Finite Group Databasebeta

Subgroups — #10604764

Order 640.

284 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4053	C₂	1	1	yes	maximal, characteristic
2	#65761	C₂	1	1	yes	maximal, characteristic
3	#65745	C₂	1	1	yes	maximal, characteristic
4	#65718	C₂	1	2	yes	maximal, complemented
5	#64894	C₂	1	2	yes	maximal, complemented
6	C₂×#709	C₂²	1	1	yes	characteristic
7	#4130	C₂²	1	2	yes	—
8	D₅⋊M₁₆	C₂²	1	2	yes	—
9	C₂×C₅×M₁₆	C₄	1	1	yes	characteristic, complemented
10	C₂²×#168	C₄	1	1	yes	characteristic, complemented
11	#4053	—	2	2	no	—
12	#4053	—	2	2	no	—
13	C₂×#676	C₄	1	1	yes	characteristic, complemented
14	C₂×#676	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₅	C₄	1	1	yes	characteristic, complemented
16	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
17	#4013	C₄	1	2	yes	complemented
18	#4011	C₄	1	2	yes	—
19	#4005	C₄	1	2	yes	complemented
20	#3999	C₄	1	2	yes	—
21	#3992	—	2	2	no	—
22	C₈×AGL(1,𝔽₅)	—	2	2	no	—
23	#1757	—	5	1	no	maximal, Sylow
24	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
27	#709	—	2	2	no	—
28	#709	—	2	2	no	—
29	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
33	C₂²⋊AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
34	#704	C₂×C₄	1	4	yes	—
35	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
36	C₄×AGL(1,𝔽₅)	—	2	2	no	—
37	#700	—	2	2	no	—
38	D₅⋊C₈	—	2	2	no	—
39	C₅×M₁₆	—	2	2	no	—
40	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×#168	—	2	1	no	—
43	C₂×#168	—	2	1	no	—
44	C₂×#168	C₂×C₄	1	1	yes	characteristic
45	C₂×#168	C₂×C₄	1	1	yes	characteristic
46	#676	—	2	2	no	—
47	#676	—	2	2	no	—
48	C₈×D₅	—	2	1	no	—
49	C₈×D₅	—	2	1	no	—
50	C₈×D₅	D₄	1	2	yes	—
51	C₈×D₅	D₄	1	2	yes	—
52	C₂×#131	—	5	1	no	—
53	#413	—	5	1	no	—
54	#410	—	5	1	no	—
55	#405	—	5	2	no	—
56	#344	—	5	2	no	—
57	C₂³×C₅	C₄²	1	1	yes	characteristic
58	C₂²×D₅	C₄²	1	1	yes	characteristic
59	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×AGL(1,𝔽₅)	—	2	4	no	—
61	C₂×C₄×C₅	—	2	1	no	—
62	C₂×C₄×C₅	—	2	1	no	—
63	C₂×C₄×C₅	C₄²	1	1	yes	characteristic
64	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²⋊D₅	—	2	2	no	—
66	C₂²⋊D₅	—	2	2	no	—
67	C₂×Dic₅	C₄²	1	1	yes	characteristic
68	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
69	D₂₀	—	2	2	no	—
70	C₄×D₅	—	2	1	no	—
71	C₄×D₅	—	2	1	no	—
72	C₄×D₅	C₂×D₄	1	2	yes	—
73	C₄×D₅	#41	1	2	yes	—
74	Dic₁₀	—	2	2	no	—
75	#170	—	2	4	no	—
76	C₅×C₈	—	2	1	no	—
77	C₅×C₈	C₂²⋊C₄	1	2	yes	—
78	#168	—	2	1	no	—
79	#168	C₂²⋊C₄	1	1	yes	characteristic
80	#168	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×#41	—	5	1	no	—
82	#131	—	10	2	no	—
83	#131	—	10	2	no	—
84	C₂×M₁₆	—	5	1	no	—
85	C₂×M₁₆	—	5	1	no	—
86	C₂×M₁₆	—	5	2	no	—
87	C₂×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
88	C₂²×C₈	—	5	1	no	—
89	C₂²×C₈	—	5	1	no	—
90	C₂²×C₈	—	5	1	no	—
91	#117	—	5	2	no	—
92	#101	—	5	2	no	—
93	#100	—	5	2	no	—
94	#99	—	5	2	no	—
95	#99	—	5	2	no	—
96	#97	—	10	2	no	—
97	C₄×C₈	—	10	2	no	—
98	C₂²×C₅	—	2	1	no	—
99	C₂²×C₅	—	4	1	no	—
100	C₂²×C₅	C₂×C₄²	1	1	yes	characteristic
101	C₂×D₅	—	2	1	no	—
102	C₂×D₅	—	4	1	no	—
103	C₂×D₅	—	4	1	no	—
104	C₂×D₅	C₄×D₄	1	2	yes	—
105	AGL(1,𝔽₅)	—	4	4	no	—
106	C₄×C₅	—	2	1	no	—
107	C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₅	#117	1	1	yes	characteristic
109	Dic₅	—	2	1	no	—
110	Dic₅	C₄×D₄	1	2	yes	—
111	#41	—	10	2	no	—
112	#41	—	10	2	no	—
113	C₂×Q₈	—	5	1	no	—
114	C₂×D₄	—	5	1	no	—
115	C₂×D₄	—	5	1	no	—
116	C₂×D₄	—	5	1	no	—
117	C₂²×C₄	—	5	1	no	—
118	C₂²×C₄	—	5	1	no	—
119	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
120	M₁₆	—	2	2	no	—
121	M₁₆	—	5	4	no	—
122	M₁₆	—	10	2	no	—
123	M₁₆	—	10	2	no	—
124	M₁₆	—	10	2	no	—
125	C₂×C₈	—	5	1	no	—
126	C₂×C₈	—	5	2	no	—
127	C₂×C₈	—	5	2	no	—
128	C₂×C₈	—	5	1	no	—
129	C₂×C₈	—	10	1	no	—
130	C₂×C₈	—	10	1	no	—
131	C₂×C₈	—	10	1	no	—
132	C₂×C₈	—	10	1	no	—
133	C₂×C₈	—	10	2	no	—
134	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
135	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
136	C₄⋊C₄	—	10	2	no	—
137	C₂²⋊C₄	—	5	4	no	—
138	C₄²	—	10	2	no	—
139	C₂×C₅	—	2	1	no	—
140	C₂×C₅	—	4	1	no	—
141	C₂×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	D₅	—	2	2	no	—
143	D₅	—	4	1	no	—
144	C₂³	—	5	1	no	—
145	C₂³	—	5	1	no	—
146	C₂³	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
147	Q₈	—	10	2	no	—
148	D₄	—	10	2	no	—
149	D₄	—	10	2	no	—
150	D₄	—	10	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	5	1	no	—
154	C₂×C₄	—	5	2	no	—
155	C₂×C₄	—	5	2	no	—
156	C₂×C₄	—	5	1	no	—
157	C₂×C₄	—	10	1	no	—
158	C₂×C₄	—	10	1	no	—
159	C₂×C₄	—	10	4	no	—
160	C₂×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
161	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	5	1	no	—
164	C₈	—	5	1	no	—
165	C₈	—	10	4	no	—
166	C₈	—	10	1	no	—
167	C₈	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
168	C₅	#1757	1	1	yes	characteristic, complemented, Sylow
169	C₂²	—	2	1	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	5	2	no	—
172	C₂²	—	10	1	no	—
173	C₂²	—	20	1	no	—
174	C₂²	—	20	1	no	—
175	C₂²	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	5	2	no	—
178	C₄	—	10	1	no	—
179	C₄	—	20	4	no	—
180	C₄	#4130	1	1	yes	characteristic
181	C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	4	1	no	—
184	C₂	—	10	2	no	—
185	C₂	—	20	1	no	—
186	C₂	C₄×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂×C₄²
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×M₁₆	C₄
socle	C₂×C₅	C₄×C₂²⋊C₄
prime radical	C₂×#709	C₂²
cosocle	C₂×C₄×D₅	C₂³
hypercenter	C₂×M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#1757
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. #65718	C₂	semidirect
5. #64894	C₂	semidirect
9. C₂×C₅×M₁₆	C₄	semidirect
10. C₂²×#168	C₄	semidirect
13. C₂×#676	C₄	semidirect
15. C₂×C₈×D₅	C₄	semidirect
17. #4013	C₄	semidirect
19. #4005	C₄	semidirect
87. C₂×M₁₆	AGL(1,𝔽₅)	semidirect
168. C₅	#1757	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4053	2
#65761	2
#65745	2
#65718	2
#64894	2
#1757	5