Finite Group Databasebeta

← #27288445

Subgroups — #27288445

Order 13568.

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

normal subgroups only

This group has 448 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#20285490C211yesmaximal, characteristic
2#20275183C212yesmaximal
3#20273234C212yesmaximal
4#20272592C211yesmaximal, characteristic
5C53×#1851C211yesmaximal, characteristic
6C2×#13133196C2212yes—
7#16810959—22no—
8#16810864—21no—
9#16810864—21no—
10#16810781—22no—
11C2×#13133135—22no—
12#16810587C2212yes—
13#16810391—22no—
14#16810359—22no—
15#16810359—22no—
16C22×M16×C53C2211yescharacteristic
17C2×C4⋊C8×C53—21no—
18C2×C22⋊C8×C53C2211yescharacteristic
19C4×C22⋊C4×C53C2211yescharacteristic
20C53×#336—21no—
21C22×Dic106D412yes—
22C2×Dic212—21no—
23C2×Dic212—21no—
24C2×#11719917—22no—
25#13133196—22no—
26#13133196—22no—
27#13133196—22no—
28#13133196—22no—
29C2×C4⋊Dic53D412yes—
30C2×#11719919—22no—
31C2×C22⋊Dic53—22no—
32#13133151—21no—
33#13133151—21no—
34#13133142—22no—
35#13133142—22no—
36#13133135—44no—
37C22⋊#88322—22no—
38C22⋊#88322—22no—
39C23×C4×C53C2311yescharacteristic
40C2×M16×C53—22no—
41C2×M16×C53—42no—
42C22×C8×C53—21no—
43C22×C8×C53D411yescharacteristic
44C22×C8×C53D411yescharacteristic
45C2×C22⋊C4×C53—21no—
46C2×C42×C53—21no—
47C4⋊C8×C53—42no—
48C22⋊C8×C53—22no—
49#95×C53D411yescharacteristic
50#95×C53D411yescharacteristic
51C24×C53#4111yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—41no—
55C22×C4×C53—41no—
56C22×C4×C53C2×D412yes—
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53#4112yes—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C2×Dic106—22no—
63C2×Dic106—22no—
64C2×Dic106—42no—
65C2×Dic106—42no—
66M16×C53—24no—
67M16×C53—42no—
68C2×C8×C53—22no—
69C2×C8×C53—21no—
70C2×C8×C53—21no—
71C2×C8×C53—41no—
72C2×C8×C53—41no—
73C2×C8×C53—41no—
74C2×C8×C53—41no—
75C22⋊C4×C53—42no—
76C42×C53—41no—
77C42×C53—41no—
78C22⋊Dic53—42no—
79C22⋊Dic53—42no—
80C4⋊Dic53—22no—
81C4⋊Dic53—22no—
82#11719919—42no—
83#11719919—42no—
84#11719917—42no—
85#11719917—42no—
86C2×#88322—42no—
87Dic212—81no—
88Dic212—81no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53#12111yescharacteristic
94C23×C53#12312yes—
95C2×C4×C53—22no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53#12011yescharacteristic
108C2×C4×C53#12112yes—
109C2×C4×C53#12411yescharacteristic
110C2×Dic53—44no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—82no—
114Dic106—42no—
115Dic106—42no—
116Dic106—82no—
117C8×C53—41no—
118C8×C53—42no—
119C8×C53—41no—
120#88322—82no—
121#10559—531nomaximal, Sylow
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—22no—
127C22×C53—81no—
128C22×C53—81no—
129C22×C53#39411yescharacteristic
130C22×C53#47311yescharacteristic
131C22×C53#47311yescharacteristic
132C4×C53—22no—
133C4×C53—21no—
134C4×C53—42no—
135C4×C53—81no—
136C4×C53B(4,𝔽2)11yescharacteristic
137C4×C53#45811yescharacteristic
138Dic53—82no—
139Dic53—82no—
140#3189—531no—
141#1901—532no—
142#1851D5311yescharacteristic
143#1508—532no—
144#1293—531no—
145C2×C53—21no—
146C2×C53—21no—
147C2×C53—42no—
148C2×C53#204411yescharacteristic
149C2×C53#204711yescharacteristic
150C2×C53#205011yescharacteristic
151C22×M16C2×D5311yescharacteristic
152C2×C22⋊Q8—532no—
153#484—1062no—
154#467—1061no—
155#467—1061no—
156#451—1062no—
157C2×C4⋊C8—21no—
158C2×#101—1062no—
159#407—532no—
160C2×C22⋊C8C2×D5311yescharacteristic
161C4×C22⋊C4C2×D5311yescharacteristic
162#343—1062no—
163#336—21no—
164#328—1062no—
165#328—1062no—
166C53#1055911yescharacteristic, complemented, Sylow
167C22×Q8—532no—
168C23×C4C22×D5311yescharacteristic
169C2×Q16—1061no—
170C2×Q16—1061no—
171C2×M16—22no—
172C2×M16—42no—
173C2×M16—1062no—
174C22×C8—21no—
175C22×C8C22⋊D5311yescharacteristic
176C22×C8C22⋊D5311yescharacteristic
177C22⋊Q8—1062no—
178C22⋊Q8—1062no—
179C22⋊Q8—1062no—
180C22⋊Q8—1062no—
181C2×C4⋊C4—532no—
182C2×C4⋊C4—1062no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—1062no—
185C2×C42—21no—
186#107—1061no—
187#107—1061no—
188C4⋊C8—42no—
189#103—1062no—
190#103—1062no—
191#101—2124no—
192C22⋊C8—22no—
193C22⋊C8—1062no—
194C22⋊C8—1062no—
195#95D21211yescharacteristic
196#95D21211yescharacteristic
197C24#1171994511yescharacteristic
198C2×Q8—1062no—
199C2×Q8—1062no—
200C2×Q8—2122no—
201C2×Q8—2122no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—1062no—
207C22×C4—1062no—
208C22×C4C2×D21211yescharacteristic
209C22×C4D4×D5312yes—
210C22×C4#1171994912yes—
211C22×C4C2×C22⋊D5311yescharacteristic
212Q16—4241no—
213Q16—4241no—
214M16—24no—
215M16—42no—
216M16—2122no—
217M16—2122no—
218C2×C8—22no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—2122no—
226C4⋊C4—1062no—
227C4⋊C4—1062no—
228C4⋊C4—2122no—
229C4⋊C4—2122no—
230C22⋊C4—42no—
231C22⋊C4—2122no—
232C22⋊C4—2122no—
233C42—41no—
234C42—41no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23#1313319111yescharacteristic
240C23#1313320212yes—
241Q8—4242no—
242Q8—2122no—
243Q8—2122no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—4242no—
257C2×C4—2124no—
258C2×C4—2122no—
259C2×C4—2122no—
260C2×C4#1313318511yescharacteristic
261C2×C4#1313319212yes—
262C2×C4#1313320611yescharacteristic
263C8—42no—
264C8—41no—
265C8—41no—
266C8—4242no—
267C22—21no—
268C22—21no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—81no—
273C22—81no—
274C22#1681054011yescharacteristic
275C22#1681089811yescharacteristic
276C22#1681089811yescharacteristic
277C4—22no—
278C4—21no—
279C4—42no—
280C4—81no—
281C4—4242no—
282C4—4242no—
283C4#1681080911yescharacteristic
284C4#1681081211yescharacteristic
285C2—21no—
286C2—21no—
287C2—42no—
288C2#2027596611yescharacteristic
289C2#2027598311yescharacteristic
290C2#2027600211yescharacteristic
subgroupG/H
centerC22#16810540
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1851C2
socleC22×C53#394
prime radicalC24×C53#41
hypercenter#1851D53
hypocenterC53#10559
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
166. C53#10559semidirect

Maximal subgroups

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

subgroupindex
#202854902
#202751832
#202732342
#202725922
C53×#18512
#1055953