Subgroups — C₅×#2605 #10616200

Order 640.

396 conjugacy classes of subgroups.

The subgroup lattice has 196 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
2	C₅×#395	C₂	1	2	yes	maximal
3	C₅×#391	C₂	1	2	yes	maximal
4	C₅×#391	C₂	1	2	yes	maximal
5	C₅×#387	C₂	1	2	yes	maximal
6	C₅×#387	C₂	1	2	yes	maximal
7	C₅×#383	C₂	1	1	yes	maximal, characteristic
8	C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₅×#376	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₅×Q₈	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	2	yes	—
14	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₄²×C₅	C₂²	1	2	yes	—
16	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₅×#95	C₂²	1	2	yes	—
20	C₅×#95	C₂²	1	1	yes	characteristic
21	C₅×#95	C₂²	1	1	yes	characteristic
22	C₅×#95	C₂²	1	2	yes	—
23	C₅×#95	C₂²	1	1	yes	characteristic
24	C₅×#95	C₂²	1	2	yes	—
25	C₅×#95	C₂²	1	1	yes	characteristic
26	C₅×#95	C₂²	1	2	yes	—
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	2	yes	—
31	#2605	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₅×Q₈	—	2	4	no	—
33	C₂×C₅×Q₈	—	2	4	no	—
34	C₂×C₅×Q₈	—	4	2	no	—
35	C₂²×C₄×C₅	C₂³	1	2	yes	—
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	1	yes	characteristic
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	—	2	4	no	—
47	C₅×C₄⋊C₄	—	4	2	no	—
48	C₄²×C₅	—	2	4	no	—
49	C₄²×C₅	—	4	1	no	—
50	C₄²×C₅	—	4	1	no	—
51	C₄²×C₅	Q₈	1	4	yes	complemented
52	#400	C₂×C₅	1	2	yes	—
53	#395	C₂×C₅	1	2	yes	—
54	#391	C₂×C₅	1	2	yes	—
55	#391	C₂×C₅	1	2	yes	—
56	#387	C₂×C₅	1	2	yes	—
57	#387	C₂×C₅	1	2	yes	—
58	#383	C₂×C₅	1	1	yes	characteristic
59	C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
60	#376	C₂×C₅	1	1	yes	characteristic
61	C₂³×C₅	C₂⁴	1	1	yes	characteristic
62	C₅×Q₈	—	4	8	no	—
63	C₂×C₄×C₅	—	2	2	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	1	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	—	2	1	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	2	no	—
83	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
84	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
85	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
86	C₂×C₄×C₅	#41	1	4	yes	—
87	C₂×C₄×C₅	#41	1	4	yes	—
88	C₂²×Q₈	C₂²×C₅	1	2	yes	—
89	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
90	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
91	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
92	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
93	C₂×C₄²	C₂²×C₅	1	2	yes	—
94	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
95	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
97	#95	C₂²×C₅	1	2	yes	—
98	#95	C₂²×C₅	1	1	yes	characteristic
99	#95	C₂²×C₅	1	1	yes	characteristic
100	#95	C₂²×C₅	1	2	yes	—
101	#95	C₂²×C₅	1	1	yes	characteristic
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	#95	C₂²×C₅	1	2	yes	—
108	#95	C₂²×C₅	1	2	yes	—
109	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
110	C₂²×C₅	C₂×#41	1	2	yes	—
111	C₂²×C₅	C₂×#41	1	2	yes	—
112	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
114	C₄×C₅	—	2	2	no	—
115	C₄×C₅	—	2	2	no	—
116	C₄×C₅	—	2	2	no	—
117	C₄×C₅	—	2	4	no	—
118	C₄×C₅	—	2	4	no	—
119	C₄×C₅	—	4	2	no	—
120	C₄×C₅	—	4	2	no	—
121	C₄×C₅	—	4	2	no	—
122	C₄×C₅	—	4	2	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	—	2	4	no	—
125	C₂×Q₈	—	4	2	no	—
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	2	yes	—
130	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
131	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
133	C₂²×C₄	C₂³×C₅	1	2	yes	—
134	C₂²×C₄	C₂³×C₅	1	2	yes	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄²	—	2	4	no	—
140	C₄²	—	4	1	no	—
141	C₄²	—	4	1	no	—
142	C₄²	C₅×Q₈	1	4	yes	complemented
143	C₂×C₅	#533	1	1	yes	characteristic
144	C₂×C₅	#533	1	1	yes	characteristic
145	C₂×C₅	#544	1	1	yes	characteristic
146	C₂×C₅	#551	1	2	yes	—
147	C₂×C₅	#556	1	2	yes	—
148	C₂³	C₂⁴×C₅	1	1	yes	characteristic
149	Q₈	—	4	8	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	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	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	—	2	2	no	—
170	C₂×C₄	C₂×C₅×Q₈	1	2	yes	—
171	C₂×C₄	C₂×C₅×Q₈	1	2	yes	—
172	C₂×C₄	C₂×C₅×Q₈	1	2	yes	—
173	C₂×C₄	C₅×#41	1	4	yes	—
174	C₂×C₄	C₅×#41	1	4	yes	—
175	C₅	#2605	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²	C₂²×C₅×Q₈	1	1	yes	characteristic
177	C₂²	C₂×C₅×#41	1	2	yes	—
178	C₂²	C₂×C₅×#41	1	2	yes	—
179	C₂²	C₅×H(5,𝔽₂)	1	1	yes	characteristic
180	C₂²	C₅×2₋¹⁺⁴	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	2	4	no	—
185	C₄	—	2	4	no	—
186	C₄	—	4	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	2	no	—
190	C₂	C₅×#533	1	1	yes	characteristic
191	C₂	C₅×#533	1	1	yes	characteristic
192	C₂	C₅×#544	1	1	yes	characteristic
193	C₂	C₅×#551	1	2	yes	—
194	C₂	C₅×#556	1	2	yes	—

	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. #2605	C₅	direct (N × H)
51. C₄²×C₅	Q₈	semidirect
142. C₄²	C₅×Q₈	semidirect
175. C₅	#2605	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×#395	2
C₅×#391	2
C₅×#387	2
C₅×#383	2
C₄×C₅×C₄⋊C₄	2
C₅×#376	2
#2605	5

Subgroups — C5×#2605 #10616200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2605 #10616200