Finite Group Databasebeta

← #27295620

Subgroups — #27295620

Order 13568.

280 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 79 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#20285343C211yesmaximal, characteristic
2#20276481C211yesmaximal, characteristic, complemented
3#20272777C212yesmaximal, complemented
4#20272755C212yesmaximal, complemented
5C53×#2166C211yesmaximal, characteristic, complemented
6#16811235C411yescharacteristic
7#16811228C411yescharacteristic
8C2×#13133157—21no—
9#16811025C2211yescharacteristic
10#16810996C2211yescharacteristic
11#16810992C411yescharacteristic
12#16810988C411yescharacteristic
13#16810869—21no—
14#16810852—21no—
15#16810725—21no—
16C4×#11719913—21no—
17#16810444—21no—
18#16810444C2212yes—
19C53×#405C2211yescharacteristic
20C53×#364C2212yes—
21C53×#349—21no—
22C53×#347—21no—
23C2×#11719913—21no—
24C2×#11719913—21no—
25#13133217C2×C411yescharacteristic
26#13133214C2×C411yescharacteristic
27#13133210—41no—
28#13133209C2×C411yescharacteristic
29#13133209C2×C411yescharacteristic
30#13133207—41no—
31#13133202—41no—
32#13133196—21no—
33#13133191—21no—
34#13133190—41no—
35C4×Dic106—21no—
36C4×D212—21no—
37#13133157—22no—
38#13133157—41no—
39C2×#11719908—41no—
40#13133151—21no—
41#13133149—21no—
42#13133142—21no—
43#13133142—21no—
44#13133139—21no—
45#13133139—21no—
46C2×M16×C53D411yescharacteristic
47C22×C8×C53D411yescharacteristic
48#117×C53C2×C411yescharacteristic
49M32×C53—22no—
50#97×C53C2×C411yescharacteristic
51#97×C53D411yescharacteristic
52C4×C8×C53D411yescharacteristic
53C4×C8×C53C2311yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C2×C22⋊D53—41no—
56C2×D212—21no—
57C2×D212—41no—
58C2×C4×D53—41no—
59C2×Dic106—21no—
60M16×C53—22no—
61C2×C8×C53—21no—
62C2×C8×C53—21no—
63C2×C8×C53C22⋊C411yescharacteristic
64C2×C8×C53C22⋊C411yescharacteristic
65C2×C8×C53C2×D411yescharacteristic
66C2×C8×C53C2×D411yescharacteristic
67C4⋊C4×C53—21no—
68C22⋊C4×C53—21no—
69C42×C53C22⋊C411yescharacteristic
70C42×C53C22×C411yescharacteristic
71C22⋊Dic53—41no—
72#11719921—41no—
73#11719921—41no—
74C4⋊Dic53—21no—
75C4⋊Dic53—21no—
76C4⋊Dic53—41no—
77#11719919—41no—
78#11719919—41no—
79#11719919—41no—
80#11719919—41no—
81C4×Dic53—41no—
82#11719913—42no—
83#11719913—41no—
84#11719913—41no—
85C16×C53—42no—
86#11719908—42no—
87C23×C53#10011yescharacteristic
88C22×D53—41no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53#10011yescharacteristic
94C2×C4×C53C2×C22⋊C411yescharacteristic
95C22⋊D53—161no—
96C2×Dic53—41no—
97C2×Dic53—41no—
98C2×Dic53—41no—
99D212—41no—
100D212—41no—
101D212—161no—
102C4×D53—161no—
103Dic106—41no—
104Dic106—41no—
105C8×C53—21no—
106C8×C53—21no—
107C8×C53#13512yes—
108C8×C53AGL(1,ℤ/8)11yescharacteristic
109C8×C53#13711yescharacteristic
110#11991—531nomaximal, Sylow
111C22×C53—21no—
112C22×C53—41no—
113C22×C53C2×#10011yescharacteristic
114C2×D53—81no—
115C2×D53—81no—
116C2×D53—81no—
117C4×C53—21no—
118C4×C53—41no—
119C4×C53—41no—
120C4×C53#41611yescharacteristic
121C4×C53#41711yescharacteristic
122Dic53—81no—
123Dic53—81no—
124Dic53—81no—
125#3174—531no—
126#2166D5311yescharacteristic, complemented
127#2153—531no—
128#1366—532no—
129#1359—532no—
130C2×C53—21no—
131C2×C53—41no—
132C2×C53#155111yescharacteristic
133D53—161no—
134#541—531no—
135#540—531no—
136C2×M32—1061no—
137#499—531no—
138#492—531no—
139#491—531no—
140#490—531no—
141#468—1061no—
142#465—1061no—
143#440—1061no—
144C4×SD16—1061no—
145#405C2×D5311yescharacteristic
146#364—532no—
147#364—1061no—
148#364C2×D5312yes—
149#349—21no—
150#347—21no—
151C53#1199111yescharacteristic, complemented, Sylow
152C2×SD16—1061no—
153C2×SD16—1061no—
154C2×M16C22⋊D5311yescharacteristic
155C22×C8D21211yescharacteristic
156C4⋊Q8—531no—
157#127—531no—
158#126—2121no—
159#125—531no—
160#125—531no—
161#125—2121no—
162#123—2121no—
163C22⋊Q8—1061no—
164#121—1061no—
165#121—2121no—
166C4×Q8—1061no—
167C4×D4—1061no—
168#117C4×D5311yescharacteristic
169M32—22no—
170M32—1062no—
171M32—2121no—
172C2×C16—2121no—
173#107—1061no—
174#106—1061no—
175#103—1061no—
176#103—1061no—
177#102—1061no—
178#102—1061no—
179#97C4×D5311yescharacteristic
180#97C22⋊D5311yescharacteristic
181C4×C8D21211yescharacteristic
182C4×C8C22×D5311yescharacteristic
183C2×Q8—1061no—
184C2×D4—1061no—
185C2×D4—2121no—
186C2×D4—2121no—
187C22×C4—2121no—
188C22×C4#1171992111yescharacteristic
189SD16—2122no—
190SD16—2121no—
191SD16—2121no—
192M16—22no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8#1171992111yescharacteristic
196C2×C8#1171992111yescharacteristic
197C2×C8C2×D21211yescharacteristic
198C2×C8C2×C22⋊D5311yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—1061no—
201C4⋊C4—1061no—
202C4⋊C4—2121no—
203C4⋊C4—2121no—
204C4⋊C4—2121no—
205C4⋊C4—2121no—
206C4⋊C4—2121no—
207C22⋊C4—21no—
208C22⋊C4—2121no—
209C22⋊C4—2121no—
210C22⋊C4—2121no—
211C42—2121no—
212C42#1171992111yescharacteristic
213C42C2×C4×D5311yescharacteristic
214C16—42no—
215C16—2122no—
216C23—2121no—
217C23#1313313311yescharacteristic
218Q8—2121no—
219Q8—2121no—
220D4—8481no—
221D4—8481no—
222D4—2121no—
223D4—2121no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—8481no—
229C2×C4—2121no—
230C2×C4—2121no—
231C2×C4—2121no—
232C2×C4#1313313311yescharacteristic
233C2×C4C2×#1171992111yescharacteristic
234C8—21no—
235C8—21no—
236C8#1313323712yes—
237C8#1313324511yescharacteristic
238C8#1313325011yescharacteristic
239C22—21no—
240C22—41no—
241C22—4241no—
242C22—4241no—
243C22—4241no—
244C22C2×#1313313311yescharacteristic
245C4—21no—
246C4—41no—
247C4—41no—
248C4—4241no—
249C4—4241no—
250C4—4241no—
251C4#1681061711yescharacteristic
252C4#1681062511yescharacteristic
253C2—21no—
254C2—41no—
255C2—8481no—
256C2#2027341111yescharacteristic
subgroupG/H
centerC2#20273411
commutatorC42×C53C22×C4
FrattiniC4×C8C22×D53
FittingC53×#2166C2
socleC2×C53#1551
prime radical#16811228C4
cosocleC4×C8×C53C23
hypercenter#2166D53
hypocenterC53#11991
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
2. #20276481C2semidirect
3. #20272777C2semidirect
4. #20272755C2semidirect
5. C53×#2166C2semidirect
126. #2166D53semidirect
151. C53#11991semidirect

Maximal subgroups

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

subgroupindex
#202853432
#202764812
#202727772
#202727552
C53×#21662
#1199153