Subgroups — C₅×#12912 #11935632

Order 1280.

462 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₅×#1374	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	—	2	4	no	—
8	C₂²×C₅×C₁₆	—	2	2	no	—
9	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
12	C₂×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
14	C₄×C₅×C₂²⋊C₄	C₄	1	1	yes	characteristic
15	C₅×C₄⋊C₁₆	—	2	4	no	—
16	C₅×C₂²⋊C₁₆	—	2	2	no	—
17	C₅×#336	C₄	1	2	yes	—
18	#12912	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	2	yes	—
25	C₂²×C₅×C₈	Q₈	1	2	yes	—
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₅×C₂²⋊C₄	C₈	1	1	yes	characteristic
28	C₂×C₅×C₂²⋊C₄	C₈	1	1	yes	characteristic
29	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₂×C₅×C₁₆	—	2	8	no	—
31	C₂×C₅×C₁₆	—	2	2	no	—
32	C₂×C₅×C₁₆	—	2	2	no	—
33	C₂×C₅×C₁₆	—	2	1	no	—
34	C₂×C₅×C₁₆	—	2	1	no	—
35	C₂×C₅×C₁₆	—	2	2	no	—
36	C₂×C₅×C₁₆	—	2	2	no	—
37	C₂×C₅×C₁₆	D₄	1	4	yes	—
38	C₅×C₂²⋊C₈	—	2	2	no	—
39	C₄×C₅×C₈	—	2	2	no	—
40	C₄×C₅×C₈	—	2	2	no	—
41	C₅×#95	C₈	1	2	yes	—
42	C₂³×C₁₆	C₂×C₅	1	1	yes	characteristic
43	C₂×C₄⋊C₁₆	C₂×C₅	1	2	yes	complemented
44	C₂×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic
45	C₈×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
46	#1374	C₂×C₅	1	2	yes	complemented
47	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	2	no	—
49	C₂²×C₄×C₅	—	2	2	no	—
50	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅	M₁₆	1	2	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	4	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	2	2	no	—
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
64	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
65	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₅×C₈	#41	1	2	yes	—
68	C₅×C₂²⋊C₄	—	2	2	no	—
69	C₅×C₂²⋊C₄	C₁₆	1	4	yes	complemented
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	—	2	2	no	—
72	C₅×C₁₆	—	2	4	no	—
73	C₅×C₁₆	—	4	1	no	—
74	C₅×C₁₆	—	4	1	no	—
75	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
76	C₂²×C₁₆	—	2	2	no	—
77	C₂²×C₁₆	—	2	4	no	—
78	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
79	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
80	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
81	C₂×C₂²⋊C₈	C₄×C₅	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
83	C₄×C₂²⋊C₄	C₄×C₅	1	1	yes	characteristic
84	C₄⋊C₁₆	—	2	4	no	—
85	C₂²⋊C₁₆	—	2	2	no	—
86	#336	C₄×C₅	1	2	yes	—
87	C₂³×C₅	—	2	2	no	—
88	C₂³×C₅	—	2	2	no	—
89	C₂³×C₅	C₄⋊C₈	1	2	yes	—
90	C₂³×C₅	C₂×C₁₆	1	2	yes	—
91	C₂³×C₅	M₃₂	1	2	yes	—
92	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	—	2	2	no	—
95	C₂×C₄×C₅	—	2	2	no	—
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
103	C₂×C₄×C₅	C₂×C₁₆	1	4	yes	complemented
104	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅	#131	1	2	yes	—
108	C₅×C₈	—	2	2	no	—
109	C₅×C₈	—	4	2	no	—
110	C₅×C₈	#120	1	1	yes	characteristic
111	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
112	C₅×C₈	#123	1	1	yes	characteristic
113	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
117	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
118	C₂²×C₈	C₅×D₄	1	2	yes	—
119	C₂²×C₈	C₅×Q₈	1	2	yes	—
120	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
121	C₂×C₂²⋊C₄	C₅×C₈	1	1	yes	characteristic
122	C₂×C₂²⋊C₄	C₅×C₈	1	1	yes	characteristic
123	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
124	C₂×C₁₆	—	2	8	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	C₅×D₄	1	4	yes	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	#95	C₅×C₈	1	2	yes	—
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	—	2	2	no	—
138	C₂²×C₅	—	2	4	no	—
139	C₂²×C₅	—	2	4	no	—
140	C₂²×C₅	C₄⋊C₁₆	1	4	yes	complemented
141	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
142	C₂²×C₅	C₈×D₄	1	2	yes	—
143	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
144	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
145	C₂²×C₅	#504	1	2	yes	—
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	—	2	4	no	—
148	C₄×C₅	—	4	2	no	—
149	C₄×C₅	#385	1	1	yes	characteristic
150	C₄×C₅	#424	1	1	yes	characteristic
151	C₄×C₅	#435	1	2	yes	—
152	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
156	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
157	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
159	C₂²×C₄	C₅×M₁₆	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
169	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
170	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
171	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
172	C₂×C₈	C₅×#41	1	2	yes	—
173	C₂²⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	C₅×C₁₆	1	4	yes	complemented
175	C₄²	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₁₆	—	2	4	no	—
178	C₁₆	—	4	1	no	—
179	C₁₆	—	4	1	no	—
180	C₂×C₅	—	2	4	no	—
181	C₂×C₅	#1850	1	1	yes	characteristic
182	C₂×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic
183	C₂×C₅	#2151	1	1	yes	characteristic
184	C₂×C₅	D₄×C₁₆	1	2	yes	—
185	C₂×C₅	#2168	1	2	yes	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	C₅×C₄⋊C₈	1	2	yes	—
189	C₂³	C₂×C₅×C₁₆	1	2	yes	—
190	C₂³	C₅×M₃₂	1	2	yes	—
191	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	C₅×C₄⋊C₈	1	2	yes	—
202	C₂×C₄	C₂×C₅×C₁₆	1	4	yes	complemented
203	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
204	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
205	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
206	C₂×C₄	C₅×#131	1	2	yes	—
207	C₈	—	2	2	no	—
208	C₈	—	4	2	no	—
209	C₈	C₅×#120	1	1	yes	characteristic
210	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
211	C₈	C₅×#123	1	1	yes	characteristic
212	C₅	#12912	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	C₅×C₄⋊C₁₆	1	4	yes	complemented
218	C₂²	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
219	C₂²	C₅×C₈×D₄	1	2	yes	—
220	C₂²	C₂²×C₅×C₁₆	1	1	yes	characteristic
221	C₂²	C₂×C₅×M₃₂	1	1	yes	characteristic
222	C₂²	C₅×#504	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	2	4	no	—
225	C₄	—	4	2	no	—
226	C₄	C₅×#385	1	1	yes	characteristic
227	C₄	C₅×#424	1	1	yes	characteristic
228	C₄	C₅×#435	1	2	yes	—
229	C₂	—	2	4	no	—
230	C₂	C₅×#1850	1	1	yes	characteristic
231	C₂	C₂×C₅×C₄⋊C₁₆	1	1	yes	characteristic
232	C₂	C₅×#2151	1	1	yes	characteristic
233	C₂	C₅×D₄×C₁₆	1	2	yes	—
234	C₂	C₅×#2168	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂²	C₂²×C₅×C₁₆
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
5. C₅×#1374	C₂	semidirect
18. #12912	C₅	direct (N × H)
43. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
46. #1374	C₂×C₅	semidirect
69. C₅×C₂²⋊C₄	C₁₆	semidirect
103. C₂×C₄×C₅	C₂×C₁₆	semidirect
140. C₂²×C₅	C₄⋊C₁₆	semidirect
174. C₂²⋊C₄	C₅×C₁₆	semidirect
202. C₂×C₄	C₂×C₅×C₁₆	semidirect
212. C₅	#12912	direct (N × H)
217. C₂²	C₅×C₄⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₁₆	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×C₂²⋊C₁₆	2
C₅×C₈×C₂²⋊C₄	2
C₅×#1374	2
#12912	5

Subgroups — C5×#12912 #11935632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12912 #11935632