Subgroups — C₅×#22237 #11944957

Order 1280.

694 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3135	C₂	1	1	yes	maximal, characteristic
2	C₅×#2961	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2932	C₂	1	1	yes	maximal, characteristic
4	C₅×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1962	C₂	1	1	yes	maximal, characteristic
6	C₅×#1929	C₂	1	4	yes	maximal, complemented
7	C₅×#1866	C₂	1	4	yes	maximal, complemented
8	C₅×#1849	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1745	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#131	C₂²	1	2	yes	—
11	C₅×#533	C₂²	1	1	yes	characteristic
12	C₅×#532	C₂²	1	1	yes	characteristic
13	C₅×#491	C₄	1	2	yes	complemented
14	C₅×#490	C₄	1	2	yes	complemented
15	C₅×#489	C₄	1	4	yes	complemented
16	C₅×#446	—	2	4	no	—
17	C₅×#435	—	2	4	no	—
18	C₂×C₅×#108	C₂²	1	4	yes	—
19	C₅×#424	C₂²	1	2	yes	—
20	C₅×#423	C₂²	1	1	yes	characteristic
21	C₅×#423	C₂²	1	1	yes	characteristic
22	C₂×C₅×#104	C₂²	1	4	yes	—
23	C₅×#416	C₂²	1	4	yes	—
24	C₅×#408	C₂²	1	2	yes	—
25	C₅×#405	C₂²	1	2	yes	—
26	C₂×C₅×#97	C₂²	1	1	yes	characteristic
27	C₂×C₅×#97	C₂²	1	1	yes	characteristic
28	C₄³×C₅	C₂²	1	1	yes	characteristic
29	C₅×#339	C₂²	1	4	yes	—
30	C₅×#336	C₂²	1	2	yes	—
31	C₅×#336	C₂²	1	2	yes	—
32	#22237	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₅×#41	C₂³	1	2	yes	—
34	C₅×#131	—	4	4	no	—
35	C₂×C₅×M₁₆	—	2	2	no	—
36	C₂×C₅×M₁₆	C₂³	1	2	yes	—
37	C₂×C₅×M₁₆	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	—	2	2	no	—
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
43	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
44	C₅×#127	C₂×C₄	1	1	yes	characteristic
45	C₅×#125	C₂×C₄	1	2	yes	—
46	C₅×#124	C₂×C₄	1	2	yes	—
47	C₅×C₂²⋊Q₈	—	2	2	no	—
48	C₅×#121	—	2	2	no	—
49	C₄×C₅×Q₈	—	2	2	no	—
50	C₄×C₅×D₄	—	2	2	no	—
51	C₅×#117	C₂³	1	2	yes	—
52	C₂×C₄²×C₅	—	2	2	no	—
53	C₂×C₄²×C₅	C₂³	1	2	yes	—
54	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₅×#108	—	4	4	no	—
56	C₅×C₄⋊C₈	—	2	4	no	—
57	C₅×C₄⋊C₈	—	2	2	no	—
58	C₅×C₄⋊C₈	—	2	2	no	—
59	C₅×#104	—	4	4	no	—
60	C₅×#103	C₂×C₄	1	8	yes	—
61	C₅×#102	C₂×C₄	1	8	yes	—
62	C₅×C₂²⋊C₈	—	2	4	no	—
63	C₅×#97	—	2	2	no	—
64	C₅×#97	C₂×C₄	1	2	yes	—
65	C₅×#97	C₂×C₄	1	2	yes	—
66	C₅×#97	D₄	1	4	yes	—
67	C₄×C₅×C₈	—	2	2	no	—
68	#3135	C₂×C₅	1	1	yes	characteristic
69	#2961	C₂×C₅	1	1	yes	characteristic, complemented
70	#2932	C₂×C₅	1	1	yes	characteristic
71	#1994	C₂×C₅	1	1	yes	characteristic, complemented
72	#1962	C₂×C₅	1	1	yes	characteristic
73	#1929	C₂×C₅	1	4	yes	complemented
74	#1866	C₂×C₅	1	4	yes	complemented
75	#1849	C₂×C₅	1	1	yes	characteristic, complemented
76	#1745	C₂×C₅	1	1	yes	characteristic, complemented
77	C₅×#41	—	2	4	no	—
78	C₅×#41	—	2	4	no	—
79	C₂×C₅×Q₈	—	2	2	no	—
80	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
81	C₂×C₅×D₄	—	2	2	no	—
82	C₂×C₅×D₄	—	2	2	no	—
83	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅	—	2	2	no	—
85	C₂²×C₄×C₅	—	2	2	no	—
86	C₂²×C₄×C₅	#41	1	2	yes	—
87	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₅×M₁₆	—	2	4	no	—
89	C₅×M₁₆	—	2	4	no	—
90	C₅×M₁₆	—	4	4	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	—	4	4	no	—
98	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
99	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
100	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	2	4	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
105	C₅×C₂²⋊C₄	—	2	2	no	—
106	C₅×C₂²⋊C₄	—	2	4	no	—
107	C₄²×C₅	—	2	2	no	—
108	C₄²×C₅	—	2	2	no	—
109	C₄²×C₅	—	2	2	no	—
110	C₄²×C₅	—	4	4	no	—
111	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅	C₂×D₄	1	2	yes	—
114	C₂×#131	C₂²×C₅	1	2	yes	—
115	#533	C₂²×C₅	1	1	yes	characteristic
116	#532	C₂²×C₅	1	1	yes	characteristic
117	#491	C₄×C₅	1	2	yes	complemented
118	#490	C₄×C₅	1	2	yes	complemented
119	#489	C₄×C₅	1	4	yes	complemented
120	#446	—	2	4	no	—
121	#435	—	2	4	no	—
122	C₂×#108	C₂²×C₅	1	4	yes	—
123	#424	C₂²×C₅	1	2	yes	—
124	#423	C₂²×C₅	1	1	yes	characteristic
125	#423	C₂²×C₅	1	1	yes	characteristic
126	C₂×#104	C₂²×C₅	1	4	yes	—
127	#416	C₂²×C₅	1	4	yes	—
128	#408	C₂²×C₅	1	2	yes	—
129	#405	C₂²×C₅	1	2	yes	—
130	C₂×#97	C₂²×C₅	1	1	yes	characteristic
131	C₂×#97	C₂²×C₅	1	1	yes	characteristic
132	C₄³	C₂²×C₅	1	1	yes	characteristic
133	#339	C₂²×C₅	1	4	yes	—
134	#336	C₂²×C₅	1	2	yes	—
135	#336	C₂²×C₅	1	2	yes	—
136	C₂³×C₅	—	2	2	no	—
137	C₂³×C₅	C₂×#41	1	1	yes	characteristic
138	C₅×Q₈	—	2	4	no	—
139	C₅×Q₈	—	8	2	no	—
140	C₅×D₄	—	2	4	no	—
141	C₅×D₄	—	4	4	no	—
142	C₅×D₄	—	8	2	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	—	4	2	no	—
149	C₂×C₄×C₅	—	4	2	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
152	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
153	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
156	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
157	C₅×C₈	—	2	4	no	—
158	C₅×C₈	—	2	2	no	—
159	C₅×C₈	—	2	2	no	—
160	C₅×C₈	—	4	2	no	—
161	C₅×C₈	—	4	2	no	—
162	C₂×#41	C₂³×C₅	1	2	yes	—
163	#131	—	4	4	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	C₂³×C₅	1	2	yes	—
166	C₂×M₁₆	C₂³×C₅	1	2	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	C₂³×C₅	1	2	yes	—
169	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
171	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
172	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
173	#127	C₂×C₄×C₅	1	1	yes	characteristic
174	#125	C₂×C₄×C₅	1	2	yes	—
175	#124	C₂×C₄×C₅	1	2	yes	—
176	C₂²⋊Q₈	—	2	2	no	—
177	#121	—	2	2	no	—
178	C₄×Q₈	—	2	2	no	—
179	C₄×D₄	—	2	2	no	—
180	#117	C₂³×C₅	1	2	yes	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	C₂³×C₅	1	2	yes	—
183	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
184	#108	—	4	4	no	—
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	2	2	no	—
188	#104	—	4	4	no	—
189	#103	C₂×C₄×C₅	1	8	yes	—
190	#102	C₂×C₄×C₅	1	8	yes	—
191	C₂²⋊C₈	—	2	4	no	—
192	#97	—	2	2	no	—
193	#97	C₂×C₄×C₅	1	2	yes	—
194	#97	C₂×C₄×C₅	1	2	yes	—
195	#97	C₅×D₄	1	4	yes	—
196	C₄×C₈	—	2	2	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	4	2	no	—
199	C₂²×C₅	—	4	4	no	—
200	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
201	C₂²×C₅	#542	1	2	yes	—
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	—	2	2	no	—
204	C₄×C₅	—	4	4	no	—
205	C₄×C₅	—	4	2	no	—
206	C₄×C₅	—	4	4	no	—
207	C₄×C₅	#520	1	2	yes	—
208	C₄×C₅	#535	1	1	yes	characteristic
209	C₄×C₅	#536	1	1	yes	characteristic
210	#41	—	2	4	no	—
211	#41	—	2	4	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₅×#41	1	2	yes	—
220	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	4	no	—
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	—	4	4	no	—
231	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
232	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
233	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	4	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	4	4	no	—
244	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
245	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
246	C₄²	C₂×C₅×D₄	1	2	yes	—
247	C₂×C₅	—	2	2	no	—
248	C₂×C₅	—	8	2	no	—
249	C₂×C₅	#2326	1	1	yes	characteristic
250	C₂×C₅	#2956	1	2	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	C₂×C₅×#41	1	1	yes	characteristic
253	Q₈	—	2	4	no	—
254	Q₈	—	8	2	no	—
255	D₄	—	2	4	no	—
256	D₄	—	4	4	no	—
257	D₄	—	8	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
267	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
268	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
270	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
271	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
272	C₈	—	2	4	no	—
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	2	no	—
277	C₅	#22237	1	1	yes	characteristic, direct product factor, Sylow
278	C₂²	—	2	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
282	C₂²	C₅×#542	1	2	yes	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	4	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	4	no	—
288	C₄	C₅×#520	1	2	yes	—
289	C₄	C₅×#535	1	1	yes	characteristic
290	C₄	C₅×#536	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	—	8	2	no	—
293	C₂	C₅×#2326	1	1	yes	characteristic
294	C₂	C₅×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2961	C₂	semidirect
4. C₅×#1994	C₂	semidirect
6. C₅×#1929	C₂	semidirect
7. C₅×#1866	C₂	semidirect
8. C₅×#1849	C₂	semidirect
9. C₅×#1745	C₂	semidirect
13. C₅×#491	C₄	semidirect
14. C₅×#490	C₄	semidirect
15. C₅×#489	C₄	semidirect
32. #22237	C₅	direct (N × H)
69. #2961	C₂×C₅	semidirect
71. #1994	C₂×C₅	semidirect
73. #1929	C₂×C₅	semidirect
74. #1866	C₂×C₅	semidirect
75. #1849	C₂×C₅	semidirect
76. #1745	C₂×C₅	semidirect
117. #491	C₄×C₅	semidirect
118. #490	C₄×C₅	semidirect
119. #489	C₄×C₅	semidirect
277. C₅	#22237	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×#3135	2
C₅×#2961	2
C₅×#2932	2
C₅×#1994	2
C₅×#1962	2
C₅×#1929	2
C₅×#1866	2
C₅×#1849	2
C₅×#1745	2
#22237	5

Subgroups — C5×#22237 #11944957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22237 #11944957