Subgroups — C₃×C₅×#1694 #14274728

Order 1920.

304 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#462	C₂	1	2	yes	maximal
3	C₃×C₅×#446	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#332	C₂	1	2	yes	maximal
6	C₅×#1694	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	—	2	2	no	—
8	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₃×C₅×#107	—	2	1	no	—
11	C₃×C₅×#107	—	2	1	no	—
12	C₃×C₅×C₄⋊C₈	—	2	1	no	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×#103	—	2	2	no	—
16	C₃×C₅×#97	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#1694	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#500	C₂×C₃	1	1	yes	characteristic
20	C₅×#462	C₂×C₃	1	2	yes	—
21	C₅×#446	C₂×C₃	1	1	yes	characteristic
22	C₅×#334	C₂×C₃	1	1	yes	characteristic
23	C₅×#332	C₂×C₃	1	2	yes	—
24	C₂×C₃×C₅×Q₈	—	2	2	no	—
25	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
26	C₃×C₅×Q₁₆	—	4	4	no	—
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	—	2	1	no	—
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	—	2	1	no	—
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
35	C₃×C₄²×C₅	—	2	1	no	—
36	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₃×#500	C₂×C₅	1	1	yes	characteristic
38	C₃×#462	C₂×C₅	1	2	yes	—
39	C₃×#446	C₂×C₅	1	1	yes	characteristic
40	C₃×#334	C₂×C₅	1	1	yes	characteristic
41	C₃×#332	C₂×C₅	1	2	yes	—
42	C₂×C₅×Q₁₆	—	2	2	no	—
43	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
44	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
45	C₅×#107	—	2	1	no	—
46	C₅×#107	—	2	1	no	—
47	C₅×C₄⋊C₈	—	2	1	no	—
48	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
49	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
50	C₅×#103	—	2	2	no	—
51	C₅×#97	—	2	1	no	—
52	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
53	#1694	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₃×C₅×Q₈	—	2	2	no	—
55	C₃×C₅×Q₈	—	4	2	no	—
56	C₃×C₅×Q₈	—	4	2	no	—
57	C₂×C₃×C₄×C₅	—	2	1	no	—
58	C₂×C₃×C₄×C₅	—	2	1	no	—
59	C₂×C₃×C₄×C₅	—	4	2	no	—
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
63	C₃×C₅×C₈	—	2	1	no	—
64	C₃×C₅×C₈	—	4	2	no	—
65	C₃×C₅×C₈	—	4	1	no	—
66	C₃×C₅×C₈	Q₁₆	1	2	yes	complemented
67	C₂×C₃×Q₁₆	—	2	2	no	—
68	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
69	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
70	C₃×#107	—	2	1	no	—
71	C₃×#107	—	2	1	no	—
72	C₃×C₄⋊C₈	—	2	1	no	—
73	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
74	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
75	C₃×#103	—	2	2	no	—
76	C₃×#97	—	2	1	no	—
77	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
78	C₂×C₅×Q₈	—	2	2	no	—
79	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
80	C₅×Q₁₆	—	4	4	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	1	no	—
83	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
85	C₅×C₄⋊C₄	—	2	1	no	—
86	C₅×C₄⋊C₄	—	2	2	no	—
87	C₅×C₄⋊C₄	—	2	2	no	—
88	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
89	C₄²×C₅	—	2	1	no	—
90	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
91	#500	C₂×C₃×C₅	1	1	yes	characteristic
92	#462	C₂×C₃×C₅	1	2	yes	—
93	#446	C₂×C₃×C₅	1	1	yes	characteristic
94	#334	C₂×C₃×C₅	1	1	yes	characteristic
95	#332	C₂×C₃×C₅	1	2	yes	—
96	C₂²×C₃×C₅	#121	1	1	yes	characteristic
97	C₃×C₄×C₅	—	2	1	no	—
98	C₃×C₄×C₅	—	4	1	no	—
99	C₃×C₄×C₅	—	4	1	no	—
100	C₃×C₄×C₅	—	8	2	no	—
101	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
102	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
103	C₃×C₄×C₅	#137	1	1	yes	characteristic
104	C₂×C₃×Q₈	—	2	2	no	—
105	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
106	C₃×Q₁₆	—	4	4	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
110	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
111	C₃×C₄⋊C₄	—	2	1	no	—
112	C₃×C₄⋊C₄	—	2	2	no	—
113	C₃×C₄⋊C₄	—	2	2	no	—
114	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
115	C₃×C₄²	—	2	1	no	—
116	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
117	C₅×Q₈	—	2	2	no	—
118	C₅×Q₈	—	4	2	no	—
119	C₅×Q₈	—	4	2	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	4	2	no	—
123	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
126	C₅×C₈	—	2	1	no	—
127	C₅×C₈	—	4	2	no	—
128	C₅×C₈	—	4	1	no	—
129	C₅×C₈	C₃×Q₁₆	1	2	yes	complemented
130	C₂×Q₁₆	—	2	2	no	—
131	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
132	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
133	#107	—	2	1	no	—
134	#107	—	2	1	no	—
135	C₄⋊C₈	—	2	1	no	—
136	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
137	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
138	#103	—	2	2	no	—
139	#97	—	2	1	no	—
140	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
141	C₂×C₃×C₅	#462	1	1	yes	characteristic
142	C₂×C₃×C₅	#469	1	1	yes	characteristic
143	C₂×C₃×C₅	#473	1	1	yes	characteristic
144	C₃×Q₈	—	2	2	no	—
145	C₃×Q₈	—	4	2	no	—
146	C₃×Q₈	—	4	2	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	4	2	no	—
150	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
153	C₃×C₈	—	2	1	no	—
154	C₃×C₈	—	4	2	no	—
155	C₃×C₈	—	4	1	no	—
156	C₃×C₈	C₅×Q₁₆	1	2	yes	complemented
157	C₂²×C₅	C₃×#121	1	1	yes	characteristic
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	4	1	no	—
160	C₄×C₅	—	4	1	no	—
161	C₄×C₅	—	8	2	no	—
162	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
163	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
164	C₄×C₅	C₃×#137	1	1	yes	characteristic
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
167	Q₁₆	—	4	4	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
171	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
176	C₄²	—	2	1	no	—
177	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
178	C₃×C₅	#1694	1	1	yes	characteristic, direct product factor, Hall
179	C₂²×C₃	C₅×#121	1	1	yes	characteristic
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	4	1	no	—
182	C₃×C₄	—	4	1	no	—
183	C₃×C₄	—	8	2	no	—
184	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
185	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
186	C₃×C₄	C₅×#137	1	1	yes	characteristic
187	C₂×C₅	C₃×#462	1	1	yes	characteristic
188	C₂×C₅	C₃×#469	1	1	yes	characteristic
189	C₂×C₅	C₃×#473	1	1	yes	characteristic
190	Q₈	—	2	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	4	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
197	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
198	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	4	2	no	—
201	C₈	—	4	1	no	—
202	C₈	C₃×C₅×Q₁₆	1	2	yes	complemented
203	C₂×C₃	C₅×#462	1	1	yes	characteristic
204	C₂×C₃	C₅×#469	1	1	yes	characteristic
205	C₂×C₃	C₅×#473	1	1	yes	characteristic
206	C₅	C₃×#1694	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	C₃×C₅×#121	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	4	1	no	—
210	C₄	—	4	1	no	—
211	C₄	—	8	2	no	—
212	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
213	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
214	C₄	C₃×C₅×#137	1	1	yes	characteristic
215	C₃	C₅×#1694	1	1	yes	characteristic, direct product factor, Sylow
216	C₂	C₃×C₅×#462	1	1	yes	characteristic
217	C₂	C₃×C₅×#469	1	1	yes	characteristic
218	C₂	C₃×C₅×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#1694	C₃	direct (N × H)
18. C₃×#1694	C₅	direct (N × H)
53. #1694	C₃×C₅	direct (N × H)
66. C₃×C₅×C₈	Q₁₆	semidirect
129. C₅×C₈	C₃×Q₁₆	semidirect
156. C₃×C₈	C₅×Q₁₆	semidirect
178. C₃×C₅	#1694	direct (N × H)
202. C₈	C₃×C₅×Q₁₆	semidirect
206. C₅	C₃×#1694	direct (N × H)
215. C₃	C₅×#1694	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#500	2
C₃×C₅×#462	2
C₃×C₅×#446	2
C₃×C₅×#334	2
C₃×C₅×#332	2
C₅×#1694	3
C₃×#1694	5

Subgroups — C3×C5×#1694 #14274728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1694 #14274728