Subgroups — C₅×#33858 #11956578

Order 1280.

668 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3415	C₂	1	2	yes	maximal, complemented
2	C₅×#3292	C₂	1	2	yes	maximal, complemented
3	C₅×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2221	C₂	1	2	yes	maximal, complemented
5	C₅×#2213	C₂	1	2	yes	maximal, complemented
6	C₅×#2192	C₂	1	4	yes	maximal, complemented
7	C₅×#2178	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#576	—	2	2	no	—
10	C₅×#576	C₂²	1	4	yes	complemented
11	C₅×#509	—	2	4	no	—
12	C₅×#509	C₂²	1	4	yes	complemented
13	C₅×#508	—	2	2	no	—
14	C₅×#508	C₂²	1	1	yes	characteristic
15	C₂×C₅×D₁₆	—	2	2	no	—
16	C₂×C₅×D₁₆	C₂²	1	2	yes	complemented
17	C₂×C₅×D₁₆	C₂²	1	2	yes	complemented
18	C₅×#504	C₂²	1	2	yes	complemented
19	C₅×#493	C₂²	1	2	yes	complemented
20	C₅×#472	—	2	2	no	—
21	C₅×#472	C₂²	1	4	yes	complemented
22	C₅×#453	—	2	4	no	—
23	C₅×#443	C₂²	1	1	yes	characteristic
24	C₅×#443	C₂²	1	2	yes	complemented
25	C₅×#368	C₂²	1	1	yes	characteristic, complemented
26	C₅×#364	C₂²	1	1	yes	characteristic, complemented
27	C₅×#361	C₂²	1	4	yes	complemented
28	C₅×#359	C₂²	1	2	yes	complemented
29	C₅×#350	C₂²	1	2	yes	complemented
30	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
31	#33858	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₅×H(5,𝔽₂)	—	2	4	no	—
33	C₅×AGL(1,ℤ/8)	—	2	4	no	—
34	C₅×AGL(1,ℤ/8)	—	2	4	no	—
35	C₅×AGL(1,ℤ/8)	—	4	4	no	—
36	C₅×AGL(1,ℤ/8)	—	4	4	no	—
37	C₅×#135	—	2	4	no	—
38	C₅×#135	—	4	2	no	—
39	C₅×#135	D₄	1	1	yes	characteristic
40	C₅×#135	C₂³	1	2	yes	—
41	C₂×C₅×D₈	—	2	4	no	—
42	C₂×C₅×D₈	—	2	2	no	—
43	C₂×C₅×D₈	—	4	2	no	—
44	C₂×C₅×D₈	C₂³	1	4	yes	—
45	C₅×#131	—	2	2	no	—
46	C₅×#131	C₂³	1	2	yes	—
47	C₅×#127	—	2	2	no	—
48	C₅×Q₃₂	D₄	1	1	yes	characteristic, complemented
49	C₅×SD₃₂	—	2	4	no	—
50	C₅×SD₃₂	D₄	1	2	yes	complemented
51	C₅×D₁₆	—	2	4	no	—
52	C₅×D₁₆	—	2	2	no	—
53	C₅×D₁₆	—	2	2	no	—
54	C₅×D₁₆	—	4	2	no	—
55	C₅×D₁₆	D₄	1	1	yes	characteristic, complemented
56	C₅×M₃₂	—	2	2	no	—
57	C₅×M₃₂	C₂³	1	2	yes	—
58	C₂×C₅×C₁₆	—	2	2	no	—
59	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
61	C₅×#108	D₄	1	1	yes	characteristic
62	C₅×#108	C₂³	1	2	yes	—
63	C₅×#104	—	2	2	no	—
64	C₅×#104	D₄	1	2	yes	—
65	C₅×#100	—	2	4	no	—
66	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
67	#3415	C₂×C₅	1	2	yes	complemented
68	#3292	C₂×C₅	1	2	yes	complemented
69	#2246	C₂×C₅	1	1	yes	characteristic, complemented
70	#2221	C₂×C₅	1	2	yes	complemented
71	#2213	C₂×C₅	1	2	yes	complemented
72	#2192	C₂×C₅	1	4	yes	complemented
73	#2178	C₂×C₅	1	1	yes	characteristic, complemented
74	#2170	C₂×C₅	1	1	yes	characteristic, complemented
75	C₅×#41	—	2	2	no	—
76	C₅×#41	—	4	4	no	—
77	C₅×#41	—	4	4	no	—
78	C₅×#41	C₂×D₄	1	2	yes	—
79	C₂×C₅×D₄	—	2	4	no	—
80	C₂×C₅×D₄	—	4	4	no	—
81	C₂×C₅×D₄	—	4	4	no	—
82	C₂×C₅×D₄	—	8	2	no	—
83	C₂×C₅×D₄	—	8	4	no	—
84	C₅×Q₁₆	D₈	1	1	yes	characteristic
85	C₅×Q₁₆	C₂×D₄	1	2	yes	—
86	C₅×SD₁₆	—	2	2	no	—
87	C₅×SD₁₆	—	4	4	no	—
88	C₅×SD₁₆	—	4	4	no	—
89	C₅×SD₁₆	D₈	1	2	yes	—
90	C₅×D₈	—	2	4	no	—
91	C₅×D₈	—	2	4	no	—
92	C₅×D₈	—	4	4	no	—
93	C₅×D₈	—	4	4	no	—
94	C₅×D₈	—	4	2	no	—
95	C₅×D₈	—	4	2	no	—
96	C₅×D₈	D₈	1	1	yes	characteristic
97	C₅×D₈	C₂×D₄	1	2	yes	—
98	C₅×M₁₆	—	2	2	no	—
99	C₅×M₁₆	—	2	2	no	—
100	C₅×M₁₆	—	4	2	no	—
101	C₅×M₁₆	C₂×D₄	1	2	yes	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	—	4	2	no	—
104	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
106	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
107	C₅×C₁₆	—	2	2	no	—
108	C₅×C₁₆	—	2	1	no	—
109	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
110	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
111	#576	—	2	2	no	—
112	#576	C₂²×C₅	1	4	yes	complemented
113	#509	—	2	4	no	—
114	#509	C₂²×C₅	1	4	yes	complemented
115	#508	—	2	2	no	—
116	#508	C₂²×C₅	1	1	yes	characteristic
117	C₂×D₁₆	—	2	2	no	—
118	C₂×D₁₆	C₂²×C₅	1	2	yes	complemented
119	C₂×D₁₆	C₂²×C₅	1	2	yes	complemented
120	#504	C₂²×C₅	1	2	yes	complemented
121	#493	C₂²×C₅	1	2	yes	complemented
122	#472	—	2	2	no	—
123	#472	C₂²×C₅	1	4	yes	complemented
124	#453	—	2	4	no	—
125	#443	C₂²×C₅	1	1	yes	characteristic
126	#443	C₂²×C₅	1	2	yes	complemented
127	#368	C₂²×C₅	1	1	yes	characteristic, complemented
128	#364	C₂²×C₅	1	1	yes	characteristic, complemented
129	#361	C₂²×C₅	1	4	yes	complemented
130	#359	C₂²×C₅	1	2	yes	complemented
131	#350	C₂²×C₅	1	2	yes	complemented
132	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
133	C₂³×C₅	—	4	4	no	—
134	C₂³×C₅	—	8	4	no	—
135	C₅×Q₈	—	2	2	no	—
136	C₅×Q₈	C₂×D₈	1	2	yes	—
137	C₅×D₄	—	2	2	no	—
138	C₅×D₄	—	2	2	no	—
139	C₅×D₄	—	4	2	no	—
140	C₅×D₄	—	4	4	no	—
141	C₅×D₄	—	4	4	no	—
142	C₅×D₄	—	8	4	no	—
143	C₅×D₄	—	8	4	no	—
144	C₅×D₄	—	16	4	no	—
145	C₅×D₄	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	4	2	no	—
149	C₂×C₄×C₅	—	8	4	no	—
150	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
151	C₅×C₈	—	2	2	no	—
152	C₅×C₈	—	4	2	no	—
153	C₅×C₈	C₂×D₈	1	1	yes	characteristic
154	C₅×C₈	C₂×D₈	1	1	yes	characteristic
155	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
156	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
157	H(5,𝔽₂)	—	2	4	no	—
158	AGL(1,ℤ/8)	—	2	4	no	—
159	AGL(1,ℤ/8)	—	2	4	no	—
160	AGL(1,ℤ/8)	—	4	4	no	—
161	AGL(1,ℤ/8)	—	4	4	no	—
162	#135	—	2	4	no	—
163	#135	—	4	2	no	—
164	#135	C₅×D₄	1	1	yes	characteristic
165	#135	C₂³×C₅	1	2	yes	—
166	C₂×D₈	—	2	4	no	—
167	C₂×D₈	—	2	2	no	—
168	C₂×D₈	—	4	2	no	—
169	C₂×D₈	C₂³×C₅	1	4	yes	—
170	#131	—	2	2	no	—
171	#131	C₂³×C₅	1	2	yes	—
172	#127	—	2	2	no	—
173	Q₃₂	C₅×D₄	1	1	yes	characteristic, complemented
174	SD₃₂	—	2	4	no	—
175	SD₃₂	C₅×D₄	1	2	yes	complemented
176	D₁₆	—	2	4	no	—
177	D₁₆	—	2	2	no	—
178	D₁₆	—	2	2	no	—
179	D₁₆	—	4	2	no	—
180	D₁₆	C₅×D₄	1	1	yes	characteristic, complemented
181	M₃₂	—	2	2	no	—
182	M₃₂	C₂³×C₅	1	2	yes	—
183	C₂×C₁₆	—	2	2	no	—
184	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
185	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
186	#108	C₅×D₄	1	1	yes	characteristic
187	#108	C₂³×C₅	1	2	yes	—
188	#104	—	2	2	no	—
189	#104	C₅×D₄	1	2	yes	—
190	#100	—	2	4	no	—
191	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	4	2	no	—
194	C₂²×C₅	—	8	4	no	—
195	C₂²×C₅	—	16	4	no	—
196	C₂²×C₅	—	32	4	no	—
197	C₂²×C₅	#576	1	1	yes	characteristic
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	4	2	no	—
201	C₄×C₅	D₄²	1	1	yes	characteristic
202	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
203	#41	—	2	2	no	—
204	#41	—	4	4	no	—
205	#41	—	4	4	no	—
206	#41	C₂×C₅×D₄	1	2	yes	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	4	4	no	—
209	C₂×D₄	—	4	4	no	—
210	C₂×D₄	—	8	2	no	—
211	C₂×D₄	—	8	4	no	—
212	Q₁₆	C₅×D₈	1	1	yes	characteristic
213	Q₁₆	C₂×C₅×D₄	1	2	yes	—
214	SD₁₆	—	2	2	no	—
215	SD₁₆	—	4	4	no	—
216	SD₁₆	—	4	4	no	—
217	SD₁₆	C₅×D₈	1	2	yes	—
218	D₈	—	2	4	no	—
219	D₈	—	2	4	no	—
220	D₈	—	4	4	no	—
221	D₈	—	4	4	no	—
222	D₈	—	4	2	no	—
223	D₈	—	4	2	no	—
224	D₈	C₅×D₈	1	1	yes	characteristic
225	D₈	C₂×C₅×D₄	1	2	yes	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	C₂×C₅×D₄	1	2	yes	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
233	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
234	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
235	C₁₆	—	2	2	no	—
236	C₁₆	—	2	1	no	—
237	C₁₆	C₂×C₅×D₄	1	1	yes	characteristic
238	C₁₆	C₂×C₅×D₄	1	1	yes	characteristic
239	C₂×C₅	—	2	1	no	—
240	C₂×C₅	—	4	2	no	—
241	C₂×C₅	—	8	2	no	—
242	C₂×C₅	—	16	4	no	—
243	C₂×C₅	D₄×D₈	1	1	yes	characteristic
244	C₂³	—	4	4	no	—
245	C₂³	—	8	4	no	—
246	Q₈	—	2	2	no	—
247	Q₈	C₂×C₅×D₈	1	2	yes	—
248	D₄	—	2	2	no	—
249	D₄	—	2	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	4	no	—
252	D₄	—	4	4	no	—
253	D₄	—	8	4	no	—
254	D₄	—	8	4	no	—
255	D₄	—	16	4	no	—
256	D₄	C₂×C₅×D₈	1	2	yes	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
262	C₈	—	2	2	no	—
263	C₈	—	4	2	no	—
264	C₈	C₂×C₅×D₈	1	1	yes	characteristic
265	C₈	C₂×C₅×D₈	1	1	yes	characteristic
266	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
267	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
268	C₅	#33858	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²	—	2	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	8	4	no	—
272	C₂²	—	16	4	no	—
273	C₂²	—	32	4	no	—
274	C₂²	C₅×#576	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	2	no	—
278	C₄	C₅×D₄²	1	1	yes	characteristic
279	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	4	2	no	—
282	C₂	—	8	2	no	—
283	C₂	—	16	4	no	—
284	C₂	C₅×D₄×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#443 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3415	C₂	semidirect
2. C₅×#3292	C₂	semidirect
3. C₅×#2246	C₂	semidirect
4. C₅×#2221	C₂	semidirect
5. C₅×#2213	C₂	semidirect
6. C₅×#2192	C₂	semidirect
7. C₅×#2178	C₂	semidirect
8. C₅×#2170	C₂	semidirect
10. C₅×#576	C₂²	semidirect
12. C₅×#509	C₂²	semidirect
16. C₂×C₅×D₁₆	C₂²	semidirect
17. C₂×C₅×D₁₆	C₂²	semidirect
18. C₅×#504	C₂²	semidirect
19. C₅×#493	C₂²	semidirect
21. C₅×#472	C₂²	semidirect
24. C₅×#443	C₂²	semidirect
25. C₅×#368	C₂²	semidirect
26. C₅×#364	C₂²	semidirect
27. C₅×#361	C₂²	semidirect
28. C₅×#359	C₂²	semidirect
29. C₅×#350	C₂²	semidirect
30. C₄×C₅×C₁₆	C₂²	semidirect
31. #33858	C₅	direct (N × H)
48. C₅×Q₃₂	D₄	semidirect
50. C₅×SD₃₂	D₄	semidirect
55. C₅×D₁₆	D₄	semidirect
67. #3415	C₂×C₅	semidirect
68. #3292	C₂×C₅	semidirect
69. #2246	C₂×C₅	semidirect
70. #2221	C₂×C₅	semidirect
71. #2213	C₂×C₅	semidirect
72. #2192	C₂×C₅	semidirect
73. #2178	C₂×C₅	semidirect
74. #2170	C₂×C₅	semidirect
112. #576	C₂²×C₅	semidirect
114. #509	C₂²×C₅	semidirect
118. C₂×D₁₆	C₂²×C₅	semidirect
119. C₂×D₁₆	C₂²×C₅	semidirect
120. #504	C₂²×C₅	semidirect
121. #493	C₂²×C₅	semidirect
123. #472	C₂²×C₅	semidirect
126. #443	C₂²×C₅	semidirect
127. #368	C₂²×C₅	semidirect
128. #364	C₂²×C₅	semidirect
129. #361	C₂²×C₅	semidirect
130. #359	C₂²×C₅	semidirect
131. #350	C₂²×C₅	semidirect
132. C₄×C₁₆	C₂²×C₅	semidirect
173. Q₃₂	C₅×D₄	semidirect
175. SD₃₂	C₅×D₄	semidirect
180. D₁₆	C₅×D₄	semidirect
268. C₅	#33858	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3415	2
C₅×#3292	2
C₅×#2246	2
C₅×#2221	2
C₅×#2213	2
C₅×#2192	2
C₅×#2178	2
C₅×#2170	2
#33858	5

Subgroups — C5×#33858 #11956578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33858 #11956578