Subgroups — #14298852

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11751037	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#65225	C₂	1	1	yes	maximal, characteristic
4	#11745980	C₂	1	4	yes	maximal
5	#10600556	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
8	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
10	#91115	—	2	2	no	—
11	C₄×C₅×C₃⋊C₈	C₄	1	4	yes	complemented
12	C₃×#3963	—	2	2	no	—
13	#69547	—	5	1	no	maximal, Hall
14	C₂×C₄×C₅×C₈	—	3	1	no	—
15	#65225	—	3	1	no	—
16	#65225	D₃	1	1	yes	characteristic
17	#64771	—	3	4	no	—
18	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
19	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₅	—	2	2	no	—
21	C₂²×C₃×Dic₅	—	2	2	no	—
22	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
23	C₃×C₄²×C₅	Q₈	1	2	yes	—
24	C₄⋊Dic₁₅	—	2	4	no	—
25	C₂×C₅×C₃⋊C₈	—	2	2	no	—
26	C₂×C₅×C₃⋊C₈	—	2	2	no	—
27	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
28	C₃×C₄⋊Dic₅	—	2	4	no	—
29	C₃×#382	—	5	1	no	—
30	#4936	—	5	1	no	—
31	C₂×C₄×C₃⋊C₈	D₅	1	1	yes	characteristic
32	#4529	—	5	4	no	—
33	C₂²×C₅×C₈	—	3	2	no	—
34	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
35	C₂×C₄⋊Dic₅	—	3	2	no	—
36	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
37	C₄×C₅×C₈	—	3	4	no	—
38	#3963	—	2	2	no	—
39	#3963	—	6	2	no	—
40	#1835	—	15	1	no	Sylow
41	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	—	2	4	no	—
43	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
44	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	#41	1	2	yes	—
47	C₂×C₃×C₄×C₅	#41	1	2	yes	—
48	C₂×Dic₁₅	—	4	2	no	—
49	C₂×Dic₁₅	—	4	4	no	—
50	C₂×C₃×Dic₅	—	4	2	no	—
51	C₂×C₃×Dic₅	—	4	4	no	—
52	C₅×C₃⋊C₈	—	2	4	no	—
53	C₂×C₃×C₄⋊C₄	—	5	2	no	—
54	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
55	C₂×C₄⋊Dic₃	—	5	2	no	—
56	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
57	C₃×#95	—	10	2	no	—
58	C₃⋊#95	—	10	2	no	—
59	C₄×C₃⋊C₈	Dic₅	1	4	yes	complemented
60	C₂²×C₄×C₅	#67	1	2	yes	—
61	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
62	C₂²×Dic₅	—	2	2	no	—
63	C₂²×Dic₅	—	6	2	no	—
64	C₂×C₅×C₈	—	3	4	no	—
65	C₂×C₅×C₈	—	6	2	no	—
66	C₂×C₅×C₈	—	6	2	no	—
67	C₄²×C₅	Dic₆	1	2	yes	—
68	C₄²×C₅	C₄×D₃	1	2	yes	—
69	C₄⋊Dic₅	—	2	4	no	—
70	C₄⋊Dic₅	—	6	4	no	—
71	C₂×C₄×C₈	—	3	1	no	—
72	#382	—	5	1	no	—
73	#382	—	15	1	no	—
74	#340	—	15	4	no	—
75	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₃×C₅	#124	1	1	yes	characteristic
77	C₂²×C₃×C₅	#124	1	1	yes	characteristic
78	C₂²×C₃×C₅	#135	1	4	yes	—
79	C₃×C₄×C₅	—	2	4	no	—
80	C₃×C₄×C₅	#117	1	2	yes	—
81	C₃×C₄×C₅	#125	1	2	yes	—
82	Dic₁₅	—	8	2	no	—
83	C₃×Dic₅	—	8	2	no	—
84	C₂²×C₃×C₄	—	10	2	no	—
85	C₂²×C₃×C₄	D₂₀	1	2	yes	—
86	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
87	C₂²×Dic₃	—	10	2	no	—
88	C₃×C₄⋊C₄	—	10	4	no	—
89	C₃×C₄²	Dic₁₀	1	2	yes	—
90	C₃×C₄²	C₂×Dic₅	1	2	yes	—
91	C₄⋊Dic₃	—	10	4	no	—
92	C₂×C₃⋊C₈	—	2	2	no	—
93	C₂×C₃⋊C₈	—	2	2	no	—
94	C₂×C₃⋊C₈	C₂×Dic₅	1	4	yes	—
95	C₂³×C₅	C₂×#67	1	1	yes	characteristic
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	Dic₃⋊C₄	1	4	yes	—
98	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
100	C₂×C₄×C₅	#237	1	2	yes	—
101	C₂×C₄×C₅	#239	1	2	yes	—
102	C₂×Dic₅	—	4	2	no	—
103	C₂×Dic₅	—	4	4	no	—
104	C₂×Dic₅	—	12	2	no	—
105	C₂×Dic₅	—	12	4	no	—
106	C₅×C₈	—	6	4	no	—
107	C₂²×C₈	—	3	2	no	—
108	C₂×C₄⋊C₄	—	5	2	no	—
109	C₂×C₄⋊C₄	—	15	2	no	—
110	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
111	C₄×C₈	—	3	4	no	—
112	#95	—	10	2	no	—
113	#95	—	30	2	no	—
114	C₂×C₃×C₅	#382	1	1	yes	characteristic
115	C₂×C₃×C₅	#427	1	2	yes	—
116	C₂×C₃×C₅	#488	1	2	yes	—
117	C₂×C₃×C₅	#488	1	2	yes	—
118	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
119	C₂×C₃×C₄	—	2	4	no	—
120	C₂×C₃×C₄	—	20	2	no	—
121	C₂×C₃×C₄	—	20	4	no	—
122	C₂×C₃×C₄	C₄⋊Dic₅	1	4	yes	—
123	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
124	C₂×C₃×C₄	#709	1	2	yes	—
125	C₂×C₃×C₄	#709	1	2	yes	—
126	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
127	C₂×Dic₃	—	20	2	no	—
128	C₂×Dic₃	—	20	4	no	—
129	C₃⋊C₈	—	2	4	no	—
130	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
131	C₂²×C₅	#936	1	1	yes	characteristic
132	C₂²×C₅	#947	1	1	yes	characteristic
133	C₂²×C₅	#950	1	4	yes	—
134	C₄×C₅	—	2	4	no	—
135	C₄×C₅	#890	1	2	yes	—
136	C₄×C₅	#892	1	2	yes	—
137	Dic₅	—	8	2	no	—
138	Dic₅	—	24	2	no	—
139	C₂²×C₄	—	10	2	no	—
140	C₂²×C₄	—	30	2	no	—
141	C₂²×C₄	#1202	1	2	yes	—
142	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
143	C₂×C₈	—	3	4	no	—
144	C₂×C₈	—	6	2	no	—
145	C₂×C₈	—	6	2	no	—
146	C₄⋊C₄	—	10	4	no	—
147	C₄⋊C₄	—	30	4	no	—
148	C₄²	D₃×Dic₅	1	2	yes	—
149	C₄²	#1204	1	2	yes	—
150	C₃×C₅	#1835	1	1	yes	characteristic, complemented, Hall
151	C₂²×C₃	#4021	1	1	yes	characteristic
152	C₂²×C₃	#4021	1	1	yes	characteristic
153	C₂²×C₃	#4050	1	4	yes	—
154	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
155	C₃×C₄	—	2	4	no	—
156	C₃×C₄	—	40	2	no	—
157	C₃×C₄	#4016	1	2	yes	—
158	C₃×C₄	#4072	1	2	yes	—
159	Dic₃	—	40	2	no	—
160	C₂×C₅	#4977	1	1	yes	characteristic
161	C₂×C₅	#4998	1	2	yes	—
162	C₂×C₅	#5056	1	2	yes	—
163	C₂×C₅	#5068	1	2	yes	—
164	C₂³	C₂×#1202	1	1	yes	characteristic
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	20	4	no	—
167	C₂×C₄	—	20	2	no	—
168	C₂×C₄	—	60	4	no	—
169	C₂×C₄	—	60	2	no	—
170	C₂×C₄	Dic₃⋊Dic₅	1	4	yes	—
171	C₂×C₄	#6765	1	2	yes	—
172	C₂×C₄	#6769	1	2	yes	—
173	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
174	C₂×C₄	C₂×#1204	1	1	yes	characteristic
175	C₈	—	6	4	no	—
176	C₂×C₃	#64986	1	2	yes	—
177	C₂×C₃	#64986	1	2	yes	—
178	C₂×C₃	#65225	1	1	yes	characteristic
179	C₂×C₃	#65395	1	2	yes	—
180	C₅	#69547	1	1	yes	characteristic, complemented, Sylow
181	C₂²	#91311	1	4	yes	—
182	C₂²	#91364	1	1	yes	characteristic
183	C₂²	#91374	1	1	yes	characteristic
184	C₂²	C₂×Dic₃⋊Dic₅	1	1	yes	characteristic
185	C₄	—	2	4	no	—
186	C₄	—	40	2	no	—
187	C₄	—	120	2	no	—
188	C₄	#91338	1	2	yes	—
189	C₄	#91381	1	2	yes	—
190	C₃	#10600556	1	1	yes	characteristic, complemented, Sylow
191	C₂	#11746891	1	2	yes	—
192	C₂	#11746933	1	2	yes	—
193	C₂	#11748451	1	2	yes	—
194	C₂	#11748619	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1202
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1835
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
11. C₄×C₅×C₃⋊C₈	C₄	semidirect
59. C₄×C₃⋊C₈	Dic₅	semidirect
150. C₃×C₅	#1835	semidirect
180. C₅	#69547	semidirect
190. C₃	#10600556	semidirect

Maximal subgroups

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

subgroup	index
#11751037	2
C₂×C₄×C₅×C₃⋊C₈	2
C₃×#65225	2
#11745980	2
#10600556	3
#69547	5