Subgroups — C₅×#2771 #10616366

Order 640.

392 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#400	C₂	1	2	yes	maximal, complemented
2	C₅×#400	C₂	1	2	yes	maximal, complemented
3	C₅×#399	C₂	1	1	yes	maximal, characteristic
4	C₅×#397	C₂	1	1	yes	maximal, characteristic
5	C₅×#389	C₂	1	2	yes	maximal, complemented
6	C₅×#388	C₂	1	2	yes	maximal
7	C₅×#387	C₂	1	2	yes	maximal, complemented
8	C₅×#385	C₂	1	2	yes	maximal
9	C₅×#381	C₂	1	1	yes	maximal, characteristic
10	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₅	C₂²	1	2	yes	—
22	C₅×#95	C₂²	1	2	yes	—
23	C₅×#95	C₂²	1	2	yes	—
24	C₅×#95	C₂²	1	2	yes	—
25	C₅×#95	C₂²	1	2	yes	—
26	C₅×#95	C₂²	1	1	yes	characteristic
27	C₅×#95	C₂²	1	2	yes	—
28	C₅×#95	C₂²	1	2	yes	—
29	C₅×#95	C₂²	1	2	yes	—
30	C₅×#95	C₂²	1	1	yes	characteristic
31	#2771	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₅	—	2	2	no	—
34	C₂²×C₄×C₅	—	4	1	no	—
35	C₂²×C₄×C₅	—	4	1	no	—
36	C₂²×C₄×C₅	C₂³	1	2	yes	—
37	C₂²×C₄×C₅	C₂³	1	2	yes	—
38	C₂²×C₄×C₅	C₂³	1	2	yes	—
39	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂³	1	2	yes	—
41	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂³	1	2	yes	—
43	C₂²×C₄×C₅	C₂³	1	2	yes	—
44	C₅×C₄⋊C₄	—	2	4	no	—
45	C₅×C₄⋊C₄	—	2	4	no	—
46	C₅×C₄⋊C₄	—	4	2	no	—
47	C₅×C₄⋊C₄	—	4	1	no	—
48	C₅×C₄⋊C₄	—	4	1	no	—
49	C₅×C₄⋊C₄	—	4	2	no	—
50	C₅×C₂²⋊C₄	—	2	4	no	—
51	C₅×C₂²⋊C₄	—	2	4	no	—
52	C₅×C₂²⋊C₄	—	4	1	no	—
53	C₅×C₂²⋊C₄	—	4	1	no	—
54	C₄²×C₅	—	4	2	no	—
55	#400	C₂×C₅	1	2	yes	complemented
56	#400	C₂×C₅	1	2	yes	complemented
57	#399	C₂×C₅	1	1	yes	characteristic
58	#397	C₂×C₅	1	1	yes	characteristic
59	#389	C₂×C₅	1	2	yes	complemented
60	#388	C₂×C₅	1	2	yes	—
61	#387	C₂×C₅	1	2	yes	complemented
62	#385	C₂×C₅	1	2	yes	—
63	#381	C₂×C₅	1	1	yes	characteristic
64	C₂³×C₅	—	2	2	no	—
65	C₂³×C₅	—	2	1	no	—
66	C₂³×C₅	—	2	1	no	—
67	C₂³×C₅	—	2	2	no	—
68	C₂³×C₅	#41	1	2	yes	—
69	C₂³×C₅	C₂⁴	1	1	yes	characteristic
70	C₂×C₄×C₅	—	2	2	no	—
71	C₂×C₄×C₅	—	2	2	no	—
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	1	no	—
83	C₂×C₄×C₅	—	2	1	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	—	4	4	no	—
92	C₂×C₄×C₅	#41	1	4	yes	—
93	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
95	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
96	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
97	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
98	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
101	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
102	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
104	C₂×C₄²	C₂²×C₅	1	2	yes	—
105	#95	C₂²×C₅	1	2	yes	—
106	#95	C₂²×C₅	1	2	yes	—
107	#95	C₂²×C₅	1	2	yes	—
108	#95	C₂²×C₅	1	2	yes	—
109	#95	C₂²×C₅	1	1	yes	characteristic
110	#95	C₂²×C₅	1	2	yes	—
111	#95	C₂²×C₅	1	2	yes	—
112	#95	C₂²×C₅	1	2	yes	—
113	#95	C₂²×C₅	1	1	yes	characteristic
114	C₂²×C₅	—	2	2	no	—
115	C₂²×C₅	—	2	4	no	—
116	C₂²×C₅	—	4	2	no	—
117	C₂²×C₅	—	4	1	no	—
118	C₂²×C₅	—	4	1	no	—
119	C₂²×C₅	C₂×#41	1	1	yes	characteristic
120	C₂²×C₅	C₂×#41	1	2	yes	—
121	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
124	C₄×C₅	—	2	4	no	—
125	C₄×C₅	—	4	2	no	—
126	C₄×C₅	—	4	2	no	—
127	C₄×C₅	—	4	2	no	—
128	C₄×C₅	—	4	2	no	—
129	C₄×C₅	—	4	2	no	—
130	C₄×C₅	—	4	1	no	—
131	C₄×C₅	—	4	1	no	—
132	C₂⁴	C₂³×C₅	1	1	yes	characteristic
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	C₂³×C₅	1	2	yes	—
137	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
138	C₂²×C₄	C₂³×C₅	1	2	yes	—
139	C₂²×C₄	C₂³×C₅	1	2	yes	—
140	C₂²×C₄	C₂³×C₅	1	2	yes	—
141	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₅	1	2	yes	—
143	C₂²×C₄	C₂³×C₅	1	2	yes	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	4	1	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₂²⋊C₄	—	2	4	no	—
151	C₂²⋊C₄	—	2	4	no	—
152	C₂²⋊C₄	—	4	1	no	—
153	C₂²⋊C₄	—	4	1	no	—
154	C₄²	—	4	2	no	—
155	C₂×C₅	—	4	2	no	—
156	C₂×C₅	#538	1	1	yes	characteristic
157	C₂×C₅	#539	1	2	yes	—
158	C₂×C₅	#552	1	2	yes	—
159	C₂×C₅	#559	1	1	yes	characteristic
160	C₂×C₅	#563	1	1	yes	characteristic
161	C₂³	—	2	2	no	—
162	C₂³	—	2	1	no	—
163	C₂³	—	2	1	no	—
164	C₂³	—	2	2	no	—
165	C₂³	C₅×#41	1	2	yes	—
166	C₂³	C₂⁴×C₅	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	C₅×#41	1	4	yes	—
190	C₅	#2771	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²	—	2	2	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	4	2	no	—
194	C₂²	—	4	1	no	—
195	C₂²	—	4	1	no	—
196	C₂²	C₂×C₅×#41	1	1	yes	characteristic
197	C₂²	C₂×C₅×#41	1	2	yes	—
198	C₂²	C₅×H(5,𝔽₂)	1	1	yes	characteristic
199	C₂²	C₅×H(5,𝔽₂)	1	1	yes	characteristic
200	C₂²	C₅×2₋¹⁺⁴	1	2	yes	—
201	C₄	—	2	4	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	1	no	—
208	C₄	—	4	1	no	—
209	C₂	—	4	2	no	—
210	C₂	C₅×#538	1	1	yes	characteristic
211	C₂	C₅×#539	1	2	yes	—
212	C₂	C₅×#552	1	2	yes	—
213	C₂	C₅×#559	1	1	yes	characteristic
214	C₂	C₅×#563	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#400	C₂	semidirect
2. C₅×#400	C₂	semidirect
5. C₅×#389	C₂	semidirect
7. C₅×#387	C₂	semidirect
31. #2771	C₅	direct (N × H)
55. #400	C₂×C₅	semidirect
56. #400	C₂×C₅	semidirect
59. #389	C₂×C₅	semidirect
61. #387	C₂×C₅	semidirect
190. C₅	#2771	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×#400	2
C₅×#399	2
C₅×#397	2
C₅×#389	2
C₅×#388	2
C₅×#387	2
C₅×#385	2
C₅×#381	2
#2771	5

Subgroups — C5×#2771 #10616366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2771 #10616366