Subgroups — C₅×#13239 #11935959

Order 1280.

502 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	C₂²×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#357	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1970	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×D₈	C₂²	1	2	yes	—
7	C₂×C₅×D₁₆	—	2	2	no	—
8	C₂×C₅×D₁₆	—	2	2	no	—
9	C₂×C₅×D₁₆	C₄	1	4	yes	complemented
10	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₅×#107	C₂²	1	2	yes	—
13	C₂×C₅×#102	—	2	2	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#390	—	2	2	no	—
16	C₅×#367	—	2	2	no	—
17	C₅×#357	—	2	4	no	—
18	C₅×#346	—	2	2	no	—
19	#13239	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₅×D₄	—	2	2	no	—
21	C₂×C₅×D₈	—	2	2	no	—
22	C₂×C₅×D₈	—	2	2	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	C₂×C₄	1	4	yes	—
27	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₂²⋊C₄	—	4	4	no	—
31	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₅×D₁₆	—	2	8	no	—
33	C₂×C₅×C₁₆	—	2	1	no	—
34	C₂×C₅×C₁₆	—	2	1	no	—
35	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₅×C₁₆	D₄	1	2	yes	—
37	C₂×C₅×C₁₆	D₄	1	2	yes	—
38	C₂×C₅×C₁₆	D₄	1	2	yes	—
39	C₅×#107	—	2	2	no	—
40	C₅×#107	—	2	2	no	—
41	C₅×#102	—	4	4	no	—
42	C₄×C₅×C₈	—	2	1	no	—
43	C₄×C₅×C₈	—	2	1	no	—
44	C₂²×D₁₆	C₂×C₅	1	1	yes	characteristic
45	C₂×#367	C₂×C₅	1	1	yes	characteristic, complemented
46	C₂×#357	C₂×C₅	1	2	yes	complemented
47	C₂×#346	C₂×C₅	1	1	yes	characteristic, complemented
48	#1970	C₂×C₅	1	2	yes	complemented
49	C₂⁴×C₅	—	4	2	no	—
50	C₂×C₅×D₄	—	2	4	no	—
51	C₂×C₅×D₄	—	4	2	no	—
52	C₂×C₅×D₄	—	4	2	no	—
53	C₂×C₅×D₄	—	4	2	no	—
54	C₂×C₅×D₄	—	4	2	no	—
55	C₂×C₅×D₄	—	4	2	no	—
56	C₂²×C₄×C₅	—	4	2	no	—
57	C₂²×C₄×C₅	D₈	1	2	yes	—
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₅×D₈	—	2	4	no	—
60	C₅×D₈	—	2	4	no	—
61	C₅×D₈	—	4	2	no	—
62	C₅×D₈	—	4	2	no	—
63	C₂×C₅×C₈	—	2	1	no	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	D₈	1	2	yes	—
66	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₅×C₈	#41	1	1	yes	characteristic
71	C₂×C₅×C₈	#41	1	1	yes	characteristic
72	C₅×C₄⋊C₄	—	4	2	no	—
73	C₅×C₄⋊C₄	—	4	2	no	—
74	C₅×C₂²⋊C₄	—	8	8	no	—
75	C₄²×C₅	—	2	1	no	—
76	C₄²×C₅	—	2	1	no	—
77	C₅×C₁₆	—	2	2	no	—
78	C₅×C₁₆	C₂²⋊C₄	1	4	yes	complemented
79	C₂²×D₈	C₂²×C₅	1	2	yes	—
80	C₂×D₁₆	—	2	2	no	—
81	C₂×D₁₆	—	2	2	no	—
82	C₂×D₁₆	C₄×C₅	1	4	yes	complemented
83	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
84	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
85	C₂×#107	C₂²×C₅	1	2	yes	—
86	C₂×#102	—	2	2	no	—
87	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
88	#390	—	2	2	no	—
89	#367	—	2	2	no	—
90	#357	—	2	4	no	—
91	#346	—	2	2	no	—
92	C₂³×C₅	—	4	4	no	—
93	C₂³×C₅	—	8	2	no	—
94	C₂³×C₅	—	8	2	no	—
95	C₂³×C₅	—	8	4	no	—
96	C₂³×C₅	—	8	4	no	—
97	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
98	C₅×D₄	—	4	4	no	—
99	C₅×D₄	—	4	4	no	—
100	C₅×D₄	—	8	2	no	—
101	C₅×D₄	—	8	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	8	2	no	—
105	C₂×C₄×C₅	—	8	4	no	—
106	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅	#121	1	1	yes	characteristic
108	C₂×C₄×C₅	#127	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₅	#135	1	1	yes	characteristic
111	C₂×C₄×C₅	#135	1	1	yes	characteristic
112	C₅×C₈	—	2	2	no	—
113	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₅×C₈	C₄×D₄	1	1	yes	characteristic
115	C₅×C₈	#121	1	1	yes	characteristic
116	C₅×C₈	#124	1	1	yes	characteristic
117	C₂²×D₄	—	2	2	no	—
118	C₂×D₈	—	2	2	no	—
119	C₂×D₈	—	2	2	no	—
120	C₂×D₈	—	2	2	no	—
121	C₂×D₈	—	2	2	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
124	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₂²⋊C₄	—	4	4	no	—
128	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
129	D₁₆	—	2	8	no	—
130	C₂×C₁₆	—	2	1	no	—
131	C₂×C₁₆	—	2	1	no	—
132	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
133	C₂×C₁₆	C₅×D₄	1	2	yes	—
134	C₂×C₁₆	C₅×D₄	1	2	yes	—
135	C₂×C₁₆	C₅×D₄	1	2	yes	—
136	#107	—	2	2	no	—
137	#107	—	2	2	no	—
138	#102	—	4	4	no	—
139	C₄×C₈	—	2	1	no	—
140	C₄×C₈	—	2	1	no	—
141	C₂²×C₅	—	8	8	no	—
142	C₂²×C₅	—	8	4	no	—
143	C₂²×C₅	—	16	4	no	—
144	C₂²×C₅	—	16	2	no	—
145	C₂²×C₅	—	16	2	no	—
146	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
147	C₂²×C₅	#466	1	1	yes	characteristic
148	C₂²×C₅	#493	1	1	yes	characteristic
149	C₂²×C₅	#509	1	4	yes	—
150	C₄×C₅	—	4	2	no	—
151	C₄×C₅	—	16	2	no	—
152	C₄×C₅	#390	1	1	yes	characteristic
153	C₄×C₅	C₄×D₈	1	1	yes	characteristic
154	C₄×C₅	#466	1	1	yes	characteristic
155	C₄×C₅	#495	1	1	yes	characteristic
156	C₂⁴	—	4	2	no	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	—	4	2	no	—
159	C₂×D₄	—	4	2	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	2	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₅×D₈	1	2	yes	—
165	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
166	D₈	—	2	4	no	—
167	D₈	—	2	4	no	—
168	D₈	—	4	2	no	—
169	D₈	—	4	2	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	C₅×D₈	1	2	yes	—
173	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
174	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
175	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
176	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
177	C₂×C₈	C₅×#41	1	1	yes	characteristic
178	C₂×C₈	C₅×#41	1	1	yes	characteristic
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	8	8	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₁₆	—	2	2	no	—
185	C₁₆	C₅×C₂²⋊C₄	1	4	yes	complemented
186	C₂×C₅	—	16	4	no	—
187	C₂×C₅	#1970	1	1	yes	characteristic
188	C₂×C₅	#2177	1	2	yes	—
189	C₂×C₅	#2218	1	2	yes	—
190	C₂×C₅	#2250	1	2	yes	—
191	C₂³	—	4	4	no	—
192	C₂³	—	8	2	no	—
193	C₂³	—	8	2	no	—
194	C₂³	—	8	4	no	—
195	C₂³	—	8	4	no	—
196	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
197	D₄	—	4	4	no	—
198	D₄	—	4	4	no	—
199	D₄	—	8	2	no	—
200	D₄	—	8	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
206	C₂×C₄	C₅×#121	1	1	yes	characteristic
207	C₂×C₄	C₅×#127	1	1	yes	characteristic
208	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
209	C₂×C₄	C₅×#135	1	1	yes	characteristic
210	C₂×C₄	C₅×#135	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₈	C₄×C₅×D₄	1	1	yes	characteristic
214	C₈	C₅×#121	1	1	yes	characteristic
215	C₈	C₅×#124	1	1	yes	characteristic
216	C₅	#13239	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	—	8	8	no	—
218	C₂²	—	8	4	no	—
219	C₂²	—	16	4	no	—
220	C₂²	—	16	2	no	—
221	C₂²	—	16	2	no	—
222	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
223	C₂²	C₅×#466	1	1	yes	characteristic
224	C₂²	C₅×#493	1	1	yes	characteristic
225	C₂²	C₅×#509	1	4	yes	—
226	C₄	—	4	2	no	—
227	C₄	—	16	2	no	—
228	C₄	C₅×#390	1	1	yes	characteristic
229	C₄	C₄×C₅×D₈	1	1	yes	characteristic
230	C₄	C₅×#466	1	1	yes	characteristic
231	C₄	C₅×#495	1	1	yes	characteristic
232	C₂	—	16	4	no	—
233	C₂	C₅×#1970	1	1	yes	characteristic
234	C₂	C₅×#2177	1	2	yes	—
235	C₂	C₅×#2218	1	2	yes	—
236	C₂	C₅×#2250	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₁₆	C₂
LCS	G ▷ C₂×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×#367	C₂	semidirect
3. C₂×C₅×#357	C₂	semidirect
4. C₂×C₅×#346	C₂	semidirect
5. C₅×#1970	C₂	semidirect
9. C₂×C₅×D₁₆	C₄	semidirect
19. #13239	C₅	direct (N × H)
45. C₂×#367	C₂×C₅	semidirect
46. C₂×#357	C₂×C₅	semidirect
47. C₂×#346	C₂×C₅	semidirect
48. #1970	C₂×C₅	semidirect
78. C₅×C₁₆	C₂²⋊C₄	semidirect
82. C₂×D₁₆	C₄×C₅	semidirect
185. C₁₆	C₅×C₂²⋊C₄	semidirect
216. C₅	#13239	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅×D₁₆	2
C₂×C₅×#367	2
C₂×C₅×#357	2
C₂×C₅×#346	2
C₅×#1970	2
#13239	5

Subgroups — C5×#13239 #11935959

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13239 #11935959