Subgroups — C₅×#71432 #14312860

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#901	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×(C₂×D₃)⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₅×#2113	—	3	1	no	maximal, Hall
6	C₂×C₅×#235	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#208	C₄	1	1	yes	characteristic
9	C₅×#901	—	2	2	no	—
10	C₅×#901	—	2	2	no	—
11	C₂×C₅×ΓL(1,𝔽₂₅)	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	—
15	C₂×C₅×C₃⋊C₁₆	—	2	2	no	—
16	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	—
17	#71432	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#131	—	3	1	no	—
19	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic, complemented
20	C₂²×C₅×C₁₆	—	3	1	no	—
21	C₅×C₂²⋊C₁₆	—	3	4	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅×#67	C₈	1	2	yes	—
24	C₅×#235	—	2	2	no	—
25	C₅×#235	—	2	2	no	—
26	C₂×C₅×D₁₂	C₈	1	1	yes	characteristic
27	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
28	C₂×C₅×Dic₆	C₈	1	1	yes	characteristic
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	C₅×#208	—	2	2	no	—
34	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₅×ΓL(1,𝔽₂₅)	—	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	C₂×#901	C₂×C₅	1	1	yes	characteristic
41	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂²×C₃⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
43	(C₂×D₃)⋊C₁₆	C₂×C₅	1	4	yes	complemented
44	C₂×C₅×#41	—	3	1	no	—
45	C₅×#131	—	6	2	no	—
46	C₅×#131	—	6	2	no	—
47	C₂×C₅×M₁₆	—	3	1	no	—
48	C₂×C₅×M₁₆	—	3	2	no	—
49	C₂²×C₅×C₈	—	3	2	no	—
50	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
51	C₅×M₃₂	—	2	2	no	—
52	C₂×C₅×C₁₆	—	3	2	no	—
53	C₂×C₅×C₁₆	—	6	2	no	—
54	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	—
55	#2113	—	3	1	no	Sylow
56	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₅×D₃	C₂×C₈	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₅×#67	—	2	4	no	—
63	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
64	C₅×D₁₂	—	2	2	no	—
65	C₄×C₅×D₃	—	2	2	no	—
66	C₄×C₅×D₃	—	2	2	no	—
67	C₅×Dic₆	—	2	2	no	—
68	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
69	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₅×C₃⋊C₈	—	2	2	no	—
72	C₂×#235	C₄×C₅	1	1	yes	characteristic
73	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
74	C₂×#208	C₄×C₅	1	1	yes	characteristic
75	#901	—	2	2	no	—
76	#901	—	2	2	no	—
77	C₂×ΓL(1,𝔽₂₅)	C₄×C₅	1	2	yes	—
78	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
79	C₃×M₃₂	—	2	2	no	—
80	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
81	C₂×C₃⋊C₁₆	—	2	2	no	—
82	C₂×C₃⋊C₁₆	C₂²×C₅	1	2	yes	—
83	C₅×#41	—	6	2	no	—
84	C₅×#41	—	6	2	no	—
85	C₂×C₅×Q₈	—	3	1	no	—
86	C₂×C₅×D₄	—	3	2	no	—
87	C₂×C₅×D₄	—	3	1	no	—
88	C₂²×C₄×C₅	—	3	2	no	—
89	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
90	C₅×M₁₆	—	6	2	no	—
91	C₅×M₁₆	—	6	4	no	—
92	C₂×C₅×C₈	—	3	2	no	—
93	C₂×C₅×C₈	—	6	2	no	—
94	C₂×C₅×C₈	—	6	2	no	—
95	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
96	C₂×C₅×C₈	D₁₂	1	2	yes	—
97	C₂×C₅×C₈	#67	1	2	yes	—
98	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
99	C₅×C₁₆	—	2	2	no	—
100	C₅×C₁₆	—	6	2	no	—
101	C₂×#131	—	3	1	no	—
102	C₂×M₃₂	C₅×D₃	1	1	yes	characteristic, complemented
103	C₂²×C₁₆	—	3	1	no	—
104	C₂²⋊C₁₆	—	3	4	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	C₂²⋊C₈	1	2	yes	—
107	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₅×D₃	—	2	2	no	—
109	C₂×C₅×D₃	—	2	4	no	—
110	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
111	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
113	C₅×Dic₃	—	2	2	no	—
114	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
115	C₂×#67	C₅×C₈	1	2	yes	—
116	#235	—	2	2	no	—
117	#235	—	2	2	no	—
118	C₂×D₁₂	C₅×C₈	1	1	yes	characteristic
119	C₂×C₄×D₃	C₂×C₄×C₅	1	2	yes	—
120	C₂×Dic₆	C₅×C₈	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
123	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
124	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
125	#208	—	2	2	no	—
126	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
127	ΓL(1,𝔽₂₅)	—	2	4	no	—
128	D₃×C₈	—	2	2	no	—
129	D₃×C₈	—	2	2	no	—
130	C₃×C₁₆	—	2	2	no	—
131	C₃⋊C₁₆	—	2	2	no	—
132	C₂³×C₅	—	3	2	no	—
133	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
134	C₅×Q₈	—	6	2	no	—
135	C₅×D₄	—	6	4	no	—
136	C₅×D₄	—	6	2	no	—
137	C₂×C₄×C₅	—	3	2	no	—
138	C₂×C₄×C₅	—	6	2	no	—
139	C₂×C₄×C₅	—	6	2	no	—
140	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
141	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	2	yes	—
142	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
143	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
144	C₅×C₈	—	6	2	no	—
145	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
146	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
147	C₅×C₈	C₂×#67	1	1	yes	characteristic
148	C₂×#41	—	3	1	no	—
149	#131	—	6	2	no	—
150	#131	—	6	2	no	—
151	C₂×M₁₆	—	3	1	no	—
152	C₂×M₁₆	—	3	2	no	—
153	C₂²×C₈	—	3	2	no	—
154	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
155	M₃₂	—	2	2	no	—
156	C₂×C₁₆	—	3	2	no	—
157	C₂×C₁₆	—	6	2	no	—
158	C₂×C₁₆	C₂×C₅×D₃	1	2	yes	—
159	C₂×C₃×C₅	—	2	2	no	—
160	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₂×C₃×C₅	#504	1	2	yes	—
162	C₅×D₃	—	4	2	no	—
163	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
164	C₂²×D₃	C₂×C₅×C₈	1	2	yes	—
165	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
166	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
167	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
168	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
169	#67	—	2	4	no	—
170	C₂×Dic₃	C₂×C₅×C₈	1	2	yes	—
171	D₁₂	—	2	2	no	—
172	C₄×D₃	—	2	2	no	—
173	C₄×D₃	—	2	2	no	—
174	Dic₆	—	2	2	no	—
175	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
176	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
177	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
178	C₃⋊C₈	—	2	2	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	6	4	no	—
181	C₂²×C₅	—	6	2	no	—
182	C₂²×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
183	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
184	C₄×C₅	—	6	2	no	—
185	C₄×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
186	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
187	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
188	#41	—	6	2	no	—
189	#41	—	6	2	no	—
190	C₂×Q₈	—	3	1	no	—
191	C₂×D₄	—	3	2	no	—
192	C₂×D₄	—	3	1	no	—
193	C₂²×C₄	—	3	2	no	—
194	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
195	M₁₆	—	6	2	no	—
196	M₁₆	—	6	4	no	—
197	C₂×C₈	—	3	2	no	—
198	C₂×C₈	—	6	2	no	—
199	C₂×C₈	—	6	2	no	—
200	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
201	C₂×C₈	C₅×D₁₂	1	2	yes	—
202	C₂×C₈	C₅×#67	1	2	yes	—
203	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
204	C₁₆	—	2	2	no	—
205	C₁₆	—	6	2	no	—
206	C₃×C₅	#2113	1	1	yes	characteristic, complemented, Hall
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	C₅×C₂²⋊C₈	1	2	yes	—
209	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
210	C₂×D₃	—	2	2	no	—
211	C₂×D₃	—	2	4	no	—
212	C₃×C₄	C₅×C₂²⋊C₈	1	2	yes	—
213	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
214	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
215	Dic₃	—	2	2	no	—
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	—	12	2	no	—
218	C₂×C₅	#4907	1	2	yes	—
219	C₂×C₅	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
220	C₂³	—	3	2	no	—
221	C₂³	C₅×D₃×C₈	1	1	yes	characteristic
222	Q₈	—	6	2	no	—
223	D₄	—	6	4	no	—
224	D₄	—	6	2	no	—
225	C₂×C₄	—	3	2	no	—
226	C₂×C₄	—	6	2	no	—
227	C₂×C₄	—	6	2	no	—
228	C₂×C₄	C₅×D₃×C₈	1	1	yes	characteristic
229	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
230	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
231	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
232	C₈	—	6	2	no	—
233	C₈	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
234	C₈	C₂×C₅×D₁₂	1	1	yes	characteristic
235	C₈	C₂×C₅×#67	1	1	yes	characteristic
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
238	C₂×C₃	C₅×#504	1	2	yes	—
239	D₃	—	4	2	no	—
240	C₅	#71432	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	6	4	no	—
243	C₂²	—	6	2	no	—
244	C₂²	C₅×(C₂×D₃)⋊C₈	1	2	yes	—
245	C₂²	C₂×C₅×D₃×C₈	1	1	yes	characteristic
246	C₄	—	6	2	no	—
247	C₄	C₅×(C₂×D₃)⋊C₈	1	2	yes	—
248	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
249	C₄	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
250	C₃	C₅×#2113	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	2	2	no	—
252	C₂	—	12	2	no	—
253	C₂	C₅×#4907	1	2	yes	—
254	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	C₂×C₅×#235	C₄
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂×C₅×M₃₂	D₃
hypocenter	C₃	C₅×#2113
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. C₂²×C₅×C₃⋊C₁₆	C₂	semidirect
4. C₅×(C₂×D₃)⋊C₁₆	C₂	semidirect
17. #71432	C₅	direct (N × H)
19. C₂×C₅×M₃₂	D₃	semidirect
41. C₂×C₃×M₃₂	C₂×C₅	semidirect
42. C₂²×C₃⋊C₁₆	C₂×C₅	semidirect
43. (C₂×D₃)⋊C₁₆	C₂×C₅	semidirect
102. C₂×M₃₂	C₅×D₃	semidirect
206. C₃×C₅	#2113	semidirect
240. C₅	#71432	direct (N × H)
250. C₃	C₅×#2113	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#901	2
C₂×C₃×C₅×M₃₂	2
C₂²×C₅×C₃⋊C₁₆	2
C₅×(C₂×D₃)⋊C₁₆	2
C₅×#2113	3
#71432	5

Subgroups — C5×#71432 #14312860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71432 #14312860