Finite Group Databasebeta

Subgroups — #10602808

Order 640.

264 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	#65521	C₂	1	2	yes	maximal
3	C₂×#3951	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#3949	C₂	1	1	yes	maximal, characteristic, complemented
5	#64774	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#41	C₂²	1	1	yes	characteristic
7	C₅×#131	—	2	2	no	—
8	C₅×#131	C₄	1	4	yes	complemented
9	C₂×C₅×M₁₆	—	2	1	no	—
10	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₈	—	2	1	no	—
12	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
14	C₂×#681	C₂²	1	2	yes	—
15	#3967	—	2	2	no	—
16	#3964	—	2	2	no	—
17	#3951	—	2	2	no	—
18	#3949	—	2	1	no	—
19	#3949	—	2	1	no	—
20	#1815	—	5	1	no	maximal, Sylow
21	C₅×#41	—	2	1	no	—
22	C₅×#41	—	2	1	no	—
23	C₅×#41	C₂×C₄	1	2	yes	—
24	C₅×#41	Q₈	1	2	yes	—
25	C₂×C₅×Q₈	D₄	1	1	yes	characteristic
26	C₂×C₅×D₄	—	2	1	no	—
27	C₂×C₅×D₄	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅	—	2	1	no	—
29	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅	—	2	2	no	—
31	C₅×M₁₆	—	2	1	no	—
32	C₅×M₁₆	—	2	4	no	—
33	C₅×M₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₅×C₈	—	2	1	no	—
35	C₂×C₅×C₈	—	2	4	no	—
36	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₅×C₈	D₄	1	2	yes	—
40	C₄⋊Dic₅	—	2	2	no	—
41	C₄⋊Dic₅	—	2	2	no	—
42	#681	—	2	2	no	—
43	#681	—	2	2	no	—
44	C₂×#168	—	2	2	no	—
45	C₂×#131	D₅	1	1	yes	characteristic
46	C₂×#108	—	5	1	no	—
47	C₂×#106	—	5	1	no	—
48	#417	—	5	2	no	—
49	#343	—	5	2	no	—
50	C₂³×C₅	—	2	1	no	—
51	C₂³×C₅	#41	1	1	yes	characteristic
52	C₅×Q₈	—	2	1	no	—
53	C₅×Q₈	C₄⋊C₄	1	2	yes	—
54	C₅×D₄	—	2	2	no	—
55	C₅×D₄	—	2	2	no	—
56	C₅×D₄	—	2	1	no	—
57	C₅×D₄	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₅	—	2	1	no	—
59	C₂×C₄×C₅	—	2	2	no	—
60	C₂×C₄×C₅	—	2	2	no	—
61	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅	#41	1	1	yes	characteristic
67	C₂×Dic₅	—	4	2	no	—
68	C₂×Dic₅	—	8	2	no	—
69	C₅×C₈	—	2	2	no	—
70	C₅×C₈	C₂²⋊C₄	1	2	yes	—
71	C₅×C₈	C₂²⋊C₄	1	2	yes	—
72	#168	—	4	2	no	—
73	C₂×#41	C₂×D₅	1	1	yes	characteristic
74	#131	—	2	2	no	—
75	#131	Dic₅	1	4	yes	complemented
76	C₂×M₁₆	—	2	1	no	—
77	C₂×M₁₆	—	5	2	no	—
78	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
79	C₂²×C₈	—	2	1	no	—
80	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
81	C₂×C₄⋊C₄	—	5	2	no	—
82	#108	—	10	2	no	—
83	#106	—	10	1	no	—
84	#106	—	10	1	no	—
85	#103	—	10	2	no	—
86	#102	—	10	2	no	—
87	C₂²×C₅	—	2	2	no	—
88	C₂²×C₅	—	2	2	no	—
89	C₂²×C₅	—	4	2	no	—
90	C₂²×C₅	#117	1	1	yes	characteristic
91	C₂²×C₅	#121	1	1	yes	characteristic
92	C₂²×C₅	C₂²⋊Q₈	1	1	yes	characteristic
93	C₄×C₅	—	2	2	no	—
94	C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₅	#121	1	1	yes	characteristic
97	C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
98	Dic₅	—	8	2	no	—
99	#41	—	2	1	no	—
100	#41	—	2	1	no	—
101	#41	Dic₁₀	1	2	yes	—
102	#41	C₂×Dic₅	1	2	yes	—
103	C₂×Q₈	D₂₀	1	1	yes	characteristic
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	D₂₀	1	1	yes	characteristic
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	10	2	no	—
108	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
109	M₁₆	—	2	1	no	—
110	M₁₆	—	2	4	no	—
111	M₁₆	—	10	2	no	—
112	M₁₆	—	10	2	no	—
113	M₁₆	C₂×Dic₅	1	2	yes	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	10	2	no	—
117	C₂×C₈	C₂×Dic₅	1	2	yes	—
118	C₂×C₈	C₂²⋊D₅	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊D₅	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊D₅	1	2	yes	—
121	C₄⋊C₄	—	10	2	no	—
122	C₄⋊C₄	—	10	2	no	—
123	C₂×C₅	—	2	1	no	—
124	C₂×C₅	—	2	1	no	—
125	C₂×C₅	—	4	2	no	—
126	C₂×C₅	#380	1	1	yes	characteristic
127	C₂×C₅	#471	1	2	yes	—
128	C₂³	—	2	1	no	—
129	C₂³	#709	1	1	yes	characteristic
130	Q₈	—	2	1	no	—
131	Q₈	C₄⋊Dic₅	1	2	yes	—
132	D₄	—	2	2	no	—
133	D₄	—	2	2	no	—
134	D₄	—	2	1	no	—
135	D₄	C₄⋊Dic₅	1	2	yes	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	20	2	no	—
140	C₂×C₄	—	40	2	no	—
141	C₂×C₄	C₂×Dic₁₀	1	1	yes	characteristic
142	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
143	C₂×C₄	#709	1	1	yes	characteristic
144	C₂×C₄	C₂²×Dic₅	1	1	yes	characteristic
145	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	20	2	no	—
149	C₈	C₂²⋊Dic₅	1	2	yes	—
150	C₈	C₂²⋊Dic₅	1	2	yes	—
151	C₅	#1815	1	1	yes	characteristic, complemented, Sylow
152	C₂²	—	2	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	2	no	—
155	C₂²	#4070	1	1	yes	characteristic
156	C₂²	#4072	1	1	yes	characteristic
157	C₂²	#4076	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	40	2	no	—
160	C₄	#4070	1	1	yes	characteristic
161	C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
162	C₄	#4076	1	1	yes	characteristic
163	C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	—	4	2	no	—
167	C₂	#65430	1	2	yes	—
168	C₂	#65501	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#4076
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₂×C₅×#131	C₂
socle	C₂²×C₅	#121
prime radical	C₂×C₅×#41	C₂²
cosocle	C₂²×C₄×C₅	C₂³
hypercenter	C₂×#131	D₅
hypocenter	C₅	#1815
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
3. C₂×#3951	C₂	semidirect
4. C₂×#3949	C₂	semidirect
5. #64774	C₂	semidirect
8. C₅×#131	C₄	semidirect
75. #131	Dic₅	semidirect
151. C₅	#1815	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#131	2
#65521	2
C₂×#3951	2
C₂×#3949	2
#64774	2
#1815	5