Finite Group Databasebeta

← #27296549

Subgroups — #27296549

Order 13568.

398 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 67 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#20286025C211yesmaximal, characteristic
2#20276632C212yesmaximal, complemented
3#20272723C212yesmaximal, complemented
4#20272655C211yesmaximal, characteristic, complemented
5C53×#2056C211yesmaximal, characteristic, complemented
6#16811173—21no—
7C2×#13133160—22no—
8#16810904—22no—
9#16810859—21no—
10#16810859C2212yes—
11#16810819—22no—
12#16810427—22no—
13C22⋊#11719908C2212yes—
14C22×Q16×C53C2211yescharacteristic
15C2×C4⋊C8×C53C2211yescharacteristic
16C2×#103×C53—21no—
17C53×#391—21no—
18C53×#340C2211yescharacteristic
19C2×#11719945—41no—
20C2×D424—21no—
21C2×D424D412yes—
22#13133217—21no—
23#13133214—21no—
24#13133205—41no—
25#13133191—22no—
26#13133191—41no—
27C4×D212—22no—
28C4×D212—41no—
29#13133160—44no—
30C2×#11719908—22no—
31#13133151—21no—
32#13133151D412yes—
33#13133139—22no—
34#13133139—22no—
35C22×Q8×C53—21no—
36C2×Q16×C53—22no—
37C2×Q16×C53—42no—
38C22×C8×C53D411yescharacteristic
39C22×C8×C53C2311yescharacteristic
40C2×C4⋊C4×C53—21no—
41C2×C42×C53D411yescharacteristic
42C4⋊C8×C53—22no—
43#103×C53—42no—
44#95×C53—41no—
45#95×C53—41no—
46C2×Q8×C53—22no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C22×C4×C53—41no—
53C22×C4×C53—41no—
54C22×C4×C53D811yescharacteristic
55C22×C4×C53D811yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C2×C22⋊D53—42no—
58#11719945—82no—
59#11719945—82no—
60C2×D212—22no—
61C2×D212—41no—
62C2×D212—41no—
63C2×C4×D53—42no—
64C2×Dic106—41no—
65Q16×C53—24no—
66Q16×C53—42no—
67C2×C8×C53—21no—
68C2×C8×C53—41no—
69C2×C8×C53—41no—
70C2×C8×C53C2×D412yes—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C42×C53—21no—
74C42×C53—21no—
75#11719921—42no—
76#11719921—42no—
77C4⋊Dic53—22no—
78C4⋊Dic53—41no—
79D424—42no—
80D424—81no—
81#11719908—42no—
82C23×C53C2×D811yescharacteristic
83C22×D53—42no—
84Q8×C53—42no—
85Q8×C53—42no—
86Q8×C53—81no—
87Q8×C53—81no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—81no—
96C2×C4×C53—81no—
97C2×C4×C53—81no—
98C2×C4×C53—81no—
99C2×C4×C53—81no—
100C2×C4×C53SD3212yescomplemented
101C2×C4×C53#12011yescharacteristic
102C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
103C22⋊D53—82no—
104C22⋊D53—82no—
105C2×Dic53—42no—
106D212—42no—
107D212—42no—
108D212—82no—
109D212—82no—
110C4×D53—84no—
111Dic106—82no—
112C8×C53—22no—
113C8×C53—41no—
114#12161—531nomaximal, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53#44711yescharacteristic
118C22×C53C2×SD3211yescharacteristic
119C22×C53#50911yescharacteristic
120C2×D53—82no—
121C2×D53—82no—
122C2×D53—82no—
123C4×C53—21no—
124C4×C53—22no—
125C4×C53—41no—
126C4×C53—82no—
127C4×C53—161no—
128C4×C53#45411yescharacteristic
129C4×C53B(4,𝔽2)11yescharacteristic
130Dic53—82no—
131#3245—531no—
132#2185—532no—
133#2056D5311yescharacteristic, complemented
134#1347—532no—
135#1322—531no—
136C2×C53—21no—
137C2×C53—21no—
138C2×C53#159811yescharacteristic
139C2×C53#218711yescharacteristic
140C2×C53#219111yescharacteristic
141D53—162no—
142C22×Q16C2×D5311yescharacteristic
143#532—1061no—
144C2×SD32—1062no—
145#474—1062no—
146#466—532no—
147#466—1061no—
148#459—1062no—
149C2×C4⋊C8C2×D5311yescharacteristic
150C2×#103—21no—
151#391—21no—
152#358—1062no—
153C22⋊C16—532no—
154#340C2×D5311yescharacteristic
155C53#1216111yescharacteristic, complemented, Sylow
156C2×#41—2121no—
157C22×Q8—21no—
158C2×Q16—22no—
159C2×Q16—42no—
160C2×D8—532no—
161C2×D8—1061no—
162C22×C8C22⋊D5311yescharacteristic
163C22×C8C22×D5311yescharacteristic
164C4⋊Q8—1061no—
165#127—1061no—
166#124—2121no—
167#121—1062no—
168#121—2121no—
169C4×D4—1062no—
170C4×D4—2121no—
171C2×C4⋊C4—21no—
172C2×C42C22⋊D5311yescharacteristic
173SD32—2124no—
174C2×C16—1062no—
175#107—532no—
176#107—1061no—
177C4⋊C8—22no—
178#103—42no—
179#102—1062no—
180#102—1062no—
181#95—41no—
182#95—41no—
183#41—4242no—
184#41—4242no—
185C2×Q8—22no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—2121no—
192C2×D4—1062no—
193C2×D4—2121no—
194C2×D4—2121no—
195C2×D4—2122no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—2122no—
199C22×C4#1171992211yescharacteristic
200C22×C4#1171992211yescharacteristic
201C22×C4C2×C22⋊D5311yescharacteristic
202Q16—24no—
203Q16—42no—
204D8—4241no—
205D8—2122no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8D4×D5312yes—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—1062no—
213C4⋊C4—2121no—
214C22⋊C4—2122no—
215C22⋊C4—2122no—
216C42—21no—
217C42—21no—
218C16—2122no—
219C23—2122no—
220C23C2×#1171992211yescharacteristic
221Q8—42no—
222Q8—42no—
223Q8—81no—
224Q8—81no—
225Q8—4242no—
226D4—4242no—
227D4—4242no—
228D4—4242no—
229D4—4242no—
230D4—2122no—
231D4—2122no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—4244no—
245C2×C4—2122no—
246C2×C4#1313316012yescomplemented
247C2×C4#1313318511yescharacteristic
248C2×C4#1313324211yescharacteristic
249C8—22no—
250C8—41no—
251C22—21no—
252C22—21no—
253C22—4242no—
254C22—4242no—
255C22—4242no—
256C22#1681075611yescharacteristic
257C22C2×#1313316011yescharacteristic
258C22#1681105811yescharacteristic
259C4—22no—
260C4—21no—
261C4—41no—
262C4—82no—
263C4—161no—
264C4—4242no—
265C4#1681079511yescharacteristic
266C4#1681080911yescharacteristic
267C2—21no—
268C2—21no—
269C2—8482no—
270C2#2027364911yescharacteristic
271C2#2027664311yescharacteristic
272C2#2027666511yescharacteristic
subgroupG/H
centerC22#16810756
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#2056C2
socleC22×C53#447
prime radical#20286025C2
hypercenter#2056D53
hypocenterC53#12161
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2056

Complements

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

NH ≅ G/Ntype
2. #20276632C2semidirect
3. #20272723C2semidirect
4. #20272655C2semidirect
5. C53×#2056C2semidirect
100. C2×C4×C53SD32semidirect
133. #2056D53semidirect
155. C53#12161semidirect
246. C2×C4#13133160semidirect

Maximal subgroups

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

subgroupindex
#202860252
#202766322
#202727232
#202726552
C53×#20562
#1216153