Subgroups — C₅×#70939 #14291451

Order 1920.

376 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5027	C₂	1	2	yes	maximal, complemented
3	C₅×#4695	C₂	1	1	yes	maximal, characteristic
4	C₅×#4485	C₂	1	2	yes	maximal, complemented
5	C₅×#4458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1668	—	3	1	no	maximal, Hall
7	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#215	—	2	2	no	—
9	C₂×C₅×D₂₄	—	2	1	no	—
10	C₂×C₅×D₂₄	—	2	1	no	—
11	C₅×#874	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊C₈	—	2	1	no	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	—	2	1	no	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#809	—	2	2	no	—
17	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	#70939	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#493	—	3	1	no	—
20	C₅×#460	—	3	2	no	—
21	C₅×C₈×Q₈	D₃	1	1	yes	characteristic, complemented
22	C₅×#335	—	3	1	no	—
23	C₅×#331	—	3	2	no	—
24	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
25	C₂×C₅×D₁₂	—	2	2	no	—
26	C₂×C₅×D₁₂	—	2	2	no	—
27	C₂×C₅×D₁₂	—	2	2	no	—
28	C₂×C₃×C₅×C₈	—	2	1	no	—
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	—	2	1	no	—
32	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
33	C₃×C₄²×C₅	—	2	1	no	—
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₅×#215	—	4	4	no	—
36	C₂×C₅×C₃⋊C₈	—	2	2	no	—
37	C₅×D₂₄	—	4	2	no	—
38	C₅×D₂₄	—	4	2	no	—
39	C₃×C₈×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
40	#5027	C₂×C₅	1	2	yes	complemented
41	#4695	C₂×C₅	1	1	yes	characteristic
42	#4485	C₂×C₅	1	2	yes	complemented
43	#4458	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂×C₅×SD₁₆	—	6	2	no	—
45	C₂×C₅×D₈	—	6	1	no	—
46	C₂×C₅×D₈	—	6	1	no	—
47	C₅×#127	—	3	2	no	—
48	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
49	C₅×C₄⋊C₈	—	2	1	no	—
50	C₅×C₄⋊C₈	—	3	2	no	—
51	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
52	C₅×#102	—	6	2	no	—
53	C₄×C₅×C₈	—	2	1	no	—
54	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
55	#1668	—	3	1	no	Sylow
56	C₂²×C₅×D₃	—	4	2	no	—
57	C₃×C₅×Q₈	—	2	1	no	—
58	C₃×C₅×Q₈	D₈	1	2	yes	complemented
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	2	1	no	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
64	C₅×D₁₂	—	4	2	no	—
65	C₅×D₁₂	—	4	2	no	—
66	C₅×D₁₂	—	4	2	no	—
67	C₅×D₁₂	—	4	2	no	—
68	C₅×D₁₂	—	8	2	no	—
69	C₃×C₅×C₈	—	2	1	no	—
70	C₃×C₅×C₈	—	4	1	no	—
71	C₃×C₅×C₈	SD₁₆	1	2	yes	complemented
72	C₅×C₃⋊C₈	—	4	2	no	—
73	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
74	C₂×#215	—	2	2	no	—
75	C₂×D₂₄	—	2	1	no	—
76	C₂×D₂₄	—	2	1	no	—
77	#874	C₂²×C₅	1	2	yes	—
78	C₃×C₄⋊C₈	—	2	1	no	—
79	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
80	C₃×C₄×C₈	—	2	1	no	—
81	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
82	#809	—	2	2	no	—
83	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
84	C₂×C₅×Q₈	D₁₂	1	1	yes	characteristic
85	C₂×C₅×D₄	—	6	2	no	—
86	C₂×C₅×D₄	—	6	2	no	—
87	C₂×C₅×D₄	—	6	2	no	—
88	C₅×SD₁₆	—	12	4	no	—
89	C₅×D₈	—	12	2	no	—
90	C₅×D₈	—	12	2	no	—
91	C₂×C₅×C₈	—	2	1	no	—
92	C₂×C₅×C₈	—	6	2	no	—
93	C₂×C₅×C₈	#67	1	1	yes	characteristic
94	C₂×C₅×C₈	#67	1	1	yes	characteristic
95	C₅×C₄⋊C₄	—	2	1	no	—
96	C₅×C₄⋊C₄	D₁₂	1	1	yes	characteristic
97	C₄²×C₅	—	2	1	no	—
98	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
99	#493	—	3	1	no	—
100	#460	—	3	2	no	—
101	C₈×Q₈	C₅×D₃	1	1	yes	characteristic, complemented
102	#335	—	3	1	no	—
103	#331	—	3	2	no	—
104	C₂²×C₃×C₅	#121	1	1	yes	characteristic
105	C₂×C₅×D₃	—	8	2	no	—
106	C₂×C₅×D₃	—	8	2	no	—
107	C₂×C₅×D₃	—	8	2	no	—
108	C₃×C₄×C₅	—	2	2	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	4	1	no	—
111	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
112	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
113	C₃×C₄×C₅	#135	1	1	yes	characteristic
114	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
116	C₂×D₁₂	—	2	2	no	—
117	C₂×D₁₂	—	2	2	no	—
118	C₂×D₁₂	—	2	2	no	—
119	C₂×C₃×C₈	—	2	1	no	—
120	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
121	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
122	C₃×C₄⋊C₄	—	2	1	no	—
123	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
124	C₃×C₄²	—	2	1	no	—
125	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
126	#215	—	4	4	no	—
127	C₂×C₃⋊C₈	—	2	2	no	—
128	D₂₄	—	4	2	no	—
129	D₂₄	—	4	2	no	—
130	C₂³×C₅	—	12	2	no	—
131	C₅×Q₈	—	2	1	no	—
132	C₅×Q₈	D₂₄	1	2	yes	complemented
133	C₅×D₄	—	12	2	no	—
134	C₅×D₄	—	12	2	no	—
135	C₅×D₄	—	12	2	no	—
136	C₅×D₄	—	12	2	no	—
137	C₅×D₄	—	24	2	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
141	C₂×C₄×C₅	#235	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
143	C₅×C₈	—	2	1	no	—
144	C₅×C₈	—	4	1	no	—
145	C₅×C₈	—	12	2	no	—
146	C₅×C₈	#215	1	2	yes	complemented
147	C₂×SD₁₆	—	6	2	no	—
148	C₂×D₈	—	6	1	no	—
149	C₂×D₈	—	6	1	no	—
150	#127	—	3	2	no	—
151	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	—	3	2	no	—
154	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
155	#102	—	6	2	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
158	C₂×C₃×C₅	#460	1	1	yes	characteristic
159	C₂×C₃×C₅	#466	1	1	yes	characteristic
160	C₂×C₃×C₅	#472	1	1	yes	characteristic
161	C₅×D₃	—	16	2	no	—
162	C₂²×D₃	—	4	2	no	—
163	C₃×Q₈	—	2	1	no	—
164	C₃×Q₈	C₅×D₈	1	2	yes	complemented
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
170	D₁₂	—	4	2	no	—
171	D₁₂	—	4	2	no	—
172	D₁₂	—	4	2	no	—
173	D₁₂	—	4	2	no	—
174	D₁₂	—	8	2	no	—
175	C₃×C₈	—	2	1	no	—
176	C₃×C₈	—	4	1	no	—
177	C₃×C₈	C₅×SD₁₆	1	2	yes	complemented
178	C₃⋊C₈	—	4	2	no	—
179	C₂²×C₅	—	24	2	no	—
180	C₂²×C₅	—	24	2	no	—
181	C₂²×C₅	—	24	2	no	—
182	C₂²×C₅	#930	1	1	yes	characteristic
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	4	1	no	—
186	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
187	C₄×C₅	#904	1	1	yes	characteristic
188	C₄×C₅	C₂×#215	1	1	yes	characteristic
189	C₄×C₅	#949	1	1	yes	characteristic
190	C₂×Q₈	C₅×D₁₂	1	1	yes	characteristic
191	C₂×D₄	—	6	2	no	—
192	C₂×D₄	—	6	2	no	—
193	C₂×D₄	—	6	2	no	—
194	SD₁₆	—	12	4	no	—
195	D₈	—	12	2	no	—
196	D₈	—	12	2	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	6	2	no	—
199	C₂×C₈	C₅×#67	1	1	yes	characteristic
200	C₂×C₈	C₅×#67	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	C₅×D₁₂	1	1	yes	characteristic
203	C₄²	—	2	1	no	—
204	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
205	C₃×C₅	#1668	1	1	yes	characteristic, complemented, Hall
206	C₂²×C₃	C₅×#121	1	1	yes	characteristic
207	C₂×D₃	—	8	2	no	—
208	C₂×D₃	—	8	2	no	—
209	C₂×D₃	—	8	2	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	4	1	no	—
213	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
214	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
215	C₃×C₄	C₅×#135	1	1	yes	characteristic
216	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂×C₅	—	48	2	no	—
218	C₂×C₅	#5027	1	1	yes	characteristic
219	C₂×C₅	#5115	1	1	yes	characteristic
220	C₂×C₅	#5142	1	1	yes	characteristic
221	C₂³	—	12	2	no	—
222	Q₈	—	2	1	no	—
223	Q₈	C₅×D₂₄	1	2	yes	complemented
224	D₄	—	12	2	no	—
225	D₄	—	12	2	no	—
226	D₄	—	12	2	no	—
227	D₄	—	12	2	no	—
228	D₄	—	24	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
232	C₂×C₄	C₅×#235	1	1	yes	characteristic
233	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	4	1	no	—
236	C₈	—	12	2	no	—
237	C₈	C₅×#215	1	2	yes	complemented
238	C₂×C₃	C₅×#460	1	1	yes	characteristic
239	C₂×C₃	C₅×#466	1	1	yes	characteristic
240	C₂×C₃	C₅×#472	1	1	yes	characteristic
241	D₃	—	16	2	no	—
242	C₅	#70939	1	1	yes	characteristic, direct product factor, Sylow
243	C₂²	—	24	2	no	—
244	C₂²	—	24	2	no	—
245	C₂²	—	24	2	no	—
246	C₂²	C₅×#930	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	1	no	—
250	C₄	C₂×C₅×D₂₄	1	1	yes	characteristic
251	C₄	C₅×#904	1	1	yes	characteristic
252	C₄	C₂×C₅×#215	1	1	yes	characteristic
253	C₄	C₅×#949	1	1	yes	characteristic
254	C₃	C₅×#1668	1	1	yes	characteristic, complemented, Sylow
255	C₂	—	48	2	no	—
256	C₂	C₅×#5027	1	1	yes	characteristic
257	C₂	C₅×#5115	1	1	yes	characteristic
258	C₂	C₅×#5142	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#930
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×C₈×Q₈	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₅×#4695	C₂
hypercenter	C₅×C₈×Q₈	D₃
hypocenter	C₃	C₅×#1668
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₈×Q₈	C₂	semidirect
2. C₅×#5027	C₂	semidirect
4. C₅×#4485	C₂	semidirect
5. C₅×#4458	C₂	semidirect
18. #70939	C₅	direct (N × H)
21. C₅×C₈×Q₈	D₃	semidirect
39. C₃×C₈×Q₈	C₂×C₅	semidirect
40. #5027	C₂×C₅	semidirect
42. #4485	C₂×C₅	semidirect
43. #4458	C₂×C₅	semidirect
58. C₃×C₅×Q₈	D₈	semidirect
71. C₃×C₅×C₈	SD₁₆	semidirect
101. C₈×Q₈	C₅×D₃	semidirect
132. C₅×Q₈	D₂₄	semidirect
146. C₅×C₈	#215	semidirect
164. C₃×Q₈	C₅×D₈	semidirect
177. C₃×C₈	C₅×SD₁₆	semidirect
205. C₃×C₅	#1668	semidirect
223. Q₈	C₅×D₂₄	semidirect
237. C₈	C₅×#215	semidirect
242. C₅	#70939	direct (N × H)
254. C₃	C₅×#1668	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₈×Q₈	2
C₅×#5027	2
C₅×#4695	2
C₅×#4485	2
C₅×#4458	2
C₅×#1668	3
#70939	5

Subgroups — C5×#70939 #14291451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70939 #14291451