Subgroups — C₅×#33334 #11956054

Order 1280.

1172 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#504	C₂	1	2	yes	maximal, complemented
3	C₅×#2113	C₂	1	6	yes	maximal, complemented
4	C₅×#2112	C₂	1	3	yes	maximal, complemented
5	C₂×C₅×C₂²⋊C₁₆	C₂	1	3	yes	maximal, complemented
6	C₂²×C₅×#41	C₄	1	1	yes	characteristic
7	C₂×C₅×#131	—	2	8	no	—
8	C₂×C₅×#131	C₄	1	1	yes	characteristic
9	C₂×C₅×#131	C₄	1	3	yes	—
10	C₂×C₅×#131	C₂²	1	3	yes	—
11	C₂×C₅×#131	C₂²	1	1	yes	characteristic
12	C₂²×C₅×M₁₆	C₄	1	3	yes	—
13	C₂³×C₅×C₈	C₂²	1	3	yes	—
14	C₅×#504	—	2	8	no	—
15	C₂×C₅×M₃₂	C₂²	1	6	yes	complemented
16	C₂²×C₅×C₁₆	C₂²	1	6	yes	complemented
17	C₅×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₅×C₂²⋊C₁₆	C₂²	1	3	yes	—
19	C₅×C₂²⋊C₁₆	C₂²	1	6	yes	complemented
20	C₅×C₂²⋊C₁₆	C₂²	1	6	yes	complemented
21	#33334	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×#41	—	2	4	no	—
23	C₂×C₅×#41	—	2	4	no	—
24	C₂×C₅×#41	C₈	1	1	yes	characteristic
25	C₂×C₅×#41	C₈	1	3	yes	—
26	C₂×C₅×#41	C₂×C₄	1	3	yes	—
27	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×Q₈	C₈	1	1	yes	characteristic
29	C₂²×C₅×D₄	C₈	1	3	yes	—
30	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
31	C₅×#131	—	2	4	no	—
32	C₅×#131	—	2	12	no	—
33	C₅×#131	—	2	12	no	—
34	C₅×#131	D₄	1	8	yes	—
35	C₂×C₅×M₁₆	—	2	12	no	—
36	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
37	C₂×C₅×M₁₆	C₂×C₄	1	6	yes	—
38	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
39	C₂²×C₅×C₈	—	2	12	no	—
40	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
41	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
42	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂³	1	3	yes	—
45	C₂²×C₅×C₈	C₂³	1	3	yes	—
46	C₅×M₃₂	—	2	12	no	—
47	C₂×C₅×C₁₆	—	2	12	no	—
48	C₂×C₅×C₁₆	C₂³	1	2	yes	—
49	C₂×C₅×C₁₆	C₂³	1	6	yes	—
50	C₂²×#131	C₂×C₅	1	1	yes	characteristic
51	C₂×#504	C₂×C₅	1	2	yes	complemented
52	#2113	C₂×C₅	1	6	yes	complemented
53	#2112	C₂×C₅	1	3	yes	complemented
54	C₂×C₂²⋊C₁₆	C₂×C₅	1	3	yes	complemented
55	C₂⁴×C₅	C₂×C₈	1	3	yes	—
56	C₅×#41	—	2	4	no	—
57	C₅×#41	—	2	12	no	—
58	C₅×#41	—	2	6	no	—
59	C₅×#41	—	2	6	no	—
60	C₅×#41	C₂²⋊C₄	1	4	yes	—
61	C₅×#41	M₁₆	1	4	yes	—
62	C₂×C₅×Q₈	—	2	4	no	—
63	C₂×C₅×Q₈	C₂×C₈	1	3	yes	—
64	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
65	C₂×C₅×D₄	—	2	12	no	—
66	C₂×C₅×D₄	C₂×C₈	1	6	yes	—
67	C₂×C₅×D₄	C₂×C₈	1	3	yes	—
68	C₂×C₅×D₄	C₂×C₈	1	3	yes	—
69	C₂²×C₄×C₅	—	2	6	no	—
70	C₂²×C₄×C₅	—	2	6	no	—
71	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
72	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
73	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
77	C₅×M₁₆	—	2	6	no	—
78	C₅×M₁₆	—	2	12	no	—
79	C₅×M₁₆	C₂²⋊C₄	1	12	yes	—
80	C₂×C₅×C₈	—	2	6	no	—
81	C₂×C₅×C₈	—	2	6	no	—
82	C₂×C₅×C₈	—	2	6	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	2	6	no	—
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
88	C₂×C₅×C₈	C₂×D₄	1	12	yes	—
89	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
90	C₅×C₁₆	—	2	2	no	—
91	C₅×C₁₆	—	2	6	no	—
92	C₂²×#41	C₄×C₅	1	1	yes	characteristic
93	C₂×#131	—	2	8	no	—
94	C₂×#131	C₄×C₅	1	1	yes	characteristic
95	C₂×#131	C₄×C₅	1	3	yes	—
96	C₂×#131	C₂²×C₅	1	3	yes	—
97	C₂×#131	C₂²×C₅	1	1	yes	characteristic
98	C₂²×M₁₆	C₄×C₅	1	3	yes	—
99	C₂³×C₈	C₂²×C₅	1	3	yes	—
100	#504	—	2	8	no	—
101	C₂×M₃₂	C₂²×C₅	1	6	yes	complemented
102	C₂²×C₁₆	C₂²×C₅	1	6	yes	complemented
103	C₂²⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
104	C₂²⋊C₁₆	C₂²×C₅	1	3	yes	—
105	C₂²⋊C₁₆	C₂²×C₅	1	6	yes	complemented
106	C₂²⋊C₁₆	C₂²×C₅	1	6	yes	complemented
107	C₂³×C₅	—	2	6	no	—
108	C₂³×C₅	—	2	6	no	—
109	C₂³×C₅	—	4	3	no	—
110	C₂³×C₅	C₂²×C₈	1	3	yes	—
111	C₂³×C₅	C₂²×C₈	1	3	yes	—
112	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
113	C₅×Q₈	—	2	6	no	—
114	C₅×Q₈	C₂²⋊C₈	1	4	yes	—
115	C₅×D₄	—	2	12	no	—
116	C₅×D₄	—	2	6	no	—
117	C₅×D₄	C₂²⋊C₈	1	12	yes	—
118	C₂×C₄×C₅	—	2	6	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	3	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	3	no	—
123	C₂×C₄×C₅	—	2	6	no	—
124	C₂×C₄×C₅	—	2	6	no	—
125	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
126	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
128	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
129	C₂×C₄×C₅	C₂×M₁₆	1	6	yes	—
130	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
131	C₅×C₈	—	2	1	no	—
132	C₅×C₈	—	2	3	no	—
133	C₅×C₈	C₂×C₂²⋊C₄	1	6	yes	—
134	C₅×C₈	C₂²×D₄	1	2	yes	—
135	C₂×#41	—	2	4	no	—
136	C₂×#41	—	2	4	no	—
137	C₂×#41	C₅×C₈	1	1	yes	characteristic
138	C₂×#41	C₅×C₈	1	3	yes	—
139	C₂×#41	C₂×C₄×C₅	1	3	yes	—
140	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
141	C₂²×Q₈	C₅×C₈	1	1	yes	characteristic
142	C₂²×D₄	C₅×C₈	1	3	yes	—
143	C₂³×C₄	C₂×C₄×C₅	1	3	yes	—
144	#131	—	2	4	no	—
145	#131	—	2	12	no	—
146	#131	—	2	12	no	—
147	#131	C₅×D₄	1	8	yes	—
148	C₂×M₁₆	—	2	12	no	—
149	C₂×M₁₆	C₂×C₄×C₅	1	3	yes	—
150	C₂×M₁₆	C₂×C₄×C₅	1	6	yes	—
151	C₂×M₁₆	C₂×C₄×C₅	1	3	yes	—
152	C₂²×C₈	—	2	12	no	—
153	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
154	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
155	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
156	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₅	1	3	yes	—
158	C₂²×C₈	C₂³×C₅	1	3	yes	—
159	M₃₂	—	2	12	no	—
160	C₂×C₁₆	—	2	12	no	—
161	C₂×C₁₆	C₂³×C₅	1	2	yes	—
162	C₂×C₁₆	C₂³×C₅	1	6	yes	—
163	C₂²×C₅	—	2	1	no	—
164	C₂²×C₅	—	2	3	no	—
165	C₂²×C₅	—	2	1	no	—
166	C₂²×C₅	—	2	3	no	—
167	C₂²×C₅	—	2	3	no	—
168	C₂²×C₅	—	2	3	no	—
169	C₂²×C₅	—	2	3	no	—
170	C₂²×C₅	—	4	12	no	—
171	C₂²×C₅	C₂×C₂²⋊C₈	1	6	yes	—
172	C₂²×C₅	#504	1	2	yes	complemented
173	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
174	C₄×C₅	—	2	1	no	—
175	C₄×C₅	—	2	3	no	—
176	C₄×C₅	C₂×C₂²⋊C₈	1	6	yes	—
177	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
178	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
179	C₂⁴	C₂×C₅×C₈	1	3	yes	—
180	#41	—	2	4	no	—
181	#41	—	2	12	no	—
182	#41	—	2	6	no	—
183	#41	—	2	6	no	—
184	#41	C₅×C₂²⋊C₄	1	4	yes	—
185	#41	C₅×M₁₆	1	4	yes	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	C₂×C₅×C₈	1	3	yes	—
188	C₂×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
189	C₂×D₄	—	2	12	no	—
190	C₂×D₄	C₂×C₅×C₈	1	6	yes	—
191	C₂×D₄	C₂×C₅×C₈	1	3	yes	—
192	C₂×D₄	C₂×C₅×C₈	1	3	yes	—
193	C₂²×C₄	—	2	6	no	—
194	C₂²×C₄	—	2	6	no	—
195	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
196	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
197	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
198	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
200	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
201	M₁₆	—	2	6	no	—
202	M₁₆	—	2	12	no	—
203	M₁₆	C₅×C₂²⋊C₄	1	12	yes	—
204	C₂×C₈	—	2	6	no	—
205	C₂×C₈	—	2	6	no	—
206	C₂×C₈	—	2	6	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	6	no	—
209	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
210	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
211	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
212	C₂×C₈	C₂×C₅×D₄	1	12	yes	—
213	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
214	C₁₆	—	2	2	no	—
215	C₁₆	—	2	6	no	—
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	—	2	6	no	—
218	C₂×C₅	—	4	3	no	—
219	C₂×C₅	C₂²×C₂²⋊C₈	1	1	yes	characteristic
220	C₂×C₅	C₂×#504	1	1	yes	characteristic
221	C₂×C₅	#3407	1	1	yes	characteristic
222	C₂³	—	2	6	no	—
223	C₂³	—	2	6	no	—
224	C₂³	—	4	3	no	—
225	C₂³	C₂²×C₅×C₈	1	3	yes	—
226	C₂³	C₂²×C₅×C₈	1	3	yes	—
227	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
228	Q₈	—	2	6	no	—
229	Q₈	C₅×C₂²⋊C₈	1	4	yes	—
230	D₄	—	2	12	no	—
231	D₄	—	2	6	no	—
232	D₄	C₅×C₂²⋊C₈	1	12	yes	—
233	C₂×C₄	—	2	6	no	—
234	C₂×C₄	—	2	6	no	—
235	C₂×C₄	—	2	6	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	3	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	3	no	—
240	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
241	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
243	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
244	C₂×C₄	C₂×C₅×M₁₆	1	6	yes	—
245	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
246	C₈	—	2	1	no	—
247	C₈	—	2	3	no	—
248	C₈	C₂×C₅×C₂²⋊C₄	1	6	yes	—
249	C₈	C₂²×C₅×D₄	1	2	yes	—
250	C₅	#33334	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²	—	2	1	no	—
252	C₂²	—	2	3	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	2	3	no	—
256	C₂²	—	2	3	no	—
257	C₂²	—	2	3	no	—
258	C₂²	—	4	12	no	—
259	C₂²	C₂×C₅×C₂²⋊C₈	1	6	yes	—
260	C₂²	C₅×#504	1	2	yes	complemented
261	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	3	no	—
264	C₄	C₂×C₅×C₂²⋊C₈	1	6	yes	—
265	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
266	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	2	6	no	—
269	C₂	—	4	3	no	—
270	C₂	C₂²×C₅×C₂²⋊C₈	1	1	yes	characteristic
271	C₂	C₂×C₅×#504	1	1	yes	characteristic
272	C₂	C₅×#3407	1	1	yes	characteristic

	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	C₂²×C₅×#41	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. C₂×C₅×#504	C₂	semidirect
3. C₅×#2113	C₂	semidirect
4. C₅×#2112	C₂	semidirect
5. C₂×C₅×C₂²⋊C₁₆	C₂	semidirect
15. C₂×C₅×M₃₂	C₂²	semidirect
16. C₂²×C₅×C₁₆	C₂²	semidirect
19. C₅×C₂²⋊C₁₆	C₂²	semidirect
20. C₅×C₂²⋊C₁₆	C₂²	semidirect
21. #33334	C₅	direct (N × H)
51. C₂×#504	C₂×C₅	semidirect
52. #2113	C₂×C₅	semidirect
53. #2112	C₂×C₅	semidirect
54. C₂×C₂²⋊C₁₆	C₂×C₅	semidirect
101. C₂×M₃₂	C₂²×C₅	semidirect
102. C₂²×C₁₆	C₂²×C₅	semidirect
105. C₂²⋊C₁₆	C₂²×C₅	semidirect
106. C₂²⋊C₁₆	C₂²×C₅	semidirect
172. C₂²×C₅	#504	semidirect
250. C₅	#33334	direct (N × H)
260. C₂²	C₅×#504	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×#131	2
C₂×C₅×#504	2
C₅×#2113	2
C₅×#2112	2
C₂×C₅×C₂²⋊C₁₆	2
#33334	5

Subgroups — C5×#33334 #11956054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33334 #11956054