Subgroups — C₅×#33459 #11956179

Order 1280.

674 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	1	yes	maximal, characteristic
2	C₅×#2951	C₂	1	2	yes	maximal, complemented
3	C₅×#2175	C₂	1	4	yes	maximal, complemented
4	C₄×C₅×Q₃₂	C₂	1	2	yes	maximal, complemented
5	C₄×C₅×D₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2139	C₂	1	2	yes	maximal, complemented
8	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#135	C₂²	1	2	yes	—
10	C₅×#520	—	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	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₅×#440	—	2	4	no	—
18	C₄×C₅×Q₁₆	C₂²	1	4	yes	—
19	C₄×C₅×D₈	C₂²	1	4	yes	—
20	C₂×C₅×#107	C₂²	1	2	yes	—
21	C₅×#417	—	2	2	no	—
22	C₅×#405	C₂²	1	1	yes	characteristic
23	C₅×#366	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	C₄×C₅×C₁₆	C₂²	1	2	yes	—
29	#33459	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	C₂×C₅×M₁₆	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	—	4	2	no	—
42	C₅×#117	D₄	1	1	yes	characteristic
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	—	4	2	no	—
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	C₅×M₃₂	—	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	—	2	2	no	—
54	C₅×#107	C₂³	1	4	yes	—
55	C₅×#103	—	2	4	no	—
56	C₅×#102	—	2	4	no	—
57	C₅×#97	D₄	1	2	yes	—
58	C₄×C₅×C₈	C₂³	1	2	yes	—
59	C₂×#508	C₂×C₅	1	1	yes	characteristic
60	#2951	C₂×C₅	1	2	yes	complemented
61	#2175	C₂×C₅	1	4	yes	complemented
62	C₄×Q₃₂	C₂×C₅	1	2	yes	complemented
63	C₄×D₁₆	C₂×C₅	1	2	yes	complemented
64	C₂×#366	C₂×C₅	1	1	yes	characteristic, complemented
65	#2139	C₂×C₅	1	2	yes	complemented
66	#2108	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	2	no	—
74	C₂²×C₄×C₅	—	4	4	no	—
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	C₅×M₁₆	—	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	2	no	—
89	C₅×C₄⋊C₄	—	4	4	no	—
90	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
91	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
92	C₅×C₂²⋊C₄	—	4	4	no	—
93	C₅×C₂²⋊C₄	D₈	1	2	yes	—
94	C₄²×C₅	—	4	4	no	—
95	C₄²×C₅	C₂×D₄	1	2	yes	—
96	C₅×C₁₆	—	2	2	no	—
97	C₅×C₁₆	C₂²×C₄	1	2	yes	—
98	C₅×C₁₆	C₂²×C₄	1	2	yes	—
99	C₂×#135	C₂²×C₅	1	2	yes	—
100	#520	—	2	2	no	—
101	#508	C₄×C₅	1	8	yes	complemented
102	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic
103	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
104	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic
105	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
106	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
107	#440	—	2	4	no	—
108	C₄×Q₁₆	C₂²×C₅	1	4	yes	—
109	C₄×D₈	C₂²×C₅	1	4	yes	—
110	C₂×#107	C₂²×C₅	1	2	yes	—
111	#417	—	2	2	no	—
112	#405	C₂²×C₅	1	1	yes	characteristic
113	#366	C₂²×C₅	1	2	yes	—
114	#366	C₂²×C₅	1	2	yes	—
115	#358	C₂²×C₅	1	4	yes	—
116	#357	C₂²×C₅	1	4	yes	—
117	#346	C₂²×C₅	1	2	yes	—
118	C₄×C₁₆	C₂²×C₅	1	2	yes	—
119	C₂³×C₅	—	4	2	no	—
120	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
121	C₅×Q₈	—	2	4	no	—
122	C₅×Q₈	—	4	2	no	—
123	C₅×D₄	—	2	4	no	—
124	C₅×D₄	—	4	8	no	—
125	C₅×D₄	—	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	—	16	4	no	—
130	C₂×C₄×C₅	—	16	2	no	—
131	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅	#135	1	2	yes	—
136	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₅×C₈	—	2	2	no	—
138	C₅×C₈	C₄×D₄	1	2	yes	—
139	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
140	C₅×C₈	C₂×#41	1	1	yes	characteristic
141	C₂×#41	—	2	2	no	—
142	#135	—	2	4	no	—
143	#135	C₂×C₄×C₅	1	8	yes	complemented
144	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
145	C₂×SD₁₆	—	2	2	no	—
146	C₂×D₈	C₂³×C₅	1	2	yes	—
147	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
149	C₄×Q₈	—	2	4	no	—
150	C₄×D₄	—	2	4	no	—
151	C₄×D₄	—	4	4	no	—
152	#117	—	4	2	no	—
153	#117	C₅×D₄	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	4	2	no	—
156	Q₃₂	C₂×C₄×C₅	1	4	yes	complemented
157	SD₃₂	C₂×C₄×C₅	1	8	yes	complemented
158	D₁₆	C₂×C₄×C₅	1	4	yes	complemented
159	M₃₂	—	2	2	no	—
160	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	complemented
161	C₂×C₁₆	C₂³×C₅	1	2	yes	—
162	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
163	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
164	#107	—	2	2	no	—
165	#107	C₂³×C₅	1	4	yes	—
166	#103	—	2	4	no	—
167	#102	—	2	4	no	—
168	#97	C₅×D₄	1	2	yes	—
169	C₄×C₈	C₂³×C₅	1	2	yes	—
170	C₂²×C₅	—	2	1	no	—
171	C₂²×C₅	—	2	1	no	—
172	C₂²×C₅	—	4	4	no	—
173	C₂²×C₅	—	8	2	no	—
174	C₂²×C₅	—	8	2	no	—
175	C₂²×C₅	C₄×D₈	1	2	yes	—
176	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	—	4	4	no	—
180	C₄×C₅	—	8	4	no	—
181	C₄×C₅	C₄×D₈	1	2	yes	—
182	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
183	C₄×C₅	C₂×#135	1	1	yes	characteristic
184	#41	—	2	8	no	—
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
193	Q₁₆	—	2	2	no	—
194	Q₁₆	C₂²×C₄×C₅	1	4	yes	—
195	SD₁₆	—	2	8	no	—
196	D₈	—	2	2	no	—
197	D₈	C₂²×C₄×C₅	1	4	yes	—
198	M₁₆	—	2	2	no	—
199	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
200	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
201	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂×C₈	C₅×#41	1	2	yes	—
203	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
208	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
209	C₂²⋊C₄	—	4	4	no	—
210	C₂²⋊C₄	C₅×D₈	1	2	yes	—
211	C₄²	—	4	4	no	—
212	C₄²	C₂×C₅×D₄	1	2	yes	—
213	C₁₆	—	2	2	no	—
214	C₁₆	C₂²×C₄×C₅	1	2	yes	—
215	C₁₆	C₂²×C₄×C₅	1	2	yes	—
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	—	8	4	no	—
218	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
219	C₂×C₅	#3415	1	1	yes	characteristic
220	C₂×C₅	#3417	1	1	yes	characteristic
221	C₂³	—	4	2	no	—
222	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
223	Q₈	—	2	4	no	—
224	Q₈	—	4	2	no	—
225	D₄	—	2	4	no	—
226	D₄	—	4	8	no	—
227	D₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	16	4	no	—
232	C₂×C₄	—	16	2	no	—
233	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
234	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
235	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
236	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
237	C₂×C₄	C₅×#135	1	2	yes	—
238	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	C₄×C₅×D₄	1	2	yes	—
241	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
242	C₈	C₂×C₅×#41	1	1	yes	characteristic
243	C₅	#33459	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	C₄×C₅×D₈	1	2	yes	—
250	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	4	no	—
255	C₄	C₄×C₅×D₈	1	2	yes	—
256	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
257	C₄	C₂×C₅×#135	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	8	4	no	—
260	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic
261	C₂	C₅×#3415	1	1	yes	characteristic
262	C₂	C₅×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2951	C₂	semidirect
3. C₅×#2175	C₂	semidirect
4. C₄×C₅×Q₃₂	C₂	semidirect
5. C₄×C₅×D₁₆	C₂	semidirect
6. C₂×C₅×#366	C₂	semidirect
7. C₅×#2139	C₂	semidirect
8. C₅×#2108	C₂	semidirect
11. C₅×#508	C₄	semidirect
29. #33459	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. #2951	C₂×C₅	semidirect
61. #2175	C₂×C₅	semidirect
62. C₄×Q₃₂	C₂×C₅	semidirect
63. C₄×D₁₆	C₂×C₅	semidirect
64. C₂×#366	C₂×C₅	semidirect
65. #2139	C₂×C₅	semidirect
66. #2108	C₂×C₅	semidirect
101. #508	C₄×C₅	semidirect
143. #135	C₂×C₄×C₅	semidirect
156. Q₃₂	C₂×C₄×C₅	semidirect
157. SD₃₂	C₂×C₄×C₅	semidirect
158. D₁₆	C₂×C₄×C₅	semidirect
160. C₂×C₁₆	C₂×C₄×C₅	semidirect
243. C₅	#33459	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
C₅×#2951	2
C₅×#2175	2
C₄×C₅×Q₃₂	2
C₄×C₅×D₁₆	2
C₂×C₅×#366	2
C₅×#2139	2
C₅×#2108	2
#33459	5

Subgroups — C5×#33459 #11956179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33459 #11956179