Subgroups — C₃×C₅×#2551 #14275585

Order 1920.

1112 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	4	yes	maximal
2	C₃×C₅×#395	C₂	1	4	yes	maximal
3	C₃×C₅×#389	C₂	1	4	yes	maximal
4	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic
5	C₃×C₄×C₅×C₄⋊C₄	C₂	1	2	yes	maximal
6	C₅×#2551	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	4	yes	—
8	C₃×C₅×C₄⋊Q₈	—	2	16	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	8	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	4	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₃×C₅×#95	C₂²	1	4	yes	—
17	C₃×C₅×#95	C₂²	1	4	yes	—
18	C₃×#2551	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	4	yes	—
20	C₅×#395	C₂×C₃	1	4	yes	—
21	C₅×#389	C₂×C₃	1	4	yes	—
22	C₅×#384	C₂×C₃	1	1	yes	characteristic
23	C₄×C₅×C₄⋊C₄	C₂×C₃	1	2	yes	—
24	C₂×C₃×C₅×Q₈	—	2	16	no	—
25	C₂×C₃×C₅×Q₈	—	4	4	no	—
26	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
31	C₃×C₅×C₄⋊C₄	—	2	16	no	—
32	C₃×C₅×C₄⋊C₄	—	2	8	no	—
33	C₃×C₅×C₄⋊C₄	—	4	2	no	—
34	C₃×C₅×C₄⋊C₄	D₄	1	8	yes	—
35	C₃×C₅×C₄⋊C₄	Q₈	1	8	yes	complemented
36	C₃×C₄²×C₅	—	2	8	no	—
37	C₃×C₄²×C₅	Q₈	1	4	yes	complemented
38	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	4	yes	—
39	C₃×#395	C₂×C₅	1	4	yes	—
40	C₃×#389	C₂×C₅	1	4	yes	—
41	C₃×#384	C₂×C₅	1	1	yes	characteristic
42	C₃×C₄×C₄⋊C₄	C₂×C₅	1	2	yes	—
43	C₂²×C₅×Q₈	C₂²×C₃	1	4	yes	—
44	C₅×C₄⋊Q₈	—	2	16	no	—
45	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
46	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	8	yes	—
47	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
50	C₂×C₄²×C₅	C₂²×C₃	1	4	yes	—
51	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
52	C₅×#95	C₂²×C₃	1	4	yes	—
53	C₅×#95	C₂²×C₃	1	4	yes	—
54	#2551	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
56	C₃×C₅×Q₈	—	4	16	no	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	2	4	no	—
61	C₂×C₃×C₄×C₅	—	2	1	no	—
62	C₂×C₃×C₄×C₅	—	2	4	no	—
63	C₂×C₃×C₄×C₅	—	2	8	no	—
64	C₂×C₃×C₄×C₅	—	2	4	no	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	8	yes	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
67	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂×Q₈	1	8	yes	—
69	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
70	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
71	C₂²×C₃×Q₈	C₂²×C₅	1	4	yes	—
72	C₃×C₄⋊Q₈	—	2	16	no	—
73	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
74	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	8	yes	—
75	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
76	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
77	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
78	C₂×C₃×C₄²	C₂²×C₅	1	4	yes	—
79	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
80	C₃×#95	C₂²×C₅	1	4	yes	—
81	C₃×#95	C₂²×C₅	1	4	yes	—
82	C₂×C₅×Q₈	—	2	16	no	—
83	C₂×C₅×Q₈	—	4	4	no	—
84	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
85	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
86	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
87	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
89	C₅×C₄⋊C₄	—	2	16	no	—
90	C₅×C₄⋊C₄	—	2	8	no	—
91	C₅×C₄⋊C₄	—	4	2	no	—
92	C₅×C₄⋊C₄	C₃×D₄	1	8	yes	—
93	C₅×C₄⋊C₄	C₃×Q₈	1	8	yes	complemented
94	C₄²×C₅	—	2	8	no	—
95	C₄²×C₅	C₃×Q₈	1	4	yes	complemented
96	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	4	yes	—
97	#395	C₂×C₃×C₅	1	4	yes	—
98	#389	C₂×C₃×C₅	1	4	yes	—
99	#384	C₂×C₃×C₅	1	1	yes	characteristic
100	C₄×C₄⋊C₄	C₂×C₃×C₅	1	2	yes	—
101	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
102	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₂²×Q₈	1	2	yes	—
104	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
105	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	8	no	—
108	C₃×C₄×C₅	—	2	8	no	—
109	C₃×C₄×C₅	—	4	4	no	—
110	C₃×C₄×C₅	C₄⋊Q₈	1	8	yes	complemented
111	C₂×C₃×Q₈	—	2	16	no	—
112	C₂×C₃×Q₈	—	4	4	no	—
113	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
114	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
115	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
117	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
118	C₃×C₄⋊C₄	—	2	16	no	—
119	C₃×C₄⋊C₄	—	2	8	no	—
120	C₃×C₄⋊C₄	—	4	2	no	—
121	C₃×C₄⋊C₄	C₅×D₄	1	8	yes	—
122	C₃×C₄⋊C₄	C₅×Q₈	1	8	yes	complemented
123	C₃×C₄²	—	2	8	no	—
124	C₃×C₄²	C₅×Q₈	1	4	yes	complemented
125	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
126	C₅×Q₈	—	4	16	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	8	no	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	C₂×C₃×D₄	1	8	yes	—
136	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
137	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
138	C₂×C₄×C₅	C₂×C₃×Q₈	1	8	yes	—
139	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
140	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
141	C₂²×Q₈	C₂²×C₃×C₅	1	4	yes	—
142	C₄⋊Q₈	—	2	16	no	—
143	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
144	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	8	yes	—
145	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
147	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
148	C₂×C₄²	C₂²×C₃×C₅	1	4	yes	—
149	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
150	#95	C₂²×C₃×C₅	1	4	yes	—
151	#95	C₂²×C₃×C₅	1	4	yes	—
152	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	2	yes	—
153	C₂×C₃×C₅	#544	1	1	yes	characteristic
154	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
155	C₂×C₃×C₅	D₄×Q₈	1	2	yes	—
156	C₂×C₃×C₅	Q₈²	1	1	yes	characteristic
157	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
158	C₃×Q₈	—	4	16	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	2	4	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	4	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	8	no	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	C₂×C₅×D₄	1	8	yes	—
168	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
169	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
170	C₂×C₃×C₄	C₂×C₅×Q₈	1	8	yes	—
171	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
172	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
173	C₂²×C₅	C₂²×C₃×D₄	1	2	yes	—
174	C₂²×C₅	C₂²×C₃×Q₈	1	2	yes	—
175	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
176	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
177	C₂²×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	—	2	8	no	—
180	C₄×C₅	—	2	8	no	—
181	C₄×C₅	—	4	4	no	—
182	C₄×C₅	C₃×C₄⋊Q₈	1	8	yes	complemented
183	C₂×Q₈	—	2	16	no	—
184	C₂×Q₈	—	4	4	no	—
185	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
186	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
187	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
189	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
190	C₄⋊C₄	—	2	16	no	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	C₃×C₅×D₄	1	8	yes	—
194	C₄⋊C₄	C₃×C₅×Q₈	1	8	yes	complemented
195	C₄²	—	2	8	no	—
196	C₄²	C₃×C₅×Q₈	1	4	yes	complemented
197	C₃×C₅	#2551	1	1	yes	characteristic, direct product factor, Hall
198	C₂²×C₃	C₂²×C₅×D₄	1	2	yes	—
199	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
200	C₂²×C₃	C₂²×C₅×Q₈	1	2	yes	—
201	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
202	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	8	no	—
205	C₃×C₄	—	2	8	no	—
206	C₃×C₄	—	4	4	no	—
207	C₃×C₄	C₅×C₄⋊Q₈	1	8	yes	complemented
208	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	2	yes	—
209	C₂×C₅	C₃×#544	1	1	yes	characteristic
210	C₂×C₅	C₃×D₄²	1	1	yes	characteristic
211	C₂×C₅	C₃×D₄×Q₈	1	2	yes	—
212	C₂×C₅	C₃×Q₈²	1	1	yes	characteristic
213	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
214	Q₈	—	4	16	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	C₂×C₃×C₅×D₄	1	8	yes	—
224	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
225	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
226	C₂×C₄	C₂×C₃×C₅×Q₈	1	8	yes	—
227	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
228	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
229	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	2	yes	—
230	C₂×C₃	C₅×#544	1	1	yes	characteristic
231	C₂×C₃	C₅×D₄²	1	1	yes	characteristic
232	C₂×C₃	C₅×D₄×Q₈	1	2	yes	—
233	C₂×C₃	C₅×Q₈²	1	1	yes	characteristic
234	C₅	C₃×#2551	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	C₂²×C₃×C₅×D₄	1	2	yes	—
236	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
237	C₂²	C₂²×C₃×C₅×Q₈	1	2	yes	—
238	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
239	C₂²	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	8	no	—
242	C₄	—	2	8	no	—
243	C₄	—	4	4	no	—
244	C₄	C₃×C₅×C₄⋊Q₈	1	8	yes	complemented
245	C₃	C₅×#2551	1	1	yes	characteristic, direct product factor, Sylow
246	C₂	C₂×C₃×C₅×C₄⋊Q₈	1	2	yes	—
247	C₂	C₃×C₅×#544	1	1	yes	characteristic
248	C₂	C₃×C₅×D₄²	1	1	yes	characteristic
249	C₂	C₃×C₅×D₄×Q₈	1	2	yes	—
250	C₂	C₃×C₅×Q₈²	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#2551	C₃	direct (N × H)
18. C₃×#2551	C₅	direct (N × H)
35. C₃×C₅×C₄⋊C₄	Q₈	semidirect
37. C₃×C₄²×C₅	Q₈	semidirect
54. #2551	C₃×C₅	direct (N × H)
93. C₅×C₄⋊C₄	C₃×Q₈	semidirect
95. C₄²×C₅	C₃×Q₈	semidirect
110. C₃×C₄×C₅	C₄⋊Q₈	semidirect
122. C₃×C₄⋊C₄	C₅×Q₈	semidirect
124. C₃×C₄²	C₅×Q₈	semidirect
182. C₄×C₅	C₃×C₄⋊Q₈	semidirect
194. C₄⋊C₄	C₃×C₅×Q₈	semidirect
196. C₄²	C₃×C₅×Q₈	semidirect
197. C₃×C₅	#2551	direct (N × H)
207. C₃×C₄	C₅×C₄⋊Q₈	semidirect
234. C₅	C₃×#2551	direct (N × H)
244. C₄	C₃×C₅×C₄⋊Q₈	semidirect
245. C₃	C₅×#2551	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×C₄⋊Q₈	2
C₃×C₅×#395	2
C₃×C₅×#389	2
C₃×C₅×#384	2
C₃×C₄×C₅×C₄⋊C₄	2
C₅×#2551	3
C₃×#2551	5

Subgroups — C3×C5×#2551 #14275585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2551 #14275585