Subgroups — C₅×#22384 #11945104

Order 1280.

578 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2402	C₂	1	1	yes	maximal, characteristic
2	C₅×#1940	C₂	1	2	yes	maximal
3	C₅×#1917	C₂	1	2	yes	maximal
4	C₅×#1916	C₂	1	2	yes	maximal
5	C₅×#1916	C₂	1	2	yes	maximal
6	C₅×#1916	C₂	1	2	yes	maximal
7	C₅×#1840	C₂	1	1	yes	maximal, characteristic
8	C₅×#1770	C₂	1	2	yes	maximal
9	C₅×#1768	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×#97	C₂²	1	1	yes	characteristic
14	C₂×C₅×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#400	C₄	1	2	yes	—
18	C₅×#400	C₄	1	2	yes	—
19	C₅×#398	C₄	1	2	yes	—
20	C₅×#398	C₄	1	2	yes	—
21	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₅×#376	C₂²	1	1	yes	characteristic
26	C₅×#336	C₂²	1	2	yes	—
27	C₅×#336	C₂²	1	2	yes	—
28	C₅×#336	C₂²	1	2	yes	—
29	C₅×#336	C₂²	1	1	yes	characteristic
30	C₅×#336	C₂²	1	2	yes	—
31	C₅×#336	C₂²	1	1	yes	characteristic
32	C₅×#336	C₂²	1	2	yes	—
33	C₅×#336	C₂²	1	2	yes	—
34	C₅×#336	C₂²	1	2	yes	—
35	#22384	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₅×C₈	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₅	C₂³	1	2	yes	—
50	C₂×C₄²×C₅	C₂³	1	2	yes	—
51	C₅×C₄⋊C₈	—	2	4	no	—
52	C₅×C₄⋊C₈	—	2	4	no	—
53	C₅×C₄⋊C₈	—	4	2	no	—
54	C₅×#97	—	4	1	no	—
55	C₅×#97	—	4	2	no	—
56	C₄×C₅×C₈	—	2	4	no	—
57	C₄×C₅×C₈	—	4	1	no	—
58	C₅×#95	C₂×C₄	1	2	yes	—
59	C₅×#95	C₂×C₄	1	2	yes	—
60	C₅×#95	C₂×C₄	1	1	yes	characteristic
61	C₅×#95	C₂×C₄	1	1	yes	characteristic
62	C₅×#95	C₂×C₄	1	2	yes	—
63	C₅×#95	C₂×C₄	1	2	yes	—
64	C₅×#95	C₂×C₄	1	2	yes	—
65	C₅×#95	C₂×C₄	1	1	yes	characteristic
66	C₅×#95	C₂×C₄	1	2	yes	—
67	C₅×#95	C₂×C₄	1	1	yes	characteristic
68	#2402	C₂×C₅	1	1	yes	characteristic
69	#1940	C₂×C₅	1	2	yes	—
70	#1917	C₂×C₅	1	2	yes	—
71	#1916	C₂×C₅	1	2	yes	—
72	#1916	C₂×C₅	1	2	yes	—
73	#1916	C₂×C₅	1	2	yes	—
74	#1840	C₂×C₅	1	1	yes	characteristic
75	#1770	C₂×C₅	1	2	yes	—
76	#1768	C₂×C₅	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	#41	1	4	yes	—
98	C₅×C₄⋊C₄	—	2	4	no	—
99	C₅×C₄⋊C₄	—	2	4	no	—
100	C₅×C₄⋊C₄	—	2	4	no	—
101	C₅×C₄⋊C₄	—	4	2	no	—
102	C₅×C₄⋊C₄	—	4	2	no	—
103	C₅×C₄⋊C₄	—	4	2	no	—
104	C₄²×C₅	—	2	2	no	—
105	C₄²×C₅	—	2	2	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	2	1	no	—
108	C₄²×C₅	—	2	2	no	—
109	C₄²×C₅	—	2	2	no	—
110	C₄²×C₅	—	2	1	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
113	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
114	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
115	C₂×#97	C₂²×C₅	1	2	yes	—
116	C₂×#97	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
119	#400	C₄×C₅	1	2	yes	—
120	#400	C₄×C₅	1	2	yes	—
121	#398	C₄×C₅	1	2	yes	—
122	#398	C₄×C₅	1	2	yes	—
123	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
124	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
125	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
126	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
127	#376	C₂²×C₅	1	1	yes	characteristic
128	#336	C₂²×C₅	1	1	yes	characteristic
129	#336	C₂²×C₅	1	2	yes	—
130	#336	C₂²×C₅	1	2	yes	—
131	#336	C₂²×C₅	1	1	yes	characteristic
132	#336	C₂²×C₅	1	2	yes	—
133	#336	C₂²×C₅	1	2	yes	—
134	#336	C₂²×C₅	1	2	yes	—
135	#336	C₂²×C₅	1	2	yes	—
136	#336	C₂²×C₅	1	2	yes	—
137	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	#131	1	4	yes	—
160	C₂×C₄×C₅	C₂×#41	1	2	yes	—
161	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
163	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
164	C₅×C₈	—	2	4	no	—
165	C₅×C₈	—	4	2	no	—
166	C₅×C₈	—	4	2	no	—
167	C₅×C₈	—	4	2	no	—
168	C₂²×C₈	C₂³×C₅	1	2	yes	—
169	C₂²×C₈	C₂³×C₅	1	2	yes	—
170	C₂²×C₈	C₂³×C₅	1	2	yes	—
171	C₂²×C₈	C₂³×C₅	1	2	yes	—
172	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
175	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
176	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
177	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
178	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
181	C₂×C₄²	C₂³×C₅	1	2	yes	—
182	C₂×C₄²	C₂³×C₅	1	2	yes	—
183	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	2	4	no	—
185	C₄⋊C₈	—	4	2	no	—
186	#97	—	4	1	no	—
187	#97	—	4	2	no	—
188	C₄×C₈	—	2	4	no	—
189	C₄×C₈	—	4	1	no	—
190	#95	C₂×C₄×C₅	1	2	yes	—
191	#95	C₂×C₄×C₅	1	2	yes	—
192	#95	C₂×C₄×C₅	1	1	yes	characteristic
193	#95	C₂×C₄×C₅	1	1	yes	characteristic
194	#95	C₂×C₄×C₅	1	2	yes	—
195	#95	C₂×C₄×C₅	1	2	yes	—
196	#95	C₂×C₄×C₅	1	1	yes	characteristic
197	#95	C₂×C₄×C₅	1	2	yes	—
198	#95	C₂×C₄×C₅	1	2	yes	—
199	#95	C₂×C₄×C₅	1	1	yes	characteristic
200	C₂²×C₅	C₄×#41	1	1	yes	characteristic
201	C₂²×C₅	#519	1	1	yes	characteristic
202	C₂²×C₅	#520	1	1	yes	characteristic
203	C₂²×C₅	C₂×#131	1	2	yes	—
204	C₂²×C₅	#568	1	2	yes	—
205	C₄×C₅	—	2	4	no	—
206	C₄×C₅	—	4	2	no	—
207	C₄×C₅	—	4	2	no	—
208	C₄×C₅	—	4	1	no	—
209	C₄×C₅	—	4	1	no	—
210	C₄×C₅	—	4	2	no	—
211	C₄×C₅	—	4	2	no	—
212	C₄×C₅	—	4	2	no	—
213	C₄×C₅	#539	1	2	yes	—
214	C₄×C₅	#541	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
223	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₅×#41	1	4	yes	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	1	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	2	no	—
250	C₂×C₅	#2336	1	1	yes	characteristic
251	C₂×C₅	#2968	1	2	yes	—
252	C₂×C₅	#2986	1	1	yes	characteristic
253	C₂×C₅	#2987	1	1	yes	characteristic
254	C₂×C₅	#2993	1	2	yes	—
255	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	C₅×#131	1	4	yes	—
278	C₂×C₄	C₂×C₅×#41	1	2	yes	—
279	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
280	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
281	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	4	2	no	—
285	C₈	—	4	2	no	—
286	C₅	#22384	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²	C₄×C₅×#41	1	1	yes	characteristic
288	C₂²	C₅×#519	1	1	yes	characteristic
289	C₂²	C₅×#520	1	1	yes	characteristic
290	C₂²	C₂×C₅×#131	1	2	yes	—
291	C₂²	C₅×#568	1	2	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	C₅×#539	1	2	yes	—
301	C₄	C₅×#541	1	2	yes	—
302	C₂	C₅×#2336	1	1	yes	characteristic
303	C₂	C₅×#2968	1	2	yes	—
304	C₂	C₅×#2986	1	1	yes	characteristic
305	C₂	C₅×#2987	1	1	yes	characteristic
306	C₂	C₅×#2993	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂³×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
35. #22384	C₅	direct (N × H)
286. C₅	#22384	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×#2402	2
C₅×#1940	2
C₅×#1917	2
C₅×#1916	2
C₅×#1840	2
C₅×#1770	2
C₅×#1768	2
#22384	5

Subgroups — C5×#22384 #11945104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22384 #11945104