Subgroups — #14294968

Order 1920.

400 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	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	#11751202	C₂	1	2	yes	maximal
3	#11746381	C₂	1	1	yes	maximal, characteristic
4	#11746378	C₂	1	1	yes	maximal, characteristic
5	#11746352	C₂	1	2	yes	maximal
6	#10600688	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#117	C₂²	1	1	yes	characteristic
10	#91815	C₂²	1	2	yes	—
11	C₂×#6710	C₂²	1	2	yes	—
12	C₃×C₅×#97	C₄	1	2	yes	complemented
13	C₃×C₄×C₅×C₈	C₄	1	2	yes	complemented
14	#91099	C₄	1	4	yes	complemented
15	C₈×Dic₁₅	C₄	1	4	yes	complemented
16	#69895	—	5	1	no	maximal, Hall
17	C₅×#405	D₃	1	1	yes	characteristic
18	#65390	—	3	2	no	—
19	#64780	—	3	1	no	—
20	#64777	—	3	1	no	—
21	#64751	—	3	2	no	—
22	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₃×C₅×M₁₆	—	2	2	no	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	—	2	1	no	—
30	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
31	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
32	#6719	—	2	2	no	—
33	C₄⋊Dic₁₅	—	2	2	no	—
34	C₄×Dic₁₅	C₂×C₄	1	4	yes	—
35	#6710	—	2	2	no	—
36	#6710	—	2	2	no	—
37	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
38	C₃×#405	D₅	1	1	yes	characteristic
39	#5101	—	5	2	no	—
40	C₃⋊#338	—	5	1	no	—
41	#4557	—	5	1	no	—
42	#4531	—	5	2	no	—
43	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	C₅×#117	C₂×D₃	1	1	yes	characteristic
46	#4072	—	3	2	no	—
47	C₂×#681	—	3	2	no	—
48	C₅×#97	Dic₃	1	2	yes	complemented
49	C₄×C₅×C₈	Dic₃	1	2	yes	complemented
50	#3947	—	3	4	no	—
51	C₈×Dic₅	—	3	4	no	—
52	#1747	—	15	1	no	Sylow
53	C₂³×C₃×C₅	#41	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	#41	1	2	yes	—
59	C₂×Dic₁₅	—	2	4	no	—
60	C₃×C₅×C₈	—	2	2	no	—
61	C₃×C₅×C₈	C₄²	1	2	yes	—
62	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
63	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
64	C₃⋊#168	—	2	4	no	—
65	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
66	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
67	C₃×#117	C₂×D₅	1	1	yes	characteristic
68	#926	—	5	2	no	—
69	C₂×#208	—	5	2	no	—
70	C₃×#97	Dic₅	1	2	yes	complemented
71	C₃×C₄×C₈	Dic₅	1	2	yes	complemented
72	#815	—	5	4	no	—
73	C₈×Dic₃	—	5	4	no	—
74	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
75	C₅×M₁₆	—	2	2	no	—
76	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
77	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
78	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
79	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₂²⋊C₄	—	2	1	no	—
83	C₄²×C₅	C₂×Dic₃	1	2	yes	—
84	C₂²⋊Dic₅	—	6	2	no	—
85	C₄⋊Dic₅	—	6	2	no	—
86	C₄×Dic₅	—	3	4	no	—
87	#681	—	6	2	no	—
88	#681	—	6	2	no	—
89	C₂×#168	—	3	4	no	—
90	#405	—	15	2	no	—
91	#405	D₁₅	1	1	yes	characteristic
92	#338	—	15	1	no	—
93	H(3,ℤ/4)	—	15	1	no	—
94	H(3,ℤ/4)	—	15	2	no	—
95	C₂²×C₃×C₅	—	4	1	no	—
96	C₂²×C₃×C₅	#117	1	2	yes	—
97	C₂²×C₃×C₅	#117	1	1	yes	characteristic
98	C₃×C₄×C₅	—	4	2	no	—
99	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
100	C₃×C₄×C₅	#117	1	2	yes	—
101	C₃×C₄×C₅	#117	1	1	yes	characteristic
102	Dic₁₅	—	4	4	no	—
103	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
104	C₃×M₁₆	—	2	2	no	—
105	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
106	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
107	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
108	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
110	C₃×C₄⋊C₄	—	2	1	no	—
111	C₃×C₂²⋊C₄	—	2	1	no	—
112	C₃×C₄²	C₂×Dic₅	1	2	yes	—
113	#217	—	10	2	no	—
114	C₄⋊Dic₃	—	10	2	no	—
115	C₄×Dic₃	—	5	4	no	—
116	#208	—	10	2	no	—
117	#208	—	10	2	no	—
118	C₂×C₃⋊C₈	—	5	4	no	—
119	C₂³×C₅	#235	1	1	yes	characteristic
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
122	C₂×C₄×C₅	#235	1	1	yes	characteristic
123	C₂×C₄×C₅	#235	1	2	yes	—
124	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
125	C₂×Dic₅	—	6	4	no	—
126	C₅×C₈	—	2	2	no	—
127	C₅×C₈	C₄×Dic₃	1	2	yes	—
128	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
129	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
130	#168	—	6	4	no	—
131	C₂×M₁₆	—	15	2	no	—
132	C₂×M₁₆	C₂×D₁₅	1	1	yes	characteristic
133	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
134	#117	—	15	2	no	—
135	#117	C₂×D₁₅	1	1	yes	characteristic
136	#97	—	15	4	no	—
137	#97	Dic₁₅	1	2	yes	complemented
138	C₄×C₈	—	15	4	no	—
139	C₄×C₈	Dic₁₅	1	2	yes	complemented
140	C₂×C₃×C₅	—	2	2	no	—
141	C₂×C₃×C₅	—	2	1	no	—
142	C₂×C₃×C₅	#376	1	1	yes	characteristic
143	C₂³×C₃	#709	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
146	C₂×C₃×C₄	#709	1	1	yes	characteristic
147	C₂×C₃×C₄	#709	1	2	yes	—
148	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
149	C₂×Dic₃	—	10	4	no	—
150	C₃×C₈	—	2	2	no	—
151	C₃×C₈	C₄×Dic₅	1	2	yes	—
152	C₃×C₈	C₄×Dic₅	1	1	yes	characteristic
153	C₃×C₈	C₄×Dic₅	1	1	yes	characteristic
154	C₃⋊C₈	—	10	4	no	—
155	C₂²×C₅	—	4	1	no	—
156	C₂²×C₅	#872	1	2	yes	—
157	C₂²×C₅	#926	1	1	yes	characteristic
158	C₄×C₅	—	4	2	no	—
159	C₄×C₅	#872	1	2	yes	—
160	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
161	C₄×C₅	#926	1	1	yes	characteristic
162	Dic₅	—	12	4	no	—
163	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
164	M₁₆	—	2	2	no	—
165	M₁₆	—	30	2	no	—
166	M₁₆	—	30	2	no	—
167	C₂×C₈	—	15	4	no	—
168	C₂×C₈	C₄×D₁₅	1	2	yes	—
169	C₂×C₈	C₄×D₁₅	1	2	yes	—
170	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
171	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
172	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	30	2	no	—
175	C₂²⋊C₄	—	2	1	no	—
176	C₂²⋊C₄	—	30	2	no	—
177	C₄²	—	15	4	no	—
178	C₄²	C₂×Dic₁₅	1	2	yes	—
179	C₃×C₅	#1747	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	—	4	1	no	—
181	C₂²×C₃	#4018	1	2	yes	—
182	C₂²×C₃	#4072	1	1	yes	characteristic
183	C₃×C₄	—	4	2	no	—
184	C₃×C₄	#4018	1	2	yes	—
185	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
186	C₃×C₄	#4072	1	1	yes	characteristic
187	Dic₃	—	20	4	no	—
188	C₂×C₅	—	2	2	no	—
189	C₂×C₅	—	2	1	no	—
190	C₂×C₅	#4932	1	1	yes	characteristic
191	C₂³	#6817	1	1	yes	characteristic
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	30	4	no	—
194	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
195	C₂×C₄	#6817	1	1	yes	characteristic
196	C₂×C₄	#6817	1	2	yes	—
197	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	30	4	no	—
200	C₈	C₄×Dic₁₅	1	1	yes	characteristic
201	C₈	C₄×Dic₁₅	1	2	yes	—
202	C₈	C₄×Dic₁₅	1	1	yes	characteristic
203	C₂×C₃	—	2	2	no	—
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	#65221	1	1	yes	characteristic
206	C₅	#69895	1	1	yes	characteristic, complemented, Sylow
207	C₂²	—	4	1	no	—
208	C₂²	#91761	1	2	yes	—
209	C₂²	#91815	1	1	yes	characteristic
210	C₄	—	4	2	no	—
211	C₄	—	60	4	no	—
212	C₄	#91761	1	2	yes	—
213	C₄	C₂×C₄×Dic₁₅	1	1	yes	characteristic
214	C₄	#91815	1	1	yes	characteristic
215	C₃	#10600688	1	1	yes	characteristic, complemented, Sylow
216	C₂	—	2	1	no	—
217	C₂	—	2	2	no	—
218	C₂	#11751033	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₁₅
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#405	C₂
socle	C₂×C₃×C₅	#376
prime radical	C₂³×C₃×C₅	#41
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#405	D₁₅
hypocenter	C₃×C₅	#1747
LCS	G ▷ C₃×C₄×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
12. C₃×C₅×#97	C₄	semidirect
13. C₃×C₄×C₅×C₈	C₄	semidirect
14. #91099	C₄	semidirect
15. C₈×Dic₁₅	C₄	semidirect
48. C₅×#97	Dic₃	semidirect
49. C₄×C₅×C₈	Dic₃	semidirect
70. C₃×#97	Dic₅	semidirect
71. C₃×C₄×C₈	Dic₅	semidirect
137. #97	Dic₁₅	semidirect
139. C₄×C₈	Dic₁₅	semidirect
179. C₃×C₅	#1747	semidirect
206. C₅	#69895	semidirect
215. C₃	#10600688	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#405	2
#11751202	2
#11746381	2
#11746378	2
#11746352	2
#10600688	3
#69895	5