	1	no	—
227	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
228	#556	C₂×C₃×C₅	1	1	yes	characteristic, complemented
229	D₄²	C₂×C₃×C₅	1	1	yes	characteristic
230	#486	C₂×C₃×C₅	1	1	yes	characteristic, complemented
231	#482	C₂×C₃×C₅	1	2	yes	complemented
232	#474	C₂×C₃×C₅	1	1	yes	characteristic, complemented
233	#468	C₂×C₃×C₅	1	2	yes	complemented
234	#460	C₂×C₃×C₅	1	1	yes	characteristic, complemented
235	#450	C₂×C₃×C₅	1	2	yes	complemented
236	C₄×SD₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
237	#436	C₂×C₃×C₅	1	1	yes	characteristic, complemented
238	#418	C₂×C₃×C₅	1	2	yes	complemented
239	C₂²×C₃×C₅	—	2	2	no	—
240	C₂²×C₃×C₅	—	2	2	no	—
241	C₂²×C₃×C₅	—	2	2	no	—
242	C₂²×C₃×C₅	—	2	2	no	—
243	C₂²×C₃×C₅	—	4	1	no	—
244	C₂²×C₃×C₅	—	4	1	no	—
245	C₂²×C₃×C₅	—	4	2	no	—
246	C₂²×C₃×C₅	—	4	2	no	—
247	C₂²×C₃×C₅	—	4	2	no	—
248	C₂²×C₃×C₅	—	8	8	no	—
249	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
250	C₃×C₄×C₅	—	2	2	no	—
251	C₃×C₄×C₅	—	2	1	no	—
252	C₃×C₄×C₅	—	4	1	no	—
253	C₃×C₄×C₅	—	4	1	no	—
254	C₃×C₄×C₅	—	4	2	no	—
255	C₃×C₄×C₅	—	4	1	no	—
256	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
257	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
258	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
259	C₂⁴×C₃	—	2	2	no	—
260	C₂×C₃×Q₈	—	2	1	no	—
261	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
262	C₂×C₃×D₄	—	2	4	no	—
263	C₂×C₃×D₄	—	2	2	no	—
264	C₂×C₃×D₄	—	2	2	no	—
265	C₂×C₃×D₄	—	2	1	no	—
266	C₂×C₃×D₄	—	2	1	no	—
267	C₂×C₃×D₄	—	4	2	no	—
268	C₂×C₃×D₄	—	4	4	no	—
269	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
270	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
271	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
272	C₂²×C₃×C₄	—	2	1	no	—
273	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
274	C₃×SD₁₆	—	2	2	no	—
275	C₃×SD₁₆	—	4	2	no	—
276	C₃×M₁₆	—	4	2	no	—
277	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
278	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
279	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
280	C₃×C₄⋊C₄	—	2	2	no	—
281	C₃×C₄⋊C₄	—	2	1	no	—
282	C₃×C₄⋊C₄	—	2	1	no	—
283	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
284	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
285	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
286	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
287	C₃×C₂²⋊C₄	—	2	2	no	—
288	C₃×C₂²⋊C₄	—	2	2	no	—
289	C₃×C₂²⋊C₄	—	2	1	no	—
290	C₃×C₂²⋊C₄	—	2	2	no	—
291	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
292	C₃×C₄²	—	2	2	no	—
293	C₃×C₄²	—	2	1	no	—
294	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
295	C₂³×C₅	—	2	1	no	—
296	C₂³×C₅	—	2	2	no	—
297	C₂³×C₅	—	4	4	no	—
298	C₂³×C₅	—	4	4	no	—
299	C₂³×C₅	—	8	2	no	—
300	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
301	C₅×Q₈	—	2	1	no	—
302	C₅×Q₈	—	2	1	no	—
303	C₅×Q₈	—	4	2	no	—
304	C₅×D₄	—	2	2	no	—
305	C₅×D₄	—	2	2	no	—
306	C₅×D₄	—	2	1	no	—
307	C₅×D₄	—	2	2	no	—
308	C₅×D₄	—	4	4	no	—
309	C₅×D₄	—	4	4	no	—
310	C₅×D₄	—	4	2	no	—
311	C₅×D₄	—	8	1	no	—
312	C₅×D₄	C₃×#41	1	2	yes	—
313	C₂×C₄×C₅	—	2	2	no	—
314	C₂×C₄×C₅	—	2	2	no	—
315	C₂×C₄×C₅	—	2	1	no	—
316	C₂×C₄×C₅	—	2	1	no	—
317	C₂×C₄×C₅	—	2	2	no	—
318	C₂×C₄×C₅	—	2	1	no	—
319	C₂×C₄×C₅	—	4	2	no	—
320	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
321	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
322	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
323	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
324	C₅×C₈	—	2	1	no	—
325	C₅×C₈	—	2	1	no	—
326	C₅×C₈	—	2	2	no	—
327	C₂²×D₄	—	2	1	no	—
328	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
329	C₂×SD₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
330	C₂×SD₁₆	C₂²×C₃×C₅	1	2	yes	complemented
331	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	complemented
332	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
333	#127	C₂²×C₃×C₅	1	1	yes	characteristic
334	#126	—	2	2	no	—
335	#124	—	2	1	no	—
336	C₂²⋊Q₈	C₂²×C₃×C₅	1	2	yes	complemented
337	#121	—	2	1	no	—
338	#121	C₂²×C₃×C₅	1	2	yes	—
339	#120	—	2	2	no	—
340	C₄×Q₈	—	2	1	no	—
341	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
342	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
343	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
344	#117	C₂²×C₃×C₅	1	2	yes	complemented
345	#107	C₂²×C₃×C₅	1	2	yes	complemented
346	#106	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
347	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
348	#103	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
349	#102	C₂²×C₃×C₅	1	1	yes	characteristic
350	#102	C₂²×C₃×C₅	1	2	yes	—
351	#102	C₂²×C₃×C₅	1	2	yes	complemented
352	#102	C₂²×C₃×C₅	1	2	yes	complemented
353	#102	C₂²×C₃×C₅	1	2	yes	complemented
354	C₂²⋊C₈	C₂²×C₃×C₅	1	2	yes	complemented
355	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
356	C₂×C₃×C₅	—	4	2	no	—
357	C₂×C₃×C₅	—	4	2	no	—
358	C₂×C₃×C₅	—	8	2	no	—
359	C₂×C₃×C₅	#546	1	1	yes	characteristic
360	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
361	C₂×C₃×C₅	#577	1	1	yes	characteristic
362	C₂³×C₃	—	2	1	no	—
363	C₂³×C₃	—	2	2	no	—
364	C₂³×C₃	—	4	4	no	—
365	C₂³×C₃	—	4	4	no	—
366	C₂³×C₃	—	8	2	no	—
367	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
368	C₃×Q₈	—	2	1	no	—
369	C₃×Q₈	—	2	1	no	—
370	C₃×Q₈	—	4	2	no	—
371	C₃×D₄	—	2	2	no	—
372	C₃×D₄	—	2	2	no	—
373	C₃×D₄	—	2	1	no	—
374	C₃×D₄	—	2	2	no	—
375	C₃×D₄	—	4	4	no	—
376	C₃×D₄	—	4	4	no	—
377	C₃×D₄	—	4	2	no	—
378	C₃×D₄	—	8	1	no	—
379	C₃×D₄	C₅×#41	1	2	yes	—
380	C₂×C₃×C₄	—	2	2	no	—
381	C₂×C₃×C₄	—	2	2	no	—
382	C₂×C₃×C₄	—	2	1	no	—
383	C₂×C₃×C₄	—	2	1	no	—
384	C₂×C₃×C₄	—	2	2	no	—
385	C₂×C₃×C₄	—	2	1	no	—
386	C₂×C₃×C₄	—	4	2	no	—
387	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
388	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
389	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
390	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
391	C₃×C₈	—	2	1	no	—
392	C₃×C₈	—	2	1	no	—
393	C₃×C₈	—	2	2	no	—
394	C₂²×C₅	—	2	2	no	—
395	C₂²×C₅	—	2	2	no	—
396	C₂²×C₅	—	2	2	no	—
397	C₂²×C₅	—	2	2	no	—
398	C₂²×C₅	—	4	1	no	—
399	C₂²×C₅	—	4	1	no	—
400	C₂²×C₅	—	4	2	no	—
401	C₂²×C₅	—	4	2	no	—
402	C₂²×C₅	—	4	2	no	—
403	C₂²×C₅	—	8	8	no	—
404	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
405	C₄×C₅	—	2	1	no	—
406	C₄×C₅	—	2	2	no	—
407	C₄×C₅	—	4	1	no	—
408	C₄×C₅	—	4	2	no	—
409	C₄×C₅	—	4	1	no	—
410	C₄×C₅	—	4	1	no	—
411	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
412	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
413	C₄×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
414	C₂⁴	—	2	2	no	—
415	C₂×Q₈	—	2	1	no	—
416	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	4	2	no	—
423	C₂×D₄	—	4	4	no	—
424	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
425	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
426	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
429	SD₁₆	—	2	2	no	—
430	SD₁₆	—	4	2	no	—
431	M₁₆	—	4	2	no	—
432	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
433	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
434	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
435	C₄⋊C₄	—	2	2	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
439	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
440	C₄⋊C₄	C₂³×C₃×C₅	1	2	yes	—
441	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
442	C₂²⋊C₄	—	2	2	no	—
443	C₂²⋊C₄	—	2	2	no	—
444	C₂²⋊C₄	—	2	1	no	—
445	C₂²⋊C₄	—	2	2	no	—
446	C₂²⋊C₄	C₃×C₅×D₄	1	2	yes	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	1	no	—
449	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
450	C₃×C₅	#3324	1	1	yes	characteristic, direct product factor, Hall
451	C₂²×C₃	—	2	2	no	—
452	C₂²×C₃	—	2	2	no	—
453	C₂²×C₃	—	2	2	no	—
454	C₂²×C₃	—	2	2	no	—
455	C₂²×C₃	—	4	1	no	—
456	C₂²×C₃	—	4	1	no	—
457	C₂²×C₃	—	4	2	no	—
458	C₂²×C₃	—	4	2	no	—
459	C₂²×C₃	—	4	2	no	—
460	C₂²×C₃	—	8	8	no	—
461	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
462	C₃×C₄	—	2	2	no	—
463	C₃×C₄	—	2	1	no	—
464	C₃×C₄	—	4	1	no	—
465	C₃×C₄	—	4	1	no	—
466	C₃×C₄	—	4	2	no	—
467	C₃×C₄	—	4	1	no	—
468	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
469	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
470	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
471	C₂×C₅	—	4	2	no	—
472	C₂×C₅	—	4	2	no	—
473	C₂×C₅	—	8	2	no	—
474	C₂×C₅	C₃×#546	1	1	yes	characteristic
475	C₂×C₅	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
476	C₂×C₅	C₃×#577	1	1	yes	characteristic
477	C₂³	—	2	1	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	4	4	no	—
480	C₂³	—	4	4	no	—
481	C₂³	—	8	2	no	—
482	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
483	Q₈	—	2	1	no	—
484	Q₈	—	2	1	no	—
485	Q₈	—	4	2	no	—
486	D₄	—	2	2	no	—
487	D₄	—	2	1	no	—
488	D₄	—	2	2	no	—
489	D₄	—	2	2	no	—
490	D₄	—	4	4	no	—
491	D₄	—	4	4	no	—
492	D₄	—	4	2	no	—
493	D₄	—	8	1	no	—
494	D₄	C₃×C₅×#41	1	2	yes	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
503	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
504	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
505	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
506	C₈	—	2	1	no	—
507	C₈	—	2	1	no	—
508	C₈	—	2	2	no	—
509	C₂×C₃	—	4	2	no	—
510	C₂×C₃	—	4	2	no	—
511	C₂×C₃	—	8	2	no	—
512	C₂×C₃	C₅×#546	1	1	yes	characteristic
513	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
514	C₂×C₃	C₅×#577	1	1	yes	characteristic
515	C₅	C₃×#3324	1	1	yes	characteristic, direct product factor, Sylow
516	C₂²	—	2	2	no	—
517	C₂²	—	2	2	no	—
518	C₂²	—	2	2	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	4	1	no	—
521	C₂²	—	4	1	no	—
522	C₂²	—	4	2	no	—
523	C₂²	—	4	2	no	—
524	C₂²	—	4	2	no	—
525	C₂²	—	8	8	no	—
526	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
527	C₄	—	2	2	no	—
528	C₄	—	2	1	no	—
529	C₄	—	4	1	no	—
530	C₄	—	4	1	no	—
531	C₄	—	4	2	no	—
532	C₄	—	4	1	no	—
533	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
534	C₄	C₂×C₃×C₅×#41	1	1	yes	characteristic
535	C₄	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
536	C₃	C₅×#3324	1	1	yes	characteristic, direct product factor, Sylow
537	C₂	—	4	2	no	—
538	C₂	—	4	2	no	—
539	C₂	—	8	2	no	—
540	C₂	C₃×C₅×#546	1	1	yes	characteristic
541	C₂	C₂×C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
542	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×D₄²	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. C₃×C₅×#556	C₂	semidirect
3. C₃×C₅×#486	C₂	semidirect
4. C₃×C₅×#482	C₂	semidirect
5. C₃×C₅×#474	C₂	semidirect
6. C₃×C₅×#468	C₂	semidirect
7. C₃×C₅×#460	C₂	semidirect
8. C₃×C₅×#450	C₂	semidirect
9. C₃×C₄×C₅×SD₁₆	C₂	semidirect
10. C₃×C₅×#436	C₂	semidirect
11. C₃×C₅×#418	C₂	semidirect
12. C₅×#3324	C₃	direct (N × H)
16. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
17. C₂×C₃×C₅×M₁₆	C₂²	semidirect
18. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
22. C₃×C₅×C₂²⋊Q₈	C₂²	semidirect
27. C₃×C₄×C₅×Q₈	C₂²	semidirect
30. C₃×C₅×#117	C₂²	semidirect
31. C₃×C₅×#107	C₂²	semidirect
32. C₃×C₅×#106	C₂²	semidirect
33. C₃×C₅×C₄⋊C₈	C₂²	semidirect
34. C₃×C₅×#103	C₂²	semidirect
37. C₃×C₅×#102	C₂²	semidirect
38. C₃×C₅×#102	C₂²	semidirect
39. C₃×C₅×#102	C₂²	semidirect
40. C₃×C₅×C₂²⋊C₈	C₂²	semidirect
41. C₃×C₄×C₅×C₈	C₂²	semidirect
42. C₃×#3324	C₅	direct (N × H)
43. C₅×#556	C₂×C₃	semidirect
45. C₅×#486	C₂×C₃	semidirect
46. C₅×#482	C₂×C₃	semidirect
47. C₅×#474	C₂×C₃	semidirect
48. C₅×#468	C₂×C₃	semidirect
49. C₅×#460	C₂×C₃	semidirect
50. C₅×#450	C₂×C₃	semidirect
51. C₄×C₅×SD₁₆	C₂×C₃	semidirect
52. C₅×#436	C₂×C₃	semidirect
53. C₅×#418	C₂×C₃	semidirect
90. C₃×#556	C₂×C₅	semidirect
92. C₃×#486	C₂×C₅	semidirect
93. C₃×#482	C₂×C₅	semidirect
94. C₃×#474	C₂×C₅	semidirect
95. C₃×#468	C₂×C₅	semidirect
96. C₃×#460	C₂×C₅	semidirect
97. C₃×#450	C₂×C₅	semidirect
98. C₃×C₄×SD₁₆	C₂×C₅	semidirect
99. C₃×#436	C₂×C₅	semidirect
100. C₃×#418	C₂×C₅	semidirect
104. C₂×C₅×SD₁₆	C₂²×C₃	semidirect
105. C₂×C₅×M₁₆	C₂²×C₃	semidirect
106. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
110. C₅×C₂²⋊Q₈	C₂²×C₃	semidirect
115. C₄×C₅×Q₈	C₂²×C₃	semidirect
118. C₅×#117	C₂²×C₃	semidirect
119. C₅×#107	C₂²×C₃	semidirect
120. C₅×#106	C₂²×C₃	semidirect
121. C₅×C₄⋊C₈	C₂²×C₃	semidirect
122. C₅×#103	C₂²×C₃	semidirect
125. C₅×#102	C₂²×C₃	semidirect
126. C₅×#102	C₂²×C₃	semidirect
127. C₅×#102	C₂²×C₃	semidirect
128. C₅×C₂²⋊C₈	C₂²×C₃	semidirect
129. C₄×C₅×C₈	C₂²×C₃	semidirect
130. #3324	C₃×C₅	direct (N × H)
166. C₂×C₃×SD₁₆	C₂²×C₅	semidirect
167. C₂×C₃×M₁₆	C₂²×C₅	semidirect
168. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
172. C₃×C₂²⋊Q₈	C₂²×C₅	semidirect
177. C₃×C₄×Q₈	C₂²×C₅	semidirect
180. C₃×#117	C₂²×C₅	semidirect
181. C₃×#107	C₂²×C₅	semidirect
182. C₃×#106	C₂²×C₅	semidirect
183. C₃×C₄⋊C₈	C₂²×C₅	semidirect
184. C₃×#103	C₂²×C₅	semidirect
187. C₃×#102	C₂²×C₅	semidirect
188. C₃×#102	C₂²×C₅	semidirect
189. C₃×#102	C₂²×C₅	semidirect
190. C₃×C₂²⋊C₈	C₂²×C₅	semidirect
191. C₃×C₄×C₈	C₂²×C₅	semidirect
228. #556	C₂×C₃×C₅	semidirect
230. #486	C₂×C₃×C₅	semidirect
231. #482	C₂×C₃×C₅	semidirect
232. #474	C₂×C₃×C₅	semidirect
233. #468	C₂×C₃×C₅	semidirect
234. #460	C₂×C₃×C₅	semidirect
235. #450	C₂×C₃×C₅	semidirect
236. C₄×SD₁₆	C₂×C₃×C₅	semidirect
237. #436	C₂×C₃×C₅	semidirect
238. #418	C₂×C₃×C₅	semidirect
330. C₂×SD₁₆	C₂²×C₃×C₅	semidirect
331. C₂×M₁₆	C₂²×C₃×C₅	semidirect
332. C₄⋊Q₈	C₂²×C₃×C₅	semidirect
336. C₂²⋊Q₈	C₂²×C₃×C₅	semidirect
341. C₄×Q₈	C₂²×C₃×C₅	semidirect
344. #117	C₂²×C₃×C₅	semidirect
345. #107	C₂²×C₃×C₅	semidirect
346. #106	C₂²×C₃×C₅	semidirect
347. C₄⋊C₈	C₂²×C₃×C₅	semidirect
348. #103	C₂²×C₃×C₅	semidirect
351. #102	C₂²×C₃×C₅	semidirect
352. #102	C₂²×C₃×C₅	semidirect
353. #102	C₂²×C₃×C₅	semidirect
354. C₂²⋊C₈	C₂²×C₃×C₅	semidirect
355. C₄×C₈	C₂²×C₃×C₅	semidirect
450. C₃×C₅	#3324	direct (N × H)
515. C₅	C₃×#3324	direct (N × H)
536. C₃	C₅×#3324	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×#556	2
C₃×C₅×D₄²	2
C₃×C₅×#486	2
C₃×C₅×#482	2
C₃×C₅×#474	2
C₃×C₅×#468	2
C₃×C₅×#460	2
C₃×C₅×#450	2
C₃×C₄×C₅×SD₁₆	2
C₃×C₅×#436	2
C₃×C₅×#418	2
C₅×#3324	3
C₃×#3324	5

Subgroups — C3×C5×#3324 #14276358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3324 #14276358