Subgroups — C₅×#21353 #11944073

Order 1280.

866 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
3	C₅×#2283	C₂	1	1	yes	maximal, characteristic
4	C₅×#1940	C₂	1	4	yes	maximal, complemented
5	C₅×#1850	C₂	1	4	yes	maximal
6	C₅×#1768	C₂	1	4	yes	maximal, complemented
7	C₂³×C₅×C₈	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₅×#431	—	2	4	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	8	yes	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×#97	—	2	4	no	—
14	C₂×C₅×#97	C₂²	1	2	yes	—
15	C₂×C₅×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₅×#383	C₄	1	2	yes	—
18	C₅×#382	C₄	1	2	yes	—
19	C₅×#381	C₄	1	2	yes	—
20	C₅×#380	C₄	1	2	yes	—
21	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₅×#376	C₂²	1	2	yes	—
24	C₅×#336	C₂²	1	8	yes	—
25	#21353	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₅×C₈	—	2	4	no	—
30	C₂²×C₅×C₈	—	2	8	no	—
31	C₂²×C₅×C₈	C₂³	1	4	yes	—
32	C₂²×C₅×C₈	C₂³	1	4	yes	—
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	4	yes	—
38	C₂×C₄²×C₅	Q₈	1	4	yes	—
39	C₂×C₄²×C₅	C₂³	1	4	yes	—
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×C₄⋊C₈	—	2	16	no	—
43	C₅×C₄⋊C₈	—	2	4	no	—
44	C₅×C₂²⋊C₈	—	2	4	no	—
45	C₅×#97	—	2	4	no	—
46	C₅×#97	—	2	4	no	—
47	C₄×C₅×C₈	—	2	4	no	—
48	C₅×#95	C₂×C₄	1	4	yes	—
49	C₅×#95	C₂×C₄	1	4	yes	—
50	C₅×#95	C₂×C₄	1	4	yes	—
51	C₅×#95	C₂×C₄	1	4	yes	—
52	C₂×#431	C₂×C₅	1	1	yes	characteristic
53	C₂²×#97	C₂×C₅	1	1	yes	characteristic
54	#2283	C₂×C₅	1	1	yes	characteristic
55	#1940	C₂×C₅	1	4	yes	complemented
56	#1850	C₂×C₅	1	4	yes	—
57	#1768	C₂×C₅	1	4	yes	complemented
58	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	—	4	2	no	—
62	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
65	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
73	C₂×C₅×C₈	—	2	16	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	2	4	no	—
76	C₂×C₅×C₈	—	2	4	no	—
77	C₂×C₅×C₈	—	2	8	no	—
78	C₂×C₅×C₈	#41	1	8	yes	—
79	C₅×C₄⋊C₄	—	4	4	no	—
80	C₅×C₂²⋊C₄	—	2	8	no	—
81	C₄²×C₅	—	2	8	no	—
82	C₄²×C₅	C₄⋊C₄	1	4	yes	—
83	C₄²×C₅	C₄⋊C₄	1	4	yes	—
84	C₄²×C₅	C₂×D₄	1	2	yes	—
85	C₄²×C₅	C₂×D₄	1	2	yes	—
86	C₄²×C₅	C₂×Q₈	1	2	yes	—
87	C₄²×C₅	C₂×Q₈	1	2	yes	—
88	C₂³×C₈	C₂²×C₅	1	2	yes	—
89	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
90	#431	—	2	4	no	—
91	C₂×C₄⋊C₈	C₂²×C₅	1	8	yes	—
92	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
93	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
94	C₂×#97	—	2	4	no	—
95	C₂×#97	C₂²×C₅	1	2	yes	—
96	C₂×#97	C₂²×C₅	1	2	yes	—
97	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
98	#383	C₄×C₅	1	2	yes	—
99	#382	C₄×C₅	1	2	yes	—
100	#381	C₄×C₅	1	2	yes	—
101	#380	C₄×C₅	1	2	yes	—
102	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
103	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
104	#376	C₂²×C₅	1	2	yes	—
105	#336	C₂²×C₅	1	8	yes	—
106	C₂³×C₅	—	2	2	no	—
107	C₂³×C₅	—	2	2	no	—
108	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
109	C₂³×C₅	#131	1	4	yes	—
110	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	8	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	2	8	no	—
120	C₂×C₄×C₅	—	2	8	no	—
121	C₂×C₄×C₅	—	2	8	no	—
122	C₂×C₄×C₅	—	4	4	no	—
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	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×#41	1	4	yes	—
131	C₅×C₈	—	2	8	no	—
132	C₅×C₈	—	4	4	no	—
133	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
134	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
135	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
136	C₂²×C₈	—	2	4	no	—
137	C₂²×C₈	—	2	8	no	—
138	C₂²×C₈	C₂³×C₅	1	4	yes	—
139	C₂²×C₈	C₂³×C₅	1	4	yes	—
140	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
141	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
142	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
143	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
144	C₂×C₄²	C₅×D₄	1	4	yes	—
145	C₂×C₄²	C₅×Q₈	1	4	yes	—
146	C₂×C₄²	C₂³×C₅	1	4	yes	—
147	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
148	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
149	C₄⋊C₈	—	2	16	no	—
150	C₄⋊C₈	—	2	4	no	—
151	C₂²⋊C₈	—	2	4	no	—
152	#97	—	2	4	no	—
153	#97	—	2	4	no	—
154	C₄×C₈	—	2	4	no	—
155	#95	C₂×C₄×C₅	1	4	yes	—
156	#95	C₂×C₄×C₅	1	4	yes	—
157	#95	C₂×C₄×C₅	1	4	yes	—
158	#95	C₂×C₄×C₅	1	4	yes	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	2	8	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	#424	1	4	yes	—
163	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
164	C₂²×C₅	C₄×#41	1	2	yes	—
165	C₂²×C₅	C₂×#131	1	2	yes	—
166	C₂²×C₅	#568	1	2	yes	—
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	—	2	2	no	—
169	C₄×C₅	—	2	2	no	—
170	C₄×C₅	—	2	2	no	—
171	C₄×C₅	—	2	2	no	—
172	C₄×C₅	—	4	8	no	—
173	C₄×C₅	#525	1	2	yes	—
174	C₄×C₅	#533	1	2	yes	—
175	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
180	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
181	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
182	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
183	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
185	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
186	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
188	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
189	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
190	C₂×C₈	—	2	16	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	8	no	—
195	C₂×C₈	C₅×#41	1	8	yes	—
196	C₄⋊C₄	—	4	4	no	—
197	C₂²⋊C₄	—	2	8	no	—
198	C₄²	—	2	8	no	—
199	C₄²	C₅×C₄⋊C₄	1	4	yes	—
200	C₄²	C₅×C₄⋊C₄	1	4	yes	—
201	C₄²	C₂×C₅×D₄	1	2	yes	—
202	C₄²	C₂×C₅×D₄	1	2	yes	—
203	C₄²	C₂×C₅×Q₈	1	2	yes	—
204	C₄²	C₂×C₅×Q₈	1	2	yes	—
205	C₂×C₅	—	2	4	no	—
206	C₂×C₅	#2296	1	1	yes	characteristic
207	C₂×C₅	C₂×#424	1	1	yes	characteristic
208	C₂×C₅	#2905	1	1	yes	characteristic
209	C₂×C₅	#2968	1	4	yes	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
213	C₂³	C₅×#131	1	4	yes	—
214	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
228	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
229	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
230	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
231	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
232	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
234	C₂×C₄	C₂×C₅×#41	1	4	yes	—
235	C₈	—	2	8	no	—
236	C₈	—	4	4	no	—
237	C₅	#21353	1	1	yes	characteristic, direct product factor, Sylow
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	8	no	—
241	C₂²	C₅×#424	1	4	yes	—
242	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
243	C₂²	C₄×C₅×#41	1	2	yes	—
244	C₂²	C₂×C₅×#131	1	2	yes	—
245	C₂²	C₅×#568	1	2	yes	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	4	8	no	—
252	C₄	C₅×#525	1	2	yes	—
253	C₄	C₅×#533	1	2	yes	—
254	C₂	—	2	4	no	—
255	C₂	C₅×#2296	1	1	yes	characteristic
256	C₂	C₂×C₅×#424	1	1	yes	characteristic
257	C₂	C₅×#2905	1	1	yes	characteristic
258	C₂	C₅×#2968	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅×#1940	C₂	semidirect
6. C₅×#1768	C₂	semidirect
25. #21353	C₅	direct (N × H)
55. #1940	C₂×C₅	semidirect
57. #1768	C₂×C₅	semidirect
237. C₅	#21353	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×#431	2
C₂²×C₅×#97	2
C₅×#2283	2
C₅×#1940	2
C₅×#1850	2
C₅×#1768	2
#21353	5

Subgroups — C5×#21353 #11944073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21353 #11944073