Subgroups — #14330066

Order 1920.

896 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	#11751081	C₂	1	4	yes	maximal
3	#11751081	C₂	1	4	yes	maximal
4	C₄⋊C₄×Dic₁₅	C₂	1	4	yes	maximal
5	#11751033	C₂	1	2	yes	maximal
6	#10614757	—	3	1	no	maximal, Hall
7	C₃×C₅×#125	C₄	1	8	yes	complemented
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₅	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₅	C₂²	1	4	yes	—
13	C₂×C₄×Dic₁₅	C₂²	1	4	yes	—
14	#91115	C₂²	1	2	yes	—
15	#91115	C₂²	1	2	yes	—
16	#91115	C₂²	1	4	yes	—
17	#91115	C₂²	1	4	yes	—
18	#91115	C₂²	1	4	yes	—
19	#83026	—	5	1	no	maximal, Hall
20	C₂×C₅×#125	D₃	1	1	yes	characteristic
21	#65270	—	3	4	no	—
22	#65270	—	3	4	no	—
23	C₄⋊C₄×Dic₅	—	3	4	no	—
24	#65221	—	3	2	no	—
25	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
26	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×Dic₁₅	C₂³	1	4	yes	—
29	C₂²×Dic₁₅	C₂³	1	4	yes	—
30	C₃×C₅×C₄⋊C₄	C₂×C₄	1	16	yes	—
31	C₃×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
32	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
33	C₄⋊Dic₁₅	—	2	8	no	—
34	C₄×Dic₁₅	—	2	8	no	—
35	C₄×Dic₁₅	—	2	8	no	—
36	C₂×C₃×#125	D₅	1	1	yes	characteristic
37	#4980	—	5	4	no	—
38	#4980	—	5	4	no	—
39	Dic₃×C₄⋊C₄	—	5	4	no	—
40	#4932	—	5	2	no	—
41	C₅×#125	Dic₃	1	8	yes	complemented
42	C₂×C₅×C₄⋊C₄	C₂×D₃	1	4	yes	—
43	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
44	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
45	C₂×C₄⋊Dic₅	—	3	4	no	—
46	C₂×C₄×Dic₅	—	3	4	no	—
47	C₂×C₄×Dic₅	—	3	4	no	—
48	#3963	—	3	2	no	—
49	#3963	—	3	4	no	—
50	#3963	—	3	2	no	—
51	#3963	—	3	4	no	—
52	#3963	—	3	4	no	—
53	#2371	—	15	1	no	Sylow
54	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	4	no	—
56	C₂×C₃×C₄×C₅	—	2	4	no	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	8	yes	—
62	C₂×Dic₁₅	—	2	4	no	—
63	C₂×Dic₁₅	—	2	4	no	—
64	C₂×Dic₁₅	—	2	4	no	—
65	C₂×Dic₁₅	—	2	4	no	—
66	C₂×Dic₁₅	—	2	4	no	—
67	C₂×Dic₁₅	#41	1	8	yes	—
68	C₃×#125	Dic₅	1	8	yes	complemented
69	C₂×C₃×C₄⋊C₄	C₂×D₅	1	4	yes	—
70	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
71	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
72	C₂×C₄⋊Dic₃	—	5	4	no	—
73	C₂×C₄×Dic₃	—	5	4	no	—
74	C₂×C₄×Dic₃	—	5	4	no	—
75	C₃⋊#95	—	5	2	no	—
76	C₃⋊#95	—	5	4	no	—
77	C₃⋊#95	—	5	2	no	—
78	C₃⋊#95	—	5	4	no	—
79	C₃⋊#95	—	5	4	no	—
80	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
82	C₂²×C₄×C₅	C₂²×D₃	1	4	yes	—
83	C₂²×Dic₅	—	3	4	no	—
84	C₂²×Dic₅	—	3	4	no	—
85	C₅×C₄⋊C₄	C₂×Dic₃	1	16	yes	—
86	C₅×C₄⋊C₄	C₂×Dic₃	1	8	yes	—
87	C₄²×C₅	C₂×Dic₃	1	4	yes	—
88	C₄⋊Dic₅	—	6	8	no	—
89	C₄×Dic₅	—	6	8	no	—
90	C₄×Dic₅	—	6	8	no	—
91	C₂×#125	D₁₅	1	1	yes	characteristic
92	#387	—	15	4	no	—
93	#387	—	15	4	no	—
94	C₄×C₄⋊C₄	—	15	4	no	—
95	#376	—	15	2	no	—
96	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
97	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
98	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
99	C₃×C₄×C₅	—	2	4	no	—
100	C₃×C₄×C₅	—	2	8	no	—
101	C₃×C₄×C₅	—	2	2	no	—
102	Dic₁₅	—	2	8	no	—
103	Dic₁₅	—	4	4	no	—
104	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
106	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
107	C₂²×Dic₃	—	5	4	no	—
108	C₂²×Dic₃	—	5	4	no	—
109	C₃×C₄⋊C₄	C₂×Dic₅	1	16	yes	—
110	C₃×C₄⋊C₄	C₂×Dic₅	1	8	yes	—
111	C₃×C₄²	C₂×Dic₅	1	4	yes	—
112	C₄⋊Dic₃	—	10	8	no	—
113	C₄×Dic₃	—	10	8	no	—
114	C₄×Dic₃	—	10	8	no	—
115	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
121	C₂×C₄×C₅	C₂²×Dic₃	1	4	yes	—
122	C₂×C₄×C₅	C₂²×Dic₃	1	8	yes	—
123	C₂×Dic₅	—	3	8	no	—
124	C₂×Dic₅	—	6	4	no	—
125	C₂×Dic₅	—	6	4	no	—
126	C₂×Dic₅	—	6	4	no	—
127	C₂×Dic₅	—	6	4	no	—
128	C₂×Dic₅	—	6	4	no	—
129	#125	Dic₁₅	1	8	yes	complemented
130	C₂×C₄⋊C₄	—	15	4	no	—
131	C₂×C₄⋊C₄	C₂×D₁₅	1	4	yes	—
132	C₂×C₄⋊C₄	C₂×D₁₅	1	2	yes	—
133	C₂×C₄²	—	15	4	no	—
134	C₂×C₄²	—	15	4	no	—
135	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
136	#95	—	15	2	no	—
137	#95	—	15	4	no	—
138	#95	—	15	2	no	—
139	#95	—	15	4	no	—
140	#95	—	15	4	no	—
141	C₂×C₃×C₅	C₄×#41	1	2	yes	—
142	C₂×C₃×C₅	#519	1	1	yes	characteristic
143	C₂×C₃×C₅	#552	1	4	yes	—
144	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
150	C₂×C₃×C₄	C₂²×Dic₅	1	4	yes	—
151	C₂×C₃×C₄	C₂²×Dic₅	1	8	yes	—
152	C₂×Dic₃	—	5	8	no	—
153	C₂×Dic₃	—	10	4	no	—
154	C₂×Dic₃	—	10	4	no	—
155	C₂×Dic₃	—	10	4	no	—
156	C₂×Dic₃	—	10	4	no	—
157	C₂×Dic₃	—	10	4	no	—
158	C₂²×C₅	#1007	1	2	yes	—
159	C₂²×C₅	D₃×#41	1	4	yes	—
160	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	2	4	no	—
163	C₄×C₅	—	2	8	no	—
164	Dic₅	—	6	8	no	—
165	Dic₅	—	12	4	no	—
166	C₂²×C₄	—	15	4	no	—
167	C₂²×C₄	—	15	4	no	—
168	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₅	1	2	yes	—
170	C₂²×C₄	C₂²×D₁₅	1	4	yes	—
171	C₄⋊C₄	—	30	8	no	—
172	C₄⋊C₄	C₂×Dic₁₅	1	16	yes	—
173	C₄⋊C₄	C₂×Dic₁₅	1	8	yes	—
174	C₄²	—	30	8	no	—
175	C₄²	—	30	8	no	—
176	C₄²	C₂×Dic₁₅	1	4	yes	—
177	C₃×C₅	#2371	1	1	yes	characteristic, complemented, Hall
178	C₂²×C₃	#4147	1	2	yes	—
179	C₂²×C₃	D₅×#41	1	4	yes	—
180	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	4	no	—
183	C₃×C₄	—	2	8	no	—
184	Dic₃	—	10	8	no	—
185	Dic₃	—	20	4	no	—
186	C₂×C₅	#5693	1	4	yes	—
187	C₂×C₅	#5812	1	1	yes	characteristic
188	C₂×C₅	Dic₃×#41	1	2	yes	—
189	C₂³	C₂³×D₁₅	1	1	yes	characteristic
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	15	8	no	—
195	C₂×C₄	—	30	4	no	—
196	C₂×C₄	—	30	4	no	—
197	C₂×C₄	—	30	4	no	—
198	C₂×C₄	—	30	4	no	—
199	C₂×C₄	—	30	4	no	—
200	C₂×C₄	C₂²×Dic₁₅	1	2	yes	—
201	C₂×C₄	C₂²×Dic₁₅	1	4	yes	—
202	C₂×C₄	C₂²×Dic₁₅	1	8	yes	—
203	C₂×C₃	#66027	1	4	yes	—
204	C₂×C₃	#66146	1	1	yes	characteristic
205	C₂×C₃	#41×Dic₅	1	2	yes	—
206	C₅	#83026	1	1	yes	characteristic, complemented, Sylow
207	C₂²	#92098	1	2	yes	—
208	C₂²	#41×D₁₅	1	4	yes	—
209	C₂²	C₂³×Dic₁₅	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	4	no	—
212	C₄	—	2	8	no	—
213	C₄	—	30	8	no	—
214	C₄	—	60	4	no	—
215	C₃	#10614757	1	1	yes	characteristic, complemented, Sylow
216	C₂	#11756452	1	4	yes	—
217	C₂	#11756571	1	1	yes	characteristic
218	C₂	#41×Dic₁₅	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₅
commutator	C₂²×C₃×C₅	C₂³×C₄
Fitting	C₂×C₃×C₅×#125	C₂
socle, prime radical, cosocle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×#125	D₁₅
hypocenter	C₃×C₅	#2371
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
7. C₃×C₅×#125	C₄	semidirect
41. C₅×#125	Dic₃	semidirect
68. C₃×#125	Dic₅	semidirect
129. #125	Dic₁₅	semidirect
177. C₃×C₅	#2371	semidirect
206. C₅	#83026	semidirect
215. C₃	#10614757	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#125	2
#11751081	2
C₄⋊C₄×Dic₁₅	2
#11751033	2
#10614757	3
#83026	5