Subgroups — C₅×#33775 #11956495

Order 1280.

582 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic
3	C₅×#2247	C₂	1	2	yes	maximal, complemented
4	C₅×#2241	C₂	1	2	yes	maximal, complemented
5	C₅×#2209	C₂	1	4	yes	maximal, complemented
6	C₅×#2188	C₂	1	4	yes	maximal
7	C₅×#2162	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅×Q₁₆	—	2	1	no	—
9	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
10	C₅×#533	—	2	1	no	—
11	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
12	C₂×C₅×Q₃₂	C₂²	1	8	yes	—
13	C₅×#500	C₂²	1	1	yes	characteristic
14	C₅×#494	—	2	1	no	—
15	C₅×#494	—	2	1	no	—
16	C₅×#494	C₂²	1	2	yes	—
17	C₅×#494	C₂²	1	2	yes	—
18	C₅×#494	C₂²	1	1	yes	characteristic
19	C₅×#487	—	2	1	no	—
20	C₅×#467	—	2	1	no	—
21	C₅×#467	C₂²	1	2	yes	—
22	C₄×C₅×Q₁₆	—	2	1	no	—
23	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₅×#358	C₂²	1	8	yes	—
27	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₅×C₁₆	C₂²	1	2	yes	—
29	#33775	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₅×Q₈	—	2	2	no	—
31	C₂×C₅×Q₁₆	—	2	1	no	—
32	C₂×C₅×Q₁₆	—	2	2	no	—
33	C₂×C₅×Q₁₆	—	2	2	no	—
34	C₂×C₅×Q₁₆	—	2	2	no	—
35	C₂×C₅×Q₁₆	—	4	4	no	—
36	C₂×C₅×Q₁₆	—	4	4	no	—
37	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
38	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊Q₈	—	2	1	no	—
42	C₅×C₄⋊Q₈	—	2	1	no	—
43	C₅×C₄⋊Q₈	—	2	2	no	—
44	C₅×C₄⋊Q₈	—	2	2	no	—
45	C₅×C₄⋊Q₈	—	4	1	no	—
46	C₅×C₄⋊Q₈	—	4	1	no	—
47	C₅×#125	—	4	1	no	—
48	C₅×C₂²⋊Q₈	—	2	2	no	—
49	C₅×C₂²⋊Q₈	—	4	1	no	—
50	C₄×C₅×Q₈	—	2	2	no	—
51	C₄×C₅×Q₈	—	4	1	no	—
52	C₅×#117	—	4	1	no	—
53	C₂×C₅×C₄⋊C₄	—	4	1	no	—
54	C₂×C₅×C₄⋊C₄	—	4	1	no	—
55	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
56	C₅×Q₃₂	—	4	8	no	—
57	C₂×C₅×C₁₆	C₂³	1	4	yes	—
58	C₅×#107	—	2	1	no	—
59	C₅×#107	C₂³	1	2	yes	—
60	C₅×#103	—	2	2	no	—
61	C₅×#103	—	2	2	no	—
62	C₄×C₅×C₈	D₄	1	1	yes	characteristic
63	C₄×C₅×C₈	D₄	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
66	#3170	C₂×C₅	1	1	yes	characteristic
67	C₂×#494	C₂×C₅	1	1	yes	characteristic
68	#2247	C₂×C₅	1	2	yes	complemented
69	#2241	C₂×C₅	1	2	yes	complemented
70	#2209	C₂×C₅	1	4	yes	complemented
71	#2188	C₂×C₅	1	4	yes	—
72	#2162	C₂×C₅	1	1	yes	characteristic
73	C₂×C₅×Q₈	—	2	2	no	—
74	C₂×C₅×Q₈	—	2	4	no	—
75	C₂×C₅×Q₈	—	4	4	no	—
76	C₂×C₅×Q₈	—	4	4	no	—
77	C₂×C₅×Q₈	—	4	1	no	—
78	C₂×C₅×Q₈	—	4	2	no	—
79	C₂²×C₄×C₅	—	4	2	no	—
80	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₅×Q₁₆	—	2	8	no	—
84	C₅×Q₁₆	—	2	4	no	—
85	C₅×Q₁₆	—	4	4	no	—
86	C₅×Q₁₆	—	4	4	no	—
87	C₅×Q₁₆	—	4	2	no	—
88	C₅×Q₁₆	—	4	4	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₅×C₄⋊C₄	—	4	2	no	—
97	C₅×C₄⋊C₄	—	4	2	no	—
98	C₅×C₄⋊C₄	—	4	2	no	—
99	C₅×C₄⋊C₄	—	4	2	no	—
100	C₅×C₄⋊C₄	—	4	1	no	—
101	C₅×C₄⋊C₄	—	4	1	no	—
102	C₅×C₄⋊C₄	—	4	2	no	—
103	C₅×C₄⋊C₄	—	4	2	no	—
104	C₅×C₄⋊C₄	—	4	1	no	—
105	C₅×C₂²⋊C₄	—	4	2	no	—
106	C₄²×C₅	—	4	2	no	—
107	C₄²×C₅	D₈	1	1	yes	characteristic
108	C₄²×C₅	D₈	1	1	yes	characteristic
109	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
111	C₅×C₁₆	—	2	4	no	—
112	C₂²×Q₁₆	—	2	1	no	—
113	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
114	#533	—	2	1	no	—
115	C₂×C₄⋊Q₈	—	2	1	no	—
116	C₂×Q₃₂	C₂²×C₅	1	8	yes	—
117	#500	C₂²×C₅	1	1	yes	characteristic
118	#494	—	2	1	no	—
119	#494	—	2	1	no	—
120	#494	C₂²×C₅	1	2	yes	—
121	#494	C₂²×C₅	1	2	yes	—
122	#494	C₂²×C₅	1	1	yes	characteristic
123	#487	—	2	1	no	—
124	#467	—	2	1	no	—
125	#467	C₂²×C₅	1	2	yes	—
126	C₄×Q₁₆	—	2	1	no	—
127	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
129	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
130	#358	C₂²×C₅	1	8	yes	—
131	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
132	C₄×C₁₆	C₂²×C₅	1	2	yes	—
133	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
134	C₅×Q₈	—	4	4	no	—
135	C₅×Q₈	—	4	2	no	—
136	C₅×Q₈	—	4	4	no	—
137	C₅×Q₈	—	4	2	no	—
138	C₅×Q₈	—	8	4	no	—
139	C₅×Q₈	—	8	2	no	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	4	2	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	—	4	4	no	—
146	C₂×C₄×C₅	—	8	4	no	—
147	C₂×C₄×C₅	Q₃₂	1	4	yes	complemented
148	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₅×C₈	—	2	1	no	—
155	C₅×C₈	—	2	1	no	—
156	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
157	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
158	C₂²×Q₈	—	2	2	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×Q₁₆	—	2	2	no	—
161	C₂×Q₁₆	—	2	2	no	—
162	C₂×Q₁₆	—	2	1	no	—
163	C₂×Q₁₆	—	4	4	no	—
164	C₂×Q₁₆	—	4	4	no	—
165	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
166	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
167	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
169	C₄⋊Q₈	—	2	1	no	—
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	—	2	1	no	—
173	C₄⋊Q₈	—	4	1	no	—
174	C₄⋊Q₈	—	4	1	no	—
175	#125	—	4	1	no	—
176	C₂²⋊Q₈	—	2	2	no	—
177	C₂²⋊Q₈	—	4	1	no	—
178	C₄×Q₈	—	2	2	no	—
179	C₄×Q₈	—	4	1	no	—
180	#117	—	4	1	no	—
181	C₂×C₄⋊C₄	—	4	1	no	—
182	C₂×C₄⋊C₄	—	4	1	no	—
183	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
184	Q₃₂	—	4	8	no	—
185	C₂×C₁₆	C₂³×C₅	1	4	yes	—
186	#107	—	2	1	no	—
187	#107	C₂³×C₅	1	2	yes	—
188	#103	—	2	2	no	—
189	#103	—	2	2	no	—
190	C₄×C₈	C₅×D₄	1	1	yes	characteristic
191	C₄×C₈	C₅×D₄	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
194	C₂²×C₅	—	2	1	no	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	C₂×Q₃₂	1	2	yes	—
197	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	8	2	no	—
201	C₄×C₅	—	8	4	no	—
202	C₄×C₅	—	8	2	no	—
203	C₄×C₅	C₂×Q₃₂	1	2	yes	—
204	C₄×C₅	C₂×Q₃₂	1	2	yes	—
205	C₄×C₅	#510	1	2	yes	—
206	C₄×C₅	#535	1	1	yes	characteristic
207	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	4	4	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×Q₈	—	4	1	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
216	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	2	8	no	—
220	Q₁₆	—	4	4	no	—
221	Q₁₆	—	4	4	no	—
222	Q₁₆	—	4	4	no	—
223	Q₁₆	—	4	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
227	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
228	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
229	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₄²	—	4	2	no	—
242	C₄²	C₅×D₈	1	1	yes	characteristic
243	C₄²	C₅×D₈	1	1	yes	characteristic
244	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
245	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
246	C₁₆	—	2	4	no	—
247	C₂×C₅	—	2	2	no	—
248	C₂×C₅	#3205	1	1	yes	characteristic
249	C₂×C₅	C₂²×Q₃₂	1	1	yes	characteristic
250	C₂×C₅	C₂×#510	1	1	yes	characteristic
251	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
252	Q₈	—	4	4	no	—
253	Q₈	—	4	2	no	—
254	Q₈	—	4	4	no	—
255	Q₈	—	4	2	no	—
256	Q₈	—	8	4	no	—
257	Q₈	—	8	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	C₅×Q₃₂	1	4	yes	complemented
266	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
267	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
268	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
270	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
275	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
276	C₅	#33775	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	C₂×C₅×Q₃₂	1	2	yes	—
280	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	—	8	2	no	—
286	C₄	C₂×C₅×Q₃₂	1	2	yes	—
287	C₄	C₂×C₅×Q₃₂	1	2	yes	—
288	C₄	C₅×#510	1	2	yes	—
289	C₄	C₅×#535	1	1	yes	characteristic
290	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	C₅×#3205	1	1	yes	characteristic
293	C₂	C₂²×C₅×Q₃₂	1	1	yes	characteristic
294	C₂	C₂×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂³×C₅	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. C₅×#2247	C₂	semidirect
4. C₅×#2241	C₂	semidirect
5. C₅×#2209	C₂	semidirect
29. #33775	C₅	direct (N × H)
68. #2247	C₂×C₅	semidirect
69. #2241	C₂×C₅	semidirect
70. #2209	C₂×C₅	semidirect
147. C₂×C₄×C₅	Q₃₂	semidirect
265. C₂×C₄	C₅×Q₃₂	semidirect
276. C₅	#33775	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×#3170	2
C₂×C₅×#494	2
C₅×#2247	2
C₅×#2241	2
C₅×#2209	2
C₅×#2188	2
C₅×#2162	2
#33775	5

Subgroups — C5×#33775 #11956495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33775 #11956495