Subgroups — C₅×#33441 #11956161

Order 1280.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#504	C₂	1	2	yes	maximal, complemented
2	C₅×#2955	C₂	1	1	yes	maximal, characteristic
3	C₅×#2171	C₂	1	4	yes	maximal, complemented
4	C₅×#2170	C₂	1	4	yes	maximal, complemented
5	C₅×#2153	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2117	C₂	1	2	yes	maximal, complemented
7	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#575	C₄	1	1	yes	characteristic
9	C₂×C₅×#131	C₂²	1	2	yes	—
10	C₅×#504	—	2	2	no	—
11	C₅×#504	—	2	2	no	—
12	C₅×#504	C₂²	1	4	yes	complemented
13	C₅×#504	C₂²	1	4	yes	complemented
14	C₂×C₅×M₃₂	—	2	2	no	—
15	C₂×C₅×M₃₂	C₂²	1	2	yes	—
16	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₅×C₁₆	—	2	2	no	—
18	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
19	C₅×#444	C₄	1	4	yes	—
20	C₅×#443	C₂²	1	4	yes	—
21	C₅×#430	C₄	1	1	yes	characteristic
22	C₅×#421	C₄	1	2	yes	—
23	C₅×#405	C₂²	1	1	yes	characteristic
24	C₅×#364	C₂²	1	4	yes	complemented
25	C₅×#350	C₂²	1	8	yes	complemented
26	C₅×#346	C₂²	1	2	yes	complemented
27	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
28	#33441	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂×C₅×#41	C₂×C₄	1	2	yes	—
30	C₅×#137	C₈	1	4	yes	—
31	C₅×AGL(1,ℤ/8)	C₈	1	4	yes	—
32	C₅×#135	C₂×C₄	1	4	yes	—
33	C₅×#131	—	2	2	no	—
34	C₅×#131	—	2	2	no	—
35	C₅×#131	C₂×C₄	1	4	yes	—
36	C₅×#131	C₂³	1	4	yes	—
37	C₂×C₅×M₁₆	—	2	2	no	—
38	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	—	2	2	no	—
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₅×#117	C₂×C₄	1	1	yes	characteristic
43	C₅×M₃₂	—	2	2	no	—
44	C₅×M₃₂	—	2	2	no	—
45	C₅×M₃₂	—	2	4	no	—
46	C₅×M₃₂	—	2	4	no	—
47	C₅×M₃₂	C₂³	1	4	yes	—
48	C₂×C₅×C₁₆	—	2	2	no	—
49	C₂×C₅×C₁₆	—	2	4	no	—
50	C₂×C₅×C₁₆	—	2	4	no	—
51	C₂×C₅×C₁₆	D₄	1	2	yes	—
52	C₂×C₅×C₁₆	D₄	1	2	yes	—
53	C₂×C₅×C₁₆	C₂³	1	2	yes	—
54	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
56	C₅×#108	C₂×C₄	1	4	yes	—
57	C₅×#104	C₂×C₄	1	8	yes	—
58	C₅×#97	C₂×C₄	1	2	yes	—
59	C₄×C₅×C₈	C₂³	1	2	yes	—
60	C₂×#504	C₂×C₅	1	2	yes	complemented
61	#2955	C₂×C₅	1	1	yes	characteristic
62	#2171	C₂×C₅	1	4	yes	complemented
63	#2170	C₂×C₅	1	4	yes	complemented
64	#2153	C₂×C₅	1	1	yes	characteristic, complemented
65	#2117	C₂×C₅	1	2	yes	complemented
66	#2108	C₂×C₅	1	1	yes	characteristic, complemented
67	C₅×#41	—	2	2	no	—
68	C₅×#41	—	2	2	no	—
69	C₅×#41	C₂×C₈	1	4	yes	—
70	C₅×#41	C₂²×C₄	1	4	yes	—
71	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
72	C₂×C₅×D₄	—	2	2	no	—
73	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₅	—	2	2	no	—
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₅×Q₁₆	C₂×C₈	1	4	yes	—
77	C₅×SD₁₆	C₂×C₈	1	8	yes	—
78	C₅×D₈	C₂×C₈	1	4	yes	—
79	C₅×M₁₆	—	2	2	no	—
80	C₅×M₁₆	—	2	4	no	—
81	C₅×M₁₆	—	2	4	no	—
82	C₅×M₁₆	C₂×C₈	1	4	yes	—
83	C₅×M₁₆	C₂²×C₄	1	4	yes	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	2	4	no	—
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₅×C₈	#41	1	2	yes	—
91	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
92	C₅×C₄⋊C₄	—	2	1	no	—
93	C₅×C₂²⋊C₄	—	2	1	no	—
94	C₄²×C₅	C₂²×C₄	1	2	yes	—
95	C₅×C₁₆	—	2	4	no	—
96	C₅×C₁₆	—	2	2	no	—
97	C₅×C₁₆	C₂×D₄	1	2	yes	—
98	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
99	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
100	#575	C₄×C₅	1	1	yes	characteristic
101	C₂×#131	C₂²×C₅	1	2	yes	—
102	#504	—	2	2	no	—
103	#504	—	2	2	no	—
104	#504	C₂²×C₅	1	4	yes	complemented
105	#504	C₂²×C₅	1	4	yes	complemented
106	C₂×M₃₂	—	2	2	no	—
107	C₂×M₃₂	C₂²×C₅	1	2	yes	—
108	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
109	C₂²×C₁₆	—	2	2	no	—
110	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
111	#444	C₄×C₅	1	4	yes	—
112	#443	C₂²×C₅	1	4	yes	—
113	#430	C₄×C₅	1	1	yes	characteristic
114	#421	C₄×C₅	1	2	yes	—
115	#405	C₂²×C₅	1	1	yes	characteristic
116	#364	C₂²×C₅	1	4	yes	complemented
117	#350	C₂²×C₅	1	8	yes	complemented
118	#346	C₂²×C₅	1	2	yes	complemented
119	C₄×C₁₆	C₂²×C₅	1	2	yes	complemented
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	#131	1	1	yes	characteristic
122	C₅×Q₈	—	2	2	no	—
123	C₅×Q₈	C₂²×C₈	1	4	yes	—
124	C₅×D₄	—	2	4	no	—
125	C₅×D₄	—	2	4	no	—
126	C₅×D₄	—	2	2	no	—
127	C₅×D₄	C₂²×C₈	1	4	yes	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₅	#131	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₅×C₈	—	2	4	no	—
137	C₅×C₈	C₄×D₄	1	2	yes	—
138	C₅×C₈	C₂²×C₈	1	4	yes	—
139	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
140	C₅×C₈	C₂×#41	1	1	yes	characteristic
141	C₂×#41	C₂×C₄×C₅	1	2	yes	—
142	#137	C₅×C₈	1	4	yes	—
143	AGL(1,ℤ/8)	C₅×C₈	1	4	yes	—
144	#135	C₂×C₄×C₅	1	4	yes	—
145	#131	—	2	2	no	—
146	#131	—	2	2	no	—
147	#131	C₂×C₄×C₅	1	4	yes	—
148	#131	C₂³×C₅	1	4	yes	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
151	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
154	#117	C₂×C₄×C₅	1	1	yes	characteristic
155	M₃₂	—	2	2	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	—	2	4	no	—
158	M₃₂	—	2	4	no	—
159	M₃₂	C₂³×C₅	1	4	yes	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	2	4	no	—
162	C₂×C₁₆	—	2	4	no	—
163	C₂×C₁₆	C₅×D₄	1	2	yes	—
164	C₂×C₁₆	C₅×D₄	1	2	yes	—
165	C₂×C₁₆	C₂³×C₅	1	2	yes	—
166	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
167	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
168	#108	C₂×C₄×C₅	1	4	yes	—
169	#104	C₂×C₄×C₅	1	8	yes	—
170	#97	C₂×C₄×C₅	1	2	yes	—
171	C₄×C₈	C₂³×C₅	1	2	yes	—
172	C₂²×C₅	—	2	4	no	—
173	C₂²×C₅	—	4	1	no	—
174	C₂²×C₅	—	4	4	no	—
175	C₂²×C₅	C₈×D₄	1	2	yes	—
176	C₂²×C₅	C₂×#131	1	1	yes	characteristic
177	C₄×C₅	—	2	4	no	—
178	C₄×C₅	—	4	2	no	—
179	C₄×C₅	C₈×D₄	1	2	yes	—
180	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
182	#41	—	2	2	no	—
183	#41	—	2	2	no	—
184	#41	C₂×C₅×C₈	1	4	yes	—
185	#41	C₂²×C₄×C₅	1	4	yes	—
186	C₂×Q₈	C₂×C₅×C₈	1	2	yes	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
191	Q₁₆	C₂×C₅×C₈	1	4	yes	—
192	SD₁₆	C₂×C₅×C₈	1	8	yes	—
193	D₈	C₂×C₅×C₈	1	4	yes	—
194	M₁₆	—	2	2	no	—
195	M₁₆	—	2	4	no	—
196	M₁₆	—	2	4	no	—
197	M₁₆	C₂×C₅×C₈	1	4	yes	—
198	M₁₆	C₂²×C₄×C₅	1	4	yes	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
203	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
204	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
205	C₂×C₈	C₅×#41	1	2	yes	—
206	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
207	C₄⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₄²	C₂²×C₄×C₅	1	2	yes	—
210	C₁₆	—	2	4	no	—
211	C₁₆	—	2	2	no	—
212	C₁₆	C₂×C₅×D₄	1	1	yes	characteristic
213	C₁₆	C₂×C₅×D₄	1	2	yes	—
214	C₁₆	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂×C₅	—	2	1	no	—
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	—	4	4	no	—
218	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
219	C₂³	—	2	2	no	—
220	C₂³	C₅×#131	1	1	yes	characteristic
221	Q₈	—	2	2	no	—
222	Q₈	C₂²×C₅×C₈	1	4	yes	—
223	D₄	—	2	4	no	—
224	D₄	—	2	4	no	—
225	D₄	—	2	2	no	—
226	D₄	C₂²×C₅×C₈	1	4	yes	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
232	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
233	C₂×C₄	C₅×#131	1	1	yes	characteristic
234	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
235	C₈	—	2	4	no	—
236	C₈	C₄×C₅×D₄	1	2	yes	—
237	C₈	C₂²×C₅×C₈	1	4	yes	—
238	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
239	C₈	C₂×C₅×#41	1	1	yes	characteristic
240	C₅	#33441	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	2	4	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	C₅×C₈×D₄	1	2	yes	—
245	C₂²	C₂×C₅×#131	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	C₅×C₈×D₄	1	2	yes	—
249	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
250	C₄	C₂³×C₅×C₈	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	4	4	no	—
254	C₂	C₂×C₅×C₈×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₁₆	C₂×D₄
commutator	C₄	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂×C₅	C₂×C₈×D₄
prime radical	C₅×#575	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₁₆ ◁ C₂²×C₅×C₁₆ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₅×#504	C₂	semidirect
3. C₅×#2171	C₂	semidirect
4. C₅×#2170	C₂	semidirect
5. C₅×#2153	C₂	semidirect
6. C₅×#2117	C₂	semidirect
7. C₅×#2108	C₂	semidirect
12. C₅×#504	C₂²	semidirect
13. C₅×#504	C₂²	semidirect
24. C₅×#364	C₂²	semidirect
25. C₅×#350	C₂²	semidirect
26. C₅×#346	C₂²	semidirect
27. C₄×C₅×C₁₆	C₂²	semidirect
28. #33441	C₅	direct (N × H)
60. C₂×#504	C₂×C₅	semidirect
62. #2171	C₂×C₅	semidirect
63. #2170	C₂×C₅	semidirect
64. #2153	C₂×C₅	semidirect
65. #2117	C₂×C₅	semidirect
66. #2108	C₂×C₅	semidirect
104. #504	C₂²×C₅	semidirect
105. #504	C₂²×C₅	semidirect
116. #364	C₂²×C₅	semidirect
117. #350	C₂²×C₅	semidirect
118. #346	C₂²×C₅	semidirect
119. C₄×C₁₆	C₂²×C₅	semidirect
240. C₅	#33441	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×#504	2
C₅×#2955	2
C₅×#2171	2
C₅×#2170	2
C₅×#2153	2
C₅×#2117	2
C₅×#2108	2
#33441	5

Subgroups — C5×#33441 #11956161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33441 #11956161