Subgroups — C₃×C₅×#2129 #14275163

Order 1920.

476 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#536	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#410	C₂	1	2	yes	maximal
3	C₃×C₅×#354	C₂	1	4	yes	maximal, complemented
4	C₅×#2129	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
6	C₂²×C₃×C₅×Q₈	C₄	1	1	yes	characteristic, complemented
7	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	complemented
8	C₃×C₅×#124	C₂²	1	2	yes	—
9	C₃×C₅×#123	—	2	2	no	—
10	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
11	C₃×C₅×#117	—	2	2	no	—
12	C₃×C₅×#117	C₄	1	1	yes	characteristic, complemented
13	C₃×C₅×#99	—	2	2	no	—
14	C₃×C₅×#99	C₂²	1	4	yes	—
15	C₃×#2129	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₅×#536	C₂×C₃	1	1	yes	characteristic
17	C₅×#410	C₂×C₃	1	2	yes	—
18	C₅×#354	C₂×C₃	1	4	yes	complemented
19	C₃×C₅×#41	—	4	2	no	—
20	C₂×C₃×C₅×Q₈	—	2	1	no	—
21	C₂×C₃×C₅×Q₈	—	4	2	no	—
22	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
25	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	—	2	1	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₃×C₅×C₄⋊C₄	—	2	4	no	—
30	C₃×C₅×C₄⋊C₄	—	2	1	no	—
31	C₃×C₅×C₄⋊C₄	—	4	2	no	—
32	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
33	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
34	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
35	C₃×C₄²×C₅	—	4	2	no	—
36	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
37	C₃×#536	C₂×C₅	1	1	yes	characteristic
38	C₃×#410	C₂×C₅	1	2	yes	—
39	C₃×#354	C₂×C₅	1	4	yes	complemented
40	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₅×Q₈	C₃×C₄	1	1	yes	characteristic, complemented
42	C₅×C₄⋊Q₈	C₃×C₄	1	2	yes	complemented
43	C₅×#124	C₂²×C₃	1	2	yes	—
44	C₅×#123	—	2	2	no	—
45	C₅×C₂²⋊Q₈	—	2	2	no	—
46	C₅×#117	—	2	2	no	—
47	C₅×#117	C₃×C₄	1	1	yes	characteristic, complemented
48	C₅×#99	—	2	2	no	—
49	C₅×#99	C₂²×C₃	1	4	yes	—
50	#2129	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
51	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
53	C₃×C₅×Q₈	—	4	1	no	—
54	C₃×C₅×Q₈	—	4	2	no	—
55	C₃×C₅×Q₈	—	4	2	no	—
56	C₃×C₅×D₄	—	4	2	no	—
57	C₃×C₅×D₄	—	4	1	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	4	2	no	—
64	C₂×C₃×C₄×C₅	—	4	4	no	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
69	C₂²×C₃×Q₈	C₄×C₅	1	1	yes	characteristic, complemented
70	C₃×C₄⋊Q₈	C₄×C₅	1	2	yes	complemented
71	C₃×#124	C₂²×C₅	1	2	yes	—
72	C₃×#123	—	2	2	no	—
73	C₃×C₂²⋊Q₈	—	2	2	no	—
74	C₃×#117	—	2	2	no	—
75	C₃×#117	C₄×C₅	1	1	yes	characteristic, complemented
76	C₃×#99	—	2	2	no	—
77	C₃×#99	C₂²×C₅	1	4	yes	—
78	C₅×#41	—	4	2	no	—
79	C₂×C₅×Q₈	—	2	1	no	—
80	C₂×C₅×Q₈	—	4	2	no	—
81	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
82	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
83	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
84	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
85	C₂²×C₄×C₅	—	2	1	no	—
86	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
88	C₅×C₄⋊C₄	—	2	1	no	—
89	C₅×C₄⋊C₄	—	2	4	no	—
90	C₅×C₄⋊C₄	—	4	2	no	—
91	C₅×C₂²⋊C₄	—	2	1	no	—
92	C₅×C₂²⋊C₄	—	2	4	no	—
93	C₅×C₂²⋊C₄	—	2	4	no	—
94	C₄²×C₅	—	4	2	no	—
95	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
96	#536	C₂×C₃×C₅	1	1	yes	characteristic
97	#410	C₂×C₃×C₅	1	2	yes	—
98	#354	C₂×C₃×C₅	1	4	yes	complemented
99	C₂²×C₃×C₅	—	2	1	no	—
100	C₂²×C₃×C₅	—	2	2	no	—
101	C₂²×C₃×C₅	—	4	1	no	—
102	C₂²×C₃×C₅	—	4	2	no	—
103	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₃×C₄×C₅	—	2	2	no	—
105	C₃×C₄×C₅	—	2	1	no	—
106	C₃×C₄×C₅	—	4	2	no	—
107	C₃×C₄×C₅	—	4	2	no	—
108	C₃×C₄×C₅	—	8	4	no	—
109	C₃×C₄×C₅	#99	1	2	yes	—
110	C₃×#41	—	4	2	no	—
111	C₂×C₃×Q₈	—	2	1	no	—
112	C₂×C₃×Q₈	—	4	2	no	—
113	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
114	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
115	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
116	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
117	C₂²×C₃×C₄	—	2	1	no	—
118	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
120	C₃×C₄⋊C₄	—	2	1	no	—
121	C₃×C₄⋊C₄	—	2	4	no	—
122	C₃×C₄⋊C₄	—	4	2	no	—
123	C₃×C₂²⋊C₄	—	2	1	no	—
124	C₃×C₂²⋊C₄	—	2	4	no	—
125	C₃×C₂²⋊C₄	—	2	4	no	—
126	C₃×C₄²	—	4	2	no	—
127	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
128	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
129	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
130	C₅×Q₈	—	4	1	no	—
131	C₅×Q₈	—	4	2	no	—
132	C₅×Q₈	—	4	2	no	—
133	C₅×D₄	—	4	2	no	—
134	C₅×D₄	—	4	1	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	4	2	no	—
141	C₂×C₄×C₅	—	4	4	no	—
142	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
143	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
145	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
146	C₂²×Q₈	C₃×C₄×C₅	1	1	yes	characteristic, complemented
147	C₄⋊Q₈	C₃×C₄×C₅	1	2	yes	complemented
148	#124	C₂²×C₃×C₅	1	2	yes	—
149	#123	—	2	2	no	—
150	C₂²⋊Q₈	—	2	2	no	—
151	#117	—	2	2	no	—
152	#117	C₃×C₄×C₅	1	1	yes	characteristic, complemented
153	#99	—	2	2	no	—
154	#99	C₂²×C₃×C₅	1	4	yes	—
155	C₂×C₃×C₅	—	2	1	no	—
156	C₂×C₃×C₅	—	4	2	no	—
157	C₂×C₃×C₅	—	4	1	no	—
158	C₂×C₃×C₅	C₂×#99	1	1	yes	characteristic
159	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
160	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
161	C₃×Q₈	—	4	1	no	—
162	C₃×Q₈	—	4	2	no	—
163	C₃×Q₈	—	4	2	no	—
164	C₃×D₄	—	4	2	no	—
165	C₃×D₄	—	4	1	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	—	4	4	no	—
173	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
174	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
176	C₂²×C₅	—	2	1	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	4	1	no	—
179	C₂²×C₅	—	4	2	no	—
180	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	4	2	no	—
184	C₄×C₅	—	4	2	no	—
185	C₄×C₅	—	8	4	no	—
186	C₄×C₅	C₃×#99	1	2	yes	—
187	#41	—	4	2	no	—
188	C₂×Q₈	—	2	1	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	C₂×C₃×C₄×C₅	1	2	yes	—
191	C₂×Q₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
192	C₂×D₄	C₃×C₅×D₄	1	2	yes	—
193	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
196	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₄²	—	4	2	no	—
204	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
205	C₃×C₅	#2129	1	1	yes	characteristic, direct product factor, Hall
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	4	1	no	—
209	C₂²×C₃	—	4	2	no	—
210	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	4	2	no	—
214	C₃×C₄	—	4	2	no	—
215	C₃×C₄	—	8	4	no	—
216	C₃×C₄	C₅×#99	1	2	yes	—
217	C₂×C₅	—	2	1	no	—
218	C₂×C₅	—	4	2	no	—
219	C₂×C₅	—	4	1	no	—
220	C₂×C₅	C₂×C₃×#99	1	1	yes	characteristic
221	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
223	Q₈	—	4	1	no	—
224	Q₈	—	4	2	no	—
225	Q₈	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
236	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	4	2	no	—
240	C₂×C₃	—	4	1	no	—
241	C₂×C₃	C₂×C₅×#99	1	1	yes	characteristic
242	C₅	C₃×#2129	1	1	yes	characteristic, direct product factor, Sylow
243	C₂²	—	2	1	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	2	no	—
247	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	1	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	2	no	—
252	C₄	—	8	4	no	—
253	C₄	C₃×C₅×#99	1	2	yes	—
254	C₃	C₅×#2129	1	1	yes	characteristic, direct product factor, Sylow
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	—	4	1	no	—
258	C₂	C₂×C₃×C₅×#99	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	C₂×#99
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×D₄	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₅×#41 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#354	C₂	semidirect
4. C₅×#2129	C₃	direct (N × H)
6. C₂²×C₃×C₅×Q₈	C₄	semidirect
7. C₃×C₅×C₄⋊Q₈	C₄	semidirect
12. C₃×C₅×#117	C₄	semidirect
15. C₃×#2129	C₅	direct (N × H)
18. C₅×#354	C₂×C₃	semidirect
39. C₃×#354	C₂×C₅	semidirect
41. C₂²×C₅×Q₈	C₃×C₄	semidirect
42. C₅×C₄⋊Q₈	C₃×C₄	semidirect
47. C₅×#117	C₃×C₄	semidirect
50. #2129	C₃×C₅	direct (N × H)
69. C₂²×C₃×Q₈	C₄×C₅	semidirect
70. C₃×C₄⋊Q₈	C₄×C₅	semidirect
75. C₃×#117	C₄×C₅	semidirect
98. #354	C₂×C₃×C₅	semidirect
146. C₂²×Q₈	C₃×C₄×C₅	semidirect
147. C₄⋊Q₈	C₃×C₄×C₅	semidirect
152. #117	C₃×C₄×C₅	semidirect
205. C₃×C₅	#2129	direct (N × H)
242. C₅	C₃×#2129	direct (N × H)
254. C₃	C₅×#2129	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×#536	2
C₃×C₅×#410	2
C₃×C₅×#354	2
C₅×#2129	3
C₃×#2129	5

Subgroups — C3×C5×#2129 #14275163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2129 #14275163