Subgroups — C₃×C₅×#1696 #14274730

Order 1920.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 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×#477	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×#333	C₂	1	2	yes	maximal
6	C₅×#1696	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#107	—	2	1	no	—
10	C₃×C₅×#107	—	2	1	no	—
11	C₃×C₅×#107	—	2	2	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×#1696	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#500	C₂×C₃	1	1	yes	characteristic
20	C₅×#477	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×#333	C₂×C₃	1	2	yes	—
24	C₂×C₃×C₅×Q₈	—	2	2	no	—
25	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	—	2	1	no	—
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₃×C₅×C₄⋊C₄	—	2	1	no	—
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
34	C₃×C₄²×C₅	—	2	1	no	—
35	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₃×#500	C₂×C₅	1	1	yes	characteristic
37	C₃×#477	C₂×C₅	1	2	yes	—
38	C₃×#446	C₂×C₅	1	1	yes	characteristic
39	C₃×#334	C₂×C₅	1	1	yes	characteristic
40	C₃×#333	C₂×C₅	1	2	yes	—
41	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
42	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
43	C₅×#107	—	2	1	no	—
44	C₅×#107	—	2	1	no	—
45	C₅×#107	—	2	2	no	—
46	C₅×C₄⋊C₈	—	2	1	no	—
47	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
48	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
49	C₅×#103	—	2	2	no	—
50	C₅×#97	—	2	1	no	—
51	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
52	#1696	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
53	C₃×C₅×Q₈	—	2	2	no	—
54	C₃×C₅×Q₈	—	8	2	no	—
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	—	4	2	no	—
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
61	C₃×C₅×C₈	—	2	1	no	—
62	C₃×C₅×C₈	—	4	1	no	—
63	C₃×C₅×C₈	—	4	2	no	—
64	C₃×C₅×C₈	Q₁₆	1	2	yes	—
65	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
66	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
67	C₃×#107	—	2	1	no	—
68	C₃×#107	—	2	1	no	—
69	C₃×#107	—	2	2	no	—
70	C₃×C₄⋊C₈	—	2	1	no	—
71	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
72	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
73	C₃×#103	—	2	2	no	—
74	C₃×#97	—	2	1	no	—
75	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
76	C₂×C₅×Q₈	—	2	2	no	—
77	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
78	C₂×C₅×C₈	—	2	1	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
82	C₅×C₄⋊C₄	—	2	1	no	—
83	C₅×C₄⋊C₄	—	2	2	no	—
84	C₅×C₄⋊C₄	—	2	2	no	—
85	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
86	C₄²×C₅	—	2	1	no	—
87	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
88	#500	C₂×C₃×C₅	1	1	yes	characteristic
89	#477	C₂×C₃×C₅	1	2	yes	—
90	#446	C₂×C₃×C₅	1	1	yes	characteristic
91	#334	C₂×C₃×C₅	1	1	yes	characteristic
92	#333	C₂×C₃×C₅	1	2	yes	—
93	C₂²×C₃×C₅	#121	1	1	yes	characteristic
94	C₃×C₄×C₅	—	2	1	no	—
95	C₃×C₄×C₅	—	4	1	no	—
96	C₃×C₄×C₅	—	4	1	no	—
97	C₃×C₄×C₅	—	8	2	no	—
98	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
99	C₃×C₄×C₅	#137	1	1	yes	characteristic
100	C₃×C₄×C₅	#137	1	2	yes	—
101	C₂×C₃×Q₈	—	2	2	no	—
102	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
107	C₃×C₄⋊C₄	—	2	1	no	—
108	C₃×C₄⋊C₄	—	2	2	no	—
109	C₃×C₄⋊C₄	—	2	2	no	—
110	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
111	C₃×C₄²	—	2	1	no	—
112	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
113	C₅×Q₈	—	2	2	no	—
114	C₅×Q₈	—	8	2	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
121	C₅×C₈	—	2	1	no	—
122	C₅×C₈	—	4	1	no	—
123	C₅×C₈	—	4	2	no	—
124	C₅×C₈	C₃×Q₁₆	1	2	yes	—
125	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
126	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
127	#107	—	2	1	no	—
128	#107	—	2	1	no	—
129	#107	—	2	2	no	—
130	C₄⋊C₈	—	2	1	no	—
131	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
132	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
133	#103	—	2	2	no	—
134	#97	—	2	1	no	—
135	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
136	C₂×C₃×C₅	#462	1	1	yes	characteristic
137	C₂×C₃×C₅	#470	1	1	yes	characteristic
138	C₂×C₃×C₅	#472	1	1	yes	characteristic
139	C₃×Q₈	—	2	2	no	—
140	C₃×Q₈	—	8	2	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	4	2	no	—
144	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
146	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
147	C₃×C₈	—	2	1	no	—
148	C₃×C₈	—	4	1	no	—
149	C₃×C₈	—	4	2	no	—
150	C₃×C₈	C₅×Q₁₆	1	2	yes	—
151	C₂²×C₅	C₃×#121	1	1	yes	characteristic
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	4	1	no	—
154	C₄×C₅	—	4	1	no	—
155	C₄×C₅	—	8	2	no	—
156	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
157	C₄×C₅	C₃×#137	1	1	yes	characteristic
158	C₄×C₅	C₃×#137	1	2	yes	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
164	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
169	C₄²	—	2	1	no	—
170	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
171	C₃×C₅	#1696	1	1	yes	characteristic, direct product factor, Hall
172	C₂²×C₃	C₅×#121	1	1	yes	characteristic
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	4	1	no	—
175	C₃×C₄	—	4	1	no	—
176	C₃×C₄	—	8	2	no	—
177	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
178	C₃×C₄	C₅×#137	1	1	yes	characteristic
179	C₃×C₄	C₅×#137	1	2	yes	—
180	C₂×C₅	C₃×#462	1	1	yes	characteristic
181	C₂×C₅	C₃×#470	1	1	yes	characteristic
182	C₂×C₅	C₃×#472	1	1	yes	characteristic
183	Q₈	—	2	2	no	—
184	Q₈	—	8	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
189	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
190	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	4	1	no	—
193	C₈	—	4	2	no	—
194	C₈	C₃×C₅×Q₁₆	1	2	yes	—
195	C₂×C₃	C₅×#462	1	1	yes	characteristic
196	C₂×C₃	C₅×#470	1	1	yes	characteristic
197	C₂×C₃	C₅×#472	1	1	yes	characteristic
198	C₅	C₃×#1696	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²	C₃×C₅×#121	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	4	1	no	—
202	C₄	—	4	1	no	—
203	C₄	—	8	2	no	—
204	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
205	C₄	C₃×C₅×#137	1	1	yes	characteristic
206	C₄	C₃×C₅×#137	1	2	yes	—
207	C₃	C₅×#1696	1	1	yes	characteristic, direct product factor, Sylow
208	C₂	C₃×C₅×#462	1	1	yes	characteristic
209	C₂	C₃×C₅×#470	1	1	yes	characteristic
210	C₂	C₃×C₅×#472	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×#1696	C₃	direct (N × H)
18. C₃×#1696	C₅	direct (N × H)
52. #1696	C₃×C₅	direct (N × H)
171. C₃×C₅	#1696	direct (N × H)
198. C₅	C₃×#1696	direct (N × H)
207. C₃	C₅×#1696	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×#477	2
C₃×C₅×#446	2
C₃×C₅×#334	2
C₃×C₅×#333	2
C₅×#1696	3
C₃×#1696	5

Subgroups — C3×C5×#1696 #14274730

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1696 #14274730