Subgroups — C₃×C₅×#2207 #14275241

Order 1920.

348 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×SD₃₂	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×Q₁₆	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×#2207	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×D₈	—	2	1	no	—
9	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
10	C₃×C₅×#127	—	2	1	no	—
11	C₃×C₄×C₅×Q₈	—	2	1	no	—
12	C₃×C₅×SD₃₂	—	2	4	no	—
13	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
14	C₃×C₅×#107	C₂²	1	1	yes	characteristic
15	C₃×C₅×#103	—	2	1	no	—
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×#2207	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×SD₃₂	C₂×C₃	1	2	yes	complemented
19	C₅×#493	C₂×C₃	1	1	yes	characteristic
20	C₄×C₅×Q₁₆	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×Q₈	—	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	C₃×C₅×Q₁₆	—	2	1	no	—
28	C₃×C₅×Q₁₆	D₄	1	2	yes	complemented
29	C₃×C₅×D₈	—	2	2	no	—
30	C₃×C₅×D₈	—	2	2	no	—
31	C₃×C₅×D₈	—	4	2	no	—
32	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₃×C₅×C₄⋊C₄	—	2	1	no	—
35	C₃×C₅×C₄⋊C₄	—	4	1	no	—
36	C₃×C₄²×C₅	—	4	1	no	—
37	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₃×C₅×C₁₆	—	2	2	no	—
39	C₂×C₃×SD₃₂	C₂×C₅	1	2	yes	complemented
40	C₃×#493	C₂×C₅	1	1	yes	characteristic
41	C₃×C₄×Q₁₆	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×Q₁₆	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₅×D₈	—	2	1	no	—
46	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
47	C₅×#127	—	2	1	no	—
48	C₄×C₅×Q₈	—	2	1	no	—
49	C₅×SD₃₂	—	2	4	no	—
50	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
51	C₅×#107	C₂²×C₃	1	1	yes	characteristic
52	C₅×#103	—	2	1	no	—
53	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
54	#2207	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₅	—	4	2	no	—
56	C₃×C₅×Q₈	—	2	2	no	—
57	C₃×C₅×Q₈	—	4	1	no	—
58	C₃×C₅×D₄	—	4	2	no	—
59	C₃×C₅×D₄	—	4	2	no	—
60	C₃×C₅×D₄	—	8	2	no	—
61	C₃×C₅×D₄	—	8	2	no	—
62	C₂×C₃×C₄×C₅	—	4	1	no	—
63	C₂×C₃×C₄×C₅	—	4	1	no	—
64	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₃×C₅×C₈	—	2	1	no	—
68	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
69	C₃×C₅×C₈	#41	1	1	yes	characteristic
70	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₃×D₈	—	2	1	no	—
72	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
73	C₃×#127	—	2	1	no	—
74	C₃×C₄×Q₈	—	2	1	no	—
75	C₃×SD₃₂	—	2	4	no	—
76	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
77	C₃×#107	C₂²×C₅	1	1	yes	characteristic
78	C₃×#103	—	2	1	no	—
79	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
80	C₂×C₅×Q₈	—	2	1	no	—
81	C₂×C₅×D₄	—	2	2	no	—
82	C₂×C₅×D₄	—	4	1	no	—
83	C₂×C₅×D₄	—	4	1	no	—
84	C₅×Q₁₆	—	2	1	no	—
85	C₅×Q₁₆	C₃×D₄	1	2	yes	complemented
86	C₅×D₈	—	2	2	no	—
87	C₅×D₈	—	2	2	no	—
88	C₅×D₈	—	4	2	no	—
89	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
91	C₅×C₄⋊C₄	—	2	1	no	—
92	C₅×C₄⋊C₄	—	4	1	no	—
93	C₄²×C₅	—	4	1	no	—
94	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
95	C₅×C₁₆	—	2	2	no	—
96	C₂×SD₃₂	C₂×C₃×C₅	1	2	yes	complemented
97	#493	C₂×C₃×C₅	1	1	yes	characteristic
98	C₄×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
99	C₄⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
100	#357	C₂×C₃×C₅	1	2	yes	complemented
101	C₂²×C₃×C₅	—	8	2	no	—
102	C₂²×C₃×C₅	—	8	2	no	—
103	C₂²×C₃×C₅	—	8	2	no	—
104	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
105	C₃×C₄×C₅	—	2	1	no	—
106	C₃×C₄×C₅	—	4	2	no	—
107	C₃×C₄×C₅	—	8	1	no	—
108	C₃×C₄×C₅	SD₃₂	1	2	yes	complemented
109	C₃×C₄×C₅	#121	1	1	yes	characteristic
110	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₃×Q₈	—	2	1	no	—
112	C₂×C₃×D₄	—	2	2	no	—
113	C₂×C₃×D₄	—	4	1	no	—
114	C₂×C₃×D₄	—	4	1	no	—
115	C₃×Q₁₆	—	2	1	no	—
116	C₃×Q₁₆	C₅×D₄	1	2	yes	complemented
117	C₃×D₈	—	2	2	no	—
118	C₃×D₈	—	2	2	no	—
119	C₃×D₈	—	4	2	no	—
120	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
122	C₃×C₄⋊C₄	—	2	1	no	—
123	C₃×C₄⋊C₄	—	4	1	no	—
124	C₃×C₄²	—	4	1	no	—
125	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
126	C₃×C₁₆	—	2	2	no	—
127	C₂³×C₅	—	4	2	no	—
128	C₅×Q₈	—	2	2	no	—
129	C₅×Q₈	—	4	1	no	—
130	C₅×D₄	—	4	2	no	—
131	C₅×D₄	—	4	2	no	—
132	C₅×D₄	—	8	2	no	—
133	C₅×D₄	—	8	2	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
137	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
139	C₅×C₈	—	2	1	no	—
140	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
141	C₅×C₈	C₃×#41	1	1	yes	characteristic
142	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
143	C₂×D₈	—	2	1	no	—
144	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
145	#127	—	2	1	no	—
146	C₄×Q₈	—	2	1	no	—
147	SD₃₂	—	2	4	no	—
148	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
149	#107	C₂²×C₃×C₅	1	1	yes	characteristic
150	#103	—	2	1	no	—
151	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
152	C₂×C₃×C₅	—	16	2	no	—
153	C₂×C₃×C₅	#459	1	1	yes	characteristic
154	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
155	C₂×C₃×C₅	#509	1	1	yes	characteristic
156	C₂³×C₃	—	4	2	no	—
157	C₃×Q₈	—	2	2	no	—
158	C₃×Q₈	—	4	1	no	—
159	C₃×D₄	—	4	2	no	—
160	C₃×D₄	—	4	2	no	—
161	C₃×D₄	—	8	2	no	—
162	C₃×D₄	—	8	2	no	—
163	C₂×C₃×C₄	—	4	1	no	—
164	C₂×C₃×C₄	—	4	1	no	—
165	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
166	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
168	C₃×C₈	—	2	1	no	—
169	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
170	C₃×C₈	C₅×#41	1	1	yes	characteristic
171	C₂²×C₅	—	8	2	no	—
172	C₂²×C₅	—	8	2	no	—
173	C₂²×C₅	—	8	2	no	—
174	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	4	2	no	—
177	C₄×C₅	—	8	1	no	—
178	C₄×C₅	C₃×SD₃₂	1	2	yes	complemented
179	C₄×C₅	C₃×#121	1	1	yes	characteristic
180	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂×Q₈	—	2	1	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	4	1	no	—
184	C₂×D₄	—	4	1	no	—
185	Q₁₆	—	2	1	no	—
186	Q₁₆	C₃×C₅×D₄	1	2	yes	complemented
187	D₈	—	2	2	no	—
188	D₈	—	2	2	no	—
189	D₈	—	4	2	no	—
190	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
191	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₄²	—	4	1	no	—
195	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
196	C₁₆	—	2	2	no	—
197	C₃×C₅	#2207	1	1	yes	characteristic, direct product factor, Hall
198	C₂²×C₃	—	8	2	no	—
199	C₂²×C₃	—	8	2	no	—
200	C₂²×C₃	—	8	2	no	—
201	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	4	2	no	—
204	C₃×C₄	—	8	1	no	—
205	C₃×C₄	C₅×SD₃₂	1	2	yes	complemented
206	C₃×C₄	C₅×#121	1	1	yes	characteristic
207	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₂×C₅	—	16	2	no	—
209	C₂×C₅	C₃×#459	1	1	yes	characteristic
210	C₂×C₅	C₂×C₃×SD₃₂	1	1	yes	characteristic
211	C₂×C₅	C₃×#509	1	1	yes	characteristic
212	C₂³	—	4	2	no	—
213	Q₈	—	2	2	no	—
214	Q₈	—	4	1	no	—
215	D₄	—	4	2	no	—
216	D₄	—	4	2	no	—
217	D₄	—	8	2	no	—
218	D₄	—	8	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
222	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
223	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
224	C₈	—	2	1	no	—
225	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
226	C₈	C₃×C₅×#41	1	1	yes	characteristic
227	C₂×C₃	—	16	2	no	—
228	C₂×C₃	C₅×#459	1	1	yes	characteristic
229	C₂×C₃	C₂×C₅×SD₃₂	1	1	yes	characteristic
230	C₂×C₃	C₅×#509	1	1	yes	characteristic
231	C₅	C₃×#2207	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²	—	8	2	no	—
233	C₂²	—	8	2	no	—
234	C₂²	—	8	2	no	—
235	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	4	2	no	—
238	C₄	—	8	1	no	—
239	C₄	C₃×C₅×SD₃₂	1	2	yes	complemented
240	C₄	C₃×C₅×#121	1	1	yes	characteristic
241	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₃	C₅×#2207	1	1	yes	characteristic, direct product factor, Sylow
243	C₂	—	16	2	no	—
244	C₂	C₃×C₅×#459	1	1	yes	characteristic
245	C₂	C₂×C₃×C₅×SD₃₂	1	1	yes	characteristic
246	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
prime radical	C₃×C₅×#493	C₂
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×SD₃₂	C₂	semidirect
3. C₃×C₄×C₅×Q₁₆	C₂	semidirect
4. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
5. C₃×C₅×#357	C₂	semidirect
6. C₅×#2207	C₃	direct (N × H)
17. C₃×#2207	C₅	direct (N × H)
18. C₂×C₅×SD₃₂	C₂×C₃	semidirect
20. C₄×C₅×Q₁₆	C₂×C₃	semidirect
21. C₅×C₄⋊C₁₆	C₂×C₃	semidirect
22. C₅×#357	C₂×C₃	semidirect
28. C₃×C₅×Q₁₆	D₄	semidirect
39. C₂×C₃×SD₃₂	C₂×C₅	semidirect
41. C₃×C₄×Q₁₆	C₂×C₅	semidirect
42. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
43. C₃×#357	C₂×C₅	semidirect
54. #2207	C₃×C₅	direct (N × H)
85. C₅×Q₁₆	C₃×D₄	semidirect
96. C₂×SD₃₂	C₂×C₃×C₅	semidirect
98. C₄×Q₁₆	C₂×C₃×C₅	semidirect
99. C₄⋊C₁₆	C₂×C₃×C₅	semidirect
100. #357	C₂×C₃×C₅	semidirect
108. C₃×C₄×C₅	SD₃₂	semidirect
116. C₃×Q₁₆	C₅×D₄	semidirect
178. C₄×C₅	C₃×SD₃₂	semidirect
186. Q₁₆	C₃×C₅×D₄	semidirect
197. C₃×C₅	#2207	direct (N × H)
205. C₃×C₄	C₅×SD₃₂	semidirect
231. C₅	C₃×#2207	direct (N × H)
239. C₄	C₃×C₅×SD₃₂	semidirect
242. C₃	C₅×#2207	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×SD₃₂	2
C₃×C₅×#493	2
C₃×C₄×C₅×Q₁₆	2
C₃×C₅×C₄⋊C₁₆	2
C₃×C₅×#357	2
C₅×#2207	3
C₃×#2207	5

Subgroups — C3×C5×#2207 #14275241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2207 #14275241