Subgroups — C₅×#11749 #11934469

Order 1280.

330 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3168	C₂	1	1	yes	maximal, characteristic
2	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1350	C₂	1	2	yes	maximal, complemented
5	C₅×#1349	C₂	1	2	yes	maximal
6	C₅×#529	—	2	1	no	—
7	C₅×#499	C₄	1	1	yes	characteristic
8	C₅×#495	C₄	1	1	yes	characteristic
9	C₅×#488	—	2	1	no	—
10	C₅×#467	C₂²	1	1	yes	characteristic
11	C₅×#466	C₂²	1	1	yes	characteristic
12	C₅×#465	—	2	1	no	—
13	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic
14	C₄×C₅×SD₁₆	—	2	1	no	—
15	C₄×C₅×D₈	C₄	1	1	yes	characteristic
16	C₂×C₅×#108	—	2	1	no	—
17	C₄×C₅×M₁₆	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×C₄⋊C₁₆	—	2	1	no	—
20	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
21	C₄×C₅×C₁₆	—	2	1	no	—
22	C₅×#341	C₂²	1	2	yes	—
23	#11749	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×SD₁₆	—	2	1	no	—
26	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₅×#126	—	4	1	no	—
31	C₅×#125	—	2	1	no	—
32	C₅×#124	—	2	1	no	—
33	C₅×#123	—	4	1	no	—
34	C₅×C₂²⋊Q₈	—	2	1	no	—
35	C₅×#121	—	2	1	no	—
36	C₄×C₅×Q₈	—	2	1	no	—
37	C₄×C₅×D₄	—	2	1	no	—
38	C₄×C₅×D₄	—	4	1	no	—
39	C₅×#117	—	4	1	no	—
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×C₅×C₁₆	—	2	2	no	—
42	C₅×#108	—	2	4	no	—
43	C₅×#107	C₂×C₄	1	1	yes	characteristic
44	C₅×#107	C₂×C₄	1	1	yes	characteristic
45	C₅×#106	—	2	1	no	—
46	C₅×#103	—	2	1	no	—
47	C₅×#103	—	2	1	no	—
48	C₅×#102	—	2	1	no	—
49	C₅×#102	—	2	1	no	—
50	C₅×#97	—	4	1	no	—
51	C₄×C₅×C₈	—	4	1	no	—
52	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₅×C₈	D₄	1	1	yes	characteristic
55	C₄×C₅×C₈	D₄	1	1	yes	characteristic
56	#3168	C₂×C₅	1	1	yes	characteristic
57	#2162	C₂×C₅	1	1	yes	characteristic, complemented
58	C₄×#108	C₂×C₅	1	1	yes	characteristic, complemented
59	#1350	C₂×C₅	1	2	yes	complemented
60	#1349	C₂×C₅	1	2	yes	—
61	C₂×C₅×Q₈	—	2	1	no	—
62	C₂×C₅×D₄	—	2	1	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂²×C₄×C₅	—	4	1	no	—
65	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₅×Q₁₆	—	4	1	no	—
69	C₅×SD₁₆	—	4	2	no	—
70	C₅×D₈	—	4	1	no	—
71	C₅×M₁₆	—	2	4	no	—
72	C₅×M₁₆	—	4	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	4	1	no	—
75	C₂×C₅×C₈	—	4	2	no	—
76	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	2	1	no	—
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₄⋊C₄	—	4	1	no	—
83	C₅×C₄⋊C₄	—	4	1	no	—
84	C₅×C₄⋊C₄	—	4	1	no	—
85	C₅×C₄⋊C₄	—	4	1	no	—
86	C₅×C₂²⋊C₄	—	4	1	no	—
87	C₅×C₂²⋊C₄	—	4	1	no	—
88	C₅×C₂²⋊C₄	—	4	1	no	—
89	C₅×C₂²⋊C₄	—	4	1	no	—
90	C₅×C₂²⋊C₄	—	4	1	no	—
91	C₄²×C₅	—	4	1	no	—
92	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₅	D₈	1	1	yes	characteristic
95	C₄²×C₅	D₈	1	1	yes	characteristic
96	C₅×C₁₆	—	4	2	no	—
97	#529	—	2	1	no	—
98	#499	C₄×C₅	1	1	yes	characteristic
99	#495	C₄×C₅	1	1	yes	characteristic
100	#488	—	2	1	no	—
101	#467	C₂²×C₅	1	1	yes	characteristic
102	#466	C₂²×C₅	1	1	yes	characteristic
103	#465	—	2	1	no	—
104	C₄×Q₁₆	C₄×C₅	1	1	yes	characteristic
105	C₄×SD₁₆	—	2	1	no	—
106	C₄×D₈	C₄×C₅	1	1	yes	characteristic
107	C₂×#108	—	2	1	no	—
108	C₄×M₁₆	—	2	1	no	—
109	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₄⋊C₁₆	—	2	1	no	—
111	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
112	C₄×C₁₆	—	2	1	no	—
113	#341	C₂²×C₅	1	2	yes	—
114	C₂³×C₅	—	4	1	no	—
115	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
116	C₅×Q₈	—	4	1	no	—
117	C₅×Q₈	—	4	1	no	—
118	C₅×D₄	—	4	1	no	—
119	C₅×D₄	—	4	1	no	—
120	C₅×D₄	—	8	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	4	1	no	—
125	C₂×C₄×C₅	—	4	1	no	—
126	C₂×C₄×C₅	—	4	1	no	—
127	C₂×C₄×C₅	—	4	1	no	—
128	C₂×C₄×C₅	—	8	2	no	—
129	C₂×C₄×C₅	#102	1	1	yes	characteristic
130	C₂×C₄×C₅	#102	1	1	yes	characteristic
131	C₂×C₄×C₅	#102	1	1	yes	characteristic
132	C₂×C₄×C₅	#102	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
135	C₅×C₈	—	2	1	no	—
136	C₅×C₈	—	2	1	no	—
137	C₅×C₈	—	2	1	no	—
138	C₅×C₈	—	4	4	no	—
139	C₅×C₈	#104	1	2	yes	—
140	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
141	C₂×SD₁₆	—	2	1	no	—
142	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
143	C₂×M₁₆	—	2	2	no	—
144	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
145	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
146	#126	—	4	1	no	—
147	#125	—	2	1	no	—
148	#124	—	2	1	no	—
149	#123	—	4	1	no	—
150	C₂²⋊Q₈	—	2	1	no	—
151	#121	—	2	1	no	—
152	C₄×Q₈	—	2	1	no	—
153	C₄×D₄	—	2	1	no	—
154	C₄×D₄	—	4	1	no	—
155	#117	—	4	1	no	—
156	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
157	C₂×C₁₆	—	2	2	no	—
158	#108	—	2	4	no	—
159	#107	C₂×C₄×C₅	1	1	yes	characteristic
160	#107	C₂×C₄×C₅	1	1	yes	characteristic
161	#106	—	2	1	no	—
162	#103	—	2	1	no	—
163	#103	—	2	1	no	—
164	#102	—	2	1	no	—
165	#102	—	2	1	no	—
166	#97	—	4	1	no	—
167	C₄×C₈	—	4	1	no	—
168	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
169	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
170	C₄×C₈	C₅×D₄	1	1	yes	characteristic
171	C₄×C₈	C₅×D₄	1	1	yes	characteristic
172	C₂²×C₅	—	2	1	no	—
173	C₂²×C₅	—	2	1	no	—
174	C₂²×C₅	—	2	1	no	—
175	C₂²×C₅	—	8	1	no	—
176	C₂²×C₅	—	8	1	no	—
177	C₂²×C₅	—	8	1	no	—
178	C₂²×C₅	C₂×#102	1	1	yes	characteristic
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	8	1	no	—
184	C₄×C₅	—	8	1	no	—
185	C₄×C₅	—	8	1	no	—
186	C₄×C₅	#359	1	2	yes	—
187	C₄×C₅	#410	1	1	yes	characteristic
188	C₄×C₅	C₂×#104	1	1	yes	characteristic
189	C₂×Q₈	—	2	1	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
194	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
196	Q₁₆	—	4	1	no	—
197	SD₁₆	—	4	2	no	—
198	D₈	—	4	1	no	—
199	M₁₆	—	2	4	no	—
200	M₁₆	—	4	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
207	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₂²⋊C₄	—	4	1	no	—
215	C₂²⋊C₄	—	4	1	no	—
216	C₂²⋊C₄	—	4	1	no	—
217	C₂²⋊C₄	—	4	1	no	—
218	C₂²⋊C₄	—	4	1	no	—
219	C₄²	—	4	1	no	—
220	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
221	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₄²	C₅×D₈	1	1	yes	characteristic
223	C₄²	C₅×D₈	1	1	yes	characteristic
224	C₁₆	—	4	2	no	—
225	C₂×C₅	—	4	1	no	—
226	C₂×C₅	—	16	1	no	—
227	C₂×C₅	#1502	1	1	yes	characteristic
228	C₂×C₅	C₂×#359	1	1	yes	characteristic
229	C₂×C₅	#2148	1	1	yes	characteristic
230	C₂³	—	4	1	no	—
231	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
232	Q₈	—	4	1	no	—
233	Q₈	—	4	1	no	—
234	D₄	—	4	1	no	—
235	D₄	—	4	1	no	—
236	D₄	—	8	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	C₅×#102	1	1	yes	characteristic
246	C₂×C₄	C₅×#102	1	1	yes	characteristic
247	C₂×C₄	C₅×#102	1	1	yes	characteristic
248	C₂×C₄	C₅×#102	1	1	yes	characteristic
249	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	4	4	no	—
255	C₈	C₅×#104	1	2	yes	—
256	C₅	#11749	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	8	1	no	—
261	C₂²	—	8	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	C₂×C₅×#102	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	8	1	no	—
269	C₄	—	8	1	no	—
270	C₄	—	8	1	no	—
271	C₄	C₅×#359	1	2	yes	—
272	C₄	C₅×#410	1	1	yes	characteristic
273	C₄	C₂×C₅×#104	1	1	yes	characteristic
274	C₂	—	4	1	no	—
275	C₂	—	16	1	no	—
276	C₂	C₅×#1502	1	1	yes	characteristic
277	C₂	C₂×C₅×#359	1	1	yes	characteristic
278	C₂	C₅×#2148	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#102
prime radical	C₅×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2162	C₂	semidirect
3. C₄×C₅×#108	C₂	semidirect
4. C₅×#1350	C₂	semidirect
23. #11749	C₅	direct (N × H)
57. #2162	C₂×C₅	semidirect
58. C₄×#108	C₂×C₅	semidirect
59. #1350	C₂×C₅	semidirect
256. C₅	#11749	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×#3168	2
C₅×#2162	2
C₄×C₅×#108	2
C₅×#1350	2
C₅×#1349	2
#11749	5

Subgroups — C5×#11749 #11934469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11749 #11934469