Subgroups — C₅×#12662 #11935382

Order 1280.

374 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#346	C₂	1	2	yes	maximal
3	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₅×#1374	C₂	1	2	yes	maximal
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
8	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₅×C₄⋊C₁₆	C₄	1	4	yes	complemented
14	C₅×#346	—	2	4	no	—
15	C₄×C₅×C₁₆	C₄	1	4	yes	complemented
16	C₅×#336	C₄	1	2	yes	—
17	#12662	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
22	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₅×C₁₆	—	2	2	no	—
25	C₂×C₅×C₁₆	—	2	1	no	—
26	C₂×C₅×C₁₆	—	2	1	no	—
27	C₂×C₅×C₁₆	—	2	2	no	—
28	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₅×C₁₆	D₄	1	2	yes	—
32	C₂×C₅×C₁₆	Q₈	1	2	yes	—
33	C₅×C₄⋊C₈	C₈	1	4	yes	—
34	C₄×C₅×C₈	—	2	2	no	—
35	C₄×C₅×C₈	—	2	2	no	—
36	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
37	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₅×#95	C₈	1	2	yes	—
39	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
40	C₂×#346	C₂×C₅	1	2	yes	—
41	C₂×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic
42	C₈×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
43	#1374	C₂×C₅	1	2	yes	—
44	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	—	2	2	no	—
49	C₂×C₅×C₈	—	2	1	no	—
50	C₂×C₅×C₈	—	2	1	no	—
51	C₂×C₅×C₈	—	2	2	no	—
52	C₂×C₅×C₈	C₄²	1	2	yes	—
53	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
54	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₅×C₈	#41	1	1	yes	characteristic
59	C₂×C₅×C₈	#41	1	1	yes	characteristic
60	C₅×C₄⋊C₄	—	2	4	no	—
61	C₄²×C₅	—	2	2	no	—
62	C₄²×C₅	—	2	2	no	—
63	C₄²×C₅	C₄²	1	2	yes	—
64	C₄²×C₅	C₂×C₈	1	2	yes	—
65	C₅×C₁₆	—	2	2	no	—
66	C₅×C₁₆	—	2	4	no	—
67	C₅×C₁₆	C₄⋊C₄	1	4	yes	complemented
68	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
69	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
70	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₄⋊C₈	C₄×C₅	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
73	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
74	C₄×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
75	C₄⋊C₁₆	C₄×C₅	1	4	yes	complemented
76	#346	—	2	4	no	—
77	C₄×C₁₆	C₄×C₅	1	4	yes	complemented
78	#336	C₄×C₅	1	2	yes	—
79	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	4	no	—
82	C₂×C₄×C₅	—	2	4	no	—
83	C₂×C₄×C₅	—	2	4	no	—
84	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
85	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
86	C₂×C₄×C₅	M₃₂	1	4	yes	—
87	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅	#131	1	1	yes	characteristic
92	C₂×C₄×C₅	#131	1	1	yes	characteristic
93	C₅×C₈	—	2	2	no	—
94	C₅×C₈	—	2	4	no	—
95	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₅×C₈	#117	1	1	yes	characteristic
97	C₅×C₈	C₄×D₄	1	1	yes	characteristic
98	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
99	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
100	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
101	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
102	C₂×C₄⋊C₄	C₅×C₈	1	2	yes	—
103	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
104	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	1	no	—
107	C₂×C₁₆	—	2	1	no	—
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
110	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
111	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
112	C₂×C₁₆	C₅×D₄	1	2	yes	—
113	C₂×C₁₆	C₅×Q₈	1	2	yes	—
114	C₄⋊C₈	C₅×C₈	1	4	yes	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
118	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
119	#95	C₅×C₈	1	2	yes	—
120	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
121	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
122	C₂²×C₅	C₈×Q₈	1	1	yes	characteristic
123	C₂²×C₅	C₂×M₃₂	1	2	yes	—
124	C₂²×C₅	#504	1	2	yes	—
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	—	4	4	no	—
127	C₄×C₅	#346	1	4	yes	complemented
128	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₅	#405	1	1	yes	characteristic
130	C₄×C₅	C₈×D₄	1	1	yes	characteristic
131	C₄×C₅	C₈×Q₈	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
133	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
135	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₄²×C₅	1	2	yes	—
141	C₂×C₈	C₂×C₅×C₈	1	4	yes	—
142	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
143	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
144	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
145	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
146	C₂×C₈	C₅×#41	1	1	yes	characteristic
147	C₂×C₈	C₅×#41	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	C₄²×C₅	1	2	yes	—
152	C₄²	C₂×C₅×C₈	1	2	yes	—
153	C₁₆	—	2	2	no	—
154	C₁₆	—	2	4	no	—
155	C₁₆	C₅×C₄⋊C₄	1	4	yes	complemented
156	C₂×C₅	C₈×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₅	C₂×#346	1	1	yes	characteristic
158	C₂×C₅	#2108	1	1	yes	characteristic
159	C₂×C₅	#2169	1	2	yes	—
160	C₂×C₅	#2183	1	2	yes	—
161	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	C₄×C₅×C₈	1	2	yes	—
167	C₂×C₄	C₄×C₅×C₈	1	2	yes	—
168	C₂×C₄	C₅×M₃₂	1	4	yes	—
169	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
170	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
171	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
172	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
173	C₂×C₄	C₅×#131	1	1	yes	characteristic
174	C₂×C₄	C₅×#131	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	4	no	—
177	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
178	C₈	C₅×#117	1	1	yes	characteristic
179	C₈	C₄×C₅×D₄	1	1	yes	characteristic
180	C₈	C₄×C₅×Q₈	1	1	yes	characteristic
181	C₅	#12662	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²	C₂×C₄×C₅×C₈	1	1	yes	characteristic
183	C₂²	C₅×C₈×D₄	1	1	yes	characteristic
184	C₂²	C₅×C₈×Q₈	1	1	yes	characteristic
185	C₂²	C₂×C₅×M₃₂	1	2	yes	—
186	C₂²	C₅×#504	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	4	4	no	—
189	C₄	C₅×#346	1	4	yes	complemented
190	C₄	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
191	C₄	C₅×#405	1	1	yes	characteristic
192	C₄	C₅×C₈×D₄	1	1	yes	characteristic
193	C₄	C₅×C₈×Q₈	1	1	yes	characteristic
194	C₂	C₅×C₈×C₄⋊C₄	1	1	yes	characteristic
195	C₂	C₂×C₅×#346	1	1	yes	characteristic
196	C₂	C₅×#2108	1	1	yes	characteristic
197	C₂	C₅×#2169	1	2	yes	—
198	C₂	C₅×#2183	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂²	C₂×C₄×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂²×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
13. C₅×C₄⋊C₁₆	C₄	semidirect
15. C₄×C₅×C₁₆	C₄	semidirect
17. #12662	C₅	direct (N × H)
67. C₅×C₁₆	C₄⋊C₄	semidirect
75. C₄⋊C₁₆	C₄×C₅	semidirect
77. C₄×C₁₆	C₄×C₅	semidirect
127. C₄×C₅	#346	semidirect
155. C₁₆	C₅×C₄⋊C₄	semidirect
181. C₅	#12662	direct (N × H)
189. C₄	C₅×#346	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#346	2
C₂×C₄×C₅×C₁₆	2
C₅×C₈×C₄⋊C₄	2
C₅×#1374	2
#12662	5

Subgroups — C5×#12662 #11935382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12662 #11935382