Finite Group Databasebeta

← #28311207

Subgroups — #28311207

Order 13568.

432 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C2×#16811534C212yesmaximal
2#20290581C212yesmaximal
3#20289812C212yesmaximal
4#20277434C211yesmaximal, characteristic
5#16811626—22no—
6C2×#13133278C2214yes—
7#16811534—24no—
8#16811503—22no—
9C2×#13133217C2211yescharacteristic
10C2×C4×Dic106C411yescharacteristic
11C2×C4×Dic106C2211yescharacteristic
12#16810529C411yescharacteristic
13#16810529C2211yescharacteristic
14#16810508—22no—
15#16810508—22no—
16#16810489C411yescharacteristic
17C53×#376C411yescharacteristic
18C2×#11719940—24no—
19#13133284—42no—
20C4⋊#88324—42no—
21#13133278—24no—
22#13133278—24no—
23C22×Dic106C2×C411yescharacteristic
24C22×Dic106C2311yescharacteristic
25#13133217—22no—
26#13133217D412yes—
27#13133217Q812yes—
28C4×Dic106—42no—
29C4×Dic106—42no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—21no—
34C2×#11719919D412yes—
35C2×C4×Dic53—21no—
36C2×C4×Dic53D412yes—
37#13133124—22no—
38#13133124—22no—
39#13133124—22no—
40C2×C42×C53C2×C411yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43#95×C53—21no—
44#95×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53C22⋊C411yescharacteristic
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—21no—
51C22×Dic53—21no—
52C22×Dic53C22⋊C412yes—
53C22×Dic53C2×D412yes—
54C2×Dic106—21no—
55C2×Dic106—41no—
56C2×Dic106—41no—
57C2×Dic106—41no—
58C2×Dic106—41no—
59C2×Dic106C4⋊C412yes—
60C2×Dic106C2×D411yescharacteristic
61C2×Dic106C2×Q811yescharacteristic
62C2×Dic106#4112yes—
63#11719940—44no—
64#11719940—44no—
65C2×#88324—44no—
66C42×C53—22no—
67C42×C53—21no—
68C42×C53—41no—
69C42×C53C4⋊C412yes—
70C4⋊Dic53—24no—
71C4⋊Dic53—24no—
72C4⋊Dic53—41no—
73C4⋊Dic53—41no—
74#11719919—42no—
75#11719919—42no—
76C4×Dic53—22no—
77C4×Dic53—22no—
78C4×Dic53—42no—
79C23×C53C2×C22⋊C411yescharacteristic
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—21no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53#9912yes—
92C2×C4×C53C2×C4⋊C411yescharacteristic
93C2×C4×C53#11711yescharacteristic
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—41no—
99C2×Dic53—41no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—41no—
103C2×Dic53#12112yes—
104C2×Dic53C22⋊Q812yes—
105Dic106—41no—
106Dic106—41no—
107Dic106—42no—
108Dic106—42no—
109Dic106—42no—
110#88324—84no—
111#12804—531nomaximal, Sylow
112C22×C53—21no—
113C22×C53—21no—
114C22×C53#38011yescharacteristic
115C22×C53C2×#9911yescharacteristic
116C22×C53#41011yescharacteristic
117C4×C53—21no—
118C4×C53—21no—
119C4×C53—22no—
120C4×C53—42no—
121C4×C53—41no—
122C4×C53—81no—
123Dic53—22no—
124Dic53—22no—
125Dic53—42no—
126Dic53—42no—
127Dic53—81no—
128#2327—531no—
129C2×#356—532no—
130#1801—532no—
131#1394—532no—
132C2×C53—21no—
133C2×C53—21no—
134C2×C53#179911yescharacteristic
135C2×C53#213311yescharacteristic
136C2×C53#213511yescharacteristic
137C2×C4⋊Q8—531no—
138C2×C4×Q8—531no—
139C2×C4×Q8—531no—
140#423—1062no—
141C2×#101—534no—
142#391—531no—
143#391—531no—
144#387—1062no—
145#387—1062no—
146#382—531no—
147#376#615711yescharacteristic
148#356—1064no—
149#343—1062no—
150C53#1280411yescharacteristic, complemented, Sylow
151C22×Q8—531no—
152C22×Q8—531no—
153C2×M16—1064no—
154C4⋊Q8—532no—
155C4⋊Q8—532no—
156C4⋊Q8—1062no—
157C4×Q8—2122no—
158C4×Q8—2122no—
159C2×C4⋊C4—532no—
160C2×C4⋊C4—531no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4—1061no—
164C2×C42—532no—
165C2×C42—1061no—
166C2×C42C2×#615711yescharacteristic
167C2×C42C2×#615711yescharacteristic
168C2×C42C2×#615711yescharacteristic
169C4⋊C8—2122no—
170C4⋊C8—2122no—
171#101—1064no—
172#101—1064no—
173#95—21no—
174#95—21no—
175#95—1062no—
176#95—1062no—
177#95—1062no—
178C2×Q8—531no—
179C2×Q8—531no—
180C2×Q8—532no—
181C2×Q8—532no—
182C2×Q8—1061no—
183C2×Q8—2121no—
184C2×Q8—2121no—
185C2×Q8—2121no—
186C2×Q8—2121no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—532no—
190C22×C4—532no—
191C22×C4—1061no—
192C22×C4—1061no—
193C22×C4C22⋊#615711yescharacteristic
194C22×C4C22⋊#615711yescharacteristic
195C22×C4C22×#615711yescharacteristic
196M16—2124no—
197M16—2124no—
198C2×C8—2124no—
199C4⋊C4—1064no—
200C4⋊C4—1064no—
201C4⋊C4—2122no—
202C4⋊C4—2121no—
203C4⋊C4—2121no—
204C4⋊C4—2122no—
205C42—22no—
206C42—21no—
207C42—41no—
208C42—1062no—
209C42—1062no—
210C42—2122no—
211C42C4⋊#615712yes—
212C23C2×C22⋊#615711yescharacteristic
213Q8—2121no—
214Q8—2121no—
215Q8—2122no—
216Q8—2122no—
217Q8—2122no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—532no—
230C2×C4—532no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—2122no—
236C2×C4—2121no—
237C2×C4—2121no—
238C2×C4—2121no—
239C2×C4—2122no—
240C2×C4#1313327612yes—
241C2×C4C2×C4⋊#615711yescharacteristic
242C2×C4#1313329611yescharacteristic
243C8—4244no—
244C22—21no—
245C22—21no—
246C22#1681156511yescharacteristic
247C22C2×#1313327611yescharacteristic
248C22#1681160211yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—42no—
253C4—41no—
254C4—81no—
255C4—1062no—
256C4—1062no—
257C4—4241no—
258C4—2122no—
259C4—2122no—
260C2—21no—
261C2—21no—
262C2#2029057611yescharacteristic
263C2#2029101211yescharacteristic
264C2#2029101911yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#376C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Dic106C23
hypercenter#376#6157
hypocenterC53#12804
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
150. C53#12804semidirect

Maximal subgroups

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

subgroupindex
C2×#168115342
#202905812
#202898122
#202774342
#1280453