Subgroups — C₂×C₃×C₈×#168 #14280307

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 160 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	C₂×C₃×C₄×#168	C₂	1	2	yes	maximal
3	C₃×C₈×#168	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₈×#168	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₈	C₄	1	2	yes	—
6	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃×#168	C₄	1	4	yes	—
8	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
9	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
10	C₃×C₄×#168	C₄	1	4	yes	—
11	C₃×C₄×#168	C₂²	1	4	yes	—
12	C₂×C₃×C₈²	—	5	1	no	maximal, Hall
13	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
14	C₂×C₄×#168	C₂×C₃	1	2	yes	—
15	C₈×#168	C₂×C₃	1	4	yes	direct product factor
16	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₅×C₈	C₈	1	8	yes	complemented
19	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
20	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
21	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₃×#168	C₈	1	16	yes	direct product factor
24	C₂×C₃×#168	C₂×C₄	1	8	yes	—
25	C₂×C₃×C₄×C₈	—	5	2	no	—
26	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
27	C₃×C₈²	—	5	4	no	—
28	C₂²×C₅×C₈	C₃×C₄	1	2	yes	—
29	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
30	C₂²×#168	C₃×C₄	1	4	yes	—
31	C₄×C₅×C₈	C₃×C₄	1	2	yes	—
32	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
33	C₄×#168	C₃×C₄	1	4	yes	—
34	C₄×#168	C₂²×C₃	1	4	yes	—
35	C₂×C₈²	—	5	1	no	Sylow
36	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
37	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
39	C₂×C₃×C₄×C₅	C₂×C₈	1	8	yes	—
40	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
41	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
42	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
43	C₃×C₅×C₈	C₂×C₈	1	8	yes	complemented
44	C₃×#168	C₂×C₈	1	16	yes	direct product factor
45	C₂²×C₃×C₈	—	5	4	no	—
46	C₂²×C₃×C₈	Dic₅	1	2	yes	—
47	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
48	C₃×C₄×C₈	—	5	4	no	—
49	C₃×C₄×C₈	—	5	4	no	—
50	C₃×C₄×C₈	Dic₅	1	2	yes	—
51	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
52	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
53	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
54	C₂×C₅×C₈	C₃×C₈	1	8	yes	complemented
55	C₂×C₅×C₈	C₂×C₃×C₄	1	4	yes	—
56	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
57	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
58	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
59	C₂×#168	C₃×C₈	1	16	yes	direct product factor
60	C₂×#168	C₂×C₃×C₄	1	8	yes	—
61	C₂×C₄×C₈	—	5	2	no	—
62	C₂×C₄×C₈	C₃×D₅	1	1	yes	characteristic
63	C₈²	—	5	4	no	—
64	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
65	C₂²×C₃×C₅	C₄×C₈	1	4	yes	—
66	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
67	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
68	C₃×C₄×C₅	C₄×C₈	1	4	yes	—
69	C₃×C₄×C₅	C₂²×C₈	1	4	yes	—
70	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
71	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
72	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
73	C₂×C₃×C₈	—	5	16	no	—
74	C₂×C₃×C₈	—	5	8	no	—
75	C₂×C₃×C₈	#168	1	8	yes	direct product factor
76	C₂×C₃×C₈	C₂×Dic₅	1	4	yes	—
77	C₃×C₄²	C₄×D₅	1	2	yes	—
78	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
79	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
80	C₂³×C₅	C₃×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
83	C₂×C₄×C₅	C₂×C₃×C₈	1	8	yes	—
84	C₂×C₄×C₅	C₂×C₃×C₈	1	4	yes	—
85	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
86	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
87	C₅×C₈	C₂×C₃×C₈	1	8	yes	complemented
88	#168	C₂×C₃×C₈	1	16	yes	direct product factor
89	C₂²×C₈	—	5	4	no	—
90	C₂²×C₈	C₃×Dic₅	1	2	yes	—
91	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
92	C₄×C₈	—	5	4	no	—
93	C₄×C₈	—	5	4	no	—
94	C₄×C₈	C₃×Dic₅	1	2	yes	—
95	C₄×C₈	C₂×C₃×D₅	1	2	yes	—
96	C₂×C₃×C₅	C₈²	1	4	yes	complemented
97	C₂×C₃×C₅	C₂×C₄×C₈	1	2	yes	—
98	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
99	C₂³×C₃	C₄×Dic₅	1	1	yes	characteristic
100	C₂×C₃×C₄	C₈×D₅	1	8	yes	—
101	C₂×C₃×C₄	C₂×#168	1	4	yes	—
102	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
103	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
104	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
105	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
106	C₃×C₈	—	5	16	no	—
107	C₃×C₈	C₂×#168	1	8	yes	direct product factor
108	C₂²×C₅	C₃×C₄×C₈	1	2	yes	—
109	C₂²×C₅	C₃×C₄×C₈	1	4	yes	—
110	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
111	C₄×C₅	C₃×C₄×C₈	1	2	yes	—
112	C₄×C₅	C₃×C₄×C₈	1	4	yes	—
113	C₄×C₅	C₂²×C₃×C₈	1	4	yes	—
114	C₄×C₅	C₂²×C₃×C₈	1	2	yes	—
115	C₂²×C₄	C₃×C₄×D₅	1	2	yes	—
116	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
117	C₂×C₈	—	5	16	no	—
118	C₂×C₈	—	5	8	no	—
119	C₂×C₈	C₃×#168	1	8	yes	direct product factor
120	C₂×C₈	C₂×C₃×Dic₅	1	4	yes	—
121	C₄²	C₃×C₄×D₅	1	2	yes	—
122	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
123	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
124	C₃×C₅	C₂×C₈²	1	1	yes	characteristic, complemented, Hall
125	C₂²×C₃	C₄×#168	1	2	yes	—
126	C₂²×C₃	C₈×Dic₅	1	4	yes	—
127	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
128	C₃×C₄	C₄×#168	1	2	yes	—
129	C₃×C₄	C₈×Dic₅	1	4	yes	—
130	C₃×C₄	C₂×C₈×D₅	1	4	yes	—
131	C₃×C₄	C₂²×#168	1	2	yes	—
132	C₂×C₅	C₃×C₈²	1	4	yes	complemented
133	C₂×C₅	C₂×C₃×C₄×C₈	1	2	yes	—
134	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
135	C₂³	C₃×C₄×Dic₅	1	1	yes	characteristic
136	C₂×C₄	C₃×C₈×D₅	1	8	yes	—
137	C₂×C₄	C₂×C₃×#168	1	4	yes	—
138	C₂×C₄	C₃×C₄×Dic₅	1	1	yes	characteristic
139	C₂×C₄	C₃×C₄×Dic₅	1	2	yes	—
140	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
141	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
142	C₈	—	5	16	no	—
143	C₈	C₂×C₃×#168	1	8	yes	direct product factor
144	C₂×C₃	C₈×#168	1	4	yes	direct product factor
145	C₂×C₃	C₂×C₄×#168	1	1	yes	characteristic
146	C₂×C₃	C₂×C₈×Dic₅	1	2	yes	—
147	C₅	C₂×C₃×C₈²	1	1	yes	characteristic, complemented, Sylow
148	C₂²	C₃×C₄×#168	1	2	yes	—
149	C₂²	C₃×C₈×Dic₅	1	4	yes	—
150	C₂²	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
151	C₄	C₃×C₄×#168	1	2	yes	—
152	C₄	C₃×C₈×Dic₅	1	4	yes	—
153	C₄	C₂×C₃×C₈×D₅	1	4	yes	—
154	C₄	C₂²×C₃×#168	1	2	yes	—
155	C₃	C₂×C₈×#168	1	1	yes	characteristic, direct product factor, Sylow
156	C₂	C₃×C₈×#168	1	4	yes	direct product factor
157	C₂	C₂×C₃×C₄×#168	1	1	yes	characteristic
158	C₂	C₂×C₃×C₈×Dic₅	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄×C₈	D₅
commutator, hypocenter	C₅	C₂×C₃×C₈²
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₄²×C₅	C₂³×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×C₈×#168	C₂	direct (N × H)
4. C₂×C₈×#168	C₃	direct (N × H)
15. C₈×#168	C₂×C₃	direct (N × H)
18. C₂×C₃×C₅×C₈	C₈	semidirect
23. C₂×C₃×#168	C₈	direct (N × H)
43. C₃×C₅×C₈	C₂×C₈	semidirect
44. C₃×#168	C₂×C₈	direct (N × H)
54. C₂×C₅×C₈	C₃×C₈	semidirect
59. C₂×#168	C₃×C₈	direct (N × H)
75. C₂×C₃×C₈	#168	direct (N × H)
87. C₅×C₈	C₂×C₃×C₈	semidirect
88. #168	C₂×C₃×C₈	direct (N × H)
96. C₂×C₃×C₅	C₈²	semidirect
107. C₃×C₈	C₂×#168	direct (N × H)
119. C₂×C₈	C₃×#168	direct (N × H)
124. C₃×C₅	C₂×C₈²	semidirect
132. C₂×C₅	C₃×C₈²	semidirect
143. C₈	C₂×C₃×#168	direct (N × H)
144. C₂×C₃	C₈×#168	direct (N × H)
147. C₅	C₂×C₃×C₈²	semidirect
155. C₃	C₂×C₈×#168	direct (N × H)
156. C₂	C₃×C₈×#168	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×C₃×C₄×C₅×C₈	2
C₂×C₃×C₄×#168	2
C₃×C₈×#168	2
C₂×C₈×#168	3
C₂×C₃×C₈²	5

Subgroups — C2×C3×C8×#168 #14280307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₈×#168 #14280307