Finite Group Databasebeta

← #27303778

Subgroups — #27303778

Order 13568.

416 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 81 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#20282708C211yesmaximal, characteristic
2#20274932C212yesmaximal
3#20274929C211yesmaximal, characteristic
4#20272841C212yesmaximal
5C53×#1394C211yesmaximal, characteristic
6#16811101C2211yescharacteristic
7#16810665—22no—
8#16810663—21no—
9C2×#13133136C2211yescharacteristic
10#16810544—22no—
11#16810543—21no—
12#16810530C2212yes—
13#16810529—21no—
14#16810527—22no—
15#16810384—22no—
16#16810383—21no—
17C2×#101×C53C2212yes—
18C53×#391C2211yescharacteristic
19C22×Dic106—21no—
20C22×Dic106—21no—
21C2×#11719917—21no—
22Q8×Dic53—41no—
23Q8×Dic53—41no—
24#13133176—21no—
25#13133176—22no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53D412yes—
28C2×#11719919—22no—
29C2×C4×Dic53D412yes—
30#13133151—42no—
31#13133150—41no—
32#13133149—42no—
33#13133148—41no—
34#13133136—21no—
35#13133136Q812yescomplemented
36#13133124—21no—
37#13133124—22no—
38#13133124—22no—
39#13133124—22no—
40C22×Q8×C53C2311yescharacteristic
41C2×M16×C53—22no—
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C42×C53D411yescharacteristic
44#101×C53—22no—
45#101×C53Q814yescomplemented
46#95×C53—22no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53C2×Q812yes—
50C2×Q8×C53C2×Q811yescharacteristic
51C2×Q8×C53#4111yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53C2×D412yes—
56C22×Dic53—22no—
57C22×Dic53—22no—
58C2×Dic106—24no—
59C2×Dic106—81no—
60C2×Dic106—81no—
61C2×Dic106—82no—
62C2×Dic106—81no—
63M16×C53—24no—
64M16×C53—42no—
65C2×C8×C53—42no—
66C4⋊C4×C53—41no—
67C42×C53—41no—
68C22⋊Dic53—41no—
69C4⋊Dic53—21no—
70C4⋊Dic53—22no—
71C4⋊Dic53—21no—
72C4⋊Dic53—22no—
73C4⋊Dic53—42no—
74C4⋊Dic53—81no—
75#11719919—82no—
76C4×Dic53—41no—
77C4×Dic53—42no—
78C4×Dic53—42no—
79#11719917—22no—
80#11719917—41no—
81C2×#88322—41no—
82C23×C53#12011yescharacteristic
83Q8×C53—41no—
84Q8×C53—42no—
85Q8×C53—41no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53—41no—
95C2×C4×C53C22⋊Q812yes—
96C2×C4×C53C22⋊Q811yescharacteristic
97C2×C4×C53C4⋊Q812yes—
98C2×C4×C53C4⋊Q811yescharacteristic
99C2×Dic53—24no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53—41no—
106C2×Dic53—41no—
107Dic106—84no—
108Dic106—84no—
109C8×C53—44no—
110#88322—42no—
111#13464—531nomaximal, Sylow
112C22×C53—21no—
113C22×C53—21no—
114C22×C53#39511yescharacteristic
115C22×C53B(4,𝔽2)11yescharacteristic
116C22×C53#45811yescharacteristic
117C4×C53—21no—
118C4×C53—22no—
119C4×C53—22no—
120C4×C53—21no—
121C4×C53—42no—
122C4×C53—81no—
123Dic53—44no—
124Dic53—82no—
125Dic53—81no—
126Dic53—81no—
127#2811—531no—
128#1857—532no—
129#1857—531no—
130#1394—532no—
131#1394D5311yescharacteristic
132C2×C53—21no—
133C2×C53—21no—
134C2×C53#206211yescharacteristic
135C2×C53#220211yescharacteristic
136C2×C53#220511yescharacteristic
137#519—531no—
138#428—1062no—
139#428—1061no—
140C2×#101—531no—
141C2×#101C2×D5312yes—
142#395—1062no—
143#395—1061no—
144#391—532no—
145#391—1062no—
146#391—1061no—
147#391C2×D5311yescharacteristic
148#339—1062no—
149#339—1061no—
150C53#1346411yescharacteristic, complemented, Sylow
151C22×Q8—1061no—
152C22×Q8—1061no—
153C22×Q8C22×D5311yescharacteristic
154C2×M16—22no—
155C2×M16—1061no—
156C4×Q8—2121no—
157C4×Q8—2121no—
158#117—1061no—
159#117—1062no—
160C2×C4⋊C4—532no—
161C2×C4⋊C4—1062no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4D21211yescharacteristic
164C2×C42—532no—
165C2×C42D21211yescharacteristic
166#107—2121no—
167#107—2122no—
168#106—2121no—
169#106—2122no—
170#101—22no—
171#101—532no—
172#101—1061no—
173#101Dic10614yescomplemented
174#95—22no—
175#95—1062no—
176#95—1061no—
177#95—1062no—
178#95—1062no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—1064no—
182C2×Q8—4241no—
183C2×Q8—4241no—
184C2×Q8—4242no—
185C2×Q8—4241no—
186C2×Q8C2×Dic10612yes—
187C2×Q8#1171994711yescharacteristic
188C2×Q8Q8×D5311yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—1062no—
192C22×C4—1062no—
193C22×C4C2×D21211yescharacteristic
194C22×C4D4×D5312yes—
195M16—24no—
196M16—42no—
197M16—1062no—
198M16—2121no—
199C2×C8—42no—
200C2×C8—2121no—
201C4⋊C4—41no—
202C4⋊C4—1061no—
203C4⋊C4—1062no—
204C4⋊C4—1061no—
205C4⋊C4—1062no—
206C4⋊C4—4242no—
207C4⋊C4—4241no—
208C4⋊C4—2122no—
209C22⋊C4—2121no—
210C42—41no—
211C42—2122no—
212C42—2121no—
213C42—2122no—
214C23#1313318711yescharacteristic
215Q8—42no—
216Q8—41no—
217Q8—41no—
218Q8—4244no—
219Q8—4244no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—1064no—
230C2×C4—2122no—
231C2×C4—2122no—
232C2×C4—2121no—
233C2×C4—2121no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4#1313319412yes—
238C2×C4#1313319711yescharacteristic
239C2×C4#1313321612yes—
240C2×C4#1313321711yescharacteristic
241C8—44no—
242C8—2122no—
243C22—21no—
244C22—21no—
245C22#1681054411yescharacteristic
246C22#1681081111yescharacteristic
247C22#1681081411yescharacteristic
248C4—22no—
249C4—21no—
250C4—22no—
251C4—21no—
252C4—42no—
253C4—81no—
254C4—4241no—
255C4—4242no—
256C4—4241no—
257C4—2124no—
258C2—21no—
259C2—21no—
260C2#2027606911yescharacteristic
261C2#2027673311yescharacteristic
262C2#2027674611yescharacteristic
subgroupG/H
centerC22#16810811
commutator, cosocleC22×Q8×C53C23
FrattiniC22×Q8C22×D53
FittingC53×#1394C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#1394D53
hypocenterC53#13464
LCSG ▷ C22×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
35. #13133136Q8semidirect
45. #101×C53Q8semidirect
150. C53#13464semidirect
173. #101Dic106semidirect

Maximal subgroups

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

subgroupindex
#202827082
#202749322
#202749292
#202728412
C53×#13942
#1346453