Subgroups — C₂×C₅×#4912 #14478263

Order 1920.

736 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×C₅×#912	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#827	C₂	1	2	yes	maximal, complemented
4	C₅×#4912	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×Dic₂₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#508	—	3	1	no	maximal, Hall
8	C₂×C₅×#237	—	2	2	no	—
9	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
10	C₂×C₅×#214	—	2	2	no	—
11	C₅×#912	—	2	4	no	—
12	C₅×#912	C₂²	1	8	yes	complemented
13	C₂×C₅×Dic₁₂	C₂²	1	2	yes	—
14	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×D₁₆	C₂²	1	4	yes	complemented
16	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₅×#827	C₂²	1	8	yes	complemented
18	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₅×Dic₂₄	C₂²	1	4	yes	complemented
20	C₅×D₃×C₁₆	C₂²	1	4	yes	complemented
21	C₂×#4912	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×#135	—	3	2	no	—
23	C₅×#508	—	3	8	no	—
24	C₂×C₅×Q₃₂	—	3	1	no	—
25	C₂×C₅×SD₃₂	—	3	2	no	—
26	C₂×C₅×D₁₆	D₃	1	1	yes	characteristic, complemented
27	C₂²×C₅×C₁₆	—	3	1	no	—
28	C₂×C₃×C₅×D₄	—	2	2	no	—
29	C₂×C₅×#67	—	4	2	no	—
30	C₂²×C₅×Dic₃	—	4	2	no	—
31	C₅×#237	—	2	8	no	—
32	C₅×#237	—	4	4	no	—
33	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
34	C₂×C₅×Dic₆	—	2	2	no	—
35	C₃×C₅×D₈	—	2	2	no	—
36	C₃×C₅×D₈	C₂³	1	4	yes	—
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₅×#214	—	2	8	no	—
39	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
40	C₅×Dic₁₂	—	2	2	no	—
41	C₅×Dic₁₂	C₂³	1	4	yes	—
42	C₅×D₃×C₈	D₄	1	2	yes	—
43	C₅×D₃×C₈	C₂³	1	2	yes	—
44	C₃×C₅×C₁₆	C₂³	1	2	yes	—
45	C₅×C₃⋊C₁₆	C₂³	1	2	yes	—
46	C₂×#912	C₂×C₅	1	2	yes	complemented
47	C₂×C₃×D₁₆	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂×#827	C₂×C₅	1	2	yes	complemented
49	#4912	C₂×C₅	1	8	yes	direct product factor
50	C₂×Dic₂₄	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×D₃×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×C₅×#41	—	6	2	no	—
53	C₅×#135	—	3	8	no	—
54	C₅×#135	—	6	4	no	—
55	C₂×C₅×Q₁₆	—	3	2	no	—
56	C₂×C₅×SD₁₆	—	6	2	no	—
57	C₂×C₅×D₈	C₂×D₃	1	2	yes	—
58	C₂²×C₅×C₈	—	3	1	no	—
59	C₅×Q₃₂	—	3	4	no	—
60	C₅×SD₃₂	—	3	8	no	—
61	C₅×D₁₆	C₂×D₃	1	4	yes	complemented
62	C₂×C₅×C₁₆	—	3	1	no	—
63	C₂×C₅×C₁₆	—	3	4	no	—
64	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
65	C₂×#508	—	3	1	no	Sylow
66	C₂³×C₃×C₅	—	4	2	no	—
67	C₂²×C₅×D₃	D₈	1	1	yes	characteristic
68	C₃×C₅×D₄	—	2	4	no	—
69	C₃×C₅×D₄	—	4	2	no	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₅×#67	—	4	8	no	—
72	C₂×C₅×Dic₃	—	4	2	no	—
73	C₂×C₅×Dic₃	—	4	8	no	—
74	C₂×C₅×Dic₃	D₈	1	1	yes	characteristic
75	C₄×C₅×D₃	D₈	1	2	yes	—
76	C₄×C₅×D₃	C₂×D₄	1	2	yes	—
77	C₅×Dic₆	—	2	4	no	—
78	C₅×Dic₆	—	4	2	no	—
79	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
81	C₅×C₃⋊C₈	C₂×D₄	1	2	yes	—
82	C₂×#237	—	2	2	no	—
83	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
84	C₂×#214	—	2	2	no	—
85	#912	—	2	4	no	—
86	#912	C₂²×C₅	1	8	yes	complemented
87	C₂×Dic₁₂	C₂²×C₅	1	2	yes	—
88	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
89	C₃×D₁₆	C₂²×C₅	1	4	yes	complemented
90	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
91	#827	C₂²×C₅	1	8	yes	complemented
92	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
93	Dic₂₄	C₂²×C₅	1	4	yes	complemented
94	D₃×C₁₆	C₂²×C₅	1	4	yes	complemented
95	C₅×#41	—	6	8	no	—
96	C₅×#41	—	12	4	no	—
97	C₂×C₅×Q₈	—	6	2	no	—
98	C₂×C₅×D₄	—	2	2	no	—
99	C₂×C₅×D₄	—	12	2	no	—
100	C₂²×C₄×C₅	—	3	1	no	—
101	C₂²×C₄×C₅	—	12	2	no	—
102	C₅×Q₁₆	—	3	4	no	—
103	C₅×Q₁₆	—	6	2	no	—
104	C₅×SD₁₆	—	6	8	no	—
105	C₅×D₈	—	2	2	no	—
106	C₅×D₈	C₂²×D₃	1	4	yes	—
107	C₂×C₅×C₈	—	3	1	no	—
108	C₂×C₅×C₈	—	3	2	no	—
109	C₂×C₅×C₈	—	3	2	no	—
110	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
111	C₅×C₁₆	—	3	2	no	—
112	C₅×C₁₆	C₂²×D₃	1	2	yes	—
113	C₂×#135	—	3	2	no	—
114	#508	—	3	8	no	—
115	C₂×Q₃₂	—	3	1	no	—
116	C₂×SD₃₂	—	3	2	no	—
117	C₂×D₁₆	C₅×D₃	1	1	yes	characteristic, complemented
118	C₂²×C₁₆	—	3	1	no	—
119	C₂²×C₃×C₅	—	4	4	no	—
120	C₂²×C₃×C₅	—	8	4	no	—
121	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
122	C₂×C₅×D₃	—	2	2	no	—
123	C₂×C₅×D₃	C₂×D₈	1	2	yes	—
124	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
125	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
126	C₅×Dic₃	—	4	4	no	—
127	C₅×Dic₃	C₂×D₈	1	2	yes	—
128	C₂×C₃×D₄	—	2	2	no	—
129	C₂×#67	—	4	2	no	—
130	C₂²×Dic₃	—	4	2	no	—
131	#237	—	2	8	no	—
132	#237	—	4	4	no	—
133	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
134	C₂×Dic₆	—	2	2	no	—
135	C₃×D₈	—	2	2	no	—
136	C₃×D₈	C₂³×C₅	1	4	yes	—
137	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
138	#214	—	2	8	no	—
139	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
140	Dic₁₂	—	2	2	no	—
141	Dic₁₂	C₂³×C₅	1	4	yes	—
142	D₃×C₈	C₅×D₄	1	2	yes	—
143	D₃×C₈	C₂³×C₅	1	2	yes	—
144	C₃×C₁₆	C₂³×C₅	1	2	yes	—
145	C₃⋊C₁₆	C₂³×C₅	1	2	yes	—
146	C₂³×C₅	—	3	1	no	—
147	C₂³×C₅	—	4	2	no	—
148	C₅×Q₈	—	6	4	no	—
149	C₅×Q₈	—	12	2	no	—
150	C₅×D₄	—	2	4	no	—
151	C₅×D₄	—	4	2	no	—
152	C₅×D₄	—	12	8	no	—
153	C₂×C₄×C₅	—	3	1	no	—
154	C₂×C₄×C₅	—	3	2	no	—
155	C₂×C₄×C₅	—	3	2	no	—
156	C₂×C₄×C₅	—	12	8	no	—
157	C₂×C₄×C₅	—	12	2	no	—
158	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
159	C₅×C₈	—	3	2	no	—
160	C₅×C₈	D₃×D₄	1	1	yes	characteristic
161	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
162	C₂×#41	—	6	2	no	—
163	#135	—	3	8	no	—
164	#135	—	6	4	no	—
165	C₂×Q₁₆	—	3	2	no	—
166	C₂×SD₁₆	—	6	2	no	—
167	C₂×D₈	C₂×C₅×D₃	1	2	yes	—
168	C₂²×C₈	—	3	1	no	—
169	Q₃₂	—	3	4	no	—
170	SD₃₂	—	3	8	no	—
171	D₁₆	C₂×C₅×D₃	1	4	yes	complemented
172	C₂×C₁₆	—	3	1	no	—
173	C₂×C₁₆	—	3	4	no	—
174	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
175	C₂×C₃×C₅	—	8	4	no	—
176	C₂×C₃×C₅	#508	1	2	yes	complemented
177	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
178	C₅×D₃	—	2	2	no	—
179	C₂³×C₃	—	4	2	no	—
180	C₂²×D₃	C₅×D₈	1	1	yes	characteristic
181	C₃×D₄	—	2	4	no	—
182	C₃×D₄	—	4	2	no	—
183	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
184	#67	—	4	8	no	—
185	C₂×Dic₃	—	4	2	no	—
186	C₂×Dic₃	—	4	8	no	—
187	C₂×Dic₃	C₅×D₈	1	1	yes	characteristic
188	C₄×D₃	C₅×D₈	1	2	yes	—
189	C₄×D₃	C₂×C₅×D₄	1	2	yes	—
190	Dic₆	—	2	4	no	—
191	Dic₆	—	4	2	no	—
192	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
193	C₃×C₈	C₂⁴×C₅	1	1	yes	characteristic
194	C₃⋊C₈	C₂×C₅×D₄	1	2	yes	—
195	C₂²×C₅	—	3	2	no	—
196	C₂²×C₅	—	4	4	no	—
197	C₂²×C₅	—	6	2	no	—
198	C₂²×C₅	—	8	4	no	—
199	C₂²×C₅	D₃×D₈	1	1	yes	characteristic
200	C₄×C₅	—	3	2	no	—
201	C₄×C₅	—	12	4	no	—
202	C₄×C₅	D₃×D₈	1	1	yes	characteristic
203	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
204	#41	—	6	8	no	—
205	#41	—	12	4	no	—
206	C₂×Q₈	—	6	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	12	2	no	—
209	C₂²×C₄	—	3	1	no	—
210	C₂²×C₄	—	12	2	no	—
211	Q₁₆	—	3	4	no	—
212	Q₁₆	—	6	2	no	—
213	SD₁₆	—	6	8	no	—
214	D₈	—	2	2	no	—
215	D₈	C₂²×C₅×D₃	1	4	yes	—
216	C₂×C₈	—	3	1	no	—
217	C₂×C₈	—	3	2	no	—
218	C₂×C₈	—	3	2	no	—
219	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
220	C₁₆	—	3	2	no	—
221	C₁₆	C₂²×C₅×D₃	1	2	yes	—
222	C₃×C₅	C₂×#508	1	1	yes	characteristic, complemented, Hall
223	C₂²×C₃	—	4	4	no	—
224	C₂²×C₃	—	8	4	no	—
225	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
226	C₂×D₃	—	2	2	no	—
227	C₂×D₃	C₂×C₅×D₈	1	2	yes	—
228	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
229	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
230	Dic₃	—	4	4	no	—
231	Dic₃	C₂×C₅×D₈	1	2	yes	—
232	C₂×C₅	—	6	2	no	—
233	C₂×C₅	—	8	4	no	—
234	C₂×C₅	#4912	1	2	yes	direct product factor
235	C₂×C₅	C₂×D₃×D₈	1	1	yes	characteristic
236	C₂³	—	3	1	no	—
237	C₂³	—	4	2	no	—
238	Q₈	—	6	4	no	—
239	Q₈	—	12	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	4	2	no	—
242	D₄	—	12	8	no	—
243	C₂×C₄	—	3	1	no	—
244	C₂×C₄	—	3	2	no	—
245	C₂×C₄	—	3	2	no	—
246	C₂×C₄	—	12	8	no	—
247	C₂×C₄	—	12	2	no	—
248	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
249	C₈	—	3	2	no	—
250	C₈	C₅×D₃×D₄	1	1	yes	characteristic
251	C₈	C₂³×C₅×D₃	1	1	yes	characteristic
252	C₂×C₃	—	8	4	no	—
253	C₂×C₃	C₅×#508	1	2	yes	complemented
254	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
255	D₃	—	2	2	no	—
256	C₅	C₂×#4912	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²	—	3	2	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	6	2	no	—
260	C₂²	—	8	4	no	—
261	C₂²	C₅×D₃×D₈	1	1	yes	characteristic
262	C₄	—	3	2	no	—
263	C₄	—	12	4	no	—
264	C₄	C₅×D₃×D₈	1	1	yes	characteristic
265	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
266	C₃	C₂×C₅×#508	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	6	2	no	—
268	C₂	—	8	4	no	—
269	C₂	C₅×#4912	1	2	yes	direct product factor
270	C₂	C₂×C₅×D₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₃×D₈
commutator, cosocle	C₃×C₈	C₂⁴×C₅
Frattini	C₈	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×D₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₈
hypercenter	C₂×C₅×D₁₆	D₃
hypocenter	C₃	C₂×C₅×#508
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#912	C₂	semidirect
2. C₂×C₃×C₅×D₁₆	C₂	semidirect
3. C₂×C₅×#827	C₂	semidirect
4. C₅×#4912	C₂	direct (N × H)
5. C₂×C₅×Dic₂₄	C₂	semidirect
6. C₂×C₅×D₃×C₁₆	C₂	semidirect
12. C₅×#912	C₂²	semidirect
15. C₃×C₅×D₁₆	C₂²	semidirect
17. C₅×#827	C₂²	semidirect
19. C₅×Dic₂₄	C₂²	semidirect
20. C₅×D₃×C₁₆	C₂²	semidirect
21. C₂×#4912	C₅	direct (N × H)
26. C₂×C₅×D₁₆	D₃	semidirect
46. C₂×#912	C₂×C₅	semidirect
47. C₂×C₃×D₁₆	C₂×C₅	semidirect
48. C₂×#827	C₂×C₅	semidirect
49. #4912	C₂×C₅	direct (N × H)
50. C₂×Dic₂₄	C₂×C₅	semidirect
51. C₂×D₃×C₁₆	C₂×C₅	semidirect
61. C₅×D₁₆	C₂×D₃	semidirect
86. #912	C₂²×C₅	semidirect
89. C₃×D₁₆	C₂²×C₅	semidirect
91. #827	C₂²×C₅	semidirect
93. Dic₂₄	C₂²×C₅	semidirect
94. D₃×C₁₆	C₂²×C₅	semidirect
117. C₂×D₁₆	C₅×D₃	semidirect
171. D₁₆	C₂×C₅×D₃	semidirect
176. C₂×C₃×C₅	#508	semidirect
222. C₃×C₅	C₂×#508	semidirect
234. C₂×C₅	#4912	direct (N × H)
253. C₂×C₃	C₅×#508	semidirect
256. C₅	C₂×#4912	direct (N × H)
266. C₃	C₂×C₅×#508	semidirect
269. C₂	C₅×#4912	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×C₅×#912	2
C₂×C₃×C₅×D₁₆	2
C₂×C₅×#827	2
C₅×#4912	2
C₂×C₅×Dic₂₄	2
C₂×C₅×D₃×C₁₆	2
C₂×C₅×#508	3
C₂×#4912	5

Subgroups — C2×C5×#4912 #14478263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4912 #14478263