Subgroups — C₃×C₄×#3944 #14312585

Order 1920.

392 conjugacy classes of subgroups.

The subgroup lattice has 200 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×#3944	C₂	1	2	yes	maximal
3	C₃×#64681	C₂	1	2	yes	maximal, complemented
4	C₃×C₄×#672	C₂	1	2	yes	maximal, complemented
5	C₄×#3944	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
9	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×#3944	C₄	1	8	yes	direct product factor
14	C₂×C₃×#672	C₂²	1	4	yes	—
15	C₃×C₄×M₃₂	—	5	1	no	maximal, Hall
16	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
17	C₂×#3944	C₂×C₃	1	2	yes	—
18	#64681	C₂×C₃	1	2	yes	complemented
19	C₄×#672	C₂×C₃	1	2	yes	complemented
20	C₂²×C₃×C₄×C₅	C₈	1	2	yes	—
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₅×C₈	C₈	1	4	yes	—
23	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₃×C₄²×C₅	C₈	1	2	yes	—
29	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₃×#672	C₂×C₄	1	8	yes	complemented
32	C₂×C₃×M₃₂	—	5	2	no	—
33	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
34	C₃×#346	—	5	2	no	—
35	C₃×C₄×C₁₆	—	5	2	no	—
36	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₃×C₄	1	1	yes	characteristic
39	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
40	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
41	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
43	#3944	C₃×C₄	1	8	yes	direct product factor
44	C₂×#672	C₂²×C₃	1	4	yes	—
45	C₄×M₃₂	—	5	1	no	Sylow
46	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	—	2	4	no	—
48	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₃×C₅×C₈	C₄²	1	2	yes	—
55	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
56	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
57	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
58	C₂²×C₃×C₈	Dic₅	1	1	yes	characteristic
59	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
60	C₂×C₃×C₄²	Dic₅	1	1	yes	characteristic
61	C₃×M₃₂	—	5	8	no	—
62	C₂×C₃×C₁₆	—	5	4	no	—
63	C₃×C₄×C₈	Dic₅	1	1	yes	characteristic
64	C₃×C₄×C₈	Dic₅	1	1	yes	characteristic
65	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
66	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₃×C₈	1	2	yes	—
68	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₃×C₈	1	4	yes	—
70	C₂×C₅×C₈	C₂×C₃×C₄	1	4	yes	—
71	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
75	C₄²×C₅	C₃×C₈	1	2	yes	—
76	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
77	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
78	#672	C₂×C₃×C₄	1	8	yes	complemented
79	C₂×M₃₂	—	5	2	no	—
80	C₂×C₄×C₈	C₃×D₅	1	1	yes	characteristic
81	#346	—	5	2	no	—
82	C₄×C₁₆	—	5	2	no	—
83	C₂²×C₃×C₅	—	2	2	no	—
84	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
85	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
86	C₃×C₄×C₅	—	2	2	no	—
87	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
88	C₃×C₄×C₅	M₃₂	1	4	yes	complemented
89	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
90	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
91	C₂²×C₃×C₄	#168	1	2	yes	—
92	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
93	C₂×C₃×C₈	#168	1	4	yes	—
94	C₂×C₃×C₈	C₄×D₅	1	4	yes	—
95	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
97	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
99	C₃×C₄²	#168	1	2	yes	—
100	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
101	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
102	C₃×C₁₆	—	5	8	no	—
103	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
106	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
107	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
109	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
110	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
111	C₅×C₈	C₃×C₄²	1	2	yes	—
112	C₅×C₈	C₂×C₃×C₈	1	2	yes	—
113	C₅×C₈	C₂×C₃×C₈	1	2	yes	—
114	C₅×C₈	C₂²×C₃×C₄	1	2	yes	—
115	C₂²×C₈	C₃×Dic₅	1	1	yes	characteristic
116	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
117	C₂×C₄²	C₃×Dic₅	1	1	yes	characteristic
118	M₃₂	—	5	8	no	—
119	C₂×C₁₆	—	5	4	no	—
120	C₄×C₈	C₃×Dic₅	1	1	yes	characteristic
121	C₄×C₈	C₃×Dic₅	1	1	yes	characteristic
122	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
123	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
124	C₂×C₃×C₅	—	2	2	no	—
125	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
126	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
127	C₂³×C₃	C₂×#168	1	1	yes	characteristic
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	C₂×#168	1	2	yes	—
130	C₂×C₃×C₄	C₂×#168	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×#168	1	2	yes	—
132	C₂×C₃×C₄	C₂×#168	1	2	yes	—
133	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
134	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
135	C₃×C₈	C₂×#168	1	2	yes	—
136	C₃×C₈	C₂×#168	1	2	yes	—
137	C₃×C₈	C₄×Dic₅	1	2	yes	—
138	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
139	C₂²×C₅	—	2	2	no	—
140	C₂²×C₅	C₃×C₄×C₈	1	2	yes	—
141	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
142	C₄×C₅	—	2	2	no	—
143	C₄×C₅	C₃×C₄×C₈	1	2	yes	—
144	C₄×C₅	C₃×M₃₂	1	4	yes	complemented
145	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
146	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
147	C₂²×C₄	C₃×#168	1	2	yes	—
148	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
149	C₂×C₈	C₃×#168	1	4	yes	—
150	C₂×C₈	C₃×C₄×D₅	1	4	yes	—
151	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
152	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
153	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
154	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
155	C₄²	C₃×#168	1	2	yes	—
156	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
157	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
158	C₁₆	—	5	8	no	—
159	C₃×C₅	C₄×M₃₂	1	1	yes	characteristic, complemented, Hall
160	C₂²×C₃	—	2	2	no	—
161	C₂²×C₃	C₄×#168	1	2	yes	—
162	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
163	C₃×C₄	—	2	2	no	—
164	C₃×C₄	C₄×#168	1	2	yes	—
165	C₃×C₄	#3944	1	4	yes	direct product factor
166	C₃×C₄	C₂²×#168	1	1	yes	characteristic
167	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
168	C₂×C₅	—	2	2	no	—
169	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
170	C₂×C₅	C₂×C₃×M₃₂	1	2	yes	—
171	C₂³	C₂×C₃×#168	1	1	yes	characteristic
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	C₂×C₃×#168	1	2	yes	—
174	C₂×C₄	C₂×C₃×#168	1	1	yes	characteristic
175	C₂×C₄	C₂×C₃×#168	1	2	yes	—
176	C₂×C₄	C₂×C₃×#168	1	2	yes	—
177	C₂×C₄	C₃×C₄×Dic₅	1	2	yes	—
178	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
179	C₈	C₂×C₃×#168	1	2	yes	—
180	C₈	C₂×C₃×#168	1	2	yes	—
181	C₈	C₃×C₄×Dic₅	1	2	yes	—
182	C₈	C₂×C₃×C₄×D₅	1	2	yes	—
183	C₂×C₃	—	2	2	no	—
184	C₂×C₃	C₂×C₄×#168	1	1	yes	characteristic
185	C₂×C₃	C₂×#3944	1	2	yes	—
186	C₅	C₃×C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
187	C₂²	—	2	2	no	—
188	C₂²	C₃×C₄×#168	1	2	yes	—
189	C₂²	C₂²×C₃×#168	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	C₃×C₄×#168	1	2	yes	—
192	C₄	C₃×#3944	1	4	yes	direct product factor
193	C₄	C₂²×C₃×#168	1	1	yes	characteristic
194	C₄	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
195	C₃	C₄×#3944	1	1	yes	characteristic, direct product factor, Sylow
196	C₂	—	2	2	no	—
197	C₂	C₂×C₃×C₄×#168	1	1	yes	characteristic
198	C₂	C₂×C₃×#3944	1	2	yes	—

	subgroup	G/H
center	C₃×C₄×C₈	C₂×D₅
commutator	C₂×C₅	C₂×C₃×C₄×C₈
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂³×C₃×C₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₅
hypocenter	C₅	C₃×C₄×M₃₂
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄×C₈ ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×#64681	C₂	semidirect
4. C₃×C₄×#672	C₂	semidirect
5. C₄×#3944	C₃	direct (N × H)
13. C₃×#3944	C₄	direct (N × H)
18. #64681	C₂×C₃	semidirect
19. C₄×#672	C₂×C₃	semidirect
31. C₃×#672	C₂×C₄	semidirect
43. #3944	C₃×C₄	direct (N × H)
78. #672	C₂×C₃×C₄	semidirect
88. C₃×C₄×C₅	M₃₂	semidirect
144. C₄×C₅	C₃×M₃₂	semidirect
159. C₃×C₅	C₄×M₃₂	semidirect
165. C₃×C₄	#3944	direct (N × H)
186. C₅	C₃×C₄×M₃₂	semidirect
192. C₄	C₃×#3944	direct (N × H)
195. C₃	C₄×#3944	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×#3944	2
C₃×#64681	2
C₃×C₄×#672	2
C₄×#3944	3
C₃×C₄×M₃₂	5

Subgroups — C3×C4×#3944 #14312585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#3944 #14312585