Subgroups — C₅×#11966 #11934686

Order 1280.

400 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3147	C₂	1	1	yes	maximal, characteristic
2	C₅×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1362	C₂	1	2	yes	maximal, complemented
5	C₅×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#135	—	2	1	no	—
7	C₂×C₅×#135	C₄	1	2	yes	—
8	C₅×#532	—	2	1	no	—
9	C₅×#495	—	2	1	no	—
10	C₅×#495	C₄	1	2	yes	—
11	C₅×#494	C₂²	1	1	yes	characteristic
12	C₅×#493	C₂²	1	1	yes	characteristic
13	C₅×#492	—	2	1	no	—
14	C₅×#423	—	2	1	no	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₅×C₂²⋊C₁₆	—	2	1	no	—
18	C₅×#346	—	2	1	no	—
19	C₅×#335	—	2	1	no	—
20	C₅×#335	C₂²	1	2	yes	—
21	#11966	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×#41	—	2	2	no	—
23	C₅×#135	—	4	4	no	—
24	C₅×#135	—	4	4	no	—
25	C₂×C₅×Q₁₆	—	2	1	no	—
26	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₅×SD₁₆	—	2	2	no	—
28	C₂×C₅×SD₁₆	—	2	2	no	—
29	C₂×C₅×D₈	—	2	1	no	—
30	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
31	C₂×C₅×M₁₆	—	4	1	no	—
32	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₅×C₄⋊Q₈	—	2	1	no	—
35	C₅×#127	—	2	1	no	—
36	C₅×#124	—	2	2	no	—
37	C₅×#121	—	4	1	no	—
38	C₅×#121	—	4	1	no	—
39	C₄×C₅×D₄	—	4	1	no	—
40	C₄×C₅×D₄	—	4	1	no	—
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₅×C₁₆	—	2	2	no	—
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₄⋊C₈	—	4	1	no	—
45	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₅×C₈	D₄	1	1	yes	characteristic
47	C₄×C₅×C₈	D₄	1	1	yes	characteristic
48	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
49	#3147	C₂×C₅	1	1	yes	characteristic
50	#2163	C₂×C₅	1	1	yes	characteristic, complemented
51	#1567	C₂×C₅	1	1	yes	characteristic, complemented
52	#1362	C₂×C₅	1	2	yes	complemented
53	#1361	C₂×C₅	1	2	yes	complemented
54	C₅×#41	—	4	4	no	—
55	C₅×#41	—	4	4	no	—
56	C₂×C₅×Q₈	—	2	2	no	—
57	C₂×C₅×D₄	—	2	2	no	—
58	C₂×C₅×D₄	—	4	2	no	—
59	C₂×C₅×D₄	—	4	1	no	—
60	C₂×C₅×D₄	—	4	1	no	—
61	C₂²×C₄×C₅	—	4	2	no	—
62	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
65	C₅×Q₁₆	—	4	2	no	—
66	C₅×Q₁₆	—	4	2	no	—
67	C₅×SD₁₆	—	4	4	no	—
68	C₅×SD₁₆	—	4	4	no	—
69	C₅×D₈	—	4	2	no	—
70	C₅×D₈	—	4	2	no	—
71	C₅×M₁₆	—	8	2	no	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	4	2	no	—
75	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
79	C₅×C₄⋊C₄	—	4	1	no	—
80	C₅×C₄⋊C₄	—	4	1	no	—
81	C₅×C₂²⋊C₄	—	4	2	no	—
82	C₅×C₂²⋊C₄	—	4	2	no	—
83	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₅	D₈	1	1	yes	characteristic
85	C₄²×C₅	SD₁₆	1	1	yes	characteristic
86	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
87	C₅×C₁₆	—	4	2	no	—
88	C₂×#135	—	2	1	no	—
89	C₂×#135	C₄×C₅	1	2	yes	—
90	#532	—	2	1	no	—
91	#495	—	2	1	no	—
92	#495	C₄×C₅	1	2	yes	—
93	#494	C₂²×C₅	1	1	yes	characteristic
94	#493	C₂²×C₅	1	1	yes	characteristic
95	#492	—	2	1	no	—
96	#423	—	2	1	no	—
97	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
98	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
99	C₂²⋊C₁₆	—	2	1	no	—
100	#346	—	2	1	no	—
101	#335	—	2	1	no	—
102	#335	C₂²×C₅	1	2	yes	—
103	C₂³×C₅	—	4	2	no	—
104	C₂³×C₅	#102	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	4	no	—
111	C₅×D₄	—	8	1	no	—
112	C₅×D₄	—	8	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	4	1	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	—	8	4	no	—
120	C₂×C₄×C₅	#102	1	1	yes	characteristic
121	C₂×C₄×C₅	#102	1	1	yes	characteristic
122	C₂×C₄×C₅	#102	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
126	C₅×C₈	—	2	1	no	—
127	C₅×C₈	—	2	1	no	—
128	C₅×C₈	—	8	2	no	—
129	C₅×C₈	#100	1	2	yes	—
130	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×#41	—	2	2	no	—
132	#135	—	4	4	no	—
133	#135	—	4	4	no	—
134	C₂×Q₁₆	—	2	1	no	—
135	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
136	C₂×SD₁₆	—	2	2	no	—
137	C₂×SD₁₆	—	2	2	no	—
138	C₂×D₈	—	2	1	no	—
139	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
140	C₂×M₁₆	—	4	1	no	—
141	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
142	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	1	no	—
144	#127	—	2	1	no	—
145	#124	—	2	2	no	—
146	#121	—	4	1	no	—
147	#121	—	4	1	no	—
148	C₄×D₄	—	4	1	no	—
149	C₄×D₄	—	4	1	no	—
150	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
151	C₂×C₁₆	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
155	C₄×C₈	C₅×D₄	1	1	yes	characteristic
156	C₄×C₈	C₅×D₄	1	1	yes	characteristic
157	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
158	C₂²×C₅	—	2	1	no	—
159	C₂²×C₅	—	2	1	no	—
160	C₂²×C₅	—	2	1	no	—
161	C₂²×C₅	—	8	2	no	—
162	C₂²×C₅	—	8	2	no	—
163	C₂²×C₅	—	8	2	no	—
164	C₂²×C₅	C₂×#102	1	1	yes	characteristic
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	8	2	no	—
170	C₄×C₅	C₂×#100	1	1	yes	characteristic
171	C₄×C₅	#418	1	1	yes	characteristic
172	C₄×C₅	#509	1	1	yes	characteristic
173	C₄×C₅	#510	1	1	yes	characteristic
174	#41	—	4	4	no	—
175	#41	—	4	4	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	1	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
184	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
185	Q₁₆	—	4	2	no	—
186	Q₁₆	—	4	2	no	—
187	SD₁₆	—	4	4	no	—
188	SD₁₆	—	4	4	no	—
189	D₈	—	4	2	no	—
190	D₈	—	4	2	no	—
191	M₁₆	—	8	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
198	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
199	C₄⋊C₄	—	4	1	no	—
200	C₄⋊C₄	—	4	1	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
204	C₄²	C₅×D₈	1	1	yes	characteristic
205	C₄²	C₅×SD₁₆	1	1	yes	characteristic
206	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
207	C₁₆	—	4	2	no	—
208	C₂×C₅	—	4	1	no	—
209	C₂×C₅	—	16	2	no	—
210	C₂×C₅	#1545	1	1	yes	characteristic
211	C₂×C₅	#2139	1	1	yes	characteristic
212	C₂×C₅	#2148	1	1	yes	characteristic
213	C₂³	—	4	2	no	—
214	C₂³	C₅×#102	1	1	yes	characteristic
215	Q₈	—	4	2	no	—
216	Q₈	—	4	2	no	—
217	D₄	—	4	2	no	—
218	D₄	—	4	2	no	—
219	D₄	—	8	2	no	—
220	D₄	—	8	4	no	—
221	D₄	—	8	1	no	—
222	D₄	—	8	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	C₅×#102	1	1	yes	characteristic
231	C₂×C₄	C₅×#102	1	1	yes	characteristic
232	C₂×C₄	C₅×#102	1	1	yes	characteristic
233	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
235	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	8	2	no	—
239	C₈	C₅×#100	1	2	yes	—
240	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
241	C₅	#11966	1	1	yes	characteristic, direct product factor, Sylow
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	C₂×C₅×#102	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	2	no	—
254	C₄	C₂×C₅×#100	1	1	yes	characteristic
255	C₄	C₅×#418	1	1	yes	characteristic
256	C₄	C₅×#509	1	1	yes	characteristic
257	C₄	C₅×#510	1	1	yes	characteristic
258	C₂	—	4	1	no	—
259	C₂	—	16	2	no	—
260	C₂	C₅×#1545	1	1	yes	characteristic
261	C₂	C₅×#2139	1	1	yes	characteristic
262	C₂	C₅×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#3147	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×#2163	C₂	semidirect
3. C₅×#1567	C₂	semidirect
4. C₅×#1362	C₂	semidirect
5. C₅×#1361	C₂	semidirect
21. #11966	C₅	direct (N × H)
50. #2163	C₂×C₅	semidirect
51. #1567	C₂×C₅	semidirect
52. #1362	C₂×C₅	semidirect
53. #1361	C₂×C₅	semidirect
241. C₅	#11966	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×#3147	2
C₅×#2163	2
C₅×#1567	2
C₅×#1362	2
C₅×#1361	2
#11966	5

Subgroups — C5×#11966 #11934686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11966 #11934686