Subgroups — C₃×Dic₅×#127 #14322255

Order 1920.

1848 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×D₄×Dic₅	C₂	1	6	yes	maximal, complemented
3	C₃×#65510	C₂	1	6	yes	maximal, complemented
4	C₃×#65224	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₅×#127	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	C₂²	1	6	yes	—
8	C₂³×C₃×Dic₅	C₂²	1	4	yes	—
9	C₃×C₅×#127	C₄	1	8	yes	complemented
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₂²⋊Dic₅	C₂²	1	12	yes	—
12	C₃×D₄×Dic₅	—	2	24	no	—
13	C₂×C₃×C₄⋊Dic₅	C₂²	1	6	yes	—
14	C₂×C₃×C₄×Dic₅	C₂²	1	3	yes	—
15	C₂×C₃×C₄×Dic₅	C₂²	1	3	yes	—
16	C₃×C₄×#127	—	5	1	no	maximal, Hall
17	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
18	C₂×D₄×Dic₅	C₂×C₃	1	6	yes	complemented
19	#65510	C₂×C₃	1	6	yes	complemented
20	#65224	C₂×C₃	1	1	yes	characteristic, complemented
21	C₄²×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂⁴×C₃×C₅	C₂³	1	4	yes	—
23	C₂×C₃×C₅×D₄	—	2	12	no	—
24	C₂×C₃×C₅×D₄	—	2	12	no	—
25	C₂×C₃×C₅×D₄	C₂×C₄	1	24	yes	complemented
26	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
27	C₂²×C₃×Dic₅	—	2	24	no	—
28	C₂²×C₃×Dic₅	C₂³	1	4	yes	—
29	C₂²×C₃×Dic₅	C₂³	1	3	yes	—
30	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
31	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
32	C₃×C₂²⋊Dic₅	—	2	24	no	—
33	C₃×C₄⋊Dic₅	—	2	12	no	—
34	C₃×C₄×Dic₅	—	2	6	no	—
35	C₃×C₄×Dic₅	D₄	1	12	yes	complemented
36	C₂×C₃×#127	D₅	1	1	yes	characteristic
37	C₂×C₃×C₄×D₄	—	5	6	no	—
38	C₃×#390	—	5	6	no	—
39	C₃×#384	—	5	1	no	—
40	C₃×C₄³	—	5	1	no	—
41	C₂²×C₅×D₄	C₂²×C₃	1	6	yes	—
42	C₂³×Dic₅	C₂²×C₃	1	4	yes	—
43	C₅×#127	C₃×C₄	1	8	yes	complemented
44	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₂²⋊Dic₅	C₂²×C₃	1	12	yes	—
46	D₄×Dic₅	—	2	24	no	—
47	C₂×C₄⋊Dic₅	C₂²×C₃	1	6	yes	—
48	C₂×C₄×Dic₅	C₂²×C₃	1	3	yes	—
49	C₂×C₄×Dic₅	C₂²×C₃	1	3	yes	—
50	C₄×#127	—	5	1	no	Sylow
51	C₂³×C₃×C₅	—	2	12	no	—
52	C₂³×C₃×C₅	—	2	12	no	—
53	C₂³×C₃×C₅	C₂²×C₄	1	8	yes	—
54	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
55	C₃×C₅×D₄	—	2	48	no	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
57	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
58	C₂×C₃×C₄×C₅	#41	1	6	yes	—
59	C₂×C₃×Dic₅	—	2	6	no	—
60	C₂×C₃×Dic₅	—	2	12	no	—
61	C₂×C₃×Dic₅	—	4	16	no	—
62	C₂×C₃×Dic₅	C₂×D₄	1	6	yes	—
63	C₂×C₃×Dic₅	C₂×D₄	1	6	yes	—
64	C₂²×C₃×D₄	C₂×D₅	1	6	yes	—
65	C₂³×C₃×C₄	—	5	4	no	—
66	C₃×#127	Dic₅	1	8	yes	direct product factor
67	C₃×C₄×D₄	—	10	24	no	—
68	C₂×C₃×C₄⋊C₄	—	5	6	no	—
69	C₂×C₃×C₂²⋊C₄	—	5	12	no	—
70	C₂×C₃×C₄²	—	5	3	no	—
71	C₂×C₃×C₄²	—	5	3	no	—
72	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
73	C₂⁴×C₅	C₂³×C₃	1	4	yes	—
74	C₂×C₅×D₄	—	2	12	no	—
75	C₂×C₅×D₄	—	2	12	no	—
76	C₂×C₅×D₄	C₂×C₃×C₄	1	24	yes	complemented
77	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
78	C₂²×Dic₅	—	2	24	no	—
79	C₂²×Dic₅	C₂³×C₃	1	4	yes	—
80	C₂²×Dic₅	C₂³×C₃	1	3	yes	—
81	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
82	C₄²×C₅	C₂×C₃×C₄	1	4	yes	complemented
83	C₂²⋊Dic₅	—	2	24	no	—
84	C₄⋊Dic₅	—	2	12	no	—
85	C₄×Dic₅	—	2	6	no	—
86	C₄×Dic₅	C₃×D₄	1	12	yes	complemented
87	C₂×#127	C₃×D₅	1	1	yes	characteristic
88	C₂×C₄×D₄	—	5	6	no	—
89	#390	—	5	6	no	—
90	#384	—	5	1	no	—
91	C₄³	—	5	1	no	—
92	C₂²×C₃×C₅	—	2	24	no	—
93	C₂²×C₃×C₅	—	4	4	no	—
94	C₂²×C₃×C₅	—	4	12	no	—
95	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
96	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
97	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
98	C₃×C₄×C₅	C₄×D₄	1	12	yes	complemented
99	C₃×Dic₅	—	2	6	no	—
100	C₃×Dic₅	—	4	4	no	—
101	C₃×Dic₅	#127	1	4	yes	direct product factor
102	C₂⁴×C₃	C₂²×D₅	1	4	yes	—
103	C₂×C₃×D₄	—	2	12	no	—
104	C₂×C₃×D₄	—	2	12	no	—
105	C₂×C₃×D₄	C₂×Dic₅	1	24	yes	complemented
106	C₂²×C₃×C₄	—	5	1	no	—
107	C₂²×C₃×C₄	—	5	3	no	—
108	C₂²×C₃×C₄	—	5	4	no	—
109	C₂²×C₃×C₄	—	10	24	no	—
110	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
111	C₃×C₄⋊C₄	—	10	12	no	—
112	C₃×C₂²⋊C₄	—	10	24	no	—
113	C₃×C₄²	—	5	12	no	—
114	C₃×C₄²	—	10	6	no	—
115	C₃×C₄²	C₂×Dic₅	1	4	yes	complemented
116	C₂³×C₅	—	2	12	no	—
117	C₂³×C₅	—	2	12	no	—
118	C₂³×C₅	C₂²×C₃×C₄	1	8	yes	—
119	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
120	C₅×D₄	—	2	48	no	—
121	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
122	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
123	C₂×C₄×C₅	C₃×#41	1	6	yes	—
124	C₂×Dic₅	—	2	6	no	—
125	C₂×Dic₅	—	2	12	no	—
126	C₂×Dic₅	—	4	16	no	—
127	C₂×Dic₅	C₂×C₃×D₄	1	6	yes	—
128	C₂×Dic₅	C₂×C₃×D₄	1	6	yes	—
129	C₂²×D₄	C₂×C₃×D₅	1	6	yes	—
130	C₂³×C₄	—	5	4	no	—
131	#127	C₃×Dic₅	1	8	yes	direct product factor
132	C₄×D₄	—	10	24	no	—
133	C₂×C₄⋊C₄	—	5	6	no	—
134	C₂×C₂²⋊C₄	—	5	12	no	—
135	C₂×C₄²	—	5	3	no	—
136	C₂×C₄²	—	5	3	no	—
137	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
138	C₂×C₃×C₅	—	4	8	no	—
139	C₂×C₃×C₅	C₂×C₄×D₄	1	3	yes	—
140	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
141	C₂×C₃×C₅	#532	1	3	yes	—
142	C₂³×C₃	—	2	12	no	—
143	C₂³×C₃	—	2	12	no	—
144	C₂³×C₃	C₂²×Dic₅	1	8	yes	—
145	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
146	C₃×D₄	—	2	48	no	—
147	C₂×C₃×C₄	—	5	6	no	—
148	C₂×C₃×C₄	—	5	6	no	—
149	C₂×C₃×C₄	—	10	6	no	—
150	C₂×C₃×C₄	—	10	12	no	—
151	C₂×C₃×C₄	—	20	16	no	—
152	C₂×C₃×C₄	D₄×D₅	1	6	yes	—
153	C₂×C₃×C₄	#711	1	6	yes	—
154	C₂×C₃×C₄	C₂²×Dic₅	1	6	yes	—
155	C₂²×C₅	—	2	24	no	—
156	C₂²×C₅	—	4	4	no	—
157	C₂²×C₅	—	4	12	no	—
158	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
159	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
160	C₂²×C₅	C₂×C₃×#41	1	3	yes	—
161	C₄×C₅	C₃×C₄×D₄	1	12	yes	complemented
162	Dic₅	—	2	6	no	—
163	Dic₅	—	4	4	no	—
164	Dic₅	C₃×#127	1	4	yes	direct product factor
165	C₂⁴	C₂²×C₃×D₅	1	4	yes	—
166	C₂×D₄	—	2	12	no	—
167	C₂×D₄	—	2	12	no	—
168	C₂×D₄	C₂×C₃×Dic₅	1	24	yes	complemented
169	C₂²×C₄	—	5	1	no	—
170	C₂²×C₄	—	5	3	no	—
171	C₂²×C₄	—	5	4	no	—
172	C₂²×C₄	—	10	24	no	—
173	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
174	C₄⋊C₄	—	10	12	no	—
175	C₂²⋊C₄	—	10	24	no	—
176	C₄²	—	5	12	no	—
177	C₄²	—	10	6	no	—
178	C₄²	C₂×C₃×Dic₅	1	4	yes	complemented
179	C₃×C₅	C₄×#127	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	—	2	24	no	—
181	C₂²×C₃	—	4	4	no	—
182	C₂²×C₃	—	4	12	no	—
183	C₂²×C₃	C₂×D₄×D₅	1	3	yes	—
184	C₂²×C₃	C₂×#711	1	3	yes	—
185	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
186	C₃×C₄	—	5	4	no	—
187	C₃×C₄	—	10	6	no	—
188	C₃×C₄	—	20	4	no	—
189	C₃×C₄	D₄×Dic₅	1	12	yes	complemented
190	C₂×C₅	—	4	8	no	—
191	C₂×C₅	C₂×C₃×C₄×D₄	1	3	yes	—
192	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
193	C₂×C₅	C₃×#532	1	3	yes	—
194	C₂³	—	2	12	no	—
195	C₂³	—	2	12	no	—
196	C₂³	C₂²×C₃×Dic₅	1	8	yes	—
197	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
198	D₄	—	2	48	no	—
199	C₂×C₄	—	5	6	no	—
200	C₂×C₄	—	5	6	no	—
201	C₂×C₄	—	10	6	no	—
202	C₂×C₄	—	10	12	no	—
203	C₂×C₄	—	20	16	no	—
204	C₂×C₄	C₃×D₄×D₅	1	6	yes	—
205	C₂×C₄	C₃×#711	1	6	yes	—
206	C₂×C₄	C₂²×C₃×Dic₅	1	6	yes	—
207	C₂×C₃	—	4	8	no	—
208	C₂×C₃	D₅×#127	1	1	yes	characteristic
209	C₂×C₃	#66050	1	3	yes	—
210	C₂×C₃	C₂×D₄×Dic₅	1	3	yes	—
211	C₅	C₃×C₄×#127	1	1	yes	characteristic, complemented, Sylow
212	C₂²	—	2	24	no	—
213	C₂²	—	4	4	no	—
214	C₂²	—	4	12	no	—
215	C₂²	C₂×C₃×D₄×D₅	1	3	yes	—
216	C₂²	C₂×C₃×#711	1	3	yes	—
217	C₂²	C₂³×C₃×Dic₅	1	1	yes	characteristic
218	C₄	—	5	4	no	—
219	C₄	—	10	6	no	—
220	C₄	—	20	4	no	—
221	C₄	C₃×D₄×Dic₅	1	12	yes	complemented
222	C₃	Dic₅×#127	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	—	4	8	no	—
224	C₂	C₃×D₅×#127	1	1	yes	characteristic
225	C₂	C₃×#66050	1	3	yes	—
226	C₂	C₂×C₃×D₄×Dic₅	1	3	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂³	C₂³×C₃×D₅
Fitting, prime radical	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂⁴
cosocle	C₂³×C₅	C₂⁴×C₃
hypercenter	C₂×C₃×#127	D₅
hypocenter	C₅	C₃×C₄×#127
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×#127

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×D₄×Dic₅	C₂	semidirect
3. C₃×#65510	C₂	semidirect
4. C₃×#65224	C₂	semidirect
5. C₃×C₄²×Dic₅	C₂	semidirect
6. Dic₅×#127	C₃	direct (N × H)
9. C₃×C₅×#127	C₄	semidirect
18. C₂×D₄×Dic₅	C₂×C₃	semidirect
19. #65510	C₂×C₃	semidirect
20. #65224	C₂×C₃	semidirect
21. C₄²×Dic₅	C₂×C₃	semidirect
25. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
31. C₃×C₄²×C₅	C₂×C₄	semidirect
35. C₃×C₄×Dic₅	D₄	semidirect
43. C₅×#127	C₃×C₄	semidirect
66. C₃×#127	Dic₅	direct (N × H)
76. C₂×C₅×D₄	C₂×C₃×C₄	semidirect
82. C₄²×C₅	C₂×C₃×C₄	semidirect
86. C₄×Dic₅	C₃×D₄	semidirect
98. C₃×C₄×C₅	C₄×D₄	semidirect
101. C₃×Dic₅	#127	direct (N × H)
105. C₂×C₃×D₄	C₂×Dic₅	semidirect
115. C₃×C₄²	C₂×Dic₅	semidirect
131. #127	C₃×Dic₅	direct (N × H)
161. C₄×C₅	C₃×C₄×D₄	semidirect
164. Dic₅	C₃×#127	direct (N × H)
168. C₂×D₄	C₂×C₃×Dic₅	semidirect
178. C₄²	C₂×C₃×Dic₅	semidirect
179. C₃×C₅	C₄×#127	semidirect
189. C₃×C₄	D₄×Dic₅	semidirect
211. C₅	C₃×C₄×#127	semidirect
221. C₄	C₃×D₄×Dic₅	semidirect
222. C₃	Dic₅×#127	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×#127	2
C₂×C₃×D₄×Dic₅	2
C₃×#65510	2
C₃×#65224	2
C₃×C₄²×Dic₅	2
Dic₅×#127	3
C₃×C₄×#127	5

Subgroups — C3×Dic5×#127 #14322255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Dic₅×#127 #14322255