Subgroups — C₃×C₅×#2185 #14275219

Order 1920.

452 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×SD₃₂	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#466	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#358	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2185	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	—	2	1	no	—
8	C₂×C₃×C₅×Q₁₆	—	2	4	no	—
9	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
10	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
11	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#121	—	2	1	no	—
14	C₃×C₅×SD₃₂	—	2	4	no	—
15	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
16	C₃×C₅×#107	C₂²	1	1	yes	characteristic
17	C₃×C₅×#102	—	2	1	no	—
18	C₃×#2185	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂²×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂×C₅×SD₃₂	C₂×C₃	1	2	yes	complemented
21	C₅×#466	C₂×C₃	1	1	yes	characteristic
22	C₅×#358	C₂×C₃	1	2	yes	complemented
23	C₅×C₂²⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₃×C₅×Q₈	—	2	4	no	—
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	4	2	no	—
27	C₂×C₃×C₅×Q₈	—	4	1	no	—
28	C₂×C₃×C₅×D₄	—	2	1	no	—
29	C₂×C₃×C₅×D₄	—	4	1	no	—
30	C₂²×C₃×C₄×C₅	—	4	1	no	—
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₃×C₅×Q₁₆	—	2	4	no	—
33	C₃×C₅×Q₁₆	—	2	2	no	—
34	C₃×C₅×Q₁₆	D₄	1	4	yes	complemented
35	C₃×C₅×D₈	—	2	2	no	—
36	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
41	C₃×C₅×C₁₆	—	2	2	no	—
42	C₂²×C₃×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂×C₃×SD₃₂	C₂×C₅	1	2	yes	complemented
44	C₃×#466	C₂×C₅	1	1	yes	characteristic
45	C₃×#358	C₂×C₅	1	2	yes	complemented
46	C₃×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
47	C₂²×C₅×Q₈	—	2	1	no	—
48	C₂×C₅×Q₁₆	—	2	4	no	—
49	C₂×C₅×Q₁₆	—	2	1	no	—
50	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
51	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
52	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₅×#121	—	2	1	no	—
54	C₅×SD₃₂	—	2	4	no	—
55	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
56	C₅×#107	C₂²×C₃	1	1	yes	characteristic
57	C₅×#102	—	2	1	no	—
58	#2185	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
59	C₂³×C₃×C₅	—	4	1	no	—
60	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
61	C₃×C₅×Q₈	—	2	4	no	—
62	C₃×C₅×Q₈	—	4	4	no	—
63	C₃×C₅×Q₈	—	4	2	no	—
64	C₃×C₅×D₄	—	4	1	no	—
65	C₃×C₅×D₄	—	4	1	no	—
66	C₃×C₅×D₄	—	8	2	no	—
67	C₂×C₃×C₄×C₅	—	2	1	no	—
68	C₂×C₃×C₄×C₅	—	2	1	no	—
69	C₂×C₃×C₄×C₅	—	4	4	no	—
70	C₂×C₃×C₄×C₅	—	4	2	no	—
71	C₂×C₃×C₄×C₅	—	4	1	no	—
72	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₃×C₅×C₈	—	2	1	no	—
75	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
76	C₂²×C₃×Q₈	—	2	1	no	—
77	C₂×C₃×Q₁₆	—	2	4	no	—
78	C₂×C₃×Q₁₆	—	2	1	no	—
79	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
80	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
81	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
82	C₃×#121	—	2	1	no	—
83	C₃×SD₃₂	—	2	4	no	—
84	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
85	C₃×#107	C₂²×C₅	1	1	yes	characteristic
86	C₃×#102	—	2	1	no	—
87	C₂×C₅×Q₈	—	2	4	no	—
88	C₂×C₅×Q₈	—	2	2	no	—
89	C₂×C₅×Q₈	—	4	2	no	—
90	C₂×C₅×Q₈	—	4	1	no	—
91	C₂×C₅×D₄	—	2	1	no	—
92	C₂×C₅×D₄	—	4	1	no	—
93	C₂²×C₄×C₅	—	4	1	no	—
94	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
95	C₅×Q₁₆	—	2	4	no	—
96	C₅×Q₁₆	—	2	2	no	—
97	C₅×Q₁₆	C₃×D₄	1	4	yes	complemented
98	C₅×D₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	2	1	no	—
103	C₅×C₂²⋊C₄	—	4	1	no	—
104	C₅×C₁₆	—	2	2	no	—
105	C₂²×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
106	C₂×SD₃₂	C₂×C₃×C₅	1	2	yes	complemented
107	#466	C₂×C₃×C₅	1	1	yes	characteristic
108	#358	C₂×C₃×C₅	1	2	yes	complemented
109	C₂²⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
110	C₂²×C₃×C₅	—	2	1	no	—
111	C₂²×C₃×C₅	—	2	1	no	—
112	C₂²×C₃×C₅	—	8	1	no	—
113	C₂²×C₃×C₅	—	8	1	no	—
114	C₂²×C₃×C₅	—	8	1	no	—
115	C₂²×C₃×C₅	SD₃₂	1	2	yes	complemented
116	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
117	C₃×C₄×C₅	—	2	1	no	—
118	C₃×C₄×C₅	—	4	4	no	—
119	C₃×C₄×C₅	—	8	1	no	—
120	C₃×C₄×C₅	#120	1	1	yes	characteristic
121	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₃×Q₈	—	2	4	no	—
123	C₂×C₃×Q₈	—	2	2	no	—
124	C₂×C₃×Q₈	—	4	2	no	—
125	C₂×C₃×Q₈	—	4	1	no	—
126	C₂×C₃×D₄	—	2	1	no	—
127	C₂×C₃×D₄	—	4	1	no	—
128	C₂²×C₃×C₄	—	4	1	no	—
129	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
130	C₃×Q₁₆	—	2	4	no	—
131	C₃×Q₁₆	—	2	2	no	—
132	C₃×Q₁₆	C₅×D₄	1	4	yes	complemented
133	C₃×D₈	—	2	2	no	—
134	C₂×C₃×C₈	—	2	2	no	—
135	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
137	C₃×C₄⋊C₄	—	2	1	no	—
138	C₃×C₂²⋊C₄	—	4	1	no	—
139	C₃×C₁₆	—	2	2	no	—
140	C₂³×C₅	—	4	1	no	—
141	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
142	C₅×Q₈	—	2	4	no	—
143	C₅×Q₈	—	4	4	no	—
144	C₅×Q₈	—	4	2	no	—
145	C₅×D₄	—	4	1	no	—
146	C₅×D₄	—	4	1	no	—
147	C₅×D₄	—	8	2	no	—
148	C₂×C₄×C₅	—	2	1	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	4	4	no	—
151	C₂×C₄×C₅	—	4	2	no	—
152	C₂×C₄×C₅	—	4	1	no	—
153	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
155	C₅×C₈	—	2	1	no	—
156	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
157	C₂²×Q₈	—	2	1	no	—
158	C₂×Q₁₆	—	2	4	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
161	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
162	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
163	#121	—	2	1	no	—
164	SD₃₂	—	2	4	no	—
165	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
166	#107	C₂²×C₃×C₅	1	1	yes	characteristic
167	#102	—	2	1	no	—
168	C₂×C₃×C₅	—	2	2	no	—
169	C₂×C₃×C₅	—	16	1	no	—
170	C₂×C₃×C₅	#447	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
172	C₂×C₃×C₅	#510	1	1	yes	characteristic
173	C₂³×C₃	—	4	1	no	—
174	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
175	C₃×Q₈	—	2	4	no	—
176	C₃×Q₈	—	4	4	no	—
177	C₃×Q₈	—	4	2	no	—
178	C₃×D₄	—	4	1	no	—
179	C₃×D₄	—	4	1	no	—
180	C₃×D₄	—	8	2	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	4	4	no	—
184	C₂×C₃×C₄	—	4	2	no	—
185	C₂×C₃×C₄	—	4	1	no	—
186	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
188	C₃×C₈	—	2	1	no	—
189	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
190	C₂²×C₅	—	2	1	no	—
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	—	8	1	no	—
193	C₂²×C₅	—	8	1	no	—
194	C₂²×C₅	—	8	1	no	—
195	C₂²×C₅	C₃×SD₃₂	1	2	yes	complemented
196	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	4	4	no	—
199	C₄×C₅	—	8	1	no	—
200	C₄×C₅	C₃×#120	1	1	yes	characteristic
201	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
210	Q₁₆	—	2	4	no	—
211	Q₁₆	—	2	2	no	—
212	Q₁₆	C₃×C₅×D₄	1	4	yes	complemented
213	D₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
216	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
217	C₄⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	4	1	no	—
219	C₁₆	—	2	2	no	—
220	C₃×C₅	#2185	1	1	yes	characteristic, direct product factor, Hall
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	8	1	no	—
224	C₂²×C₃	—	8	1	no	—
225	C₂²×C₃	—	8	1	no	—
226	C₂²×C₃	C₅×SD₃₂	1	2	yes	complemented
227	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	4	4	no	—
230	C₃×C₄	—	8	1	no	—
231	C₃×C₄	C₅×#120	1	1	yes	characteristic
232	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
233	C₂×C₅	—	2	2	no	—
234	C₂×C₅	—	16	1	no	—
235	C₂×C₅	C₃×#447	1	1	yes	characteristic
236	C₂×C₅	C₂×C₃×SD₃₂	1	1	yes	characteristic
237	C₂×C₅	C₃×#510	1	1	yes	characteristic
238	C₂³	—	4	1	no	—
239	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
240	Q₈	—	2	4	no	—
241	Q₈	—	4	4	no	—
242	Q₈	—	4	2	no	—
243	D₄	—	4	1	no	—
244	D₄	—	4	1	no	—
245	D₄	—	8	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
252	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
255	C₂×C₃	—	2	2	no	—
256	C₂×C₃	—	16	1	no	—
257	C₂×C₃	C₅×#447	1	1	yes	characteristic
258	C₂×C₃	C₂×C₅×SD₃₂	1	1	yes	characteristic
259	C₂×C₃	C₅×#510	1	1	yes	characteristic
260	C₅	C₃×#2185	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	8	1	no	—
264	C₂²	—	8	1	no	—
265	C₂²	—	8	1	no	—
266	C₂²	C₃×C₅×SD₃₂	1	2	yes	complemented
267	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	4	4	no	—
270	C₄	—	8	1	no	—
271	C₄	C₃×C₅×#120	1	1	yes	characteristic
272	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₃	C₅×#2185	1	1	yes	characteristic, direct product factor, Sylow
274	C₂	—	2	2	no	—
275	C₂	—	16	1	no	—
276	C₂	C₃×C₅×#447	1	1	yes	characteristic
277	C₂	C₂×C₃×C₅×SD₃₂	1	1	yes	characteristic
278	C₂	C₃×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₃×C₅×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×Q₁₆	C₂	semidirect
2. C₂×C₃×C₅×SD₃₂	C₂	semidirect
4. C₃×C₅×#358	C₂	semidirect
5. C₃×C₅×C₂²⋊C₁₆	C₂	semidirect
6. C₅×#2185	C₃	direct (N × H)
18. C₃×#2185	C₅	direct (N × H)
19. C₂²×C₅×Q₁₆	C₂×C₃	semidirect
20. C₂×C₅×SD₃₂	C₂×C₃	semidirect
22. C₅×#358	C₂×C₃	semidirect
23. C₅×C₂²⋊C₁₆	C₂×C₃	semidirect
34. C₃×C₅×Q₁₆	D₄	semidirect
42. C₂²×C₃×Q₁₆	C₂×C₅	semidirect
43. C₂×C₃×SD₃₂	C₂×C₅	semidirect
45. C₃×#358	C₂×C₅	semidirect
46. C₃×C₂²⋊C₁₆	C₂×C₅	semidirect
58. #2185	C₃×C₅	direct (N × H)
97. C₅×Q₁₆	C₃×D₄	semidirect
105. C₂²×Q₁₆	C₂×C₃×C₅	semidirect
106. C₂×SD₃₂	C₂×C₃×C₅	semidirect
108. #358	C₂×C₃×C₅	semidirect
109. C₂²⋊C₁₆	C₂×C₃×C₅	semidirect
115. C₂²×C₃×C₅	SD₃₂	semidirect
132. C₃×Q₁₆	C₅×D₄	semidirect
195. C₂²×C₅	C₃×SD₃₂	semidirect
212. Q₁₆	C₃×C₅×D₄	semidirect
220. C₃×C₅	#2185	direct (N × H)
226. C₂²×C₃	C₅×SD₃₂	semidirect
260. C₅	C₃×#2185	direct (N × H)
266. C₂²	C₃×C₅×SD₃₂	semidirect
273. C₃	C₅×#2185	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₅×Q₁₆	2
C₂×C₃×C₅×SD₃₂	2
C₃×C₅×#466	2
C₃×C₅×#358	2
C₃×C₅×C₂²⋊C₁₆	2
C₅×#2185	3
C₃×#2185	5

Subgroups — C3×C5×#2185 #14275219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2185 #14275219