Subgroups — C₃×C₅×#1995 #14275029

Order 1920.

488 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#339	C₂	1	4	yes	maximal
5	C₅×#1995	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
7	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	complemented
8	C₃×C₅×#125	C₄	1	2	yes	complemented
9	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
10	C₃×C₄×C₅×Q₈	—	2	2	no	—
11	C₃×C₅×#117	C₂²	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	—	2	2	no	—
15	C₃×C₅×#95	—	2	2	no	—
16	C₃×#1995	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#533	C₂×C₃	1	1	yes	characteristic
18	C₅×#423	C₂×C₃	1	1	yes	characteristic
19	C₅×#376	C₂×C₃	1	1	yes	characteristic
20	C₅×#339	C₂×C₃	1	4	yes	—
21	C₂×C₃×C₅×Q₈	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₃×C₅×M₁₆	—	2	4	no	—
26	C₂×C₃×C₅×C₈	—	2	2	no	—
27	C₃×C₅×C₄⋊C₄	—	2	4	no	—
28	C₃×C₅×C₄⋊C₄	—	2	2	no	—
29	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
31	C₃×C₄²×C₅	—	2	2	no	—
32	C₃×C₄²×C₅	—	2	2	no	—
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄²×C₅	D₄	1	2	yes	—
36	C₃×C₄²×C₅	Q₈	1	4	yes	complemented
37	C₃×#533	C₂×C₅	1	1	yes	characteristic
38	C₃×#423	C₂×C₅	1	1	yes	characteristic
39	C₃×#376	C₂×C₅	1	1	yes	characteristic
40	C₃×#339	C₂×C₅	1	4	yes	—
41	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
42	C₅×C₄⋊Q₈	C₃×C₄	1	2	yes	complemented
43	C₅×#125	C₃×C₄	1	2	yes	complemented
44	C₅×C₂²⋊Q₈	—	2	2	no	—
45	C₄×C₅×Q₈	—	2	2	no	—
46	C₅×#117	C₂²×C₃	1	2	yes	—
47	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
48	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
49	C₅×C₄⋊C₈	—	2	2	no	—
50	C₅×#95	—	2	2	no	—
51	#1995	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
52	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
53	C₃×C₅×Q₈	—	4	4	no	—
54	C₂×C₃×C₄×C₅	—	2	4	no	—
55	C₂×C₃×C₄×C₅	—	2	4	no	—
56	C₂×C₃×C₄×C₅	—	4	2	no	—
57	C₂×C₃×C₄×C₅	—	4	4	no	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
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×Q₈	1	2	yes	—
63	C₂×C₃×C₄×C₅	#41	1	2	yes	—
64	C₃×C₅×C₈	—	4	2	no	—
65	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
66	C₃×C₄⋊Q₈	C₄×C₅	1	2	yes	complemented
67	C₃×#125	C₄×C₅	1	2	yes	complemented
68	C₃×C₂²⋊Q₈	—	2	2	no	—
69	C₃×C₄×Q₈	—	2	2	no	—
70	C₃×#117	C₂²×C₅	1	2	yes	—
71	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
72	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
73	C₃×C₄⋊C₈	—	2	2	no	—
74	C₃×#95	—	2	2	no	—
75	C₂×C₅×Q₈	—	2	2	no	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
78	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
79	C₅×M₁₆	—	2	4	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₅×C₄⋊C₄	—	2	4	no	—
82	C₅×C₄⋊C₄	—	2	2	no	—
83	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
84	C₅×C₂²⋊C₄	—	2	2	no	—
85	C₄²×C₅	—	2	2	no	—
86	C₄²×C₅	—	2	2	no	—
87	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
88	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
89	C₄²×C₅	C₃×D₄	1	2	yes	—
90	C₄²×C₅	C₃×Q₈	1	4	yes	complemented
91	#533	C₂×C₃×C₅	1	1	yes	characteristic
92	#423	C₂×C₃×C₅	1	1	yes	characteristic
93	#376	C₂×C₃×C₅	1	1	yes	characteristic
94	#339	C₂×C₃×C₅	1	4	yes	—
95	C₂²×C₃×C₅	—	2	2	no	—
96	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
98	C₃×C₄×C₅	—	2	2	no	—
99	C₃×C₄×C₅	—	2	2	no	—
100	C₃×C₄×C₅	—	4	4	no	—
101	C₃×C₄×C₅	—	4	4	no	—
102	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
103	C₃×C₄×C₅	#124	1	1	yes	characteristic
104	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₃×Q₈	—	2	2	no	—
106	C₂²×C₃×C₄	—	2	2	no	—
107	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
108	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
109	C₃×M₁₆	—	2	4	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₃×C₄⋊C₄	—	2	4	no	—
112	C₃×C₄⋊C₄	—	2	2	no	—
113	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
114	C₃×C₂²⋊C₄	—	2	2	no	—
115	C₃×C₄²	—	2	2	no	—
116	C₃×C₄²	—	2	2	no	—
117	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
118	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
119	C₃×C₄²	C₅×D₄	1	2	yes	—
120	C₃×C₄²	C₅×Q₈	1	4	yes	complemented
121	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
122	C₅×Q₈	—	4	4	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
132	C₂×C₄×C₅	C₃×#41	1	2	yes	—
133	C₅×C₈	—	4	2	no	—
134	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
135	C₄⋊Q₈	C₃×C₄×C₅	1	2	yes	complemented
136	#125	C₃×C₄×C₅	1	2	yes	complemented
137	C₂²⋊Q₈	—	2	2	no	—
138	C₄×Q₈	—	2	2	no	—
139	#117	C₂²×C₃×C₅	1	2	yes	—
140	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
141	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
142	C₄⋊C₈	—	2	2	no	—
143	#95	—	2	2	no	—
144	C₂×C₃×C₅	—	2	2	no	—
145	C₂×C₃×C₅	#391	1	1	yes	characteristic
146	C₂×C₃×C₅	#421	1	2	yes	—
147	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
148	C₃×Q₈	—	4	4	no	—
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	—	4	4	no	—
153	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
154	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
155	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
158	C₂×C₃×C₄	C₅×#41	1	2	yes	—
159	C₃×C₈	—	4	2	no	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	—	4	4	no	—
166	C₄×C₅	—	4	4	no	—
167	C₄×C₅	C₃×C₄×Q₈	1	2	yes	—
168	C₄×C₅	C₃×#124	1	1	yes	characteristic
169	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
170	C₂×Q₈	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
173	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
174	M₁₆	—	2	4	no	—
175	C₂×C₈	—	2	2	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
183	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
184	C₄²	C₃×C₅×D₄	1	2	yes	—
185	C₄²	C₃×C₅×Q₈	1	4	yes	complemented
186	C₃×C₅	#1995	1	1	yes	characteristic, direct product factor, Hall
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	4	4	no	—
193	C₃×C₄	—	4	4	no	—
194	C₃×C₄	C₄×C₅×Q₈	1	2	yes	—
195	C₃×C₄	C₅×#124	1	1	yes	characteristic
196	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	C₃×#391	1	1	yes	characteristic
199	C₂×C₅	C₃×#421	1	2	yes	—
200	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
201	Q₈	—	4	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
207	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
208	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
209	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
210	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
211	C₂×C₄	C₃×C₅×#41	1	2	yes	—
212	C₈	—	4	2	no	—
213	C₂×C₃	—	2	2	no	—
214	C₂×C₃	C₅×#391	1	1	yes	characteristic
215	C₂×C₃	C₅×#421	1	2	yes	—
216	C₅	C₃×#1995	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	—	2	2	no	—
218	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
219	C₂²	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	4	no	—
223	C₄	—	4	4	no	—
224	C₄	C₃×C₄×C₅×Q₈	1	2	yes	—
225	C₄	C₃×C₅×#124	1	1	yes	characteristic
226	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
227	C₃	C₅×#1995	1	1	yes	characteristic, direct product factor, Sylow
228	C₂	—	2	2	no	—
229	C₂	C₃×C₅×#391	1	1	yes	characteristic
230	C₂	C₃×C₅×#421	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅×#1995	C₃	direct (N × H)
7. C₃×C₅×C₄⋊Q₈	C₄	semidirect
8. C₃×C₅×#125	C₄	semidirect
16. C₃×#1995	C₅	direct (N × H)
36. C₃×C₄²×C₅	Q₈	semidirect
42. C₅×C₄⋊Q₈	C₃×C₄	semidirect
43. C₅×#125	C₃×C₄	semidirect
51. #1995	C₃×C₅	direct (N × H)
66. C₃×C₄⋊Q₈	C₄×C₅	semidirect
67. C₃×#125	C₄×C₅	semidirect
90. C₄²×C₅	C₃×Q₈	semidirect
120. C₃×C₄²	C₅×Q₈	semidirect
135. C₄⋊Q₈	C₃×C₄×C₅	semidirect
136. #125	C₃×C₄×C₅	semidirect
185. C₄²	C₃×C₅×Q₈	semidirect
186. C₃×C₅	#1995	direct (N × H)
216. C₅	C₃×#1995	direct (N × H)
227. C₃	C₅×#1995	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×#533	2
C₃×C₅×#423	2
C₃×C₅×#376	2
C₃×C₅×#339	2
C₅×#1995	3
C₃×#1995	5

Subgroups — C3×C5×#1995 #14275029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1995 #14275029