Subgroups — C₃×C₅×#2188 #14275222

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 264 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
2	C₂×C₃×C₅×Q₃₂	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#467	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
6	C₅×#2188	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	1	yes	characteristic
11	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
14	C₃×C₅×Q₃₂	—	2	4	no	—
15	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
16	C₃×C₅×#107	C₂²	1	1	yes	characteristic
17	C₃×C₅×#103	—	2	1	no	—
18	C₃×#2188	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂²×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic
20	C₂×C₅×Q₃₂	C₂×C₃	1	2	yes	complemented
21	C₅×#467	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
24	C₂×C₃×C₅×Q₈	—	2	4	no	—
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₂×C₃×C₅×Q₈	—	4	2	no	—
28	C₂×C₃×C₅×Q₈	—	4	1	no	—
29	C₂²×C₃×C₄×C₅	—	4	1	no	—
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₃×C₅×Q₁₆	—	2	4	no	—
32	C₃×C₅×Q₁₆	—	2	2	no	—
33	C₃×C₅×Q₁₆	—	2	2	no	—
34	C₃×C₅×Q₁₆	D₄	1	4	yes	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	—	4	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
43	C₂×C₃×Q₃₂	C₂×C₅	1	2	yes	complemented
44	C₃×#467	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
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	1	yes	characteristic
51	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
52	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₅×C₂²⋊Q₈	—	2	1	no	—
54	C₅×Q₃₂	—	2	4	no	—
55	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
56	C₅×#107	C₂²×C₃	1	1	yes	characteristic
57	C₅×#103	—	2	1	no	—
58	#2188	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
59	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
60	C₃×C₅×Q₈	—	2	4	no	—
61	C₃×C₅×Q₈	—	4	4	no	—
62	C₃×C₅×Q₈	—	4	2	no	—
63	C₃×C₅×Q₈	—	4	1	no	—
64	C₃×C₅×Q₈	—	4	1	no	—
65	C₂×C₃×C₄×C₅	—	2	1	no	—
66	C₂×C₃×C₄×C₅	—	2	1	no	—
67	C₂×C₃×C₄×C₅	—	4	4	no	—
68	C₂×C₃×C₄×C₅	—	4	2	no	—
69	C₂×C₃×C₄×C₅	—	4	1	no	—
70	C₂×C₃×C₄×C₅	—	4	1	no	—
71	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₃×C₅×C₈	—	2	1	no	—
74	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
75	C₂²×C₃×Q₈	—	2	1	no	—
76	C₂×C₃×Q₁₆	—	2	4	no	—
77	C₂×C₃×Q₁₆	—	2	1	no	—
78	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
80	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
81	C₃×C₂²⋊Q₈	—	2	1	no	—
82	C₃×Q₃₂	—	2	4	no	—
83	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
84	C₃×#107	C₂²×C₅	1	1	yes	characteristic
85	C₃×#103	—	2	1	no	—
86	C₂×C₅×Q₈	—	2	4	no	—
87	C₂×C₅×Q₈	—	2	2	no	—
88	C₂×C₅×Q₈	—	2	1	no	—
89	C₂×C₅×Q₈	—	4	2	no	—
90	C₂×C₅×Q₈	—	4	1	no	—
91	C₂²×C₄×C₅	—	4	1	no	—
92	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
93	C₅×Q₁₆	—	2	4	no	—
94	C₅×Q₁₆	—	2	2	no	—
95	C₅×Q₁₆	—	2	2	no	—
96	C₅×Q₁₆	C₃×D₄	1	4	yes	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
100	C₅×C₄⋊C₄	—	2	1	no	—
101	C₅×C₄⋊C₄	—	4	1	no	—
102	C₅×C₂²⋊C₄	—	4	1	no	—
103	C₅×C₁₆	—	2	2	no	—
104	C₂²×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic
105	C₂×Q₃₂	C₂×C₃×C₅	1	2	yes	complemented
106	#467	C₂×C₃×C₅	1	1	yes	characteristic
107	#358	C₂×C₃×C₅	1	2	yes	complemented
108	C₂²⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
109	C₂²×C₃×C₅	—	2	1	no	—
110	C₂²×C₃×C₅	—	2	1	no	—
111	C₂²×C₃×C₅	Q₃₂	1	2	yes	complemented
112	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	—	4	4	no	—
115	C₃×C₄×C₅	—	8	1	no	—
116	C₃×C₄×C₅	—	8	1	no	—
117	C₃×C₄×C₅	#120	1	1	yes	characteristic
118	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₃×Q₈	—	2	4	no	—
120	C₂×C₃×Q₈	—	2	2	no	—
121	C₂×C₃×Q₈	—	2	1	no	—
122	C₂×C₃×Q₈	—	4	2	no	—
123	C₂×C₃×Q₈	—	4	1	no	—
124	C₂²×C₃×C₄	—	4	1	no	—
125	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
126	C₃×Q₁₆	—	2	4	no	—
127	C₃×Q₁₆	—	2	2	no	—
128	C₃×Q₁₆	—	2	2	no	—
129	C₃×Q₁₆	C₅×D₄	1	4	yes	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
132	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
133	C₃×C₄⋊C₄	—	2	1	no	—
134	C₃×C₄⋊C₄	—	4	1	no	—
135	C₃×C₂²⋊C₄	—	4	1	no	—
136	C₃×C₁₆	—	2	2	no	—
137	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
138	C₅×Q₈	—	2	4	no	—
139	C₅×Q₈	—	4	4	no	—
140	C₅×Q₈	—	4	2	no	—
141	C₅×Q₈	—	4	1	no	—
142	C₅×Q₈	—	4	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	4	4	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	1	no	—
148	C₂×C₄×C₅	—	4	1	no	—
149	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
151	C₅×C₈	—	2	1	no	—
152	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
153	C₂²×Q₈	—	2	1	no	—
154	C₂×Q₁₆	—	2	4	no	—
155	C₂×Q₁₆	—	2	1	no	—
156	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
157	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
158	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
159	C₂²⋊Q₈	—	2	1	no	—
160	Q₃₂	—	2	4	no	—
161	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
162	#107	C₂²×C₃×C₅	1	1	yes	characteristic
163	#103	—	2	1	no	—
164	C₂×C₃×C₅	—	2	2	no	—
165	C₂×C₃×C₅	#447	1	1	yes	characteristic
166	C₂×C₃×C₅	C₂×Q₃₂	1	1	yes	characteristic
167	C₂×C₃×C₅	#510	1	1	yes	characteristic
168	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
169	C₃×Q₈	—	2	4	no	—
170	C₃×Q₈	—	4	4	no	—
171	C₃×Q₈	—	4	2	no	—
172	C₃×Q₈	—	4	1	no	—
173	C₃×Q₈	—	4	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	4	4	no	—
177	C₂×C₃×C₄	—	4	2	no	—
178	C₂×C₃×C₄	—	4	1	no	—
179	C₂×C₃×C₄	—	4	1	no	—
180	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
182	C₃×C₈	—	2	1	no	—
183	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
184	C₂²×C₅	—	2	1	no	—
185	C₂²×C₅	—	2	1	no	—
186	C₂²×C₅	C₃×Q₃₂	1	2	yes	complemented
187	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	4	4	no	—
190	C₄×C₅	—	8	1	no	—
191	C₄×C₅	—	8	1	no	—
192	C₄×C₅	C₃×#120	1	1	yes	characteristic
193	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×Q₈	—	4	2	no	—
198	C₂×Q₈	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
201	Q₁₆	—	2	4	no	—
202	Q₁₆	—	2	2	no	—
203	Q₁₆	—	2	2	no	—
204	Q₁₆	C₃×C₅×D₄	1	4	yes	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
207	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	1	no	—
211	C₁₆	—	2	2	no	—
212	C₃×C₅	#2188	1	1	yes	characteristic, direct product factor, Hall
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	C₅×Q₃₂	1	2	yes	complemented
216	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	4	4	no	—
219	C₃×C₄	—	8	1	no	—
220	C₃×C₄	—	8	1	no	—
221	C₃×C₄	C₅×#120	1	1	yes	characteristic
222	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₂×C₅	—	2	2	no	—
224	C₂×C₅	C₃×#447	1	1	yes	characteristic
225	C₂×C₅	C₂×C₃×Q₃₂	1	1	yes	characteristic
226	C₂×C₅	C₃×#510	1	1	yes	characteristic
227	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
228	Q₈	—	2	4	no	—
229	Q₈	—	4	4	no	—
230	Q₈	—	4	2	no	—
231	Q₈	—	4	1	no	—
232	Q₈	—	4	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
240	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
243	C₂×C₃	—	2	2	no	—
244	C₂×C₃	C₅×#447	1	1	yes	characteristic
245	C₂×C₃	C₂×C₅×Q₃₂	1	1	yes	characteristic
246	C₂×C₃	C₅×#510	1	1	yes	characteristic
247	C₅	C₃×#2188	1	1	yes	characteristic, direct product factor, Sylow
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	C₃×C₅×Q₃₂	1	2	yes	complemented
251	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	1	no	—
255	C₄	—	8	1	no	—
256	C₄	C₃×C₅×#120	1	1	yes	characteristic
257	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₃	C₅×#2188	1	1	yes	characteristic, direct product factor, Sylow
259	C₂	—	2	2	no	—
260	C₂	C₃×C₅×#447	1	1	yes	characteristic
261	C₂	C₂×C₃×C₅×Q₃₂	1	1	yes	characteristic
262	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	C₂³×C₃×C₅	D₈
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
2. C₂×C₃×C₅×Q₃₂	C₂	semidirect
4. C₃×C₅×#358	C₂	semidirect
6. C₅×#2188	C₃	direct (N × H)
18. C₃×#2188	C₅	direct (N × H)
20. C₂×C₅×Q₃₂	C₂×C₃	semidirect
22. C₅×#358	C₂×C₃	semidirect
43. C₂×C₃×Q₃₂	C₂×C₅	semidirect
45. C₃×#358	C₂×C₅	semidirect
58. #2188	C₃×C₅	direct (N × H)
105. C₂×Q₃₂	C₂×C₃×C₅	semidirect
107. #358	C₂×C₃×C₅	semidirect
111. C₂²×C₃×C₅	Q₃₂	semidirect
186. C₂²×C₅	C₃×Q₃₂	semidirect
212. C₃×C₅	#2188	direct (N × H)
215. C₂²×C₃	C₅×Q₃₂	semidirect
247. C₅	C₃×#2188	direct (N × H)
250. C₂²	C₃×C₅×Q₃₂	semidirect
258. C₃	C₅×#2188	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×Q₃₂	2
C₃×C₅×#467	2
C₃×C₅×#358	2
C₃×C₅×C₂²⋊C₁₆	2
C₅×#2188	3
C₃×#2188	5

Subgroups — C3×C5×#2188 #14275222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2188 #14275222