Subgroups — C₅×#21969 #11944689

Order 1280.

734 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2920	C₂	1	2	yes	maximal
2	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#1950	C₂	1	8	yes	maximal, complemented
4	C₅×#1810	C₂	1	4	yes	maximal
5	C₂³×C₅×C₈	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₅×#432	—	2	2	no	—
8	C₅×#432	—	2	2	no	—
9	C₅×#431	—	2	2	no	—
10	C₅×#431	—	2	2	no	—
11	C₂×C₅×#108	C₂²	1	8	yes	—
12	C₅×#423	C₂²	1	4	yes	—
13	C₂×C₅×C₄⋊C₈	C₄	1	4	yes	—
14	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
15	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
16	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
17	C₅×#407	C₂²	1	4	yes	—
18	C₄×C₅×M₁₆	C₂²	1	4	yes	—
19	C₅×#341	C₂²	1	8	yes	—
20	#21969	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
22	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₅×M₁₆	C₂³	1	8	yes	—
25	C₂²×C₅×C₈	—	4	4	no	—
26	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₅×C₈	C₂³	1	4	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	D₄	1	2	yes	—
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	Q₈	1	2	yes	—
36	C₂×C₄²×C₅	C₂³	1	2	yes	—
37	C₅×#108	—	4	8	no	—
38	C₅×C₄⋊C₈	—	2	4	no	—
39	C₅×C₄⋊C₈	—	2	4	no	—
40	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
41	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
42	C₅×C₂²⋊C₈	—	2	4	no	—
43	C₅×C₂²⋊C₈	—	2	4	no	—
44	C₅×#97	—	2	4	no	—
45	C₄×C₅×C₈	—	2	4	no	—
46	#2920	C₂×C₅	1	2	yes	—
47	C₂²×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
48	#1950	C₂×C₅	1	8	yes	complemented
49	#1810	C₂×C₅	1	4	yes	—
50	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₅	—	2	4	no	—
52	C₂²×C₄×C₅	—	2	2	no	—
53	C₂²×C₄×C₅	—	2	2	no	—
54	C₂²×C₄×C₅	—	4	2	no	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
69	C₅×M₁₆	—	2	8	no	—
70	C₅×M₁₆	—	2	8	no	—
71	C₂×C₅×C₈	—	2	8	no	—
72	C₂×C₅×C₈	—	2	8	no	—
73	C₂×C₅×C₈	—	4	4	no	—
74	C₂×C₅×C₈	—	4	4	no	—
75	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
76	C₄²×C₅	C₄⋊C₄	1	2	yes	—
77	C₄²×C₅	C₄⋊C₄	1	2	yes	—
78	C₄²×C₅	C₄⋊C₄	1	2	yes	—
79	C₄²×C₅	C₄⋊C₄	1	2	yes	—
80	C₄²×C₅	C₂²×C₄	1	2	yes	—
81	C₄²×C₅	C₂²×C₄	1	2	yes	—
82	C₄²×C₅	C₂×D₄	1	2	yes	—
83	C₄²×C₅	C₂×D₄	1	2	yes	—
84	C₂³×C₈	C₂²×C₅	1	2	yes	—
85	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
86	#432	—	2	2	no	—
87	#432	—	2	2	no	—
88	#431	—	2	2	no	—
89	#431	—	2	2	no	—
90	C₂×#108	C₂²×C₅	1	8	yes	—
91	#423	C₂²×C₅	1	4	yes	—
92	C₂×C₄⋊C₈	C₄×C₅	1	4	yes	—
93	C₂×C₄⋊C₈	C₄×C₅	1	2	yes	—
94	C₂×C₄⋊C₈	C₄×C₅	1	2	yes	—
95	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
96	#407	C₂²×C₅	1	4	yes	—
97	C₄×M₁₆	C₂²×C₅	1	4	yes	—
98	#341	C₂²×C₅	1	8	yes	—
99	C₂³×C₅	—	2	2	no	—
100	C₂³×C₅	—	2	2	no	—
101	C₂³×C₅	—	2	2	no	—
102	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	4	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	4	no	—
115	C₂×C₄×C₅	—	2	4	no	—
116	C₂×C₄×C₅	—	4	1	no	—
117	C₂×C₄×C₅	—	4	1	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	#108	1	4	yes	complemented
120	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
129	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
130	C₅×C₈	—	2	8	no	—
131	C₅×C₈	—	4	8	no	—
132	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
133	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
134	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
135	C₂×M₁₆	C₂³×C₅	1	8	yes	—
136	C₂²×C₈	—	4	4	no	—
137	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
138	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
139	C₂²×C₈	C₂³×C₅	1	4	yes	—
140	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
141	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
142	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
143	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
144	C₂×C₄²	C₅×D₄	1	2	yes	—
145	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
146	C₂×C₄²	C₅×Q₈	1	2	yes	—
147	C₂×C₄²	C₂³×C₅	1	2	yes	—
148	#108	—	4	8	no	—
149	C₄⋊C₈	—	2	4	no	—
150	C₄⋊C₈	—	2	4	no	—
151	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
152	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
153	C₂²⋊C₈	—	2	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	#97	—	2	4	no	—
156	C₄×C₈	—	2	4	no	—
157	C₂²×C₅	—	2	8	no	—
158	C₂²×C₅	—	2	1	no	—
159	C₂²×C₅	—	2	1	no	—
160	C₂²×C₅	—	2	1	no	—
161	C₂²×C₅	—	2	1	no	—
162	C₂²×C₅	—	2	1	no	—
163	C₂²×C₅	—	2	1	no	—
164	C₂²×C₅	C₂×#108	1	2	yes	—
165	C₂²×C₅	#430	1	2	yes	—
166	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
167	C₂²×C₅	#544	1	2	yes	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	—	2	2	no	—
171	C₄×C₅	—	2	2	no	—
172	C₄×C₅	—	2	2	no	—
173	C₄×C₅	C₂×#108	1	4	yes	—
174	C₄×C₅	#518	1	1	yes	characteristic
175	C₄×C₅	#519	1	1	yes	characteristic
176	C₄×C₅	#537	1	2	yes	—
177	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
184	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
185	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
186	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
189	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
194	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
195	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
196	M₁₆	—	2	8	no	—
197	M₁₆	—	2	8	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	—	2	8	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
203	C₄²	C₅×C₄⋊C₄	1	2	yes	—
204	C₄²	C₅×C₄⋊C₄	1	2	yes	—
205	C₄²	C₅×C₄⋊C₄	1	2	yes	—
206	C₄²	C₅×C₄⋊C₄	1	2	yes	—
207	C₄²	C₂²×C₄×C₅	1	2	yes	—
208	C₄²	C₂²×C₄×C₅	1	2	yes	—
209	C₄²	C₂×C₅×D₄	1	2	yes	—
210	C₄²	C₂×C₅×D₄	1	2	yes	—
211	C₂×C₅	—	2	2	no	—
212	C₂×C₅	—	2	2	no	—
213	C₂×C₅	—	2	2	no	—
214	C₂×C₅	#2316	1	1	yes	characteristic
215	C₂×C₅	C₂²×#108	1	1	yes	characteristic
216	C₂×C₅	C₂×#430	1	1	yes	characteristic
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
221	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
222	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₅×#108	1	4	yes	complemented
238	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
239	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
240	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
241	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
242	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
243	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
244	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
246	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
247	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
248	C₈	—	2	8	no	—
249	C₈	—	4	8	no	—
250	C₅	#21969	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²	—	2	8	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	C₂×C₅×#108	1	2	yes	—
259	C₂²	C₅×#430	1	2	yes	—
260	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
261	C₂²	C₅×#544	1	2	yes	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	C₂×C₅×#108	1	4	yes	—
268	C₄	C₅×#518	1	1	yes	characteristic
269	C₄	C₅×#519	1	1	yes	characteristic
270	C₄	C₅×#537	1	2	yes	—
271	C₂	—	2	2	no	—
272	C₂	—	2	2	no	—
273	C₂	—	2	2	no	—
274	C₂	C₅×#2316	1	1	yes	characteristic
275	C₂	C₂²×C₅×#108	1	1	yes	characteristic
276	C₂	C₂×C₅×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1950	C₂	semidirect
20. #21969	C₅	direct (N × H)
48. #1950	C₂×C₅	semidirect
119. C₂×C₄×C₅	#108	semidirect
237. C₂×C₄	C₅×#108	semidirect
250. C₅	#21969	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#2920	2
C₂²×C₅×C₄⋊C₈	2
C₅×#1950	2
C₅×#1810	2
#21969	5

Subgroups — C5×#21969 #11944689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21969 #11944689