Subgroups — C₃×C₅×#2206 #14275240

Order 1920.

372 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	C₂×C₃×C₅×D₁₆	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#357	C₂	1	2	yes	maximal, complemented
6	C₅×#2206	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×D₈	—	2	1	no	—
8	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
9	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₃×C₅×#127	—	2	1	no	—
11	C₃×C₄×C₅×D₄	—	2	1	no	—
12	C₃×C₅×D₁₆	—	2	4	no	—
13	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
14	C₃×C₅×#107	C₂²	1	1	yes	characteristic
15	C₃×C₅×#102	—	2	1	no	—
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×#2206	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×D₁₆	C₂×C₃	1	2	yes	complemented
19	C₅×#493	C₂×C₃	1	1	yes	characteristic, complemented
20	C₄×C₅×D₈	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
22	C₅×#357	C₂×C₃	1	2	yes	complemented
23	C₂×C₃×C₅×D₄	—	2	1	no	—
24	C₂×C₃×C₅×D₄	—	2	2	no	—
25	C₂×C₃×C₅×D₄	—	4	1	no	—
26	C₂×C₃×C₅×D₄	—	4	1	no	—
27	C₂²×C₃×C₄×C₅	—	4	1	no	—
28	C₃×C₅×D₈	—	2	2	no	—
29	C₃×C₅×D₈	—	2	1	no	—
30	C₃×C₅×D₈	—	2	2	no	—
31	C₃×C₅×D₈	—	4	2	no	—
32	C₃×C₅×D₈	D₄	1	2	yes	complemented
33	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₃×C₅×C₄⋊C₄	—	2	1	no	—
36	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
37	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₃×C₅×C₁₆	—	2	2	no	—
39	C₂×C₃×D₁₆	C₂×C₅	1	2	yes	complemented
40	C₃×#493	C₂×C₅	1	1	yes	characteristic, complemented
41	C₃×C₄×D₈	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×#357	C₂×C₅	1	2	yes	complemented
44	C₂×C₅×D₈	—	2	1	no	—
45	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
46	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
47	C₅×#127	—	2	1	no	—
48	C₄×C₅×D₄	—	2	1	no	—
49	C₅×D₁₆	—	2	4	no	—
50	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
51	C₅×#107	C₂²×C₃	1	1	yes	characteristic
52	C₅×#102	—	2	1	no	—
53	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
54	#2206	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₅	—	4	1	no	—
56	C₂³×C₃×C₅	—	4	2	no	—
57	C₃×C₅×D₄	—	2	2	no	—
58	C₃×C₅×D₄	—	4	2	no	—
59	C₃×C₅×D₄	—	4	1	no	—
60	C₃×C₅×D₄	—	4	2	no	—
61	C₃×C₅×D₄	—	8	2	no	—
62	C₃×C₅×D₄	—	8	2	no	—
63	C₂×C₃×C₄×C₅	—	4	1	no	—
64	C₂×C₃×C₄×C₅	—	8	2	no	—
65	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₃×C₅×C₈	—	2	1	no	—
69	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₃×C₅×C₈	#41	1	1	yes	characteristic
71	C₂×C₃×D₈	—	2	1	no	—
72	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
73	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
74	C₃×#127	—	2	1	no	—
75	C₃×C₄×D₄	—	2	1	no	—
76	C₃×D₁₆	—	2	4	no	—
77	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
78	C₃×#107	C₂²×C₅	1	1	yes	characteristic
79	C₃×#102	—	2	1	no	—
80	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
81	C₂×C₅×D₄	—	2	1	no	—
82	C₂×C₅×D₄	—	2	2	no	—
83	C₂×C₅×D₄	—	4	1	no	—
84	C₂×C₅×D₄	—	4	1	no	—
85	C₂²×C₄×C₅	—	4	1	no	—
86	C₅×D₈	—	2	2	no	—
87	C₅×D₈	—	2	1	no	—
88	C₅×D₈	—	2	2	no	—
89	C₅×D₈	—	4	2	no	—
90	C₅×D₈	C₃×D₄	1	2	yes	complemented
91	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
93	C₅×C₄⋊C₄	—	2	1	no	—
94	C₅×C₂²⋊C₄	—	4	1	no	—
95	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
96	C₅×C₁₆	—	2	2	no	—
97	C₂×D₁₆	C₂×C₃×C₅	1	2	yes	complemented
98	#493	C₂×C₃×C₅	1	1	yes	characteristic, complemented
99	C₄×D₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
100	C₄⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
101	#357	C₂×C₃×C₅	1	2	yes	complemented
102	C₂²×C₃×C₅	—	4	2	no	—
103	C₂²×C₃×C₅	—	8	1	no	—
104	C₂²×C₃×C₅	—	8	1	no	—
105	C₂²×C₃×C₅	—	8	2	no	—
106	C₂²×C₃×C₅	—	8	2	no	—
107	C₂²×C₃×C₅	—	8	2	no	—
108	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	8	1	no	—
111	C₃×C₄×C₅	D₁₆	1	2	yes	complemented
112	C₃×C₄×C₅	#121	1	1	yes	characteristic
113	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₃×D₄	—	2	1	no	—
115	C₂×C₃×D₄	—	2	2	no	—
116	C₂×C₃×D₄	—	4	1	no	—
117	C₂×C₃×D₄	—	4	1	no	—
118	C₂²×C₃×C₄	—	4	1	no	—
119	C₃×D₈	—	2	2	no	—
120	C₃×D₈	—	2	1	no	—
121	C₃×D₈	—	2	2	no	—
122	C₃×D₈	—	4	2	no	—
123	C₃×D₈	C₅×D₄	1	2	yes	complemented
124	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
126	C₃×C₄⋊C₄	—	2	1	no	—
127	C₃×C₂²⋊C₄	—	4	1	no	—
128	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
129	C₃×C₁₆	—	2	2	no	—
130	C₂³×C₅	—	4	1	no	—
131	C₂³×C₅	—	4	2	no	—
132	C₅×D₄	—	2	2	no	—
133	C₅×D₄	—	4	2	no	—
134	C₅×D₄	—	4	1	no	—
135	C₅×D₄	—	4	2	no	—
136	C₅×D₄	—	8	2	no	—
137	C₅×D₄	—	8	2	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	8	2	no	—
140	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
141	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
143	C₅×C₈	—	2	1	no	—
144	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
145	C₅×C₈	C₃×#41	1	1	yes	characteristic
146	C₂×D₈	—	2	1	no	—
147	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
148	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
149	#127	—	2	1	no	—
150	C₄×D₄	—	2	1	no	—
151	D₁₆	—	2	4	no	—
152	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
153	#107	C₂²×C₃×C₅	1	1	yes	characteristic
154	#102	—	2	1	no	—
155	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
156	C₂×C₃×C₅	—	8	2	no	—
157	C₂×C₃×C₅	—	16	2	no	—
158	C₂×C₃×C₅	#459	1	1	yes	characteristic
159	C₂×C₃×C₅	C₂×D₁₆	1	1	yes	characteristic
160	C₂×C₃×C₅	#509	1	1	yes	characteristic
161	C₂³×C₃	—	4	1	no	—
162	C₂³×C₃	—	4	2	no	—
163	C₃×D₄	—	2	2	no	—
164	C₃×D₄	—	4	2	no	—
165	C₃×D₄	—	4	1	no	—
166	C₃×D₄	—	4	2	no	—
167	C₃×D₄	—	8	2	no	—
168	C₃×D₄	—	8	2	no	—
169	C₂×C₃×C₄	—	4	1	no	—
170	C₂×C₃×C₄	—	8	2	no	—
171	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
172	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
174	C₃×C₈	—	2	1	no	—
175	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
176	C₃×C₈	C₅×#41	1	1	yes	characteristic
177	C₂²×C₅	—	4	2	no	—
178	C₂²×C₅	—	8	1	no	—
179	C₂²×C₅	—	8	1	no	—
180	C₂²×C₅	—	8	2	no	—
181	C₂²×C₅	—	8	2	no	—
182	C₂²×C₅	—	8	2	no	—
183	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	8	1	no	—
186	C₄×C₅	C₃×D₁₆	1	2	yes	complemented
187	C₄×C₅	C₃×#121	1	1	yes	characteristic
188	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	D₈	—	2	2	no	—
195	D₈	—	2	1	no	—
196	D₈	—	2	2	no	—
197	D₈	—	4	2	no	—
198	D₈	C₃×C₅×D₄	1	2	yes	complemented
199	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
200	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	4	1	no	—
203	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
204	C₁₆	—	2	2	no	—
205	C₃×C₅	#2206	1	1	yes	characteristic, direct product factor, Hall
206	C₂²×C₃	—	4	2	no	—
207	C₂²×C₃	—	8	1	no	—
208	C₂²×C₃	—	8	1	no	—
209	C₂²×C₃	—	8	2	no	—
210	C₂²×C₃	—	8	2	no	—
211	C₂²×C₃	—	8	2	no	—
212	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	8	1	no	—
215	C₃×C₄	C₅×D₁₆	1	2	yes	complemented
216	C₃×C₄	C₅×#121	1	1	yes	characteristic
217	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₂×C₅	—	8	2	no	—
219	C₂×C₅	—	16	2	no	—
220	C₂×C₅	C₃×#459	1	1	yes	characteristic
221	C₂×C₅	C₂×C₃×D₁₆	1	1	yes	characteristic
222	C₂×C₅	C₃×#509	1	1	yes	characteristic
223	C₂³	—	4	1	no	—
224	C₂³	—	4	2	no	—
225	D₄	—	2	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	1	no	—
228	D₄	—	4	2	no	—
229	D₄	—	8	2	no	—
230	D₄	—	8	2	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
234	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
235	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
238	C₈	C₃×C₅×#41	1	1	yes	characteristic
239	C₂×C₃	—	8	2	no	—
240	C₂×C₃	—	16	2	no	—
241	C₂×C₃	C₅×#459	1	1	yes	characteristic
242	C₂×C₃	C₂×C₅×D₁₆	1	1	yes	characteristic
243	C₂×C₃	C₅×#509	1	1	yes	characteristic
244	C₅	C₃×#2206	1	1	yes	characteristic, direct product factor, Sylow
245	C₂²	—	4	2	no	—
246	C₂²	—	8	1	no	—
247	C₂²	—	8	1	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	8	1	no	—
254	C₄	C₃×C₅×D₁₆	1	2	yes	complemented
255	C₄	C₃×C₅×#121	1	1	yes	characteristic
256	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₃	C₅×#2206	1	1	yes	characteristic, direct product factor, Sylow
258	C₂	—	8	2	no	—
259	C₂	—	16	2	no	—
260	C₂	C₃×C₅×#459	1	1	yes	characteristic
261	C₂	C₂×C₃×C₅×D₁₆	1	1	yes	characteristic
262	C₂	C₃×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×D₁₆	C₂	semidirect
2. C₃×C₅×#493	C₂	semidirect
3. C₃×C₄×C₅×D₈	C₂	semidirect
4. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
5. C₃×C₅×#357	C₂	semidirect
6. C₅×#2206	C₃	direct (N × H)
17. C₃×#2206	C₅	direct (N × H)
18. C₂×C₅×D₁₆	C₂×C₃	semidirect
19. C₅×#493	C₂×C₃	semidirect
20. C₄×C₅×D₈	C₂×C₃	semidirect
21. C₅×C₄⋊C₁₆	C₂×C₃	semidirect
22. C₅×#357	C₂×C₃	semidirect
32. C₃×C₅×D₈	D₄	semidirect
39. C₂×C₃×D₁₆	C₂×C₅	semidirect
40. C₃×#493	C₂×C₅	semidirect
41. C₃×C₄×D₈	C₂×C₅	semidirect
42. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
43. C₃×#357	C₂×C₅	semidirect
54. #2206	C₃×C₅	direct (N × H)
90. C₅×D₈	C₃×D₄	semidirect
97. C₂×D₁₆	C₂×C₃×C₅	semidirect
98. #493	C₂×C₃×C₅	semidirect
99. C₄×D₈	C₂×C₃×C₅	semidirect
100. C₄⋊C₁₆	C₂×C₃×C₅	semidirect
101. #357	C₂×C₃×C₅	semidirect
111. C₃×C₄×C₅	D₁₆	semidirect
123. C₃×D₈	C₅×D₄	semidirect
186. C₄×C₅	C₃×D₁₆	semidirect
198. D₈	C₃×C₅×D₄	semidirect
205. C₃×C₅	#2206	direct (N × H)
215. C₃×C₄	C₅×D₁₆	semidirect
244. C₅	C₃×#2206	direct (N × H)
254. C₄	C₃×C₅×D₁₆	semidirect
257. C₃	C₅×#2206	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×D₁₆	2
C₃×C₅×#493	2
C₃×C₄×C₅×D₈	2
C₃×C₅×C₄⋊C₁₆	2
C₃×C₅×#357	2
C₅×#2206	3
C₃×#2206	5

Subgroups — C3×C5×#2206 #14275240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2206 #14275240