Subgroups — C₅×#22695 #11945415

Order 1280.

808 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×#2957	C₂	1	2	yes	maximal
2	C₂×C₅×#444	C₂	1	1	yes	maximal, characteristic
3	C₅×#2916	C₂	1	2	yes	maximal, complemented
4	C₅×#1953	C₂	1	2	yes	maximal, complemented
5	C₅×#1861	C₂	1	4	yes	maximal, complemented
6	C₅×#1849	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1814	C₂	1	2	yes	maximal, complemented
8	C₅×#1773	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#135	C₂²	1	1	yes	characteristic
10	C₅×#568	C₂²	1	4	yes	—
11	C₂×C₅×#131	C₂²	1	2	yes	—
12	C₅×#444	—	2	4	no	—
13	C₅×#444	C₄	1	8	yes	complemented
14	C₅×#443	—	2	4	no	—
15	C₅×#430	—	2	4	no	—
16	C₅×#430	C₂²	1	4	yes	—
17	C₂×C₅×#108	—	2	4	no	—
18	C₂×C₅×#108	C₂²	1	2	yes	—
19	C₂×C₅×#108	C₂²	1	2	yes	—
20	C₂×C₅×#108	C₂²	1	1	yes	characteristic
21	C₂×C₅×#108	C₂²	1	1	yes	characteristic
22	C₅×#427	C₂²	1	1	yes	characteristic
23	C₅×#423	C₂²	1	2	yes	—
24	C₅×#421	C₂²	1	4	yes	—
25	C₂×C₅×#104	C₂²	1	2	yes	—
26	C₅×#405	C₂²	1	2	yes	—
27	C₂×C₅×#97	C₂²	1	1	yes	characteristic
28	C₅×#344	C₂²	1	4	yes	—
29	C₅×#339	C₂²	1	2	yes	—
30	#22695	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₅×#41	C₂³	1	2	yes	—
32	C₅×#135	C₂×C₄	1	4	yes	—
33	C₅×#135	D₄	1	4	yes	—
34	C₂×C₅×Q₁₆	Q₈	1	1	yes	characteristic
35	C₂×C₅×SD₁₆	Q₈	1	2	yes	—
36	C₂×C₅×D₈	Q₈	1	1	yes	characteristic
37	C₅×#131	—	2	2	no	—
38	C₅×#131	—	2	2	no	—
39	C₅×#131	—	2	8	no	—
40	C₅×#131	—	4	4	no	—
41	C₅×#131	C₂×C₄	1	4	yes	—
42	C₅×#131	C₂×C₄	1	4	yes	—
43	C₂×C₅×M₁₆	—	2	2	no	—
44	C₂×C₅×M₁₆	—	2	8	no	—
45	C₂×C₅×M₁₆	C₂³	1	4	yes	—
46	C₂×C₅×M₁₆	C₂³	1	2	yes	—
47	C₂×C₅×M₁₆	C₂³	1	2	yes	—
48	C₂²×C₅×C₈	—	2	2	no	—
49	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₅×#117	C₂³	1	2	yes	—
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₅×#108	—	2	8	no	—
54	C₅×#108	—	2	2	no	—
55	C₅×#108	—	2	2	no	—
56	C₅×#108	—	2	2	no	—
57	C₅×#108	—	2	2	no	—
58	C₅×#108	C₂×C₄	1	4	yes	—
59	C₅×#108	D₄	1	4	yes	—
60	C₅×#107	—	2	1	no	—
61	C₅×#106	—	2	1	no	—
62	C₅×C₄⋊C₈	—	4	2	no	—
63	C₅×#104	—	2	8	no	—
64	C₅×#104	C₂×C₄	1	8	yes	—
65	C₅×#97	—	2	2	no	—
66	C₅×#97	C₂×C₄	1	4	yes	—
67	C₄×C₅×C₈	—	2	2	no	—
68	#2957	C₂×C₅	1	2	yes	—
69	C₂×#444	C₂×C₅	1	1	yes	characteristic
70	#2916	C₂×C₅	1	2	yes	complemented
71	#1953	C₂×C₅	1	2	yes	complemented
72	#1861	C₂×C₅	1	4	yes	complemented
73	#1849	C₂×C₅	1	1	yes	characteristic, complemented
74	#1814	C₂×C₅	1	2	yes	complemented
75	#1773	C₂×C₅	1	1	yes	characteristic, complemented
76	C₅×#41	—	2	2	no	—
77	C₅×#41	—	2	2	no	—
78	C₅×#41	C₂²×C₄	1	4	yes	—
79	C₅×#41	C₂×D₄	1	4	yes	—
80	C₂×C₅×Q₈	C₂×Q₈	1	2	yes	—
81	C₂×C₅×D₄	—	2	2	no	—
82	C₂×C₅×D₄	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₅	—	2	2	no	—
84	C₂²×C₄×C₅	—	2	1	no	—
85	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₅×Q₁₆	C₄⋊C₄	1	4	yes	—
87	C₅×SD₁₆	C₄⋊C₄	1	8	yes	—
88	C₅×D₈	C₄⋊C₄	1	4	yes	—
89	C₅×M₁₆	—	2	2	no	—
90	C₅×M₁₆	—	2	2	no	—
91	C₅×M₁₆	—	2	4	no	—
92	C₅×M₁₆	—	2	4	no	—
93	C₅×M₁₆	—	2	4	no	—
94	C₅×M₁₆	—	2	4	no	—
95	C₅×M₁₆	—	4	8	no	—
96	C₅×M₁₆	C₂²×C₄	1	4	yes	—
97	C₅×M₁₆	C₂×D₄	1	4	yes	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	2	4	no	—
101	C₂×C₅×C₈	—	2	4	no	—
102	C₂×C₅×C₈	—	2	4	no	—
103	C₂×C₅×C₈	—	4	8	no	—
104	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
107	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
108	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₅×C₈	#41	1	1	yes	characteristic
111	C₂×C₅×C₈	#41	1	1	yes	characteristic
112	C₅×C₄⋊C₄	—	2	2	no	—
113	C₅×C₂²⋊C₄	—	2	2	no	—
114	C₄²×C₅	—	2	1	no	—
115	C₄²×C₅	—	2	2	no	—
116	C₄²×C₅	C₂²×C₄	1	2	yes	—
117	C₂×#135	C₂²×C₅	1	1	yes	characteristic
118	#568	C₂²×C₅	1	4	yes	—
119	C₂×#131	C₂²×C₅	1	2	yes	—
120	#444	—	2	4	no	—
121	#444	C₄×C₅	1	8	yes	complemented
122	#443	—	2	4	no	—
123	#430	—	2	4	no	—
124	#430	C₂²×C₅	1	4	yes	—
125	C₂×#108	—	2	4	no	—
126	C₂×#108	C₂²×C₅	1	2	yes	—
127	C₂×#108	C₂²×C₅	1	2	yes	—
128	C₂×#108	C₂²×C₅	1	1	yes	characteristic
129	C₂×#108	C₂²×C₅	1	1	yes	characteristic
130	#427	C₂²×C₅	1	1	yes	characteristic
131	#423	C₂²×C₅	1	2	yes	—
132	#421	C₂²×C₅	1	4	yes	—
133	C₂×#104	C₂²×C₅	1	2	yes	—
134	#405	C₂²×C₅	1	2	yes	—
135	C₂×#97	C₂²×C₅	1	1	yes	characteristic
136	#344	C₂²×C₅	1	4	yes	—
137	#339	C₂²×C₅	1	2	yes	—
138	C₂³×C₅	—	2	2	no	—
139	C₂³×C₅	2₋¹⁺⁴	1	1	yes	characteristic
140	C₅×Q₈	—	2	2	no	—
141	C₅×Q₈	C₂×C₄⋊C₄	1	4	yes	—
142	C₅×D₄	—	2	4	no	—
143	C₅×D₄	—	2	4	no	—
144	C₅×D₄	—	2	2	no	—
145	C₅×D₄	C₂×C₄⋊C₄	1	4	yes	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	4	no	—
148	C₂×C₄×C₅	—	2	4	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	—	8	1	no	—
152	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
158	C₅×C₈	—	2	4	no	—
159	C₅×C₈	—	2	4	no	—
160	C₅×C₈	—	4	4	no	—
161	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
162	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
163	C₅×C₈	C₄×D₄	1	2	yes	—
164	C₅×C₈	C₄×D₄	1	2	yes	—
165	C₂×#41	C₂³×C₅	1	2	yes	—
166	#135	C₂×C₄×C₅	1	4	yes	—
167	#135	C₅×D₄	1	4	yes	—
168	C₂×Q₁₆	C₅×Q₈	1	1	yes	characteristic
169	C₂×SD₁₆	C₅×Q₈	1	2	yes	—
170	C₂×D₈	C₅×Q₈	1	1	yes	characteristic
171	#131	—	2	2	no	—
172	#131	—	2	2	no	—
173	#131	—	2	8	no	—
174	#131	—	4	4	no	—
175	#131	C₂×C₄×C₅	1	4	yes	—
176	#131	C₂×C₄×C₅	1	4	yes	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	8	no	—
179	C₂×M₁₆	C₂³×C₅	1	4	yes	—
180	C₂×M₁₆	C₂³×C₅	1	2	yes	—
181	C₂×M₁₆	C₂³×C₅	1	2	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
185	#117	C₂³×C₅	1	2	yes	—
186	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
187	#108	—	2	8	no	—
188	#108	—	2	2	no	—
189	#108	—	2	2	no	—
190	#108	—	2	2	no	—
191	#108	—	2	2	no	—
192	#108	C₂×C₄×C₅	1	4	yes	—
193	#108	C₅×D₄	1	4	yes	—
194	#107	—	2	1	no	—
195	#106	—	2	1	no	—
196	C₄⋊C₈	—	4	2	no	—
197	#104	—	2	8	no	—
198	#104	C₂×C₄×C₅	1	8	yes	—
199	#97	—	2	2	no	—
200	#97	C₂×C₄×C₅	1	4	yes	—
201	C₄×C₈	—	2	2	no	—
202	C₂²×C₅	—	2	4	no	—
203	C₂²×C₅	—	4	1	no	—
204	C₂²×C₅	—	8	2	no	—
205	C₂²×C₅	#520	1	1	yes	characteristic
206	C₂²×C₅	#547	1	1	yes	characteristic
207	C₂²×C₅	D₄×Q₈	1	1	yes	characteristic
208	C₄×C₅	—	2	4	no	—
209	C₄×C₅	—	4	2	no	—
210	C₄×C₅	—	8	2	no	—
211	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
212	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
213	C₄×C₅	D₄²	1	1	yes	characteristic
214	C₄×C₅	#554	1	1	yes	characteristic
215	#41	—	2	2	no	—
216	#41	—	2	2	no	—
217	#41	C₂²×C₄×C₅	1	4	yes	—
218	#41	C₂×C₅×D₄	1	4	yes	—
219	C₂×Q₈	C₂×C₅×Q₈	1	2	yes	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	C₂×C₅×Q₈	1	2	yes	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
225	Q₁₆	C₅×C₄⋊C₄	1	4	yes	—
226	SD₁₆	C₅×C₄⋊C₄	1	8	yes	—
227	D₈	C₅×C₄⋊C₄	1	4	yes	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	2	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	8	no	—
235	M₁₆	C₂²×C₄×C₅	1	4	yes	—
236	M₁₆	C₂×C₅×D₄	1	4	yes	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	8	no	—
243	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
244	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
245	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
246	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
247	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
248	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
249	C₂×C₈	C₅×#41	1	1	yes	characteristic
250	C₂×C₈	C₅×#41	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	2	no	—
255	C₄²	C₂²×C₄×C₅	1	2	yes	—
256	C₂×C₅	—	2	1	no	—
257	C₂×C₅	—	2	1	no	—
258	C₂×C₅	—	2	1	no	—
259	C₂×C₅	—	4	4	no	—
260	C₂×C₅	D₄×C₄⋊C₄	1	1	yes	characteristic
261	C₂³	—	2	2	no	—
262	C₂³	C₅×2₋¹⁺⁴	1	1	yes	characteristic
263	Q₈	—	2	2	no	—
264	Q₈	C₂×C₅×C₄⋊C₄	1	4	yes	—
265	D₄	—	2	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	2	2	no	—
268	D₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
279	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
280	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	4	4	no	—
284	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
285	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
286	C₈	C₄×C₅×D₄	1	2	yes	—
287	C₈	C₄×C₅×D₄	1	2	yes	—
288	C₅	#22695	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	C₅×#520	1	1	yes	characteristic
293	C₂²	C₅×#547	1	1	yes	characteristic
294	C₂²	C₅×D₄×Q₈	1	1	yes	characteristic
295	C₄	—	2	4	no	—
296	C₄	—	4	2	no	—
297	C₄	—	8	2	no	—
298	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
299	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
300	C₄	C₅×D₄²	1	1	yes	characteristic
301	C₄	C₅×#554	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	4	no	—
306	C₂	C₅×D₄×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	D₄²
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂×C₅	D₄×C₄⋊C₄
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2916	C₂	semidirect
4. C₅×#1953	C₂	semidirect
5. C₅×#1861	C₂	semidirect
6. C₅×#1849	C₂	semidirect
7. C₅×#1814	C₂	semidirect
8. C₅×#1773	C₂	semidirect
13. C₅×#444	C₄	semidirect
30. #22695	C₅	direct (N × H)
70. #2916	C₂×C₅	semidirect
71. #1953	C₂×C₅	semidirect
72. #1861	C₂×C₅	semidirect
73. #1849	C₂×C₅	semidirect
74. #1814	C₂×C₅	semidirect
75. #1773	C₂×C₅	semidirect
121. #444	C₄×C₅	semidirect
288. C₅	#22695	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×#2957	2
C₂×C₅×#444	2
C₅×#2916	2
C₅×#1953	2
C₅×#1861	2
C₅×#1849	2
C₅×#1814	2
C₅×#1773	2
#22695	5

Subgroups — C5×#22695 #11945415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22695 #11945415