Subgroups — C₃²×D₅×#127 #72231548

Order 2880.

2988 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	C₂×C₃²×D₄×D₅	C₂	1	6	yes	maximal, complemented
2	C₃²×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#4086	C₂	1	6	yes	maximal, complemented
4	C₃²×#4020	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₄²×D₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×D₅×#127	C₃	1	4	yes	maximal, direct product factor
7	C₂³×C₃²×D₅	C₂²	1	4	yes	—
8	C₂×C₃²×C₅×D₄	C₂²	1	6	yes	complemented
9	C₂×C₃²×C₂²⋊D₅	C₂²	1	12	yes	complemented
10	C₃²×D₄×D₅	—	2	24	no	—
11	C₂×C₃²×D₂₀	C₂²	1	6	yes	complemented
12	C₂×C₃²×C₄×D₅	C₂²	1	3	yes	—
13	C₃²×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
14	C₃²×C₄×Dic₅	C₂²	1	3	yes	complemented
15	C₂×C₃²×#127	—	5	1	no	maximal, Hall
16	C₂×C₃×D₄×D₅	C₂×C₃	1	24	yes	complemented
17	C₃×C₅×#127	C₂×C₃	1	4	yes	complemented
18	C₃×#4086	C₂×C₃	1	24	yes	complemented
19	C₃×#4020	C₂×C₃	1	4	yes	complemented
20	C₃×C₄²×D₅	C₂×C₃	1	4	yes	complemented
21	C₂³×C₃²×C₅	C₂³	1	4	yes	—
22	C₂²×C₃²×D₅	—	2	24	no	—
23	C₂²×C₃²×D₅	C₂³	1	4	yes	—
24	C₂²×C₃²×D₅	C₂³	1	1	yes	characteristic
25	C₃²×C₅×D₄	—	2	12	no	—
26	C₂×C₃²×C₄×C₅	C₂³	1	3	yes	—
27	C₃²×C₂²⋊D₅	—	2	24	no	—
28	C₂×C₃²×Dic₅	C₂³	1	3	yes	—
29	C₃²×D₂₀	—	2	12	no	—
30	C₃²×C₄×D₅	D₄	1	12	yes	complemented
31	D₅×#127	C₃²	1	1	yes	characteristic, direct product factor, Hall
32	C₂²×C₃²×D₄	—	5	6	no	—
33	C₃²×#127	—	5	6	no	—
34	C₃²×#127	—	5	1	no	—
35	C₃²×#127	D₅	1	1	yes	characteristic, direct product factor
36	C₂×C₃²×C₄²	—	5	1	no	—
37	C₂³×C₃×D₅	C₂²×C₃	1	16	yes	—
38	C₂×C₃×C₅×D₄	C₂²×C₃	1	24	yes	complemented
39	C₂×C₃×C₂²⋊D₅	C₂²×C₃	1	48	yes	complemented
40	C₃×D₄×D₅	—	2	96	no	—
41	C₂×C₃×D₂₀	C₂²×C₃	1	24	yes	complemented
42	C₂×C₃×C₄×D₅	C₂²×C₃	1	12	yes	—
43	C₃×C₄²×C₅	C₂²×C₃	1	4	yes	complemented
44	C₃×C₄×Dic₅	C₂²×C₃	1	12	yes	complemented
45	C₂×C₃×#127	—	5	4	no	—
46	C₂²×C₃²×C₅	—	2	12	no	—
47	C₂²×C₃²×C₅	C₂⁴	1	1	yes	characteristic
48	C₂×C₃²×D₅	—	2	12	no	—
49	C₂×C₃²×D₅	—	4	16	no	—
50	C₂×C₃²×D₅	C₂×D₄	1	6	yes	—
51	C₃²×C₄×C₅	C₂×D₄	1	6	yes	complemented
52	C₃²×Dic₅	C₂×D₄	1	6	yes	complemented
53	C₂×D₄×D₅	C₂×C₃²	1	6	yes	complemented
54	C₅×#127	C₂×C₃²	1	1	yes	characteristic, complemented
55	#4086	C₂×C₃²	1	6	yes	complemented
56	#4020	C₂×C₃²	1	1	yes	characteristic, complemented
57	C₄²×D₅	C₂×C₃²	1	1	yes	characteristic, complemented
58	C₂⁴×C₃²	—	5	4	no	—
59	C₂×C₃²×D₄	—	5	12	no	—
60	C₂×C₃²×D₄	—	5	6	no	—
61	C₂×C₃²×D₄	—	10	24	no	—
62	C₂×C₃²×D₄	C₂×D₅	1	6	yes	complemented
63	C₂²×C₃²×C₄	—	5	3	no	—
64	C₃²×C₄²	—	5	3	no	—
65	C₃²×C₄²	C₂×D₅	1	1	yes	characteristic, complemented
66	C₂³×C₃×C₅	C₂³×C₃	1	16	yes	—
67	C₂²×C₃×D₅	—	2	96	no	—
68	C₂²×C₃×D₅	C₂³×C₃	1	16	yes	—
69	C₂²×C₃×D₅	C₂³×C₃	1	4	yes	—
70	C₃×C₅×D₄	—	2	48	no	—
71	C₂×C₃×C₄×C₅	C₂³×C₃	1	12	yes	—
72	C₃×C₂²⋊D₅	—	2	96	no	—
73	C₂×C₃×Dic₅	C₂³×C₃	1	12	yes	—
74	C₃×D₂₀	—	2	48	no	—
75	C₃×C₄×D₅	C₃×D₄	1	48	yes	complemented
76	C₂²×C₃×D₄	—	5	24	no	—
77	C₃×#127	—	5	24	no	—
78	C₃×#127	—	5	4	no	—
79	C₃×#127	C₃×D₅	1	4	yes	direct product factor
80	C₂×C₃×C₄²	—	5	4	no	—
81	C₂×C₃²×C₅	—	4	4	no	—
82	C₂×C₃²×C₅	C₂²×D₄	1	3	yes	—
83	C₃²×D₅	—	4	4	no	—
84	C₃²×D₅	#127	1	4	yes	direct product factor
85	C₂³×D₅	C₂²×C₃²	1	4	yes	—
86	C₂×C₅×D₄	C₂²×C₃²	1	6	yes	complemented
87	C₂×C₂²⋊D₅	C₂²×C₃²	1	12	yes	complemented
88	D₄×D₅	—	2	24	no	—
89	C₂×D₂₀	C₂²×C₃²	1	6	yes	complemented
90	C₂×C₄×D₅	C₂²×C₃²	1	3	yes	—
91	C₄²×C₅	C₂²×C₃²	1	1	yes	characteristic, complemented
92	C₄×Dic₅	C₂²×C₃²	1	3	yes	complemented
93	C₂³×C₃²	—	5	1	no	—
94	C₂³×C₃²	—	5	4	no	—
95	C₂³×C₃²	—	10	24	no	—
96	C₂³×C₃²	C₂²×D₅	1	4	yes	—
97	C₃²×D₄	—	2	12	no	—
98	C₃²×D₄	—	10	24	no	—
99	C₃²×D₄	—	10	12	no	—
100	C₂×C₃²×C₄	—	5	3	no	—
101	C₂×C₃²×C₄	—	5	12	no	—
102	C₂×C₃²×C₄	C₂²×D₅	1	3	yes	—
103	C₂×#127	—	5	1	no	Sylow
104	C₂²×C₃×C₅	—	2	48	no	—
105	C₂²×C₃×C₅	C₂⁴×C₃	1	4	yes	—
106	C₂×C₃×D₅	—	2	48	no	—
107	C₂×C₃×D₅	—	4	64	no	—
108	C₂×C₃×D₅	C₂×C₃×D₄	1	24	yes	—
109	C₃×C₄×C₅	C₂×C₃×D₄	1	24	yes	complemented
110	C₃×Dic₅	C₂×C₃×D₄	1	24	yes	complemented
111	C₂⁴×C₃	—	5	16	no	—
112	C₂×C₃×D₄	—	5	48	no	—
113	C₂×C₃×D₄	—	5	24	no	—
114	C₂×C₃×D₄	—	10	96	no	—
115	C₂×C₃×D₄	C₂×C₃×D₅	1	24	yes	complemented
116	C₂²×C₃×C₄	—	5	12	no	—
117	C₃×C₄²	—	5	12	no	—
118	C₃×C₄²	C₂×C₃×D₅	1	4	yes	complemented
119	C₃²×C₅	C₂×#127	1	1	yes	characteristic, complemented, Hall
120	C₂³×C₅	C₂³×C₃²	1	4	yes	—
121	C₂²×D₅	—	2	24	no	—
122	C₂²×D₅	C₂³×C₃²	1	4	yes	—
123	C₂²×D₅	C₂³×C₃²	1	1	yes	characteristic
124	C₅×D₄	—	2	12	no	—
125	C₂×C₄×C₅	C₂³×C₃²	1	3	yes	—
126	C₂²⋊D₅	—	2	24	no	—
127	C₂×Dic₅	C₂³×C₃²	1	3	yes	—
128	D₂₀	—	2	12	no	—
129	C₄×D₅	C₃²×D₄	1	12	yes	complemented
130	C₂²×C₃²	—	2	12	no	—
131	C₂²×C₃²	—	5	6	no	—
132	C₂²×C₃²	—	10	12	no	—
133	C₂²×C₃²	—	20	16	no	—
134	C₂²×C₃²	C₂³×D₅	1	1	yes	characteristic
135	C₃²×C₄	—	5	6	no	—
136	C₃²×C₄	D₄×D₅	1	6	yes	complemented
137	C₂²×D₄	—	5	6	no	—
138	#127	—	5	6	no	—
139	#127	—	5	1	no	—
140	#127	C₃²×D₅	1	1	yes	characteristic, direct product factor
141	C₂×C₄²	—	5	1	no	—
142	C₂×C₃×C₅	—	4	16	no	—
143	C₂×C₃×C₅	C₂²×C₃×D₄	1	12	yes	—
144	C₃×D₅	—	4	16	no	—
145	C₃×D₅	C₃×#127	1	16	yes	direct product factor
146	C₂³×C₃	—	5	4	no	—
147	C₂³×C₃	—	5	16	no	—
148	C₂³×C₃	—	10	96	no	—
149	C₂³×C₃	C₂²×C₃×D₅	1	16	yes	—
150	C₃×D₄	—	2	48	no	—
151	C₃×D₄	—	10	96	no	—
152	C₃×D₄	—	10	48	no	—
153	C₂×C₃×C₄	—	5	12	no	—
154	C₂×C₃×C₄	—	5	48	no	—
155	C₂×C₃×C₄	C₂²×C₃×D₅	1	12	yes	—
156	C₂²×C₅	—	2	12	no	—
157	C₂²×C₅	C₂⁴×C₃²	1	1	yes	characteristic
158	C₂×D₅	—	2	12	no	—
159	C₂×D₅	—	4	16	no	—
160	C₂×D₅	C₂×C₃²×D₄	1	6	yes	—
161	C₄×C₅	C₂×C₃²×D₄	1	6	yes	complemented
162	Dic₅	C₂×C₃²×D₄	1	6	yes	complemented
163	C₂×C₃²	—	4	4	no	—
164	C₂×C₃²	—	5	4	no	—
165	C₂×C₃²	—	20	4	no	—
166	C₂×C₃²	C₂×D₄×D₅	1	3	yes	—
167	C₂⁴	—	5	4	no	—
168	C₂×D₄	—	5	12	no	—
169	C₂×D₄	—	5	6	no	—
170	C₂×D₄	—	10	24	no	—
171	C₂×D₄	C₂×C₃²×D₅	1	6	yes	complemented
172	C₂²×C₄	—	5	3	no	—
173	C₄²	—	5	3	no	—
174	C₄²	C₂×C₃²×D₅	1	1	yes	characteristic, complemented
175	C₃×C₅	C₂×C₃×#127	1	4	yes	complemented
176	C₂²×C₃	—	2	48	no	—
177	C₂²×C₃	—	5	24	no	—
178	C₂²×C₃	—	10	48	no	—
179	C₂²×C₃	—	20	64	no	—
180	C₂²×C₃	C₂³×C₃×D₅	1	4	yes	—
181	C₃×C₄	—	5	24	no	—
182	C₃×C₄	C₃×D₄×D₅	1	24	yes	complemented
183	C₂×C₅	—	4	4	no	—
184	C₂×C₅	C₂²×C₃²×D₄	1	3	yes	—
185	D₅	—	4	4	no	—
186	D₅	C₃²×#127	1	4	yes	direct product factor
187	C₃²	D₅×#127	1	1	yes	characteristic, direct product factor, Sylow
188	C₂³	—	5	1	no	—
189	C₂³	—	5	4	no	—
190	C₂³	—	10	24	no	—
191	C₂³	C₂²×C₃²×D₅	1	4	yes	—
192	D₄	—	2	12	no	—
193	D₄	—	10	24	no	—
194	D₄	—	10	12	no	—
195	C₂×C₄	—	5	3	no	—
196	C₂×C₄	—	5	12	no	—
197	C₂×C₄	C₂²×C₃²×D₅	1	3	yes	—
198	C₂×C₃	—	4	16	no	—
199	C₂×C₃	—	5	16	no	—
200	C₂×C₃	—	20	16	no	—
201	C₂×C₃	C₂×C₃×D₄×D₅	1	12	yes	—
202	C₅	C₂×C₃²×#127	1	1	yes	characteristic, complemented, Sylow
203	C₂²	—	2	12	no	—
204	C₂²	—	5	6	no	—
205	C₂²	—	10	12	no	—
206	C₂²	—	20	16	no	—
207	C₂²	C₂³×C₃²×D₅	1	1	yes	characteristic
208	C₄	—	5	6	no	—
209	C₄	C₃²×D₄×D₅	1	6	yes	complemented
210	C₃	C₃×D₅×#127	1	4	yes	direct product factor
211	C₂	—	4	4	no	—
212	C₂	—	5	4	no	—
213	C₂	—	20	4	no	—
214	C₂	C₂×C₃²×D₄×D₅	1	3	yes	—

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₅
Fitting	C₃²×C₅×#127	C₂
socle	C₂²×C₃²×C₅	C₂⁴
hypercenter	C₃²×#127	D₅
hypocenter	C₅	C₂×C₃²×#127
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×D₄×D₅	C₂	semidirect
2. C₃²×C₅×#127	C₂	semidirect
3. C₃²×#4086	C₂	semidirect
4. C₃²×#4020	C₂	semidirect
5. C₃²×C₄²×D₅	C₂	semidirect
6. C₃×D₅×#127	C₃	direct (N × H)
8. C₂×C₃²×C₅×D₄	C₂²	semidirect
9. C₂×C₃²×C₂²⋊D₅	C₂²	semidirect
11. C₂×C₃²×D₂₀	C₂²	semidirect
13. C₃²×C₄²×C₅	C₂²	semidirect
14. C₃²×C₄×Dic₅	C₂²	semidirect
16. C₂×C₃×D₄×D₅	C₂×C₃	semidirect
17. C₃×C₅×#127	C₂×C₃	semidirect
18. C₃×#4086	C₂×C₃	semidirect
19. C₃×#4020	C₂×C₃	semidirect
20. C₃×C₄²×D₅	C₂×C₃	semidirect
30. C₃²×C₄×D₅	D₄	semidirect
31. D₅×#127	C₃²	direct (N × H)
35. C₃²×#127	D₅	direct (N × H)
38. C₂×C₃×C₅×D₄	C₂²×C₃	semidirect
39. C₂×C₃×C₂²⋊D₅	C₂²×C₃	semidirect
41. C₂×C₃×D₂₀	C₂²×C₃	semidirect
43. C₃×C₄²×C₅	C₂²×C₃	semidirect
44. C₃×C₄×Dic₅	C₂²×C₃	semidirect
51. C₃²×C₄×C₅	C₂×D₄	semidirect
52. C₃²×Dic₅	C₂×D₄	semidirect
53. C₂×D₄×D₅	C₂×C₃²	semidirect
54. C₅×#127	C₂×C₃²	semidirect
55. #4086	C₂×C₃²	semidirect
56. #4020	C₂×C₃²	semidirect
57. C₄²×D₅	C₂×C₃²	semidirect
62. C₂×C₃²×D₄	C₂×D₅	semidirect
65. C₃²×C₄²	C₂×D₅	semidirect
75. C₃×C₄×D₅	C₃×D₄	semidirect
79. C₃×#127	C₃×D₅	direct (N × H)
84. C₃²×D₅	#127	direct (N × H)
86. C₂×C₅×D₄	C₂²×C₃²	semidirect
87. C₂×C₂²⋊D₅	C₂²×C₃²	semidirect
89. C₂×D₂₀	C₂²×C₃²	semidirect
91. C₄²×C₅	C₂²×C₃²	semidirect
92. C₄×Dic₅	C₂²×C₃²	semidirect
109. C₃×C₄×C₅	C₂×C₃×D₄	semidirect
110. C₃×Dic₅	C₂×C₃×D₄	semidirect
115. C₂×C₃×D₄	C₂×C₃×D₅	semidirect
118. C₃×C₄²	C₂×C₃×D₅	semidirect
119. C₃²×C₅	C₂×#127	semidirect
129. C₄×D₅	C₃²×D₄	semidirect
136. C₃²×C₄	D₄×D₅	semidirect
140. #127	C₃²×D₅	direct (N × H)
145. C₃×D₅	C₃×#127	direct (N × H)
161. C₄×C₅	C₂×C₃²×D₄	semidirect
162. Dic₅	C₂×C₃²×D₄	semidirect
171. C₂×D₄	C₂×C₃²×D₅	semidirect
174. C₄²	C₂×C₃²×D₅	semidirect
175. C₃×C₅	C₂×C₃×#127	semidirect
182. C₃×C₄	C₃×D₄×D₅	semidirect
186. D₅	C₃²×#127	direct (N × H)
187. C₃²	D₅×#127	direct (N × H)
202. C₅	C₂×C₃²×#127	semidirect
209. C₄	C₃²×D₄×D₅	semidirect
210. C₃	C₃×D₅×#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×C₃²×D₄×D₅	2
C₃²×C₅×#127	2
C₃²×#4086	2
C₃²×#4020	2
C₃²×C₄²×D₅	2
C₃×D₅×#127	3
C₂×C₃²×#127	5

Subgroups — C32×D5×#127 #72231548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₅×#127 #72231548