Subgroups — C₃×C₅×#1998 #14275032

Order 1920.

456 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×#125	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#343	C₂	1	4	yes	maximal
5	C₅×#1998	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₃×C₅×#125	—	2	1	no	—
9	C₃×C₅×#125	—	2	1	no	—
10	C₃×C₅×#125	—	2	2	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₄	1	4	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	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×#97	—	2	1	no	—
16	C₃×C₄×C₅×C₈	—	2	1	no	—
17	C₃×#1998	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#125	C₂×C₃	1	1	yes	characteristic
19	C₅×#423	C₂×C₃	1	1	yes	characteristic
20	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic
21	C₅×#343	C₂×C₃	1	4	yes	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	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	4	no	—
27	C₃×C₅×M₁₆	Q₈	1	4	yes	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₃×C₅×C₄⋊C₄	—	2	4	no	—
31	C₃×C₅×C₄⋊C₄	—	2	4	no	—
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	D₄	1	2	yes	—
37	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic
38	C₃×#423	C₂×C₅	1	1	yes	characteristic
39	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
40	C₃×#343	C₂×C₅	1	4	yes	—
41	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
42	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
43	C₅×#125	—	2	1	no	—
44	C₅×#125	—	2	1	no	—
45	C₅×#125	—	2	2	no	—
46	C₂×C₅×C₄⋊C₄	C₃×C₄	1	4	yes	—
47	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
48	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
49	C₅×C₄⋊C₈	—	2	2	no	—
50	C₅×#97	—	2	1	no	—
51	C₄×C₅×C₈	—	2	1	no	—
52	#1998	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
53	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	—	2	4	no	—
55	C₂×C₃×C₄×C₅	—	4	2	no	—
56	C₂×C₃×C₄×C₅	—	4	4	no	—
57	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
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	C₂×Q₈	1	2	yes	—
62	C₂×C₃×C₄×C₅	#41	1	2	yes	—
63	C₃×C₅×C₈	—	2	4	no	—
64	C₃×C₅×C₈	—	4	2	no	—
65	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
66	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
67	C₃×#125	—	2	1	no	—
68	C₃×#125	—	2	1	no	—
69	C₃×#125	—	2	2	no	—
70	C₂×C₃×C₄⋊C₄	C₄×C₅	1	4	yes	—
71	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
72	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
73	C₃×C₄⋊C₈	—	2	2	no	—
74	C₃×#97	—	2	1	no	—
75	C₃×C₄×C₈	—	2	1	no	—
76	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
77	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
78	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
79	C₅×M₁₆	—	2	2	no	—
80	C₅×M₁₆	—	2	4	no	—
81	C₅×M₁₆	C₃×Q₈	1	4	yes	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₅×C₄⋊C₄	—	2	4	no	—
85	C₅×C₄⋊C₄	—	2	4	no	—
86	C₅×C₄⋊C₄	—	2	2	no	—
87	C₅×C₄⋊C₄	—	2	2	no	—
88	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
89	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
90	C₄²×C₅	C₃×D₄	1	2	yes	—
91	C₂×#125	C₂×C₃×C₅	1	1	yes	characteristic
92	#423	C₂×C₃×C₅	1	1	yes	characteristic
93	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic
94	#343	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	C₄×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	C₂²⋊Q₈	1	2	yes	—
102	C₃×C₄×C₅	#124	1	1	yes	characteristic
103	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
105	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
106	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
107	C₃×M₁₆	—	2	2	no	—
108	C₃×M₁₆	—	2	4	no	—
109	C₃×M₁₆	C₅×Q₈	1	4	yes	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₃×C₄⋊C₄	—	2	4	no	—
113	C₃×C₄⋊C₄	—	2	4	no	—
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₄⋊C₄	—	2	2	no	—
116	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
117	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
118	C₃×C₄²	C₅×D₄	1	2	yes	—
119	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	—	2	4	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	—	4	4	no	—
123	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
124	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
125	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
128	C₂×C₄×C₅	C₃×#41	1	2	yes	—
129	C₅×C₈	—	2	4	no	—
130	C₅×C₈	—	4	2	no	—
131	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
132	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
133	#125	—	2	1	no	—
134	#125	—	2	1	no	—
135	#125	—	2	2	no	—
136	C₂×C₄⋊C₄	C₃×C₄×C₅	1	4	yes	—
137	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
138	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
139	C₄⋊C₈	—	2	2	no	—
140	#97	—	2	1	no	—
141	C₄×C₈	—	2	1	no	—
142	C₂×C₃×C₅	—	2	2	no	—
143	C₂×C₃×C₅	#391	1	1	yes	characteristic
144	C₂×C₃×C₅	#413	1	2	yes	—
145	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	4	2	no	—
148	C₂×C₃×C₄	—	4	4	no	—
149	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
150	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
151	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
154	C₂×C₃×C₄	C₅×#41	1	2	yes	—
155	C₃×C₈	—	2	4	no	—
156	C₃×C₈	—	4	2	no	—
157	C₂²×C₅	—	2	2	no	—
158	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₅	C₃×C₄×Q₈	1	2	yes	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	4	4	no	—
163	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
164	C₄×C₅	C₃×#124	1	1	yes	characteristic
165	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
167	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
168	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
169	M₁₆	—	2	2	no	—
170	M₁₆	—	2	4	no	—
171	M₁₆	C₃×C₅×Q₈	1	4	yes	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
179	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
180	C₄²	C₃×C₅×D₄	1	2	yes	—
181	C₃×C₅	#1998	1	1	yes	characteristic, direct product factor, Hall
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
184	C₂²×C₃	C₄×C₅×Q₈	1	2	yes	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	2	2	no	—
187	C₃×C₄	—	4	4	no	—
188	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
189	C₃×C₄	C₅×#124	1	1	yes	characteristic
190	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
191	C₂×C₅	—	2	2	no	—
192	C₂×C₅	C₃×#391	1	1	yes	characteristic
193	C₂×C₅	C₃×#413	1	2	yes	—
194	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
199	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
200	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
201	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
202	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
203	C₂×C₄	C₃×C₅×#41	1	2	yes	—
204	C₈	—	2	4	no	—
205	C₈	—	4	2	no	—
206	C₂×C₃	—	2	2	no	—
207	C₂×C₃	C₅×#391	1	1	yes	characteristic
208	C₂×C₃	C₅×#413	1	2	yes	—
209	C₅	C₃×#1998	1	1	yes	characteristic, direct product factor, Sylow
210	C₂²	—	2	2	no	—
211	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
212	C₂²	C₃×C₄×C₅×Q₈	1	2	yes	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	4	no	—
216	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
217	C₄	C₃×C₅×#124	1	1	yes	characteristic
218	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
219	C₃	C₅×#1998	1	1	yes	characteristic, direct product factor, Sylow
220	C₂	—	2	2	no	—
221	C₂	C₃×C₅×#391	1	1	yes	characteristic
222	C₂	C₃×C₅×#413	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₅×#1998	C₃	direct (N × H)
17. C₃×#1998	C₅	direct (N × H)
52. #1998	C₃×C₅	direct (N × H)
181. C₃×C₅	#1998	direct (N × H)
209. C₅	C₃×#1998	direct (N × H)
219. C₃	C₅×#1998	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×#125	2
C₃×C₅×#423	2
C₃×C₄×C₅×M₁₆	2
C₃×C₅×#343	2
C₅×#1998	3
C₃×#1998	5

Subgroups — C3×C5×#1998 #14275032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1998 #14275032