Finite Group Databasebeta

← #27303777

Subgroups — #27303777

Order 13568.

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

normal subgroups only

This group has 416 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#20282708C211yesmaximal, characteristic
2#20274930C212yesmaximal
3#20272842C211yesmaximal, characteristic
4#20272841C212yesmaximal
5C53×#1857C211yesmaximal, characteristic
6#16810666—22no—
7C2×#13133136C2212yes—
8#16810544—22no—
9#16810543—21no—
10#16810530C2212yes—
11#16810529—21no—
12#16810527—22no—
13#16810382—22no—
14C53×#519C2211yescharacteristic
15C53×#428—21no—
16C2×#101×C53C2211yescharacteristic
17C53×#391C2211yescharacteristic
18C53×#339—21no—
19C22×Dic106—21no—
20C22×Dic106—21no—
21C2×#11719917—22no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53D412yes—
24C2×#11719919—22no—
25C2×C4×Dic53D412yes—
26#13133150—42no—
27#13133148—42no—
28#13133136—22no—
29#13133136Q814yescomplemented
30#13133124—21no—
31#13133124—22no—
32#13133124—22no—
33#13133124—22no—
34C22×Q8×C53C2311yescharacteristic
35C2×M16×C53—21no—
36C4×Q8×C53—41no—
37C4×Q8×C53—41no—
38#117×C53—21no—
39#117×C53—22no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C42×C53D411yescharacteristic
42#107×C53—41no—
43#106×C53—41no—
44#101×C53—21no—
45#101×C53Q812yescomplemented
46#95×C53—22no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53C2×Q811yescharacteristic
50C2×Q8×C53C2×Q812yes—
51C2×Q8×C53#4111yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53C2×D412yes—
55C22×C4×C53C2×D411yescharacteristic
56C22×Dic53—22no—
57C22×Dic53—22no—
58C2×Dic106—24no—
59C2×Dic106—81no—
60C2×Dic106—81no—
61C2×Dic106—82no—
62C2×Dic106—81no—
63M16×C53—22no—
64M16×C53—41no—
65C2×C8×C53—41no—
66C4⋊C4×C53—21no—
67C4⋊C4×C53—21no—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—22no—
70C4⋊C4×C53—41no—
71C22⋊C4×C53—41no—
72C42×C53—42no—
73C42×C53—41no—
74C42×C53—41no—
75C4⋊Dic53—42no—
76C4⋊Dic53—81no—
77#11719919—82no—
78C4×Dic53—42no—
79#11719917—24no—
80#11719917—42no—
81C2×#88322—42no—
82C23×C53#12011yescharacteristic
83Q8×C53—42no—
84Q8×C53—41no—
85Q8×C53—41no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53C22⋊Q812yes—
98C2×C4×C53C22⋊Q811yescharacteristic
99C2×C4×C53C4⋊Q811yescharacteristic
100C2×C4×C53C4⋊Q812yes—
101C2×Dic53—24no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107Dic106—84no—
108Dic106—84no—
109C8×C53—42no—
110#88322—44no—
111#13464—531nomaximal, Sylow
112C22×C53—21no—
113C22×C53—21no—
114C22×C53#39511yescharacteristic
115C22×C53B(4,𝔽2)11yescharacteristic
116C22×C53#45811yescharacteristic
117C4×C53—22no—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—22no—
121C4×C53—42no—
122C4×C53—81no—
123C4×C53—81no—
124C4×C53—81no—
125Dic53—44no—
126Dic53—82no—
127#2811—531no—
128#1857—532no—
129#1857D5311yescharacteristic
130#1394—532no—
131#1394—531no—
132C2×C53—21no—
133C2×C53—21no—
134C2×C53#206211yescharacteristic
135C2×C53#220211yescharacteristic
136C2×C53#220511yescharacteristic
137#519C2×D5311yescharacteristic
138#428—21no—
139#428—1062no—
140C2×#101—532no—
141C2×#101C2×D5311yescharacteristic
142#395—1062no—
143#395—1061no—
144#391—532no—
145#391—1062no—
146#391—1061no—
147#391C2×D5311yescharacteristic
148#339—21no—
149#339—1062no—
150C53#1346411yescharacteristic, complemented, Sylow
151C22×Q8—1061no—
152C22×Q8—1061no—
153C22×Q8C22×D5311yescharacteristic
154C2×M16—21no—
155C2×M16—1062no—
156C4×Q8—41no—
157C4×Q8—41no—
158#117—22no—
159#117—21no—
160C2×C4⋊C4—532no—
161C2×C4⋊C4—1062no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4C22⋊D5311yescharacteristic
164C2×C42—532no—
165C2×C42C22⋊D5311yescharacteristic
166#107—41no—
167#107—2122no—
168#106—41no—
169#106—2122no—
170#101—21no—
171#101—534no—
172#101—1062no—
173#101Dic10612yescomplemented
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×Dic10611yescharacteristic
187C2×Q8#1171994511yescharacteristic
188C2×Q8Q8×D5312yes—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—1062no—
192C22×C4—1062no—
193C22×C4D4×D5312yes—
194C22×C4C2×C22⋊D5311yescharacteristic
195M16—22no—
196M16—41no—
197M16—1064no—
198M16—2122no—
199C2×C8—41no—
200C2×C8—2122no—
201C4⋊C4—21no—
202C4⋊C4—22no—
203C4⋊C4—21no—
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—4242no—
207C4⋊C4—4241no—
208C4⋊C4—2122no—
209C22⋊C4—41no—
210C42—41no—
211C42—41no—
212C42—42no—
213C42—2122no—
214C23#1313318511yescharacteristic
215Q8—41no—
216Q8—41no—
217Q8—42no—
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—41no—
230C2×C4—41no—
231C2×C4—1064no—
232C2×C4—2122no—
233C2×C4—2122no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4#1313319512yes—
238C2×C4#1313319611yescharacteristic
239C2×C4#1313321511yescharacteristic
240C2×C4#1313321612yes—
241C8—42no—
242C8—2124no—
243C22—21no—
244C22—21no—
245C22#1681054311yescharacteristic
246C22#1681080911yescharacteristic
247C22#1681081211yescharacteristic
248C4—21no—
249C4—22no—
250C4—21no—
251C4—22no—
252C4—42no—
253C4—81no—
254C4—81no—
255C4—81no—
256C4—4242no—
257C4—2124no—
258C2—21no—
259C2—21no—
260C2#2027606811yescharacteristic
261C2#2027673111yescharacteristic
262C2#2027674411yescharacteristic
subgroupG/H
centerC22#16810809
commutator, cosocleC22×Q8×C53C23
FrattiniC22×Q8C22×D53
FittingC53×#1857C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#1857D53
hypocenterC53#13464
LCSG ▷ C22×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1857

Complements

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

NH ≅ G/Ntype
29. #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
#202749302
#202728422
#202728412
C53×#18572
#1346453