Subgroups — C₅×#33451 #11956171

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 252 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	1	yes	maximal, characteristic
2	C₄×C₅×#135	C₂	1	2	yes	maximal, complemented
3	C₅×#2177	C₂	1	2	yes	maximal, complemented
4	C₅×#2176	C₂	1	2	yes	maximal, complemented
5	C₅×#2175	C₂	1	4	yes	maximal, complemented
6	C₅×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2138	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#135	C₂²	1	2	yes	—
10	C₄×C₅×#41	—	2	2	no	—
11	C₅×#508	C₄	1	8	yes	complemented
12	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₄×C₅×Q₁₆	C₂²	1	4	yes	—
18	C₄×C₅×SD₁₆	—	2	4	no	—
19	C₄×C₅×D₈	C₂²	1	4	yes	—
20	C₅×#427	C₂²	1	2	yes	—
21	C₅×#416	—	2	2	no	—
22	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₅×#367	C₂²	1	2	yes	—
24	C₅×#366	C₂²	1	2	yes	—
25	C₅×#358	C₂²	1	4	yes	—
26	C₅×#357	C₂²	1	4	yes	—
27	C₅×#346	C₂²	1	2	yes	—
28	C₅×#346	C₂²	1	2	yes	—
29	#33451	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×#41	—	2	2	no	—
31	C₅×#135	—	2	4	no	—
32	C₅×#135	C₂×C₄	1	8	yes	complemented
33	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
34	C₂×C₅×SD₁₆	—	2	2	no	—
35	C₂×C₅×D₈	C₂³	1	2	yes	—
36	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₄×C₅×Q₈	—	2	4	no	—
39	C₄×C₅×D₄	—	2	4	no	—
40	C₄×C₅×D₄	—	4	4	no	—
41	C₅×#117	—	2	2	no	—
42	C₅×#117	—	4	2	no	—
43	C₂×C₄²×C₅	—	4	2	no	—
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₅×Q₃₂	C₂×C₄	1	4	yes	complemented
46	C₅×SD₃₂	C₂×C₄	1	8	yes	complemented
47	C₅×D₁₆	C₂×C₄	1	4	yes	complemented
48	C₂×C₅×C₁₆	—	2	2	no	—
49	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
50	C₂×C₅×C₁₆	C₂³	1	2	yes	—
51	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
53	C₅×#107	C₂³	1	4	yes	—
54	C₅×#106	—	2	2	no	—
55	C₅×#103	—	2	4	no	—
56	C₅×#102	—	2	4	no	—
57	C₄×C₅×C₈	D₄	1	2	yes	—
58	C₄×C₅×C₈	C₂³	1	2	yes	—
59	C₂×#508	C₂×C₅	1	1	yes	characteristic
60	C₄×#135	C₂×C₅	1	2	yes	complemented
61	#2177	C₂×C₅	1	2	yes	complemented
62	#2176	C₂×C₅	1	2	yes	complemented
63	#2175	C₂×C₅	1	4	yes	complemented
64	#2158	C₂×C₅	1	1	yes	characteristic, complemented
65	#2138	C₂×C₅	1	2	yes	complemented
66	C₂×#346	C₂×C₅	1	1	yes	characteristic, complemented
67	C₅×#41	—	2	8	no	—
68	C₅×#41	—	4	4	no	—
69	C₂×C₅×Q₈	—	2	2	no	—
70	C₂×C₅×D₄	—	2	2	no	—
71	C₂×C₅×D₄	—	4	2	no	—
72	C₂²×C₄×C₅	—	4	2	no	—
73	C₂²×C₄×C₅	—	4	4	no	—
74	C₂²×C₄×C₅	D₈	1	2	yes	—
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₅×Q₁₆	—	2	2	no	—
77	C₅×Q₁₆	C₂²×C₄	1	4	yes	—
78	C₅×SD₁₆	—	2	8	no	—
79	C₅×D₈	—	2	2	no	—
80	C₅×D₈	C₂²×C₄	1	4	yes	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
84	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₅×C₈	#41	1	2	yes	—
86	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	—	4	4	no	—
89	C₅×C₂²⋊C₄	—	4	2	no	—
90	C₅×C₂²⋊C₄	—	4	4	no	—
91	C₄²×C₅	—	4	4	no	—
92	C₄²×C₅	—	4	2	no	—
93	C₄²×C₅	D₈	1	2	yes	—
94	C₄²×C₅	C₂×D₄	1	2	yes	—
95	C₅×C₁₆	—	2	2	no	—
96	C₅×C₁₆	C₂²×C₄	1	2	yes	—
97	C₅×C₁₆	C₂²×C₄	1	2	yes	—
98	C₂×#135	C₂²×C₅	1	2	yes	—
99	C₄×#41	—	2	2	no	—
100	#508	C₄×C₅	1	8	yes	complemented
101	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic
102	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
103	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic
104	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
105	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
106	C₄×Q₁₆	C₂²×C₅	1	4	yes	—
107	C₄×SD₁₆	—	2	4	no	—
108	C₄×D₈	C₂²×C₅	1	4	yes	—
109	#427	C₂²×C₅	1	2	yes	—
110	#416	—	2	2	no	—
111	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
112	#367	C₂²×C₅	1	2	yes	—
113	#366	C₂²×C₅	1	2	yes	—
114	#358	C₂²×C₅	1	4	yes	—
115	#357	C₂²×C₅	1	4	yes	—
116	#346	C₂²×C₅	1	2	yes	—
117	#346	C₂²×C₅	1	2	yes	—
118	C₂³×C₅	—	4	2	no	—
119	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
120	C₅×Q₈	—	2	4	no	—
121	C₅×Q₈	—	4	2	no	—
122	C₅×D₄	—	2	4	no	—
123	C₅×D₄	—	4	8	no	—
124	C₅×D₄	—	4	2	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	—	4	2	no	—
127	C₂×C₄×C₅	—	4	8	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	—	8	8	no	—
130	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
132	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅	#135	1	2	yes	—
134	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
135	C₅×C₈	—	2	2	no	—
136	C₅×C₈	C₄×D₄	1	2	yes	—
137	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
138	C₅×C₈	C₂×#41	1	1	yes	characteristic
139	C₂×#41	—	2	2	no	—
140	#135	—	2	4	no	—
141	#135	C₂×C₄×C₅	1	8	yes	complemented
142	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
143	C₂×SD₁₆	—	2	2	no	—
144	C₂×D₈	C₂³×C₅	1	2	yes	—
145	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
147	C₄×Q₈	—	2	4	no	—
148	C₄×D₄	—	2	4	no	—
149	C₄×D₄	—	4	4	no	—
150	#117	—	2	2	no	—
151	#117	—	4	2	no	—
152	C₂×C₄²	—	4	2	no	—
153	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
154	Q₃₂	C₂×C₄×C₅	1	4	yes	complemented
155	SD₃₂	C₂×C₄×C₅	1	8	yes	complemented
156	D₁₆	C₂×C₄×C₅	1	4	yes	complemented
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	complemented
159	C₂×C₁₆	C₂³×C₅	1	2	yes	—
160	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
161	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
162	#107	C₂³×C₅	1	4	yes	—
163	#106	—	2	2	no	—
164	#103	—	2	4	no	—
165	#102	—	2	4	no	—
166	C₄×C₈	C₅×D₄	1	2	yes	—
167	C₄×C₈	C₂³×C₅	1	2	yes	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	4	4	no	—
170	C₂²×C₅	—	8	4	no	—
171	C₂²×C₅	C₄×D₈	1	2	yes	—
172	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
173	C₄×C₅	—	2	4	no	—
174	C₄×C₅	—	4	4	no	—
175	C₄×C₅	—	8	4	no	—
176	C₄×C₅	C₄×D₈	1	2	yes	—
177	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
178	C₄×C₅	C₂×#135	1	1	yes	characteristic
179	#41	—	2	8	no	—
180	#41	—	4	4	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	4	no	—
186	C₂²×C₄	C₅×D₈	1	2	yes	—
187	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
188	Q₁₆	—	2	2	no	—
189	Q₁₆	C₂²×C₄×C₅	1	4	yes	—
190	SD₁₆	—	2	8	no	—
191	D₈	—	2	2	no	—
192	D₈	C₂²×C₄×C₅	1	4	yes	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
195	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
196	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
197	C₂×C₈	C₅×#41	1	2	yes	—
198	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	4	no	—
203	C₄²	—	4	2	no	—
204	C₄²	—	4	4	no	—
205	C₄²	C₅×D₈	1	2	yes	—
206	C₄²	C₂×C₅×D₄	1	2	yes	—
207	C₁₆	—	2	2	no	—
208	C₁₆	C₂²×C₄×C₅	1	2	yes	—
209	C₁₆	C₂²×C₄×C₅	1	2	yes	—
210	C₂×C₅	—	2	2	no	—
211	C₂×C₅	—	8	4	no	—
212	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
213	C₂×C₅	#3414	1	2	yes	—
214	C₂³	—	4	2	no	—
215	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
216	Q₈	—	2	4	no	—
217	Q₈	—	4	2	no	—
218	D₄	—	2	4	no	—
219	D₄	—	4	8	no	—
220	D₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	8	8	no	—
226	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
227	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
228	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
229	C₂×C₄	C₅×#135	1	2	yes	—
230	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	C₄×C₅×D₄	1	2	yes	—
233	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
234	C₈	C₂×C₅×#41	1	1	yes	characteristic
235	C₅	#33451	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²	—	2	2	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	8	4	no	—
239	C₂²	C₄×C₅×D₈	1	2	yes	—
240	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	4	4	no	—
243	C₄	—	8	4	no	—
244	C₄	C₄×C₅×D₈	1	2	yes	—
245	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
246	C₄	C₂×C₅×#135	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	8	4	no	—
249	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic
250	C₂	C₅×#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂²×C₅	C₂²×D₈
prime radical	C₂×C₅×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
2. C₄×C₅×#135	C₂	semidirect
3. C₅×#2177	C₂	semidirect
4. C₅×#2176	C₂	semidirect
5. C₅×#2175	C₂	semidirect
6. C₅×#2158	C₂	semidirect
7. C₅×#2138	C₂	semidirect
8. C₂×C₅×#346	C₂	semidirect
11. C₅×#508	C₄	semidirect
29. #33451	C₅	direct (N × H)
32. C₅×#135	C₂×C₄	semidirect
45. C₅×Q₃₂	C₂×C₄	semidirect
46. C₅×SD₃₂	C₂×C₄	semidirect
47. C₅×D₁₆	C₂×C₄	semidirect
49. C₂×C₅×C₁₆	C₂×C₄	semidirect
60. C₄×#135	C₂×C₅	semidirect
61. #2177	C₂×C₅	semidirect
62. #2176	C₂×C₅	semidirect
63. #2175	C₂×C₅	semidirect
64. #2158	C₂×C₅	semidirect
65. #2138	C₂×C₅	semidirect
66. C₂×#346	C₂×C₅	semidirect
100. #508	C₄×C₅	semidirect
141. #135	C₂×C₄×C₅	semidirect
154. Q₃₂	C₂×C₄×C₅	semidirect
155. SD₃₂	C₂×C₄×C₅	semidirect
156. D₁₆	C₂×C₄×C₅	semidirect
158. C₂×C₁₆	C₂×C₄×C₅	semidirect
235. C₅	#33451	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×#508	2
C₄×C₅×#135	2
C₅×#2177	2
C₅×#2176	2
C₅×#2175	2
C₅×#2158	2
C₅×#2138	2
C₂×C₅×#346	2
#33451	5

Subgroups — C5×#33451 #11956171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33451 #11956171