Subgroups — #10605448

Order 640.

452 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4036	C₂	1	2	yes	maximal
2	C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#65274	C₂	1	2	yes	maximal
4	#65224	C₂	1	2	yes	maximal
5	#65224	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
7	#65217	C₂	1	2	yes	maximal
8	#64950	C₂	1	4	yes	maximal
9	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₅	C₂²	1	2	yes	—
12	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₅	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₅	C₂²	1	4	yes	—
20	C₂×#683	C₂²	1	4	yes	—
21	C₂×C₄×Dic₅	C₂²	1	2	yes	—
22	#4036	—	2	8	no	—
23	C₅×#95	C₂²	1	2	yes	—
24	#3963	C₂²	1	2	yes	—
25	#3963	C₂²	1	4	yes	—
26	#2552	—	5	1	no	maximal, Sylow
27	C₂²×C₄×C₅	C₂³	1	2	yes	—
28	C₂²×C₄×C₅	C₂³	1	2	yes	—
29	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅	C₂³	1	2	yes	—
33	C₂²×Dic₅	C₂³	1	2	yes	—
34	C₂²×Dic₅	C₂³	1	4	yes	—
35	C₅×C₄⋊C₄	D₄	1	4	yes	—
36	C₅×C₄⋊C₄	Q₈	1	4	yes	—
37	C₄²×C₅	—	2	4	no	—
38	C₄²×C₅	Q₈	1	4	yes	—
39	C₄⋊Dic₅	—	2	8	no	—
40	C₄⋊Dic₅	—	2	4	no	—
41	C₄⋊Dic₅	—	2	2	no	—
42	C₄⋊Dic₅	—	2	4	no	—
43	C₄⋊Dic₅	—	2	2	no	—
44	C₄⋊Dic₅	—	4	2	no	—
45	C₄⋊Dic₅	—	4	4	no	—
46	#683	—	2	8	no	—
47	C₄×Dic₅	—	2	4	no	—
48	C₂×#125	—	5	2	no	—
49	#398	—	5	4	no	—
50	#389	—	5	2	no	—
51	#389	—	5	2	no	—
52	#384	—	5	2	no	—
53	#384	—	5	1	no	—
54	C₄×C₄⋊C₄	—	5	1	no	—
55	C₄×C₄⋊C₄	D₅	1	1	yes	characteristic
56	C₂³×C₅	C₂⁴	1	1	yes	characteristic
57	C₂×C₄×C₅	—	2	2	no	—
58	C₂×C₄×C₅	—	2	2	no	—
59	C₂×C₄×C₅	—	2	1	no	—
60	C₂×C₄×C₅	—	2	2	no	—
61	C₂×C₄×C₅	—	2	1	no	—
62	C₂×C₄×C₅	—	2	2	no	—
63	C₂×C₄×C₅	C₂×D₄	1	4	yes	—
64	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
66	C₂×C₄×C₅	C₂×Q₈	1	4	yes	—
67	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
69	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₅	#41	1	2	yes	—
71	C₂×C₄×C₅	#41	1	2	yes	—
72	C₂×Dic₅	—	2	2	no	—
73	C₂×Dic₅	—	2	2	no	—
74	C₂×Dic₅	—	2	4	no	—
75	C₂×Dic₅	—	2	2	no	—
76	C₂×Dic₅	—	2	2	no	—
77	C₂×Dic₅	—	2	4	no	—
78	C₂×Dic₅	—	2	2	no	—
79	C₂×Dic₅	—	2	4	no	—
80	C₂×Dic₅	—	2	2	no	—
81	#125	—	10	8	no	—
82	C₂×C₄⋊C₄	—	5	2	no	—
83	C₂×C₄⋊C₄	—	5	4	no	—
84	C₂×C₄⋊C₄	—	5	4	no	—
85	C₂×C₄⋊C₄	—	5	1	no	—
86	C₂×C₄⋊C₄	—	5	1	no	—
87	C₂×C₄⋊C₄	—	5	2	no	—
88	C₂×C₄⋊C₄	—	5	4	no	—
89	C₂×C₄⋊C₄	—	5	2	no	—
90	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
91	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
92	C₂×C₄²	—	5	2	no	—
93	C₂×C₄²	C₂×D₅	1	2	yes	—
94	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
95	#95	—	5	2	no	—
96	#95	—	5	4	no	—
97	#95	C₂×D₅	1	2	yes	—
98	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
99	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
100	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
101	C₂²×C₅	C₂×#41	1	1	yes	characteristic
102	C₂²×C₅	C₂×#41	1	1	yes	characteristic
103	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
104	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
105	C₄×C₅	—	2	2	no	—
106	C₄×C₅	—	2	4	no	—
107	C₄×C₅	—	2	4	no	—
108	C₄×C₅	#125	1	4	yes	complemented
109	C₄×C₅	C₄⋊Q₈	1	4	yes	—
110	Dic₅	—	4	2	no	—
111	Dic₅	—	4	2	no	—
112	Dic₅	—	4	4	no	—
113	C₂²×C₄	—	5	2	no	—
114	C₂²×C₄	—	5	2	no	—
115	C₂²×C₄	—	5	4	no	—
116	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₅	1	2	yes	—
118	C₂²×C₄	C₂²×D₅	1	2	yes	—
119	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
120	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
121	C₄⋊C₄	—	10	8	no	—
122	C₄⋊C₄	—	10	8	no	—
123	C₄⋊C₄	—	10	4	no	—
124	C₄⋊C₄	—	10	2	no	—
125	C₄⋊C₄	—	10	2	no	—
126	C₄⋊C₄	—	10	4	no	—
127	C₄⋊C₄	—	20	4	no	—
128	C₄⋊C₄	—	20	2	no	—
129	C₄⋊C₄	Dic₁₀	1	4	yes	—
130	C₄⋊C₄	D₂₀	1	4	yes	—
131	C₄²	—	2	4	no	—
132	C₄²	—	10	4	no	—
133	C₄²	Dic₁₀	1	4	yes	—
134	C₂×C₅	C₂×#125	1	1	yes	characteristic
135	C₂×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₅	#544	1	1	yes	characteristic
137	C₂×C₅	#547	1	1	yes	characteristic
138	C₂×C₅	#550	1	1	yes	characteristic
139	C₂×C₅	#554	1	1	yes	characteristic
140	C₂×C₅	#557	1	1	yes	characteristic
141	C₂³	C₂³×D₅	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	10	2	no	—
149	C₂×C₄	—	10	2	no	—
150	C₂×C₄	—	10	4	no	—
151	C₂×C₄	—	10	2	no	—
152	C₂×C₄	—	10	2	no	—
153	C₂×C₄	—	10	4	no	—
154	C₂×C₄	—	10	2	no	—
155	C₂×C₄	—	10	4	no	—
156	C₂×C₄	—	10	2	no	—
157	C₂×C₄	C₂×Dic₁₀	1	2	yes	—
158	C₂×C₄	C₂×Dic₁₀	1	4	yes	—
159	C₂×C₄	C₂×Dic₁₀	1	2	yes	—
160	C₂×C₄	C₂×Dic₁₀	1	2	yes	—
161	C₂×C₄	C₂×Dic₁₀	1	2	yes	—
162	C₂×C₄	C₂×D₂₀	1	4	yes	—
163	C₂×C₄	C₂×D₂₀	1	2	yes	—
164	C₂×C₄	#711	1	2	yes	—
165	C₂×C₄	#713	1	2	yes	—
166	C₅	#2552	1	1	yes	characteristic, complemented, Sylow
167	C₂²	C₂²×Dic₁₀	1	1	yes	characteristic
168	C₂²	C₂²×Dic₁₀	1	1	yes	characteristic
169	C₂²	C₂²×D₂₀	1	1	yes	characteristic
170	C₂²	C₂×#711	1	1	yes	characteristic
171	C₂²	C₂×#713	1	1	yes	characteristic
172	C₂²	#4149	1	1	yes	characteristic
173	C₂²	#4150	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	2	4	no	—
176	C₄	—	2	4	no	—
177	C₄	—	20	2	no	—
178	C₄	—	20	2	no	—
179	C₄	—	20	4	no	—
180	C₄	#4015	1	4	yes	—
181	C₄	#4036	1	4	yes	complemented
182	C₂	C₂×#4015	1	1	yes	characteristic
183	C₂	C₂×#4036	1	1	yes	characteristic
184	C₂	#65853	1	1	yes	characteristic
185	C₂	#65877	1	1	yes	characteristic
186	C₂	#65889	1	1	yes	characteristic
187	C₂	#65904	1	1	yes	characteristic
188	C₂	#65911	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂³×C₅	C₂⁴
Fitting	C₄×C₅×C₄⋊C₄	C₂
hypercenter	C₄×C₄⋊C₄	D₅
hypocenter	C₅	#2552
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
108. C₄×C₅	#125	semidirect
166. C₅	#2552	semidirect
181. C₄	#4036	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#4036	2
C₄×C₅×C₄⋊C₄	2
#65274	2
#65224	2
C₄×C₄⋊Dic₅	2
#65217	2
#64950	2
#2552	5