Subgroups — C₂×C₅×#1361 #11934647

Order 1280.

518 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×C₅×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1361	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₅×D₈	—	2	1	no	—
6	C₂²×C₅×D₈	C₄	1	2	yes	—
7	C₂×C₅×#127	—	2	1	no	—
8	C₂²×C₅×C₁₆	—	2	1	no	—
9	C₅×#493	—	2	1	no	—
10	C₅×#493	—	2	1	no	—
11	C₅×#493	C₄	1	2	yes	—
12	C₅×#493	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×C₄⋊C₁₆	—	2	1	no	—
16	C₅×C₄⋊C₁₆	C₂²	1	2	yes	complemented
17	C₅×#335	—	2	1	no	—
18	C₅×#335	C₂²	1	2	yes	complemented
19	C₂×#1361	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₅×D₄	—	2	2	no	—
21	C₂²×C₅×D₄	—	4	1	no	—
22	C₂²×C₅×D₄	—	4	1	no	—
23	C₂×C₅×D₈	—	2	2	no	—
24	C₂×C₅×D₈	—	2	2	no	—
25	C₂×C₅×D₈	—	2	2	no	—
26	C₂×C₅×D₈	—	4	4	no	—
27	C₂×C₅×D₈	—	4	2	no	—
28	C₂×C₅×D₈	—	4	2	no	—
29	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
30	C₂²×C₅×C₈	—	4	1	no	—
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₅×#127	—	2	2	no	—
34	C₅×#127	—	2	2	no	—
35	C₅×#127	—	4	1	no	—
36	C₅×#127	—	4	1	no	—
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₅×C₁₆	—	2	2	no	—
39	C₂×C₅×C₁₆	—	4	2	no	—
40	C₅×C₄⋊C₈	—	2	2	no	—
41	C₅×C₄⋊C₈	—	4	1	no	—
42	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×C₈	D₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₂×#493	C₂×C₅	1	1	yes	characteristic
47	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂×#335	C₂×C₅	1	1	yes	characteristic, complemented
49	#1361	C₂×C₅	1	4	yes	direct product factor
50	C₂⁴×C₅	—	4	2	no	—
51	C₂×C₅×D₄	—	2	4	no	—
52	C₂×C₅×D₄	—	4	4	no	—
53	C₂×C₅×D₄	—	4	4	no	—
54	C₂×C₅×D₄	—	4	2	no	—
55	C₂×C₅×D₄	—	4	2	no	—
56	C₂×C₅×D₄	—	4	2	no	—
57	C₂×C₅×D₄	—	4	2	no	—
58	C₂×C₅×D₄	—	4	2	no	—
59	C₂×C₅×D₄	—	8	4	no	—
60	C₂×C₅×D₄	—	8	2	no	—
61	C₂×C₅×D₄	—	8	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×D₈	—	4	4	no	—
66	C₅×D₈	—	4	4	no	—
67	C₅×D₈	—	4	4	no	—
68	C₅×D₈	—	4	2	no	—
69	C₅×D₈	—	4	2	no	—
70	C₂×C₅×C₈	—	2	2	no	—
71	C₂×C₅×C₈	—	4	2	no	—
72	C₂×C₅×C₈	—	8	2	no	—
73	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	D₈	1	2	yes	—
76	C₂×C₅×C₈	SD₁₆	1	2	yes	—
77	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₅	D₈	1	1	yes	characteristic
81	C₄²×C₅	SD₁₆	1	1	yes	characteristic
82	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
83	C₅×C₁₆	—	4	2	no	—
84	C₂²×D₈	—	2	1	no	—
85	C₂²×D₈	C₄×C₅	1	2	yes	—
86	C₂×#127	—	2	1	no	—
87	C₂²×C₁₆	—	2	1	no	—
88	#493	—	2	1	no	—
89	#493	—	2	1	no	—
90	#493	C₄×C₅	1	2	yes	—
91	#493	C₂²×C₅	1	2	yes	—
92	C₂×C₄⋊C₈	—	2	1	no	—
93	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	C₄⋊C₁₆	—	2	1	no	—
95	C₄⋊C₁₆	C₂²×C₅	1	2	yes	complemented
96	#335	—	2	1	no	—
97	#335	C₂²×C₅	1	2	yes	complemented
98	C₂³×C₅	—	4	4	no	—
99	C₂³×C₅	—	8	4	no	—
100	C₂³×C₅	—	8	4	no	—
101	C₂³×C₅	—	8	4	no	—
102	C₂³×C₅	#102	1	1	yes	characteristic
103	C₅×D₄	—	4	4	no	—
104	C₅×D₄	—	4	4	no	—
105	C₅×D₄	—	8	4	no	—
106	C₅×D₄	—	8	4	no	—
107	C₅×D₄	—	8	4	no	—
108	C₅×D₄	—	8	4	no	—
109	C₅×D₄	—	8	2	no	—
110	C₅×D₄	—	8	2	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	#100	1	2	yes	—
113	C₂×C₄×C₅	#102	1	1	yes	characteristic
114	C₂×C₄×C₅	#102	1	1	yes	characteristic
115	C₂×C₄×C₅	#102	1	2	yes	—
116	C₂×C₄×C₅	#102	1	1	yes	characteristic
117	C₂×C₄×C₅	D₁₆	1	2	yes	—
118	C₂×C₄×C₅	SD₃₂	1	2	yes	—
119	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
122	C₅×C₈	—	2	1	no	—
123	C₅×C₈	—	2	1	no	—
124	C₅×C₈	—	8	2	no	—
125	C₅×C₈	#102	1	2	yes	—
126	C₅×C₈	C₂×D₈	1	1	yes	characteristic
127	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
128	C₂²×D₄	—	2	2	no	—
129	C₂²×D₄	—	4	1	no	—
130	C₂²×D₄	—	4	1	no	—
131	C₂×D₈	—	2	2	no	—
132	C₂×D₈	—	2	2	no	—
133	C₂×D₈	—	2	2	no	—
134	C₂×D₈	—	4	4	no	—
135	C₂×D₈	—	4	2	no	—
136	C₂×D₈	—	4	2	no	—
137	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
138	C₂²×C₈	—	4	1	no	—
139	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
141	#127	—	2	2	no	—
142	#127	—	2	2	no	—
143	#127	—	4	1	no	—
144	#127	—	4	1	no	—
145	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	4	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	4	1	no	—
150	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
151	C₄×C₈	C₅×D₄	1	1	yes	characteristic
152	C₄×C₈	C₅×D₄	1	1	yes	characteristic
153	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
154	C₂²×C₅	—	8	4	no	—
155	C₂²×C₅	—	8	4	no	—
156	C₂²×C₅	—	8	4	no	—
157	C₂²×C₅	—	16	8	no	—
158	C₂²×C₅	#331	1	2	yes	—
159	C₂²×C₅	#357	1	2	yes	—
160	C₂²×C₅	#361	1	2	yes	—
161	C₂²×C₅	C₂×#102	1	1	yes	characteristic
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	#331	1	2	yes	—
165	C₄×C₅	#357	1	2	yes	—
166	C₄×C₅	C₂×#100	1	1	yes	characteristic
167	C₄×C₅	C₂×#102	1	1	yes	characteristic
168	C₄×C₅	C₂×D₁₆	1	1	yes	characteristic
169	C₄×C₅	C₂×SD₃₂	1	1	yes	characteristic
170	C₂⁴	—	4	2	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	4	2	no	—
173	C₂×D₄	—	4	2	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	4	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	8	4	no	—
180	C₂×D₄	—	8	2	no	—
181	C₂×D₄	—	8	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	D₈	—	4	4	no	—
186	D₈	—	4	4	no	—
187	D₈	—	4	2	no	—
188	D₈	—	4	2	no	—
189	D₈	—	4	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	8	2	no	—
193	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₈	C₅×D₈	1	2	yes	—
196	C₂×C₈	C₅×SD₁₆	1	2	yes	—
197	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
198	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
199	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₄²	C₅×D₈	1	1	yes	characteristic
201	C₄²	C₅×SD₁₆	1	1	yes	characteristic
202	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
203	C₁₆	—	4	2	no	—
204	C₂×C₅	—	16	4	no	—
205	C₂×C₅	#1361	1	4	yes	direct product factor
206	C₂×C₅	C₂×#331	1	1	yes	characteristic
207	C₂×C₅	C₂×#357	1	1	yes	characteristic
208	C₂×C₅	C₂×#361	1	1	yes	characteristic
209	C₂³	—	4	4	no	—
210	C₂³	—	8	4	no	—
211	C₂³	—	8	4	no	—
212	C₂³	—	8	4	no	—
213	C₂³	C₅×#102	1	1	yes	characteristic
214	D₄	—	4	4	no	—
215	D₄	—	4	4	no	—
216	D₄	—	8	4	no	—
217	D₄	—	8	4	no	—
218	D₄	—	8	4	no	—
219	D₄	—	8	4	no	—
220	D₄	—	8	2	no	—
221	D₄	—	8	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	C₅×#100	1	2	yes	—
224	C₂×C₄	C₅×#102	1	1	yes	characteristic
225	C₂×C₄	C₅×#102	1	1	yes	characteristic
226	C₂×C₄	C₅×#102	1	2	yes	—
227	C₂×C₄	C₅×#102	1	1	yes	characteristic
228	C₂×C₄	C₅×D₁₆	1	2	yes	—
229	C₂×C₄	C₅×SD₃₂	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×#102	1	2	yes	—
237	C₈	C₂×C₅×D₈	1	1	yes	characteristic
238	C₈	C₂×C₅×SD₁₆	1	1	yes	characteristic
239	C₅	C₂×#1361	1	1	yes	characteristic, direct product factor, Sylow
240	C₂²	—	8	4	no	—
241	C₂²	—	8	4	no	—
242	C₂²	—	8	4	no	—
243	C₂²	—	16	8	no	—
244	C₂²	C₅×#331	1	2	yes	—
245	C₂²	C₅×#357	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	C₅×#331	1	2	yes	—
251	C₄	C₅×#357	1	2	yes	—
252	C₄	C₂×C₅×#100	1	1	yes	characteristic
253	C₄	C₂×C₅×#102	1	1	yes	characteristic
254	C₄	C₂×C₅×D₁₆	1	1	yes	characteristic
255	C₄	C₂×C₅×SD₃₂	1	1	yes	characteristic
256	C₂	—	16	4	no	—
257	C₂	C₅×#1361	1	4	yes	direct product factor
258	C₂	C₂×C₅×#331	1	1	yes	characteristic
259	C₂	C₂×C₅×#357	1	1	yes	characteristic
260	C₂	C₂×C₅×#361	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂×C₅×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×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×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₅×#335	C₂	semidirect
4. C₅×#1361	C₂	direct (N × H)
16. C₅×C₄⋊C₁₆	C₂²	semidirect
18. C₅×#335	C₂²	semidirect
19. C₂×#1361	C₅	direct (N × H)
47. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
48. C₂×#335	C₂×C₅	semidirect
49. #1361	C₂×C₅	direct (N × H)
95. C₄⋊C₁₆	C₂²×C₅	semidirect
97. #335	C₂²×C₅	semidirect
205. C₂×C₅	#1361	direct (N × H)
239. C₅	C₂×#1361	direct (N × H)
257. C₂	C₅×#1361	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×#493	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#335	2
C₅×#1361	2
C₂×#1361	5

Subgroups — C2×C5×#1361 #11934647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1361 #11934647