Subgroups — C₅×#21968 #11944688

Order 1280.

754 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	C₂×C₅×#424	C₂	1	1	yes	maximal, characteristic
2	C₅×#2314	C₂	1	2	yes	maximal
3	C₅×#1949	C₂	1	4	yes	maximal, complemented
4	C₅×#1947	C₂	1	4	yes	maximal, complemented
5	C₅×#1809	C₂	1	2	yes	maximal
6	C₅×#1808	C₂	1	2	yes	maximal
7	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#117	C₂²	1	1	yes	characteristic
10	C₂×C₅×#107	C₂²	1	2	yes	—
11	C₂×C₅×#107	C₂²	1	2	yes	—
12	C₂×C₅×#107	C₂²	1	2	yes	—
13	C₂×C₅×#106	C₂²	1	2	yes	—
14	C₅×#424	C₄	1	8	yes	complemented
15	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
16	C₅×#388	—	2	4	no	—
17	C₅×#387	—	2	4	no	—
18	C₅×#384	C₂²	1	2	yes	—
19	C₅×#382	C₂²	1	2	yes	—
20	C₅×#380	C₂²	1	4	yes	—
21	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
22	C₅×#340	C₂²	1	4	yes	—
23	C₅×#340	C₂²	1	4	yes	—
24	#21968	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₅×M₁₆	—	4	1	no	—
27	C₂×C₅×M₁₆	—	4	1	no	—
28	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
29	C₂²×C₅×C₈	—	4	1	no	—
30	C₂²×C₅×C₈	—	4	1	no	—
31	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₅×C₈	C₂³	1	2	yes	—
33	C₂²×C₅×C₈	C₂³	1	2	yes	—
34	C₅×#117	C₂×C₄	1	4	yes	—
35	C₅×#117	Q₈	1	4	yes	—
36	C₂×C₅×C₄⋊C₄	—	2	4	no	—
37	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
38	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
39	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
41	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
42	C₂×C₄²×C₅	—	2	4	no	—
43	C₂×C₄²×C₅	C₂³	1	2	yes	—
44	C₅×#107	—	4	2	no	—
45	C₅×#107	—	4	2	no	—
46	C₅×#107	—	4	2	no	—
47	C₅×#106	—	4	2	no	—
48	C₅×C₄⋊C₈	C₂×C₄	1	16	yes	—
49	C₅×#95	—	2	4	no	—
50	C₅×#95	—	2	4	no	—
51	C₅×#95	—	2	4	no	—
52	C₂×#424	C₂×C₅	1	1	yes	characteristic
53	#2314	C₂×C₅	1	2	yes	—
54	#1949	C₂×C₅	1	4	yes	complemented
55	#1947	C₂×C₅	1	4	yes	complemented
56	#1809	C₂×C₅	1	2	yes	—
57	#1808	C₂×C₅	1	2	yes	—
58	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	2	1	no	—
61	C₂²×C₄×C₅	—	2	1	no	—
62	C₂²×C₄×C₅	—	2	1	no	—
63	C₂²×C₄×C₅	—	2	4	no	—
64	C₂²×C₄×C₅	—	2	4	no	—
65	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
70	C₅×M₁₆	—	4	4	no	—
71	C₂×C₅×C₈	—	2	2	no	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	4	4	no	—
76	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
77	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
78	C₅×C₄⋊C₄	—	2	4	no	—
79	C₅×C₄⋊C₄	—	2	4	no	—
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	—	2	4	no	—
82	C₅×C₄⋊C₄	—	8	4	no	—
83	C₅×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
84	C₅×C₂²⋊C₄	—	8	4	no	—
85	C₅×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
86	C₄²×C₅	—	8	4	no	—
87	C₄²×C₅	C₂²×C₄	1	4	yes	—
88	C₄²×C₅	C₂×Q₈	1	4	yes	—
89	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
90	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
91	C₂×#117	C₂²×C₅	1	1	yes	characteristic
92	C₂×#107	C₂²×C₅	1	2	yes	—
93	C₂×#107	C₂²×C₅	1	2	yes	—
94	C₂×#107	C₂²×C₅	1	2	yes	—
95	C₂×#106	C₂²×C₅	1	2	yes	—
96	#424	C₄×C₅	1	8	yes	complemented
97	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
98	#388	—	2	4	no	—
99	#387	—	2	4	no	—
100	#384	C₂²×C₅	1	2	yes	—
101	#382	C₂²×C₅	1	2	yes	—
102	#380	C₂²×C₅	1	4	yes	—
103	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
104	#340	C₂²×C₅	1	4	yes	—
105	#340	C₂²×C₅	1	4	yes	—
106	C₂³×C₅	—	2	1	no	—
107	C₂³×C₅	—	2	1	no	—
108	C₂³×C₅	—	2	2	no	—
109	C₂³×C₅	—	2	2	no	—
110	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
111	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	4	no	—
115	C₂×C₄×C₅	—	2	4	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	4	4	no	—
121	C₂×C₄×C₅	—	4	4	no	—
122	C₂×C₄×C₅	—	4	4	no	—
123	C₂×C₄×C₅	—	4	4	no	—
124	C₂×C₄×C₅	—	4	4	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
127	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
134	C₅×C₈	—	2	4	no	—
135	C₅×C₈	—	2	4	no	—
136	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
137	C₂×M₁₆	—	4	1	no	—
138	C₂×M₁₆	—	4	1	no	—
139	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
140	C₂²×C₈	—	4	1	no	—
141	C₂²×C₈	—	4	1	no	—
142	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
143	C₂²×C₈	C₂³×C₅	1	2	yes	—
144	C₂²×C₈	C₂³×C₅	1	2	yes	—
145	#117	C₂×C₄×C₅	1	4	yes	—
146	#117	C₅×Q₈	1	4	yes	—
147	C₂×C₄⋊C₄	—	2	4	no	—
148	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
149	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
150	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
151	C₂×C₂²⋊C₄	—	2	4	no	—
152	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
153	C₂×C₄²	—	2	4	no	—
154	C₂×C₄²	C₂³×C₅	1	2	yes	—
155	#107	—	4	2	no	—
156	#107	—	4	2	no	—
157	#107	—	4	2	no	—
158	#106	—	4	2	no	—
159	C₄⋊C₈	C₂×C₄×C₅	1	16	yes	—
160	#95	—	2	4	no	—
161	#95	—	2	4	no	—
162	#95	—	2	4	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	4	no	—
165	C₂²×C₅	—	4	1	no	—
166	C₂²×C₅	—	4	1	no	—
167	C₂²×C₅	—	4	2	no	—
168	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×C₅	#537	1	1	yes	characteristic
170	C₂²×C₅	#537	1	1	yes	characteristic
171	C₂²×C₅	#576	1	2	yes	—
172	C₂²×C₅	#578	1	2	yes	—
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	2	8	no	—
175	C₄×C₅	—	8	4	no	—
176	C₄×C₅	—	8	4	no	—
177	C₄×C₅	#518	1	1	yes	characteristic
178	C₄×C₅	#519	1	1	yes	characteristic
179	C₄×C₅	#544	1	1	yes	characteristic
180	C₄×C₅	#544	1	1	yes	characteristic
181	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
189	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
191	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
192	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
193	M₁₆	—	4	4	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
200	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	8	4	no	—
206	C₄⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
207	C₂²⋊C₄	—	8	4	no	—
208	C₂²⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
209	C₄²	—	8	4	no	—
210	C₄²	C₂²×C₄×C₅	1	4	yes	—
211	C₄²	C₂×C₅×Q₈	1	4	yes	—
212	C₂×C₅	—	4	2	no	—
213	C₂×C₅	#2316	1	1	yes	characteristic
214	C₂×C₅	#2913	1	2	yes	—
215	C₂×C₅	#3259	1	1	yes	characteristic
216	C₂×C₅	#3261	1	1	yes	characteristic
217	C₂×C₅	#3262	1	2	yes	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
223	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₂×C₅×C₄⋊C₄	1	8	yes	—
239	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
240	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
242	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
244	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
245	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
246	C₈	—	2	4	no	—
247	C₈	—	2	4	no	—
248	C₅	#21968	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
255	C₂²	C₅×#537	1	1	yes	characteristic
256	C₂²	C₅×#537	1	1	yes	characteristic
257	C₂²	C₅×#576	1	2	yes	—
258	C₂²	C₅×#578	1	2	yes	—
259	C₄	—	2	2	no	—
260	C₄	—	2	8	no	—
261	C₄	—	8	4	no	—
262	C₄	—	8	4	no	—
263	C₄	C₅×#518	1	1	yes	characteristic
264	C₄	C₅×#519	1	1	yes	characteristic
265	C₄	C₅×#544	1	1	yes	characteristic
266	C₄	C₅×#544	1	1	yes	characteristic
267	C₂	—	4	2	no	—
268	C₂	C₅×#2316	1	1	yes	characteristic
269	C₂	C₅×#2913	1	2	yes	—
270	C₂	C₅×#3259	1	1	yes	characteristic
271	C₂	C₅×#3261	1	1	yes	characteristic
272	C₂	C₅×#3262	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1949	C₂	semidirect
4. C₅×#1947	C₂	semidirect
14. C₅×#424	C₄	semidirect
24. #21968	C₅	direct (N × H)
54. #1949	C₂×C₅	semidirect
55. #1947	C₂×C₅	semidirect
96. #424	C₄×C₅	semidirect
248. C₅	#21968	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×#424	2
C₅×#2314	2
C₅×#1949	2
C₅×#1947	2
C₅×#1809	2
C₅×#1808	2
#21968	5

Subgroups — C5×#21968 #11944688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21968 #11944688