Finite Group Databasebeta

Subgroups — #10604858

Order 640.

220 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 62 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	C₂×#3996	C₂	1	1	yes	maximal, characteristic, complemented
3	D₅⋊M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#64909	C₂	1	2	yes	maximal, complemented
5	#64905	C₂	1	2	yes	maximal, complemented
6	D₅×#41	—	2	1	no	—
7	C₂×#700	—	2	1	no	—
8	C₅×#135	C₄	1	1	yes	characteristic
9	#4096	—	2	1	no	—
10	#4065	C₂²	1	1	yes	characteristic
11	D₅×Q₁₆	C₄	1	1	yes	characteristic
12	#4062	—	2	1	no	—
13	D₅×SD₁₆	—	2	1	no	—
14	#4058	C₂²	1	1	yes	characteristic
15	D₅×D₈	C₄	1	1	yes	characteristic
16	#4050	C₄	1	1	yes	characteristic
17	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
18	#3998	—	2	1	no	—
19	#3996	—	2	1	no	—
20	#3996	C₂²	1	2	yes	complemented
21	#3990	—	2	1	no	—
22	D₅⋊C₁₆	C₂²	1	2	yes	complemented
23	#2143	—	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	C₂×#170	—	4	1	no	—
36	#700	—	2	2	no	—
37	#700	—	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	C₂×M₃₂	—	5	1	no	—
57	C₂×#108	—	5	1	no	—
58	#359	—	5	2	no	—
59	#359	—	5	2	no	—
60	C₂²×D₅	—	4	1	no	—
61	C₂²×D₅	SD₁₆	1	1	yes	characteristic
62	C₅×Q₈	—	2	1	no	—
63	C₅×D₄	—	2	1	no	—
64	C₅×D₄	—	4	1	no	—
65	C₂×C₄×C₅	—	4	1	no	—
66	C₂×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²⋊D₅	—	4	1	no	—
68	C₂²⋊D₅	—	4	1	no	—
69	C₂²⋊D₅	—	4	1	no	—
70	C₂×Dic₅	—	4	1	no	—
71	C₂×Dic₅	D₈	1	1	yes	characteristic
72	D₂₀	—	2	1	no	—
73	D₂₀	—	4	1	no	—
74	C₄×D₅	—	4	1	no	—
75	C₄×D₅	—	4	1	no	—
76	C₄×D₅	—	4	1	no	—
77	C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₄×D₅	D₈	1	1	yes	characteristic
79	C₄×D₅	SD₁₆	1	1	yes	characteristic
80	C₄×D₅	C₂×D₄	1	1	yes	characteristic
81	Dic₁₀	—	2	1	no	—
82	Dic₁₀	—	4	1	no	—
83	#170	—	4	2	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
95	C₂×Q₁₆	—	5	1	no	—
96	C₂×SD₁₆	—	10	1	no	—
97	C₂×D₈	—	5	1	no	—
98	C₂×M₁₆	—	10	1	no	—
99	C₂²×C₈	—	5	1	no	—
100	M₃₂	—	10	1	no	—
101	M₃₂	—	10	1	no	—
102	C₂×C₁₆	—	5	2	no	—
103	#108	—	5	2	no	—
104	#108	—	10	1	no	—
105	C₂²×C₅	—	4	1	no	—
106	C₂²×C₅	#102	1	1	yes	characteristic
107	C₂×D₅	—	4	1	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×SD₁₆	1	1	yes	characteristic
112	C₄×C₅	—	4	1	no	—
113	C₄×C₅	#102	1	1	yes	characteristic
114	C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	Dic₅	—	4	1	no	—
116	Dic₅	#102	1	1	yes	characteristic
117	Dic₅	C₂×D₈	1	1	yes	characteristic
118	#41	—	2	1	no	—
119	#41	—	10	1	no	—
120	#41	—	10	1	no	—
121	#41	—	10	1	no	—
122	#41	—	20	1	no	—
123	#41	—	20	1	no	—
124	C₂×Q₈	—	10	1	no	—
125	C₂×D₄	—	10	1	no	—
126	C₂×D₄	—	20	1	no	—
127	C₂²×C₄	—	5	1	no	—
128	C₂²×C₄	—	20	1	no	—
129	Q₁₆	—	5	1	no	—
130	Q₁₆	—	10	1	no	—
131	Q₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
132	SD₁₆	—	2	1	no	—
133	SD₁₆	—	10	1	no	—
134	SD₁₆	—	10	1	no	—
135	SD₁₆	—	10	1	no	—
136	D₈	—	5	1	no	—
137	D₈	—	10	1	no	—
138	D₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
139	M₁₆	—	10	2	no	—
140	M₁₆	—	20	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	—	5	1	no	—
146	C₂×C₈	—	20	1	no	—
147	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
148	C₁₆	—	10	2	no	—
149	C₂×C₅	—	2	1	no	—
150	C₂×C₅	—	8	1	no	—
151	C₂×C₅	C₂×#102	1	1	yes	characteristic
152	D₅	—	2	1	no	—
153	D₅	—	2	1	no	—
154	D₅	—	8	1	no	—
155	C₂³	—	5	1	no	—
156	C₂³	—	20	1	no	—
157	Q₈	—	2	1	no	—
158	Q₈	—	10	1	no	—
159	Q₈	—	20	1	no	—
160	D₄	—	2	1	no	—
161	D₄	—	4	1	no	—
162	D₄	—	10	1	no	—
163	D₄	—	20	1	no	—
164	D₄	—	20	1	no	—
165	D₄	—	20	1	no	—
166	D₄	—	20	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	5	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	—	20	1	no	—
174	C₂×C₄	—	20	1	no	—
175	C₂×C₄	—	20	1	no	—
176	C₂×C₄	—	20	1	no	—
177	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
178	C₈	—	5	1	no	—
179	C₈	—	5	1	no	—
180	C₈	—	20	2	no	—
181	C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
183	C₅	#2143	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	4	1	no	—
185	C₂²	—	5	1	no	—
186	C₂²	—	5	1	no	—
187	C₂²	—	20	1	no	—
188	C₂²	—	20	1	no	—
189	C₂²	—	40	2	no	—
190	C₂²	#4007	1	1	yes	characteristic
191	C₄	—	4	1	no	—
192	C₄	—	5	1	no	—
193	C₄	—	5	1	no	—
194	C₄	—	20	1	no	—
195	C₄	#4007	1	1	yes	characteristic
196	C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	8	1	no	—
199	C₂	—	10	1	no	—
200	C₂	—	10	1	no	—
201	C₂	—	40	1	no	—
202	C₂	C₂×#4007	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#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	#2143
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
2. C₂×#3996	C₂	semidirect
3. D₅⋊M₃₂	C₂	semidirect
4. #64909	C₂	semidirect
5. #64905	C₂	semidirect
20. #3996	C₂²	semidirect
22. D₅⋊C₁₆	C₂²	semidirect
183. C₅	#2143	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
C₂×#3996	2
D₅⋊M₃₂	2
#64909	2
#64905	2
#2143	5