Subgroups — C₅×#33491 #11956211

Order 1280.

584 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2929	C₂	1	1	yes	maximal, characteristic
2	C₅×#2168	C₂	1	4	yes	maximal, complemented
3	C₅×D₄×C₁₆	C₂	1	4	yes	maximal, complemented
4	C₅×#2164	C₂	1	2	yes	maximal, complemented
5	C₅×#2113	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#131	C₂²	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×#525	C₄	1	1	yes	characteristic
10	C₂×C₅×M₃₂	C₂²	1	2	yes	complemented
11	C₂²×C₅×C₁₆	C₂²	1	4	yes	—
12	C₂²×C₅×C₁₆	C₂²	1	2	yes	complemented
13	C₅×#435	C₄	1	4	yes	—
14	C₅×C₈×D₄	C₂²	1	4	yes	—
15	C₅×#424	C₄	1	1	yes	characteristic
16	C₅×#408	C₄	1	1	yes	characteristic
17	C₅×#407	C₄	1	1	yes	characteristic
18	C₅×#405	C₂²	1	1	yes	characteristic
19	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
20	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	—
21	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
22	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
23	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
24	C₅×#346	C₂²	1	2	yes	complemented
25	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
26	#33491	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₅×#131	—	4	2	no	—
30	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	—	4	2	no	—
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₅×#123	C₈	1	2	yes	—
40	C₅×C₂²⋊Q₈	C₈	1	2	yes	—
41	C₅×#121	C₈	1	2	yes	—
42	C₅×#120	C₈	1	2	yes	—
43	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
44	C₅×#117	C₂×C₄	1	1	yes	characteristic
45	C₅×M₃₂	—	4	2	no	—
46	C₂×C₅×C₁₆	—	4	4	no	—
47	C₂×C₅×C₁₆	—	4	2	no	—
48	C₂×C₅×C₁₆	C₂³	1	4	yes	—
49	C₂×C₅×C₁₆	C₂³	1	4	yes	—
50	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
51	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
52	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
53	C₅×#97	C₂×C₄	1	2	yes	—
54	C₄×C₅×C₈	C₂³	1	2	yes	—
55	#2929	C₂×C₅	1	1	yes	characteristic
56	#2168	C₂×C₅	1	4	yes	complemented
57	D₄×C₁₆	C₂×C₅	1	4	yes	complemented
58	#2164	C₂×C₅	1	2	yes	complemented
59	#2113	C₂×C₅	1	2	yes	complemented
60	C₂×C₂²⋊C₁₆	C₂×C₅	1	2	yes	complemented
61	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
62	C₅×#41	—	4	1	no	—
63	C₅×#41	—	4	1	no	—
64	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
65	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
66	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
67	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	—	4	1	no	—
69	C₂²×C₄×C₅	—	4	1	no	—
70	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
75	C₅×M₁₆	—	4	1	no	—
76	C₅×M₁₆	—	4	2	no	—
77	C₅×M₁₆	—	4	1	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	4	1	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
89	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
90	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
91	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
92	C₅×C₂²⋊C₄	C₂×C₈	1	4	yes	—
93	C₅×C₂²⋊C₄	C₂×C₈	1	4	yes	—
94	C₄²×C₅	C₂²×C₄	1	2	yes	—
95	C₅×C₁₆	—	2	4	no	—
96	C₅×C₁₆	—	2	4	no	—
97	C₂×#131	C₂²×C₅	1	1	yes	characteristic
98	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
99	#525	C₄×C₅	1	1	yes	characteristic
100	C₂×M₃₂	C₂²×C₅	1	2	yes	complemented
101	C₂²×C₁₆	C₂²×C₅	1	4	yes	—
102	C₂²×C₁₆	C₂²×C₅	1	2	yes	complemented
103	#435	C₄×C₅	1	4	yes	—
104	C₈×D₄	C₂²×C₅	1	4	yes	—
105	#424	C₄×C₅	1	1	yes	characteristic
106	#408	C₄×C₅	1	1	yes	characteristic
107	#407	C₄×C₅	1	1	yes	characteristic
108	#405	C₂²×C₅	1	1	yes	characteristic
109	C₄⋊C₁₆	C₂²×C₅	1	4	yes	complemented
110	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	—
111	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
112	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
113	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	complemented
114	#346	C₂²×C₅	1	2	yes	complemented
115	C₄×C₁₆	C₂²×C₅	1	2	yes	complemented
116	C₂³×C₅	—	2	4	no	—
117	C₂³×C₅	—	4	1	no	—
118	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
119	C₂³×C₅	C₂²×C₈	1	2	yes	—
120	C₂³×C₅	C₂²×C₈	1	2	yes	—
121	C₅×Q₈	—	4	1	no	—
122	C₅×D₄	—	2	8	no	—
123	C₅×D₄	—	4	1	no	—
124	C₅×D₄	—	4	2	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
134	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
138	C₅×C₈	—	2	2	no	—
139	C₅×C₈	—	2	1	no	—
140	C₅×C₈	—	2	2	no	—
141	C₅×C₈	—	2	2	no	—
142	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
143	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
144	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
145	#131	—	4	2	no	—
146	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
148	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
149	C₂²×C₈	—	4	2	no	—
150	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
151	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
152	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₅	1	2	yes	—
154	C₂²×C₈	C₂³×C₅	1	2	yes	—
155	#123	C₅×C₈	1	2	yes	—
156	C₂²⋊Q₈	C₅×C₈	1	2	yes	—
157	#121	C₅×C₈	1	2	yes	—
158	#120	C₅×C₈	1	2	yes	—
159	C₄×D₄	C₂×C₄×C₅	1	4	yes	—
160	#117	C₂×C₄×C₅	1	1	yes	characteristic
161	M₃₂	—	4	2	no	—
162	C₂×C₁₆	—	4	4	no	—
163	C₂×C₁₆	—	4	2	no	—
164	C₂×C₁₆	C₂³×C₅	1	4	yes	—
165	C₂×C₁₆	C₂³×C₅	1	4	yes	—
166	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
167	C₂²⋊C₈	C₂×C₄×C₅	1	4	yes	—
168	C₂²⋊C₈	C₂×C₄×C₅	1	4	yes	—
169	#97	C₂×C₄×C₅	1	2	yes	—
170	C₄×C₈	C₂³×C₅	1	2	yes	—
171	C₂²×C₅	—	2	2	no	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	2	4	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	4	4	no	—
178	C₂²×C₅	#504	1	4	yes	—
179	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
180	C₄×C₅	—	2	4	no	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	#518	1	1	yes	characteristic
185	C₄×C₅	#568	1	1	yes	characteristic
186	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
187	#41	—	4	1	no	—
188	#41	—	4	1	no	—
189	C₂×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
190	C₂×D₄	C₂×C₅×C₈	1	4	yes	—
191	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
192	C₂×D₄	C₂×C₅×C₈	1	1	yes	characteristic
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
196	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
199	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
200	M₁₆	—	4	2	no	—
201	M₁₆	—	4	1	no	—
202	M₁₆	—	4	1	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
210	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
213	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
214	C₄⋊C₄	C₂×C₅×C₈	1	2	yes	—
215	C₄⋊C₄	C₂×C₅×C₈	1	4	yes	—
216	C₂²⋊C₄	C₂×C₅×C₈	1	2	yes	—
217	C₂²⋊C₄	C₂×C₅×C₈	1	4	yes	—
218	C₂²⋊C₄	C₂×C₅×C₈	1	4	yes	—
219	C₄²	C₂²×C₄×C₅	1	2	yes	—
220	C₁₆	—	2	4	no	—
221	C₁₆	—	2	4	no	—
222	C₂×C₅	—	2	4	no	—
223	C₂×C₅	—	4	2	no	—
224	C₂×C₅	—	4	1	no	—
225	C₂×C₅	#2972	1	1	yes	characteristic
226	C₂×C₅	C₂×#504	1	2	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	—	4	1	no	—
229	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
230	C₂³	C₂²×C₅×C₈	1	2	yes	—
231	C₂³	C₂²×C₅×C₈	1	2	yes	—
232	Q₈	—	4	1	no	—
233	D₄	—	2	8	no	—
234	D₄	—	4	1	no	—
235	D₄	—	4	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
245	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
246	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
247	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
248	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	2	2	no	—
253	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
254	C₅	#33491	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₅×#504	1	4	yes	—
263	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	C₅×#518	1	1	yes	characteristic
269	C₄	C₅×#568	1	1	yes	characteristic
270	C₂	—	2	4	no	—
271	C₂	—	4	1	no	—
272	C₂	—	4	2	no	—
273	C₂	C₅×#2972	1	1	yes	characteristic
274	C₂	C₂×C₅×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₅×C₈	C₂⁴
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂²×C₅	C₂³×C₈
prime radical	C₅×#525	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2168	C₂	semidirect
3. C₅×D₄×C₁₆	C₂	semidirect
4. C₅×#2164	C₂	semidirect
5. C₅×#2113	C₂	semidirect
6. C₂×C₅×C₂²⋊C₁₆	C₂	semidirect
10. C₂×C₅×M₃₂	C₂²	semidirect
12. C₂²×C₅×C₁₆	C₂²	semidirect
19. C₅×C₄⋊C₁₆	C₂²	semidirect
23. C₅×C₂²⋊C₁₆	C₂²	semidirect
24. C₅×#346	C₂²	semidirect
25. C₄×C₅×C₁₆	C₂²	semidirect
26. #33491	C₅	direct (N × H)
56. #2168	C₂×C₅	semidirect
57. D₄×C₁₆	C₂×C₅	semidirect
58. #2164	C₂×C₅	semidirect
59. #2113	C₂×C₅	semidirect
60. C₂×C₂²⋊C₁₆	C₂×C₅	semidirect
100. C₂×M₃₂	C₂²×C₅	semidirect
102. C₂²×C₁₆	C₂²×C₅	semidirect
109. C₄⋊C₁₆	C₂²×C₅	semidirect
113. C₂²⋊C₁₆	C₂²×C₅	semidirect
114. #346	C₂²×C₅	semidirect
115. C₄×C₁₆	C₂²×C₅	semidirect
254. C₅	#33491	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×#2929	2
C₅×#2168	2
C₅×D₄×C₁₆	2
C₅×#2164	2
C₅×#2113	2
C₂×C₅×C₂²⋊C₁₆	2
#33491	5

Subgroups — C5×#33491 #11956211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33491 #11956211