Subgroups — C₅×#3181 #10616776

Order 640.

366 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×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#529	C₂	1	1	yes	maximal, characteristic
3	C₅×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#467	C₂	1	2	yes	maximal, complemented
8	C₅×#465	C₂	1	2	yes	maximal
9	C₅×#441	C₂	1	2	yes	maximal, complemented
10	C₅×#440	C₂	1	2	yes	maximal, complemented
11	C₂×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₅×SD₁₆	C₂²	1	2	yes	—
14	C₂²×C₅×C₈	C₂²	1	2	yes	—
15	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₅×#126	—	2	1	no	—
18	C₅×#125	—	2	1	no	—
19	C₅×#125	C₂²	1	1	yes	characteristic
20	C₅×#124	C₂²	1	1	yes	characteristic
21	C₅×#123	—	2	1	no	—
22	C₅×C₂²⋊Q₈	—	2	1	no	—
23	C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₅×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₅×#121	C₂²	1	1	yes	characteristic
26	C₄×C₅×Q₈	—	2	1	no	—
27	C₄×C₅×Q₈	C₂²	1	2	yes	—
28	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
29	C₄×C₅×D₄	—	2	1	no	—
30	C₄×C₅×D₄	C₂²	1	1	yes	characteristic
31	C₅×#117	—	2	1	no	—
32	C₅×#117	—	2	1	no	—
33	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
34	C₅×#107	C₂²	1	2	yes	—
35	C₅×#106	C₂²	1	2	yes	—
36	C₅×#103	C₂²	1	2	yes	—
37	C₅×#103	C₂²	1	2	yes	—
38	C₅×#103	C₂²	1	2	yes	—
39	C₅×#102	C₂²	1	2	yes	—
40	C₅×#97	C₂²	1	2	yes	—
41	C₅×#97	C₂²	1	1	yes	characteristic
42	C₅×#97	C₂²	1	1	yes	characteristic
43	#3181	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₅×Q₈	—	2	1	no	—
45	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
46	C₂×C₅×Q₈	C₂³	1	2	yes	—
47	C₂×C₅×D₄	—	2	1	no	—
48	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅	—	2	1	no	—
50	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₅×Q₁₆	—	2	4	no	—
54	C₅×SD₁₆	—	2	4	no	—
55	C₂×C₅×C₈	—	2	4	no	—
56	C₂×C₅×C₈	C₂³	1	2	yes	—
57	C₂×C₅×C₈	C₂³	1	2	yes	—
58	C₅×C₄⋊C₄	—	2	1	no	—
59	C₅×C₄⋊C₄	—	2	1	no	—
60	C₅×C₄⋊C₄	—	2	1	no	—
61	C₅×C₄⋊C₄	—	2	1	no	—
62	C₅×C₄⋊C₄	—	2	1	no	—
63	C₅×C₄⋊C₄	—	2	1	no	—
64	C₅×C₄⋊C₄	—	2	2	no	—
65	C₅×C₄⋊C₄	—	2	2	no	—
66	C₅×C₄⋊C₄	—	2	1	no	—
67	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₅×C₄⋊C₄	C₂³	1	2	yes	—
70	C₅×C₂²⋊C₄	—	2	2	no	—
71	C₅×C₂²⋊C₄	—	2	1	no	—
72	C₅×C₂²⋊C₄	—	2	1	no	—
73	C₅×C₂²⋊C₄	—	2	1	no	—
74	C₅×C₂²⋊C₄	—	2	1	no	—
75	C₅×C₂²⋊C₄	—	2	1	no	—
76	C₄²×C₅	—	2	1	no	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	D₄	1	1	yes	characteristic
79	C₄²×C₅	D₄	1	1	yes	characteristic
80	C₄²×C₅	C₂³	1	1	yes	characteristic
81	C₄²×C₅	C₂³	1	1	yes	characteristic
82	#533	C₂×C₅	1	1	yes	characteristic, complemented
83	#529	C₂×C₅	1	1	yes	characteristic
84	#501	C₂×C₅	1	1	yes	characteristic, complemented
85	#497	C₂×C₅	1	1	yes	characteristic, complemented
86	#491	C₂×C₅	1	1	yes	characteristic, complemented
87	#489	C₂×C₅	1	1	yes	characteristic, complemented
88	#467	C₂×C₅	1	2	yes	complemented
89	#465	C₂×C₅	1	2	yes	—
90	#441	C₂×C₅	1	2	yes	complemented
91	#440	C₂×C₅	1	2	yes	complemented
92	C₂×#97	C₂×C₅	1	1	yes	characteristic, complemented
93	C₂³×C₅	—	2	1	no	—
94	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
95	C₅×Q₈	—	2	2	no	—
96	C₅×Q₈	—	2	2	no	—
97	C₅×Q₈	—	2	2	no	—
98	C₅×Q₈	—	4	2	no	—
99	C₅×D₄	—	2	2	no	—
100	C₅×D₄	—	4	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	4	1	no	—
109	C₂×C₄×C₅	—	4	2	no	—
110	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
116	C₅×C₈	—	2	2	no	—
117	C₅×C₈	#41	1	4	yes	—
118	C₂×Q₁₆	C₂²×C₅	1	2	yes	—
119	C₂×SD₁₆	C₂²×C₅	1	2	yes	—
120	C₂²×C₈	C₂²×C₅	1	2	yes	—
121	C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
122	C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
123	#126	—	2	1	no	—
124	#125	—	2	1	no	—
125	#125	C₂²×C₅	1	1	yes	characteristic
126	#124	C₂²×C₅	1	1	yes	characteristic
127	#123	—	2	1	no	—
128	C₂²⋊Q₈	—	2	1	no	—
129	C₂²⋊Q₈	C₂²×C₅	1	1	yes	characteristic
130	C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
131	#121	C₂²×C₅	1	1	yes	characteristic
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
134	C₄×Q₈	C₂²×C₅	1	2	yes	—
135	C₄×D₄	—	2	1	no	—
136	C₄×D₄	C₂²×C₅	1	1	yes	characteristic
137	#117	—	2	1	no	—
138	#117	—	2	1	no	—
139	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
140	#107	C₂²×C₅	1	2	yes	—
141	#106	C₂²×C₅	1	2	yes	—
142	#103	C₂²×C₅	1	2	yes	—
143	#103	C₂²×C₅	1	2	yes	—
144	#103	C₂²×C₅	1	2	yes	—
145	#102	C₂²×C₅	1	2	yes	—
146	#97	C₂²×C₅	1	2	yes	—
147	#97	C₂²×C₅	1	1	yes	characteristic
148	#97	C₂²×C₅	1	1	yes	characteristic
149	C₂²×C₅	—	2	1	no	—
150	C₂²×C₅	—	2	1	no	—
151	C₂²×C₅	—	4	1	no	—
152	C₂²×C₅	—	4	1	no	—
153	C₂²×C₅	—	4	1	no	—
154	C₂²×C₅	#137	1	2	yes	—
155	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
156	C₄×C₅	—	2	1	no	—
157	C₄×C₅	—	2	1	no	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	—	4	1	no	—
162	C₄×C₅	—	4	2	no	—
163	C₄×C₅	—	4	1	no	—
164	C₄×C₅	—	4	1	no	—
165	C₄×C₅	—	4	2	no	—
166	C₄×C₅	C₂×#41	1	2	yes	—
167	C₂×Q₈	—	2	1	no	—
168	C₂×Q₈	C₂³×C₅	1	1	yes	characteristic
169	C₂×Q₈	C₂³×C₅	1	2	yes	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	C₂³×C₅	1	1	yes	characteristic
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	C₅×D₄	1	1	yes	characteristic
174	C₂²×C₄	C₅×D₄	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
176	Q₁₆	—	2	4	no	—
177	SD₁₆	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	C₂³×C₅	1	2	yes	—
180	C₂×C₈	C₂³×C₅	1	2	yes	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	C₂³×C₅	1	2	yes	—
191	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
192	C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	2	1	no	—
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	2	no	—
201	C₄²	C₅×D₄	1	1	yes	characteristic
202	C₄²	C₅×D₄	1	1	yes	characteristic
203	C₄²	C₂³×C₅	1	1	yes	characteristic
204	C₄²	C₂³×C₅	1	1	yes	characteristic
205	C₂×C₅	—	2	2	no	—
206	C₂×C₅	—	8	1	no	—
207	C₂×C₅	#532	1	1	yes	characteristic
208	C₂×C₅	C₂×#137	1	1	yes	characteristic
209	C₂×C₅	#575	1	1	yes	characteristic
210	C₂³	—	2	1	no	—
211	C₂³	C₂×C₅×D₄	1	1	yes	characteristic
212	Q₈	—	2	2	no	—
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	4	2	no	—
216	D₄	—	2	2	no	—
217	D₄	—	4	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
228	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
229	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
230	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
231	C₂×C₄	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂×C₄	C₂⁴×C₅	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	C₅×#41	1	4	yes	—
235	C₅	#3181	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	C₅×#137	1	2	yes	—
242	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	1	no	—
252	C₄	—	4	2	no	—
253	C₄	C₂×C₅×#41	1	2	yes	—
254	C₂	—	2	2	no	—
255	C₂	—	8	1	no	—
256	C₂	C₅×#532	1	1	yes	characteristic
257	C₂	C₂×C₅×#137	1	1	yes	characteristic
258	C₂	C₅×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₅×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
1. C₅×#533	C₂	semidirect
3. C₅×#501	C₂	semidirect
4. C₅×#497	C₂	semidirect
5. C₅×#491	C₂	semidirect
6. C₅×#489	C₂	semidirect
7. C₅×#467	C₂	semidirect
9. C₅×#441	C₂	semidirect
10. C₅×#440	C₂	semidirect
11. C₂×C₅×#97	C₂	semidirect
43. #3181	C₅	direct (N × H)
82. #533	C₂×C₅	semidirect
84. #501	C₂×C₅	semidirect
85. #497	C₂×C₅	semidirect
86. #491	C₂×C₅	semidirect
87. #489	C₂×C₅	semidirect
88. #467	C₂×C₅	semidirect
90. #441	C₂×C₅	semidirect
91. #440	C₂×C₅	semidirect
92. C₂×#97	C₂×C₅	semidirect
235. C₅	#3181	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×#533	2
C₅×#529	2
C₅×#501	2
C₅×#497	2
C₅×#491	2
C₅×#489	2
C₅×#467	2
C₅×#465	2
C₅×#441	2
C₅×#440	2
C₂×C₅×#97	2
#3181	5

Subgroups — C5×#3181 #10616776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3181 #10616776