Finite Group Databasebeta

Subgroups — #10604855

Order 640.

228 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 64 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	D₅×#135	C₂	1	1	yes	maximal, characteristic
2	#65721	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₅⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#64908	C₂	1	2	yes	maximal, complemented
5	#64904	C₂	1	2	yes	maximal, complemented
6	D₅×#41	—	2	1	no	—
7	#4130	—	2	1	no	—
8	C₅×#135	C₄	1	1	yes	characteristic, complemented
9	#4096	—	2	1	no	—
10	#4065	C₄	1	1	yes	characteristic, complemented
11	D₅×Q₁₆	C₂²	1	1	yes	characteristic
12	#4062	—	2	1	no	—
13	D₅×SD₁₆	—	2	1	no	—
14	#4058	C₄	1	1	yes	characteristic, complemented
15	D₅×D₈	C₂²	1	1	yes	characteristic
16	#4050	C₄	1	1	yes	characteristic, complemented
17	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
18	C₂×#674	—	2	1	no	—
19	#3994	C₂²	1	2	yes	—
20	#3993	—	2	1	no	—
21	D₅⋊C₁₆	—	2	1	no	—
22	D₅⋊C₁₆	C₂²	1	2	yes	—
23	#2138	—	5	1	no	maximal, Sylow
24	C₅×#41	—	2	1	no	—
25	#713	—	2	1	no	—
26	Q₈×D₅	—	2	1	no	—
27	#711	—	2	1	no	—
28	#711	—	4	1	no	—
29	D₄×D₅	—	2	1	no	—
30	D₄×D₅	—	4	1	no	—
31	#709	—	2	1	no	—
32	#709	—	4	1	no	—
33	C₂×C₄×D₅	—	4	1	no	—
34	C₂×C₄×D₅	D₄	1	1	yes	characteristic
35	C₂²⋊AGL(1,𝔽₅)	—	4	1	no	—
36	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
37	C₄×AGL(1,𝔽₅)	—	4	1	no	—
38	C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
39	C₅×SD₁₆	—	2	1	no	—
40	C₅×D₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	#689	—	2	1	no	—
43	#688	—	2	1	no	—
44	#687	—	2	1	no	—
45	#686	—	2	1	no	—
46	C₂×#168	D₄	1	1	yes	characteristic
47	Dic₂₀	C₂×C₄	1	1	yes	characteristic
48	D₄₀	C₂×C₄	1	1	yes	characteristic
49	#677	—	2	1	no	—
50	C₈×D₅	C₂×C₄	1	1	yes	characteristic
51	C₈×D₅	D₄	1	1	yes	characteristic
52	C₈×D₅	D₄	1	1	yes	characteristic
53	C₈×D₅	C₂³	1	1	yes	characteristic
54	#674	—	2	2	no	—
55	C₂×#135	—	5	1	no	—
56	C₂²×C₁₆	—	5	1	no	—
57	#427	—	5	1	no	—
58	#358	—	5	2	no	—
59	#357	—	5	2	no	—
60	C₂²×D₅	—	4	1	no	—
61	C₂²×D₅	D₈	1	1	yes	characteristic
62	C₂×AGL(1,𝔽₅)	—	4	2	no	—
63	C₅×Q₈	—	2	1	no	—
64	C₅×D₄	—	2	1	no	—
65	C₅×D₄	—	4	1	no	—
66	C₂×C₄×C₅	—	4	1	no	—
67	C₂×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²⋊D₅	—	4	1	no	—
69	C₂²⋊D₅	—	4	1	no	—
70	C₂²⋊D₅	—	4	1	no	—
71	C₂×Dic₅	—	4	1	no	—
72	C₂×Dic₅	SD₁₆	1	1	yes	characteristic
73	D₂₀	—	2	1	no	—
74	D₂₀	—	4	1	no	—
75	C₄×D₅	—	4	1	no	—
76	C₄×D₅	—	4	1	no	—
77	C₄×D₅	—	4	1	no	—
78	C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₄×D₅	D₈	1	1	yes	characteristic
80	C₄×D₅	SD₁₆	1	1	yes	characteristic
81	C₄×D₅	C₂×D₄	1	1	yes	characteristic
82	Dic₁₀	—	2	1	no	—
83	Dic₁₀	—	4	1	no	—
84	C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	#168	C₂²⋊C₄	1	1	yes	characteristic
87	#168	C₂×D₄	1	1	yes	characteristic
88	C₂×#41	—	10	1	no	—
89	#135	—	5	1	no	—
90	#135	—	5	1	no	—
91	#135	—	5	1	no	—
92	#135	—	10	1	no	—
93	#135	—	10	1	no	—
94	#135	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
95	C₂×Q₁₆	—	5	1	no	—
96	C₂×SD₁₆	—	10	1	no	—
97	C₂×D₈	—	5	1	no	—
98	C₂²×C₈	—	5	1	no	—
99	#117	—	10	1	no	—
100	C₂×C₁₆	—	5	2	no	—
101	C₂×C₁₆	—	10	1	no	—
102	C₂×C₁₆	—	10	1	no	—
103	#107	—	5	2	no	—
104	#106	—	10	1	no	—
105	C₂²×C₅	—	4	1	no	—
106	C₂²×C₅	#102	1	1	yes	characteristic
107	C₂×D₅	—	2	2	no	—
108	C₂×D₅	—	4	1	no	—
109	C₂×D₅	—	8	2	no	—
110	C₂×D₅	#102	1	1	yes	characteristic
111	C₂×D₅	C₂×D₈	1	1	yes	characteristic
112	AGL(1,𝔽₅)	—	8	2	no	—
113	C₄×C₅	—	4	1	no	—
114	C₄×C₅	#102	1	1	yes	characteristic
115	C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	Dic₅	—	4	1	no	—
117	Dic₅	#102	1	1	yes	characteristic
118	Dic₅	C₂×SD₁₆	1	1	yes	characteristic
119	#41	—	2	1	no	—
120	#41	—	10	1	no	—
121	#41	—	10	1	no	—
122	#41	—	10	1	no	—
123	#41	—	20	1	no	—
124	#41	—	20	1	no	—
125	C₂×Q₈	—	10	1	no	—
126	C₂×D₄	—	10	1	no	—
127	C₂×D₄	—	20	1	no	—
128	C₂²×C₄	—	5	1	no	—
129	C₂²×C₄	—	20	1	no	—
130	Q₁₆	—	5	1	no	—
131	Q₁₆	—	10	1	no	—
132	Q₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
133	SD₁₆	—	2	1	no	—
134	SD₁₆	—	10	1	no	—
135	SD₁₆	—	10	1	no	—
136	SD₁₆	—	10	1	no	—
137	D₈	—	5	1	no	—
138	D₈	—	10	1	no	—
139	D₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₂×C₈	—	5	1	no	—
141	C₂×C₈	—	5	1	no	—
142	C₂×C₈	—	5	1	no	—
143	C₂×C₈	—	5	1	no	—
144	C₂×C₈	—	5	1	no	—
145	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
146	C₄⋊C₄	—	10	2	no	—
147	C₂²⋊C₄	—	20	1	no	—
148	C₄²	—	20	1	no	—
149	C₁₆	—	10	2	no	—
150	C₂×C₅	—	2	1	no	—
151	C₂×C₅	—	8	1	no	—
152	C₂×C₅	C₂×#102	1	1	yes	characteristic
153	D₅	—	2	1	no	—
154	D₅	—	8	1	no	—
155	D₅	#508	1	2	yes	—
156	C₂³	—	5	1	no	—
157	C₂³	—	20	1	no	—
158	Q₈	—	2	1	no	—
159	Q₈	—	10	1	no	—
160	Q₈	—	20	1	no	—
161	D₄	—	2	1	no	—
162	D₄	—	4	1	no	—
163	D₄	—	10	1	no	—
164	D₄	—	20	1	no	—
165	D₄	—	20	1	no	—
166	D₄	—	20	1	no	—
167	D₄	—	20	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	5	1	no	—
170	C₂×C₄	—	5	1	no	—
171	C₂×C₄	—	5	1	no	—
172	C₂×C₄	—	5	1	no	—
173	C₂×C₄	—	5	1	no	—
174	C₂×C₄	—	20	1	no	—
175	C₂×C₄	—	20	1	no	—
176	C₂×C₄	—	20	1	no	—
177	C₂×C₄	—	20	1	no	—
178	C₂×C₄	—	20	2	no	—
179	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
180	C₈	—	5	1	no	—
181	C₈	—	5	1	no	—
182	C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
183	C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
184	C₅	#2138	1	1	yes	characteristic, complemented, Sylow
185	C₂²	—	4	1	no	—
186	C₂²	—	5	1	no	—
187	C₂²	—	5	1	no	—
188	C₂²	—	10	2	no	—
189	C₂²	—	20	1	no	—
190	C₂²	—	40	2	no	—
191	C₂²	#4007	1	1	yes	characteristic
192	C₄	—	4	1	no	—
193	C₄	—	5	1	no	—
194	C₄	—	5	1	no	—
195	C₄	—	20	1	no	—
196	C₄	—	40	2	no	—
197	C₄	#4007	1	1	yes	characteristic
198	C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	5	2	no	—
201	C₂	—	8	1	no	—
202	C₂	—	10	1	no	—
203	C₂	—	40	1	no	—
204	C₂	C₂×#4007	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#4007
commutator	C₅×C₈	C₂²×C₄
Frattini	C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#135	C₄
socle	C₂×C₅	C₂×#102
prime radical	D₅×#135	C₂
cosocle	C₈×D₅	C₂³
hypercenter	#135	AGL(1,𝔽₅)
hypocenter	C₅	#2138
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
2. #65721	C₂	semidirect
3. C₂×D₅⋊C₁₆	C₂	semidirect
4. #64908	C₂	semidirect
5. #64904	C₂	semidirect
8. C₅×#135	C₄	semidirect
10. #4065	C₄	semidirect
14. #4058	C₄	semidirect
16. #4050	C₄	semidirect
94. #135	AGL(1,𝔽₅)	semidirect
184. C₅	#2138	semidirect

Maximal subgroups

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

subgroup	index
D₅×#135	2
#65721	2
C₂×D₅⋊C₁₆	2
#64908	2
#64904	2
#2138	5