Subgroups — C₃×C₅×#1953 #14274987

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#430	C₂	1	2	yes	maximal
2	C₂×C₃×C₅×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×#108	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#344	C₂	1	2	yes	maximal
6	C₅×#1953	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×#117	C₂²	1	1	yes	characteristic
12	C₃×C₅×#108	—	2	4	no	—
13	C₃×C₅×#108	—	2	1	no	—
14	C₃×C₅×#108	—	2	1	no	—
15	C₃×C₅×#108	C₄	1	4	yes	complemented
16	C₃×C₅×#97	—	2	1	no	—
17	C₃×C₄×C₅×C₈	—	2	1	no	—
18	C₃×#1953	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#430	C₂×C₃	1	2	yes	—
20	C₂×C₅×#108	C₂×C₃	1	1	yes	characteristic
21	C₂×C₅×#108	C₂×C₃	1	1	yes	characteristic
22	C₅×#405	C₂×C₃	1	1	yes	characteristic
23	C₅×#344	C₂×C₃	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₃×C₅×M₁₆	—	2	2	no	—
26	C₃×C₅×M₁₆	—	2	1	no	—
27	C₃×C₅×M₁₆	—	2	2	no	—
28	C₃×C₅×M₁₆	—	2	2	no	—
29	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
30	C₃×C₅×M₁₆	D₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	—	2	1	no	—
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₂²⋊C₄	Q₈	1	2	yes	—
40	C₃×C₄²×C₅	—	2	1	no	—
41	C₃×#430	C₂×C₅	1	2	yes	—
42	C₂×C₃×#108	C₂×C₅	1	1	yes	characteristic
43	C₂×C₃×#108	C₂×C₅	1	1	yes	characteristic
44	C₃×#405	C₂×C₅	1	1	yes	characteristic
45	C₃×#344	C₂×C₅	1	2	yes	—
46	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
47	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
48	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
49	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₅×#117	C₂²×C₃	1	1	yes	characteristic
51	C₅×#108	—	2	4	no	—
52	C₅×#108	—	2	1	no	—
53	C₅×#108	—	2	1	no	—
54	C₅×#108	C₃×C₄	1	4	yes	complemented
55	C₅×#97	—	2	1	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	#1953	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
58	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
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	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
66	C₃×C₅×C₈	—	2	4	no	—
67	C₃×C₅×C₈	—	2	2	no	—
68	C₃×C₅×C₈	—	4	2	no	—
69	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
70	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
71	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
72	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
73	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
74	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
75	C₃×#117	C₂²×C₅	1	1	yes	characteristic
76	C₃×#108	—	2	4	no	—
77	C₃×#108	—	2	1	no	—
78	C₃×#108	—	2	1	no	—
79	C₃×#108	C₄×C₅	1	4	yes	complemented
80	C₃×#97	—	2	1	no	—
81	C₃×C₄×C₈	—	2	1	no	—
82	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
83	C₅×M₁₆	—	2	2	no	—
84	C₅×M₁₆	—	2	1	no	—
85	C₅×M₁₆	—	2	2	no	—
86	C₅×M₁₆	—	2	2	no	—
87	C₅×M₁₆	C₂×C₃×C₄	1	4	yes	—
88	C₅×M₁₆	C₃×D₄	1	2	yes	—
89	C₂×C₅×C₈	—	2	1	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
93	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
96	C₅×C₄⋊C₄	—	2	1	no	—
97	C₅×C₂²⋊C₄	C₃×Q₈	1	2	yes	—
98	C₄²×C₅	—	2	1	no	—
99	#430	C₂×C₃×C₅	1	2	yes	—
100	C₂×#108	C₂×C₃×C₅	1	1	yes	characteristic
101	C₂×#108	C₂×C₃×C₅	1	1	yes	characteristic
102	#405	C₂×C₃×C₅	1	1	yes	characteristic
103	#344	C₂×C₃×C₅	1	2	yes	—
104	C₂²×C₃×C₅	—	4	1	no	—
105	C₂²×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
106	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
108	C₃×C₄×C₅	—	4	2	no	—
109	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
111	C₃×C₄×C₅	#121	1	1	yes	characteristic
112	C₃×C₄×C₅	#125	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
114	C₃×M₁₆	—	2	2	no	—
115	C₃×M₁₆	—	2	1	no	—
116	C₃×M₁₆	—	2	2	no	—
117	C₃×M₁₆	—	2	2	no	—
118	C₃×M₁₆	C₂×C₄×C₅	1	4	yes	—
119	C₃×M₁₆	C₅×D₄	1	2	yes	—
120	C₂×C₃×C₈	—	2	1	no	—
121	C₂×C₃×C₈	—	2	2	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
124	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
125	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
126	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
127	C₃×C₄⋊C₄	—	2	1	no	—
128	C₃×C₂²⋊C₄	C₅×Q₈	1	2	yes	—
129	C₃×C₄²	—	2	1	no	—
130	C₂³×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
137	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
138	C₅×C₈	—	2	4	no	—
139	C₅×C₈	—	2	2	no	—
140	C₅×C₈	—	4	2	no	—
141	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
142	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
143	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
144	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
145	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
146	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
147	#117	C₂²×C₃×C₅	1	1	yes	characteristic
148	#108	—	2	4	no	—
149	#108	—	2	1	no	—
150	#108	—	2	1	no	—
151	#108	C₃×C₄×C₅	1	4	yes	complemented
152	#97	—	2	1	no	—
153	C₄×C₈	—	2	1	no	—
154	C₂×C₃×C₅	—	2	1	no	—
155	C₂×C₃×C₅	—	2	1	no	—
156	C₂×C₃×C₅	—	2	1	no	—
157	C₂×C₃×C₅	#389	1	1	yes	characteristic
158	C₂³×C₃	C₂×C₅×Q₈	1	1	yes	characteristic
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
164	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
165	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
166	C₃×C₈	—	2	4	no	—
167	C₃×C₈	—	2	2	no	—
168	C₃×C₈	—	4	2	no	—
169	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
170	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
171	C₂²×C₅	—	4	1	no	—
172	C₂²×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
173	C₂²×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
174	C₂²×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
175	C₄×C₅	—	4	2	no	—
176	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
177	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
178	C₄×C₅	C₃×#121	1	1	yes	characteristic
179	C₄×C₅	C₃×#125	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
181	M₁₆	—	2	2	no	—
182	M₁₆	—	2	1	no	—
183	M₁₆	—	2	2	no	—
184	M₁₆	—	2	2	no	—
185	M₁₆	C₂×C₃×C₄×C₅	1	4	yes	—
186	M₁₆	C₃×C₅×D₄	1	2	yes	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
191	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
192	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
193	C₂×C₈	C₃×C₅×Q₈	1	2	yes	—
194	C₄⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	C₃×C₅×Q₈	1	2	yes	—
196	C₄²	—	2	1	no	—
197	C₃×C₅	#1953	1	1	yes	characteristic, direct product factor, Hall
198	C₂²×C₃	—	4	1	no	—
199	C₂²×C₃	C₄×C₅×Q₈	1	1	yes	characteristic
200	C₂²×C₃	C₅×C₂²⋊Q₈	1	1	yes	characteristic
201	C₂²×C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic
202	C₃×C₄	—	4	2	no	—
203	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
204	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
205	C₃×C₄	C₅×#121	1	1	yes	characteristic
206	C₃×C₄	C₅×#125	1	1	yes	characteristic
207	C₂×C₅	—	2	1	no	—
208	C₂×C₅	—	2	1	no	—
209	C₂×C₅	—	2	1	no	—
210	C₂×C₅	C₃×#389	1	1	yes	characteristic
211	C₂³	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
214	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
215	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
216	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
217	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
218	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
219	C₈	—	2	4	no	—
220	C₈	—	2	2	no	—
221	C₈	—	4	2	no	—
222	C₈	C₃×C₅×C₄⋊C₄	1	2	yes	—
223	C₈	C₃×C₅×C₄⋊C₄	1	2	yes	—
224	C₂×C₃	—	2	1	no	—
225	C₂×C₃	—	2	1	no	—
226	C₂×C₃	—	2	1	no	—
227	C₂×C₃	C₅×#389	1	1	yes	characteristic
228	C₅	C₃×#1953	1	1	yes	characteristic, direct product factor, Sylow
229	C₂²	—	4	1	no	—
230	C₂²	C₃×C₄×C₅×Q₈	1	1	yes	characteristic
231	C₂²	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
232	C₂²	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
233	C₄	—	4	2	no	—
234	C₄	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
235	C₄	C₃×C₄×C₅×D₄	1	1	yes	characteristic
236	C₄	C₃×C₅×#121	1	1	yes	characteristic
237	C₄	C₃×C₅×#125	1	1	yes	characteristic
238	C₃	C₅×#1953	1	1	yes	characteristic, direct product factor, Sylow
239	C₂	—	2	1	no	—
240	C₂	—	2	1	no	—
241	C₂	—	2	1	no	—
242	C₂	C₃×C₅×#389	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	#121
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle	C₂×C₃×C₅	#389
prime radical	C₂³×C₃×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#1953	C₃	direct (N × H)
15. C₃×C₅×#108	C₄	semidirect
18. C₃×#1953	C₅	direct (N × H)
54. C₅×#108	C₃×C₄	semidirect
57. #1953	C₃×C₅	direct (N × H)
79. C₃×#108	C₄×C₅	semidirect
151. #108	C₃×C₄×C₅	semidirect
197. C₃×C₅	#1953	direct (N × H)
228. C₅	C₃×#1953	direct (N × H)
238. C₃	C₅×#1953	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×#430	2
C₂×C₃×C₅×#108	2
C₃×C₅×#405	2
C₃×C₅×#344	2
C₅×#1953	3
C₃×#1953	5

Subgroups — C3×C5×#1953 #14274987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1953 #14274987