Subgroups — C₅×#12637 #11935357

Order 1280.

526 conjugacy classes of subgroups.

The subgroup lattice has 254 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×#357	C₂	1	2	yes	maximal
3	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×D₄	—	2	1	no	—
8	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
9	C₄×C₅×D₈	—	2	2	no	—
10	C₄×C₅×D₈	C₄	1	4	yes	complemented
11	C₂×C₅×#107	C₂²	1	1	yes	characteristic
12	C₂×C₅×#107	C₂²	1	2	yes	—
13	C₂×C₅×#102	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₅×C₄⋊C₄	—	2	1	no	—
16	C₅×#357	C₄	1	8	yes	complemented
17	C₅×#346	—	2	2	no	—
18	C₅×#340	—	2	1	no	—
19	#12637	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₅×D₄	—	2	1	no	—
21	C₂³×C₄×C₅	—	4	1	no	—
22	C₂×C₅×D₈	—	2	1	no	—
23	C₂×C₅×D₈	—	2	2	no	—
24	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
25	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₄×C₅×D₄	—	2	4	no	—
29	C₄×C₅×D₄	—	4	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	—	2	1	no	—
32	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
33	C₂×C₄²×C₅	—	4	1	no	—
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₅×C₁₆	—	2	2	no	—
36	C₂×C₅×C₁₆	—	2	2	no	—
37	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
38	C₅×#107	—	2	1	no	—
39	C₅×#107	—	2	2	no	—
40	C₅×#107	C₂×C₄	1	2	yes	—
41	C₅×#107	C₂×C₄	1	4	yes	complemented
42	C₅×#102	—	2	4	no	—
43	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
44	C₄×C₅×C₈	D₄	1	2	yes	—
45	C₅×#95	—	4	1	no	—
46	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic
47	C₂×#357	C₂×C₅	1	2	yes	—
48	C₂×#346	C₂×C₅	1	1	yes	characteristic, complemented
49	C₄×#107	C₂×C₅	1	1	yes	characteristic, complemented
50	#1380	C₂×C₅	1	2	yes	complemented
51	C₂⁴×C₅	—	4	1	no	—
52	C₂×C₅×D₄	—	2	2	no	—
53	C₂×C₅×D₄	—	2	4	no	—
54	C₂×C₅×D₄	—	4	1	no	—
55	C₂×C₅×D₄	—	4	2	no	—
56	C₂²×C₄×C₅	—	4	2	no	—
57	C₂²×C₄×C₅	—	4	1	no	—
58	C₂²×C₄×C₅	—	4	4	no	—
59	C₂²×C₄×C₅	—	8	4	no	—
60	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₅×D₈	—	2	2	no	—
64	C₅×D₈	—	2	4	no	—
65	C₅×D₈	C₄²	1	4	yes	complemented
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
68	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₅×C₈	#41	1	2	yes	—
72	C₅×C₄⋊C₄	—	2	2	no	—
73	C₅×C₄⋊C₄	—	2	4	no	—
74	C₅×C₄⋊C₄	—	4	1	no	—
75	C₅×C₄⋊C₄	—	4	2	no	—
76	C₅×C₂²⋊C₄	—	4	4	no	—
77	C₄²×C₅	—	8	2	no	—
78	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₅	SD₁₆	1	2	yes	—
80	C₅×C₁₆	—	2	4	no	—
81	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
82	C₂×C₄×D₄	—	2	1	no	—
83	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
84	C₄×D₈	—	2	2	no	—
85	C₄×D₈	C₄×C₅	1	4	yes	complemented
86	C₂×#107	C₂²×C₅	1	1	yes	characteristic
87	C₂×#107	C₂²×C₅	1	2	yes	—
88	C₂×#102	—	2	1	no	—
89	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
90	C₄×C₄⋊C₄	—	2	1	no	—
91	#357	C₄×C₅	1	8	yes	complemented
92	#346	—	2	2	no	—
93	#340	—	2	1	no	—
94	C₂³×C₅	—	4	2	no	—
95	C₂³×C₅	—	4	4	no	—
96	C₂³×C₅	—	8	2	no	—
97	C₂³×C₅	—	8	1	no	—
98	C₂³×C₅	—	8	1	no	—
99	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
100	C₅×D₄	—	2	4	no	—
101	C₅×D₄	—	4	2	no	—
102	C₅×D₄	—	4	4	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	4	2	no	—
107	C₂×C₄×C₅	—	4	4	no	—
108	C₂×C₄×C₅	—	8	8	no	—
109	C₂×C₄×C₅	—	8	2	no	—
110	C₂×C₄×C₅	—	8	2	no	—
111	C₂×C₄×C₅	—	8	2	no	—
112	C₂×C₄×C₅	#102	1	2	yes	—
113	C₂×C₄×C₅	#102	1	2	yes	—
114	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₅	#135	1	2	yes	—
118	C₅×C₈	—	2	2	no	—
119	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
120	C₅×C₈	#117	1	1	yes	characteristic
121	C₅×C₈	C₄×D₄	1	2	yes	—
122	C₂²×D₄	—	2	1	no	—
123	C₂³×C₄	—	4	1	no	—
124	C₂×D₈	—	2	1	no	—
125	C₂×D₈	—	2	2	no	—
126	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
127	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
128	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
129	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
130	C₄×D₄	—	2	4	no	—
131	C₄×D₄	—	4	2	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₂²⋊C₄	—	4	1	no	—
135	C₂×C₄²	—	4	1	no	—
136	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	complemented
140	#107	—	2	1	no	—
141	#107	—	2	2	no	—
142	#107	C₂×C₄×C₅	1	2	yes	—
143	#107	C₂×C₄×C₅	1	4	yes	complemented
144	#102	—	2	4	no	—
145	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
146	C₄×C₈	C₅×D₄	1	2	yes	—
147	#95	—	4	1	no	—
148	C₂²×C₅	—	4	4	no	—
149	C₂²×C₅	—	8	2	no	—
150	C₂²×C₅	—	8	2	no	—
151	C₂²×C₅	—	8	4	no	—
152	C₂²×C₅	—	8	4	no	—
153	C₂²×C₅	C₂×#102	1	1	yes	characteristic
154	C₂²×C₅	C₄×D₈	1	2	yes	—
155	C₂²×C₅	#509	1	1	yes	characteristic
156	C₂²×C₅	#509	1	2	yes	—
157	C₂²×C₅	#510	1	1	yes	characteristic
158	C₄×C₅	—	2	2	no	—
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	—	8	2	no	—
161	C₄×C₅	—	8	4	no	—
162	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
163	C₄×C₅	#416	1	1	yes	characteristic
164	C₄×C₅	C₄×SD₁₆	1	2	yes	—
165	C₂⁴	—	4	1	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	—	4	1	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	—	4	4	no	—
173	C₂²×C₄	—	8	4	no	—
174	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
175	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
177	D₈	—	2	2	no	—
178	D₈	—	2	4	no	—
179	D₈	C₄²×C₅	1	4	yes	complemented
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
182	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
183	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
184	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
185	C₂×C₈	C₅×#41	1	2	yes	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	—	4	4	no	—
191	C₄²	—	8	2	no	—
192	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
193	C₄²	C₅×SD₁₆	1	2	yes	—
194	C₁₆	—	2	4	no	—
195	C₂×C₅	—	8	4	no	—
196	C₂×C₅	C₄×#102	1	1	yes	characteristic
197	C₂×C₅	#2139	1	1	yes	characteristic
198	C₂×C₅	#2140	1	1	yes	characteristic
199	C₂×C₅	#2175	1	2	yes	—
200	C₂×C₅	#2177	1	2	yes	—
201	C₂³	—	4	2	no	—
202	C₂³	—	4	4	no	—
203	C₂³	—	8	2	no	—
204	C₂³	—	8	1	no	—
205	C₂³	—	8	1	no	—
206	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
207	D₄	—	2	4	no	—
208	D₄	—	4	2	no	—
209	D₄	—	4	4	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	8	8	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	C₅×#102	1	2	yes	—
220	C₂×C₄	C₅×#102	1	2	yes	—
221	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
223	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
224	C₂×C₄	C₅×#135	1	2	yes	—
225	C₈	—	2	2	no	—
226	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
227	C₈	C₅×#117	1	1	yes	characteristic
228	C₈	C₄×C₅×D₄	1	2	yes	—
229	C₅	#12637	1	1	yes	characteristic, direct product factor, Sylow
230	C₂²	—	4	4	no	—
231	C₂²	—	8	2	no	—
232	C₂²	—	8	2	no	—
233	C₂²	—	8	4	no	—
234	C₂²	—	8	4	no	—
235	C₂²	C₂×C₅×#102	1	1	yes	characteristic
236	C₂²	C₄×C₅×D₈	1	2	yes	—
237	C₂²	C₅×#509	1	1	yes	characteristic
238	C₂²	C₅×#509	1	2	yes	—
239	C₂²	C₅×#510	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	8	4	no	—
244	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
245	C₄	C₅×#416	1	1	yes	characteristic
246	C₄	C₄×C₅×SD₁₆	1	2	yes	—
247	C₂	—	8	4	no	—
248	C₂	C₄×C₅×#102	1	1	yes	characteristic
249	C₂	C₅×#2139	1	1	yes	characteristic
250	C₂	C₅×#2140	1	1	yes	characteristic
251	C₂	C₅×#2175	1	2	yes	—
252	C₂	C₅×#2177	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
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
3. C₂×C₅×#346	C₂	semidirect
4. C₄×C₅×#107	C₂	semidirect
5. C₅×#1380	C₂	semidirect
10. C₄×C₅×D₈	C₄	semidirect
16. C₅×#357	C₄	semidirect
19. #12637	C₅	direct (N × H)
37. C₂×C₅×C₁₆	C₂×C₄	semidirect
41. C₅×#107	C₂×C₄	semidirect
48. C₂×#346	C₂×C₅	semidirect
49. C₄×#107	C₂×C₅	semidirect
50. #1380	C₂×C₅	semidirect
65. C₅×D₈	C₄²	semidirect
85. C₄×D₈	C₄×C₅	semidirect
91. #357	C₄×C₅	semidirect
139. C₂×C₁₆	C₂×C₄×C₅	semidirect
143. #107	C₂×C₄×C₅	semidirect
179. D₈	C₄²×C₅	semidirect
229. C₅	#12637	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×#357	2
C₂×C₅×#346	2
C₄×C₅×#107	2
C₅×#1380	2
#12637	5

Subgroups — C5×#12637 #11935357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12637 #11935357