Subgroups — C₅×#71431 #14312823

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×(C₂×D₃)⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₅×#2112	—	3	1	no	maximal, Hall
6	C₂²×C₄×C₅×D₃	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₅×D₃×C₈	—	2	2	no	—
10	C₂×C₅×D₃×C₈	—	2	2	no	—
11	C₂×C₅×D₃×C₈	C₄	1	2	yes	—
12	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
13	C₃×C₅×M₃₂	—	2	2	no	—
14	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	complemented
15	C₅×#812	—	2	2	no	—
16	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	complemented
17	#71431	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂³×C₅×C₈	—	3	1	no	—
19	C₂×C₅×M₃₂	—	3	1	no	—
20	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic, complemented
21	C₅×C₂²⋊C₁₆	—	3	4	no	—
22	C₂³×C₅×D₃	C₈	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×Dic₃	C₈	1	1	yes	characteristic
25	C₂×C₄×C₅×D₃	—	2	2	no	—
26	C₂×C₄×C₅×D₃	—	2	2	no	—
27	C₂×C₄×C₅×D₃	C₈	1	2	yes	—
28	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₅×C₃⋊C₈	—	2	2	no	—
34	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₅×D₃×C₈	—	2	4	no	—
36	C₅×D₃×C₈	—	2	2	no	—
37	C₅×D₃×C₈	—	2	2	no	—
38	C₃×C₅×C₁₆	—	2	2	no	—
39	C₅×C₃⋊C₁₆	—	2	2	no	—
40	C₂²×D₃×C₈	C₂×C₅	1	1	yes	characteristic
41	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂×#812	C₂×C₅	1	1	yes	characteristic, complemented
43	(C₂×D₃)⋊C₁₆	C₂×C₅	1	4	yes	complemented
44	C₂³×C₄×C₅	—	3	1	no	—
45	C₂²×C₅×C₈	—	3	1	no	—
46	C₂²×C₅×C₈	—	3	2	no	—
47	C₂²×C₅×C₈	—	3	2	no	—
48	C₂²×C₅×C₈	—	6	2	no	—
49	C₂²×C₅×C₈	—	6	2	no	—
50	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
51	C₅×M₃₂	—	2	2	no	—
52	C₅×M₃₂	—	6	2	no	—
53	C₂×C₅×C₁₆	—	3	2	no	—
54	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	complemented
55	#2112	—	3	1	no	Sylow
56	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₅×D₃	—	2	2	no	—
58	C₂²×C₅×D₃	—	2	4	no	—
59	C₂²×C₅×D₃	C₂×C₈	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₅×Dic₃	—	2	2	no	—
65	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
66	C₄×C₅×D₃	—	2	4	no	—
67	C₄×C₅×D₃	—	2	2	no	—
68	C₄×C₅×D₃	—	2	2	no	—
69	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₅×C₃⋊C₈	—	2	2	no	—
73	C₂²×C₄×D₃	C₄×C₅	1	1	yes	characteristic
74	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
75	C₂²×C₃⋊C₈	C₄×C₅	1	1	yes	characteristic
76	C₂×D₃×C₈	—	2	2	no	—
77	C₂×D₃×C₈	—	2	2	no	—
78	C₂×D₃×C₈	C₄×C₅	1	2	yes	—
79	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
80	C₃×M₃₂	—	2	2	no	—
81	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	complemented
82	#812	—	2	2	no	—
83	C₂×C₃⋊C₁₆	C₂²×C₅	1	2	yes	complemented
84	C₂⁴×C₅	—	3	1	no	—
85	C₂²×C₄×C₅	—	3	1	no	—
86	C₂²×C₄×C₅	—	3	2	no	—
87	C₂²×C₄×C₅	—	3	2	no	—
88	C₂²×C₄×C₅	—	6	2	no	—
89	C₂²×C₄×C₅	—	6	2	no	—
90	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
91	C₂×C₅×C₈	—	3	2	no	—
92	C₂×C₅×C₈	—	6	2	no	—
93	C₂×C₅×C₈	—	6	4	no	—
94	C₂×C₅×C₈	—	6	2	no	—
95	C₂×C₅×C₈	—	6	2	no	—
96	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
97	C₂×C₅×C₈	D₁₂	1	2	yes	—
98	C₂×C₅×C₈	#67	1	2	yes	—
99	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
100	C₅×C₁₆	—	2	2	no	—
101	C₅×C₁₆	—	6	2	no	—
102	C₂³×C₈	—	3	1	no	—
103	C₂×M₃₂	—	3	1	no	—
104	C₂×M₃₂	C₅×D₃	1	1	yes	characteristic, complemented
105	C₂²⋊C₁₆	—	3	4	no	—
106	C₂²×C₃×C₅	—	2	2	no	—
107	C₂²×C₃×C₅	C₂²⋊C₈	1	2	yes	—
108	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₅×D₃	—	2	8	no	—
110	C₂×C₅×D₃	—	2	2	no	—
111	C₂×C₅×D₃	—	2	2	no	—
112	C₂×C₅×D₃	M₃₂	1	4	yes	complemented
113	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
114	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
116	C₅×Dic₃	—	2	2	no	—
117	C₂³×D₃	C₅×C₈	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
119	C₂²×Dic₃	C₅×C₈	1	1	yes	characteristic
120	C₂×C₄×D₃	—	2	2	no	—
121	C₂×C₄×D₃	—	2	2	no	—
122	C₂×C₄×D₃	C₅×C₈	1	2	yes	—
123	C₂×C₄×D₃	C₂×C₄×C₅	1	2	yes	—
124	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
126	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
127	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
128	C₂×C₃⋊C₈	—	2	2	no	—
129	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
130	D₃×C₈	—	2	4	no	—
131	D₃×C₈	—	2	2	no	—
132	D₃×C₈	—	2	2	no	—
133	C₃×C₁₆	—	2	2	no	—
134	C₃⋊C₁₆	—	2	2	no	—
135	C₂³×C₅	—	3	2	no	—
136	C₂³×C₅	—	6	4	no	—
137	C₂³×C₅	—	6	2	no	—
138	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
139	C₂×C₄×C₅	—	3	2	no	—
140	C₂×C₄×C₅	—	6	2	no	—
141	C₂×C₄×C₅	—	6	4	no	—
142	C₂×C₄×C₅	—	6	2	no	—
143	C₂×C₄×C₅	—	6	2	no	—
144	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
145	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	2	yes	—
146	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
147	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
148	C₅×C₈	—	6	2	no	—
149	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
150	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
151	C₅×C₈	C₂×#67	1	1	yes	characteristic
152	C₂³×C₄	—	3	1	no	—
153	C₂²×C₈	—	3	1	no	—
154	C₂²×C₈	—	3	2	no	—
155	C₂²×C₈	—	3	2	no	—
156	C₂²×C₈	—	6	2	no	—
157	C₂²×C₈	—	6	2	no	—
158	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
159	M₃₂	—	2	2	no	—
160	M₃₂	—	6	2	no	—
161	C₂×C₁₆	—	3	2	no	—
162	C₂×C₁₆	C₂×C₅×D₃	1	2	yes	complemented
163	C₂×C₃×C₅	—	2	2	no	—
164	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
165	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	—
166	C₅×D₃	—	2	4	no	—
167	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
168	C₂²×D₃	—	2	2	no	—
169	C₂²×D₃	—	2	4	no	—
170	C₂²×D₃	C₂×C₅×C₈	1	2	yes	—
171	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
172	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
173	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
174	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
175	C₂×Dic₃	—	2	2	no	—
176	C₂×Dic₃	C₂×C₅×C₈	1	2	yes	—
177	C₄×D₃	—	2	2	no	—
178	C₄×D₃	—	2	2	no	—
179	C₄×D₃	—	2	4	no	—
180	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
181	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
182	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
183	C₃⋊C₈	—	2	2	no	—
184	C₂²×C₅	—	2	2	no	—
185	C₂²×C₅	—	3	4	no	—
186	C₂²×C₅	—	6	8	no	—
187	C₂²×C₅	—	6	2	no	—
188	C₂²×C₅	—	6	2	no	—
189	C₂²×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
190	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
191	C₄×C₅	—	6	2	no	—
192	C₄×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
193	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
194	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
195	C₂⁴	—	3	1	no	—
196	C₂²×C₄	—	3	1	no	—
197	C₂²×C₄	—	3	2	no	—
198	C₂²×C₄	—	3	2	no	—
199	C₂²×C₄	—	6	2	no	—
200	C₂²×C₄	—	6	2	no	—
201	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
202	C₂×C₈	—	3	2	no	—
203	C₂×C₈	—	6	2	no	—
204	C₂×C₈	—	6	4	no	—
205	C₂×C₈	—	6	2	no	—
206	C₂×C₈	—	6	2	no	—
207	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
208	C₂×C₈	C₅×D₁₂	1	2	yes	—
209	C₂×C₈	C₅×#67	1	2	yes	—
210	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
211	C₁₆	—	2	2	no	—
212	C₁₆	—	6	2	no	—
213	C₃×C₅	#2112	1	1	yes	characteristic, complemented, Hall
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	C₅×C₂²⋊C₈	1	2	yes	—
216	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
217	C₂×D₃	—	2	8	no	—
218	C₂×D₃	—	2	2	no	—
219	C₂×D₃	—	2	2	no	—
220	C₂×D₃	C₅×M₃₂	1	4	yes	complemented
221	C₃×C₄	C₅×C₂²⋊C₈	1	2	yes	—
222	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
224	Dic₃	—	2	2	no	—
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	6	4	no	—
227	C₂×C₅	D₃×M₃₂	1	2	yes	—
228	C₂×C₅	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
229	C₂³	—	3	2	no	—
230	C₂³	—	6	4	no	—
231	C₂³	—	6	2	no	—
232	C₂³	C₅×D₃×C₈	1	1	yes	characteristic
233	C₂×C₄	—	3	2	no	—
234	C₂×C₄	—	6	2	no	—
235	C₂×C₄	—	6	2	no	—
236	C₂×C₄	—	6	2	no	—
237	C₂×C₄	—	6	4	no	—
238	C₂×C₄	C₅×D₃×C₈	1	1	yes	characteristic
239	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
240	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
241	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
242	C₈	—	6	2	no	—
243	C₈	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
244	C₈	C₂×C₅×D₁₂	1	1	yes	characteristic
245	C₈	C₂×C₅×#67	1	1	yes	characteristic
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
248	C₂×C₃	C₂×C₅×M₃₂	1	2	yes	—
249	D₃	—	2	4	no	—
250	C₅	#71431	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²	—	2	2	no	—
252	C₂²	—	3	4	no	—
253	C₂²	—	6	8	no	—
254	C₂²	—	6	2	no	—
255	C₂²	—	6	2	no	—
256	C₂²	C₅×(C₂×D₃)⋊C₈	1	2	yes	—
257	C₂²	C₂×C₅×D₃×C₈	1	1	yes	characteristic
258	C₄	—	6	2	no	—
259	C₄	C₅×(C₂×D₃)⋊C₈	1	2	yes	—
260	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
261	C₄	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
262	C₃	C₅×#2112	1	1	yes	characteristic, complemented, Sylow
263	C₂	—	2	2	no	—
264	C₂	—	6	4	no	—
265	C₂	C₅×D₃×M₃₂	1	2	yes	—
266	C₂	C₂×C₅×(C₂×D₃)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂³×C₅×D₃	C₈
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂×C₅×M₃₂	D₃
hypocenter	C₃	C₅×#2112
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂×C₅×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₂×C₅×#812	C₂	semidirect
4. C₅×(C₂×D₃)⋊C₁₆	C₂	semidirect
14. C₂×C₃×C₅×C₁₆	C₂²	semidirect
16. C₂×C₅×C₃⋊C₁₆	C₂²	semidirect
17. #71431	C₅	direct (N × H)
20. C₂×C₅×M₃₂	D₃	semidirect
41. C₂×C₃×M₃₂	C₂×C₅	semidirect
42. C₂×#812	C₂×C₅	semidirect
43. (C₂×D₃)⋊C₁₆	C₂×C₅	semidirect
54. C₂×C₅×C₁₆	C₂×D₃	semidirect
81. C₂×C₃×C₁₆	C₂²×C₅	semidirect
83. C₂×C₃⋊C₁₆	C₂²×C₅	semidirect
104. C₂×M₃₂	C₅×D₃	semidirect
112. C₂×C₅×D₃	M₃₂	semidirect
162. C₂×C₁₆	C₂×C₅×D₃	semidirect
213. C₃×C₅	#2112	semidirect
220. C₂×D₃	C₅×M₃₂	semidirect
250. C₅	#71431	direct (N × H)
262. C₃	C₅×#2112	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×D₃×C₈	2
C₂×C₃×C₅×M₃₂	2
C₂×C₅×#812	2
C₅×(C₂×D₃)⋊C₁₆	2
C₅×#2112	3
#71431	5

Subgroups — C5×#71431 #14312823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71431 #14312823