Subgroups — C₃×C₄×C₅×#127 #14275340

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×C₄×C₅×D₄	C₂	1	6	yes	maximal, complemented
3	C₃×C₅×#390	C₂	1	6	yes	maximal, complemented
4	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄³×C₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₅×#127	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	C₂²	1	6	yes	—
8	C₂³×C₃×C₄×C₅	C₂²	1	4	yes	—
9	C₃×C₅×#127	C₄	1	8	yes	direct product factor
10	C₃×C₄×C₅×D₄	—	2	24	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
12	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	12	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
16	C₃×C₄×#127	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
18	C₂×C₄×C₅×D₄	C₂×C₃	1	6	yes	complemented
19	C₅×#390	C₂×C₃	1	6	yes	complemented
20	C₅×#384	C₂×C₃	1	1	yes	characteristic, complemented
21	C₄³×C₅	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	—	2	24	no	—
27	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	—	2	12	no	—
32	C₃×C₅×C₂²⋊C₄	—	2	24	no	—
33	C₃×C₄²×C₅	—	2	6	no	—
34	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
35	C₃×C₄²×C₅	D₄	1	12	yes	complemented
36	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic
37	C₂×C₃×C₄×D₄	C₂×C₅	1	6	yes	complemented
38	C₃×#390	C₂×C₅	1	6	yes	complemented
39	C₃×#384	C₂×C₅	1	1	yes	characteristic, complemented
40	C₃×C₄³	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂²×C₅×D₄	C₂²×C₃	1	6	yes	—
42	C₂³×C₄×C₅	C₂²×C₃	1	4	yes	—
43	C₅×#127	C₃×C₄	1	8	yes	direct product factor
44	C₄×C₅×D₄	—	2	24	no	—
45	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	6	yes	—
46	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
47	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	—
48	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	—
50	C₄×#127	C₃×C₅	1	1	yes	characteristic, direct product factor, 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	—	2	6	no	—
57	C₂×C₃×C₄×C₅	—	2	12	no	—
58	C₂×C₃×C₄×C₅	—	4	16	no	—
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
60	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
63	C₂×C₃×C₄×C₅	#41	1	6	yes	—
64	C₂²×C₃×D₄	C₂²×C₅	1	6	yes	—
65	C₂³×C₃×C₄	C₂²×C₅	1	4	yes	—
66	C₃×#127	C₄×C₅	1	8	yes	direct product factor
67	C₃×C₄×D₄	—	2	24	no	—
68	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	6	yes	—
69	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
70	C₂×C₃×C₄²	C₂²×C₅	1	3	yes	—
71	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
72	C₂×C₃×C₄²	C₂²×C₅	1	3	yes	—
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	—	2	24	no	—
78	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
79	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
80	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
81	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
82	C₅×C₄⋊C₄	—	2	12	no	—
83	C₅×C₂²⋊C₄	—	2	24	no	—
84	C₄²×C₅	—	2	6	no	—
85	C₄²×C₅	C₂×C₃×C₄	1	4	yes	complemented
86	C₄²×C₅	C₃×D₄	1	12	yes	complemented
87	C₂×#127	C₂×C₃×C₅	1	1	yes	characteristic
88	C₂×C₄×D₄	C₂×C₃×C₅	1	6	yes	complemented
89	#390	C₂×C₃×C₅	1	6	yes	complemented
90	#384	C₂×C₃×C₅	1	1	yes	characteristic, complemented
91	C₄³	C₂×C₃×C₅	1	1	yes	characteristic, complemented
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	—	2	6	no	—
99	C₃×C₄×C₅	—	4	4	no	—
100	C₃×C₄×C₅	C₄×D₄	1	12	yes	complemented
101	C₃×C₄×C₅	#127	1	4	yes	direct product factor
102	C₂⁴×C₃	C₂³×C₅	1	4	yes	—
103	C₂×C₃×D₄	—	2	12	no	—
104	C₂×C₃×D₄	—	2	12	no	—
105	C₂×C₃×D₄	C₂×C₄×C₅	1	24	yes	complemented
106	C₂²×C₃×C₄	—	2	24	no	—
107	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
108	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
109	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
110	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
111	C₃×C₄⋊C₄	—	2	12	no	—
112	C₃×C₂²⋊C₄	—	2	24	no	—
113	C₃×C₄²	—	2	6	no	—
114	C₃×C₄²	C₂×C₄×C₅	1	4	yes	complemented
115	C₃×C₄²	C₅×D₄	1	12	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	—	2	6	no	—
122	C₂×C₄×C₅	—	2	12	no	—
123	C₂×C₄×C₅	—	4	16	no	—
124	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
125	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
126	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
127	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
128	C₂×C₄×C₅	C₃×#41	1	6	yes	—
129	C₂²×D₄	C₂²×C₃×C₅	1	6	yes	—
130	C₂³×C₄	C₂²×C₃×C₅	1	4	yes	—
131	#127	C₃×C₄×C₅	1	8	yes	direct product factor
132	C₄×D₄	—	2	24	no	—
133	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	6	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	12	yes	—
135	C₂×C₄²	C₂²×C₃×C₅	1	3	yes	—
136	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
137	C₂×C₄²	C₂²×C₃×C₅	1	3	yes	—
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×C₄×C₅	1	8	yes	—
145	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
146	C₃×D₄	—	2	48	no	—
147	C₂×C₃×C₄	—	2	6	no	—
148	C₂×C₃×C₄	—	2	12	no	—
149	C₂×C₃×C₄	—	4	16	no	—
150	C₂×C₃×C₄	C₂²×C₄×C₅	1	6	yes	—
151	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
152	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
153	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
154	C₂×C₃×C₄	C₅×#41	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	—	2	6	no	—
162	C₄×C₅	—	4	4	no	—
163	C₄×C₅	C₃×C₄×D₄	1	12	yes	complemented
164	C₄×C₅	C₃×#127	1	4	yes	direct product factor
165	C₂⁴	C₂³×C₃×C₅	1	4	yes	—
166	C₂×D₄	—	2	12	no	—
167	C₂×D₄	—	2	12	no	—
168	C₂×D₄	C₂×C₃×C₄×C₅	1	24	yes	complemented
169	C₂²×C₄	—	2	24	no	—
170	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
171	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
172	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
173	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
174	C₄⋊C₄	—	2	12	no	—
175	C₂²⋊C₄	—	2	24	no	—
176	C₄²	—	2	6	no	—
177	C₄²	C₂×C₃×C₄×C₅	1	4	yes	complemented
178	C₄²	C₃×C₅×D₄	1	12	yes	complemented
179	C₃×C₅	C₄×#127	1	1	yes	characteristic, direct product factor, Hall
180	C₂²×C₃	—	2	24	no	—
181	C₂²×C₃	—	4	4	no	—
182	C₂²×C₃	—	4	12	no	—
183	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
184	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
185	C₂²×C₃	C₂×C₅×#41	1	3	yes	—
186	C₃×C₄	—	2	6	no	—
187	C₃×C₄	—	4	4	no	—
188	C₃×C₄	C₄×C₅×D₄	1	12	yes	complemented
189	C₃×C₄	C₅×#127	1	4	yes	direct product factor
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×C₄×C₅	1	8	yes	—
197	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
198	D₄	—	2	48	no	—
199	C₂×C₄	—	2	6	no	—
200	C₂×C₄	—	2	12	no	—
201	C₂×C₄	—	4	16	no	—
202	C₂×C₄	C₂²×C₃×C₄×C₅	1	6	yes	—
203	C₂×C₄	C₂×C₃×C₅×D₄	1	6	yes	—
204	C₂×C₄	C₂×C₃×C₅×D₄	1	6	yes	—
205	C₂×C₄	C₂×C₃×C₅×D₄	1	6	yes	—
206	C₂×C₄	C₃×C₅×#41	1	6	yes	—
207	C₂×C₃	—	4	8	no	—
208	C₂×C₃	C₂×C₄×C₅×D₄	1	3	yes	—
209	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
210	C₂×C₃	C₅×#532	1	3	yes	—
211	C₅	C₃×C₄×#127	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²	—	2	24	no	—
213	C₂²	—	4	4	no	—
214	C₂²	—	4	12	no	—
215	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
216	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
217	C₂²	C₂×C₃×C₅×#41	1	3	yes	—
218	C₄	—	2	6	no	—
219	C₄	—	4	4	no	—
220	C₄	C₃×C₄×C₅×D₄	1	12	yes	complemented
221	C₄	C₃×C₅×#127	1	4	yes	direct product factor
222	C₃	C₄×C₅×#127	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	—	4	8	no	—
224	C₂	C₂×C₃×C₄×C₅×D₄	1	3	yes	—
225	C₂	C₂×C₃×C₅×#127	1	1	yes	characteristic
226	C₂	C₃×C₅×#532	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×C₅×D₄	C₂	semidirect
3. C₃×C₅×#390	C₂	semidirect
4. C₃×C₅×#384	C₂	semidirect
5. C₃×C₄³×C₅	C₂	semidirect
6. C₄×C₅×#127	C₃	direct (N × H)
9. C₃×C₅×#127	C₄	direct (N × H)
16. C₃×C₄×#127	C₅	direct (N × H)
18. C₂×C₄×C₅×D₄	C₂×C₃	semidirect
19. C₅×#390	C₂×C₃	semidirect
20. C₅×#384	C₂×C₃	semidirect
21. C₄³×C₅	C₂×C₃	semidirect
25. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
34. C₃×C₄²×C₅	C₂×C₄	semidirect
35. C₃×C₄²×C₅	D₄	semidirect
37. C₂×C₃×C₄×D₄	C₂×C₅	semidirect
38. C₃×#390	C₂×C₅	semidirect
39. C₃×#384	C₂×C₅	semidirect
40. C₃×C₄³	C₂×C₅	semidirect
43. C₅×#127	C₃×C₄	direct (N × H)
50. C₄×#127	C₃×C₅	direct (N × H)
66. C₃×#127	C₄×C₅	direct (N × H)
76. C₂×C₅×D₄	C₂×C₃×C₄	semidirect
85. C₄²×C₅	C₂×C₃×C₄	semidirect
86. C₄²×C₅	C₃×D₄	semidirect
88. C₂×C₄×D₄	C₂×C₃×C₅	semidirect
89. #390	C₂×C₃×C₅	semidirect
90. #384	C₂×C₃×C₅	semidirect
91. C₄³	C₂×C₃×C₅	semidirect
100. C₃×C₄×C₅	C₄×D₄	semidirect
101. C₃×C₄×C₅	#127	direct (N × H)
105. C₂×C₃×D₄	C₂×C₄×C₅	semidirect
114. C₃×C₄²	C₂×C₄×C₅	semidirect
115. C₃×C₄²	C₅×D₄	semidirect
131. #127	C₃×C₄×C₅	direct (N × H)
163. C₄×C₅	C₃×C₄×D₄	semidirect
164. C₄×C₅	C₃×#127	direct (N × H)
168. C₂×D₄	C₂×C₃×C₄×C₅	semidirect
177. C₄²	C₂×C₃×C₄×C₅	semidirect
178. C₄²	C₃×C₅×D₄	semidirect
179. C₃×C₅	C₄×#127	direct (N × H)
188. C₃×C₄	C₄×C₅×D₄	semidirect
189. C₃×C₄	C₅×#127	direct (N × H)
211. C₅	C₃×C₄×#127	direct (N × H)
220. C₄	C₃×C₄×C₅×D₄	semidirect
221. C₄	C₃×C₅×#127	direct (N × H)
222. C₃	C₄×C₅×#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×C₄×C₅×D₄	2
C₃×C₅×#390	2
C₃×C₅×#384	2
C₃×C₄³×C₅	2
C₄×C₅×#127	3
C₃×C₄×#127	5

Subgroups — C3×C4×C5×#127 #14275340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₅×#127 #14275340