Subgroups — C₅×#12824 #11935544

Order 1280.

414 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×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#360	C₂	1	2	yes	maximal
4	C₅×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1383	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#135	C₂²	1	1	yes	characteristic
7	C₂×C₅×#131	—	2	1	no	—
8	C₂×C₅×M₃₂	C₂²	1	2	yes	—
9	C₅×#443	—	2	2	no	—
10	C₅×#443	C₄	1	4	yes	complemented
11	C₂×C₅×#108	C₂²	1	1	yes	characteristic
12	C₂×C₅×#107	—	2	1	no	—
13	C₂×C₅×#106	C₂²	1	2	yes	—
14	C₂×C₅×#104	—	2	1	no	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#382	—	2	1	no	—
17	C₅×#364	—	2	2	no	—
18	C₅×#360	—	2	4	no	—
19	#12824	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂×C₅×#41	—	2	1	no	—
21	C₅×#135	—	2	1	no	—
22	C₅×#135	—	2	1	no	—
23	C₅×#135	C₂×C₄	1	2	yes	—
24	C₅×#135	D₄	1	2	yes	—
25	C₂×C₅×Q₁₆	Q₈	1	1	yes	characteristic
26	C₂×C₅×SD₁₆	—	2	1	no	—
27	C₂×C₅×D₈	Q₈	1	1	yes	characteristic
28	C₅×#131	—	2	4	no	—
29	C₅×#131	—	4	2	no	—
30	C₂×C₅×M₁₆	—	2	1	no	—
31	C₂×C₅×M₁₆	—	4	1	no	—
32	C₂²×C₅×C₈	—	4	1	no	—
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂×C₅×C₄⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₅×M₃₂	—	2	2	no	—
38	C₅×M₃₂	—	2	2	no	—
39	C₂×C₅×C₁₆	—	2	2	no	—
40	C₅×#108	—	2	1	no	—
41	C₅×#108	C₂×C₄	1	2	yes	—
42	C₅×#107	—	4	2	no	—
43	C₅×#106	—	2	2	no	—
44	C₅×#106	—	2	2	no	—
45	C₅×#104	—	2	4	no	—
46	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₅×C₈	D₄	1	2	yes	—
48	C₅×#95	—	4	2	no	—
49	C₂×#443	C₂×C₅	1	1	yes	characteristic
50	C₂×#364	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×#360	C₂×C₅	1	2	yes	—
52	#1846	C₂×C₅	1	1	yes	characteristic, complemented
53	#1383	C₂×C₅	1	2	yes	complemented
54	C₅×#41	—	2	2	no	—
55	C₅×#41	—	2	2	no	—
56	C₅×#41	—	4	1	no	—
57	C₅×#41	—	4	1	no	—
58	C₂×C₅×Q₈	—	2	1	no	—
59	C₂×C₅×D₄	—	2	1	no	—
60	C₂×C₅×D₄	—	4	1	no	—
61	C₂²×C₄×C₅	—	2	1	no	—
62	C₂²×C₄×C₅	—	4	1	no	—
63	C₂²×C₄×C₅	—	4	2	no	—
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₅×Q₁₆	—	2	1	no	—
66	C₅×Q₁₆	C₄⋊C₄	1	2	yes	—
67	C₅×SD₁₆	—	2	2	no	—
68	C₅×SD₁₆	—	2	2	no	—
69	C₅×D₈	—	2	1	no	—
70	C₅×D₈	C₄⋊C₄	1	2	yes	—
71	C₅×M₁₆	—	2	2	no	—
72	C₅×M₁₆	—	4	1	no	—
73	C₅×M₁₆	—	4	4	no	—
74	C₂×C₅×C₈	—	4	1	no	—
75	C₂×C₅×C₈	—	4	4	no	—
76	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₅×C₈	D₈	1	2	yes	—
78	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
79	C₂×C₅×C₈	SD₁₆	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₅×C₈	#41	1	1	yes	characteristic
85	C₂×C₅×C₈	#41	1	1	yes	characteristic
86	C₅×C₄⋊C₄	—	4	2	no	—
87	C₅×C₄⋊C₄	—	4	2	no	—
88	C₄²×C₅	—	2	1	no	—
89	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
90	C₅×C₁₆	—	4	2	no	—
91	C₂×#135	C₂²×C₅	1	1	yes	characteristic
92	C₂×#131	—	2	1	no	—
93	C₂×M₃₂	C₂²×C₅	1	2	yes	—
94	#443	—	2	2	no	—
95	#443	C₄×C₅	1	4	yes	complemented
96	C₂×#108	C₂²×C₅	1	1	yes	characteristic
97	C₂×#107	—	2	1	no	—
98	C₂×#106	C₂²×C₅	1	2	yes	—
99	C₂×#104	—	2	1	no	—
100	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
101	#382	—	2	1	no	—
102	#364	—	2	2	no	—
103	#360	—	2	4	no	—
104	C₂³×C₅	—	4	1	no	—
105	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₅×Q₈	—	2	2	no	—
107	C₅×Q₈	—	4	1	no	—
108	C₅×D₄	—	2	2	no	—
109	C₅×D₄	—	4	2	no	—
110	C₅×D₄	—	4	2	no	—
111	C₅×D₄	—	4	1	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	4	2	no	—
114	C₂×C₄×C₅	—	4	1	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	8	4	no	—
118	C₂×C₄×C₅	—	8	2	no	—
119	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	#121	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅	#137	1	1	yes	characteristic
125	C₅×C₈	—	4	2	no	—
126	C₅×C₈	#102	1	2	yes	—
127	C₅×C₈	#102	1	2	yes	—
128	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₅×C₈	#117	1	1	yes	characteristic
130	C₅×C₈	#121	1	1	yes	characteristic
131	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×#41	—	2	1	no	—
133	#135	—	2	1	no	—
134	#135	—	2	1	no	—
135	#135	C₂×C₄×C₅	1	2	yes	—
136	#135	C₅×D₄	1	2	yes	—
137	C₂×Q₁₆	C₅×Q₈	1	1	yes	characteristic
138	C₂×SD₁₆	—	2	1	no	—
139	C₂×D₈	C₅×Q₈	1	1	yes	characteristic
140	#131	—	2	4	no	—
141	#131	—	4	2	no	—
142	C₂×M₁₆	—	2	1	no	—
143	C₂×M₁₆	—	4	1	no	—
144	C₂²×C₈	—	4	1	no	—
145	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
149	M₃₂	—	2	2	no	—
150	M₃₂	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	#108	—	2	1	no	—
153	#108	C₂×C₄×C₅	1	2	yes	—
154	#107	—	4	2	no	—
155	#106	—	2	2	no	—
156	#106	—	2	2	no	—
157	#104	—	2	4	no	—
158	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
159	C₄×C₈	C₅×D₄	1	2	yes	—
160	#95	—	4	2	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	4	2	no	—
163	C₂²×C₅	—	8	2	no	—
164	C₂²×C₅	#417	1	1	yes	characteristic
165	C₂²×C₅	#459	1	1	yes	characteristic
166	C₂²×C₅	#476	1	1	yes	characteristic
167	C₄×C₅	—	4	2	no	—
168	C₄×C₅	—	4	2	no	—
169	C₄×C₅	—	16	2	no	—
170	C₄×C₅	#380	1	1	yes	characteristic
171	C₄×C₅	C₂×#102	1	1	yes	characteristic
172	C₄×C₅	#461	1	1	yes	characteristic
173	C₄×C₅	#474	1	1	yes	characteristic
174	#41	—	2	2	no	—
175	#41	—	2	2	no	—
176	#41	—	4	1	no	—
177	#41	—	4	1	no	—
178	C₂×Q₈	—	2	1	no	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	4	1	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
185	Q₁₆	—	2	1	no	—
186	Q₁₆	C₅×C₄⋊C₄	1	2	yes	—
187	SD₁₆	—	2	2	no	—
188	SD₁₆	—	2	2	no	—
189	D₈	—	2	1	no	—
190	D₈	C₅×C₄⋊C₄	1	2	yes	—
191	M₁₆	—	2	2	no	—
192	M₁₆	—	4	1	no	—
193	M₁₆	—	4	4	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
197	C₂×C₈	C₅×D₈	1	2	yes	—
198	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
199	C₂×C₈	C₅×SD₁₆	1	1	yes	characteristic
200	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
201	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
204	C₂×C₈	C₅×#41	1	1	yes	characteristic
205	C₂×C₈	C₅×#41	1	1	yes	characteristic
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
210	C₁₆	—	4	2	no	—
211	C₂×C₅	—	2	1	no	—
212	C₂×C₅	—	2	1	no	—
213	C₂×C₅	—	8	2	no	—
214	C₂×C₅	#1802	1	1	yes	characteristic
215	C₂×C₅	#2231	1	2	yes	—
216	C₂³	—	4	1	no	—
217	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
218	Q₈	—	2	2	no	—
219	Q₈	—	4	1	no	—
220	D₄	—	2	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₅×#121	1	1	yes	characteristic
233	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
234	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
235	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
236	C₂×C₄	C₅×#137	1	1	yes	characteristic
237	C₈	—	4	2	no	—
238	C₈	C₅×#102	1	2	yes	—
239	C₈	C₅×#102	1	2	yes	—
240	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
241	C₈	C₅×#117	1	1	yes	characteristic
242	C₈	C₅×#121	1	1	yes	characteristic
243	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
244	C₅	#12824	1	1	yes	characteristic, direct product factor, Sylow
245	C₂²	—	2	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	C₅×#417	1	1	yes	characteristic
249	C₂²	C₅×#459	1	1	yes	characteristic
250	C₂²	C₅×#476	1	1	yes	characteristic
251	C₄	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	16	2	no	—
254	C₄	C₅×#380	1	1	yes	characteristic
255	C₄	C₂×C₅×#102	1	1	yes	characteristic
256	C₄	C₅×#461	1	1	yes	characteristic
257	C₄	C₅×#474	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	8	2	no	—
261	C₂	C₅×#1802	1	1	yes	characteristic
262	C₂	C₅×#2231	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#476
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#364	C₂	semidirect
4. C₅×#1846	C₂	semidirect
5. C₅×#1383	C₂	semidirect
10. C₅×#443	C₄	semidirect
19. #12824	C₅	direct (N × H)
50. C₂×#364	C₂×C₅	semidirect
52. #1846	C₂×C₅	semidirect
53. #1383	C₂×C₅	semidirect
95. #443	C₄×C₅	semidirect
244. C₅	#12824	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×#443	2
C₂×C₅×#364	2
C₂×C₅×#360	2
C₅×#1846	2
C₅×#1383	2
#12824	5

Subgroups — C5×#12824 #11935544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12824 #11935544