Subgroups — C₂×C₃×C₅×#492 #14276177

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×SD₁₆	C₂	1	4	yes	maximal, complemented
2	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#492	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#492	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×D₄	—	2	2	no	—
9	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
10	C₂×C₃×C₅×SD₁₆	—	2	16	no	—
11	C₂×C₃×C₅×SD₁₆	C₂²	1	16	yes	complemented
12	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
13	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
14	C₃×C₅×C₄⋊Q₈	C₂²	1	4	yes	complemented
15	C₃×C₅×#127	—	2	2	no	—
16	C₃×C₅×#127	C₂²	1	4	yes	—
17	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
18	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₃×C₄×C₅×C₈	C₂²	1	4	yes	complemented
20	C₂×C₃×#492	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂²×C₅×SD₁₆	C₂×C₃	1	4	yes	complemented
22	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
24	C₅×#492	C₂×C₃	1	8	yes	direct product factor
25	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂⁴×C₃×C₅	—	2	2	no	—
27	C₂×C₃×C₅×Q₈	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	8	no	—
29	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
30	C₂×C₃×C₅×D₄	—	2	8	no	—
31	C₂×C₃×C₅×D₄	—	2	8	no	—
32	C₂×C₃×C₅×D₄	—	2	2	no	—
33	C₂×C₃×C₅×D₄	—	4	8	no	—
34	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
35	C₂²×C₃×C₄×C₅	—	2	2	no	—
36	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₃×C₅×SD₁₆	—	2	32	no	—
39	C₂×C₃×C₅×C₈	D₄	1	8	yes	complemented
40	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
41	C₃×C₅×C₄⋊C₄	—	2	8	no	—
42	C₃×C₄²×C₅	D₄	1	2	yes	—
43	C₃×C₄²×C₅	C₂³	1	2	yes	—
44	C₂²×C₃×SD₁₆	C₂×C₅	1	4	yes	complemented
45	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
46	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic
47	C₃×#492	C₂×C₅	1	8	yes	direct product factor
48	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
50	C₂²×C₅×D₄	—	2	2	no	—
51	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
52	C₂×C₅×SD₁₆	—	2	16	no	—
53	C₂×C₅×SD₁₆	C₂²×C₃	1	16	yes	complemented
54	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
55	C₅×C₄⋊Q₈	—	2	2	no	—
56	C₅×C₄⋊Q₈	C₂²×C₃	1	4	yes	complemented
57	C₅×#127	—	2	2	no	—
58	C₅×#127	C₂²×C₃	1	4	yes	—
59	C₂×C₅×C₄⋊C₄	—	2	2	no	—
60	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂²×C₃	1	4	yes	complemented
62	C₂×#492	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
63	C₂³×C₃×C₅	—	2	4	no	—
64	C₂³×C₃×C₅	—	4	4	no	—
65	C₂³×C₃×C₅	—	4	8	no	—
66	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
67	C₃×C₅×Q₈	—	2	8	no	—
68	C₃×C₅×Q₈	—	4	4	no	—
69	C₃×C₅×D₄	—	2	8	no	—
70	C₃×C₅×D₄	—	4	4	no	—
71	C₃×C₅×D₄	—	4	16	no	—
72	C₂×C₃×C₄×C₅	—	2	4	no	—
73	C₂×C₃×C₄×C₅	—	4	4	no	—
74	C₂×C₃×C₄×C₅	SD₁₆	1	8	yes	complemented
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
78	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	C₃×C₅×C₈	C₂×D₄	1	8	yes	complemented
80	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
81	C₂²×C₃×D₄	—	2	2	no	—
82	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
83	C₂×C₃×SD₁₆	—	2	16	no	—
84	C₂×C₃×SD₁₆	C₂²×C₅	1	16	yes	complemented
85	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
86	C₃×C₄⋊Q₈	—	2	2	no	—
87	C₃×C₄⋊Q₈	C₂²×C₅	1	4	yes	complemented
88	C₃×#127	—	2	2	no	—
89	C₃×#127	C₂²×C₅	1	4	yes	—
90	C₂×C₃×C₄⋊C₄	—	2	2	no	—
91	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
92	C₃×C₄×C₈	C₂²×C₅	1	4	yes	complemented
93	C₂⁴×C₅	—	2	2	no	—
94	C₂×C₅×Q₈	—	2	2	no	—
95	C₂×C₅×Q₈	—	2	8	no	—
96	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
97	C₂×C₅×D₄	—	2	8	no	—
98	C₂×C₅×D₄	—	2	2	no	—
99	C₂×C₅×D₄	—	2	8	no	—
100	C₂×C₅×D₄	—	4	8	no	—
101	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
102	C₂²×C₄×C₅	—	2	2	no	—
103	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
104	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
105	C₅×SD₁₆	—	2	32	no	—
106	C₂×C₅×C₈	C₃×D₄	1	8	yes	complemented
107	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
108	C₅×C₄⋊C₄	—	2	8	no	—
109	C₄²×C₅	C₃×D₄	1	2	yes	—
110	C₄²×C₅	C₂³×C₃	1	2	yes	—
111	C₂²×SD₁₆	C₂×C₃×C₅	1	4	yes	complemented
112	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
113	C₂×#127	C₂×C₃×C₅	1	1	yes	characteristic
114	#492	C₂×C₃×C₅	1	8	yes	direct product factor
115	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
116	C₂²×C₃×C₅	—	4	4	no	—
117	C₂²×C₃×C₅	—	4	8	no	—
118	C₂²×C₃×C₅	—	8	8	no	—
119	C₂²×C₃×C₅	#127	1	2	yes	—
120	C₂²×C₃×C₅	C₂×SD₁₆	1	4	yes	—
121	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
122	C₃×C₄×C₅	—	4	4	no	—
123	C₃×C₄×C₅	#127	1	2	yes	—
124	C₃×C₄×C₅	C₂×SD₁₆	1	8	yes	complemented
125	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
126	C₂⁴×C₃	—	2	2	no	—
127	C₂×C₃×Q₈	—	2	2	no	—
128	C₂×C₃×Q₈	—	2	8	no	—
129	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
130	C₂×C₃×D₄	—	2	8	no	—
131	C₂×C₃×D₄	—	2	2	no	—
132	C₂×C₃×D₄	—	2	8	no	—
133	C₂×C₃×D₄	—	4	8	no	—
134	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
135	C₂²×C₃×C₄	—	2	2	no	—
136	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
137	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
138	C₃×SD₁₆	—	2	32	no	—
139	C₂×C₃×C₈	C₅×D₄	1	8	yes	complemented
140	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
141	C₃×C₄⋊C₄	—	2	8	no	—
142	C₃×C₄²	C₅×D₄	1	2	yes	—
143	C₃×C₄²	C₂³×C₅	1	2	yes	—
144	C₂³×C₅	—	2	4	no	—
145	C₂³×C₅	—	4	4	no	—
146	C₂³×C₅	—	4	8	no	—
147	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
148	C₅×Q₈	—	2	8	no	—
149	C₅×Q₈	—	4	4	no	—
150	C₅×D₄	—	2	8	no	—
151	C₅×D₄	—	4	4	no	—
152	C₅×D₄	—	4	16	no	—
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	—	4	4	no	—
155	C₂×C₄×C₅	C₃×SD₁₆	1	8	yes	complemented
156	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
158	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
159	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
160	C₅×C₈	C₂×C₃×D₄	1	8	yes	complemented
161	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
162	C₂²×D₄	—	2	2	no	—
163	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
164	C₂×SD₁₆	—	2	16	no	—
165	C₂×SD₁₆	C₂²×C₃×C₅	1	16	yes	complemented
166	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	C₂²×C₃×C₅	1	4	yes	complemented
169	#127	—	2	2	no	—
170	#127	C₂²×C₃×C₅	1	4	yes	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
173	C₄×C₈	C₂²×C₃×C₅	1	4	yes	complemented
174	C₂×C₃×C₅	—	8	4	no	—
175	C₂×C₃×C₅	#492	1	4	yes	direct product factor
176	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
177	C₂×C₃×C₅	C₂²×SD₁₆	1	2	yes	—
178	C₂³×C₃	—	2	4	no	—
179	C₂³×C₃	—	4	4	no	—
180	C₂³×C₃	—	4	8	no	—
181	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
182	C₃×Q₈	—	2	8	no	—
183	C₃×Q₈	—	4	4	no	—
184	C₃×D₄	—	2	8	no	—
185	C₃×D₄	—	4	4	no	—
186	C₃×D₄	—	4	16	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	4	4	no	—
189	C₂×C₃×C₄	C₅×SD₁₆	1	8	yes	complemented
190	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
192	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
193	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
194	C₃×C₈	C₂×C₅×D₄	1	8	yes	complemented
195	C₂²×C₅	—	4	8	no	—
196	C₂²×C₅	—	4	4	no	—
197	C₂²×C₅	—	8	8	no	—
198	C₂²×C₅	C₃×#127	1	2	yes	—
199	C₂²×C₅	C₂×C₃×SD₁₆	1	4	yes	—
200	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
201	C₄×C₅	—	4	4	no	—
202	C₄×C₅	C₃×#127	1	2	yes	—
203	C₄×C₅	C₂×C₃×SD₁₆	1	8	yes	complemented
204	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
205	C₂⁴	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	8	no	—
208	C₂×Q₈	C₂³×C₃×C₅	1	4	yes	—
209	C₂×D₄	—	2	8	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	8	no	—
212	C₂×D₄	—	4	8	no	—
213	C₂×D₄	C₂³×C₃×C₅	1	4	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
216	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
217	SD₁₆	—	2	32	no	—
218	C₂×C₈	C₃×C₅×D₄	1	8	yes	complemented
219	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄²	C₃×C₅×D₄	1	2	yes	—
222	C₄²	C₂³×C₃×C₅	1	2	yes	—
223	C₃×C₅	C₂×#492	1	1	yes	characteristic, direct product factor, Hall
224	C₂²×C₃	—	4	4	no	—
225	C₂²×C₃	—	4	8	no	—
226	C₂²×C₃	—	8	8	no	—
227	C₂²×C₃	C₅×#127	1	2	yes	—
228	C₂²×C₃	C₂×C₅×SD₁₆	1	4	yes	—
229	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
230	C₃×C₄	—	4	4	no	—
231	C₃×C₄	C₅×#127	1	2	yes	—
232	C₃×C₄	C₂×C₅×SD₁₆	1	8	yes	complemented
233	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
234	C₂×C₅	—	8	4	no	—
235	C₂×C₅	C₃×#492	1	4	yes	direct product factor
236	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
237	C₂×C₅	C₂²×C₃×SD₁₆	1	2	yes	—
238	C₂³	—	2	4	no	—
239	C₂³	—	4	4	no	—
240	C₂³	—	4	8	no	—
241	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
242	Q₈	—	2	8	no	—
243	Q₈	—	4	4	no	—
244	D₄	—	2	8	no	—
245	D₄	—	4	4	no	—
246	D₄	—	4	16	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	C₃×C₅×SD₁₆	1	8	yes	complemented
250	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
251	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
252	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
253	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
254	C₈	C₂×C₃×C₅×D₄	1	8	yes	complemented
255	C₂×C₃	—	8	4	no	—
256	C₂×C₃	C₅×#492	1	4	yes	direct product factor
257	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
258	C₂×C₃	C₂²×C₅×SD₁₆	1	2	yes	—
259	C₅	C₂×C₃×#492	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	4	4	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	8	8	no	—
263	C₂²	C₃×C₅×#127	1	2	yes	—
264	C₂²	C₂×C₃×C₅×SD₁₆	1	4	yes	—
265	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
266	C₄	—	4	4	no	—
267	C₄	C₃×C₅×#127	1	2	yes	—
268	C₄	C₂×C₃×C₅×SD₁₆	1	8	yes	complemented
269	C₄	C₂²×C₃×C₅×D₄	1	2	yes	—
270	C₃	C₂×C₅×#492	1	1	yes	characteristic, direct product factor, Sylow
271	C₂	—	8	4	no	—
272	C₂	C₃×C₅×#492	1	4	yes	direct product factor
273	C₂	C₂×C₃×C₅×#127	1	1	yes	characteristic
274	C₂	C₂²×C₃×C₅×SD₁₆	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
prime radical	C₂×C₃×C₅×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×SD₁₆	C₂	semidirect
2. C₂×C₃×C₅×C₄⋊Q₈	C₂	semidirect
4. C₃×C₅×#492	C₂	direct (N × H)
5. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
6. C₂×C₅×#492	C₃	direct (N × H)
11. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
14. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
19. C₃×C₄×C₅×C₈	C₂²	semidirect
20. C₂×C₃×#492	C₅	direct (N × H)
21. C₂²×C₅×SD₁₆	C₂×C₃	semidirect
22. C₂×C₅×C₄⋊Q₈	C₂×C₃	semidirect
24. C₅×#492	C₂×C₃	direct (N × H)
25. C₂×C₄×C₅×C₈	C₂×C₃	semidirect
39. C₂×C₃×C₅×C₈	D₄	semidirect
44. C₂²×C₃×SD₁₆	C₂×C₅	semidirect
45. C₂×C₃×C₄⋊Q₈	C₂×C₅	semidirect
47. C₃×#492	C₂×C₅	direct (N × H)
48. C₂×C₃×C₄×C₈	C₂×C₅	semidirect
53. C₂×C₅×SD₁₆	C₂²×C₃	semidirect
56. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
61. C₄×C₅×C₈	C₂²×C₃	semidirect
62. C₂×#492	C₃×C₅	direct (N × H)
74. C₂×C₃×C₄×C₅	SD₁₆	semidirect
79. C₃×C₅×C₈	C₂×D₄	semidirect
84. C₂×C₃×SD₁₆	C₂²×C₅	semidirect
87. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
92. C₃×C₄×C₈	C₂²×C₅	semidirect
106. C₂×C₅×C₈	C₃×D₄	semidirect
111. C₂²×SD₁₆	C₂×C₃×C₅	semidirect
112. C₂×C₄⋊Q₈	C₂×C₃×C₅	semidirect
114. #492	C₂×C₃×C₅	direct (N × H)
115. C₂×C₄×C₈	C₂×C₃×C₅	semidirect
124. C₃×C₄×C₅	C₂×SD₁₆	semidirect
139. C₂×C₃×C₈	C₅×D₄	semidirect
155. C₂×C₄×C₅	C₃×SD₁₆	semidirect
160. C₅×C₈	C₂×C₃×D₄	semidirect
165. C₂×SD₁₆	C₂²×C₃×C₅	semidirect
168. C₄⋊Q₈	C₂²×C₃×C₅	semidirect
173. C₄×C₈	C₂²×C₃×C₅	semidirect
175. C₂×C₃×C₅	#492	direct (N × H)
189. C₂×C₃×C₄	C₅×SD₁₆	semidirect
194. C₃×C₈	C₂×C₅×D₄	semidirect
203. C₄×C₅	C₂×C₃×SD₁₆	semidirect
218. C₂×C₈	C₃×C₅×D₄	semidirect
223. C₃×C₅	C₂×#492	direct (N × H)
232. C₃×C₄	C₂×C₅×SD₁₆	semidirect
235. C₂×C₅	C₃×#492	direct (N × H)
249. C₂×C₄	C₃×C₅×SD₁₆	semidirect
254. C₈	C₂×C₃×C₅×D₄	semidirect
256. C₂×C₃	C₅×#492	direct (N × H)
259. C₅	C₂×C₃×#492	direct (N × H)
268. C₄	C₂×C₃×C₅×SD₁₆	semidirect
270. C₃	C₂×C₅×#492	direct (N × H)
272. C₂	C₃×C₅×#492	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₅×SD₁₆	2
C₂×C₃×C₅×C₄⋊Q₈	2
C₂×C₃×C₅×#127	2
C₃×C₅×#492	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₅×#492	3
C₂×C₃×#492	5

Subgroups — C2×C3×C5×#492 #14276177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#492 #14276177