Subgroups — C₃×C₅×#1556 #14274590

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 272 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
2	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#332	C₂	1	4	yes	maximal, complemented
5	C₅×#1556	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	—	2	1	no	—
7	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
8	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
9	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
12	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
16	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
17	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
18	C₃×C₅×#97	—	2	1	no	—
19	C₃×C₅×#97	—	2	1	no	—
20	C₃×#1556	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
22	C₅×#432	C₂×C₃	1	1	yes	characteristic
23	C₅×#432	C₂×C₃	1	1	yes	characteristic
24	C₅×#332	C₂×C₃	1	4	yes	complemented
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	2	2	no	—
27	C₂×C₃×C₅×Q₈	—	4	4	no	—
28	C₂²×C₃×C₄×C₅	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
43	C₃×#432	C₂×C₅	1	1	yes	characteristic
44	C₃×#432	C₂×C₅	1	1	yes	characteristic
45	C₃×#332	C₂×C₅	1	4	yes	complemented
46	C₂²×C₅×Q₈	—	2	1	no	—
47	C₅×C₄⋊Q₈	—	2	1	no	—
48	C₅×C₄⋊Q₈	—	2	1	no	—
49	C₅×C₄⋊Q₈	C₃×C₄	1	2	yes	—
50	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
51	C₂×C₅×C₄⋊C₄	—	2	1	no	—
52	C₂×C₅×C₄⋊C₄	C₃×C₄	1	2	yes	—
53	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
54	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
55	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
56	C₅×C₂²⋊C₈	—	2	1	no	—
57	C₅×C₂²⋊C₈	—	2	1	no	—
58	C₅×#97	—	2	1	no	—
59	C₅×#97	—	2	1	no	—
60	#1556	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
61	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
62	C₃×C₅×Q₈	—	4	4	no	—
63	C₃×C₅×Q₈	—	4	4	no	—
64	C₂×C₃×C₄×C₅	—	2	2	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	4	no	—
68	C₂×C₃×C₄×C₅	—	4	4	no	—
69	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₃×C₅×C₈	—	4	2	no	—
76	C₃×C₅×C₈	—	4	2	no	—
77	C₂²×C₃×Q₈	—	2	1	no	—
78	C₃×C₄⋊Q₈	—	2	1	no	—
79	C₃×C₄⋊Q₈	—	2	1	no	—
80	C₃×C₄⋊Q₈	C₄×C₅	1	2	yes	—
81	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
82	C₂×C₃×C₄⋊C₄	—	2	1	no	—
83	C₂×C₃×C₄⋊C₄	C₄×C₅	1	2	yes	—
84	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
85	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
86	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
87	C₃×C₂²⋊C₈	—	2	1	no	—
88	C₃×C₂²⋊C₈	—	2	1	no	—
89	C₃×#97	—	2	1	no	—
90	C₃×#97	—	2	1	no	—
91	C₂×C₅×Q₈	—	2	2	no	—
92	C₂×C₅×Q₈	—	2	2	no	—
93	C₂×C₅×Q₈	—	4	4	no	—
94	C₂²×C₄×C₅	—	2	2	no	—
95	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
104	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
105	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
106	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
107	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
108	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
109	#432	C₂×C₃×C₅	1	1	yes	characteristic
110	#432	C₂×C₃×C₅	1	1	yes	characteristic
111	#332	C₂×C₃×C₅	1	4	yes	complemented
112	C₂²×C₃×C₅	—	2	1	no	—
113	C₂²×C₃×C₅	—	2	1	no	—
114	C₂²×C₃×C₅	#101	1	2	yes	—
115	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₃×C₄×C₅	—	2	1	no	—
117	C₃×C₄×C₅	—	2	1	no	—
118	C₃×C₄×C₅	—	2	1	no	—
119	C₃×C₄×C₅	—	2	1	no	—
120	C₃×C₄×C₅	—	4	4	no	—
121	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
122	C₃×C₄×C₅	#137	1	2	yes	—
123	C₂×C₃×Q₈	—	2	2	no	—
124	C₂×C₃×Q₈	—	2	2	no	—
125	C₂×C₃×Q₈	—	4	4	no	—
126	C₂²×C₃×C₄	—	2	2	no	—
127	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₃×C₄⋊C₄	—	2	2	no	—
133	C₃×C₄⋊C₄	—	2	2	no	—
134	C₃×C₄⋊C₄	—	2	2	no	—
135	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
136	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
137	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
138	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
139	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
140	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
141	C₅×Q₈	—	4	4	no	—
142	C₅×Q₈	—	4	4	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	4	4	no	—
148	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
154	C₅×C₈	—	4	2	no	—
155	C₅×C₈	—	4	2	no	—
156	C₂²×Q₈	—	2	1	no	—
157	C₄⋊Q₈	—	2	1	no	—
158	C₄⋊Q₈	—	2	1	no	—
159	C₄⋊Q₈	C₃×C₄×C₅	1	2	yes	—
160	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₃×C₄×C₅	1	2	yes	—
163	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
164	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
165	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
166	C₂²⋊C₈	—	2	1	no	—
167	C₂²⋊C₈	—	2	1	no	—
168	#97	—	2	1	no	—
169	#97	—	2	1	no	—
170	C₂×C₃×C₅	—	2	2	no	—
171	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
172	C₂×C₃×C₅	#418	1	1	yes	characteristic
173	C₂×C₃×C₅	#419	1	1	yes	characteristic
174	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
175	C₃×Q₈	—	4	4	no	—
176	C₃×Q₈	—	4	4	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	4	4	no	—
182	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
188	C₃×C₈	—	4	2	no	—
189	C₃×C₈	—	4	2	no	—
190	C₂²×C₅	—	2	1	no	—
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	C₃×#101	1	2	yes	—
193	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	4	4	no	—
199	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
200	C₄×C₅	C₃×#137	1	2	yes	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
206	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
207	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
214	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
215	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
216	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
217	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
218	C₃×C₅	#1556	1	1	yes	characteristic, direct product factor, Hall
219	C₂²×C₃	—	2	1	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	C₅×#101	1	2	yes	—
222	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	4	4	no	—
228	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
229	C₃×C₄	C₅×#137	1	2	yes	—
230	C₂×C₅	—	2	2	no	—
231	C₂×C₅	C₂×C₃×#101	1	1	yes	characteristic
232	C₂×C₅	C₃×#418	1	1	yes	characteristic
233	C₂×C₅	C₃×#419	1	1	yes	characteristic
234	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
235	Q₈	—	4	4	no	—
236	Q₈	—	4	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
246	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
247	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
248	C₈	—	4	2	no	—
249	C₈	—	4	2	no	—
250	C₂×C₃	—	2	2	no	—
251	C₂×C₃	C₂×C₅×#101	1	1	yes	characteristic
252	C₂×C₃	C₅×#418	1	1	yes	characteristic
253	C₂×C₃	C₅×#419	1	1	yes	characteristic
254	C₅	C₃×#1556	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	C₃×C₅×#101	1	2	yes	—
258	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	4	no	—
264	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
265	C₄	C₃×C₅×#137	1	2	yes	—
266	C₃	C₅×#1556	1	1	yes	characteristic, direct product factor, Sylow
267	C₂	—	2	2	no	—
268	C₂	C₂×C₃×C₅×#101	1	1	yes	characteristic
269	C₂	C₃×C₅×#418	1	1	yes	characteristic
270	C₂	C₃×C₅×#419	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
4. C₃×C₅×#332	C₂	semidirect
5. C₅×#1556	C₃	direct (N × H)
20. C₃×#1556	C₅	direct (N × H)
24. C₅×#332	C₂×C₃	semidirect
45. C₃×#332	C₂×C₅	semidirect
60. #1556	C₃×C₅	direct (N × H)
111. #332	C₂×C₃×C₅	semidirect
218. C₃×C₅	#1556	direct (N × H)
254. C₅	C₃×#1556	direct (N × H)
266. C₃	C₅×#1556	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×#432	2
C₃×C₅×#332	2
C₅×#1556	3
C₃×#1556	5

Subgroups — C3×C5×#1556 #14274590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1556 #14274590