Subgroups — C₅²×D₃×#127 #72231093

Order 4800.

3984 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#940	C₂	1	6	yes	maximal, complemented
2	C₅²×#874	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅²×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×C₅²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅²×D₃×D₄	C₂	1	6	yes	maximal, complemented
6	C₂×C₅²×#127	—	3	1	no	maximal, Hall
7	C₂³×C₅²×D₃	C₂²	1	4	yes	—
8	C₂×C₃×C₅²×D₄	C₂²	1	6	yes	complemented
9	C₂×C₅²×#67	C₂²	1	12	yes	complemented
10	C₅²×D₃×D₄	—	2	24	no	—
11	C₂×C₅²×D₁₂	C₂²	1	6	yes	complemented
12	C₂×C₄×C₅²×D₃	C₂²	1	3	yes	—
13	C₃×C₄²×C₅²	C₂²	1	1	yes	characteristic, complemented
14	C₄×C₅²×Dic₃	C₂²	1	3	yes	complemented
15	C₅×D₃×#127	C₅	1	6	yes	maximal, direct product factor
16	C₂²×C₅²×D₄	—	3	6	no	—
17	C₅²×#127	—	3	6	no	—
18	C₅²×#127	—	3	1	no	—
19	C₅²×#127	D₃	1	1	yes	characteristic, direct product factor
20	C₂×C₄²×C₅²	—	3	1	no	—
21	C₂³×C₃×C₅²	C₂³	1	4	yes	—
22	C₂²×C₅²×D₃	—	2	24	no	—
23	C₂²×C₅²×D₃	C₂³	1	4	yes	—
24	C₂²×C₅²×D₃	C₂³	1	1	yes	characteristic
25	C₃×C₅²×D₄	—	2	12	no	—
26	C₂×C₃×C₄×C₅²	C₂³	1	3	yes	—
27	C₅²×#67	—	2	24	no	—
28	C₂×C₅²×Dic₃	C₂³	1	3	yes	—
29	C₅²×D₁₂	—	2	12	no	—
30	C₄×C₅²×D₃	D₄	1	12	yes	complemented
31	C₂×C₅×D₃×D₄	C₂×C₅	1	36	yes	complemented
32	C₃×C₅×#127	C₂×C₅	1	6	yes	complemented
33	C₅×#940	C₂×C₅	1	36	yes	complemented
34	C₅×#874	C₂×C₅	1	6	yes	complemented
35	C₄²×C₅×D₃	C₂×C₅	1	6	yes	complemented
36	C₂⁴×C₅²	—	3	4	no	—
37	C₂×C₅²×D₄	—	3	12	no	—
38	C₂×C₅²×D₄	—	3	6	no	—
39	C₂×C₅²×D₄	—	6	24	no	—
40	C₂×C₅²×D₄	C₂×D₃	1	6	yes	complemented
41	C₂²×C₄×C₅²	—	3	3	no	—
42	C₄²×C₅²	—	3	3	no	—
43	C₄²×C₅²	C₂×D₃	1	1	yes	characteristic, complemented
44	C₂×C₅×#127	—	3	6	no	—
45	C₂²×C₃×C₅²	—	2	12	no	—
46	C₂²×C₃×C₅²	C₂⁴	1	1	yes	characteristic
47	C₂×C₅²×D₃	—	2	12	no	—
48	C₂×C₅²×D₃	—	4	16	no	—
49	C₂×C₅²×D₃	C₂×D₄	1	6	yes	—
50	C₃×C₄×C₅²	C₂×D₄	1	6	yes	complemented
51	C₅²×Dic₃	C₂×D₄	1	6	yes	complemented
52	C₂³×C₅×D₃	C₂²×C₅	1	24	yes	—
53	C₂×C₃×C₅×D₄	C₂²×C₅	1	36	yes	complemented
54	C₂×C₅×#67	C₂²×C₅	1	72	yes	complemented
55	C₅×D₃×D₄	—	2	144	no	—
56	C₂×C₅×D₁₂	C₂²×C₅	1	36	yes	complemented
57	C₂×C₄×C₅×D₃	C₂²×C₅	1	18	yes	—
58	C₃×C₄²×C₅	C₂²×C₅	1	6	yes	complemented
59	C₄×C₅×Dic₃	C₂²×C₅	1	18	yes	complemented
60	C₂³×C₅²	—	3	1	no	—
61	C₂³×C₅²	—	3	4	no	—
62	C₂³×C₅²	—	6	24	no	—
63	C₂³×C₅²	C₂²×D₃	1	4	yes	—
64	C₅²×D₄	—	2	12	no	—
65	C₅²×D₄	—	6	24	no	—
66	C₅²×D₄	—	6	12	no	—
67	C₂×C₄×C₅²	—	3	3	no	—
68	C₂×C₄×C₅²	—	3	12	no	—
69	C₂×C₄×C₅²	C₂²×D₃	1	3	yes	—
70	D₃×#127	C₅²	1	1	yes	characteristic, direct product factor, Hall
71	C₂²×C₅×D₄	—	3	36	no	—
72	C₅×#127	—	3	36	no	—
73	C₅×#127	—	3	6	no	—
74	C₅×#127	C₅×D₃	1	6	yes	direct product factor
75	C₂×C₄²×C₅	—	3	6	no	—
76	C₂×C₃×C₅²	—	4	4	no	—
77	C₂×C₃×C₅²	C₂²×D₄	1	3	yes	—
78	C₅²×D₃	—	4	4	no	—
79	C₅²×D₃	#127	1	4	yes	direct product factor
80	C₂³×C₃×C₅	C₂³×C₅	1	24	yes	—
81	C₂²×C₅×D₃	—	2	144	no	—
82	C₂²×C₅×D₃	C₂³×C₅	1	24	yes	—
83	C₂²×C₅×D₃	C₂³×C₅	1	6	yes	—
84	C₃×C₅×D₄	—	2	72	no	—
85	C₂×C₃×C₄×C₅	C₂³×C₅	1	18	yes	—
86	C₅×#67	—	2	144	no	—
87	C₂×C₅×Dic₃	C₂³×C₅	1	18	yes	—
88	C₅×D₁₂	—	2	72	no	—
89	C₄×C₅×D₃	C₅×D₄	1	72	yes	complemented
90	C₂²×C₅²	—	2	12	no	—
91	C₂²×C₅²	—	3	6	no	—
92	C₂²×C₅²	—	6	12	no	—
93	C₂²×C₅²	—	12	16	no	—
94	C₂²×C₅²	C₂³×D₃	1	1	yes	characteristic
95	C₄×C₅²	—	3	6	no	—
96	C₄×C₅²	D₃×D₄	1	6	yes	complemented
97	C₂×D₃×D₄	C₂×C₅²	1	6	yes	complemented
98	C₃×#127	C₂×C₅²	1	1	yes	characteristic, complemented
99	#940	C₂×C₅²	1	6	yes	complemented
100	#874	C₂×C₅²	1	1	yes	characteristic, complemented
101	C₄²×D₃	C₂×C₅²	1	1	yes	characteristic, complemented
102	C₂⁴×C₅	—	3	24	no	—
103	C₂×C₅×D₄	—	3	72	no	—
104	C₂×C₅×D₄	—	3	36	no	—
105	C₂×C₅×D₄	—	6	144	no	—
106	C₂×C₅×D₄	C₂×C₅×D₃	1	36	yes	complemented
107	C₂²×C₄×C₅	—	3	18	no	—
108	C₄²×C₅	—	3	18	no	—
109	C₄²×C₅	C₂×C₅×D₃	1	6	yes	complemented
110	C₃×C₅²	C₂×#127	1	1	yes	characteristic, complemented, Hall
111	C₂×#127	—	3	1	no	Sylow
112	C₂²×C₃×C₅	—	2	72	no	—
113	C₂²×C₃×C₅	C₂⁴×C₅	1	6	yes	—
114	C₂×C₅×D₃	—	2	72	no	—
115	C₂×C₅×D₃	—	4	96	no	—
116	C₂×C₅×D₃	C₂×C₅×D₄	1	36	yes	—
117	C₃×C₄×C₅	C₂×C₅×D₄	1	36	yes	complemented
118	C₅×Dic₃	C₂×C₅×D₄	1	36	yes	complemented
119	C₂×C₅²	—	3	4	no	—
120	C₂×C₅²	—	4	4	no	—
121	C₂×C₅²	—	12	4	no	—
122	C₂×C₅²	C₂×D₃×D₄	1	3	yes	—
123	C₂³×D₃	C₂²×C₅²	1	4	yes	—
124	C₂×C₃×D₄	C₂²×C₅²	1	6	yes	complemented
125	C₂×#67	C₂²×C₅²	1	12	yes	complemented
126	D₃×D₄	—	2	24	no	—
127	C₂×D₁₂	C₂²×C₅²	1	6	yes	complemented
128	C₂×C₄×D₃	C₂²×C₅²	1	3	yes	—
129	C₃×C₄²	C₂²×C₅²	1	1	yes	characteristic, complemented
130	C₄×Dic₃	C₂²×C₅²	1	3	yes	complemented
131	C₂³×C₅	—	3	6	no	—
132	C₂³×C₅	—	3	24	no	—
133	C₂³×C₅	—	6	144	no	—
134	C₂³×C₅	C₂²×C₅×D₃	1	24	yes	—
135	C₅×D₄	—	2	72	no	—
136	C₅×D₄	—	6	144	no	—
137	C₅×D₄	—	6	72	no	—
138	C₂×C₄×C₅	—	3	18	no	—
139	C₂×C₄×C₅	—	3	72	no	—
140	C₂×C₄×C₅	C₂²×C₅×D₃	1	18	yes	—
141	C₂²×D₄	—	3	6	no	—
142	#127	—	3	6	no	—
143	#127	—	3	1	no	—
144	#127	C₅²×D₃	1	1	yes	characteristic, direct product factor
145	C₂×C₄²	—	3	1	no	—
146	C₂×C₃×C₅	—	4	24	no	—
147	C₂×C₃×C₅	C₂²×C₅×D₄	1	18	yes	—
148	C₅×D₃	—	4	24	no	—
149	C₅×D₃	C₅×#127	1	24	yes	direct product factor
150	C₅²	D₃×#127	1	1	yes	characteristic, direct product factor, Sylow
151	C₂³×C₃	C₂³×C₅²	1	4	yes	—
152	C₂²×D₃	—	2	24	no	—
153	C₂²×D₃	C₂³×C₅²	1	4	yes	—
154	C₂²×D₃	C₂³×C₅²	1	1	yes	characteristic
155	C₃×D₄	—	2	12	no	—
156	C₂×C₃×C₄	C₂³×C₅²	1	3	yes	—
157	#67	—	2	24	no	—
158	C₂×Dic₃	C₂³×C₅²	1	3	yes	—
159	D₁₂	—	2	12	no	—
160	C₄×D₃	C₅²×D₄	1	12	yes	complemented
161	C₂²×C₅	—	2	72	no	—
162	C₂²×C₅	—	3	36	no	—
163	C₂²×C₅	—	6	72	no	—
164	C₂²×C₅	—	12	96	no	—
165	C₂²×C₅	C₂³×C₅×D₃	1	6	yes	—
166	C₄×C₅	—	3	36	no	—
167	C₄×C₅	C₅×D₃×D₄	1	36	yes	complemented
168	C₂⁴	—	3	4	no	—
169	C₂×D₄	—	3	12	no	—
170	C₂×D₄	—	3	6	no	—
171	C₂×D₄	—	6	24	no	—
172	C₂×D₄	C₂×C₅²×D₃	1	6	yes	complemented
173	C₂²×C₄	—	3	3	no	—
174	C₄²	—	3	3	no	—
175	C₄²	C₂×C₅²×D₃	1	1	yes	characteristic, complemented
176	C₃×C₅	C₂×C₅×#127	1	6	yes	complemented
177	C₂²×C₃	—	2	12	no	—
178	C₂²×C₃	C₂⁴×C₅²	1	1	yes	characteristic
179	C₂×D₃	—	2	12	no	—
180	C₂×D₃	—	4	16	no	—
181	C₂×D₃	C₂×C₅²×D₄	1	6	yes	—
182	C₃×C₄	C₂×C₅²×D₄	1	6	yes	complemented
183	Dic₃	C₂×C₅²×D₄	1	6	yes	complemented
184	C₂×C₅	—	3	24	no	—
185	C₂×C₅	—	4	24	no	—
186	C₂×C₅	—	12	24	no	—
187	C₂×C₅	C₂×C₅×D₃×D₄	1	18	yes	—
188	C₂³	—	3	1	no	—
189	C₂³	—	3	4	no	—
190	C₂³	—	6	24	no	—
191	C₂³	C₂²×C₅²×D₃	1	4	yes	—
192	D₄	—	2	12	no	—
193	D₄	—	6	24	no	—
194	D₄	—	6	12	no	—
195	C₂×C₄	—	3	3	no	—
196	C₂×C₄	—	3	12	no	—
197	C₂×C₄	C₂²×C₅²×D₃	1	3	yes	—
198	C₂×C₃	—	4	4	no	—
199	C₂×C₃	C₂²×C₅²×D₄	1	3	yes	—
200	D₃	—	4	4	no	—
201	D₃	C₅²×#127	1	4	yes	direct product factor
202	C₅	C₅×D₃×#127	1	6	yes	direct product factor
203	C₂²	—	2	12	no	—
204	C₂²	—	3	6	no	—
205	C₂²	—	6	12	no	—
206	C₂²	—	12	16	no	—
207	C₂²	C₂³×C₅²×D₃	1	1	yes	characteristic
208	C₄	—	3	6	no	—
209	C₄	C₅²×D₃×D₄	1	6	yes	complemented
210	C₃	C₂×C₅²×#127	1	1	yes	characteristic, complemented, Sylow
211	C₂	—	3	4	no	—
212	C₂	—	4	4	no	—
213	C₂	—	12	4	no	—
214	C₂	C₂×C₅²×D₃×D₄	1	3	yes	—

	subgroup	G/H
center	C₂²×C₅²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅²
Frattini	C₂²	C₂³×C₅²×D₃
Fitting	C₃×C₅²×#127	C₂
socle	C₂²×C₃×C₅²	C₂⁴
hypercenter	C₅²×#127	D₃
hypocenter	C₃	C₂×C₅²×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅² ◁ C₅²×#127

Complements

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

N	H ≅ G/N	type
1. C₅²×#940	C₂	semidirect
2. C₅²×#874	C₂	semidirect
3. C₃×C₅²×#127	C₂	semidirect
4. C₄²×C₅²×D₃	C₂	semidirect
5. C₂×C₅²×D₃×D₄	C₂	semidirect
8. C₂×C₃×C₅²×D₄	C₂²	semidirect
9. C₂×C₅²×#67	C₂²	semidirect
11. C₂×C₅²×D₁₂	C₂²	semidirect
13. C₃×C₄²×C₅²	C₂²	semidirect
14. C₄×C₅²×Dic₃	C₂²	semidirect
15. C₅×D₃×#127	C₅	direct (N × H)
19. C₅²×#127	D₃	direct (N × H)
30. C₄×C₅²×D₃	D₄	semidirect
31. C₂×C₅×D₃×D₄	C₂×C₅	semidirect
32. C₃×C₅×#127	C₂×C₅	semidirect
33. C₅×#940	C₂×C₅	semidirect
34. C₅×#874	C₂×C₅	semidirect
35. C₄²×C₅×D₃	C₂×C₅	semidirect
40. C₂×C₅²×D₄	C₂×D₃	semidirect
43. C₄²×C₅²	C₂×D₃	semidirect
50. C₃×C₄×C₅²	C₂×D₄	semidirect
51. C₅²×Dic₃	C₂×D₄	semidirect
53. C₂×C₃×C₅×D₄	C₂²×C₅	semidirect
54. C₂×C₅×#67	C₂²×C₅	semidirect
56. C₂×C₅×D₁₂	C₂²×C₅	semidirect
58. C₃×C₄²×C₅	C₂²×C₅	semidirect
59. C₄×C₅×Dic₃	C₂²×C₅	semidirect
70. D₃×#127	C₅²	direct (N × H)
74. C₅×#127	C₅×D₃	direct (N × H)
79. C₅²×D₃	#127	direct (N × H)
89. C₄×C₅×D₃	C₅×D₄	semidirect
96. C₄×C₅²	D₃×D₄	semidirect
97. C₂×D₃×D₄	C₂×C₅²	semidirect
98. C₃×#127	C₂×C₅²	semidirect
99. #940	C₂×C₅²	semidirect
100. #874	C₂×C₅²	semidirect
101. C₄²×D₃	C₂×C₅²	semidirect
106. C₂×C₅×D₄	C₂×C₅×D₃	semidirect
109. C₄²×C₅	C₂×C₅×D₃	semidirect
110. C₃×C₅²	C₂×#127	semidirect
117. C₃×C₄×C₅	C₂×C₅×D₄	semidirect
118. C₅×Dic₃	C₂×C₅×D₄	semidirect
124. C₂×C₃×D₄	C₂²×C₅²	semidirect
125. C₂×#67	C₂²×C₅²	semidirect
127. C₂×D₁₂	C₂²×C₅²	semidirect
129. C₃×C₄²	C₂²×C₅²	semidirect
130. C₄×Dic₃	C₂²×C₅²	semidirect
144. #127	C₅²×D₃	direct (N × H)
149. C₅×D₃	C₅×#127	direct (N × H)
150. C₅²	D₃×#127	direct (N × H)
160. C₄×D₃	C₅²×D₄	semidirect
167. C₄×C₅	C₅×D₃×D₄	semidirect
172. C₂×D₄	C₂×C₅²×D₃	semidirect
175. C₄²	C₂×C₅²×D₃	semidirect
176. C₃×C₅	C₂×C₅×#127	semidirect
182. C₃×C₄	C₂×C₅²×D₄	semidirect
183. Dic₃	C₂×C₅²×D₄	semidirect
201. D₃	C₅²×#127	direct (N × H)
202. C₅	C₅×D₃×#127	direct (N × H)
209. C₄	C₅²×D₃×D₄	semidirect
210. C₃	C₂×C₅²×#127	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#940	2
C₅²×#874	2
C₃×C₅²×#127	2
C₄²×C₅²×D₃	2
C₂×C₅²×D₃×D₄	2
C₂×C₅²×#127	3
C₅×D₃×#127	5

Subgroups — C52×D3×#127 #72231093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×D₃×#127 #72231093