Finite Group Databasebeta

← #28298560

Subgroups — #28298560

Order 13568.

248 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 58 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
1#20290599C211yesmaximal, characteristic
2#20290597C211yesmaximal, characteristic
3#20276328C211yesmaximal, characteristic
4D53⋊#108—21no—
5#16811633—21no—
6#16811609—21no—
7#16811602—21no—
8#16811492C412yescomplemented
9#16811491C412yes—
10C2×#13133223C2211yescharacteristic
11C2×#13133157C411yescharacteristic, complemented
12#16810402—21no—
13#16810402—21no—
14C22×C16×C53C411yescharacteristic, complemented
15D53⋊M16—22no—
16#13133296—22no—
17#13133288—41no—
18#13133287—41no—
19#13133286—41no—
20#13133285—41no—
21#13133278—41no—
22#13133277—41no—
23#13133276—41no—
24#13133275—41no—
25C2×#11719945D411yescharacteristic
26#13133223—41no—
27#13133223—41no—
28C2×#11719917Q811yescharacteristic
29C2×#11719912C2×C411yescharacteristic
30C2×#11719912C2×C411yescharacteristic
31C2×C8×D53D411yescharacteristic
32C2×C8×D53D411yescharacteristic
33#13133157—41no—
34C2×#11719908—21no—
35C22×C8×C53C2×C411yescharacteristic
36C2×C16×C53—21no—
37C2×C16×C53—41no—
38C22×C4×C53C4⋊C411yescharacteristic
39C2×C22⋊D53D811yescharacteristic
40C2×C22⋊D53Q1611yescharacteristic
41#11719945—41no—
42#11719945—41no—
43C2×D212SD1611yescharacteristic
44C2×C4×D53C22⋊C411yescharacteristic
45C2×C4×D53C22⋊C411yescharacteristic
46C2×Dic106SD1611yescharacteristic
47C22⋊#6157—42no—
48#11719940—42no—
49C4⋊#6157—42no—
50C4×#6157—42no—
51#11719936—42no—
52D53⋊C8—42no—
53C2×C8×C53—21no—
54C2×C8×C53—21no—
55C2×C8×C53C4211yescharacteristic
56C2×C8×C53C22⋊C411yescharacteristic
57#11719917—41no—
58C2×#88322C4⋊C411yescharacteristic
59C2×#88322C4⋊C411yescharacteristic
60#11719912—41no—
61#11719912—41no—
62C8×D53—21no—
63C8×D53—21no—
64C8×D53—21no—
65C8×D53—21no—
66C16×C53—41no—
67#11719908—41no—
68C23×C53#10711yescharacteristic
69C22×D53#10211yescharacteristic
70C22×D53#10311yescharacteristic
71C2×#6157—82no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53#9511yescharacteristic
75C2×C4×C53#10611yescharacteristic
76C22⋊D53—41no—
77C22⋊D53—41no—
78C2×Dic53#10211yescharacteristic
79C2×Dic53#10311yescharacteristic
80D212—41no—
81C4×D53—21no—
82C4×D53—21no—
83C4×D53—21no—
84C4×D53—21no—
85Dic106—41no—
86#88324—82no—
87C8×C53—21no—
88C8×C53#9911yescharacteristic
89C8×C53#9911yescharacteristic
90#88322—21no—
91#88322—21no—
92#7227—531nomaximal, Sylow
93C22×C53—21no—
94C22×C53—41no—
95C22×C53#34011yescharacteristic
96C2×D53—21no—
97C2×D53—21no—
98C2×D53—41no—
99C2×D53—41no—
100#6157—162no—
101C4×C53—21no—
102C4×C53H(3,ℤ/4)11yescharacteristic
103C4×C53#34211yescharacteristic
104Dic53—21no—
105Dic53—21no—
106#2113—531no—
107#1813—531no—
108#1812—531no—
109C2×C53—21no—
110C2×C53—41no—
111C2×C53#128911yescharacteristic
112D53—41no—
113D53—41no—
114C2×#131—531no—
115C2×M32—531no—
116C22×C16#615711yescharacteristic, complemented
117C2×#108—1061no—
118#427—1061no—
119#413—1061no—
120#410—1061no—
121C22⋊C16—1061no—
122C22⋊C16—1061no—
123#338—532no—
124H(3,ℤ/4)—532no—
125C53#722711yescharacteristic, complemented, Sylow
126C2×#41—531no—
127#131—2121no—
128#131—2121no—
129C2×M16—531no—
130C2×M16—531no—
131C2×M16—531no—
132C2×M16—1062no—
133C22×C8—531no—
134C22×C8—531no—
135C22×C8C2×#615711yescharacteristic
136#117—1062no—
137M32—2121no—
138C2×C16—21no—
139C2×C16—41no—
140C2×C16—1061no—
141#108—2121no—
142#108—2121no—
143#107—2121no—
144#106—2121no—
145#101—2121no—
146#100—2121no—
147#99—2121no—
148#99—2121no—
149#41—2121no—
150#41—2121no—
151C2×Q8—531no—
152C2×D4—531no—
153C2×D4—531no—
154C2×D4—531no—
155C22×C4—531no—
156C22×C4—531no—
157C22×C4C4⋊#615711yescharacteristic
158M16—2121no—
159M16—2121no—
160M16—2121no—
161M16—2122no—
162M16—2122no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—531no—
166C2×C8—531no—
167C2×C8—1061no—
168C2×C8—1061no—
169C2×C8—1061no—
170C2×C8—1061no—
171C2×C8—2122no—
172C2×C8C4×#615711yescharacteristic
173C2×C8C22⋊#615711yescharacteristic
174C4⋊C4—2122no—
175C22⋊C4—2122no—
176C42—2122no—
177C16—41no—
178C16—2121no—
179C23—531no—
180C23—531no—
181C23#1313328611yescharacteristic
182Q8—2121no—
183D4—2121no—
184D4—2121no—
185D4—2121no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—531no—
189C2×C4—531no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—4242no—
195C2×C4(C2×#6157)⋊C411yescharacteristic
196C2×C4#1313328511yescharacteristic
197C8—21no—
198C8—1061no—
199C8—1061no—
200C8—4242no—
201C8#1313327511yescharacteristic
202C8#1313327611yescharacteristic
203C22—21no—
204C22—41no—
205C22—1061no—
206C22—1061no—
207C22—2121no—
208C22—2121no—
209C22#1681149711yescharacteristic
210C4—21no—
211C4—8482no—
212C4—1061no—
213C4—1061no—
214C4#1681149111yescharacteristic
215C4#1681150111yescharacteristic
216C2—21no—
217C2—41no—
218C2—2121no—
219C2—2121no—
220C2#2028957811yescharacteristic
subgroupG/H
centerC4#16811491
commutatorC2×C8×C53C42
FrattiniC22×C8C2×#6157
FittingC22×C16×C53C4
socleC2×C53#1289
prime radicalC2×#11719945D4
cosocleC2×#13133223C22
hypercenterC22×C16#6157
hypocenterC53#7227
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C8 ◁ C22×C16

Complements

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

NH ≅ G/Ntype
8. #16811492C4semidirect
11. C2×#13133157C4semidirect
14. C22×C16×C53C4semidirect
116. C22×C16#6157semidirect
125. C53#7227semidirect

Maximal subgroups

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

subgroupindex
#202905992
#202905972
#202763282
#722753