Subgroups — C₅×#8634 #11931354

Order 1280.

388 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	C₅×#2931	C₂	1	1	yes	maximal, characteristic
2	C₅×#1568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1536	C₂	1	2	yes	maximal
5	C₅×#1307	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₄	1	2	yes	—
7	C₅×#532	C₂²	1	1	yes	characteristic
8	C₅×#436	—	2	1	no	—
9	C₅×#436	—	2	1	no	—
10	C₅×C₈×D₄	—	2	1	no	—
11	C₅×C₈×D₄	—	2	1	no	—
12	C₅×#432	C₂²	1	2	yes	—
13	C₅×#432	C₂²	1	2	yes	—
14	C₅×#423	C₂²	1	1	yes	characteristic
15	C₅×#408	C₄	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#335	—	2	1	no	—
18	C₅×#335	—	2	1	no	—
19	C₅×#334	—	2	1	no	—
20	C₅×#334	—	2	1	no	—
21	C₅×#329	—	2	4	no	—
22	#8634	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂×C₅×#41	C₂×C₄	1	2	yes	—
24	C₅×#131	—	4	4	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₅×C₄⋊Q₈	Q₈	1	1	yes	characteristic
31	C₅×#127	Q₈	1	1	yes	characteristic
32	C₅×#124	D₄	1	2	yes	—
33	C₅×#121	—	2	1	no	—
34	C₅×#121	—	2	1	no	—
35	C₄×C₅×D₄	—	2	1	no	—
36	C₄×C₅×D₄	—	2	1	no	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×C₄⋊C₈	—	2	1	no	—
39	C₅×C₄⋊C₈	—	2	1	no	—
40	C₅×C₄⋊C₈	—	2	2	no	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₂²⋊C₈	—	2	2	no	—
43	C₅×C₂²⋊C₈	—	2	2	no	—
44	C₅×C₂²⋊C₈	—	2	2	no	—
45	C₅×C₂²⋊C₈	—	2	2	no	—
46	C₅×#97	—	2	2	no	—
47	C₅×#97	—	2	2	no	—
48	C₄×C₅×C₈	D₄	1	1	yes	characteristic
49	C₄×C₅×C₈	D₄	1	1	yes	characteristic
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	#2931	C₂×C₅	1	1	yes	characteristic
53	#1568	C₂×C₅	1	1	yes	characteristic, complemented
54	#1568	C₂×C₅	1	1	yes	characteristic, complemented
55	#1536	C₂×C₅	1	2	yes	—
56	#1307	C₂×C₅	1	2	yes	complemented
57	C₅×#41	—	4	2	no	—
58	C₅×#41	—	4	2	no	—
59	C₂×C₅×Q₈	C₄⋊C₄	1	2	yes	—
60	C₂×C₅×D₄	—	2	2	no	—
61	C₂×C₅×D₄	—	2	1	no	—
62	C₂×C₅×D₄	—	2	1	no	—
63	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	#41	1	1	yes	characteristic
67	C₂²×C₄×C₅	#41	1	1	yes	characteristic
68	C₅×M₁₆	—	4	4	no	—
69	C₅×M₁₆	—	4	2	no	—
70	C₂×C₅×C₈	—	2	2	no	—
71	C₂×C₅×C₈	—	2	2	no	—
72	C₂×C₅×C₈	—	4	1	no	—
73	C₂×C₅×C₈	—	4	4	no	—
74	C₂×C₅×C₈	—	4	2	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	C₂²⋊C₄	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	2	1	no	—
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₂²⋊C₄	—	2	2	no	—
83	C₅×C₂²⋊C₄	—	2	2	no	—
84	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
88	C₂×#131	C₄×C₅	1	2	yes	—
89	#532	C₂²×C₅	1	1	yes	characteristic
90	#436	—	2	1	no	—
91	#436	—	2	1	no	—
92	C₈×D₄	—	2	1	no	—
93	C₈×D₄	—	2	1	no	—
94	#432	C₂²×C₅	1	2	yes	—
95	#432	C₂²×C₅	1	2	yes	—
96	#423	C₂²×C₅	1	1	yes	characteristic
97	#408	C₄×C₅	1	2	yes	—
98	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
99	#335	—	2	1	no	—
100	#335	—	2	1	no	—
101	#334	—	2	1	no	—
102	#334	—	2	1	no	—
103	#329	—	2	4	no	—
104	C₂³×C₅	—	2	2	no	—
105	C₂³×C₅	#117	1	1	yes	characteristic
106	C₅×Q₈	—	4	2	no	—
107	C₅×D₄	—	4	2	no	—
108	C₅×D₄	—	4	2	no	—
109	C₅×D₄	—	4	2	no	—
110	C₅×D₄	—	4	4	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	4	1	no	—
115	C₂×C₄×C₅	—	4	1	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	#121	1	1	yes	characteristic
121	C₂×C₄×C₅	#121	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	—	2	1	no	—
126	C₅×C₈	—	2	1	no	—
127	C₅×C₈	—	4	2	no	—
128	C₅×C₈	—	8	2	no	—
129	C₅×C₈	—	8	2	no	—
130	C₅×C₈	#100	1	2	yes	—
131	C₂×#41	C₂×C₄×C₅	1	2	yes	—
132	#131	—	4	4	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
137	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
138	C₄⋊Q₈	C₅×Q₈	1	1	yes	characteristic
139	#127	C₅×Q₈	1	1	yes	characteristic
140	#124	C₅×D₄	1	2	yes	—
141	#121	—	2	1	no	—
142	#121	—	2	1	no	—
143	C₄×D₄	—	2	1	no	—
144	C₄×D₄	—	2	1	no	—
145	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
146	C₄⋊C₈	—	2	1	no	—
147	C₄⋊C₈	—	2	1	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₂²⋊C₈	—	2	2	no	—
151	C₂²⋊C₈	—	2	2	no	—
152	C₂²⋊C₈	—	2	2	no	—
153	C₂²⋊C₈	—	2	2	no	—
154	#97	—	2	2	no	—
155	#97	—	2	2	no	—
156	C₄×C₈	C₅×D₄	1	1	yes	characteristic
157	C₄×C₈	C₅×D₄	1	1	yes	characteristic
158	C₄×C₈	C₅×D₄	1	1	yes	characteristic
159	C₄×C₈	C₅×D₄	1	1	yes	characteristic
160	C₂²×C₅	—	2	1	no	—
161	C₂²×C₅	—	2	1	no	—
162	C₂²×C₅	—	2	1	no	—
163	C₂²×C₅	—	4	2	no	—
164	C₂²×C₅	—	4	2	no	—
165	C₂²×C₅	—	4	2	no	—
166	C₂²×C₅	#380	1	1	yes	characteristic
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	—	2	1	no	—
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	—	4	2	no	—
173	C₄×C₅	C₂×#100	1	1	yes	characteristic
174	C₄×C₅	#413	1	1	yes	characteristic
175	#41	—	4	2	no	—
176	#41	—	4	2	no	—
177	C₂×Q₈	C₅×C₄⋊C₄	1	2	yes	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₂²×C₄	C₅×#41	1	1	yes	characteristic
185	C₂²×C₄	C₅×#41	1	1	yes	characteristic
186	M₁₆	—	4	4	no	—
187	M₁₆	—	4	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	4	4	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
203	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
204	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
205	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
206	C₂×C₅	—	4	1	no	—
207	C₂×C₅	—	8	2	no	—
208	C₂×C₅	#1800	1	1	yes	characteristic
209	C₂×C₅	#1814	1	1	yes	characteristic
210	C₂×C₅	#1814	1	1	yes	characteristic
211	C₂³	—	2	2	no	—
212	C₂³	C₅×#117	1	1	yes	characteristic
213	Q₈	—	4	2	no	—
214	D₄	—	4	2	no	—
215	D₄	—	4	2	no	—
216	D₄	—	4	2	no	—
217	D₄	—	4	4	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
226	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₅×#121	1	1	yes	characteristic
228	C₂×C₄	C₅×#121	1	1	yes	characteristic
229	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
230	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	4	2	no	—
235	C₈	—	8	2	no	—
236	C₈	—	8	2	no	—
237	C₈	C₅×#100	1	2	yes	—
238	C₅	#8634	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	C₅×#380	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	4	2	no	—
252	C₄	C₂×C₅×#100	1	1	yes	characteristic
253	C₄	C₅×#413	1	1	yes	characteristic
254	C₂	—	4	1	no	—
255	C₂	—	8	2	no	—
256	C₂	C₅×#1800	1	1	yes	characteristic
257	C₂	C₅×#1814	1	1	yes	characteristic
258	C₂	C₅×#1814	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1568	C₂	semidirect
3. C₅×#1568	C₂	semidirect
5. C₅×#1307	C₂	semidirect
22. #8634	C₅	direct (N × H)
53. #1568	C₂×C₅	semidirect
54. #1568	C₂×C₅	semidirect
56. #1307	C₂×C₅	semidirect
238. C₅	#8634	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×#2931	2
C₅×#1568	2
C₅×#1536	2
C₅×#1307	2
#8634	5

Subgroups — C5×#8634 #11931354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8634 #11931354