Subgroups — C₃×C₅×#2098 #14275132

Order 1920.

420 conjugacy classes of subgroups.

The subgroup lattice has 252 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	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#424	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#421	C₂	1	2	yes	maximal
4	C₃×C₅×#408	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2098	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
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	2	yes	—
12	C₃×C₅×#108	—	2	2	no	—
13	C₃×C₅×C₄⋊C₈	—	2	4	no	—
14	C₃×C₅×#104	—	2	4	no	—
15	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
16	C₃×#2098	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#430	C₂×C₃	1	1	yes	characteristic, complemented
18	C₅×#424	C₂×C₃	1	2	yes	complemented
19	C₅×#421	C₂×C₃	1	2	yes	—
20	C₅×#408	C₂×C₃	1	1	yes	characteristic
21	C₅×H(3,ℤ/4)	C₂×C₃	1	1	yes	characteristic, complemented
22	C₃×C₅×#41	—	2	2	no	—
23	C₃×C₅×#41	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	Q₈	1	1	yes	characteristic
25	C₂×C₃×C₅×D₄	—	2	1	no	—
26	C₂×C₃×C₅×D₄	Q₈	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	—	2	1	no	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₃×C₅×M₁₆	—	2	2	no	—
30	C₃×C₅×M₁₆	—	2	2	no	—
31	C₃×C₅×M₁₆	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	1	no	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
38	C₃×C₄²×C₅	D₄	1	4	yes	—
39	C₃×#430	C₂×C₅	1	1	yes	characteristic, complemented
40	C₃×#424	C₂×C₅	1	2	yes	complemented
41	C₃×#421	C₂×C₅	1	2	yes	—
42	C₃×#408	C₂×C₅	1	1	yes	characteristic
43	C₃×H(3,ℤ/4)	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
46	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
47	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₅×#117	C₂²×C₃	1	2	yes	—
49	C₅×#108	—	2	2	no	—
50	C₅×C₄⋊C₈	—	2	4	no	—
51	C₅×#104	—	2	4	no	—
52	C₅×C₂²⋊C₈	—	2	2	no	—
53	#2098	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₂³×C₃×C₅	—	2	1	no	—
55	C₂³×C₃×C₅	#41	1	1	yes	characteristic
56	C₃×C₅×Q₈	—	2	2	no	—
57	C₃×C₅×D₄	—	2	2	no	—
58	C₃×C₅×D₄	—	4	2	no	—
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	2	4	no	—
61	C₂×C₃×C₄×C₅	—	4	1	no	—
62	C₂×C₃×C₄×C₅	—	4	1	no	—
63	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	#41	1	2	yes	—
67	C₃×C₅×C₈	—	2	2	no	—
68	C₃×C₅×C₈	—	4	1	no	—
69	C₃×C₅×C₈	—	4	2	no	—
70	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
72	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
73	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
74	C₃×#117	C₂²×C₅	1	2	yes	—
75	C₃×#108	—	2	2	no	—
76	C₃×C₄⋊C₈	—	2	4	no	—
77	C₃×#104	—	2	4	no	—
78	C₃×C₂²⋊C₈	—	2	2	no	—
79	C₅×#41	—	2	2	no	—
80	C₅×#41	—	2	2	no	—
81	C₂×C₅×Q₈	C₃×Q₈	1	1	yes	characteristic
82	C₂×C₅×D₄	—	2	1	no	—
83	C₂×C₅×D₄	C₃×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₅	—	2	1	no	—
85	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
86	C₅×M₁₆	—	2	2	no	—
87	C₅×M₁₆	—	2	2	no	—
88	C₅×M₁₆	—	2	2	no	—
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	C₃×D₄	1	2	yes	—
93	C₅×C₄⋊C₄	—	2	2	no	—
94	C₅×C₂²⋊C₄	—	2	2	no	—
95	C₄²×C₅	C₃×D₄	1	4	yes	—
96	#430	C₂×C₃×C₅	1	1	yes	characteristic, complemented
97	#424	C₂×C₃×C₅	1	2	yes	complemented
98	#421	C₂×C₃×C₅	1	2	yes	—
99	#408	C₂×C₃×C₅	1	1	yes	characteristic
100	H(3,ℤ/4)	C₂×C₃×C₅	1	1	yes	characteristic, complemented
101	C₂²×C₃×C₅	—	4	1	no	—
102	C₂²×C₃×C₅	—	4	2	no	—
103	C₂²×C₃×C₅	—	4	1	no	—
104	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂²×C₃×C₅	#123	1	2	yes	—
106	C₃×C₄×C₅	—	4	4	no	—
107	C₃×C₄×C₅	—	4	1	no	—
108	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
109	C₃×C₄×C₅	#123	1	1	yes	characteristic
110	C₃×C₄×C₅	#127	1	1	yes	characteristic
111	C₃×#41	—	2	2	no	—
112	C₃×#41	—	2	2	no	—
113	C₂×C₃×Q₈	C₅×Q₈	1	1	yes	characteristic
114	C₂×C₃×D₄	—	2	1	no	—
115	C₂×C₃×D₄	C₅×Q₈	1	1	yes	characteristic
116	C₂²×C₃×C₄	—	2	1	no	—
117	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
118	C₃×M₁₆	—	2	2	no	—
119	C₃×M₁₆	—	2	2	no	—
120	C₃×M₁₆	—	2	2	no	—
121	C₂×C₃×C₈	—	2	1	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	—	2	2	no	—
124	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₂²⋊C₄	—	2	2	no	—
127	C₃×C₄²	C₅×D₄	1	4	yes	—
128	C₂³×C₅	—	2	1	no	—
129	C₂³×C₅	C₃×#41	1	1	yes	characteristic
130	C₅×Q₈	—	2	2	no	—
131	C₅×D₄	—	2	2	no	—
132	C₅×D₄	—	4	2	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
138	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₃×#41	1	2	yes	—
141	C₅×C₈	—	2	2	no	—
142	C₅×C₈	—	4	1	no	—
143	C₅×C₈	—	4	2	no	—
144	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
145	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
146	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
147	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
148	#117	C₂²×C₃×C₅	1	2	yes	—
149	#108	—	2	2	no	—
150	C₄⋊C₈	—	2	4	no	—
151	#104	—	2	4	no	—
152	C₂²⋊C₈	—	2	2	no	—
153	C₂×C₃×C₅	—	2	2	no	—
154	C₂×C₃×C₅	—	2	1	no	—
155	C₂×C₃×C₅	—	8	1	no	—
156	C₂×C₃×C₅	#399	1	1	yes	characteristic
157	C₂³×C₃	—	2	1	no	—
158	C₂³×C₃	C₅×#41	1	1	yes	characteristic
159	C₃×Q₈	—	2	2	no	—
160	C₃×D₄	—	2	2	no	—
161	C₃×D₄	—	4	2	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	—	4	1	no	—
165	C₂×C₃×C₄	—	4	1	no	—
166	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
167	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
169	C₂×C₃×C₄	C₅×#41	1	2	yes	—
170	C₃×C₈	—	2	2	no	—
171	C₃×C₈	—	4	1	no	—
172	C₃×C₈	—	4	2	no	—
173	C₂²×C₅	—	4	1	no	—
174	C₂²×C₅	—	4	2	no	—
175	C₂²×C₅	—	4	1	no	—
176	C₂²×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
177	C₂²×C₅	C₃×#123	1	2	yes	—
178	C₄×C₅	—	4	4	no	—
179	C₄×C₅	—	4	1	no	—
180	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
181	C₄×C₅	C₃×#123	1	1	yes	characteristic
182	C₄×C₅	C₃×#127	1	1	yes	characteristic
183	#41	—	2	2	no	—
184	#41	—	2	2	no	—
185	C₂×Q₈	C₃×C₅×Q₈	1	1	yes	characteristic
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	C₃×C₅×Q₈	1	1	yes	characteristic
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
190	M₁₆	—	2	2	no	—
191	M₁₆	—	2	2	no	—
192	M₁₆	—	2	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
197	C₄⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₄²	C₃×C₅×D₄	1	4	yes	—
200	C₃×C₅	#2098	1	1	yes	characteristic, direct product factor, Hall
201	C₂²×C₃	—	4	1	no	—
202	C₂²×C₃	—	4	2	no	—
203	C₂²×C₃	—	4	1	no	—
204	C₂²×C₃	C₅×C₂²⋊Q₈	1	1	yes	characteristic
205	C₂²×C₃	C₅×#123	1	2	yes	—
206	C₃×C₄	—	4	4	no	—
207	C₃×C₄	—	4	1	no	—
208	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
209	C₃×C₄	C₅×#123	1	1	yes	characteristic
210	C₃×C₄	C₅×#127	1	1	yes	characteristic
211	C₂×C₅	—	2	2	no	—
212	C₂×C₅	—	2	1	no	—
213	C₂×C₅	—	8	1	no	—
214	C₂×C₅	C₃×#399	1	1	yes	characteristic
215	C₂³	—	2	1	no	—
216	C₂³	C₃×C₅×#41	1	1	yes	characteristic
217	Q₈	—	2	2	no	—
218	D₄	—	2	2	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
225	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
226	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
227	C₂×C₄	C₃×C₅×#41	1	2	yes	—
228	C₈	—	2	2	no	—
229	C₈	—	4	1	no	—
230	C₈	—	4	2	no	—
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	8	1	no	—
234	C₂×C₃	C₅×#399	1	1	yes	characteristic
235	C₅	C₃×#2098	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²	—	4	1	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
240	C₂²	C₃×C₅×#123	1	2	yes	—
241	C₄	—	4	4	no	—
242	C₄	—	4	1	no	—
243	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
244	C₄	C₃×C₅×#123	1	1	yes	characteristic
245	C₄	C₃×C₅×#127	1	1	yes	characteristic
246	C₃	C₅×#2098	1	1	yes	characteristic, direct product factor, Sylow
247	C₂	—	2	2	no	—
248	C₂	—	2	1	no	—
249	C₂	—	8	1	no	—
250	C₂	C₃×C₅×#399	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle	C₂×C₃×C₅	#399
prime radical	C₂×C₃×C₅×#41	C₂²
LCS	G ▷ C₂²×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
1. C₃×C₅×#430	C₂	semidirect
2. C₃×C₅×#424	C₂	semidirect
5. C₃×C₅×H(3,ℤ/4)	C₂	semidirect
6. C₅×#2098	C₃	direct (N × H)
16. C₃×#2098	C₅	direct (N × H)
17. C₅×#430	C₂×C₃	semidirect
18. C₅×#424	C₂×C₃	semidirect
21. C₅×H(3,ℤ/4)	C₂×C₃	semidirect
39. C₃×#430	C₂×C₅	semidirect
40. C₃×#424	C₂×C₅	semidirect
43. C₃×H(3,ℤ/4)	C₂×C₅	semidirect
53. #2098	C₃×C₅	direct (N × H)
96. #430	C₂×C₃×C₅	semidirect
97. #424	C₂×C₃×C₅	semidirect
100. H(3,ℤ/4)	C₂×C₃×C₅	semidirect
200. C₃×C₅	#2098	direct (N × H)
235. C₅	C₃×#2098	direct (N × H)
246. C₃	C₅×#2098	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
C₃×C₅×#424	2
C₃×C₅×#421	2
C₃×C₅×#408	2
C₃×C₅×H(3,ℤ/4)	2
C₅×#2098	3
C₃×#2098	5

Subgroups — C3×C5×#2098 #14275132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2098 #14275132