Subgroups — C₅×#13242 #11935962

Order 1280.

418 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#508	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#359	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1974	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#135	C₂²	1	2	yes	—
7	C₅×#508	—	2	2	no	—
8	C₅×#508	—	2	2	no	—
9	C₂×C₅×Q₃₂	C₄	1	1	yes	characteristic, complemented
10	C₂×C₅×SD₃₂	C₄	1	2	yes	complemented
11	C₂×C₅×D₁₆	C₄	1	1	yes	characteristic, complemented
12	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₅×#108	C₂²	1	2	yes	—
15	C₂×C₅×#104	—	2	2	no	—
16	C₅×#408	—	2	2	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#368	—	2	2	no	—
19	C₅×#359	—	2	4	no	—
20	C₅×#346	—	2	2	no	—
21	#13242	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×#41	—	2	2	no	—
23	C₅×#135	—	2	2	no	—
24	C₅×#135	—	2	2	no	—
25	C₅×#135	—	2	2	no	—
26	C₅×#135	—	2	2	no	—
27	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅×SD₁₆	—	2	2	no	—
29	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
30	C₂×C₅×M₁₆	—	2	2	no	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₅×Q₃₂	—	2	2	no	—
35	C₅×SD₃₂	—	2	4	no	—
36	C₅×D₁₆	—	2	2	no	—
37	C₂×C₅×C₁₆	—	2	1	no	—
38	C₂×C₅×C₁₆	—	2	1	no	—
39	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅×C₁₆	D₄	1	2	yes	—
42	C₂×C₅×C₁₆	D₄	1	2	yes	—
43	C₂×C₅×C₁₆	D₄	1	2	yes	—
44	C₅×#108	—	2	2	no	—
45	C₅×#108	—	2	2	no	—
46	C₅×#104	—	4	4	no	—
47	C₅×C₂²⋊C₈	—	4	4	no	—
48	C₄×C₅×C₈	—	2	1	no	—
49	C₄×C₅×C₈	—	2	1	no	—
50	C₂×#508	C₂×C₅	1	1	yes	characteristic
51	C₂×#368	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×#359	C₂×C₅	1	2	yes	complemented
53	C₂×#346	C₂×C₅	1	1	yes	characteristic, complemented
54	#1974	C₂×C₅	1	2	yes	complemented
55	C₅×#41	—	4	2	no	—
56	C₅×#41	—	4	2	no	—
57	C₅×#41	—	4	2	no	—
58	C₅×#41	—	4	2	no	—
59	C₂×C₅×Q₈	—	2	2	no	—
60	C₂×C₅×D₄	—	2	2	no	—
61	C₂×C₅×D₄	—	4	2	no	—
62	C₂²×C₄×C₅	—	2	1	no	—
63	C₂²×C₄×C₅	—	4	2	no	—
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₅×Q₁₆	—	2	2	no	—
66	C₅×Q₁₆	—	2	2	no	—
67	C₅×SD₁₆	—	4	2	no	—
68	C₅×SD₁₆	—	4	2	no	—
69	C₅×D₈	—	2	2	no	—
70	C₅×D₈	—	2	2	no	—
71	C₅×M₁₆	—	4	2	no	—
72	C₅×M₁₆	—	4	2	no	—
73	C₂×C₅×C₈	—	2	1	no	—
74	C₂×C₅×C₈	—	2	1	no	—
75	C₂×C₅×C₈	—	4	2	no	—
76	C₂×C₅×C₈	D₈	1	2	yes	—
77	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₅×C₈	#41	1	1	yes	characteristic
82	C₂×C₅×C₈	#41	1	1	yes	characteristic
83	C₄²×C₅	—	2	1	no	—
84	C₄²×C₅	D₈	1	2	yes	—
85	C₅×C₁₆	—	2	2	no	—
86	C₅×C₁₆	C₂²⋊C₄	1	2	yes	—
87	C₅×C₁₆	C₂²⋊C₄	1	2	yes	—
88	C₂×#135	C₂²×C₅	1	2	yes	—
89	#508	—	2	2	no	—
90	#508	—	2	2	no	—
91	C₂×Q₃₂	C₄×C₅	1	1	yes	characteristic, complemented
92	C₂×SD₃₂	C₄×C₅	1	2	yes	complemented
93	C₂×D₁₆	C₄×C₅	1	1	yes	characteristic, complemented
94	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
95	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
96	C₂×#108	C₂²×C₅	1	2	yes	—
97	C₂×#104	—	2	2	no	—
98	#408	—	2	2	no	—
99	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
100	#368	—	2	2	no	—
101	#359	—	2	4	no	—
102	#346	—	2	2	no	—
103	C₂³×C₅	—	4	2	no	—
104	C₂³×C₅	#135	1	1	yes	characteristic
105	C₅×Q₈	—	4	2	no	—
106	C₅×Q₈	—	4	2	no	—
107	C₅×D₄	—	4	2	no	—
108	C₅×D₄	—	4	2	no	—
109	C₅×D₄	—	8	2	no	—
110	C₅×D₄	—	8	2	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	4	2	no	—
113	C₂×C₄×C₅	—	4	2	no	—
114	C₂×C₄×C₅	—	8	2	no	—
115	C₂×C₄×C₅	—	8	2	no	—
116	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅	#121	1	1	yes	characteristic
118	C₂×C₄×C₅	#127	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅	#135	1	1	yes	characteristic
122	C₅×C₈	—	2	2	no	—
123	C₅×C₈	—	8	2	no	—
124	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₅×C₈	C₄×D₄	1	1	yes	characteristic
126	C₅×C₈	#121	1	1	yes	characteristic
127	C₅×C₈	#124	1	1	yes	characteristic
128	C₂×#41	—	2	2	no	—
129	#135	—	2	2	no	—
130	#135	—	2	2	no	—
131	#135	—	2	2	no	—
132	#135	—	2	2	no	—
133	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
134	C₂×SD₁₆	—	2	2	no	—
135	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
136	C₂×M₁₆	—	2	2	no	—
137	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
138	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
139	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
140	Q₃₂	—	2	2	no	—
141	SD₃₂	—	2	4	no	—
142	D₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	—	2	1	no	—
145	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂×C₁₆	C₅×D₄	1	2	yes	—
148	C₂×C₁₆	C₅×D₄	1	2	yes	—
149	C₂×C₁₆	C₅×D₄	1	2	yes	—
150	#108	—	2	2	no	—
151	#108	—	2	2	no	—
152	#104	—	4	4	no	—
153	C₂²⋊C₈	—	4	4	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₂²×C₅	—	2	2	no	—
157	C₂²×C₅	—	8	4	no	—
158	C₂²×C₅	—	8	2	no	—
159	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
160	C₂²×C₅	#466	1	1	yes	characteristic
161	C₂²×C₅	#495	1	1	yes	characteristic
162	C₄×C₅	—	4	2	no	—
163	C₄×C₅	—	8	2	no	—
164	C₄×C₅	#390	1	1	yes	characteristic
165	C₄×C₅	C₄×D₈	1	1	yes	characteristic
166	C₄×C₅	#466	1	1	yes	characteristic
167	C₄×C₅	#493	1	1	yes	characteristic
168	#41	—	4	2	no	—
169	#41	—	4	2	no	—
170	#41	—	4	2	no	—
171	#41	—	4	2	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
178	Q₁₆	—	2	2	no	—
179	Q₁₆	—	2	2	no	—
180	SD₁₆	—	4	2	no	—
181	SD₁₆	—	4	2	no	—
182	D₈	—	2	2	no	—
183	D₈	—	2	2	no	—
184	M₁₆	—	4	2	no	—
185	M₁₆	—	4	2	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	C₅×D₈	1	2	yes	—
190	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
191	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
192	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
193	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
194	C₂×C₈	C₅×#41	1	1	yes	characteristic
195	C₂×C₈	C₅×#41	1	1	yes	characteristic
196	C₄²	—	2	1	no	—
197	C₄²	C₅×D₈	1	2	yes	—
198	C₁₆	—	2	2	no	—
199	C₁₆	C₅×C₂²⋊C₄	1	2	yes	—
200	C₁₆	C₅×C₂²⋊C₄	1	2	yes	—
201	C₂×C₅	—	2	1	no	—
202	C₂×C₅	—	2	1	no	—
203	C₂×C₅	—	16	2	no	—
204	C₂×C₅	#1970	1	1	yes	characteristic
205	C₂×C₅	#2179	1	2	yes	—
206	C₂³	—	4	2	no	—
207	C₂³	C₅×#135	1	1	yes	characteristic
208	Q₈	—	4	2	no	—
209	Q₈	—	4	2	no	—
210	D₄	—	4	2	no	—
211	D₄	—	4	2	no	—
212	D₄	—	8	2	no	—
213	D₄	—	8	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
220	C₂×C₄	C₅×#121	1	1	yes	characteristic
221	C₂×C₄	C₅×#127	1	1	yes	characteristic
222	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
223	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
224	C₂×C₄	C₅×#135	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	8	2	no	—
227	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
228	C₈	C₄×C₅×D₄	1	1	yes	characteristic
229	C₈	C₅×#121	1	1	yes	characteristic
230	C₈	C₅×#124	1	1	yes	characteristic
231	C₅	#13242	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²	—	2	2	no	—
233	C₂²	—	8	4	no	—
234	C₂²	—	8	2	no	—
235	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
236	C₂²	C₅×#466	1	1	yes	characteristic
237	C₂²	C₅×#495	1	1	yes	characteristic
238	C₄	—	4	2	no	—
239	C₄	—	8	2	no	—
240	C₄	C₅×#390	1	1	yes	characteristic
241	C₄	C₄×C₅×D₈	1	1	yes	characteristic
242	C₄	C₅×#466	1	1	yes	characteristic
243	C₄	C₅×#493	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	—	16	2	no	—
247	C₂	C₅×#1970	1	1	yes	characteristic
248	C₂	C₅×#2179	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₄×D₈
prime radical	C₂×C₅×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#368	C₂	semidirect
3. C₂×C₅×#359	C₂	semidirect
4. C₂×C₅×#346	C₂	semidirect
5. C₅×#1974	C₂	semidirect
9. C₂×C₅×Q₃₂	C₄	semidirect
10. C₂×C₅×SD₃₂	C₄	semidirect
11. C₂×C₅×D₁₆	C₄	semidirect
21. #13242	C₅	direct (N × H)
51. C₂×#368	C₂×C₅	semidirect
52. C₂×#359	C₂×C₅	semidirect
53. C₂×#346	C₂×C₅	semidirect
54. #1974	C₂×C₅	semidirect
91. C₂×Q₃₂	C₄×C₅	semidirect
92. C₂×SD₃₂	C₄×C₅	semidirect
93. C₂×D₁₆	C₄×C₅	semidirect
231. C₅	#13242	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×#508	2
C₂×C₅×#368	2
C₂×C₅×#359	2
C₂×C₅×#346	2
C₅×#1974	2
#13242	5

Subgroups — C5×#13242 #11935962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13242 #11935962