Subgroups — C₂×C₃×C₄×#672 #14312510

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 168 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	C₂²×C₃×#672	C₂	1	2	yes	maximal
3	C₃×C₄×#672	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×#672	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
8	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
9	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
10	C₂×C₃×#672	C₄	1	8	yes	direct product factor
11	C₂×C₃×#672	C₂²	1	4	yes	—
12	C₂×C₃×C₄×C₁₆	—	5	1	no	maximal, Hall
13	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
14	C₂²×#672	C₂×C₃	1	2	yes	—
15	C₄×#672	C₂×C₃	1	4	yes	direct product factor
16	C₂²×C₃×C₄×C₅	C₈	1	2	yes	—
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₅×C₈	C₈	1	4	yes	—
19	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
23	C₃×C₄²×C₅	C₈	1	2	yes	—
24	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₃×#672	C₂×C₄	1	8	yes	direct product factor
26	C₂²×C₃×C₁₆	—	5	2	no	—
27	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
28	C₃×C₄×C₁₆	—	5	4	no	—
29	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₃×C₄	1	1	yes	characteristic
32	C₄×C₅×C₈	C₃×C₄	1	2	yes	—
33	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
34	C₂×#672	C₃×C₄	1	8	yes	direct product factor
35	C₂×#672	C₂²×C₃	1	4	yes	—
36	C₂×C₄×C₁₆	—	5	1	no	Sylow
37	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₁₆	1	8	yes	complemented
39	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
40	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
42	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
43	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₃×C₅×C₈	C₄²	1	2	yes	—
45	C₃×C₅×C₈	C₂×C₈	1	4	yes	—
46	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
47	C₂²×C₃×C₈	Dic₅	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
49	C₂×C₃×C₄²	Dic₅	1	1	yes	characteristic
50	C₂×C₃×C₁₆	—	5	8	no	—
51	C₂×C₃×C₁₆	—	5	4	no	—
52	C₃×C₄×C₈	Dic₅	1	2	yes	—
53	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
54	C₂²×C₄×C₅	C₃×C₈	1	2	yes	—
55	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₃×C₈	1	4	yes	—
57	C₂×C₅×C₈	C₂×C₃×C₄	1	4	yes	—
58	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
60	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
61	C₄²×C₅	C₃×C₈	1	2	yes	—
62	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
63	#672	C₂×C₃×C₄	1	8	yes	direct product factor
64	C₂²×C₁₆	—	5	2	no	—
65	C₂×C₄×C₈	C₃×D₅	1	1	yes	characteristic
66	C₄×C₁₆	—	5	4	no	—
67	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
68	C₂²×C₃×C₅	C₂×C₁₆	1	4	yes	—
69	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
70	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
71	C₃×C₄×C₅	C₂×C₁₆	1	8	yes	complemented
72	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
73	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
74	C₂²×C₃×C₄	#168	1	2	yes	—
75	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
76	C₂×C₃×C₈	#168	1	4	yes	—
77	C₂×C₃×C₈	C₄×D₅	1	4	yes	—
78	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
79	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
81	C₃×C₄²	#168	1	2	yes	—
82	C₃×C₄²	C₂×Dic₅	1	2	yes	—
83	C₃×C₁₆	—	5	8	no	—
84	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅	C₃×C₁₆	1	8	yes	complemented
86	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
87	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
89	C₂×C₄×C₅	C₂×C₃×C₈	1	4	yes	—
90	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
91	C₅×C₈	C₃×C₄²	1	2	yes	—
92	C₅×C₈	C₂×C₃×C₈	1	4	yes	—
93	C₅×C₈	C₂²×C₃×C₄	1	2	yes	—
94	C₂²×C₈	C₃×Dic₅	1	1	yes	characteristic
95	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
96	C₂×C₄²	C₃×Dic₅	1	1	yes	characteristic
97	C₂×C₁₆	—	5	8	no	—
98	C₂×C₁₆	—	5	4	no	—
99	C₄×C₈	C₃×Dic₅	1	2	yes	—
100	C₄×C₈	C₂×C₃×D₅	1	2	yes	—
101	C₂×C₃×C₅	C₄×C₁₆	1	4	yes	complemented
102	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂×C₃×C₅	C₂²×C₁₆	1	2	yes	—
104	C₂³×C₃	C₂×#168	1	1	yes	characteristic
105	C₂×C₃×C₄	#672	1	8	yes	direct product factor
106	C₂×C₃×C₄	C₂×#168	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂×#168	1	2	yes	—
108	C₂×C₃×C₄	C₂×#168	1	4	yes	—
109	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
110	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
111	C₃×C₈	C₂×#168	1	4	yes	—
112	C₃×C₈	C₄×Dic₅	1	2	yes	—
113	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
114	C₂²×C₅	C₃×C₄×C₈	1	2	yes	—
115	C₂²×C₅	C₂×C₃×C₁₆	1	4	yes	—
116	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
117	C₄×C₅	C₃×C₄×C₈	1	2	yes	—
118	C₄×C₅	C₂×C₃×C₁₆	1	8	yes	complemented
119	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
120	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
121	C₂²×C₄	C₃×#168	1	2	yes	—
122	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
123	C₂×C₈	C₃×#168	1	4	yes	—
124	C₂×C₈	C₃×C₄×D₅	1	4	yes	—
125	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
126	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
127	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
128	C₄²	C₃×#168	1	2	yes	—
129	C₄²	C₂×C₃×Dic₅	1	2	yes	—
130	C₁₆	—	5	8	no	—
131	C₃×C₅	C₂×C₄×C₁₆	1	1	yes	characteristic, complemented, Hall
132	C₂²×C₃	C₄×#168	1	2	yes	—
133	C₂²×C₃	C₂×#672	1	4	yes	—
134	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
135	C₃×C₄	C₄×#168	1	2	yes	—
136	C₃×C₄	C₂×#672	1	8	yes	direct product factor
137	C₃×C₄	C₂²×#168	1	1	yes	characteristic
138	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
139	C₂×C₅	C₃×C₄×C₁₆	1	4	yes	complemented
140	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
141	C₂×C₅	C₂²×C₃×C₁₆	1	2	yes	—
142	C₂³	C₂×C₃×#168	1	1	yes	characteristic
143	C₂×C₄	C₃×#672	1	8	yes	direct product factor
144	C₂×C₄	C₂×C₃×#168	1	1	yes	characteristic
145	C₂×C₄	C₂×C₃×#168	1	2	yes	—
146	C₂×C₄	C₂×C₃×#168	1	4	yes	—
147	C₂×C₄	C₃×C₄×Dic₅	1	2	yes	—
148	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
149	C₈	C₂×C₃×#168	1	4	yes	—
150	C₈	C₃×C₄×Dic₅	1	2	yes	—
151	C₈	C₂×C₃×C₄×D₅	1	2	yes	—
152	C₂×C₃	C₄×#672	1	4	yes	direct product factor
153	C₂×C₃	C₂×C₄×#168	1	1	yes	characteristic
154	C₂×C₃	C₂²×#672	1	2	yes	—
155	C₅	C₂×C₃×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
156	C₂²	C₃×C₄×#168	1	2	yes	—
157	C₂²	C₂×C₃×#672	1	4	yes	—
158	C₂²	C₂²×C₃×#168	1	1	yes	characteristic
159	C₄	C₃×C₄×#168	1	2	yes	—
160	C₄	C₂×C₃×#672	1	8	yes	direct product factor
161	C₄	C₂²×C₃×#168	1	1	yes	characteristic
162	C₄	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
163	C₃	C₂×C₄×#672	1	1	yes	characteristic, direct product factor, Sylow
164	C₂	C₃×C₄×#672	1	4	yes	direct product factor
165	C₂	C₂×C₃×C₄×#168	1	1	yes	characteristic
166	C₂	C₂²×C₃×#672	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄×C₈	D₅
commutator, hypocenter	C₅	C₂×C₃×C₄×C₁₆
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₅×C₈	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×C₄×#672	C₂	direct (N × H)
4. C₂×C₄×#672	C₃	direct (N × H)
10. C₂×C₃×#672	C₄	direct (N × H)
15. C₄×#672	C₂×C₃	direct (N × H)
25. C₃×#672	C₂×C₄	direct (N × H)
34. C₂×#672	C₃×C₄	direct (N × H)
38. C₂×C₃×C₄×C₅	C₁₆	semidirect
63. #672	C₂×C₃×C₄	direct (N × H)
71. C₃×C₄×C₅	C₂×C₁₆	semidirect
85. C₂×C₄×C₅	C₃×C₁₆	semidirect
101. C₂×C₃×C₅	C₄×C₁₆	semidirect
105. C₂×C₃×C₄	#672	direct (N × H)
118. C₄×C₅	C₂×C₃×C₁₆	semidirect
131. C₃×C₅	C₂×C₄×C₁₆	semidirect
136. C₃×C₄	C₂×#672	direct (N × H)
139. C₂×C₅	C₃×C₄×C₁₆	semidirect
143. C₂×C₄	C₃×#672	direct (N × H)
152. C₂×C₃	C₄×#672	direct (N × H)
155. C₅	C₂×C₃×C₄×C₁₆	semidirect
160. C₄	C₂×C₃×#672	direct (N × H)
163. C₃	C₂×C₄×#672	direct (N × H)
164. C₂	C₃×C₄×#672	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
C₂²×C₃×#672	2
C₃×C₄×#672	2
C₂×C₄×#672	3
C₂×C₃×C₄×C₁₆	5

Subgroups — C2×C3×C4×#672 #14312510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×#672 #14312510