Subgroups — C₃×C₅×#2208 #14275242

Order 1920.

332 conjugacy classes of subgroups.

The subgroup lattice has 244 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
2	C₃×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#358	C₂	1	2	yes	maximal, complemented
6	C₅×#2208	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
8	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
11	C₃×C₄×C₅×D₄	—	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×#102	—	2	1	no	—
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×#2208	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×SD₃₂	C₂×C₃	1	2	yes	—
19	C₅×#494	C₂×C₃	1	1	yes	characteristic, complemented
20	C₄×C₅×D₈	C₂×C₃	1	1	yes	characteristic
21	C₅×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
22	C₅×#358	C₂×C₃	1	2	yes	complemented
23	C₂×C₃×C₅×Q₈	—	2	2	no	—
24	C₂×C₃×C₅×D₄	—	2	1	no	—
25	C₂²×C₃×C₄×C₅	—	4	1	no	—
26	C₃×C₅×Q₁₆	—	2	2	no	—
27	C₃×C₅×Q₁₆	—	2	2	no	—
28	C₃×C₅×Q₁₆	—	4	2	no	—
29	C₃×C₅×D₈	—	2	1	no	—
30	C₃×C₅×D₈	D₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₃×C₅×C₄⋊C₄	—	2	1	no	—
34	C₃×C₅×C₄⋊C₄	—	4	1	no	—
35	C₃×C₅×C₄⋊C₄	—	4	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×SD₃₂	C₂×C₅	1	2	yes	—
40	C₃×#494	C₂×C₅	1	1	yes	characteristic, complemented
41	C₃×C₄×D₈	C₂×C₅	1	1	yes	characteristic
42	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×#358	C₂×C₅	1	2	yes	complemented
44	C₂×C₅×Q₁₆	—	2	1	no	—
45	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
46	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
47	C₅×C₄⋊Q₈	—	2	1	no	—
48	C₄×C₅×D₄	—	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×#102	—	2	1	no	—
53	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
54	#2208	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₅	—	4	1	no	—
56	C₃×C₅×Q₈	—	4	2	no	—
57	C₃×C₅×Q₈	—	4	2	no	—
58	C₃×C₅×D₄	—	2	2	no	—
59	C₃×C₅×D₄	—	4	1	no	—
60	C₂×C₃×C₄×C₅	—	4	1	no	—
61	C₂×C₃×C₄×C₅	—	4	2	no	—
62	C₂×C₃×C₄×C₅	—	8	2	no	—
63	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₃×C₅×C₈	—	2	1	no	—
67	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
68	C₃×C₅×C₈	#41	1	1	yes	characteristic
69	C₂×C₃×Q₁₆	—	2	1	no	—
70	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
71	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
72	C₃×C₄⋊Q₈	—	2	1	no	—
73	C₃×C₄×D₄	—	2	1	no	—
74	C₃×SD₃₂	—	2	4	no	—
75	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
76	C₃×#107	C₂²×C₅	1	1	yes	characteristic
77	C₃×#102	—	2	1	no	—
78	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₅×Q₈	—	2	2	no	—
80	C₂×C₅×D₄	—	2	1	no	—
81	C₂²×C₄×C₅	—	4	1	no	—
82	C₅×Q₁₆	—	2	2	no	—
83	C₅×Q₁₆	—	2	2	no	—
84	C₅×Q₁₆	—	4	2	no	—
85	C₅×D₈	—	2	1	no	—
86	C₅×D₈	C₃×D₄	1	2	yes	—
87	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
89	C₅×C₄⋊C₄	—	2	1	no	—
90	C₅×C₄⋊C₄	—	4	1	no	—
91	C₅×C₄⋊C₄	—	4	1	no	—
92	C₅×C₂²⋊C₄	—	4	1	no	—
93	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
94	C₅×C₁₆	—	2	2	no	—
95	C₂×SD₃₂	C₂×C₃×C₅	1	2	yes	—
96	#494	C₂×C₃×C₅	1	1	yes	characteristic, complemented
97	C₄×D₈	C₂×C₃×C₅	1	1	yes	characteristic
98	C₄⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
99	#358	C₂×C₃×C₅	1	2	yes	complemented
100	C₂²×C₃×C₅	—	4	2	no	—
101	C₂²×C₃×C₅	—	8	1	no	—
102	C₂²×C₃×C₅	—	8	1	no	—
103	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
104	C₃×C₄×C₅	—	2	1	no	—
105	C₃×C₄×C₅	—	8	2	no	—
106	C₃×C₄×C₅	—	8	1	no	—
107	C₃×C₄×C₅	SD₃₂	1	2	yes	complemented
108	C₃×C₄×C₅	#121	1	1	yes	characteristic
109	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₃×Q₈	—	2	2	no	—
111	C₂×C₃×D₄	—	2	1	no	—
112	C₂²×C₃×C₄	—	4	1	no	—
113	C₃×Q₁₆	—	2	2	no	—
114	C₃×Q₁₆	—	2	2	no	—
115	C₃×Q₁₆	—	4	2	no	—
116	C₃×D₈	—	2	1	no	—
117	C₃×D₈	C₅×D₄	1	2	yes	—
118	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
120	C₃×C₄⋊C₄	—	2	1	no	—
121	C₃×C₄⋊C₄	—	4	1	no	—
122	C₃×C₄⋊C₄	—	4	1	no	—
123	C₃×C₂²⋊C₄	—	4	1	no	—
124	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
125	C₃×C₁₆	—	2	2	no	—
126	C₂³×C₅	—	4	1	no	—
127	C₅×Q₈	—	4	2	no	—
128	C₅×Q₈	—	4	2	no	—
129	C₅×D₄	—	2	2	no	—
130	C₅×D₄	—	4	1	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	8	2	no	—
134	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
137	C₅×C₈	—	2	1	no	—
138	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
139	C₅×C₈	C₃×#41	1	1	yes	characteristic
140	C₂×Q₁₆	—	2	1	no	—
141	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
142	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	1	no	—
144	C₄×D₄	—	2	1	no	—
145	SD₃₂	—	2	4	no	—
146	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
147	#107	C₂²×C₃×C₅	1	1	yes	characteristic
148	#102	—	2	1	no	—
149	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
150	C₂×C₃×C₅	—	8	2	no	—
151	C₂×C₃×C₅	#459	1	1	yes	characteristic
152	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
153	C₂×C₃×C₅	#510	1	1	yes	characteristic
154	C₂³×C₃	—	4	1	no	—
155	C₃×Q₈	—	4	2	no	—
156	C₃×Q₈	—	4	2	no	—
157	C₃×D₄	—	2	2	no	—
158	C₃×D₄	—	4	1	no	—
159	C₂×C₃×C₄	—	4	1	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	—	8	2	no	—
162	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
163	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
165	C₃×C₈	—	2	1	no	—
166	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
167	C₃×C₈	C₅×#41	1	1	yes	characteristic
168	C₂²×C₅	—	4	2	no	—
169	C₂²×C₅	—	8	1	no	—
170	C₂²×C₅	—	8	1	no	—
171	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
172	C₄×C₅	—	2	1	no	—
173	C₄×C₅	—	8	1	no	—
174	C₄×C₅	—	8	2	no	—
175	C₄×C₅	C₃×SD₃₂	1	2	yes	complemented
176	C₄×C₅	C₃×#121	1	1	yes	characteristic
177	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₂×Q₈	—	2	2	no	—
179	C₂×D₄	—	2	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	Q₁₆	—	2	2	no	—
182	Q₁₆	—	2	2	no	—
183	Q₁₆	—	4	2	no	—
184	D₈	—	2	1	no	—
185	D₈	C₃×C₅×D₄	1	2	yes	—
186	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
187	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	4	1	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₂²⋊C₄	—	4	1	no	—
192	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
193	C₁₆	—	2	2	no	—
194	C₃×C₅	#2208	1	1	yes	characteristic, direct product factor, Hall
195	C₂²×C₃	—	4	2	no	—
196	C₂²×C₃	—	8	1	no	—
197	C₂²×C₃	—	8	1	no	—
198	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	8	2	no	—
201	C₃×C₄	—	8	1	no	—
202	C₃×C₄	C₅×SD₃₂	1	2	yes	complemented
203	C₃×C₄	C₅×#121	1	1	yes	characteristic
204	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₂×C₅	—	8	2	no	—
206	C₂×C₅	C₃×#459	1	1	yes	characteristic
207	C₂×C₅	C₂×C₃×SD₃₂	1	1	yes	characteristic
208	C₂×C₅	C₃×#510	1	1	yes	characteristic
209	C₂³	—	4	1	no	—
210	Q₈	—	4	2	no	—
211	Q₈	—	4	2	no	—
212	D₄	—	2	2	no	—
213	D₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
218	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
219	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
222	C₈	C₃×C₅×#41	1	1	yes	characteristic
223	C₂×C₃	—	8	2	no	—
224	C₂×C₃	C₅×#459	1	1	yes	characteristic
225	C₂×C₃	C₂×C₅×SD₃₂	1	1	yes	characteristic
226	C₂×C₃	C₅×#510	1	1	yes	characteristic
227	C₅	C₃×#2208	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²	—	4	2	no	—
229	C₂²	—	8	1	no	—
230	C₂²	—	8	1	no	—
231	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	8	2	no	—
234	C₄	—	8	1	no	—
235	C₄	C₃×C₅×SD₃₂	1	2	yes	complemented
236	C₄	C₃×C₅×#121	1	1	yes	characteristic
237	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
238	C₃	C₅×#2208	1	1	yes	characteristic, direct product factor, Sylow
239	C₂	—	8	2	no	—
240	C₂	C₃×C₅×#459	1	1	yes	characteristic
241	C₂	C₂×C₃×C₅×SD₃₂	1	1	yes	characteristic
242	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	C₂×C₃×C₅×D₈	C₂²
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
2. C₃×C₅×#494	C₂	semidirect
4. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
5. C₃×C₅×#358	C₂	semidirect
6. C₅×#2208	C₃	direct (N × H)
17. C₃×#2208	C₅	direct (N × H)
19. C₅×#494	C₂×C₃	semidirect
21. C₅×C₄⋊C₁₆	C₂×C₃	semidirect
22. C₅×#358	C₂×C₃	semidirect
40. C₃×#494	C₂×C₅	semidirect
42. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
43. C₃×#358	C₂×C₅	semidirect
54. #2208	C₃×C₅	direct (N × H)
96. #494	C₂×C₃×C₅	semidirect
98. C₄⋊C₁₆	C₂×C₃×C₅	semidirect
99. #358	C₂×C₃×C₅	semidirect
107. C₃×C₄×C₅	SD₃₂	semidirect
175. C₄×C₅	C₃×SD₃₂	semidirect
194. C₃×C₅	#2208	direct (N × H)
202. C₃×C₄	C₅×SD₃₂	semidirect
227. C₅	C₃×#2208	direct (N × H)
235. C₄	C₃×C₅×SD₃₂	semidirect
238. C₃	C₅×#2208	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×#494	2
C₃×C₄×C₅×D₈	2
C₃×C₅×C₄⋊C₁₆	2
C₃×C₅×#358	2
C₅×#2208	3
C₃×#2208	5

Subgroups — C3×C5×#2208 #14275242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2208 #14275242