Subgroups — C₅×#13152 #11935872

Order 1280.

324 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#359	C₂	1	1	yes	maximal, characteristic
2	C₅×#2138	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1974	C₂	1	1	yes	maximal, characteristic
5	C₅×#1945	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₅×#108	C₂²	1	1	yes	characteristic
12	C₂×C₅×#108	C₂²	1	1	yes	characteristic
13	C₅×#427	C₂²	1	1	yes	characteristic
14	C₅×#424	—	2	1	no	—
15	C₂×C₅×#104	—	2	1	no	—
16	C₅×#408	—	2	1	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#359	—	2	2	no	—
19	C₅×#358	C₄	1	2	yes	complemented
20	C₅×#357	C₄	1	2	yes	complemented
21	C₅×#346	—	2	2	no	—
22	C₅×#339	—	2	1	no	—
23	#13152	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×#41	—	2	1	no	—
25	C₅×#135	—	2	1	no	—
26	C₅×#135	—	2	1	no	—
27	C₅×#135	—	2	1	no	—
28	C₅×#135	—	2	1	no	—
29	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×SD₁₆	—	2	1	no	—
31	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×M₁₆	—	2	1	no	—
33	C₂×C₅×M₁₆	—	2	1	no	—
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×#117	—	2	1	no	—
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₅×C₁₆	—	2	1	no	—
39	C₂×C₅×C₁₆	—	2	1	no	—
40	C₂×C₅×C₁₆	—	2	1	no	—
41	C₂×C₅×C₁₆	—	2	1	no	—
42	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₅×C₁₆	D₄	1	2	yes	—
44	C₅×#108	—	2	1	no	—
45	C₅×#108	—	2	1	no	—
46	C₅×#108	—	2	1	no	—
47	C₅×#108	—	2	1	no	—
48	C₅×#107	C₂×C₄	1	2	yes	—
49	C₅×#106	—	2	1	no	—
50	C₅×C₄⋊C₈	—	4	2	no	—
51	C₅×#104	—	4	2	no	—
52	C₅×C₂²⋊C₈	—	4	2	no	—
53	C₄×C₅×C₈	—	2	1	no	—
54	C₄×C₅×C₈	—	2	1	no	—
55	C₂×#359	C₂×C₅	1	1	yes	characteristic
56	#2138	C₂×C₅	1	1	yes	characteristic
57	C₂×#346	C₂×C₅	1	1	yes	characteristic, complemented
58	#1974	C₂×C₅	1	1	yes	characteristic
59	#1945	C₂×C₅	1	1	yes	characteristic, complemented
60	#1382	C₂×C₅	1	1	yes	characteristic, complemented
61	#1381	C₂×C₅	1	1	yes	characteristic, complemented
62	C₅×#41	—	4	1	no	—
63	C₅×#41	—	4	1	no	—
64	C₅×#41	—	4	1	no	—
65	C₅×#41	—	4	1	no	—
66	C₂×C₅×Q₈	—	2	1	no	—
67	C₂×C₅×D₄	—	2	1	no	—
68	C₂×C₅×D₄	—	4	1	no	—
69	C₂²×C₄×C₅	—	2	1	no	—
70	C₂²×C₄×C₅	—	4	1	no	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₅×Q₁₆	—	2	1	no	—
73	C₅×Q₁₆	—	2	1	no	—
74	C₅×SD₁₆	—	4	1	no	—
75	C₅×SD₁₆	—	4	1	no	—
76	C₅×D₈	—	2	1	no	—
77	C₅×D₈	—	2	1	no	—
78	C₅×M₁₆	—	4	1	no	—
79	C₅×M₁₆	—	4	1	no	—
80	C₅×M₁₆	—	4	1	no	—
81	C₅×M₁₆	—	4	1	no	—
82	C₂×C₅×C₈	—	2	1	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	4	1	no	—
85	C₂×C₅×C₈	—	4	1	no	—
86	C₂×C₅×C₈	D₈	1	2	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	#41	1	1	yes	characteristic
90	C₂×C₅×C₈	#41	1	1	yes	characteristic
91	C₂×C₅×C₈	#41	1	1	yes	characteristic
92	C₂×C₅×C₈	#41	1	1	yes	characteristic
93	C₅×C₄⋊C₄	—	2	2	no	—
94	C₅×C₂²⋊C₄	—	4	1	no	—
95	C₄²×C₅	—	2	1	no	—
96	C₄²×C₅	—	4	1	no	—
97	C₄²×C₅	SD₁₆	1	2	yes	—
98	C₅×C₁₆	—	2	2	no	—
99	C₅×C₁₆	—	2	2	no	—
100	C₂×#135	C₂²×C₅	1	1	yes	characteristic
101	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
102	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
103	C₂×#108	C₂²×C₅	1	1	yes	characteristic
104	C₂×#108	C₂²×C₅	1	1	yes	characteristic
105	#427	C₂²×C₅	1	1	yes	characteristic
106	#424	—	2	1	no	—
107	C₂×#104	—	2	1	no	—
108	#408	—	2	1	no	—
109	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
110	#359	—	2	2	no	—
111	#358	C₄×C₅	1	2	yes	complemented
112	#357	C₄×C₅	1	2	yes	complemented
113	#346	—	2	2	no	—
114	#339	—	2	1	no	—
115	C₂³×C₅	—	4	1	no	—
116	C₂³×C₅	#135	1	1	yes	characteristic
117	C₅×Q₈	—	4	1	no	—
118	C₅×Q₈	—	4	1	no	—
119	C₅×D₄	—	4	1	no	—
120	C₅×D₄	—	4	1	no	—
121	C₅×D₄	—	8	1	no	—
122	C₅×D₄	—	8	1	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	4	2	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	—	4	1	no	—
127	C₂×C₄×C₅	—	8	1	no	—
128	C₂×C₄×C₅	—	8	1	no	—
129	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	#121	1	1	yes	characteristic
131	C₂×C₄×C₅	#124	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₅	#135	1	1	yes	characteristic
135	C₅×C₈	—	2	2	no	—
136	C₅×C₈	—	8	1	no	—
137	C₅×C₈	—	8	1	no	—
138	C₅×C₈	#117	1	1	yes	characteristic
139	C₅×C₈	C₄×D₄	1	1	yes	characteristic
140	C₅×C₈	#123	1	1	yes	characteristic
141	C₅×C₈	#126	1	1	yes	characteristic
142	C₂×#41	—	2	1	no	—
143	#135	—	2	1	no	—
144	#135	—	2	1	no	—
145	#135	—	2	1	no	—
146	#135	—	2	1	no	—
147	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂×SD₁₆	—	2	1	no	—
149	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
150	C₂×M₁₆	—	2	1	no	—
151	C₂×M₁₆	—	2	1	no	—
152	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
154	#117	—	2	1	no	—
155	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
156	C₂×C₁₆	—	2	1	no	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	2	1	no	—
159	C₂×C₁₆	—	2	1	no	—
160	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
161	C₂×C₁₆	C₅×D₄	1	2	yes	—
162	#108	—	2	1	no	—
163	#108	—	2	1	no	—
164	#108	—	2	1	no	—
165	#108	—	2	1	no	—
166	#107	C₂×C₄×C₅	1	2	yes	—
167	#106	—	2	1	no	—
168	C₄⋊C₈	—	4	2	no	—
169	#104	—	4	2	no	—
170	C₂²⋊C₈	—	4	2	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	—	2	1	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	8	2	no	—
175	C₂²×C₅	—	8	1	no	—
176	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
177	C₂²×C₅	#465	1	1	yes	characteristic
178	C₂²×C₅	#488	1	1	yes	characteristic
179	C₄×C₅	—	4	2	no	—
180	C₄×C₅	—	8	2	no	—
181	C₄×C₅	—	8	1	no	—
182	C₄×C₅	#388	1	1	yes	characteristic
183	C₄×C₅	C₄×SD₁₆	1	1	yes	characteristic
184	C₄×C₅	#466	1	1	yes	characteristic
185	C₄×C₅	#486	1	1	yes	characteristic
186	#41	—	4	1	no	—
187	#41	—	4	1	no	—
188	#41	—	4	1	no	—
189	#41	—	4	1	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
196	Q₁₆	—	2	1	no	—
197	Q₁₆	—	2	1	no	—
198	SD₁₆	—	4	1	no	—
199	SD₁₆	—	4	1	no	—
200	D₈	—	2	1	no	—
201	D₈	—	2	1	no	—
202	M₁₆	—	4	1	no	—
203	M₁₆	—	4	1	no	—
204	M₁₆	—	4	1	no	—
205	M₁₆	—	4	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	C₅×D₈	1	2	yes	—
211	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
212	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂×C₈	C₅×#41	1	1	yes	characteristic
214	C₂×C₈	C₅×#41	1	1	yes	characteristic
215	C₂×C₈	C₅×#41	1	1	yes	characteristic
216	C₂×C₈	C₅×#41	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	4	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	4	1	no	—
221	C₄²	C₅×SD₁₆	1	2	yes	—
222	C₁₆	—	2	2	no	—
223	C₁₆	—	2	2	no	—
224	C₂×C₅	—	2	1	no	—
225	C₂×C₅	—	2	1	no	—
226	C₂×C₅	—	16	1	no	—
227	C₂×C₅	#1934	1	1	yes	characteristic
228	C₂×C₅	#2179	1	2	yes	—
229	C₂³	—	4	1	no	—
230	C₂³	C₅×#135	1	1	yes	characteristic
231	Q₈	—	4	1	no	—
232	Q₈	—	4	1	no	—
233	D₄	—	4	1	no	—
234	D₄	—	4	1	no	—
235	D₄	—	8	1	no	—
236	D₄	—	8	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	—	8	1	no	—
243	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
244	C₂×C₄	C₅×#121	1	1	yes	characteristic
245	C₂×C₄	C₅×#124	1	1	yes	characteristic
246	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
247	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
248	C₂×C₄	C₅×#135	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	8	1	no	—
251	C₈	—	8	1	no	—
252	C₈	C₅×#117	1	1	yes	characteristic
253	C₈	C₄×C₅×D₄	1	1	yes	characteristic
254	C₈	C₅×#123	1	1	yes	characteristic
255	C₈	C₅×#126	1	1	yes	characteristic
256	C₅	#13152	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²	—	2	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	1	no	—
260	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
261	C₂²	C₅×#465	1	1	yes	characteristic
262	C₂²	C₅×#488	1	1	yes	characteristic
263	C₄	—	4	2	no	—
264	C₄	—	8	2	no	—
265	C₄	—	8	1	no	—
266	C₄	C₅×#388	1	1	yes	characteristic
267	C₄	C₄×C₅×SD₁₆	1	1	yes	characteristic
268	C₄	C₅×#466	1	1	yes	characteristic
269	C₄	C₅×#486	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	16	1	no	—
273	C₂	C₅×#1934	1	1	yes	characteristic
274	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×#135	C₂²
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
3. C₂×C₅×#346	C₂	semidirect
5. C₅×#1945	C₂	semidirect
6. C₅×#1382	C₂	semidirect
7. C₅×#1381	C₂	semidirect
19. C₅×#358	C₄	semidirect
20. C₅×#357	C₄	semidirect
23. #13152	C₅	direct (N × H)
57. C₂×#346	C₂×C₅	semidirect
59. #1945	C₂×C₅	semidirect
60. #1382	C₂×C₅	semidirect
61. #1381	C₂×C₅	semidirect
111. #358	C₄×C₅	semidirect
112. #357	C₄×C₅	semidirect
256. C₅	#13152	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×#359	2
C₅×#2138	2
C₂×C₅×#346	2
C₅×#1974	2
C₅×#1945	2
C₅×#1382	2
C₅×#1381	2
#13152	5

Subgroups — C5×#13152 #11935872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13152 #11935872