Subgroups — #10611045

Order 640.

356 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	C₂×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₅×#432	C₂	1	1	yes	maximal, characteristic
3	#65215	C₂	1	1	yes	maximal, characteristic
4	#65117	C₂	1	4	yes	maximal, complemented
5	#65008	C₂	1	2	yes	maximal
6	#65007	C₂	1	2	yes	maximal
7	#64990	C₂	1	4	yes	maximal, complemented
8	C₂²×Dic₁₀	C₄	1	1	yes	characteristic
9	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅	C₄	1	1	yes	characteristic
11	C₂×#683	C₄	1	1	yes	characteristic
12	C₂×#683	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₀	C₄	1	2	yes	—
16	C₄×Dic₁₀	C₄	1	2	yes	—
17	C₄×Dic₁₀	C₂²	1	2	yes	—
18	C₄×Dic₁₀	C₂²	1	2	yes	—
19	C₅×C₄⋊C₈	C₂²	1	2	yes	—
20	C₅×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₅×#97	C₂²	1	2	yes	—
22	C₂²⋊#168	C₂²	1	2	yes	—
23	#3947	C₂²	1	4	yes	—
24	#3946	C₂²	1	4	yes	—
25	#3946	C₂²	1	4	yes	—
26	C₈×Dic₅	C₂²	1	4	yes	—
27	C₄⋊#168	C₂²	1	2	yes	—
28	#3935	C₂²	1	2	yes	—
29	#2983	—	5	1	no	maximal, Sylow
30	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₂×Dic₁₀	—	2	4	no	—
38	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
39	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
40	C₂×C₅×C₈	C₂³	1	4	yes	—
41	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
47	#683	C₂×C₄	1	2	yes	—
48	#683	C₂×C₄	1	2	yes	—
49	#683	C₂×C₄	1	2	yes	—
50	#683	C₂×C₄	1	2	yes	—
51	C₄×Dic₅	C₂×C₄	1	2	yes	—
52	C₄×Dic₅	C₂×C₄	1	2	yes	—
53	C₄×Dic₅	C₂³	1	2	yes	—
54	C₄×Dic₅	C₂³	1	2	yes	—
55	C₂×#168	C₂³	1	4	yes	—
56	C₂×C₄×Q₈	—	5	1	no	—
57	#446	—	5	4	no	—
58	#446	—	5	4	no	—
59	#432	—	5	1	no	—
60	#432	—	5	2	no	—
61	#432	D₅	1	1	yes	characteristic
62	#431	—	5	2	no	—
63	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₅	—	2	2	no	—
65	C₂×C₄×C₅	—	2	2	no	—
66	C₂×C₄×C₅	—	4	1	no	—
67	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
73	C₂×Dic₅	—	2	2	no	—
74	C₂×Dic₅	—	4	1	no	—
75	C₂×Dic₅	—	4	1	no	—
76	C₂×Dic₅	M₁₆	1	4	yes	—
77	C₂×Dic₅	C₂²×C₄	1	2	yes	—
78	C₂×Dic₅	C₂²×C₄	1	2	yes	—
79	C₂×Dic₅	C₂²×C₄	1	2	yes	—
80	C₂×Dic₅	C₂²×C₄	1	2	yes	—
81	Dic₁₀	—	2	4	no	—
82	Dic₁₀	—	2	4	no	—
83	C₅×C₈	—	2	4	no	—
84	#168	—	2	4	no	—
85	C₂²×Q₈	—	5	1	no	—
86	C₄×Q₈	—	5	2	no	—
87	C₄×Q₈	—	5	2	no	—
88	C₄×Q₈	—	5	2	no	—
89	C₄×Q₈	—	5	2	no	—
90	C₂×C₄⋊C₄	—	5	1	no	—
91	C₂×C₄⋊C₄	—	5	1	no	—
92	C₂×C₄⋊C₄	—	5	1	no	—
93	C₂×C₄²	—	5	1	no	—
94	C₂×C₄²	—	5	1	no	—
95	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
96	C₄⋊C₈	—	5	4	no	—
97	C₄⋊C₈	—	5	4	no	—
98	C₄⋊C₈	—	5	2	no	—
99	C₄⋊C₈	C₂×D₅	1	2	yes	—
100	C₂²⋊C₈	—	5	2	no	—
101	C₂²⋊C₈	C₂×D₅	1	2	yes	—
102	#97	—	5	4	no	—
103	#97	—	5	2	no	—
104	#97	C₂×D₅	1	2	yes	—
105	C₄×C₈	—	5	4	no	—
106	C₂²×C₅	—	2	1	no	—
107	C₂²×C₅	—	2	1	no	—
108	C₂²×C₅	C₂×M₁₆	1	2	yes	—
109	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
110	C₄×C₅	—	2	1	no	—
111	C₄×C₅	—	2	1	no	—
112	C₄×C₅	—	2	1	no	—
113	C₄×C₅	—	2	1	no	—
114	C₄×C₅	—	2	1	no	—
115	C₄×C₅	2₋¹⁺⁴	1	2	yes	—
116	Dic₅	—	2	2	no	—
117	Dic₅	—	2	2	no	—
118	Dic₅	—	2	2	no	—
119	Dic₅	C₂×M₁₆	1	4	yes	—
120	C₂×Q₈	—	5	2	no	—
121	C₂×Q₈	—	5	2	no	—
122	C₂×Q₈	—	10	4	no	—
123	C₂²×C₄	—	5	1	no	—
124	C₂²×C₄	—	5	1	no	—
125	C₂²×C₄	—	5	1	no	—
126	C₂²×C₄	—	5	1	no	—
127	C₂²×C₄	C₄×D₅	1	1	yes	characteristic
128	C₂²×C₄	C₄×D₅	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
130	C₂×C₈	—	5	4	no	—
131	C₂×C₈	C₂²×D₅	1	4	yes	—
132	C₄⋊C₄	—	5	2	no	—
133	C₄⋊C₄	—	5	2	no	—
134	C₄⋊C₄	—	5	2	no	—
135	C₄⋊C₄	—	5	2	no	—
136	C₄⋊C₄	—	5	2	no	—
137	C₄⋊C₄	—	5	2	no	—
138	C₄²	—	5	2	no	—
139	C₄²	—	5	2	no	—
140	C₄²	—	5	2	no	—
141	C₄²	—	5	2	no	—
142	C₄²	C₄×D₅	1	1	yes	characteristic
143	C₄²	C₄×D₅	1	1	yes	characteristic
144	C₄²	C₂²×D₅	1	1	yes	characteristic
145	C₄²	C₂²×D₅	1	1	yes	characteristic
146	C₂×C₅	—	2	2	no	—
147	C₂×C₅	#519	1	1	yes	characteristic
148	C₂×C₅	C₂²×M₁₆	1	1	yes	characteristic
149	C₂×C₅	#568	1	1	yes	characteristic
150	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
151	Q₈	—	10	4	no	—
152	Q₈	—	10	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	5	2	no	—
157	C₂×C₄	—	5	4	no	—
158	C₂×C₄	—	5	2	no	—
159	C₂×C₄	—	5	2	no	—
160	C₂×C₄	—	5	2	no	—
161	C₂×C₄	—	10	2	no	—
162	C₂×C₄	—	20	1	no	—
163	C₂×C₄	—	20	1	no	—
164	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
169	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
170	C₈	—	2	4	no	—
171	C₈	—	10	4	no	—
172	C₅	#2983	1	1	yes	characteristic, complemented, Sylow
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	D₅×M₁₆	1	2	yes	—
176	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	5	4	no	—
183	C₄	—	10	2	no	—
184	C₄	—	10	2	no	—
185	C₄	—	10	2	no	—
186	C₄	#4150	1	2	yes	—
187	C₂	—	2	2	no	—
188	C₂	#65850	1	1	yes	characteristic
189	C₂	C₂×D₅×M₁₆	1	1	yes	characteristic
190	C₂	#66079	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#432	C₂
prime radical	C₂³×C₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#432	D₅
hypocenter	C₅	#2983
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
4. #65117	C₂	semidirect
7. #64990	C₂	semidirect
172. C₅	#2983	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₀	2
C₅×#432	2
#65215	2
#65117	2
#65008	2
#65007	2
#64990	2
#2983	5