Subgroups — C₅×#2769 #10616364

Order 640.

344 conjugacy classes of subgroups.

The subgroup lattice has 192 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	1	yes	maximal, characteristic
2	C₅×#400	C₂	1	2	yes	maximal
3	C₅×#398	C₂	1	2	yes	maximal
4	C₅×#398	C₂	1	1	yes	maximal, characteristic
5	C₅×#389	C₂	1	2	yes	maximal
6	C₅×#387	C₂	1	2	yes	maximal
7	C₅×#387	C₂	1	2	yes	maximal
8	C₅×#383	C₂	1	2	yes	maximal
9	C₅×#382	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
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	1	yes	characteristic
14	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₅	C₂²	1	2	yes	—
20	C₅×#95	C₂²	1	2	yes	—
21	C₅×#95	C₂²	1	2	yes	—
22	C₅×#95	C₂²	1	1	yes	characteristic
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	2	yes	—
27	C₅×#95	C₂²	1	1	yes	characteristic
28	C₅×#95	C₂²	1	2	yes	—
29	C₅×#95	C₂²	1	2	yes	—
30	C₅×#95	C₂²	1	2	yes	—
31	#2769	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₄×C₅	C₂³	1	2	yes	—
33	C₂²×C₄×C₅	C₂³	1	2	yes	—
34	C₂²×C₄×C₅	C₂³	1	2	yes	—
35	C₂²×C₄×C₅	C₂³	1	2	yes	—
36	C₂²×C₄×C₅	C₂³	1	2	yes	—
37	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂³	1	2	yes	—
41	C₅×C₄⋊C₄	—	2	4	no	—
42	C₅×C₄⋊C₄	—	2	4	no	—
43	C₅×C₄⋊C₄	—	2	4	no	—
44	C₅×C₄⋊C₄	—	4	1	no	—
45	C₅×C₄⋊C₄	—	4	1	no	—
46	C₅×C₄⋊C₄	—	4	2	no	—
47	C₅×C₄⋊C₄	—	4	1	no	—
48	C₅×C₄⋊C₄	—	4	1	no	—
49	C₄²×C₅	—	2	2	no	—
50	C₄²×C₅	—	2	4	no	—
51	#400	C₂×C₅	1	2	yes	—
52	#400	C₂×C₅	1	1	yes	characteristic
53	#398	C₂×C₅	1	2	yes	—
54	#398	C₂×C₅	1	1	yes	characteristic
55	#389	C₂×C₅	1	2	yes	—
56	#387	C₂×C₅	1	2	yes	—
57	#387	C₂×C₅	1	2	yes	—
58	#383	C₂×C₅	1	2	yes	—
59	#382	C₂×C₅	1	1	yes	characteristic
60	C₂³×C₅	C₂⁴	1	1	yes	characteristic
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	2	1	no	—
63	C₂×C₄×C₅	—	2	1	no	—
64	C₂×C₄×C₅	—	2	2	no	—
65	C₂×C₄×C₅	—	2	2	no	—
66	C₂×C₄×C₅	—	2	2	no	—
67	C₂×C₄×C₅	—	2	2	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	—	2	2	no	—
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	1	no	—
77	C₂×C₄×C₅	—	2	1	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	2	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	#41	1	2	yes	—
85	C₂×C₄×C₅	#41	1	4	yes	—
86	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
87	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
88	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
89	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
92	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
95	C₂×C₄²	C₂²×C₅	1	2	yes	—
96	#95	C₂²×C₅	1	2	yes	—
97	#95	C₂²×C₅	1	2	yes	—
98	#95	C₂²×C₅	1	1	yes	characteristic
99	#95	C₂²×C₅	1	2	yes	—
100	#95	C₂²×C₅	1	2	yes	—
101	#95	C₂²×C₅	1	2	yes	—
102	#95	C₂²×C₅	1	2	yes	—
103	#95	C₂²×C₅	1	1	yes	characteristic
104	#95	C₂²×C₅	1	2	yes	—
105	#95	C₂²×C₅	1	2	yes	—
106	#95	C₂²×C₅	1	2	yes	—
107	C₂²×C₅	C₂×#41	1	1	yes	characteristic
108	C₂²×C₅	C₂×#41	1	2	yes	—
109	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
112	C₄×C₅	—	2	2	no	—
113	C₄×C₅	—	2	4	no	—
114	C₄×C₅	—	4	2	no	—
115	C₄×C₅	—	4	2	no	—
116	C₄×C₅	—	4	2	no	—
117	C₄×C₅	—	4	2	no	—
118	C₄×C₅	—	4	2	no	—
119	C₄×C₅	—	4	1	no	—
120	C₄×C₅	—	4	1	no	—
121	C₂²×C₄	C₂³×C₅	1	2	yes	—
122	C₂²×C₄	C₂³×C₅	1	2	yes	—
123	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
124	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
125	C₂²×C₄	C₂³×C₅	1	2	yes	—
126	C₂²×C₄	C₂³×C₅	1	2	yes	—
127	C₂²×C₄	C₂³×C₅	1	2	yes	—
128	C₂²×C₄	C₂³×C₅	1	2	yes	—
129	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	4	1	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	2	4	no	—
140	C₂×C₅	#539	1	1	yes	characteristic
141	C₂×C₅	#541	1	2	yes	—
142	C₂×C₅	#552	1	2	yes	—
143	C₂×C₅	#555	1	1	yes	characteristic
144	C₂×C₅	#563	1	1	yes	characteristic
145	C₂³	C₂⁴×C₅	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	C₅×#41	1	2	yes	—
170	C₂×C₄	C₅×#41	1	4	yes	—
171	C₅	#2769	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²	C₂×C₅×#41	1	1	yes	characteristic
173	C₂²	C₂×C₅×#41	1	2	yes	—
174	C₂²	C₅×H(5,𝔽₂)	1	1	yes	characteristic
175	C₂²	C₅×2₋¹⁺⁴	1	1	yes	characteristic
176	C₂²	C₅×2₋¹⁺⁴	1	2	yes	—
177	C₄	—	2	2	no	—
178	C₄	—	2	4	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₂	C₅×#539	1	1	yes	characteristic
187	C₂	C₅×#541	1	2	yes	—
188	C₂	C₅×#552	1	2	yes	—
189	C₂	C₅×#555	1	1	yes	characteristic
190	C₂	C₅×#563	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
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
31. #2769	C₅	direct (N × H)
171. C₅	#2769	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×#398	2
C₅×#389	2
C₅×#387	2
C₅×#383	2
C₅×#382	2
#2769	5

Subgroups — C5×#2769 #10616364

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2769 #10616364