Finite Group Databasebeta

← #27286822

Subgroups — #27286822

Order 13568.

504 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 105 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#20285866C211yesmaximal, characteristic
2#20285859C211yesmaximal, characteristic
3#20275692C211yesmaximal, characteristic
4#20273400C211yesmaximal, characteristic
5#20273387C211yesmaximal, characteristic
6#20272548C211yesmaximal, characteristic
7C53×#2862C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16811026—21no—
10#16811019—21no—
11#16810975—21no—
12#16810975—21no—
13#16810958—22no—
14#16810953—22no—
15#16810915—22no—
16#16810905—22no—
17#16810676C2211yescharacteristic
18#16810676C2211yescharacteristic
19#16810647C2211yescharacteristic
20#16810383—22no—
21#16810369—22no—
22#16810369—22no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×C4⋊Q8×C53C2211yescharacteristic
25C53×#391—22no—
26C53×#391—22no—
27C53×#382—22no—
28C43×C53C2211yescharacteristic
29C2×#11719917—21no—
30#13133217D411yescharacteristic
31#13133217D411yescharacteristic
32#13133209—21no—
33#13133196—21no—
34#13133196—21no—
35C4×Dic106—21no—
36C4×Dic106—21no—
37#13133176—21no—
38#13133150—24no—
39#13133148—24no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322D412yes—
43C4⋊#88322D412yes—
44#13133140—24no—
45#13133140—24no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C4×#88322—21no—
49C4×#88322—21no—
50C22×Q8×C53—21no—
51C22×Q8×C53—21no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53D412yes—
63C2×C4⋊C4×C53D412yes—
64C2×C42×C53—21no—
65C2×C42×C53—21no—
66C2×C42×C53D412yes—
67C2×C42×C53C2311yescharacteristic
68#95×C53—28no—
69C2×Q8×C53—44no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C2×Q8×C53—41no—
74C2×Q8×C53—42no—
75C2×Q8×C53—42no—
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C2×Dic106—21no—
83C2×Dic106—21no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C42×C53—24no—
93C42×C53—22no—
94C42×C53—22no—
95C42×C53—42no—
96C42×C53C2×D411yescharacteristic
97C42×C53C2×D411yescharacteristic
98C42×C53C2×D411yescharacteristic
99C42×C53#4111yescharacteristic
100C22⋊Dic53—41no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53—21no—
104C4⋊Dic53—21no—
105#11719919—41no—
106#11719919—41no—
107C4×Dic53—41no—
108#11719917—42no—
109C2×#88322—22no—
110C2×#88322—22no—
111C2×#88322—41no—
112C23×C53#12011yescharacteristic
113Q8×C53—82no—
114Q8×C53—82no—
115Q8×C53—82no—
116Q8×C53—81no—
117Q8×C53—81no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—24no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—44no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—44no—
127C2×C4×C53#12011yescharacteristic
128C2×C4×C53#12011yescharacteristic
129C2×C4×C53#12111yescharacteristic
130C2×C4×C53#12111yescharacteristic
131C2×C4×C53#12111yescharacteristic
132C2×C4×C53#12711yescharacteristic
133C2×C4×C53#13512yes—
134C2×C4×C53#13712yes—
135C2×Dic53—41no—
136C2×Dic53—41no—
137Dic106—42no—
138Dic106—42no—
139#88322—42no—
140#88322—42no—
141#88322—81no—
142#10296—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53#39211yescharacteristic
146C22×C53#45212yes—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—44no—
150C4×C53—82no—
151C4×C53—82no—
152C4×C53#45412yes—
153C4×C53#45612yes—
154C4×C53#46412yes—
155C4×C53#49511yescharacteristic
156C4×C53#49711yescharacteristic
157Dic53—81no—
158Dic53—81no—
159#3230—531no—
160#3229—531no—
161#2862D5311yescharacteristic
162#1994—531no—
163#1549—531no—
164#1546—531no—
165#1276—531no—
166C2×C53—22no—
167C2×C53#200311yescharacteristic
168C2×C53#200511yescharacteristic
169C2×C53#201511yescharacteristic
170#533—531no—
171C2×C4⋊Q8C2×D5311yescharacteristic
172C2×C4⋊Q8C2×D5311yescharacteristic
173#500—1061no—
174#498—1061no—
175#487—1061no—
176#487—1061no—
177#484—1062no—
178#483—1062no—
179#477—1062no—
180#475—1062no—
181#431—531no—
182#431—531no—
183#423—531no—
184#391—22no—
185#391—22no—
186#382—22no—
187C43C2×D5311yescharacteristic
188#339—1062no—
189#332—1062no—
190#332—1062no—
191C53#1029611yescharacteristic, complemented, Sylow
192C22×Q8—21no—
193C22×Q8—21no—
194C2×M16—1061no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—531no—
204C4⋊Q8—531no—
205#125—1061no—
206C22⋊Q8—1061no—
207C22⋊Q8—1061no—
208C4×Q8—1061no—
209C4×Q8—1061no—
210#117—1061no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C22⋊D5312yes—
214C2×C4⋊C4C22⋊D5312yes—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42C22⋊D5312yes—
218C2×C42C22×D5311yescharacteristic
219#107—1064no—
220#106—1064no—
221C4⋊C8—532no—
222C4⋊C8—532no—
223C4⋊C8—1061no—
224C4⋊C8—1061no—
225#103—1064no—
226#103—1064no—
227C22⋊C8—1061no—
228C22⋊C8—1061no—
229C4×C8—1061no—
230C4×C8—1061no—
231#95—28no—
232C2×Q8—44no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—1061no—
240C2×Q8—1061no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4C2×C22⋊D5311yescharacteristic
245C22×C4C2×C22⋊D5311yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247M16—2122no—
248C2×C8—1062no—
249C2×C8—1062no—
250C2×C8—2121no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—1061no—
260C4⋊C4—1061no—
261C4⋊C4—1061no—
262C4⋊C4—1061no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C22⋊C4—2121no—
266C42—24no—
267C42—22no—
268C42—22no—
269C42—42no—
270C42—2121no—
271C42D4×D5311yescharacteristic
272C42D4×D5311yescharacteristic
273C42D4×D5311yescharacteristic
274C42#1171994711yescharacteristic
275C23#1313318611yescharacteristic
276Q8—82no—
277Q8—81no—
278Q8—82no—
279Q8—82no—
280Q8—81no—
281Q8—2122no—
282Q8—2122no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4#1313318511yescharacteristic
295C2×C4#1313318511yescharacteristic
296C2×C4#1313318811yescharacteristic
297C2×C4#1313318911yescharacteristic
298C2×C4#1313318911yescharacteristic
299C2×C4#1313321311yescharacteristic
300C2×C4#1313323712yes—
301C2×C4#1313324712yes—
302C8—4241no—
303C8—2122no—
304C8—2122no—
305C22—21no—
306C22—21no—
307C22#1681053111yescharacteristic
308C22#1681078512yes—
309C4—21no—
310C4—21no—
311C4—44no—
312C4—82no—
313C4—82no—
314C4—4241no—
315C4—4241no—
316C4#1681079512yes—
317C4#1681080512yes—
318C4#1681084212yes—
319C4#1681100511yescharacteristic
320C4#1681101811yescharacteristic
321C2—22no—
322C2#2027573711yescharacteristic
323C2#2027575111yescharacteristic
324C2#2027580911yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2862C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#2862D53
hypocenterC53#10296
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
191. C53#10296semidirect

Maximal subgroups

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

subgroupindex
#202858662
#202858592
#202756922
#202734002
#202733872
#202725482
C53×#28622
#1029653