Finite Group Databasebeta

Subgroups — #10610989

Order 640.

458 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 177 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	#66082	C₂	1	1	yes	maximal, characteristic
2	#66079	C₂	1	2	yes	maximal, complemented
3	C₅×#430	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#3957	C₂	1	2	yes	maximal, complemented
5	M₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	#65184	C₂	1	4	yes	maximal, complemented
7	#65183	C₂	1	4	yes	maximal, complemented
8	C₂×#709	C₂²	1	2	yes	—
9	C₂×C₅×M₁₆	C₂²	1	2	yes	—
10	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×#681	C₂²	1	1	yes	characteristic
13	#4055	C₄	1	4	yes	complemented
14	#4054	C₄	1	4	yes	complemented
15	#4053	C₂²	1	8	yes	complemented
16	D₅×M₁₆	—	2	4	no	—
17	#4050	C₂²	1	4	yes	—
18	#4047	—	2	4	no	—
19	C₅×#108	C₂²	1	4	yes	complemented
20	#3957	C₂²	1	4	yes	complemented
21	#3957	C₂²	1	4	yes	complemented
22	#3947	C₂²	1	2	yes	complemented
23	C₈×Dic₅	C₂²	1	2	yes	complemented
24	#2956	—	5	1	no	maximal, Sylow
25	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₅	—	2	2	no	—
27	C₂²×Dic₅	—	2	1	no	—
28	#709	—	2	2	no	—
29	#709	—	2	2	no	—
30	#709	C₂×C₄	1	4	yes	—
31	#709	C₂³	1	4	yes	—
32	C₂×D₂₀	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₅	—	2	2	no	—
34	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
35	C₅×M₁₆	—	2	2	no	—
36	C₅×M₁₆	C₂×C₄	1	4	yes	—
37	C₅×M₁₆	C₂³	1	4	yes	—
38	C₂×C₅×C₈	—	2	2	no	—
39	C₂×C₅×C₈	C₂³	1	2	yes	—
40	C₄×Dic₅	—	2	1	no	—
41	C₄×Dic₅	C₂³	1	1	yes	characteristic
42	C₄×Dic₅	C₂³	1	1	yes	characteristic
43	#681	D₄	1	4	yes	—
44	C₂×#168	C₂³	1	2	yes	—
45	Dic₂₀	C₂×C₄	1	4	yes	complemented
46	D₄₀	C₂×C₄	1	4	yes	complemented
47	#677	C₂×C₄	1	8	yes	complemented
48	#676	—	2	8	no	—
49	C₈×D₅	—	2	8	no	—
50	#575	—	5	1	no	—
51	#568	—	5	2	no	—
52	#444	—	5	4	no	—
53	#443	—	5	4	no	—
54	#430	D₅	1	1	yes	characteristic, complemented
55	C₂×#104	—	5	2	no	—
56	C₄×M₁₆	—	5	1	no	—
57	C₂³×C₅	#41	1	1	yes	characteristic
58	C₂²×D₅	—	2	2	no	—
59	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅	#41	1	1	yes	characteristic
62	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
63	C₂²⋊D₅	—	2	4	no	—
64	C₂²⋊D₅	—	2	4	no	—
65	C₂×Dic₅	—	2	2	no	—
66	C₂×Dic₅	—	2	1	no	—
67	C₂×Dic₅	—	2	1	no	—
68	C₂×Dic₅	—	4	2	no	—
69	D₂₀	—	2	2	no	—
70	D₂₀	C₂²×C₄	1	4	yes	—
71	C₄×D₅	—	2	4	no	—
72	C₄×D₅	—	2	4	no	—
73	Dic₁₀	—	2	2	no	—
74	Dic₁₀	C₂²×C₄	1	4	yes	—
75	C₅×C₈	—	2	4	no	—
76	C₅×C₈	C₂²×C₄	1	4	yes	—
77	#168	C₂×D₄	1	4	yes	—
78	C₂×#41	—	5	2	no	—
79	#137	—	5	4	no	—
80	AGL(1,ℤ/8)	—	5	4	no	—
81	#135	—	5	4	no	—
82	#131	—	5	8	no	—
83	#131	—	10	4	no	—
84	C₂×M₁₆	—	5	1	no	—
85	C₂×M₁₆	—	10	4	no	—
86	C₂×M₁₆	C₂×D₅	1	2	yes	—
87	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
88	C₂×C₄²	—	5	1	no	—
89	#108	C₂×D₅	1	4	yes	complemented
90	#104	—	5	4	no	—
91	#104	—	5	4	no	—
92	#97	—	5	2	no	—
93	C₄×C₈	—	5	2	no	—
94	C₂²×C₅	—	4	1	no	—
95	C₂²×C₅	C₄×D₄	1	2	yes	—
96	C₂²×C₅	C₂×#41	1	1	yes	characteristic
97	C₂×D₅	—	2	4	no	—
98	C₂×D₅	—	8	2	no	—
99	C₄×C₅	C₄×D₄	1	2	yes	—
100	C₄×C₅	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
102	Dic₅	—	2	2	no	—
103	Dic₅	—	2	4	no	—
104	Dic₅	—	4	1	no	—
105	#41	—	5	4	no	—
106	#41	—	5	4	no	—
107	#41	—	10	2	no	—
108	#41	—	10	2	no	—
109	C₂×Q₈	—	5	2	no	—
110	C₂×D₄	—	5	2	no	—
111	C₂×D₄	—	10	2	no	—
112	C₂²×C₄	—	10	2	no	—
113	C₂²×C₄	—	10	1	no	—
114	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
115	Q₁₆	—	5	4	no	—
116	SD₁₆	—	5	8	no	—
117	D₈	—	5	4	no	—
118	M₁₆	—	2	2	no	—
119	M₁₆	—	5	4	no	—
120	M₁₆	—	10	8	no	—
121	M₁₆	C₄×D₅	1	4	yes	—
122	M₁₆	C₂²×D₅	1	4	yes	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	5	2	no	—
125	C₂×C₈	—	10	8	no	—
126	C₂×C₈	C₂²×D₅	1	2	yes	—
127	C₄²	—	5	1	no	—
128	C₄²	—	5	1	no	—
129	C₄²	—	10	1	no	—
130	C₂×C₅	—	2	1	no	—
131	C₂×C₅	—	2	2	no	—
132	C₂×C₅	C₂×C₄×D₄	1	1	yes	characteristic
133	D₅	—	4	4	no	—
134	C₂³	—	10	2	no	—
135	C₂³	#713	1	1	yes	characteristic
136	Q₈	—	5	4	no	—
137	Q₈	—	10	2	no	—
138	D₄	—	5	4	no	—
139	D₄	—	10	4	no	—
140	D₄	—	10	4	no	—
141	D₄	—	10	2	no	—
142	C₂×C₄	—	10	1	no	—
143	C₂×C₄	—	10	2	no	—
144	C₂×C₄	—	10	4	no	—
145	C₂×C₄	—	10	4	no	—
146	C₂×C₄	—	10	1	no	—
147	C₂×C₄	—	20	2	no	—
148	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
149	C₂×C₄	D₄×D₅	1	2	yes	—
150	C₂×C₄	#713	1	1	yes	characteristic
151	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
152	C₈	—	2	4	no	—
153	C₈	—	5	4	no	—
154	C₈	C₂×C₄×D₅	1	4	yes	—
155	C₅	#2956	1	1	yes	characteristic, complemented, Sylow
156	C₂²	—	4	1	no	—
157	C₂²	—	10	4	no	—
158	C₂²	—	40	2	no	—
159	C₂²	#4039	1	2	yes	—
160	C₂²	C₂×#713	1	1	yes	characteristic
161	C₄	—	10	2	no	—
162	C₄	—	10	4	no	—
163	C₄	—	20	1	no	—
164	C₄	#4039	1	2	yes	—
165	C₄	C₂²×C₄×D₅	1	1	yes	characteristic
166	C₄	C₂×D₄×D₅	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	2	no	—
169	C₂	—	20	4	no	—
170	C₂	C₂×#4039	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₅
commutator	C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅
Fitting	C₅×#430	C₂
socle	C₂×C₅	C₂×C₄×D₄
prime radical	#66082	C₂
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#430	D₅
hypocenter	C₅	#2956
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

Complements

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

N	H ≅ G/N	type
2. #66079	C₂	semidirect
3. C₅×#430	C₂	semidirect
4. C₂×#3957	C₂	semidirect
5. M₁₆×Dic₅	C₂	semidirect
6. #65184	C₂	semidirect
7. #65183	C₂	semidirect
13. #4055	C₄	semidirect
14. #4054	C₄	semidirect
15. #4053	C₂²	semidirect
19. C₅×#108	C₂²	semidirect
20. #3957	C₂²	semidirect
21. #3957	C₂²	semidirect
22. #3947	C₂²	semidirect
23. C₈×Dic₅	C₂²	semidirect
45. Dic₂₀	C₂×C₄	semidirect
46. D₄₀	C₂×C₄	semidirect
47. #677	C₂×C₄	semidirect
54. #430	D₅	semidirect
89. #108	C₂×D₅	semidirect
155. C₅	#2956	semidirect

Maximal subgroups

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

subgroup	index
#66082	2
#66079	2
C₅×#430	2
C₂×#3957	2
M₁₆×Dic₅	2
#65184	2
#65183	2
#2956	5