Subgroups — #10605446

Order 640.

556 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4034	C₂	1	2	yes	maximal
2	C₂×#4015	C₂	1	2	yes	maximal
3	C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#65274	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	#64945	C₂	1	4	yes	maximal
9	C₂²×Dic₁₀	C₂²	1	4	yes	—
10	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₅	C₂²	1	2	yes	—
13	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₅	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
20	C₂×#683	C₂²	1	4	yes	—
21	C₂×C₄×Dic₅	C₂²	1	2	yes	—
22	#4034	—	2	8	no	—
23	#4015	—	2	8	no	—
24	C₅×#95	C₂²	1	2	yes	—
25	#3963	C₂²	1	2	yes	—
26	#3963	C₂²	1	4	yes	—
27	#2551	—	5	1	no	maximal, Sylow
28	C₂²×C₄×C₅	C₂³	1	2	yes	—
29	C₂²×C₄×C₅	C₂³	1	2	yes	—
30	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×Dic₅	C₂³	1	4	yes	—
34	C₂²×Dic₅	C₂³	1	2	yes	—
35	C₂²×Dic₅	C₂³	1	2	yes	—
36	C₂×Dic₁₀	—	2	8	no	—
37	C₂×Dic₁₀	—	2	8	no	—
38	C₂×Dic₁₀	—	4	4	no	—
39	C₅×C₄⋊C₄	D₄	1	4	yes	—
40	C₅×C₄⋊C₄	Q₈	1	4	yes	complemented
41	C₄²×C₅	—	2	4	no	—
42	C₄²×C₅	Q₈	1	4	yes	complemented
43	C₄⋊Dic₅	—	2	4	no	—
44	C₄⋊Dic₅	—	2	4	no	—
45	C₄⋊Dic₅	—	2	8	no	—
46	C₄⋊Dic₅	—	4	2	no	—
47	C₄⋊Dic₅	D₄	1	4	yes	—
48	C₄⋊Dic₅	Q₈	1	4	yes	complemented
49	#683	—	2	8	no	—
50	C₄×Dic₅	—	2	4	no	—
51	C₂×C₄⋊Q₈	—	5	2	no	—
52	C₂×C₄⋊Q₈	—	5	2	no	—
53	#395	—	5	4	no	—
54	#389	—	5	2	no	—
55	#389	—	5	2	no	—
56	#384	—	5	1	no	—
57	C₄×C₄⋊C₄	—	5	1	no	—
58	C₄×C₄⋊C₄	D₅	1	1	yes	characteristic
59	C₂³×C₅	C₂⁴	1	1	yes	characteristic
60	C₂×C₄×C₅	—	2	2	no	—
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	2	1	no	—
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	—	2	1	no	—
65	C₂×C₄×C₅	—	2	2	no	—
66	C₂×C₄×C₅	C₂×D₄	1	4	yes	—
67	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₅	C₂×Q₈	1	4	yes	—
71	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
72	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
75	C₂×Dic₅	—	2	2	no	—
76	C₂×Dic₅	—	2	2	no	—
77	C₂×Dic₅	—	2	2	no	—
78	C₂×Dic₅	—	2	2	no	—
79	C₂×Dic₅	—	2	4	no	—
80	C₂×Dic₅	—	2	4	no	—
81	C₂×Dic₅	—	2	4	no	—
82	C₂×Dic₅	C₂×D₄	1	4	yes	—
83	C₂×Dic₅	C₂×Q₈	1	4	yes	—
84	Dic₁₀	—	4	16	no	—
85	C₂²×Q₈	—	5	4	no	—
86	C₄⋊Q₈	—	10	8	no	—
87	C₄⋊Q₈	—	10	8	no	—
88	C₂×C₄⋊C₄	—	5	1	no	—
89	C₂×C₄⋊C₄	—	5	1	no	—
90	C₂×C₄⋊C₄	—	5	2	no	—
91	C₂×C₄⋊C₄	—	5	2	no	—
92	C₂×C₄⋊C₄	—	5	4	no	—
93	C₂×C₄⋊C₄	—	5	4	no	—
94	C₂×C₄⋊C₄	—	5	2	no	—
95	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
96	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
97	C₂×C₄²	—	5	2	no	—
98	C₂×C₄²	C₂×D₅	1	2	yes	—
99	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
100	#95	—	5	2	no	—
101	#95	—	5	4	no	—
102	#95	C₂×D₅	1	2	yes	—
103	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
104	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
105	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
106	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
107	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
108	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
109	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
110	C₄×C₅	—	2	2	no	—
111	C₄×C₅	—	2	4	no	—
112	C₄×C₅	—	2	4	no	—
113	C₄×C₅	C₄⋊Q₈	1	4	yes	complemented
114	C₄×C₅	C₄⋊Q₈	1	4	yes	complemented
115	Dic₅	—	2	4	no	—
116	Dic₅	—	2	4	no	—
117	Dic₅	—	4	4	no	—
118	C₂×Q₈	—	10	8	no	—
119	C₂×Q₈	—	10	8	no	—
120	C₂×Q₈	—	20	4	no	—
121	C₂²×C₄	—	5	2	no	—
122	C₂²×C₄	—	5	2	no	—
123	C₂²×C₄	—	5	4	no	—
124	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₅	1	2	yes	—
126	C₂²×C₄	C₂²×D₅	1	2	yes	—
127	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
129	C₄⋊C₄	—	5	4	no	—
130	C₄⋊C₄	—	5	4	no	—
131	C₄⋊C₄	—	10	8	no	—
132	C₄⋊C₄	—	10	8	no	—
133	C₄⋊C₄	—	10	4	no	—
134	C₄⋊C₄	—	10	4	no	—
135	C₄⋊C₄	—	20	2	no	—
136	C₄⋊C₄	Dic₁₀	1	4	yes	complemented
137	C₄⋊C₄	D₂₀	1	4	yes	—
138	C₄²	—	2	4	no	—
139	C₄²	—	10	4	no	—
140	C₄²	Dic₁₀	1	4	yes	complemented
141	C₂×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
142	C₂×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	C₂×C₅	#544	1	1	yes	characteristic
144	C₂×C₅	D₄²	1	1	yes	characteristic
145	C₂×C₅	D₄×Q₈	1	1	yes	characteristic
146	C₂×C₅	D₄×Q₈	1	1	yes	characteristic
147	C₂×C₅	Q₈²	1	1	yes	characteristic
148	C₂³	C₂³×D₅	1	1	yes	characteristic
149	Q₈	—	20	16	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	5	4	no	—
157	C₂×C₄	—	5	4	no	—
158	C₂×C₄	—	10	2	no	—
159	C₂×C₄	—	10	2	no	—
160	C₂×C₄	—	10	2	no	—
161	C₂×C₄	—	10	2	no	—
162	C₂×C₄	—	10	4	no	—
163	C₂×C₄	—	10	4	no	—
164	C₂×C₄	—	10	4	no	—
165	C₂×C₄	C₂×Dic₁₀	1	2	yes	—
166	C₂×C₄	C₂×Dic₁₀	1	4	yes	—
167	C₂×C₄	C₂×Dic₁₀	1	2	yes	—
168	C₂×C₄	C₂×Dic₁₀	1	2	yes	—
169	C₂×C₄	C₂×Dic₁₀	1	2	yes	—
170	C₂×C₄	C₂×D₂₀	1	4	yes	—
171	C₂×C₄	C₂×D₂₀	1	2	yes	—
172	C₂×C₄	D₄×D₅	1	2	yes	—
173	C₂×C₄	Q₈×D₅	1	2	yes	—
174	C₅	#2551	1	1	yes	characteristic, complemented, Sylow
175	C₂²	C₂²×Dic₁₀	1	1	yes	characteristic
176	C₂²	C₂²×Dic₁₀	1	1	yes	characteristic
177	C₂²	C₂²×D₂₀	1	1	yes	characteristic
178	C₂²	C₂×D₄×D₅	1	1	yes	characteristic
179	C₂²	C₂×Q₈×D₅	1	1	yes	characteristic
180	C₂²	#4149	1	1	yes	characteristic
181	C₂²	#4150	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	4	no	—
184	C₄	—	2	4	no	—
185	C₄	—	10	4	no	—
186	C₄	—	10	4	no	—
187	C₄	—	20	4	no	—
188	C₄	#4015	1	4	yes	complemented
189	C₄	#4034	1	4	yes	complemented
190	C₂	C₂×#4015	1	1	yes	characteristic
191	C₂	C₂×#4034	1	1	yes	characteristic
192	C₂	#65853	1	1	yes	characteristic
193	C₂	D₄×Dic₁₀	1	1	yes	characteristic
194	C₂	D₄×D₂₀	1	1	yes	characteristic
195	C₂	Q₈×Dic₁₀	1	1	yes	characteristic
196	C₂	Q₈×D₂₀	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	#2551
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
40. C₅×C₄⋊C₄	Q₈	semidirect
42. C₄²×C₅	Q₈	semidirect
48. C₄⋊Dic₅	Q₈	semidirect
113. C₄×C₅	C₄⋊Q₈	semidirect
114. C₄×C₅	C₄⋊Q₈	semidirect
136. C₄⋊C₄	Dic₁₀	semidirect
140. C₄²	Dic₁₀	semidirect
174. C₅	#2551	semidirect
188. C₄	#4015	semidirect
189. C₄	#4034	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4034	2
C₂×#4015	2
C₄×C₅×C₄⋊C₄	2
#65274	2
#65224	2
C₄×C₄⋊Dic₅	2
#65217	2
#64945	2
#2551	5