Finite Group Databasebeta

← #72218244

Subgroups — #72218244

Order 3360.

272 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(C5⋊AGL(1,𝔽7))C211yesmaximal, characteristic, complemented
2ΓL(1,𝔽8)×#709C211yesmaximal, characteristic, complemented
3C5×#1136⋊C3C211yesmaximal, characteristic, complemented
4#10587849⋊C3C212yesmaximal, complemented
5#10587843⋊C3C212yesmaximal, complemented
6#11764804C311yesmaximal, characteristic, complemented, Hall
7C22×C5⋊AGL(1,𝔽7)—21no—
8C2×C4×C5×ΓL(1,𝔽8)C2211yescharacteristic
9C4⋊(C5⋊AGL(1,𝔽7))—21no—
10C4⋊(C5⋊AGL(1,𝔽7))C2212yes—
11ΓL(1,𝔽8)×C22⋊D5—21no—
12ΓL(1,𝔽8)×D20C2211yescharacteristic, complemented
13C4×D5×ΓL(1,𝔽8)—21no—
14ΓL(1,𝔽8)×Dic10C2211yescharacteristic, complemented
15C5×ΓL(1,𝔽8)⋊C8C2212yescomplemented
16#6512⋊C3—51nomaximal, Hall
17C2×D140C2×C311yescharacteristic, complemented
18C7×#709C2×C311yescharacteristic, complemented
19C5×#1136C2×C311yescharacteristic, complemented
20#10587849C2×C312yescomplemented
21#10587843C2×C312yescomplemented
22C3×#4054—71nomaximal, Hall
23C22×C5×ΓL(1,𝔽8)D411yescharacteristic
24C2×C5⋊AGL(1,𝔽7)—22no—
25C2×C5⋊AGL(1,𝔽7)—42no—
26C2×D5×ΓL(1,𝔽8)—21no—
27C4×C5×ΓL(1,𝔽8)D411yescharacteristic
28C4×C5×ΓL(1,𝔽8)C2311yescharacteristic
29Dic5×ΓL(1,𝔽8)—21no—
30#41×ΓL(1,𝔽8)—51no—
31C2×C4⋊AGL(1,𝔽7)—51no—
32#1143⋊C3—52no—
33#1141⋊C3—52no—
34#1136⋊C3D511yescharacteristic, complemented
35C22×D35—21no—
36C2×C4×C5×C7C22×C311yescharacteristic
37D140—21no—
38D140C22×C312yes—
39C7×C22⋊D5—21no—
40C7×D20C22×C311yescharacteristic, complemented
41C4×C7×D5—21no—
42C7×Dic10C22×C311yescharacteristic, complemented
43C5×C7⋊C8C22×C312yescomplemented
44C3×#709—71no—
45C2×C3×D20—71no—
46C3×C5×M16—71no—
47C3×D40—72no—
48C3×#677—72no—
49#6512—51noHall
50C2×C5×ΓL(1,𝔽8)—21no—
51C2×C5×ΓL(1,𝔽8)C2×D411yescharacteristic
52C5⋊AGL(1,𝔽7)—42no—
53D5×ΓL(1,𝔽8)—41no—
54C22×AGL(1,𝔽7)—101no—
55Q8×ΓL(1,𝔽8)—51no—
56D4×ΓL(1,𝔽8)—51no—
57D4×ΓL(1,𝔽8)—101no—
58C2×C4×ΓL(1,𝔽8)—101no—
59C2×C4×ΓL(1,𝔽8)C2×D511yescharacteristic
60C4⋊AGL(1,𝔽7)—52no—
61C4⋊AGL(1,𝔽7)—101no—
62ΓL(1,𝔽8)⋊C8C2×D512yescomplemented
63#4054—71noHall
64C22×C5×C7C3×D411yescharacteristic
65C2×D35—22no—
66C2×D35—42no—
67C2×C7×D5—21no—
68C4×C5×C7C3×D411yescharacteristic
69C4×C5×C7C23×C311yescharacteristic
70C7×Dic5—21no—
71C22×C3×D5—141no—
72C2×C3×C4×C5—71no—
73C3×C22⋊D5—141no—
74C3×D20—72no—
75C3×D20—71no—
76C3×D20—141no—
77C3×C4×D5—141no—
78C3×Dic10—71no—
79C3×C5×C8—72no—
80C7×#41—51no—
81C2×D28—51no—
82#1143—52no—
83#1141—52no—
84#1136C3×D511yescharacteristic, complemented
85C5×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
86C3×AGL(1,ℤ/8)—351noHall
87C22×ΓL(1,𝔽8)—101no—
88C22×ΓL(1,𝔽8)D2011yescharacteristic
89C2×AGL(1,𝔽7)—102no—
90C2×AGL(1,𝔽7)—202no—
91C4×ΓL(1,𝔽8)—101no—
92C4×ΓL(1,𝔽8)D2011yescharacteristic
93C4×ΓL(1,𝔽8)C22×D511yescharacteristic
94#709AGL(1,𝔽7)11yescharacteristic, complemented
95C2×D20—71no—
96C5×M16—71no—
97D40—72no—
98#677—72no—
99C2×C5×C7—21no—
100C2×C5×C7C2×C3×D411yescharacteristic
101D35—42no—
102C7×D5—41no—
103C22×C3×C5—71no—
104C2×C3×D5—141no—
105C2×C3×D5—142no—
106C2×C3×D5—282no—
107C3×C4×C5—71no—
108C3×C4×C5—71no—
109C3×Dic5—141no—
110C22×D7—101no—
111C7×Q8—51no—
112C7×D4—51no—
113C7×D4—101no—
114C2×C4×C7—101no—
115C2×C4×C7C2×C3×D511yescharacteristic
116D28—52no—
117D28—101no—
118C7⋊C8C2×C3×D512yescomplemented
119C3×#41—351no—
120C2×C3×D4—351no—
121C3×SD16—352no—
122C3×D8—352no—
123C3×M16—71no—
124C2×ΓL(1,𝔽8)—21no—
125C2×ΓL(1,𝔽8)—201no—
126C2×ΓL(1,𝔽8)C2×D2011yescharacteristic
127AGL(1,𝔽7)—202no—
128C22×D5—141no—
129C2×C4×C5C2×AGL(1,𝔽7)11yescharacteristic
130C22⋊D5—21no—
131D20—72no—
132D20—141no—
133D20C2×AGL(1,𝔽7)11yescharacteristic, complemented
134C4×D5—21no—
135Dic10C2×AGL(1,𝔽7)11yescharacteristic, complemented
136C5×C8—72no—
137C5×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138AGL(1,ℤ/8)—351noSylow
139C2×C3×C5—71no—
140C2×C3×C5—141no—
141C3×D5—281no—
142C3×D5—282no—
143C22×C7—101no—
144C22×C7C3×D2011yescharacteristic
145C2×D7—102no—
146C2×D7—202no—
147C4×C7—101no—
148C4×C7C3×D2011yescharacteristic
149C4×C7C22×C3×D511yescharacteristic
150C23×C3—701no—
151C3×Q8—351no—
152C3×D4—351no—
153C3×D4—352no—
154C3×D4—701no—
155C3×D4—701no—
156C2×C3×C4—71no—
157C2×C3×C4—701no—
158C3×C8—72no—
159ΓL(1,𝔽8)#405411yescharacteristic, complemented, Hall
160C22×C5(C22⋊D7)⋊C311yescharacteristic
161C2×D5—21no—
162C2×D5—142no—
163C2×D5—282no—
164C4×C5(C22⋊D7)⋊C311yescharacteristic
165C4×C5C22×AGL(1,𝔽7)11yescharacteristic
166Dic5—21no—
167#41—51no—
168C2×D4—351no—
169SD16—352no—
170D8—352no—
171M16—71no—
172C3×C5—71noHall
173C2×C7—21no—
174C2×C7—201no—
175C2×C7C2×C3×D2011yescharacteristic
176D7—202no—
177C22×C3—71no—
178C22×C3—701no—
179C22×C3—702no—
180C22×C3—1402no—
181C3×C4—71no—
182C3×C4—71no—
183C3×C4—701no—
184C2×C5—21no—
185C2×C5C2×(C22⋊D7)⋊C311yescharacteristic
186D5—41no—
187D5—282no—
188C23—701no—
189Q8—51no—
190D4—51no—
191D4—101no—
192D4—352no—
193D4—701no—
194C2×C4—101no—
195C2×C4D5×AGL(1,𝔽7)11yescharacteristic
196C8—72no—
197C7C3×#405411yescharacteristic, complemented, Sylow
198C2×C3—71no—
199C2×C3—141no—
200C2×C3—1401no—
201C2×C3—1402no—
202C5#6512⋊C311yescharacteristic, complemented, Sylow
203C22—101no—
204C22—702no—
205C22—1402no—
206C22#63307⋊C311yescharacteristic
207C4—101no—
208C4#63307⋊C311yescharacteristic
209C4C2×D5×AGL(1,𝔽7)11yescharacteristic
210C3—71noSylow
211C2—21no—
212C2—201no—
213C2—1402no—
214C2C2×#63307⋊C311yescharacteristic
subgroupG/H
centerC2C2×#63307⋊C3
commutator, cosocleC4×C5×C7C23×C3
FrattiniC4C2×D5×AGL(1,𝔽7)
FittingC2×C4×C5×C7C22×C3
socleC2×C5×C7C2×C3×D4
hypercenterC2×C4D5×AGL(1,𝔽7)
hypocenterC5×C7C3×AGL(1,ℤ/8)
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊(C5⋊AGL(1,𝔽7))C2semidirect
2. ΓL(1,𝔽8)×#709C2semidirect
3. C5×#1136⋊C3C2semidirect
4. #10587849⋊C3C2semidirect
5. #10587843⋊C3C2semidirect
6. #11764804C3semidirect
12. ΓL(1,𝔽8)×D20C22semidirect
14. ΓL(1,𝔽8)×Dic10C22semidirect
15. C5×ΓL(1,𝔽8)⋊C8C22semidirect
17. C2×D140C2×C3semidirect
18. C7×#709C2×C3semidirect
19. C5×#1136C2×C3semidirect
20. #10587849C2×C3semidirect
21. #10587843C2×C3semidirect
34. #1136⋊C3D5semidirect
40. C7×D20C22×C3semidirect
42. C7×Dic10C22×C3semidirect
43. C5×C7⋊C8C22×C3semidirect
62. ΓL(1,𝔽8)⋊C8C2×D5semidirect
84. #1136C3×D5semidirect
85. C5×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
94. #709AGL(1,𝔽7)semidirect
118. C7⋊C8C2×C3×D5semidirect
133. D20C2×AGL(1,𝔽7)semidirect
135. Dic10C2×AGL(1,𝔽7)semidirect
137. C5×C7C3×AGL(1,ℤ/8)semidirect
159. ΓL(1,𝔽8)#4054semidirect
197. C7C3×#4054semidirect
202. C5#6512⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C5⋊AGL(1,𝔽7))2
ΓL(1,𝔽8)×#7092
C5×#1136⋊C32
#10587849⋊C32
#10587843⋊C32
#117648043
#6512⋊C35
C3×#40547