Subgroups — C₅×#22247 #11944967

Order 1280.

790 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₅×#1990	C₂	1	2	yes	maximal
4	C₅×#1918	C₂	1	8	yes	maximal
5	C₅×#1834	C₂	1	1	yes	maximal, characteristic
6	C₅×#1744	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×C₅×Q₈	—	2	2	no	—
9	C₅×#500	C₄	1	8	yes	complemented
10	C₂×C₅×#107	C₂²	1	4	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#391	—	2	2	no	—
14	C₅×#389	—	2	2	no	—
15	C₅×#389	C₂²	1	4	yes	—
16	C₅×#384	C₂²	1	2	yes	—
17	C₄×C₅×C₄⋊C₄	—	2	2	no	—
18	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
19	C₄³×C₅	C₂²	1	1	yes	characteristic
20	C₅×#340	C₂²	1	4	yes	—
21	C₅×#340	C₂²	1	8	yes	—
22	C₅×#336	C₂²	1	4	yes	—
23	#22247	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×Q₈	—	2	2	no	—
25	C₂²×C₅×C₈	C₂³	1	2	yes	—
26	C₂²×C₅×C₈	C₂³	1	2	yes	—
27	C₅×C₄⋊Q₈	—	2	4	no	—
28	C₅×C₄⋊Q₈	C₂×C₄	1	8	yes	—
29	C₄×C₅×Q₈	—	4	8	no	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	—	2	2	no	—
32	C₂×C₅×C₄⋊C₄	—	2	2	no	—
33	C₂×C₅×C₄⋊C₄	—	2	4	no	—
34	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
35	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
36	C₂×C₄²×C₅	—	2	4	no	—
37	C₂×C₄²×C₅	D₄	1	2	yes	—
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	2	yes	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×#107	C₂×C₄	1	16	yes	—
43	C₄×C₅×C₈	—	2	2	no	—
44	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
45	C₅×#95	—	2	4	no	—
46	C₅×#95	—	2	4	no	—
47	C₂×#500	C₂×C₅	1	1	yes	characteristic
48	C₄×C₄⋊Q₈	C₂×C₅	1	2	yes	—
49	#1990	C₂×C₅	1	2	yes	—
50	#1918	C₂×C₅	1	8	yes	—
51	#1834	C₂×C₅	1	1	yes	characteristic
52	#1744	C₂×C₅	1	1	yes	characteristic
53	C₂×C₅×Q₈	—	2	8	no	—
54	C₂×C₅×Q₈	—	4	4	no	—
55	C₂×C₅×Q₈	—	4	4	no	—
56	C₂²×C₄×C₅	—	2	4	no	—
57	C₂²×C₄×C₅	—	2	4	no	—
58	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
63	C₂×C₅×C₈	—	2	4	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	4	no	—
66	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
67	C₅×C₄⋊C₄	—	2	8	no	—
68	C₅×C₄⋊C₄	—	2	4	no	—
69	C₅×C₄⋊C₄	—	2	8	no	—
70	C₅×C₄⋊C₄	—	4	8	no	—
71	C₅×C₄⋊C₄	—	4	4	no	—
72	C₅×C₄⋊C₄	—	4	4	no	—
73	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
74	C₄²×C₅	—	2	2	no	—
75	C₄²×C₅	—	2	4	no	—
76	C₄²×C₅	—	2	4	no	—
77	C₄²×C₅	—	4	8	no	—
78	C₄²×C₅	D₈	1	4	yes	—
79	C₄²×C₅	C₂²×C₄	1	2	yes	—
80	C₄²×C₅	#41	1	2	yes	—
81	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
82	C₂×C₄×Q₈	—	2	2	no	—
83	#500	C₄×C₅	1	8	yes	complemented
84	C₂×#107	C₂²×C₅	1	4	yes	—
85	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
87	#391	—	2	2	no	—
88	#389	—	2	2	no	—
89	#389	C₂²×C₅	1	4	yes	—
90	#384	C₂²×C₅	1	2	yes	—
91	C₄×C₄⋊C₄	—	2	2	no	—
92	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
93	C₄³	C₂²×C₅	1	1	yes	characteristic
94	#340	C₂²×C₅	1	4	yes	—
95	#340	C₂²×C₅	1	8	yes	—
96	#336	C₂²×C₅	1	4	yes	—
97	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
98	C₅×Q₈	—	4	16	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	8	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	4	4	no	—
106	C₂×C₄×C₅	—	4	4	no	—
107	C₂×C₄×C₅	—	4	4	no	—
108	C₂×C₄×C₅	—	4	4	no	—
109	C₂×C₄×C₅	—	4	4	no	—
110	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
113	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₅	#135	1	2	yes	—
115	C₂×C₄×C₅	#137	1	4	yes	—
116	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₅	2₋¹⁺⁴	1	4	yes	—
119	C₅×C₈	—	2	4	no	—
120	C₅×C₈	—	4	2	no	—
121	C₂²×Q₈	—	2	2	no	—
122	C₂²×C₈	C₂³×C₅	1	2	yes	—
123	C₂²×C₈	C₂³×C₅	1	2	yes	—
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	—
126	C₄×Q₈	—	4	8	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	4	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
132	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
133	C₂×C₄²	—	2	4	no	—
134	C₂×C₄²	C₅×D₄	1	2	yes	—
135	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
136	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
137	C₂×C₄²	C₂³×C₅	1	2	yes	—
138	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
139	#107	C₂×C₄×C₅	1	16	yes	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
142	#95	—	2	4	no	—
143	#95	—	2	4	no	—
144	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂²×C₅	#536	1	2	yes	—
146	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
147	C₂²×C₅	C₂×#135	1	1	yes	characteristic
148	C₂²×C₅	C₂×#137	1	2	yes	—
149	C₄×C₅	—	2	4	no	—
150	C₄×C₅	—	4	8	no	—
151	C₄×C₅	—	4	2	no	—
152	C₄×C₅	—	8	4	no	—
153	C₄×C₅	C₄×D₈	1	4	yes	—
154	C₄×C₅	#441	1	4	yes	—
155	C₄×C₅	#519	1	2	yes	—
156	C₄×C₅	#541	1	2	yes	—
157	C₂×Q₈	—	2	8	no	—
158	C₂×Q₈	—	4	4	no	—
159	C₂×Q₈	—	4	4	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
163	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
164	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
166	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
171	C₄⋊C₄	—	2	8	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	8	no	—
174	C₄⋊C₄	—	4	8	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	4	no	—
180	C₄²	—	2	4	no	—
181	C₄²	—	4	8	no	—
182	C₄²	C₅×D₈	1	4	yes	—
183	C₄²	C₂²×C₄×C₅	1	2	yes	—
184	C₄²	C₅×#41	1	2	yes	—
185	C₂×C₅	#2327	1	1	yes	characteristic
186	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
187	C₂×C₅	C₂×#441	1	1	yes	characteristic
188	C₂×C₅	#3226	1	2	yes	—
189	C₂×C₅	#3231	1	2	yes	—
190	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
191	Q₈	—	4	16	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
204	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
205	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
206	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
207	C₂×C₄	C₅×#135	1	2	yes	—
208	C₂×C₄	C₅×#137	1	4	yes	—
209	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
210	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
211	C₂×C₄	C₅×2₋¹⁺⁴	1	4	yes	—
212	C₈	—	2	4	no	—
213	C₈	—	4	2	no	—
214	C₅	#22247	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
216	C₂²	C₅×#536	1	2	yes	—
217	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
218	C₂²	C₂×C₅×#135	1	1	yes	characteristic
219	C₂²	C₂×C₅×#137	1	2	yes	—
220	C₄	—	2	4	no	—
221	C₄	—	4	8	no	—
222	C₄	—	4	2	no	—
223	C₄	—	8	4	no	—
224	C₄	C₄×C₅×D₈	1	4	yes	—
225	C₄	C₅×#441	1	4	yes	—
226	C₄	C₅×#519	1	2	yes	—
227	C₄	C₅×#541	1	2	yes	—
228	C₂	C₅×#2327	1	1	yes	characteristic
229	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic
230	C₂	C₂×C₅×#441	1	1	yes	characteristic
231	C₂	C₅×#3226	1	2	yes	—
232	C₂	C₅×#3231	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₅×#500	C₄	semidirect
23. #22247	C₅	direct (N × H)
83. #500	C₄×C₅	semidirect
214. C₅	#22247	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×#500	2
C₄×C₅×C₄⋊Q₈	2
C₅×#1990	2
C₅×#1918	2
C₅×#1834	2
C₅×#1744	2
#22247	5

Subgroups — C5×#22247 #11944967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22247 #11944967