Subgroups — C₅×#12908 #11935628

Order 1280.

350 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#367	C₂	1	2	yes	maximal
2	C₂×C₅×#366	C₂	1	2	yes	maximal
3	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic
4	C₅×#1846	C₂	1	1	yes	maximal, characteristic
5	C₅×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
7	C₂×C₅×#107	—	2	1	no	—
8	C₂×C₅×#107	C₂²	1	2	yes	—
9	C₂×C₅×#107	C₂²	1	1	yes	characteristic
10	C₂×C₅×#107	C₂²	1	1	yes	characteristic
11	C₂×C₅×#106	—	2	1	no	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#384	—	2	1	no	—
14	C₅×#382	—	2	1	no	—
15	C₅×#367	—	2	4	no	—
16	C₅×#366	—	2	4	no	—
17	C₅×#346	C₄	1	4	yes	complemented
18	#12908	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₅×C₄⋊C₄	—	2	1	no	—
22	C₂×C₅×C₄⋊C₄	—	2	1	no	—
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₅×C₄⋊C₄	—	4	2	no	—
25	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
26	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₅×C₁₆	D₄	1	4	yes	—
28	C₂×C₅×C₁₆	Q₈	1	4	yes	—
29	C₅×#107	—	2	2	no	—
30	C₅×#107	—	2	2	no	—
31	C₅×#107	—	2	2	no	—
32	C₅×#107	—	2	2	no	—
33	C₅×#107	—	4	2	no	—
34	C₅×#106	—	4	2	no	—
35	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₄×C₅×C₈	Q₈	1	2	yes	—
37	C₅×#95	—	4	2	no	—
38	C₂×#367	C₂×C₅	1	2	yes	—
39	C₂×#366	C₂×C₅	1	2	yes	—
40	C₂×#346	C₂×C₅	1	1	yes	characteristic
41	#1846	C₂×C₅	1	1	yes	characteristic
42	#1845	C₂×C₅	1	1	yes	characteristic
43	C₂²×C₄×C₅	—	4	2	no	—
44	C₂²×C₄×C₅	—	4	1	no	—
45	C₂²×C₄×C₅	—	4	1	no	—
46	C₂²×C₄×C₅	D₈	1	2	yes	—
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₅×C₈	D₈	1	2	yes	—
50	C₂×C₅×C₈	Q₁₆	1	2	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
53	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
54	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₅×C₄⋊C₄	—	4	1	no	—
56	C₅×C₄⋊C₄	—	4	1	no	—
57	C₅×C₄⋊C₄	—	4	2	no	—
58	C₅×C₄⋊C₄	—	4	1	no	—
59	C₅×C₄⋊C₄	—	4	1	no	—
60	C₅×C₄⋊C₄	—	4	2	no	—
61	C₅×C₄⋊C₄	—	8	4	no	—
62	C₄²×C₅	C₄⋊C₄	1	2	yes	—
63	C₄²×C₅	Q₁₆	1	2	yes	—
64	C₅×C₁₆	C₄⋊C₄	1	8	yes	complemented
65	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
66	C₂×#107	—	2	1	no	—
67	C₂×#107	C₂²×C₅	1	2	yes	—
68	C₂×#107	C₂²×C₅	1	1	yes	characteristic
69	C₂×#107	C₂²×C₅	1	1	yes	characteristic
70	C₂×#106	—	2	1	no	—
71	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
72	#384	—	2	1	no	—
73	#382	—	2	1	no	—
74	#367	—	2	4	no	—
75	#366	—	2	4	no	—
76	#346	C₄×C₅	1	4	yes	complemented
77	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	8	1	no	—
81	C₂×C₄×C₅	—	8	2	no	—
82	C₂×C₄×C₅	—	8	1	no	—
83	C₂×C₄×C₅	—	8	4	no	—
84	C₂×C₄×C₅	—	8	2	no	—
85	C₂×C₄×C₅	—	8	2	no	—
86	C₂×C₄×C₅	#107	1	4	yes	—
87	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅	#127	1	1	yes	characteristic
89	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
93	C₅×C₈	#107	1	4	yes	—
94	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
95	C₅×C₈	C₄⋊Q₈	1	2	yes	—
96	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
97	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	2	1	no	—
99	C₂×C₄⋊C₄	—	2	1	no	—
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	4	2	no	—
102	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
103	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
104	C₂×C₁₆	C₅×D₄	1	4	yes	—
105	C₂×C₁₆	C₅×Q₈	1	4	yes	—
106	#107	—	2	2	no	—
107	#107	—	2	2	no	—
108	#107	—	2	2	no	—
109	#107	—	2	2	no	—
110	#107	—	4	2	no	—
111	#106	—	4	2	no	—
112	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
113	C₄×C₈	C₅×Q₈	1	2	yes	—
114	#95	—	4	2	no	—
115	C₂²×C₅	C₂×#107	1	1	yes	characteristic
116	C₂²×C₅	#493	1	1	yes	characteristic
117	C₂²×C₅	#500	1	1	yes	characteristic
118	C₂²×C₅	#509	1	2	yes	—
119	C₂²×C₅	#510	1	2	yes	—
120	C₄×C₅	—	2	4	no	—
121	C₄×C₅	—	16	2	no	—
122	C₄×C₅	—	16	1	no	—
123	C₄×C₅	—	16	1	no	—
124	C₄×C₅	#384	1	1	yes	characteristic
125	C₄×C₅	C₂×#107	1	1	yes	characteristic
126	C₄×C₅	#494	1	1	yes	characteristic
127	C₄×C₅	#500	1	1	yes	characteristic
128	C₂²×C₄	—	4	1	no	—
129	C₂²×C₄	—	4	1	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	C₅×D₈	1	2	yes	—
132	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
133	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
134	C₂×C₈	C₅×D₈	1	2	yes	—
135	C₂×C₈	C₅×Q₁₆	1	2	yes	—
136	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
137	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
138	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
139	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
140	C₄⋊C₄	—	4	1	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	1	no	—
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	8	4	no	—
147	C₄²	C₅×C₄⋊C₄	1	2	yes	—
148	C₄²	C₅×Q₁₆	1	2	yes	—
149	C₁₆	C₅×C₄⋊C₄	1	8	yes	complemented
150	C₂×C₅	#1845	1	1	yes	characteristic
151	C₂×C₅	#2159	1	2	yes	—
152	C₂×C₅	#2250	1	1	yes	characteristic
153	C₂×C₅	#2251	1	1	yes	characteristic
154	C₂×C₅	#2255	1	2	yes	—
155	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	8	1	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	C₅×#107	1	4	yes	—
165	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄	C₅×#127	1	1	yes	characteristic
167	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
168	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
169	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
170	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
171	C₈	C₅×#107	1	4	yes	—
172	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
173	C₈	C₅×C₄⋊Q₈	1	2	yes	—
174	C₅	#12908	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²	C₂×C₅×#107	1	1	yes	characteristic
176	C₂²	C₅×#493	1	1	yes	characteristic
177	C₂²	C₅×#500	1	1	yes	characteristic
178	C₂²	C₅×#509	1	2	yes	—
179	C₂²	C₅×#510	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	16	2	no	—
182	C₄	—	16	1	no	—
183	C₄	—	16	1	no	—
184	C₄	C₅×#384	1	1	yes	characteristic
185	C₄	C₂×C₅×#107	1	1	yes	characteristic
186	C₄	C₅×#494	1	1	yes	characteristic
187	C₄	C₅×#500	1	1	yes	characteristic
188	C₂	C₅×#1845	1	1	yes	characteristic
189	C₂	C₅×#2159	1	2	yes	—
190	C₂	C₅×#2250	1	1	yes	characteristic
191	C₂	C₅×#2251	1	1	yes	characteristic
192	C₂	C₅×#2255	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₅×#346	C₄	semidirect
18. #12908	C₅	direct (N × H)
64. C₅×C₁₆	C₄⋊C₄	semidirect
76. #346	C₄×C₅	semidirect
149. C₁₆	C₅×C₄⋊C₄	semidirect
174. C₅	#12908	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#367	2
C₂×C₅×#366	2
C₂×C₅×#346	2
C₅×#1846	2
C₅×#1845	2
#12908	5

Subgroups — C5×#12908 #11935628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12908 #11935628