Subgroups — C₅×#10842 #11933562

Order 1280.

950 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#382	C₂	1	2	yes	maximal
3	C₂×C₅×#340	C₂	1	4	yes	maximal
4	C₂³×C₅×C₈	C₂²	1	2	yes	—
5	C₂²×C₅×C₄⋊C₄	C₂²	1	4	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	—	2	4	no	—
8	C₂×C₄×C₅×C₈	C₄	1	4	yes	complemented
9	C₅×#382	—	2	4	no	—
10	C₅×#382	C₄	1	8	yes	complemented
11	C₂×C₅×#95	—	2	4	no	—
12	C₅×#340	—	2	8	no	—
13	#10842	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₄×C₅	—	2	4	no	—
15	C₂³×C₄×C₅	D₄	1	2	yes	—
16	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
17	C₂²×C₅×C₈	—	2	4	no	—
18	C₂²×C₅×C₈	—	2	4	no	—
19	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₅×C₈	D₄	1	4	yes	—
21	C₂²×C₅×C₈	Q₈	1	4	yes	—
22	C₂×C₅×C₄⋊C₄	—	2	8	no	—
23	C₂×C₅×C₄⋊C₄	—	2	8	no	—
24	C₂×C₅×C₄⋊C₄	—	2	4	no	—
25	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅	D₄	1	4	yes	—
29	C₂×C₄²×C₅	Q₈	1	2	yes	—
30	C₄×C₅×C₈	—	2	8	no	—
31	C₅×#95	—	2	16	no	—
32	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
33	C₂×#382	C₂×C₅	1	2	yes	—
34	C₂×#340	C₂×C₅	1	4	yes	—
35	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅	—	2	8	no	—
37	C₂²×C₄×C₅	—	2	8	no	—
38	C₂²×C₄×C₅	—	4	4	no	—
39	C₂²×C₄×C₅	—	4	4	no	—
40	C₂²×C₄×C₅	—	4	16	no	—
41	C₂²×C₄×C₅	C₂²⋊C₄	1	8	yes	—
42	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₅	#41	1	4	yes	—
50	C₂×C₅×C₈	—	2	4	no	—
51	C₂×C₅×C₈	—	2	4	no	—
52	C₂×C₅×C₈	—	2	8	no	—
53	C₂×C₅×C₈	—	2	8	no	—
54	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
55	C₅×C₄⋊C₄	—	2	16	no	—
56	C₅×C₄⋊C₄	—	4	8	no	—
57	C₄²×C₅	C₄²	1	4	yes	—
58	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
59	C₄²×C₅	C₄⋊C₄	1	8	yes	—
60	C₂³×C₈	C₂²×C₅	1	2	yes	—
61	C₂²×C₄⋊C₄	C₂²×C₅	1	4	yes	—
62	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
63	C₂×C₄×C₈	—	2	4	no	—
64	C₂×C₄×C₈	C₄×C₅	1	4	yes	complemented
65	#382	—	2	4	no	—
66	#382	C₄×C₅	1	8	yes	complemented
67	C₂×#95	—	2	4	no	—
68	#340	—	2	8	no	—
69	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
70	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂³×C₅	#124	1	2	yes	—
72	C₂³×C₅	#127	1	1	yes	characteristic
73	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
74	C₂³×C₅	#135	1	8	yes	—
75	C₂×C₄×C₅	—	2	16	no	—
76	C₂×C₄×C₅	—	2	8	no	—
77	C₂×C₄×C₅	—	4	8	no	—
78	C₂×C₄×C₅	—	4	16	no	—
79	C₂×C₄×C₅	—	8	8	no	—
80	C₂×C₄×C₅	—	8	8	no	—
81	C₂×C₄×C₅	#95	1	8	yes	—
82	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₅	#117	1	2	yes	—
87	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
88	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
89	C₂×C₄×C₅	#121	1	4	yes	—
90	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
91	C₂×C₄×C₅	#125	1	2	yes	—
92	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
93	C₅×C₈	—	2	8	no	—
94	C₂³×C₄	—	2	4	no	—
95	C₂³×C₄	C₅×D₄	1	2	yes	—
96	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
97	C₂²×C₈	—	2	4	no	—
98	C₂²×C₈	—	2	4	no	—
99	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
100	C₂²×C₈	C₅×D₄	1	4	yes	—
101	C₂²×C₈	C₅×Q₈	1	4	yes	—
102	C₂×C₄⋊C₄	—	2	8	no	—
103	C₂×C₄⋊C₄	—	2	8	no	—
104	C₂×C₄⋊C₄	—	2	4	no	—
105	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
106	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
107	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
108	C₂×C₄²	C₅×D₄	1	4	yes	—
109	C₂×C₄²	C₅×Q₈	1	2	yes	—
110	C₄×C₈	—	2	8	no	—
111	#95	—	2	16	no	—
112	C₂²×C₅	C₂×#95	1	1	yes	characteristic
113	C₂²×C₅	#382	1	1	yes	characteristic
114	C₂²×C₅	#384	1	1	yes	characteristic
115	C₂²×C₅	#390	1	2	yes	—
116	C₂²×C₅	#391	1	2	yes	—
117	C₂²×C₅	#416	1	8	yes	—
118	C₂²×C₅	#427	1	4	yes	—
119	C₂²×C₅	#488	1	8	yes	—
120	C₂²×C₅	#495	1	4	yes	—
121	C₂²×C₅	#499	1	4	yes	—
122	C₄×C₅	—	2	8	no	—
123	C₄×C₅	—	8	8	no	—
124	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
125	C₄×C₅	#380	1	2	yes	—
126	C₄×C₅	#389	1	4	yes	—
127	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
128	C₂²×C₄	—	2	8	no	—
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	—	4	4	no	—
131	C₂²×C₄	—	4	4	no	—
132	C₂²×C₄	—	4	16	no	—
133	C₂²×C₄	C₅×C₂²⋊C₄	1	8	yes	—
134	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
135	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
136	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
137	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
138	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
139	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
141	C₂²×C₄	C₅×#41	1	4	yes	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
147	C₄⋊C₄	—	2	16	no	—
148	C₄⋊C₄	—	4	8	no	—
149	C₄²	C₄²×C₅	1	4	yes	—
150	C₄²	C₅×C₂²⋊C₄	1	4	yes	—
151	C₄²	C₅×C₄⋊C₄	1	8	yes	—
152	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
153	C₂×C₅	#1735	1	2	yes	—
154	C₂×C₅	#1835	1	2	yes	—
155	C₂×C₅	#1846	1	2	yes	—
156	C₂×C₅	#1958	1	4	yes	—
157	C₂×C₅	#1991	1	4	yes	—
158	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
159	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
160	C₂³	C₅×#124	1	2	yes	—
161	C₂³	C₅×#127	1	1	yes	characteristic
162	C₂³	C₅×C₄⋊Q₈	1	1	yes	characteristic
163	C₂³	C₅×#135	1	8	yes	—
164	C₂×C₄	—	2	16	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	4	8	no	—
167	C₂×C₄	—	4	16	no	—
168	C₂×C₄	—	8	8	no	—
169	C₂×C₄	—	8	8	no	—
170	C₂×C₄	C₅×#95	1	8	yes	—
171	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
172	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
174	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
175	C₂×C₄	C₅×#117	1	2	yes	—
176	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
177	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
178	C₂×C₄	C₅×#121	1	4	yes	—
179	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
180	C₂×C₄	C₅×#125	1	2	yes	—
181	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
182	C₈	—	2	8	no	—
183	C₅	#10842	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²	C₂×C₅×#95	1	1	yes	characteristic
185	C₂²	C₅×#382	1	1	yes	characteristic
186	C₂²	C₅×#384	1	1	yes	characteristic
187	C₂²	C₅×#390	1	2	yes	—
188	C₂²	C₅×#391	1	2	yes	—
189	C₂²	C₅×#416	1	8	yes	—
190	C₂²	C₅×#427	1	4	yes	—
191	C₂²	C₅×#488	1	8	yes	—
192	C₂²	C₅×#495	1	4	yes	—
193	C₂²	C₅×#499	1	4	yes	—
194	C₄	—	2	8	no	—
195	C₄	—	8	8	no	—
196	C₄	C₄×C₅×C₄⋊C₄	1	2	yes	—
197	C₄	C₅×#380	1	2	yes	—
198	C₄	C₅×#389	1	4	yes	—
199	C₂	C₅×C₄⋊#95	1	1	yes	characteristic
200	C₂	C₅×#1735	1	2	yes	—
201	C₂	C₅×#1835	1	2	yes	—
202	C₂	C₅×#1846	1	2	yes	—
203	C₂	C₅×#1958	1	4	yes	—
204	C₂	C₅×#1991	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂×C₄×C₅×C₈	C₄	semidirect
10. C₅×#382	C₄	semidirect
13. #10842	C₅	direct (N × H)
64. C₂×C₄×C₈	C₄×C₅	semidirect
66. #382	C₄×C₅	semidirect
183. C₅	#10842	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₈	2
C₂×C₅×#382	2
C₂×C₅×#340	2
#10842	5

Subgroups — C5×#10842 #11933562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10842 #11933562