Subgroups — C₃×C₅×#1676 #14274710

Order 1920.

288 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×#477	C₂	1	2	yes	maximal
3	C₃×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#333	C₂	1	2	yes	maximal
6	C₅×#1676	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×C₄×C₅×C₈	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#1676	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×C₈×Q₈	C₂×C₃	1	1	yes	characteristic
22	C₅×#335	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×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
39	C₃×#335	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	C₄×C₅×C₈	—	2	1	no	—
51	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
52	#1676	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
53	C₃×C₅×Q₈	—	2	1	no	—
54	C₃×C₅×Q₈	—	8	2	no	—
55	C₃×C₅×Q₈	Q₁₆	1	2	yes	—
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	—	2	1	no	—
58	C₂×C₃×C₄×C₅	—	4	2	no	—
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
62	C₃×C₅×C₈	—	2	1	no	—
63	C₃×C₅×C₈	—	4	2	no	—
64	C₃×C₅×C₈	—	4	1	no	—
65	C₃×C₅×C₈	Q₁₆	1	2	yes	—
66	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
67	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
68	C₃×#107	—	2	1	no	—
69	C₃×#107	—	2	1	no	—
70	C₃×#107	—	2	2	no	—
71	C₃×C₄⋊C₈	—	2	1	no	—
72	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
73	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
74	C₃×#103	—	2	2	no	—
75	C₃×C₄×C₈	—	2	1	no	—
76	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
77	C₂×C₅×Q₈	—	2	2	no	—
78	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
83	C₅×C₄⋊C₄	—	2	1	no	—
84	C₅×C₄⋊C₄	—	2	2	no	—
85	C₅×C₄⋊C₄	—	2	2	no	—
86	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
87	C₄²×C₅	—	2	1	no	—
88	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
89	#500	C₂×C₃×C₅	1	1	yes	characteristic
90	#477	C₂×C₃×C₅	1	2	yes	—
91	C₈×Q₈	C₂×C₃×C₅	1	1	yes	characteristic
92	#335	C₂×C₃×C₅	1	1	yes	characteristic
93	#333	C₂×C₃×C₅	1	2	yes	—
94	C₂²×C₃×C₅	#121	1	1	yes	characteristic
95	C₃×C₄×C₅	—	2	2	no	—
96	C₃×C₄×C₅	—	2	1	no	—
97	C₃×C₄×C₅	—	4	1	no	—
98	C₃×C₄×C₅	—	8	2	no	—
99	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
100	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
101	C₃×C₄×C₅	#135	1	1	yes	characteristic
102	C₃×C₄×C₅	#137	1	1	yes	characteristic
103	C₂×C₃×Q₈	—	2	2	no	—
104	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
108	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
109	C₃×C₄⋊C₄	—	2	1	no	—
110	C₃×C₄⋊C₄	—	2	2	no	—
111	C₃×C₄⋊C₄	—	2	2	no	—
112	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
113	C₃×C₄²	—	2	1	no	—
114	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
115	C₅×Q₈	—	2	1	no	—
116	C₅×Q₈	—	8	2	no	—
117	C₅×Q₈	C₃×Q₁₆	1	2	yes	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	4	2	no	—
121	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	—	4	2	no	—
126	C₅×C₈	—	4	1	no	—
127	C₅×C₈	C₃×Q₁₆	1	2	yes	—
128	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
129	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
130	#107	—	2	1	no	—
131	#107	—	2	1	no	—
132	#107	—	2	2	no	—
133	C₄⋊C₈	—	2	1	no	—
134	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
135	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
136	#103	—	2	2	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
139	C₂×C₃×C₅	#462	1	1	yes	characteristic
140	C₂×C₃×C₅	#467	1	1	yes	characteristic
141	C₂×C₃×C₅	#472	1	1	yes	characteristic
142	C₃×Q₈	—	2	1	no	—
143	C₃×Q₈	—	8	2	no	—
144	C₃×Q₈	C₅×Q₁₆	1	2	yes	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	4	2	no	—
148	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
151	C₃×C₈	—	2	1	no	—
152	C₃×C₈	—	4	2	no	—
153	C₃×C₈	—	4	1	no	—
154	C₃×C₈	C₅×Q₁₆	1	2	yes	—
155	C₂²×C₅	C₃×#121	1	1	yes	characteristic
156	C₄×C₅	—	2	2	no	—
157	C₄×C₅	—	2	1	no	—
158	C₄×C₅	—	4	1	no	—
159	C₄×C₅	—	8	2	no	—
160	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
161	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
162	C₄×C₅	C₃×#135	1	1	yes	characteristic
163	C₄×C₅	C₃×#137	1	1	yes	characteristic
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
169	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
174	C₄²	—	2	1	no	—
175	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
176	C₃×C₅	#1676	1	1	yes	characteristic, direct product factor, Hall
177	C₂²×C₃	C₅×#121	1	1	yes	characteristic
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	4	1	no	—
181	C₃×C₄	—	8	2	no	—
182	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
183	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
184	C₃×C₄	C₅×#135	1	1	yes	characteristic
185	C₃×C₄	C₅×#137	1	1	yes	characteristic
186	C₂×C₅	C₃×#462	1	1	yes	characteristic
187	C₂×C₅	C₃×#467	1	1	yes	characteristic
188	C₂×C₅	C₃×#472	1	1	yes	characteristic
189	Q₈	—	2	1	no	—
190	Q₈	—	8	2	no	—
191	Q₈	C₃×C₅×Q₁₆	1	2	yes	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
196	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
197	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	4	2	no	—
200	C₈	—	4	1	no	—
201	C₈	C₃×C₅×Q₁₆	1	2	yes	—
202	C₂×C₃	C₅×#462	1	1	yes	characteristic
203	C₂×C₃	C₅×#467	1	1	yes	characteristic
204	C₂×C₃	C₅×#472	1	1	yes	characteristic
205	C₅	C₃×#1676	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²	C₃×C₅×#121	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	1	no	—
209	C₄	—	4	1	no	—
210	C₄	—	8	2	no	—
211	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
212	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
213	C₄	C₃×C₅×#135	1	1	yes	characteristic
214	C₄	C₃×C₅×#137	1	1	yes	characteristic
215	C₃	C₅×#1676	1	1	yes	characteristic, direct product factor, Sylow
216	C₂	C₃×C₅×#462	1	1	yes	characteristic
217	C₂	C₃×C₅×#467	1	1	yes	characteristic
218	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×#1676	C₃	direct (N × H)
18. C₃×#1676	C₅	direct (N × H)
52. #1676	C₃×C₅	direct (N × H)
176. C₃×C₅	#1676	direct (N × H)
205. C₅	C₃×#1676	direct (N × H)
215. C₃	C₅×#1676	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×C₈×Q₈	2
C₃×C₅×#335	2
C₃×C₅×#333	2
C₅×#1676	3
C₃×#1676	5

Subgroups — C3×C5×#1676 #14274710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1676 #14274710