Subgroups — C₃×C₅×#2155 #14275189

Order 1920.

288 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#108	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#365	C₂	1	4	yes	maximal, complemented
5	C₅×#2155	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×M₁₆	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₃×C₅×#117	—	2	1	no	—
9	C₃×C₅×M₃₂	C₄	1	2	yes	complemented
10	C₃×C₅×M₃₂	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₁₆	C₄	1	2	yes	complemented
12	C₃×C₅×#108	—	2	1	no	—
13	C₃×C₅×#108	C₂²	1	2	yes	—
14	C₃×C₅×#107	—	2	1	no	—
15	C₃×C₅×#106	C₂²	1	2	yes	—
16	C₃×#2155	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic
18	C₂×C₅×#108	C₂×C₃	1	1	yes	characteristic
19	C₅×#427	C₂×C₃	1	1	yes	characteristic
20	C₅×#365	C₂×C₃	1	4	yes	complemented
21	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
22	C₃×C₅×M₁₆	—	2	2	no	—
23	C₃×C₅×M₁₆	—	4	1	no	—
24	C₂×C₃×C₅×C₈	—	4	1	no	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
33	C₃×C₄²×C₅	—	4	1	no	—
34	C₃×C₅×C₁₆	C₂×C₄	1	4	yes	—
35	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic
36	C₂×C₃×#108	C₂×C₅	1	1	yes	characteristic
37	C₃×#427	C₂×C₅	1	1	yes	characteristic
38	C₃×#365	C₂×C₅	1	4	yes	complemented
39	C₂×C₅×M₁₆	—	2	1	no	—
40	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
41	C₅×#117	—	2	1	no	—
42	C₅×M₃₂	C₃×C₄	1	2	yes	complemented
43	C₅×M₃₂	C₂²×C₃	1	2	yes	—
44	C₂×C₅×C₁₆	C₃×C₄	1	2	yes	complemented
45	C₅×#108	—	2	1	no	—
46	C₅×#108	C₂²×C₃	1	2	yes	—
47	C₅×#107	—	2	1	no	—
48	C₅×#106	C₂²×C₃	1	2	yes	—
49	#2155	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	SD₁₆	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	4	2	no	—
52	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
58	C₃×C₅×C₈	—	4	2	no	—
59	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
60	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
61	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃×M₁₆	—	2	1	no	—
64	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
65	C₃×#117	—	2	1	no	—
66	C₃×M₃₂	C₄×C₅	1	2	yes	complemented
67	C₃×M₃₂	C₂²×C₅	1	2	yes	—
68	C₂×C₃×C₁₆	C₄×C₅	1	2	yes	complemented
69	C₃×#108	—	2	1	no	—
70	C₃×#108	C₂²×C₅	1	2	yes	—
71	C₃×#107	—	2	1	no	—
72	C₃×#106	C₂²×C₅	1	2	yes	—
73	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
74	C₅×M₁₆	—	2	2	no	—
75	C₅×M₁₆	—	4	1	no	—
76	C₂×C₅×C₈	—	4	1	no	—
77	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₃×Q₈	1	1	yes	characteristic
81	C₂×C₅×C₈	C₃×Q₈	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
83	C₅×C₄⋊C₄	—	2	2	no	—
84	C₅×C₂²⋊C₄	—	4	1	no	—
85	C₄²×C₅	—	4	1	no	—
86	C₅×C₁₆	C₂×C₃×C₄	1	4	yes	—
87	C₂×M₃₂	C₂×C₃×C₅	1	1	yes	characteristic
88	C₂×#108	C₂×C₃×C₅	1	1	yes	characteristic
89	#427	C₂×C₃×C₅	1	1	yes	characteristic
90	#365	C₂×C₃×C₅	1	4	yes	complemented
91	C₂²×C₃×C₅	—	4	1	no	—
92	C₂²×C₃×C₅	#106	1	1	yes	characteristic
93	C₂²×C₃×C₅	#106	1	1	yes	characteristic
94	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
95	C₃×C₄×C₅	—	8	2	no	—
96	C₃×C₄×C₅	#106	1	1	yes	characteristic
97	C₃×C₄×C₅	#106	1	1	yes	characteristic
98	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
101	C₃×M₁₆	—	2	2	no	—
102	C₃×M₁₆	—	4	1	no	—
103	C₂×C₃×C₈	—	4	1	no	—
104	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
107	C₂×C₃×C₈	C₅×Q₈	1	1	yes	characteristic
108	C₂×C₃×C₈	C₅×Q₈	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
110	C₃×C₄⋊C₄	—	2	2	no	—
111	C₃×C₂²⋊C₄	—	4	1	no	—
112	C₃×C₄²	—	4	1	no	—
113	C₃×C₁₆	C₂×C₄×C₅	1	4	yes	—
114	C₂³×C₅	C₃×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅	C₃×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅	C₃×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅	C₃×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
122	C₅×C₈	—	4	2	no	—
123	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
124	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
125	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
126	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
127	C₂×M₁₆	—	2	1	no	—
128	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
129	#117	—	2	1	no	—
130	M₃₂	C₃×C₄×C₅	1	2	yes	complemented
131	M₃₂	C₂²×C₃×C₅	1	2	yes	—
132	C₂×C₁₆	C₃×C₄×C₅	1	2	yes	complemented
133	#108	—	2	1	no	—
134	#108	C₂²×C₃×C₅	1	2	yes	—
135	#107	—	2	1	no	—
136	#106	C₂²×C₃×C₅	1	2	yes	—
137	C₂×C₃×C₅	—	2	1	no	—
138	C₂×C₃×C₅	—	2	1	no	—
139	C₂×C₃×C₅	—	2	1	no	—
140	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
141	C₂³×C₃	C₅×SD₁₆	1	1	yes	characteristic
142	C₂×C₃×C₄	—	4	2	no	—
143	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₃×C₄	C₅×SD₁₆	1	1	yes	characteristic
146	C₂×C₃×C₄	C₅×SD₁₆	1	1	yes	characteristic
147	C₂×C₃×C₄	C₅×SD₁₆	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
149	C₃×C₈	—	4	2	no	—
150	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
151	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
152	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
153	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
154	C₂²×C₅	—	4	1	no	—
155	C₂²×C₅	C₃×#106	1	1	yes	characteristic
156	C₂²×C₅	C₃×#106	1	1	yes	characteristic
157	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
158	C₄×C₅	—	8	2	no	—
159	C₄×C₅	C₃×#106	1	1	yes	characteristic
160	C₄×C₅	C₃×#106	1	1	yes	characteristic
161	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
163	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
164	M₁₆	—	2	2	no	—
165	M₁₆	—	4	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
168	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
169	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
170	C₂×C₈	C₃×C₅×Q₈	1	1	yes	characteristic
171	C₂×C₈	C₃×C₅×Q₈	1	1	yes	characteristic
172	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	—	4	1	no	—
175	C₄²	—	4	1	no	—
176	C₁₆	C₂×C₃×C₄×C₅	1	4	yes	—
177	C₃×C₅	#2155	1	1	yes	characteristic, direct product factor, Hall
178	C₂²×C₃	—	4	1	no	—
179	C₂²×C₃	C₅×#106	1	1	yes	characteristic
180	C₂²×C₃	C₅×#106	1	1	yes	characteristic
181	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
182	C₃×C₄	—	8	2	no	—
183	C₃×C₄	C₅×#106	1	1	yes	characteristic
184	C₃×C₄	C₅×#106	1	1	yes	characteristic
185	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
186	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
187	C₂×C₅	—	2	1	no	—
188	C₂×C₅	—	2	1	no	—
189	C₂×C₅	—	2	1	no	—
190	C₂×C₅	C₂×C₃×#106	1	1	yes	characteristic
191	C₂³	C₃×C₅×SD₁₆	1	1	yes	characteristic
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₄	C₃×C₅×SD₁₆	1	1	yes	characteristic
196	C₂×C₄	C₃×C₅×SD₁₆	1	1	yes	characteristic
197	C₂×C₄	C₃×C₅×SD₁₆	1	1	yes	characteristic
198	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
199	C₈	—	4	2	no	—
200	C₈	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
201	C₈	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
202	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
203	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	—	2	1	no	—
206	C₂×C₃	—	2	1	no	—
207	C₂×C₃	C₂×C₅×#106	1	1	yes	characteristic
208	C₅	C₃×#2155	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²	—	4	1	no	—
210	C₂²	C₃×C₅×#106	1	1	yes	characteristic
211	C₂²	C₃×C₅×#106	1	1	yes	characteristic
212	C₂²	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
213	C₄	—	8	2	no	—
214	C₄	C₃×C₅×#106	1	1	yes	characteristic
215	C₄	C₃×C₅×#106	1	1	yes	characteristic
216	C₄	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
217	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
218	C₃	C₅×#2155	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	C₂×C₃×C₅×#106	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	#106
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂×C₃×C₅	C₂×#106
prime radical	C₂³×C₃×C₅	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#365	C₂	semidirect
5. C₅×#2155	C₃	direct (N × H)
9. C₃×C₅×M₃₂	C₄	semidirect
11. C₂×C₃×C₅×C₁₆	C₄	semidirect
16. C₃×#2155	C₅	direct (N × H)
20. C₅×#365	C₂×C₃	semidirect
38. C₃×#365	C₂×C₅	semidirect
42. C₅×M₃₂	C₃×C₄	semidirect
44. C₂×C₅×C₁₆	C₃×C₄	semidirect
49. #2155	C₃×C₅	direct (N × H)
66. C₃×M₃₂	C₄×C₅	semidirect
68. C₂×C₃×C₁₆	C₄×C₅	semidirect
90. #365	C₂×C₃×C₅	semidirect
130. M₃₂	C₃×C₄×C₅	semidirect
132. C₂×C₁₆	C₃×C₄×C₅	semidirect
177. C₃×C₅	#2155	direct (N × H)
208. C₅	C₃×#2155	direct (N × H)
218. C₃	C₅×#2155	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×M₃₂	2
C₂×C₃×C₅×#108	2
C₃×C₅×#427	2
C₃×C₅×#365	2
C₅×#2155	3
C₃×#2155	5

Subgroups — C3×C5×#2155 #14275189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2155 #14275189