Subgroups — C₃×C₅×#1994 #14275028

Order 1920.

568 conjugacy classes of subgroups.

The subgroup lattice has 236 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×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#339	C₂	1	4	yes	maximal
5	C₅×#1994	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	—	2	2	no	—
13	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	—	2	2	no	—
16	C₃×#1994	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	C₄³×C₅	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	8	no	—
32	C₃×C₄²×C₅	—	2	2	no	—
33	C₃×C₄²×C₅	—	2	2	no	—
34	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	D₄	1	2	yes	—
37	C₃×C₄²×C₅	Q₈	1	4	yes	complemented
38	C₃×#533	C₂×C₅	1	1	yes	characteristic
39	C₃×#423	C₂×C₅	1	1	yes	characteristic
40	C₃×C₄³	C₂×C₅	1	1	yes	characteristic
41	C₃×#339	C₂×C₅	1	4	yes	—
42	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
43	C₅×C₄⋊Q₈	C₃×C₄	1	2	yes	complemented
44	C₅×#125	C₃×C₄	1	2	yes	complemented
45	C₅×C₂²⋊Q₈	—	2	2	no	—
46	C₄×C₅×Q₈	—	2	2	no	—
47	C₅×#117	C₂²×C₃	1	2	yes	—
48	C₂×C₄²×C₅	—	2	2	no	—
49	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
50	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
51	C₅×C₄⋊C₈	—	2	2	no	—
52	#1994	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
53	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
54	C₃×C₅×Q₈	—	4	4	no	—
55	C₂×C₃×C₄×C₅	—	2	8	no	—
56	C₂×C₃×C₄×C₅	—	2	4	no	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
64	C₂×C₃×C₄×C₅	#41	1	2	yes	—
65	C₃×C₅×C₈	—	4	2	no	—
66	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
67	C₃×C₄⋊Q₈	C₄×C₅	1	2	yes	complemented
68	C₃×#125	C₄×C₅	1	2	yes	complemented
69	C₃×C₂²⋊Q₈	—	2	2	no	—
70	C₃×C₄×Q₈	—	2	2	no	—
71	C₃×#117	C₂²×C₅	1	2	yes	—
72	C₂×C₃×C₄²	—	2	2	no	—
73	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
74	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
75	C₃×C₄⋊C₈	—	2	2	no	—
76	C₂×C₅×Q₈	—	2	2	no	—
77	C₂²×C₄×C₅	—	2	2	no	—
78	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
79	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
80	C₅×M₁₆	—	2	4	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₅×C₄⋊C₄	—	2	4	no	—
83	C₅×C₄⋊C₄	—	2	2	no	—
84	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
85	C₅×C₂²⋊C₄	—	2	2	no	—
86	C₄²×C₅	—	2	8	no	—
87	C₄²×C₅	—	2	2	no	—
88	C₄²×C₅	—	2	2	no	—
89	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
90	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
91	C₄²×C₅	C₃×D₄	1	2	yes	—
92	C₄²×C₅	C₃×Q₈	1	4	yes	complemented
93	#533	C₂×C₃×C₅	1	1	yes	characteristic
94	#423	C₂×C₃×C₅	1	1	yes	characteristic
95	C₄³	C₂×C₃×C₅	1	1	yes	characteristic
96	#339	C₂×C₃×C₅	1	4	yes	—
97	C₂²×C₃×C₅	—	2	2	no	—
98	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
100	C₃×C₄×C₅	—	2	8	no	—
101	C₃×C₄×C₅	—	2	2	no	—
102	C₃×C₄×C₅	—	4	4	no	—
103	C₃×C₄×C₅	#104	1	4	yes	—
104	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
105	C₃×C₄×C₅	#124	1	1	yes	characteristic
106	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
107	C₂×C₃×Q₈	—	2	2	no	—
108	C₂²×C₃×C₄	—	2	2	no	—
109	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
110	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
111	C₃×M₁₆	—	2	4	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₃×C₄⋊C₄	—	2	4	no	—
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
116	C₃×C₂²⋊C₄	—	2	2	no	—
117	C₃×C₄²	—	2	8	no	—
118	C₃×C₄²	—	2	2	no	—
119	C₃×C₄²	—	2	2	no	—
120	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
121	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
122	C₃×C₄²	C₅×D₄	1	2	yes	—
123	C₃×C₄²	C₅×Q₈	1	4	yes	complemented
124	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
125	C₅×Q₈	—	4	4	no	—
126	C₂×C₄×C₅	—	2	8	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
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×Q₈	1	2	yes	—
135	C₂×C₄×C₅	C₃×#41	1	2	yes	—
136	C₅×C₈	—	4	2	no	—
137	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
138	C₄⋊Q₈	C₃×C₄×C₅	1	2	yes	complemented
139	#125	C₃×C₄×C₅	1	2	yes	complemented
140	C₂²⋊Q₈	—	2	2	no	—
141	C₄×Q₈	—	2	2	no	—
142	#117	C₂²×C₃×C₅	1	2	yes	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
145	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
146	C₄⋊C₈	—	2	2	no	—
147	C₂×C₃×C₅	—	2	2	no	—
148	C₂×C₃×C₅	#391	1	1	yes	characteristic
149	C₂×C₃×C₅	C₂×#104	1	2	yes	—
150	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
151	C₃×Q₈	—	4	4	no	—
152	C₂×C₃×C₄	—	2	8	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
157	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
158	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
161	C₂×C₃×C₄	C₅×#41	1	2	yes	—
162	C₃×C₈	—	4	2	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
166	C₄×C₅	—	2	8	no	—
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	—	4	4	no	—
169	C₄×C₅	C₃×#104	1	4	yes	—
170	C₄×C₅	C₃×C₄×Q₈	1	2	yes	—
171	C₄×C₅	C₃×#124	1	1	yes	characteristic
172	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
173	C₂×Q₈	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
176	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
177	M₁₆	—	2	4	no	—
178	C₂×C₈	—	2	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₄²	—	2	8	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
187	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
188	C₄²	C₃×C₅×D₄	1	2	yes	—
189	C₄²	C₃×C₅×Q₈	1	4	yes	complemented
190	C₃×C₅	#1994	1	1	yes	characteristic, direct product factor, Hall
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
193	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
194	C₃×C₄	—	2	8	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	4	4	no	—
197	C₃×C₄	C₅×#104	1	4	yes	—
198	C₃×C₄	C₄×C₅×Q₈	1	2	yes	—
199	C₃×C₄	C₅×#124	1	1	yes	characteristic
200	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
201	C₂×C₅	—	2	2	no	—
202	C₂×C₅	C₃×#391	1	1	yes	characteristic
203	C₂×C₅	C₂×C₃×#104	1	2	yes	—
204	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
205	Q₈	—	4	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
211	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
212	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
213	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
214	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
215	C₂×C₄	C₃×C₅×#41	1	2	yes	—
216	C₈	—	4	2	no	—
217	C₂×C₃	—	2	2	no	—
218	C₂×C₃	C₅×#391	1	1	yes	characteristic
219	C₂×C₃	C₂×C₅×#104	1	2	yes	—
220	C₅	C₃×#1994	1	1	yes	characteristic, direct product factor, Sylow
221	C₂²	—	2	2	no	—
222	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₂²	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
224	C₄	—	2	8	no	—
225	C₄	—	2	2	no	—
226	C₄	—	4	4	no	—
227	C₄	C₃×C₅×#104	1	4	yes	—
228	C₄	C₃×C₄×C₅×Q₈	1	2	yes	—
229	C₄	C₃×C₅×#124	1	1	yes	characteristic
230	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
231	C₃	C₅×#1994	1	1	yes	characteristic, direct product factor, Sylow
232	C₂	—	2	2	no	—
233	C₂	C₃×C₅×#391	1	1	yes	characteristic
234	C₂	C₂×C₃×C₅×#104	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×#1994	C₃	direct (N × H)
7. C₃×C₅×C₄⋊Q₈	C₄	semidirect
8. C₃×C₅×#125	C₄	semidirect
16. C₃×#1994	C₅	direct (N × H)
37. C₃×C₄²×C₅	Q₈	semidirect
43. C₅×C₄⋊Q₈	C₃×C₄	semidirect
44. C₅×#125	C₃×C₄	semidirect
52. #1994	C₃×C₅	direct (N × H)
67. C₃×C₄⋊Q₈	C₄×C₅	semidirect
68. C₃×#125	C₄×C₅	semidirect
92. C₄²×C₅	C₃×Q₈	semidirect
123. C₃×C₄²	C₅×Q₈	semidirect
138. C₄⋊Q₈	C₃×C₄×C₅	semidirect
139. #125	C₃×C₄×C₅	semidirect
189. C₄²	C₃×C₅×Q₈	semidirect
190. C₃×C₅	#1994	direct (N × H)
220. C₅	C₃×#1994	direct (N × H)
231. C₃	C₅×#1994	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×C₄³×C₅	2
C₃×C₅×#339	2
C₅×#1994	3
C₃×#1994	5

Subgroups — C3×C5×#1994 #14275028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1994 #14275028