Subgroups — #14294925

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	#11751033	C₂	1	1	yes	maximal, characteristic
3	C₂×#91087	C₂	1	1	yes	maximal, characteristic
4	#11746371	C₂	1	4	yes	maximal
5	#10600544	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅	C₂²	1	2	yes	—
9	C₂²×C₃⋊#168	C₂²	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₄	1	4	yes	complemented
11	#91115	C₄	1	4	yes	—
12	#91087	C₄	1	4	yes	complemented
13	#69751	—	5	1	no	maximal, Hall
14	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
15	#65221	—	3	1	no	—
16	C₂×#3935	—	3	1	no	—
17	#64770	—	3	4	no	—
18	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₅	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₅×C₈	—	2	4	no	—
22	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
23	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
24	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₅	—	2	4	no	—
26	C₂×C₃⋊#168	—	2	4	no	—
27	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
29	#4932	—	5	1	no	—
30	C₂×#803	—	5	1	no	—
31	#4550	—	5	4	no	—
32	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
33	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
34	C₂×C₄×Dic₅	—	3	2	no	—
35	C₂²×#168	—	3	2	no	—
36	C₄×C₅×C₈	Dic₃	1	4	yes	complemented
37	#3963	—	3	4	no	—
38	#3935	—	3	4	no	—
39	#1746	—	15	1	no	Sylow
40	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	—	2	4	no	—
42	C₂×C₃×C₄×C₅	C₄²	1	4	yes	—
43	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	#41	1	4	yes	—
46	C₂×Dic₁₅	—	2	4	no	—
47	C₂×Dic₁₅	—	2	8	no	—
48	C₃×C₅×C₈	—	2	4	no	—
49	C₃⋊#168	—	2	4	no	—
50	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
51	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
52	C₂×C₄×Dic₃	—	5	2	no	—
53	C₂²×C₃⋊C₈	—	5	2	no	—
54	C₃×C₄×C₈	Dic₅	1	4	yes	complemented
55	C₃⋊#95	—	5	4	no	—
56	#803	—	5	4	no	—
57	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
58	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
59	C₂²×Dic₅	—	3	4	no	—
60	C₂×C₅×C₈	—	2	4	no	—
61	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	—
62	C₄²×C₅	C₄×D₃	1	2	yes	—
63	C₄²×C₅	C₂×Dic₃	1	2	yes	—
64	C₄×Dic₅	—	6	4	no	—
65	C₂×#168	—	3	4	no	—
66	C₂×#168	—	6	4	no	—
67	C₂×#97	—	15	1	no	—
68	C₂×C₄×C₈	D₁₅	1	1	yes	characteristic
69	#376	—	15	1	no	—
70	#336	—	15	4	no	—
71	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
72	C₂²×C₃×C₅	#117	1	2	yes	—
73	C₂²×C₃×C₅	#131	1	4	yes	—
74	C₃×C₄×C₅	—	2	4	no	—
75	C₃×C₄×C₅	#117	1	2	yes	—
76	C₃×C₄×C₅	#117	1	2	yes	—
77	Dic₁₅	—	4	4	no	—
78	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
79	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
80	C₂²×Dic₃	—	5	4	no	—
81	C₂×C₃×C₈	—	2	4	no	—
82	C₂×C₃×C₈	C₂×Dic₅	1	4	yes	—
83	C₃×C₄²	C₄×D₅	1	2	yes	—
84	C₃×C₄²	C₂×Dic₅	1	2	yes	—
85	C₄×Dic₃	—	10	4	no	—
86	C₂×C₃⋊C₈	—	5	4	no	—
87	C₂×C₃⋊C₈	—	10	4	no	—
88	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	C₄×Dic₃	1	4	yes	—
91	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
92	C₂×C₄×C₅	#235	1	4	yes	—
93	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
94	C₂×Dic₅	—	6	4	no	—
95	C₂×Dic₅	—	6	8	no	—
96	C₅×C₈	—	2	4	no	—
97	#168	—	6	4	no	—
98	C₂²×C₈	—	15	2	no	—
99	C₂²×C₈	C₂×D₁₅	1	2	yes	—
100	C₂×C₄²	—	15	2	no	—
101	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
102	#97	—	15	4	no	—
103	C₄×C₈	Dic₁₅	1	4	yes	complemented
104	#95	—	15	4	no	—
105	C₂×C₃×C₅	#376	1	1	yes	characteristic
106	C₂×C₃×C₅	#405	1	2	yes	—
107	C₂×C₃×C₅	#433	1	4	yes	—
108	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
109	C₂×C₃×C₄	—	2	4	no	—
110	C₂×C₃×C₄	C₄×Dic₅	1	4	yes	—
111	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
112	C₂×C₃×C₄	#709	1	4	yes	—
113	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
114	C₂×Dic₃	—	10	4	no	—
115	C₂×Dic₃	—	10	8	no	—
116	C₃×C₈	—	2	4	no	—
117	C₃⋊C₈	—	10	4	no	—
118	C₂²×C₅	#872	1	2	yes	—
119	C₂²×C₅	#901	1	4	yes	—
120	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
121	C₄×C₅	—	2	4	no	—
122	C₄×C₅	#872	1	2	yes	—
123	C₄×C₅	#926	1	2	yes	—
124	Dic₅	—	12	4	no	—
125	C₂²×C₄	—	15	4	no	—
126	C₂²×C₄	C₄×D₁₅	1	2	yes	—
127	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	15	4	no	—
130	C₂×C₈	—	30	4	no	—
131	C₂×C₈	C₂×Dic₁₅	1	4	yes	—
132	C₄²	—	30	4	no	—
133	C₄²	C₄×D₁₅	1	2	yes	—
134	C₄²	C₂×Dic₁₅	1	2	yes	—
135	C₃×C₅	#1746	1	1	yes	characteristic, complemented, Hall
136	C₂²×C₃	#4018	1	2	yes	—
137	C₂²×C₃	#4047	1	4	yes	—
138	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
139	C₃×C₄	—	2	4	no	—
140	C₃×C₄	#4018	1	2	yes	—
141	C₃×C₄	#4072	1	2	yes	—
142	Dic₃	—	20	4	no	—
143	C₂×C₅	#4690	1	4	yes	—
144	C₂×C₅	#4932	1	1	yes	characteristic
145	C₂×C₅	#5101	1	2	yes	—
146	C₂³	C₂×C₄×D₁₅	1	1	yes	characteristic
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	30	4	no	—
149	C₂×C₄	—	30	8	no	—
150	C₂×C₄	C₄×Dic₁₅	1	4	yes	—
151	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
152	C₂×C₄	#6817	1	4	yes	—
153	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
154	C₈	—	2	4	no	—
155	C₈	—	30	4	no	—
156	C₂×C₃	#64979	1	4	yes	—
157	C₂×C₃	#65221	1	1	yes	characteristic
158	C₂×C₃	#65390	1	2	yes	—
159	C₅	#69751	1	1	yes	characteristic, complemented, Sylow
160	C₂²	#91761	1	2	yes	—
161	C₂²	#91790	1	4	yes	—
162	C₂²	C₂×C₄×Dic₁₅	1	1	yes	characteristic
163	C₄	—	2	4	no	—
164	C₄	—	60	4	no	—
165	C₄	#91761	1	2	yes	—
166	C₄	#91815	1	2	yes	—
167	C₃	#10600544	1	1	yes	characteristic, complemented, Sylow
168	C₂	#11750791	1	4	yes	—
169	C₂	#11751033	1	1	yes	characteristic
170	C₂	#11751202	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅
commutator	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄×C₈	D₁₅
hypocenter	C₃×C₅	#1746
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
10. C₃×C₄×C₅×C₈	C₄	semidirect
12. #91087	C₄	semidirect
36. C₄×C₅×C₈	Dic₃	semidirect
54. C₃×C₄×C₈	Dic₅	semidirect
103. C₄×C₈	Dic₁₅	semidirect
135. C₃×C₅	#1746	semidirect
159. C₅	#69751	semidirect
167. C₃	#10600544	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×C₈	2
#11751033	2
C₂×#91087	2
#11746371	2
#10600544	3
#69751	5