Subgroups — C₃×C₅×#1834 #14274868

Order 1920.

496 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	C₃×C₅×#384	C₂	1	2	yes	maximal
3	C₃×C₅×#340	C₂	1	4	yes	maximal
4	C₅×#1834	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
6	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
7	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₅×C₈	C₄	1	4	yes	complemented
11	C₃×#1834	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
13	C₅×#384	C₂×C₃	1	2	yes	—
14	C₅×#340	C₂×C₃	1	4	yes	—
15	C₂²×C₃×C₄×C₅	—	2	4	no	—
16	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
17	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
18	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂×C₃×C₅×C₈	—	2	4	no	—
20	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
21	C₃×C₅×C₄⋊C₄	—	2	8	no	—
22	C₃×C₅×C₄⋊C₄	—	4	4	no	—
23	C₃×C₅×C₄⋊C₄	—	4	4	no	—
24	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₃×C₄²×C₅	Q₈	1	2	yes	—
26	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
27	C₃×#384	C₂×C₅	1	2	yes	—
28	C₃×#340	C₂×C₅	1	4	yes	—
29	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	—	2	2	no	—
32	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
33	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
34	C₄×C₅×C₈	C₃×C₄	1	4	yes	complemented
35	#1834	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
36	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₅	—	4	4	no	—
38	C₂×C₃×C₄×C₅	—	4	4	no	—
39	C₂×C₃×C₄×C₅	—	4	4	no	—
40	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
41	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
42	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
43	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
44	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
45	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	#41	1	4	yes	—
48	C₃×C₅×C₈	—	2	4	no	—
49	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
50	C₂×C₃×C₄⋊C₄	—	2	2	no	—
51	C₂×C₃×C₄⋊C₄	—	2	2	no	—
52	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
53	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
54	C₃×C₄×C₈	C₄×C₅	1	4	yes	complemented
55	C₂²×C₄×C₅	—	2	4	no	—
56	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
59	C₂×C₅×C₈	—	2	4	no	—
60	C₂×C₅×C₈	C₂×C₃×C₄	1	4	yes	—
61	C₅×C₄⋊C₄	—	2	8	no	—
62	C₅×C₄⋊C₄	—	4	4	no	—
63	C₅×C₄⋊C₄	—	4	4	no	—
64	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
65	C₄²×C₅	C₃×Q₈	1	2	yes	—
66	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
67	#384	C₂×C₃×C₅	1	2	yes	—
68	#340	C₂×C₃×C₅	1	4	yes	—
69	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₅	#124	1	2	yes	—
71	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
72	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
73	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
74	C₃×C₄×C₅	—	8	4	no	—
75	C₃×C₄×C₅	#106	1	4	yes	—
76	C₃×C₄×C₅	#107	1	4	yes	—
77	C₃×C₄×C₅	#117	1	2	yes	—
78	C₃×C₄×C₅	#125	1	2	yes	—
79	C₂²×C₃×C₄	—	2	4	no	—
80	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
83	C₂×C₃×C₈	—	2	4	no	—
84	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
85	C₃×C₄⋊C₄	—	2	8	no	—
86	C₃×C₄⋊C₄	—	4	4	no	—
87	C₃×C₄⋊C₄	—	4	4	no	—
88	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
89	C₃×C₄²	C₅×Q₈	1	2	yes	—
90	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅	—	4	4	no	—
92	C₂×C₄×C₅	—	4	4	no	—
93	C₂×C₄×C₅	—	4	4	no	—
94	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
97	C₂×C₄×C₅	C₃×SD₁₆	1	4	yes	—
98	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
99	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅	C₃×#41	1	4	yes	—
102	C₅×C₈	—	2	4	no	—
103	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
107	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
108	C₄×C₈	C₃×C₄×C₅	1	4	yes	complemented
109	C₂×C₃×C₅	#382	1	1	yes	characteristic
110	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
111	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
112	C₂×C₃×C₅	#486	1	2	yes	—
113	C₂×C₃×C₅	#487	1	2	yes	—
114	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
115	C₂×C₃×C₄	—	4	4	no	—
116	C₂×C₃×C₄	—	4	4	no	—
117	C₂×C₃×C₄	—	4	4	no	—
118	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
119	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
120	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
121	C₂×C₃×C₄	C₅×SD₁₆	1	4	yes	—
122	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
123	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
125	C₂×C₃×C₄	C₅×#41	1	4	yes	—
126	C₃×C₈	—	2	4	no	—
127	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₅	C₃×#124	1	2	yes	—
129	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
130	C₂²×C₅	C₂×C₃×SD₁₆	1	2	yes	—
131	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
132	C₄×C₅	—	8	4	no	—
133	C₄×C₅	C₃×#106	1	4	yes	—
134	C₄×C₅	C₃×#107	1	4	yes	—
135	C₄×C₅	C₃×#117	1	2	yes	—
136	C₄×C₅	C₃×#125	1	2	yes	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
139	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
140	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	C₂×C₃×C₄×C₅	1	4	yes	—
143	C₄⋊C₄	—	2	8	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
147	C₄²	C₃×C₅×Q₈	1	2	yes	—
148	C₃×C₅	#1834	1	1	yes	characteristic, direct product factor, Hall
149	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₃	C₅×#124	1	2	yes	—
151	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
152	C₂²×C₃	C₂×C₅×SD₁₆	1	2	yes	—
153	C₂²×C₃	C₂×C₅×Q₁₆	1	1	yes	characteristic
154	C₃×C₄	—	8	4	no	—
155	C₃×C₄	C₅×#106	1	4	yes	—
156	C₃×C₄	C₅×#107	1	4	yes	—
157	C₃×C₄	C₅×#117	1	2	yes	—
158	C₃×C₄	C₅×#125	1	2	yes	—
159	C₂×C₅	C₃×#382	1	1	yes	characteristic
160	C₂×C₅	C₂×C₃×#106	1	1	yes	characteristic
161	C₂×C₅	C₂×C₃×#107	1	1	yes	characteristic
162	C₂×C₅	C₃×#486	1	2	yes	—
163	C₂×C₅	C₃×#487	1	2	yes	—
164	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
169	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
170	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
171	C₂×C₄	C₃×C₅×SD₁₆	1	4	yes	—
172	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
173	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
174	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
175	C₂×C₄	C₃×C₅×#41	1	4	yes	—
176	C₈	—	2	4	no	—
177	C₂×C₃	C₅×#382	1	1	yes	characteristic
178	C₂×C₃	C₂×C₅×#106	1	1	yes	characteristic
179	C₂×C₃	C₂×C₅×#107	1	1	yes	characteristic
180	C₂×C₃	C₅×#486	1	2	yes	—
181	C₂×C₃	C₅×#487	1	2	yes	—
182	C₅	C₃×#1834	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
184	C₂²	C₃×C₅×#124	1	2	yes	—
185	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
186	C₂²	C₂×C₃×C₅×SD₁₆	1	2	yes	—
187	C₂²	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	C₃×C₅×#106	1	4	yes	—
190	C₄	C₃×C₅×#107	1	4	yes	—
191	C₄	C₃×C₅×#117	1	2	yes	—
192	C₄	C₃×C₅×#125	1	2	yes	—
193	C₃	C₅×#1834	1	1	yes	characteristic, direct product factor, Sylow
194	C₂	C₃×C₅×#382	1	1	yes	characteristic
195	C₂	C₂×C₃×C₅×#106	1	1	yes	characteristic
196	C₂	C₂×C₃×C₅×#107	1	1	yes	characteristic
197	C₂	C₃×C₅×#486	1	2	yes	—
198	C₂	C₃×C₅×#487	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1834	C₃	direct (N × H)
10. C₃×C₄×C₅×C₈	C₄	semidirect
11. C₃×#1834	C₅	direct (N × H)
34. C₄×C₅×C₈	C₃×C₄	semidirect
35. #1834	C₃×C₅	direct (N × H)
54. C₃×C₄×C₈	C₄×C₅	semidirect
108. C₄×C₈	C₃×C₄×C₅	semidirect
148. C₃×C₅	#1834	direct (N × H)
182. C₅	C₃×#1834	direct (N × H)
193. C₃	C₅×#1834	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
C₃×C₅×#384	2
C₃×C₅×#340	2
C₅×#1834	3
C₃×#1834	5

Subgroups — C3×C5×#1834 #14274868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1834 #14274868