Subgroups — C₃²×#4027 #13087807

Order 1440.

564 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×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×C₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#685	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#683	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4027	C₃	1	4	yes	maximal, direct product factor
9	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×D₅	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₃²×C₂²⋊D₅	C₄	1	4	yes	complemented
14	C₂×C₃²×Dic₅	—	2	2	no	—
15	C₂×C₃²×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×C₃²×Dic₅	C₂²	1	1	yes	characteristic
17	C₂×C₃²×Dic₅	C₂²	1	1	yes	characteristic
18	C₃²×C₄×D₅	—	2	2	no	—
19	C₃²×C₄×D₄	—	5	1	no	maximal, Hall
20	C₂×C₃×C₂²⋊D₅	C₂×C₃	1	4	yes	—
21	C₂²×C₃×Dic₅	C₂×C₃	1	4	yes	complemented
22	C₂×C₃×C₄×D₅	C₂×C₃	1	4	yes	complemented
23	C₃×C₅×C₂²⋊C₄	C₂×C₃	1	4	yes	complemented
24	C₃×#685	C₂×C₃	1	4	yes	complemented
25	C₃×#683	C₂×C₃	1	4	yes	complemented
26	C₃×C₄×Dic₅	C₂×C₃	1	4	yes	complemented
27	C₂²×C₃²×C₅	—	2	1	no	—
28	C₂²×C₃²×C₅	—	2	1	no	—
29	C₂²×C₃²×C₅	C₂×C₄	1	2	yes	complemented
30	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₃²×D₅	—	2	1	no	—
32	C₂×C₃²×D₅	—	2	1	no	—
33	C₂×C₃²×D₅	C₂×C₄	1	2	yes	complemented
34	C₃²×C₄×C₅	—	2	1	no	—
35	C₃²×C₄×C₅	—	2	1	no	—
36	C₃²×Dic₅	—	2	1	no	—
37	C₃²×Dic₅	C₂×C₄	1	2	yes	complemented
38	C₃²×Dic₅	D₄	1	2	yes	complemented
39	#4027	C₃²	1	1	yes	characteristic, direct product factor, Hall
40	C₂×C₃²×D₄	—	5	1	no	—
41	C₂²×C₃²×C₄	—	5	1	no	—
42	C₂²×C₃²×C₄	—	5	1	no	—
43	C₃²×C₄⋊C₄	—	5	1	no	—
44	C₃²×C₂²⋊C₄	—	5	1	no	—
45	C₃²×C₂²⋊C₄	D₅	1	1	yes	characteristic, complemented
46	C₃²×C₄²	—	5	1	no	—
47	C₂³×C₃×C₅	C₂²×C₃	1	4	yes	—
48	C₂²×C₃×D₅	C₂²×C₃	1	4	yes	—
49	C₂×C₃×C₄×C₅	C₂²×C₃	1	4	yes	—
50	C₂×C₃×C₄×C₅	C₂²×C₃	1	4	yes	—
51	C₃×C₂²⋊D₅	C₃×C₄	1	16	yes	complemented
52	C₂×C₃×Dic₅	—	2	8	no	—
53	C₂×C₃×Dic₅	C₂²×C₃	1	4	yes	—
54	C₂×C₃×Dic₅	C₂²×C₃	1	4	yes	—
55	C₂×C₃×Dic₅	C₂²×C₃	1	4	yes	—
56	C₃×C₄×D₅	—	2	8	no	—
57	C₃×C₄×D₄	—	5	4	no	—
58	C₂×C₃²×C₅	—	2	2	no	—
59	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃²×C₅	#41	1	1	yes	characteristic
62	C₃²×D₅	—	2	2	no	—
63	C₂×C₂²⋊D₅	C₂×C₃²	1	1	yes	characteristic
64	C₂²×Dic₅	C₂×C₃²	1	1	yes	characteristic, complemented
65	C₂×C₄×D₅	C₂×C₃²	1	1	yes	characteristic, complemented
66	C₅×C₂²⋊C₄	C₂×C₃²	1	1	yes	characteristic, complemented
67	#685	C₂×C₃²	1	1	yes	characteristic, complemented
68	#683	C₂×C₃²	1	1	yes	characteristic, complemented
69	C₄×Dic₅	C₂×C₃²	1	1	yes	characteristic, complemented
70	C₂³×C₃²	—	5	1	no	—
71	C₂³×C₃²	C₂×D₅	1	1	yes	characteristic
72	C₃²×D₄	—	5	4	no	—
73	C₂×C₃²×C₄	—	5	1	no	—
74	C₂×C₃²×C₄	—	5	1	no	—
75	C₂×C₃²×C₄	—	5	1	no	—
76	C₂×C₃²×C₄	—	10	2	no	—
77	C₂×C₃²×C₄	—	10	2	no	—
78	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
80	C₂²×C₃×C₅	—	2	4	no	—
81	C₂²×C₃×C₅	—	2	4	no	—
82	C₂²×C₃×C₅	C₂×C₃×C₄	1	8	yes	complemented
83	C₂²×C₃×C₅	C₂³×C₃	1	4	yes	—
84	C₂×C₃×D₅	—	2	4	no	—
85	C₂×C₃×D₅	—	2	4	no	—
86	C₂×C₃×D₅	C₂×C₃×C₄	1	8	yes	complemented
87	C₃×C₄×C₅	—	2	4	no	—
88	C₃×C₄×C₅	—	2	4	no	—
89	C₃×Dic₅	—	2	4	no	—
90	C₃×Dic₅	C₂×C₃×C₄	1	8	yes	complemented
91	C₃×Dic₅	C₃×D₄	1	8	yes	complemented
92	C₂×C₃×D₄	—	5	4	no	—
93	C₂²×C₃×C₄	—	5	4	no	—
94	C₂²×C₃×C₄	—	5	4	no	—
95	C₃×C₄⋊C₄	—	5	4	no	—
96	C₃×C₂²⋊C₄	—	5	4	no	—
97	C₃×C₂²⋊C₄	C₃×D₅	1	4	yes	complemented
98	C₃×C₄²	—	5	4	no	—
99	C₃²×C₅	C₄×D₄	1	1	yes	characteristic, complemented, Hall
100	C₂³×C₅	C₂²×C₃²	1	1	yes	characteristic
101	C₂²×D₅	C₂²×C₃²	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂²×C₃²	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂²×C₃²	1	1	yes	characteristic
104	C₂²⋊D₅	C₃²×C₄	1	4	yes	complemented
105	C₂×Dic₅	—	2	2	no	—
106	C₂×Dic₅	C₂²×C₃²	1	1	yes	characteristic
107	C₂×Dic₅	C₂²×C₃²	1	1	yes	characteristic
108	C₂×Dic₅	C₂²×C₃²	1	1	yes	characteristic
109	C₄×D₅	—	2	2	no	—
110	C₂²×C₃²	—	2	1	no	—
111	C₂²×C₃²	—	2	1	no	—
112	C₂²×C₃²	—	5	2	no	—
113	C₂²×C₃²	—	10	1	no	—
114	C₂²×C₃²	—	10	1	no	—
115	C₂²×C₃²	C₄×D₅	1	2	yes	complemented
116	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
117	C₃²×C₄	—	2	1	no	—
118	C₃²×C₄	—	2	1	no	—
119	C₃²×C₄	—	5	2	no	—
120	C₃²×C₄	—	5	2	no	—
121	C₃²×C₄	—	10	1	no	—
122	C₄×D₄	—	5	1	no	Sylow
123	C₂×C₃×C₅	—	2	8	no	—
124	C₂×C₃×C₅	C₂²×C₃×C₄	1	4	yes	—
125	C₂×C₃×C₅	C₂×C₃×D₄	1	4	yes	—
126	C₂×C₃×C₅	C₃×#41	1	4	yes	—
127	C₃×D₅	—	2	8	no	—
128	C₂³×C₃	—	5	4	no	—
129	C₂³×C₃	C₂×C₃×D₅	1	4	yes	—
130	C₃×D₄	—	5	16	no	—
131	C₂×C₃×C₄	—	5	4	no	—
132	C₂×C₃×C₄	—	5	4	no	—
133	C₂×C₃×C₄	—	5	4	no	—
134	C₂×C₃×C₄	—	10	8	no	—
135	C₂×C₃×C₄	—	10	8	no	—
136	C₂×C₃×C₄	C₂×C₃×D₅	1	4	yes	—
137	C₂×C₃×C₄	C₂×C₃×D₅	1	4	yes	—
138	C₂²×C₅	—	2	1	no	—
139	C₂²×C₅	—	2	1	no	—
140	C₂²×C₅	C₂×C₃²×C₄	1	2	yes	complemented
141	C₂²×C₅	C₂³×C₃²	1	1	yes	characteristic
142	C₂×D₅	—	2	1	no	—
143	C₂×D₅	—	2	1	no	—
144	C₂×D₅	C₂×C₃²×C₄	1	2	yes	complemented
145	C₄×C₅	—	2	1	no	—
146	C₄×C₅	—	2	1	no	—
147	Dic₅	—	2	1	no	—
148	Dic₅	C₂×C₃²×C₄	1	2	yes	complemented
149	Dic₅	C₃²×D₄	1	2	yes	complemented
150	C₂×C₃²	—	2	2	no	—
151	C₂×C₃²	—	10	2	no	—
152	C₂×C₃²	C₂×C₄×D₅	1	1	yes	characteristic
153	C₂×C₃²	D₄×D₅	1	1	yes	characteristic
154	C₂×C₃²	#711	1	1	yes	characteristic
155	C₂×D₄	—	5	1	no	—
156	C₂²×C₄	—	5	1	no	—
157	C₂²×C₄	—	5	1	no	—
158	C₄⋊C₄	—	5	1	no	—
159	C₂²⋊C₄	—	5	1	no	—
160	C₂²⋊C₄	C₃²×D₅	1	1	yes	characteristic, complemented
161	C₄²	—	5	1	no	—
162	C₃×C₅	C₃×C₄×D₄	1	4	yes	complemented
163	C₂²×C₃	—	2	4	no	—
164	C₂²×C₃	—	2	4	no	—
165	C₂²×C₃	—	5	8	no	—
166	C₂²×C₃	—	10	4	no	—
167	C₂²×C₃	—	10	4	no	—
168	C₂²×C₃	C₃×C₄×D₅	1	8	yes	complemented
169	C₂²×C₃	C₂²×C₃×D₅	1	4	yes	—
170	C₃×C₄	—	2	4	no	—
171	C₃×C₄	—	2	4	no	—
172	C₃×C₄	—	5	8	no	—
173	C₃×C₄	—	5	8	no	—
174	C₃×C₄	—	10	4	no	—
175	C₂×C₅	—	2	2	no	—
176	C₂×C₅	C₂²×C₃²×C₄	1	1	yes	characteristic
177	C₂×C₅	C₂×C₃²×D₄	1	1	yes	characteristic
178	C₂×C₅	C₃²×#41	1	1	yes	characteristic
179	D₅	—	2	2	no	—
180	C₃²	#4027	1	1	yes	characteristic, direct product factor, Sylow
181	C₂³	—	5	1	no	—
182	C₂³	C₂×C₃²×D₅	1	1	yes	characteristic
183	D₄	—	5	4	no	—
184	C₂×C₄	—	5	1	no	—
185	C₂×C₄	—	5	1	no	—
186	C₂×C₄	—	5	1	no	—
187	C₂×C₄	—	10	2	no	—
188	C₂×C₄	—	10	2	no	—
189	C₂×C₄	C₂×C₃²×D₅	1	1	yes	characteristic
190	C₂×C₄	C₂×C₃²×D₅	1	1	yes	characteristic
191	C₂×C₃	—	2	8	no	—
192	C₂×C₃	—	10	8	no	—
193	C₂×C₃	C₂×C₃×C₄×D₅	1	4	yes	—
194	C₂×C₃	C₃×D₄×D₅	1	4	yes	—
195	C₂×C₃	C₃×#711	1	4	yes	—
196	C₅	C₃²×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	5	2	no	—
200	C₂²	—	10	1	no	—
201	C₂²	—	10	1	no	—
202	C₂²	C₃²×C₄×D₅	1	2	yes	complemented
203	C₂²	C₂²×C₃²×D₅	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	5	2	no	—
207	C₄	—	5	2	no	—
208	C₄	—	10	1	no	—
209	C₃	C₃×#4027	1	4	yes	direct product factor
210	C₂	—	2	2	no	—
211	C₂	—	10	2	no	—
212	C₂	C₂×C₃²×C₄×D₅	1	1	yes	characteristic
213	C₂	C₃²×D₄×D₅	1	1	yes	characteristic
214	C₂	C₃²×#711	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂²×D₅
commutator	C₂×C₅	C₂²×C₃²×C₄
Frattini	C₂²	C₂²×C₃²×D₅
Fitting	C₃²×C₅×C₂²⋊C₄	C₂
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂×C₃²×C₂²⋊D₅	C₂
cosocle	C₂²×C₅	C₂³×C₃²
hypercenter	C₃²×C₂²⋊C₄	D₅
hypocenter	C₅	C₃²×C₄×D₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×Dic₅	C₂	semidirect
3. C₂×C₃²×C₄×D₅	C₂	semidirect
4. C₃²×C₅×C₂²⋊C₄	C₂	semidirect
5. C₃²×#685	C₂	semidirect
6. C₃²×#683	C₂	semidirect
7. C₃²×C₄×Dic₅	C₂	semidirect
8. C₃×#4027	C₃	direct (N × H)
13. C₃²×C₂²⋊D₅	C₄	semidirect
21. C₂²×C₃×Dic₅	C₂×C₃	semidirect
22. C₂×C₃×C₄×D₅	C₂×C₃	semidirect
23. C₃×C₅×C₂²⋊C₄	C₂×C₃	semidirect
24. C₃×#685	C₂×C₃	semidirect
25. C₃×#683	C₂×C₃	semidirect
26. C₃×C₄×Dic₅	C₂×C₃	semidirect
29. C₂²×C₃²×C₅	C₂×C₄	semidirect
33. C₂×C₃²×D₅	C₂×C₄	semidirect
37. C₃²×Dic₅	C₂×C₄	semidirect
38. C₃²×Dic₅	D₄	semidirect
39. #4027	C₃²	direct (N × H)
45. C₃²×C₂²⋊C₄	D₅	semidirect
51. C₃×C₂²⋊D₅	C₃×C₄	semidirect
64. C₂²×Dic₅	C₂×C₃²	semidirect
65. C₂×C₄×D₅	C₂×C₃²	semidirect
66. C₅×C₂²⋊C₄	C₂×C₃²	semidirect
67. #685	C₂×C₃²	semidirect
68. #683	C₂×C₃²	semidirect
69. C₄×Dic₅	C₂×C₃²	semidirect
82. C₂²×C₃×C₅	C₂×C₃×C₄	semidirect
86. C₂×C₃×D₅	C₂×C₃×C₄	semidirect
90. C₃×Dic₅	C₂×C₃×C₄	semidirect
91. C₃×Dic₅	C₃×D₄	semidirect
97. C₃×C₂²⋊C₄	C₃×D₅	semidirect
99. C₃²×C₅	C₄×D₄	semidirect
104. C₂²⋊D₅	C₃²×C₄	semidirect
115. C₂²×C₃²	C₄×D₅	semidirect
140. C₂²×C₅	C₂×C₃²×C₄	semidirect
144. C₂×D₅	C₂×C₃²×C₄	semidirect
148. Dic₅	C₂×C₃²×C₄	semidirect
149. Dic₅	C₃²×D₄	semidirect
160. C₂²⋊C₄	C₃²×D₅	semidirect
162. C₃×C₅	C₃×C₄×D₄	semidirect
168. C₂²×C₃	C₃×C₄×D₅	semidirect
180. C₃²	#4027	direct (N × H)
196. C₅	C₃²×C₄×D₄	semidirect
202. C₂²	C₃²×C₄×D₅	semidirect
209. C₃	C₃×#4027	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×C₂²⋊D₅	2
C₂²×C₃²×Dic₅	2
C₂×C₃²×C₄×D₅	2
C₃²×C₅×C₂²⋊C₄	2
C₃²×#685	2
C₃²×#683	2
C₃²×C₄×Dic₅	2
C₃×#4027	3
C₃²×C₄×D₄	5

Subgroups — C32×#4027 #13087807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4027 #13087807