Subgroups — C₅×#11856 #11934576

Order 1280.

378 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1834	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1347	C₂	1	4	yes	maximal, complemented
5	C₅×#532	—	2	1	no	—
6	C₅×#500	C₄	1	1	yes	characteristic, complemented
7	C₅×#493	C₄	1	1	yes	characteristic, complemented
8	C₅×#486	—	2	1	no	—
9	C₅×#466	—	2	1	no	—
10	C₅×#466	C₂²	1	2	yes	—
11	C₄×C₅×D₈	—	2	1	no	—
12	C₄×C₅×D₈	C₄	1	2	yes	complemented
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#384	—	2	1	no	—
15	C₅×C₄⋊C₁₆	—	2	1	no	—
16	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
17	C₄×C₅×C₁₆	—	2	1	no	—
18	C₅×#340	—	2	1	no	—
19	C₅×#340	C₂²	1	2	yes	—
20	#11856	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×#41	—	4	1	no	—
22	C₂×C₅×D₈	—	2	1	no	—
23	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₅×C₄⋊Q₈	—	2	1	no	—
27	C₅×#127	—	2	1	no	—
28	C₅×#124	—	4	1	no	—
29	C₅×#121	—	2	2	no	—
30	C₅×#121	—	4	1	no	—
31	C₄×C₅×D₄	—	2	2	no	—
32	C₄×C₅×D₄	—	4	1	no	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	4	1	no	—
35	C₂×C₅×C₄⋊C₄	—	4	1	no	—
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₅×C₁₆	—	2	2	no	—
38	C₅×#107	—	2	1	no	—
39	C₅×#107	C₂×C₄	1	2	yes	—
40	C₅×#102	—	2	2	no	—
41	C₅×#102	—	2	2	no	—
42	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	#3169	C₂×C₅	1	1	yes	characteristic
47	#2162	C₂×C₅	1	1	yes	characteristic, complemented
48	#1834	C₂×C₅	1	1	yes	characteristic, complemented
49	#1347	C₂×C₅	1	4	yes	complemented
50	C₅×#41	—	8	2	no	—
51	C₅×#41	—	8	2	no	—
52	C₂×C₅×Q₈	—	4	1	no	—
53	C₂×C₅×D₄	—	2	2	no	—
54	C₂×C₅×D₄	—	4	2	no	—
55	C₂×C₅×D₄	—	4	1	no	—
56	C₂×C₅×D₄	—	4	1	no	—
57	C₂²×C₄×C₅	—	4	2	no	—
58	C₂²×C₄×C₅	—	4	2	no	—
59	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₅×D₈	—	4	2	no	—
63	C₅×D₈	—	4	2	no	—
64	C₂×C₅×C₈	—	4	1	no	—
65	C₂×C₅×C₈	—	4	1	no	—
66	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₅×C₄⋊C₄	—	2	2	no	—
71	C₅×C₄⋊C₄	—	4	2	no	—
72	C₅×C₄⋊C₄	—	4	2	no	—
73	C₅×C₄⋊C₄	—	4	1	no	—
74	C₅×C₄⋊C₄	—	8	2	no	—
75	C₅×C₄⋊C₄	—	8	2	no	—
76	C₅×C₂²⋊C₄	—	4	2	no	—
77	C₅×C₂²⋊C₄	—	4	2	no	—
78	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₅	SD₁₆	1	1	yes	characteristic
81	C₄²×C₅	SD₁₆	1	1	yes	characteristic
82	C₅×C₁₆	—	4	2	no	—
83	#532	—	2	1	no	—
84	#500	C₄×C₅	1	1	yes	characteristic, complemented
85	#493	C₄×C₅	1	1	yes	characteristic, complemented
86	#486	—	2	1	no	—
87	#466	—	2	1	no	—
88	#466	C₂²×C₅	1	2	yes	—
89	C₄×D₈	—	2	1	no	—
90	C₄×D₈	C₄×C₅	1	2	yes	complemented
91	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
92	#384	—	2	1	no	—
93	C₄⋊C₁₆	—	2	1	no	—
94	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
95	C₄×C₁₆	—	2	1	no	—
96	#340	—	2	1	no	—
97	#340	C₂²×C₅	1	2	yes	—
98	C₂³×C₅	—	4	2	no	—
99	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
100	C₅×Q₈	—	8	2	no	—
101	C₅×D₄	—	4	2	no	—
102	C₅×D₄	—	4	2	no	—
103	C₅×D₄	—	8	2	no	—
104	C₅×D₄	—	8	2	no	—
105	C₅×D₄	—	8	2	no	—
106	C₅×D₄	—	8	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	4	2	no	—
111	C₂×C₄×C₅	—	8	4	no	—
112	C₂×C₄×C₅	—	8	2	no	—
113	C₂×C₄×C₅	—	8	2	no	—
114	C₂×C₄×C₅	—	8	2	no	—
115	C₂×C₄×C₅	#102	1	1	yes	characteristic
116	C₂×C₄×C₅	#102	1	1	yes	characteristic
117	C₂×C₄×C₅	#102	1	1	yes	characteristic
118	C₂×C₄×C₅	#102	1	1	yes	characteristic
119	C₂×C₄×C₅	D₁₆	1	2	yes	—
120	C₂×C₄×C₅	SD₃₂	1	2	yes	—
121	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
123	C₅×C₈	—	2	1	no	—
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	—	2	1	no	—
126	C₅×C₈	—	2	1	no	—
127	C₂×#41	—	4	1	no	—
128	C₂×D₈	—	2	1	no	—
129	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
130	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
131	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
132	C₄⋊Q₈	—	2	1	no	—
133	#127	—	2	1	no	—
134	#124	—	4	1	no	—
135	#121	—	2	2	no	—
136	#121	—	4	1	no	—
137	C₄×D₄	—	2	2	no	—
138	C₄×D₄	—	4	1	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
143	C₂×C₁₆	—	2	2	no	—
144	#107	—	2	1	no	—
145	#107	C₂×C₄×C₅	1	2	yes	—
146	#102	—	2	2	no	—
147	#102	—	2	2	no	—
148	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
149	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
150	C₄×C₈	C₅×D₄	1	1	yes	characteristic
151	C₄×C₈	C₅×D₄	1	1	yes	characteristic
152	C₂²×C₅	—	2	1	no	—
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	—	8	2	no	—
155	C₂²×C₅	—	8	2	no	—
156	C₂²×C₅	—	8	2	no	—
157	C₂²×C₅	C₂×#102	1	1	yes	characteristic
158	C₂²×C₅	C₂×D₁₆	1	1	yes	characteristic
159	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	8	2	no	—
163	C₄×C₅	—	16	2	no	—
164	C₄×C₅	#357	1	2	yes	—
165	C₄×C₅	#357	1	2	yes	—
166	C₄×C₅	#360	1	2	yes	—
167	C₄×C₅	#410	1	1	yes	characteristic
168	C₄×C₅	#421	1	1	yes	characteristic
169	#41	—	8	2	no	—
170	#41	—	8	2	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	4	2	no	—
174	C₂×D₄	—	4	1	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
179	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
181	D₈	—	4	2	no	—
182	D₈	—	4	2	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
188	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	1	no	—
193	C₄⋊C₄	—	8	2	no	—
194	C₄⋊C₄	—	8	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
198	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
199	C₄²	C₅×SD₁₆	1	1	yes	characteristic
200	C₄²	C₅×SD₁₆	1	1	yes	characteristic
201	C₁₆	—	4	2	no	—
202	C₂×C₅	—	2	1	no	—
203	C₂×C₅	—	2	1	no	—
204	C₂×C₅	—	16	2	no	—
205	C₂×C₅	#1517	1	1	yes	characteristic
206	C₂×C₅	C₂×#357	1	1	yes	characteristic
207	C₂×C₅	C₂×#360	1	1	yes	characteristic
208	C₂³	—	4	2	no	—
209	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
210	Q₈	—	8	2	no	—
211	D₄	—	4	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	8	2	no	—
214	D₄	—	8	2	no	—
215	D₄	—	8	2	no	—
216	D₄	—	8	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	4	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	C₅×#102	1	1	yes	characteristic
226	C₂×C₄	C₅×#102	1	1	yes	characteristic
227	C₂×C₄	C₅×#102	1	1	yes	characteristic
228	C₂×C₄	C₅×#102	1	1	yes	characteristic
229	C₂×C₄	C₅×D₁₆	1	2	yes	—
230	C₂×C₄	C₅×SD₃₂	1	2	yes	—
231	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₅	#11856	1	1	yes	characteristic, direct product factor, Sylow
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	C₂×C₅×#102	1	1	yes	characteristic
244	C₂²	C₂×C₅×D₁₆	1	1	yes	characteristic
245	C₂²	C₂×C₅×SD₃₂	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	8	2	no	—
249	C₄	—	16	2	no	—
250	C₄	C₅×#357	1	2	yes	—
251	C₄	C₅×#357	1	2	yes	—
252	C₄	C₅×#360	1	2	yes	—
253	C₄	C₅×#410	1	1	yes	characteristic
254	C₄	C₅×#421	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	16	2	no	—
258	C₂	C₅×#1517	1	1	yes	characteristic
259	C₂	C₂×C₅×#357	1	1	yes	characteristic
260	C₂	C₂×C₅×#360	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₅×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
2. C₅×#2162	C₂	semidirect
3. C₅×#1834	C₂	semidirect
4. C₅×#1347	C₂	semidirect
6. C₅×#500	C₄	semidirect
7. C₅×#493	C₄	semidirect
12. C₄×C₅×D₈	C₄	semidirect
20. #11856	C₅	direct (N × H)
47. #2162	C₂×C₅	semidirect
48. #1834	C₂×C₅	semidirect
49. #1347	C₂×C₅	semidirect
84. #500	C₄×C₅	semidirect
85. #493	C₄×C₅	semidirect
90. C₄×D₈	C₄×C₅	semidirect
237. C₅	#11856	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3169	2
C₅×#2162	2
C₅×#1834	2
C₅×#1347	2
#11856	5

Subgroups — C5×#11856 #11934576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11856 #11934576