Subgroups — C₅×#13278 #11935998

Order 1280.

358 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3160	C₂	1	1	yes	maximal, characteristic
2	C₅×#2150	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#106	C₂	1	1	yes	maximal, characteristic
4	C₅×#1386	C₂	1	4	yes	maximal
5	C₅×#533	—	2	1	no	—
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₅×#500	C₄	1	1	yes	characteristic, complemented
8	C₅×#500	C₄	1	1	yes	characteristic, complemented
9	C₅×#499	—	2	1	no	—
10	C₅×#499	C₄	1	2	yes	complemented
11	C₅×#498	—	2	1	no	—
12	C₅×#427	—	2	1	no	—
13	C₅×#427	C₂²	1	2	yes	—
14	C₂×C₅×#106	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₄×C₅×C₄⋊C₄	—	2	1	no	—
17	C₅×C₄⋊C₁₆	—	2	1	no	—
18	C₅×C₄⋊C₁₆	—	2	1	no	—
19	C₅×#340	—	2	1	no	—
20	#13278	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
23	C₅×C₄⋊Q₈	—	2	1	no	—
24	C₅×C₄⋊Q₈	—	2	1	no	—
25	C₅×#125	—	2	2	no	—
26	C₅×C₂²⋊Q₈	—	4	1	no	—
27	C₅×C₂²⋊Q₈	—	4	1	no	—
28	C₄×C₅×Q₈	—	4	1	no	—
29	C₄×C₅×Q₈	—	4	1	no	—
30	C₅×#117	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	—	2	2	no	—
32	C₂×C₄²×C₅	—	4	1	no	—
33	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₅×M₃₂	—	2	4	no	—
35	C₂×C₅×C₁₆	—	2	2	no	—
36	C₅×#107	—	2	1	no	—
37	C₅×#107	—	2	1	no	—
38	C₅×#107	C₂×C₄	1	2	yes	—
39	C₅×#107	C₂×C₄	1	2	yes	—
40	C₅×#106	—	2	2	no	—
41	C₅×#106	—	2	2	no	—
42	C₅×#106	—	2	2	no	—
43	C₅×#106	Q₈	1	4	yes	complemented
44	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
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	C₅×#95	—	4	1	no	—
49	#3160	C₂×C₅	1	1	yes	characteristic
50	#2150	C₂×C₅	1	1	yes	characteristic
51	C₄×#106	C₂×C₅	1	1	yes	characteristic
52	#1386	C₂×C₅	1	4	yes	—
53	C₂×C₅×Q₈	—	4	1	no	—
54	C₂×C₅×Q₈	—	4	1	no	—
55	C₂²×C₄×C₅	—	4	2	no	—
56	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
65	C₂×C₅×C₈	#41	1	2	yes	—
66	C₅×C₄⋊C₄	—	2	2	no	—
67	C₅×C₄⋊C₄	—	2	2	no	—
68	C₅×C₄⋊C₄	—	2	4	no	—
69	C₅×C₄⋊C₄	—	4	2	no	—
70	C₅×C₄⋊C₄	—	4	2	no	—
71	C₅×C₄⋊C₄	—	4	1	no	—
72	C₅×C₄⋊C₄	—	4	1	no	—
73	C₅×C₄⋊C₄	—	4	2	no	—
74	C₅×C₂²⋊C₄	—	4	2	no	—
75	C₄²×C₅	—	4	2	no	—
76	C₄²×C₅	—	8	2	no	—
77	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₅	SD₁₆	1	1	yes	characteristic
80	C₄²×C₅	SD₁₆	1	1	yes	characteristic
81	C₅×C₁₆	—	4	2	no	—
82	#533	—	2	1	no	—
83	C₂×M₃₂	C₂²×C₅	1	2	yes	—
84	#500	C₄×C₅	1	1	yes	characteristic, complemented
85	#500	C₄×C₅	1	1	yes	characteristic, complemented
86	#499	—	2	1	no	—
87	#499	C₄×C₅	1	2	yes	complemented
88	#498	—	2	1	no	—
89	#427	—	2	1	no	—
90	#427	C₂²×C₅	1	2	yes	—
91	C₂×#106	C₂²×C₅	1	2	yes	—
92	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
93	C₄×C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₁₆	—	2	1	no	—
95	C₄⋊C₁₆	—	2	1	no	—
96	#340	—	2	1	no	—
97	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
98	C₅×Q₈	—	8	2	no	—
99	C₅×Q₈	—	8	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	4	1	no	—
102	C₂×C₄×C₅	—	4	2	no	—
103	C₂×C₄×C₅	—	4	2	no	—
104	C₂×C₄×C₅	—	4	4	no	—
105	C₂×C₄×C₅	—	8	2	no	—
106	C₂×C₄×C₅	—	8	2	no	—
107	C₂×C₄×C₅	#102	1	1	yes	characteristic
108	C₂×C₄×C₅	#102	1	1	yes	characteristic
109	C₂×C₄×C₅	#102	1	1	yes	characteristic
110	C₂×C₄×C₅	#102	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₅	#137	1	2	yes	—
115	C₅×C₈	—	2	1	no	—
116	C₅×C₈	—	2	1	no	—
117	C₅×C₈	C₄×Q₈	1	2	yes	—
118	C₅×C₈	#124	1	1	yes	characteristic
119	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
120	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
121	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
122	C₄⋊Q₈	—	2	1	no	—
123	C₄⋊Q₈	—	2	1	no	—
124	#125	—	2	2	no	—
125	C₂²⋊Q₈	—	4	1	no	—
126	C₂²⋊Q₈	—	4	1	no	—
127	C₄×Q₈	—	4	1	no	—
128	C₄×Q₈	—	4	1	no	—
129	#117	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄²	—	4	1	no	—
132	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
133	M₃₂	—	2	4	no	—
134	C₂×C₁₆	—	2	2	no	—
135	#107	—	2	1	no	—
136	#107	—	2	1	no	—
137	#107	C₂×C₄×C₅	1	2	yes	—
138	#107	C₂×C₄×C₅	1	2	yes	—
139	#106	—	2	2	no	—
140	#106	—	2	2	no	—
141	#106	—	2	2	no	—
142	#106	C₅×Q₈	1	4	yes	complemented
143	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
144	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
145	C₄×C₈	C₅×D₄	1	1	yes	characteristic
146	C₄×C₈	C₅×D₄	1	1	yes	characteristic
147	#95	—	4	1	no	—
148	C₂²×C₅	—	2	1	no	—
149	C₂²×C₅	—	2	1	no	—
150	C₂²×C₅	C₂×#102	1	1	yes	characteristic
151	C₂²×C₅	#474	1	2	yes	—
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	2	1	no	—
154	C₄×C₅	—	2	1	no	—
155	C₄×C₅	—	8	4	no	—
156	C₄×C₅	—	8	2	no	—
157	C₄×C₅	—	8	2	no	—
158	C₄×C₅	#360	1	2	yes	—
159	C₄×C₅	#391	1	1	yes	characteristic
160	C₄×C₅	#419	1	1	yes	characteristic
161	C₄×C₅	#475	1	2	yes	—
162	C₂×Q₈	—	4	1	no	—
163	C₂×Q₈	—	4	1	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
166	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
172	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
174	C₂×C₈	C₅×#41	1	2	yes	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	—	8	2	no	—
186	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
187	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
188	C₄²	C₅×SD₁₆	1	1	yes	characteristic
189	C₄²	C₅×SD₁₆	1	1	yes	characteristic
190	C₁₆	—	4	2	no	—
191	C₂×C₅	—	2	2	no	—
192	C₂×C₅	#1988	1	1	yes	characteristic
193	C₂×C₅	C₂×#360	1	1	yes	characteristic
194	C₂×C₅	#2145	1	1	yes	characteristic
195	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
196	Q₈	—	8	2	no	—
197	Q₈	—	8	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	C₅×#102	1	1	yes	characteristic
206	C₂×C₄	C₅×#102	1	1	yes	characteristic
207	C₂×C₄	C₅×#102	1	1	yes	characteristic
208	C₂×C₄	C₅×#102	1	1	yes	characteristic
209	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
211	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
212	C₂×C₄	C₅×#137	1	2	yes	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	C₄×C₅×Q₈	1	2	yes	—
216	C₈	C₅×#124	1	1	yes	characteristic
217	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
218	C₅	#13278	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₂×C₅×#102	1	1	yes	characteristic
222	C₂²	C₅×#474	1	2	yes	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	4	no	—
227	C₄	—	8	2	no	—
228	C₄	—	8	2	no	—
229	C₄	C₅×#360	1	2	yes	—
230	C₄	C₅×#391	1	1	yes	characteristic
231	C₄	C₅×#419	1	1	yes	characteristic
232	C₄	C₅×#475	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	C₅×#1988	1	1	yes	characteristic
235	C₂	C₂×C₅×#360	1	1	yes	characteristic
236	C₂	C₅×#2145	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	C₂³×C₅	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ 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
7. C₅×#500	C₄	semidirect
8. C₅×#500	C₄	semidirect
10. C₅×#499	C₄	semidirect
20. #13278	C₅	direct (N × H)
43. C₅×#106	Q₈	semidirect
84. #500	C₄×C₅	semidirect
85. #500	C₄×C₅	semidirect
87. #499	C₄×C₅	semidirect
142. #106	C₅×Q₈	semidirect
218. C₅	#13278	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×#3160	2
C₅×#2150	2
C₄×C₅×#106	2
C₅×#1386	2
#13278	5

Subgroups — C5×#13278 #11935998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13278 #11935998