Subgroups — C₃×C₅×#1674 #14274708

Order 1920.

312 conjugacy classes of subgroups.

The subgroup lattice has 228 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×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#332	C₂	1	2	yes	maximal
6	C₅×#1674	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×C₄×C₅×C₈	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#1674	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×C₈×Q₈	C₂×C₃	1	1	yes	characteristic
22	C₅×#335	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×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
40	C₃×#335	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	C₄×C₅×C₈	—	2	1	no	—
52	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
53	#1674	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₃×C₅×Q₈	—	2	1	no	—
55	C₃×C₅×Q₈	—	4	2	no	—
56	C₃×C₅×Q₈	—	4	2	no	—
57	C₃×C₅×Q₈	D₈	1	2	yes	—
58	C₂×C₃×C₄×C₅	—	2	1	no	—
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
64	C₃×C₅×C₈	—	2	1	no	—
65	C₃×C₅×C₈	—	4	2	no	—
66	C₃×C₅×C₈	—	4	1	no	—
67	C₃×C₅×C₈	Q₁₆	1	2	yes	complemented
68	C₂×C₃×Q₁₆	—	2	2	no	—
69	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
70	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
71	C₃×#107	—	2	1	no	—
72	C₃×#107	—	2	1	no	—
73	C₃×C₄⋊C₈	—	2	1	no	—
74	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
75	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
76	C₃×#103	—	2	2	no	—
77	C₃×C₄×C₈	—	2	1	no	—
78	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₅×Q₈	—	2	2	no	—
80	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
81	C₅×Q₁₆	—	4	4	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
86	C₅×C₄⋊C₄	—	2	1	no	—
87	C₅×C₄⋊C₄	—	2	2	no	—
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
90	C₄²×C₅	—	2	1	no	—
91	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
92	#500	C₂×C₃×C₅	1	1	yes	characteristic
93	#462	C₂×C₃×C₅	1	2	yes	—
94	C₈×Q₈	C₂×C₃×C₅	1	1	yes	characteristic
95	#335	C₂×C₃×C₅	1	1	yes	characteristic
96	#332	C₂×C₃×C₅	1	2	yes	—
97	C₂²×C₃×C₅	#121	1	1	yes	characteristic
98	C₃×C₄×C₅	—	2	2	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	4	1	no	—
101	C₃×C₄×C₅	—	8	2	no	—
102	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
103	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
104	C₃×C₄×C₅	#135	1	1	yes	characteristic
105	C₃×C₄×C₅	#137	1	1	yes	characteristic
106	C₂×C₃×Q₈	—	2	2	no	—
107	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
108	C₃×Q₁₆	—	4	4	no	—
109	C₂×C₃×C₈	—	2	1	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
112	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
113	C₃×C₄⋊C₄	—	2	1	no	—
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₄⋊C₄	—	2	2	no	—
116	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
117	C₃×C₄²	—	2	1	no	—
118	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
119	C₅×Q₈	—	2	1	no	—
120	C₅×Q₈	—	4	2	no	—
121	C₅×Q₈	—	4	2	no	—
122	C₅×Q₈	C₃×D₈	1	2	yes	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
129	C₅×C₈	—	2	1	no	—
130	C₅×C₈	—	4	2	no	—
131	C₅×C₈	—	4	1	no	—
132	C₅×C₈	C₃×Q₁₆	1	2	yes	complemented
133	C₂×Q₁₆	—	2	2	no	—
134	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
135	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
136	#107	—	2	1	no	—
137	#107	—	2	1	no	—
138	C₄⋊C₈	—	2	1	no	—
139	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
140	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
141	#103	—	2	2	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
144	C₂×C₃×C₅	#462	1	1	yes	characteristic
145	C₂×C₃×C₅	#466	1	1	yes	characteristic
146	C₂×C₃×C₅	#473	1	1	yes	characteristic
147	C₃×Q₈	—	2	1	no	—
148	C₃×Q₈	—	4	2	no	—
149	C₃×Q₈	—	4	2	no	—
150	C₃×Q₈	C₅×D₈	1	2	yes	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	4	2	no	—
154	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
157	C₃×C₈	—	2	1	no	—
158	C₃×C₈	—	4	2	no	—
159	C₃×C₈	—	4	1	no	—
160	C₃×C₈	C₅×Q₁₆	1	2	yes	complemented
161	C₂²×C₅	C₃×#121	1	1	yes	characteristic
162	C₄×C₅	—	2	2	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	4	1	no	—
165	C₄×C₅	—	8	2	no	—
166	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
167	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
168	C₄×C₅	C₃×#135	1	1	yes	characteristic
169	C₄×C₅	C₃×#137	1	1	yes	characteristic
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
172	Q₁₆	—	4	4	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
176	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
181	C₄²	—	2	1	no	—
182	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
183	C₃×C₅	#1674	1	1	yes	characteristic, direct product factor, Hall
184	C₂²×C₃	C₅×#121	1	1	yes	characteristic
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	4	1	no	—
188	C₃×C₄	—	8	2	no	—
189	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
190	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
191	C₃×C₄	C₅×#135	1	1	yes	characteristic
192	C₃×C₄	C₅×#137	1	1	yes	characteristic
193	C₂×C₅	C₃×#462	1	1	yes	characteristic
194	C₂×C₅	C₃×#466	1	1	yes	characteristic
195	C₂×C₅	C₃×#473	1	1	yes	characteristic
196	Q₈	—	2	1	no	—
197	Q₈	—	4	2	no	—
198	Q₈	—	4	2	no	—
199	Q₈	C₃×C₅×D₈	1	2	yes	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
204	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
205	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	4	2	no	—
208	C₈	—	4	1	no	—
209	C₈	C₃×C₅×Q₁₆	1	2	yes	complemented
210	C₂×C₃	C₅×#462	1	1	yes	characteristic
211	C₂×C₃	C₅×#466	1	1	yes	characteristic
212	C₂×C₃	C₅×#473	1	1	yes	characteristic
213	C₅	C₃×#1674	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	C₃×C₅×#121	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	4	1	no	—
218	C₄	—	8	2	no	—
219	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
220	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
221	C₄	C₃×C₅×#135	1	1	yes	characteristic
222	C₄	C₃×C₅×#137	1	1	yes	characteristic
223	C₃	C₅×#1674	1	1	yes	characteristic, direct product factor, Sylow
224	C₂	C₃×C₅×#462	1	1	yes	characteristic
225	C₂	C₃×C₅×#466	1	1	yes	characteristic
226	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×#1674	C₃	direct (N × H)
18. C₃×#1674	C₅	direct (N × H)
53. #1674	C₃×C₅	direct (N × H)
67. C₃×C₅×C₈	Q₁₆	semidirect
132. C₅×C₈	C₃×Q₁₆	semidirect
160. C₃×C₈	C₅×Q₁₆	semidirect
183. C₃×C₅	#1674	direct (N × H)
209. C₈	C₃×C₅×Q₁₆	semidirect
213. C₅	C₃×#1674	direct (N × H)
223. C₃	C₅×#1674	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×C₈×Q₈	2
C₃×C₅×#335	2
C₃×C₅×#332	2
C₅×#1674	3
C₃×#1674	5

Subgroups — C3×C5×#1674 #14274708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1674 #14274708