Subgroups — C₅×#33849 #11956569

Order 1280.

676 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#508	C₂	1	2	yes	maximal, complemented
2	C₅×#3283	C₂	1	2	yes	maximal, complemented
3	C₅×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2216	C₂	1	2	yes	maximal, complemented
5	C₅×#2208	C₂	1	2	yes	maximal, complemented
6	C₅×#2189	C₂	1	4	yes	maximal, complemented
7	C₄×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#135	C₂²	1	4	yes	—
10	C₅×#547	—	2	2	no	—
11	C₅×#508	—	2	8	no	—
12	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
13	C₂×C₅×Q₃₂	C₂²	1	2	yes	—
14	C₂×C₅×SD₃₂	C₂²	1	4	yes	—
15	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
17	C₅×#494	C₂²	1	2	yes	—
18	C₅×#467	C₂²	1	4	yes	—
19	C₅×#461	—	2	2	no	—
20	C₅×#452	—	2	4	no	—
21	C₄×C₅×D₈	C₂²	1	2	yes	—
22	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
23	C₅×#366	C₂²	1	1	yes	characteristic
24	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₅×#358	C₂²	1	4	yes	—
26	C₅×#357	C₂²	1	2	yes	—
27	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
29	#33849	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×#41	—	2	4	no	—
31	C₅×#135	—	2	8	no	—
32	C₅×#135	—	4	4	no	—
33	C₂×C₅×Q₁₆	—	2	2	no	—
34	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
35	C₂×C₅×SD₁₆	—	2	4	no	—
36	C₂×C₅×D₈	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₅×C₄⋊Q₈	—	2	2	no	—
39	C₅×#123	—	4	4	no	—
40	C₅×C₂²⋊Q₈	—	2	4	no	—
41	C₅×#121	—	4	2	no	—
42	C₄×C₅×D₄	—	2	2	no	—
43	C₄×C₅×D₄	D₄	1	1	yes	characteristic
44	C₂×C₅×C₄⋊C₄	—	4	2	no	—
45	C₅×Q₃₂	—	2	4	no	—
46	C₅×Q₃₂	—	4	2	no	—
47	C₅×SD₃₂	—	2	8	no	—
48	C₅×D₁₆	D₄	1	4	yes	complemented
49	C₂×C₅×C₁₆	—	2	4	no	—
50	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₅×C₁₆	C₂³	1	2	yes	—
52	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
53	C₅×#107	C₂³	1	2	yes	—
54	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
55	C₅×#103	—	2	4	no	—
56	C₅×#102	—	2	2	no	—
57	C₅×C₂²⋊C₈	D₄	1	2	yes	—
58	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
59	C₂×#508	C₂×C₅	1	2	yes	complemented
60	#3283	C₂×C₅	1	2	yes	complemented
61	#2247	C₂×C₅	1	1	yes	characteristic, complemented
62	#2216	C₂×C₅	1	2	yes	complemented
63	#2208	C₂×C₅	1	2	yes	complemented
64	#2189	C₂×C₅	1	4	yes	complemented
65	C₄×D₁₆	C₂×C₅	1	1	yes	characteristic, complemented
66	D₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
67	C₅×#41	—	4	8	no	—
68	C₅×#41	—	8	4	no	—
69	C₂×C₅×Q₈	—	2	4	no	—
70	C₂×C₅×D₄	—	2	2	no	—
71	C₂×C₅×D₄	—	4	4	no	—
72	C₂×C₅×D₄	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	—	4	2	no	—
74	C₂²×C₄×C₅	—	4	4	no	—
75	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₅×Q₁₆	—	2	4	no	—
77	C₅×Q₁₆	—	2	4	no	—
78	C₅×Q₁₆	—	8	2	no	—
79	C₅×SD₁₆	—	4	8	no	—
80	C₅×D₈	—	2	2	no	—
81	C₅×D₈	C₂×D₄	1	4	yes	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
85	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	2	2	no	—
88	C₅×C₄⋊C₄	—	4	2	no	—
89	C₅×C₄⋊C₄	—	4	4	no	—
90	C₅×C₄⋊C₄	—	4	2	no	—
91	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
92	C₅×C₂²⋊C₄	—	4	4	no	—
93	C₅×C₂²⋊C₄	—	4	2	no	—
94	C₅×C₂²⋊C₄	D₈	1	2	yes	—
95	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
96	C₅×C₁₆	—	2	2	no	—
97	C₅×C₁₆	—	2	1	no	—
98	C₅×C₁₆	C₂×D₄	1	2	yes	—
99	C₂×#135	C₂²×C₅	1	4	yes	—
100	#547	—	2	2	no	—
101	#508	—	2	8	no	—
102	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
103	C₂×Q₃₂	C₂²×C₅	1	2	yes	—
104	C₂×SD₃₂	C₂²×C₅	1	4	yes	—
105	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic
106	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
107	#494	C₂²×C₅	1	2	yes	—
108	#467	C₂²×C₅	1	4	yes	—
109	#461	—	2	2	no	—
110	#452	—	2	4	no	—
111	C₄×D₈	C₂²×C₅	1	2	yes	—
112	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
113	#366	C₂²×C₅	1	1	yes	characteristic
114	C₄⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
115	#358	C₂²×C₅	1	4	yes	—
116	#357	C₂²×C₅	1	2	yes	—
117	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
118	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic
119	C₂³×C₅	—	4	2	no	—
120	C₂³×C₅	C₂×D₈	1	2	yes	—
121	C₅×Q₈	—	4	4	no	—
122	C₅×Q₈	—	4	4	no	—
123	C₅×D₄	—	2	4	no	—
124	C₅×D₄	—	2	2	no	—
125	C₅×D₄	—	4	2	no	—
126	C₅×D₄	—	8	8	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	8	8	no	—
132	C₂×C₄×C₅	—	8	4	no	—
133	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₅×C₈	—	2	2	no	—
138	C₅×C₈	—	2	1	no	—
139	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
140	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×#41	—	2	4	no	—
142	#135	—	2	8	no	—
143	#135	—	4	4	no	—
144	C₂×Q₁₆	—	2	2	no	—
145	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
146	C₂×SD₁₆	—	2	4	no	—
147	C₂×D₈	C₂³×C₅	1	2	yes	—
148	C₂²×C₈	C₂³×C₅	1	2	yes	—
149	C₄⋊Q₈	—	2	2	no	—
150	#123	—	4	4	no	—
151	C₂²⋊Q₈	—	2	4	no	—
152	#121	—	4	2	no	—
153	C₄×D₄	—	2	2	no	—
154	C₄×D₄	C₅×D₄	1	1	yes	characteristic
155	C₂×C₄⋊C₄	—	4	2	no	—
156	Q₃₂	—	2	4	no	—
157	Q₃₂	—	4	2	no	—
158	SD₃₂	—	2	8	no	—
159	D₁₆	C₅×D₄	1	4	yes	complemented
160	C₂×C₁₆	—	2	4	no	—
161	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
162	C₂×C₁₆	C₂³×C₅	1	2	yes	—
163	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
164	#107	C₂³×C₅	1	2	yes	—
165	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
166	#103	—	2	4	no	—
167	#102	—	2	2	no	—
168	C₂²⋊C₈	C₅×D₄	1	2	yes	—
169	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	—	2	2	no	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	4	4	no	—
174	C₂²×C₅	—	8	2	no	—
175	C₂²×C₅	—	8	2	no	—
176	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
177	C₄×C₅	—	2	1	no	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	—	8	2	no	—
180	C₄×C₅	—	8	4	no	—
181	C₄×C₅	#508	1	2	yes	complemented
182	C₄×C₅	D₄²	1	1	yes	characteristic
183	C₄×C₅	#576	1	1	yes	characteristic
184	#41	—	4	8	no	—
185	#41	—	8	4	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	4	no	—
189	C₂×D₄	C₅×D₈	1	1	yes	characteristic
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
193	Q₁₆	—	2	4	no	—
194	Q₁₆	—	2	4	no	—
195	Q₁₆	—	8	2	no	—
196	SD₁₆	—	4	8	no	—
197	D₈	—	2	2	no	—
198	D₈	C₂×C₅×D₄	1	4	yes	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
203	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
209	C₂²⋊C₄	—	4	4	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	C₅×D₈	1	2	yes	—
212	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
213	C₁₆	—	2	2	no	—
214	C₁₆	—	2	1	no	—
215	C₁₆	C₂×C₅×D₄	1	2	yes	—
216	C₂×C₅	—	4	2	no	—
217	C₂×C₅	—	8	4	no	—
218	C₂×C₅	D₄×D₈	1	1	yes	characteristic
219	C₂×C₅	C₂×#508	1	1	yes	characteristic
220	C₂×C₅	#3417	1	1	yes	characteristic
221	C₂³	—	4	2	no	—
222	C₂³	C₂×C₅×D₈	1	2	yes	—
223	Q₈	—	4	4	no	—
224	Q₈	—	4	4	no	—
225	D₄	—	2	2	no	—
226	D₄	—	2	4	no	—
227	D₄	—	4	2	no	—
228	D₄	—	8	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	8	8	no	—
234	C₂×C₄	—	8	4	no	—
235	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
236	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
237	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	2	1	no	—
241	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
242	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
243	C₅	#33849	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	4	no	—
254	C₄	—	8	2	no	—
255	C₄	C₅×#508	1	2	yes	complemented
256	C₄	C₅×D₄²	1	1	yes	characteristic
257	C₄	C₅×#576	1	1	yes	characteristic
258	C₂	—	4	2	no	—
259	C₂	—	8	4	no	—
260	C₂	C₅×D₄×D₈	1	1	yes	characteristic
261	C₂	C₂×C₅×#508	1	1	yes	characteristic
262	C₂	C₅×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄ ◁ C₅×C₈×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#508	C₂	semidirect
2. C₅×#3283	C₂	semidirect
3. C₅×#2247	C₂	semidirect
4. C₅×#2216	C₂	semidirect
5. C₅×#2208	C₂	semidirect
6. C₅×#2189	C₂	semidirect
7. C₄×C₅×D₁₆	C₂	semidirect
8. C₅×D₄×C₁₆	C₂	semidirect
29. #33849	C₅	direct (N × H)
48. C₅×D₁₆	D₄	semidirect
59. C₂×#508	C₂×C₅	semidirect
60. #3283	C₂×C₅	semidirect
61. #2247	C₂×C₅	semidirect
62. #2216	C₂×C₅	semidirect
63. #2208	C₂×C₅	semidirect
64. #2189	C₂×C₅	semidirect
65. C₄×D₁₆	C₂×C₅	semidirect
66. D₄×C₁₆	C₂×C₅	semidirect
159. D₁₆	C₅×D₄	semidirect
181. C₄×C₅	#508	semidirect
243. C₅	#33849	direct (N × H)
255. C₄	C₅×#508	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#508	2
C₅×#3283	2
C₅×#2247	2
C₅×#2216	2
C₅×#2208	2
C₅×#2189	2
C₄×C₅×D₁₆	2
C₅×D₄×C₁₆	2
#33849	5

Subgroups — C5×#33849 #11956569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33849 #11956569