Subgroups — C₂×C₅×#1333 #11934315

Order 1280.

462 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	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1333	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₅×D₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×D₄	—	2	1	no	—
7	C₂²×C₅×C₁₆	—	2	1	no	—
8	C₄×C₅×D₈	—	2	1	no	—
9	C₄×C₅×D₈	—	2	1	no	—
10	C₄×C₅×D₈	C₄	1	2	yes	—
11	C₄×C₅×D₈	C₂²	1	2	yes	—
12	C₂×C₅×#107	C₄	1	1	yes	characteristic
13	C₂×C₅×C₄⋊C₈	—	2	1	no	—
14	C₂×C₅×#102	—	2	1	no	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#346	—	2	1	no	—
17	C₅×#346	C₂²	1	2	yes	complemented
18	C₅×#334	—	2	1	no	—
19	C₅×#334	C₂²	1	2	yes	complemented
20	C₂×#1333	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×D₄	—	2	1	no	—
22	C₂³×C₄×C₅	—	4	1	no	—
23	C₂×C₅×D₈	—	2	1	no	—
24	C₂×C₅×D₈	—	2	2	no	—
25	C₂×C₅×D₈	C₈	1	4	yes	complemented
26	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	—	4	1	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₄×C₅×D₄	—	2	2	no	—
31	C₄×C₅×D₄	—	2	2	no	—
32	C₄×C₅×D₄	—	4	1	no	—
33	C₄×C₅×D₄	—	4	1	no	—
34	C₂×C₅×C₄⋊C₄	—	2	1	no	—
35	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₅×C₁₆	—	2	4	no	—
38	C₂×C₅×C₁₆	—	2	2	no	—
39	C₅×#107	—	2	1	no	—
40	C₅×#107	C₂×C₄	1	2	yes	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₄⋊C₈	—	4	1	no	—
43	C₅×#102	—	2	2	no	—
44	C₅×#102	—	2	2	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	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic
50	C₂×#346	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×#334	C₂×C₅	1	1	yes	characteristic, complemented
52	#1333	C₂×C₅	1	4	yes	direct product factor
53	C₂⁴×C₅	—	4	1	no	—
54	C₂×C₅×D₄	—	2	4	no	—
55	C₂×C₅×D₄	—	2	2	no	—
56	C₂×C₅×D₄	—	4	1	no	—
57	C₂×C₅×D₄	—	4	2	no	—
58	C₂²×C₄×C₅	—	4	1	no	—
59	C₂²×C₄×C₅	—	4	2	no	—
60	C₂²×C₄×C₅	—	4	2	no	—
61	C₂²×C₄×C₅	—	8	4	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×D₈	—	2	4	no	—
66	C₅×D₈	—	2	2	no	—
67	C₅×D₈	C₂×C₈	1	4	yes	complemented
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	—	4	2	no	—
70	C₂×C₅×C₈	—	8	2	no	—
71	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
74	C₂×C₅×C₈	M₁₆	1	2	yes	—
75	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
77	C₅×C₄⋊C₄	—	2	2	no	—
78	C₅×C₄⋊C₄	—	4	1	no	—
79	C₅×C₂²⋊C₄	—	4	2	no	—
80	C₅×C₂²⋊C₄	—	4	2	no	—
81	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₅	D₈	1	1	yes	characteristic
83	C₄²×C₅	SD₁₆	1	1	yes	characteristic
84	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
85	C₅×C₁₆	—	2	4	no	—
86	C₂²×D₈	C₄×C₅	1	1	yes	characteristic
87	C₂×C₄×D₄	—	2	1	no	—
88	C₂²×C₁₆	—	2	1	no	—
89	C₄×D₈	—	2	1	no	—
90	C₄×D₈	—	2	1	no	—
91	C₄×D₈	C₄×C₅	1	2	yes	—
92	C₄×D₈	C₂²×C₅	1	2	yes	—
93	C₂×#107	C₄×C₅	1	1	yes	characteristic
94	C₂×C₄⋊C₈	—	2	1	no	—
95	C₂×#102	—	2	1	no	—
96	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
97	#346	—	2	1	no	—
98	#346	C₂²×C₅	1	2	yes	complemented
99	#334	—	2	1	no	—
100	#334	C₂²×C₅	1	2	yes	complemented
101	C₂³×C₅	—	4	4	no	—
102	C₂³×C₅	—	4	2	no	—
103	C₂³×C₅	—	8	2	no	—
104	C₂³×C₅	—	8	2	no	—
105	C₂³×C₅	#102	1	1	yes	characteristic
106	C₅×D₄	—	2	4	no	—
107	C₅×D₄	—	4	4	no	—
108	C₅×D₄	—	4	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	4	2	no	—
112	C₂×C₄×C₅	—	8	4	no	—
113	C₂×C₄×C₅	—	8	4	no	—
114	C₂×C₄×C₅	—	8	2	no	—
115	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
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	#104	1	2	yes	—
120	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×D₈	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	—	8	2	no	—
126	C₅×C₈	C₂²⋊C₈	1	2	yes	—
127	C₅×C₈	C₂²×C₈	1	1	yes	characteristic
128	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
129	C₂²×D₄	—	2	1	no	—
130	C₂³×C₄	—	4	1	no	—
131	C₂×D₈	—	2	1	no	—
132	C₂×D₈	—	2	2	no	—
133	C₂×D₈	C₅×C₈	1	4	yes	complemented
134	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
135	C₂²×C₈	—	4	1	no	—
136	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
137	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
138	C₄×D₄	—	2	2	no	—
139	C₄×D₄	—	2	2	no	—
140	C₄×D₄	—	4	1	no	—
141	C₄×D₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	2	1	no	—
143	C₂×C₂²⋊C₄	—	4	1	no	—
144	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
145	C₂×C₁₆	—	2	4	no	—
146	C₂×C₁₆	—	2	2	no	—
147	#107	—	2	1	no	—
148	#107	C₂×C₄×C₅	1	2	yes	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	4	1	no	—
151	#102	—	2	2	no	—
152	#102	—	2	2	no	—
153	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
154	C₄×C₈	C₅×D₄	1	1	yes	characteristic
155	C₄×C₈	C₅×D₄	1	1	yes	characteristic
156	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
157	C₂²×C₅	—	4	4	no	—
158	C₂²×C₅	—	8	8	no	—
159	C₂²×C₅	—	8	2	no	—
160	C₂²×C₅	—	8	2	no	—
161	C₂²×C₅	D₄⋊C₈	1	2	yes	—
162	C₂²×C₅	#360	1	2	yes	—
163	C₂²×C₅	#361	1	2	yes	—
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	D₄⋊C₈	1	2	yes	—
171	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
172	C₄×C₅	C₂×#104	1	1	yes	characteristic
173	C₂⁴	—	4	1	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	1	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂²×C₄	—	4	1	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	8	4	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	D₈	—	2	2	no	—
186	D₈	—	2	4	no	—
187	D₈	C₂×C₅×C₈	1	4	yes	complemented
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	8	2	no	—
191	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
192	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
194	C₂×C₈	C₅×M₁₆	1	2	yes	—
195	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
196	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	4	1	no	—
199	C₂²⋊C₄	—	4	2	no	—
200	C₂²⋊C₄	—	4	2	no	—
201	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
202	C₄²	C₅×D₈	1	1	yes	characteristic
203	C₄²	C₅×SD₁₆	1	1	yes	characteristic
204	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
205	C₁₆	—	2	4	no	—
206	C₂×C₅	—	8	4	no	—
207	C₂×C₅	#1333	1	4	yes	direct product factor
208	C₂×C₅	C₂×D₄⋊C₈	1	1	yes	characteristic
209	C₂×C₅	C₂×#360	1	1	yes	characteristic
210	C₂×C₅	C₂×#361	1	1	yes	characteristic
211	C₂³	—	4	4	no	—
212	C₂³	—	4	2	no	—
213	C₂³	—	8	2	no	—
214	C₂³	—	8	2	no	—
215	C₂³	C₅×#102	1	1	yes	characteristic
216	D₄	—	2	4	no	—
217	D₄	—	4	4	no	—
218	D₄	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	4	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
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×#104	1	2	yes	—
230	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₂×C₅×D₈	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	—	8	2	no	—
236	C₈	C₅×C₂²⋊C₈	1	2	yes	—
237	C₈	C₂²×C₅×C₈	1	1	yes	characteristic
238	C₈	C₂×C₅×M₁₆	1	1	yes	characteristic
239	C₅	C₂×#1333	1	1	yes	characteristic, direct product factor, Sylow
240	C₂²	—	4	4	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	—	8	8	no	—
244	C₂²	C₅×D₄⋊C₈	1	2	yes	—
245	C₂²	C₅×#360	1	2	yes	—
246	C₂²	C₅×#361	1	2	yes	—
247	C₂²	C₂×C₅×#102	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	8	2	no	—
253	C₄	C₅×D₄⋊C₈	1	2	yes	—
254	C₄	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
255	C₄	C₂×C₅×#104	1	1	yes	characteristic
256	C₂	—	8	4	no	—
257	C₂	C₅×#1333	1	4	yes	direct product factor
258	C₂	C₂×C₅×D₄⋊C₈	1	1	yes	characteristic
259	C₂	C₂×C₅×#360	1	1	yes	characteristic
260	C₂	C₂×C₅×#361	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	#102
commutator	C₈	C₂²×C₅×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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. C₂×C₅×#346	C₂	semidirect
3. C₂×C₅×#334	C₂	semidirect
4. C₅×#1333	C₂	direct (N × H)
17. C₅×#346	C₂²	semidirect
19. C₅×#334	C₂²	semidirect
20. C₂×#1333	C₅	direct (N × H)
25. C₂×C₅×D₈	C₈	semidirect
50. C₂×#346	C₂×C₅	semidirect
51. C₂×#334	C₂×C₅	semidirect
52. #1333	C₂×C₅	direct (N × H)
67. C₅×D₈	C₂×C₈	semidirect
98. #346	C₂²×C₅	semidirect
100. #334	C₂²×C₅	semidirect
133. C₂×D₈	C₅×C₈	semidirect
187. D₈	C₂×C₅×C₈	semidirect
207. C₂×C₅	#1333	direct (N × H)
239. C₅	C₂×#1333	direct (N × H)
257. C₂	C₅×#1333	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×C₄×C₅×D₈	2
C₂×C₅×#346	2
C₂×C₅×#334	2
C₅×#1333	2
C₂×#1333	5

Subgroups — C2×C5×#1333 #11934315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1333 #11934315