Subgroups — C₅×#13138 #11935858

Order 1280.

354 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#365	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#361	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#361	C₂	1	1	yes	maximal, characteristic
4	C₅×#1972	C₂	1	1	yes	maximal, characteristic
5	C₅×#1778	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1377	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₅×#108	C₂²	1	1	yes	characteristic
12	C₂×C₅×#108	C₂²	1	1	yes	characteristic
13	C₂×C₅×#106	C₂²	1	1	yes	characteristic
14	C₂×C₅×#102	—	2	1	no	—
15	C₄×C₅×M₁₆	—	2	1	no	—
16	C₂×C₅×#97	C₂²	1	1	yes	characteristic
17	C₅×#390	—	2	1	no	—
18	C₅×#365	—	2	2	no	—
19	C₅×#361	—	2	2	no	—
20	C₅×#361	C₄	1	4	yes	complemented
21	C₅×#341	—	2	1	no	—
22	#13138	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×D₄	—	2	1	no	—
24	C₂×C₅×D₈	—	2	1	no	—
25	C₂×C₅×D₈	—	2	1	no	—
26	C₂×C₅×D₈	—	2	1	no	—
27	C₂×C₅×D₈	—	2	1	no	—
28	C₂×C₅×D₈	—	2	1	no	—
29	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
30	C₂×C₅×M₁₆	—	2	1	no	—
31	C₂×C₅×M₁₆	—	2	1	no	—
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₅×C₄⋊C₄	—	2	1	no	—
35	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₅×M₃₂	—	2	1	no	—
38	C₅×M₃₂	—	2	1	no	—
39	C₅×M₃₂	—	2	1	no	—
40	C₅×M₃₂	C₂×C₄	1	2	yes	—
41	C₂×C₅×C₁₆	—	2	1	no	—
42	C₂×C₅×C₁₆	—	2	1	no	—
43	C₅×#108	—	2	1	no	—
44	C₅×#108	—	2	1	no	—
45	C₅×#108	C₂×C₄	1	2	yes	—
46	C₅×#108	D₄	1	2	yes	—
47	C₅×#106	—	2	1	no	—
48	C₅×#106	—	2	1	no	—
49	C₅×#102	—	4	2	no	—
50	C₅×#97	—	2	1	no	—
51	C₅×#97	—	2	1	no	—
52	C₅×#97	—	4	1	no	—
53	C₄×C₅×C₈	—	4	1	no	—
54	C₂×#365	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×#361	C₂×C₅	1	1	yes	characteristic
56	C₂×#361	C₂×C₅	1	1	yes	characteristic
57	#1972	C₂×C₅	1	1	yes	characteristic
58	#1778	C₂×C₅	1	1	yes	characteristic, complemented
59	#1383	C₂×C₅	1	1	yes	characteristic, complemented
60	#1377	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂⁴×C₅	—	4	1	no	—
62	C₂×C₅×D₄	—	2	2	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂×C₅×D₄	—	4	1	no	—
65	C₂×C₅×D₄	—	4	1	no	—
66	C₂×C₅×D₄	—	4	1	no	—
67	C₂×C₅×D₄	—	4	1	no	—
68	C₂²×C₄×C₅	—	4	1	no	—
69	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₅×D₈	—	2	2	no	—
72	C₅×D₈	—	2	2	no	—
73	C₅×D₈	—	4	1	no	—
74	C₅×D₈	—	4	1	no	—
75	C₅×M₁₆	—	2	2	no	—
76	C₅×M₁₆	—	2	2	no	—
77	C₅×M₁₆	—	4	1	no	—
78	C₅×M₁₆	—	4	1	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	—	2	1	no	—
82	C₂×C₅×C₈	—	4	1	no	—
83	C₂×C₅×C₈	—	4	1	no	—
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂×C₅×C₈	#41	1	1	yes	characteristic
87	C₂×C₅×C₈	#41	1	1	yes	characteristic
88	C₂×C₅×C₈	#41	1	1	yes	characteristic
89	C₂×C₅×C₈	#41	1	1	yes	characteristic
90	C₅×C₄⋊C₄	—	4	1	no	—
91	C₅×C₄⋊C₄	—	4	1	no	—
92	C₅×C₂²⋊C₄	—	8	4	no	—
93	C₄²×C₅	—	2	1	no	—
94	C₄²×C₅	—	2	1	no	—
95	C₅×C₁₆	—	2	2	no	—
96	C₅×C₁₆	—	4	1	no	—
97	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
98	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
99	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
100	C₂×#108	C₂²×C₅	1	1	yes	characteristic
101	C₂×#108	C₂²×C₅	1	1	yes	characteristic
102	C₂×#106	C₂²×C₅	1	1	yes	characteristic
103	C₂×#102	—	2	1	no	—
104	C₄×M₁₆	—	2	1	no	—
105	C₂×#97	C₂²×C₅	1	1	yes	characteristic
106	#390	—	2	1	no	—
107	#365	—	2	2	no	—
108	#361	—	2	2	no	—
109	#361	C₄×C₅	1	4	yes	complemented
110	#341	—	2	1	no	—
111	C₂³×C₅	—	4	2	no	—
112	C₂³×C₅	—	8	2	no	—
113	C₂³×C₅	—	8	2	no	—
114	C₂³×C₅	—	8	1	no	—
115	C₂³×C₅	—	8	1	no	—
116	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
117	C₅×D₄	—	4	2	no	—
118	C₅×D₄	—	4	2	no	—
119	C₅×D₄	—	8	1	no	—
120	C₅×D₄	—	8	1	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	4	2	no	—
123	C₂×C₄×C₅	—	8	2	no	—
124	C₂×C₄×C₅	—	8	1	no	—
125	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅	#121	1	1	yes	characteristic
127	C₂×C₄×C₅	#123	1	1	yes	characteristic
128	C₂×C₄×C₅	#135	1	1	yes	characteristic
129	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₅×C₈	—	4	2	no	—
132	C₅×C₈	—	4	2	no	—
133	C₅×C₈	—	4	1	no	—
134	C₅×C₈	#117	1	1	yes	characteristic
135	C₅×C₈	C₄×D₄	1	1	yes	characteristic
136	C₅×C₈	#124	1	1	yes	characteristic
137	C₅×C₈	#126	1	1	yes	characteristic
138	C₂²×D₄	—	2	1	no	—
139	C₂×D₈	—	2	1	no	—
140	C₂×D₈	—	2	1	no	—
141	C₂×D₈	—	2	1	no	—
142	C₂×D₈	—	2	1	no	—
143	C₂×D₈	—	2	1	no	—
144	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
145	C₂×M₁₆	—	2	1	no	—
146	C₂×M₁₆	—	2	1	no	—
147	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₂²⋊C₄	—	4	2	no	—
151	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
152	M₃₂	—	2	1	no	—
153	M₃₂	—	2	1	no	—
154	M₃₂	—	2	1	no	—
155	M₃₂	C₂×C₄×C₅	1	2	yes	—
156	C₂×C₁₆	—	2	1	no	—
157	C₂×C₁₆	—	2	1	no	—
158	#108	—	2	1	no	—
159	#108	—	2	1	no	—
160	#108	C₂×C₄×C₅	1	2	yes	—
161	#108	C₅×D₄	1	2	yes	—
162	#106	—	2	1	no	—
163	#106	—	2	1	no	—
164	#102	—	4	2	no	—
165	#97	—	2	1	no	—
166	#97	—	2	1	no	—
167	#97	—	4	1	no	—
168	C₄×C₈	—	4	1	no	—
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	8	4	no	—
171	C₂²×C₅	—	8	2	no	—
172	C₂²×C₅	—	16	1	no	—
173	C₂²×C₅	—	16	1	no	—
174	C₂²×C₅	—	16	2	no	—
175	C₂²×C₅	C₄×SD₁₆	1	1	yes	characteristic
176	C₂²×C₅	#460	1	1	yes	characteristic
177	C₂²×C₅	#481	1	1	yes	characteristic
178	C₄×C₅	—	4	2	no	—
179	C₄×C₅	—	16	1	no	—
180	C₄×C₅	#388	1	1	yes	characteristic
181	C₄×C₅	#442	1	1	yes	characteristic
182	C₄×C₅	#463	1	1	yes	characteristic
183	C₄×C₅	#482	1	1	yes	characteristic
184	C₂⁴	—	4	1	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
193	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	D₈	—	2	2	no	—
195	D₈	—	2	2	no	—
196	D₈	—	4	1	no	—
197	D₈	—	4	1	no	—
198	M₁₆	—	2	2	no	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	4	1	no	—
201	M₁₆	—	4	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
208	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂×C₈	C₅×#41	1	1	yes	characteristic
210	C₂×C₈	C₅×#41	1	1	yes	characteristic
211	C₂×C₈	C₅×#41	1	1	yes	characteristic
212	C₂×C₈	C₅×#41	1	1	yes	characteristic
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₂²⋊C₄	—	8	4	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	1	no	—
218	C₁₆	—	2	2	no	—
219	C₁₆	—	4	1	no	—
220	C₂×C₅	—	2	1	no	—
221	C₂×C₅	—	2	1	no	—
222	C₂×C₅	—	16	2	no	—
223	C₂×C₅	#1926	1	1	yes	characteristic
224	C₂×C₅	#2238	1	2	yes	—
225	C₂³	—	4	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	8	1	no	—
229	C₂³	—	8	1	no	—
230	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	D₄	—	8	1	no	—
234	D₄	—	8	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	1	no	—
239	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
240	C₂×C₄	C₅×#121	1	1	yes	characteristic
241	C₂×C₄	C₅×#123	1	1	yes	characteristic
242	C₂×C₄	C₅×#135	1	1	yes	characteristic
243	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
245	C₈	—	4	2	no	—
246	C₈	—	4	2	no	—
247	C₈	—	4	1	no	—
248	C₈	C₅×#117	1	1	yes	characteristic
249	C₈	C₄×C₅×D₄	1	1	yes	characteristic
250	C₈	C₅×#124	1	1	yes	characteristic
251	C₈	C₅×#126	1	1	yes	characteristic
252	C₅	#13138	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²	—	2	2	no	—
254	C₂²	—	8	4	no	—
255	C₂²	—	8	2	no	—
256	C₂²	—	16	1	no	—
257	C₂²	—	16	1	no	—
258	C₂²	—	16	2	no	—
259	C₂²	C₄×C₅×SD₁₆	1	1	yes	characteristic
260	C₂²	C₅×#460	1	1	yes	characteristic
261	C₂²	C₅×#481	1	1	yes	characteristic
262	C₄	—	4	2	no	—
263	C₄	—	16	1	no	—
264	C₄	C₅×#388	1	1	yes	characteristic
265	C₄	C₅×#442	1	1	yes	characteristic
266	C₄	C₅×#463	1	1	yes	characteristic
267	C₄	C₅×#482	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	16	2	no	—
271	C₂	C₅×#1926	1	1	yes	characteristic
272	C₂	C₅×#2238	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#481
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#365	C₂	semidirect
5. C₅×#1778	C₂	semidirect
6. C₅×#1383	C₂	semidirect
7. C₅×#1377	C₂	semidirect
20. C₅×#361	C₄	semidirect
22. #13138	C₅	direct (N × H)
54. C₂×#365	C₂×C₅	semidirect
58. #1778	C₂×C₅	semidirect
59. #1383	C₂×C₅	semidirect
60. #1377	C₂×C₅	semidirect
109. #361	C₄×C₅	semidirect
252. C₅	#13138	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×#365	2
C₂×C₅×#361	2
C₅×#1972	2
C₅×#1778	2
C₅×#1383	2
C₅×#1377	2
#13138	5

Subgroups — C5×#13138 #11935858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13138 #11935858