Finite Group Databasebeta

← #27288032

Subgroups — #27288032

Order 13568.

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

normal subgroups only

This group has 372 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#20285665C211yesmaximal, characteristic
2#20275559C211yesmaximal, characteristic
3#20273503C211yesmaximal, characteristic
4#20273452C211yesmaximal, characteristic
5#20273397C211yesmaximal, characteristic
6#20273137C211yesmaximal, characteristic
7C53×#1299C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9C2×#13133217C2211yescharacteristic
10#16811003—21no—
11#16810974—21no—
12#16810834—21no—
13#16810834—21no—
14#16810781—22no—
15#16810676C2211yescharacteristic
16#16810647C2211yescharacteristic
17C2×#13133135—22no—
18#16810375—21no—
19#16810373—21no—
20#16810371—22no—
21#16810370—21no—
22#16810368—21no—
23#16810355—22no—
24C53×#431C2211yescharacteristic
25C53×#423C2211yescharacteristic
26C4×M16×C53C2211yescharacteristic
27C22×Dic106D412yes—
28C2×Dic212—22no—
29C2×Dic212—22no—
30C2×#11719917—21no—
31#13133217—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35#13133217D411yescharacteristic
36#13133217D411yescharacteristic
37#13133209—21no—
38#13133196—21no—
39#13133196—21no—
40C4×Dic106—21no—
41C4×Dic106—21no—
42#13133176—21no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322D412yes—
48#13133142—22no—
49#13133142—22no—
50#13133135—44no—
51C22⋊#88322—21no—
52C4×#88322—21no—
53C2×M16×C53—21no—
54C2×M16×C53—21no—
55C2×C42×C53C2311yescharacteristic
56C4⋊C8×C53—21no—
57C4⋊C8×C53—21no—
58C4⋊C8×C53D411yescharacteristic
59C4⋊C8×C53D411yescharacteristic
60C22⋊C8×C53—21no—
61#97×C53—21no—
62C4×C8×C53—21no—
63C4×C8×C53—21no—
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53#4111yescharacteristic
66C22×C4×C53#4111yescharacteristic
67C22×Dic53—22no—
68C2×Dic106—22no—
69C2×Dic106—21no—
70C2×Dic106—21no—
71C2×Dic106—22no—
72C2×Dic106—42no—
73C2×Dic106—42no—
74M16×C53—22no—
75M16×C53—41no—
76M16×C53—81no—
77C2×C8×C53—21no—
78C2×C8×C53—21no—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C42×C53C2×D411yescharacteristic
82C42×C53C2×D411yescharacteristic
83C42×C53C2×D411yescharacteristic
84C42×C53#4111yescharacteristic
85C22⋊Dic53—41no—
86C4⋊Dic53—21no—
87C4⋊Dic53—21no—
88C4⋊Dic53—21no—
89C4⋊Dic53—21no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93C4⋊Dic53—41no—
94#11719919—41no—
95#11719919—41no—
96C4×Dic53—41no—
97#11719917—42no—
98C2×#88322—22no—
99C2×#88322—41no—
100Dic212—82no—
101Dic212—82no—
102C23×C53#12311yescharacteristic
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—22no—
108C2×C4×C53#12011yescharacteristic
109C2×C4×C53#12111yescharacteristic
110C2×C4×C53#12111yescharacteristic
111C2×C4×C53#12111yescharacteristic
112C2×C4×C53#12311yescharacteristic
113C2×C4×C53#12411yescharacteristic
114C2×Dic53—44no—
115C2×Dic53—42no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118Dic106—42no—
119Dic106—41no—
120Dic106—42no—
121Dic106—41no—
122Dic106—41no—
123Dic106—41no—
124Dic106—82no—
125C8×C53—42no—
126C8×C53—41no—
127C8×C53—41no—
128C8×C53—81no—
129#88322—42no—
130#88322—81no—
131#10493—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53#39411yescharacteristic
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—21no—
139C4×C53#45311yescharacteristic
140C4×C53#45511yescharacteristic
141C4×C53#47111yescharacteristic
142C4×C53#47311yescharacteristic
143C4×C53#47312yes—
144Dic53—81no—
145Dic53—82no—
146Dic53—81no—
147#3209—531no—
148#1966—531no—
149#1574—531no—
150#1561—531no—
151#1548—531no—
152#1485—531no—
153#1299D5311yescharacteristic
154C2×C53—41no—
155C2×C53#203711yescharacteristic
156C2×C53#204711yescharacteristic
157C2×C53#205211yescharacteristic
158#533—531no—
159C2×C4⋊Q8—531no—
160#494—1061no—
161#487—1061no—
162#462—1061no—
163#462—1061no—
164#451—1062no—
165#431—531no—
166#431C2×D5311yescharacteristic
167#423—531no—
168#423C2×D5311yescharacteristic
169C2×#101—1062no—
170C4×M16C2×D5311yescharacteristic
171#335—1061no—
172#334—1061no—
173#333—1062no—
174#332—1061no—
175#332—1061no—
176Q8⋊C8—1062no—
177C53#1049311yescharacteristic, complemented, Sylow
178C22×Q8—532no—
179C2×Q16—1062no—
180C2×Q16—1062no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—1061no—
184C4⋊Q8—531no—
185C4⋊Q8—531no—
186C4⋊Q8—1061no—
187C4⋊Q8—1061no—
188C4⋊Q8—1061no—
189C4⋊Q8—1061no—
190#125—1061no—
191C22⋊Q8—1061no—
192C22⋊Q8—1061no—
193C4×Q8—1061no—
194C4×Q8—1061no—
195#117—1061no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4—1061no—
198C2×C42C22×D5311yescharacteristic
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—532no—
202C4⋊C8—1061no—
203C4⋊C8—1061no—
204C4⋊C8D21211yescharacteristic
205C4⋊C8D21211yescharacteristic
206#103—1062no—
207#103—1062no—
208#101—2124no—
209C22⋊C8—21no—
210C22⋊C8—1061no—
211#97—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—1061no—
215C2×Q8—1062no—
216C2×Q8—1061no—
217C2×Q8—1062no—
218C2×Q8—1061no—
219C2×Q8—2122no—
220C2×Q8—2122no—
221C22×C4—1062no—
222C22×C4#1171994511yescharacteristic
223C22×C4D4×D5311yescharacteristic
224C22×C4#1171994911yescharacteristic
225Q16—4242no—
226Q16—4242no—
227M16—22no—
228M16—41no—
229M16—81no—
230M16—2122no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—1062no—
236C2×C8—2121no—
237C4⋊C4—1061no—
238C4⋊C4—1061no—
239C4⋊C4—1061no—
240C4⋊C4—1061no—
241C4⋊C4—2121no—
242C4⋊C4—2121no—
243C4⋊C4—2121no—
244C4⋊C4—2121no—
245C4⋊C4—2121no—
246C4⋊C4—2121no—
247C22⋊C4—2121no—
248C42—2121no—
249C42C2×D21211yescharacteristic
250C42#1171994511yescharacteristic
251C42D4×D5311yescharacteristic
252C42D4×D5311yescharacteristic
253C23#1313320211yescharacteristic
254Q8—4242no—
255Q8—2121no—
256Q8—2122no—
257Q8—2121no—
258Q8—2122no—
259Q8—2121no—
260Q8—2121no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—2124no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2122no—
270C2×C4#1313318711yescharacteristic
271C2×C4#1313319011yescharacteristic
272C2×C4#1313319011yescharacteristic
273C2×C4#1313319211yescharacteristic
274C2×C4#1313320211yescharacteristic
275C2×C4#1313320511yescharacteristic
276C8—42no—
277C8—41no—
278C8—41no—
279C8—81no—
280C8—4241no—
281C8—2122no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22#1681054111yescharacteristic
286C4—21no—
287C4—21no—
288C4—21no—
289C4—4241no—
290C4—4242no—
291C4—4241no—
292C4#1681079211yescharacteristic
293C4#1681080211yescharacteristic
294C4#1681088511yescharacteristic
295C4#1681089711yescharacteristic
296C4#1681089912yes—
297C2—41no—
298C2#2027592811yescharacteristic
299C2#2027598411yescharacteristic
300C2#2027601711yescharacteristic
subgroupG/H
centerC22#16810541
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1299C2
socleC22×C53#394
prime radicalC23×C53#123
hypercenter#1299D53
hypocenterC53#10493
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1299

Complements

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

NH ≅ G/Ntype
177. C53#10493semidirect

Maximal subgroups

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

subgroupindex
#202856652
#202755592
#202735032
#202734522
#202733972
#202731372
C53×#12992
#1049353