Finite Group Databasebeta

← #27286803

Subgroups — #27286803

Order 13568.

588 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 113 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#20285663C211yesmaximal, characteristic
3#20275692C211yesmaximal, characteristic
4#20273400C211yesmaximal, characteristic
5#20273396C211yesmaximal, characteristic
6#20272548C211yesmaximal, characteristic
7C53×#2868C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16811026—21no—
10#16811002—21no—
11#16810975—21no—
12#16810973—21no—
13#16810958—22no—
14#16810915—22no—
15#16810830—22no—
16#16810779—22no—
17#16810676C2211yescharacteristic
18#16810676C2211yescharacteristic
19#16810647C2211yescharacteristic
20#16810383—22no—
21#16810372—22no—
22#16810369—22no—
23C2×C4⋊Q8×C53—22no—
24C2×C4⋊Q8×C53C2211yescharacteristic
25C2×C4⋊Q8×C53C2211yescharacteristic
26C53×#384—22no—
27C53×#384—22no—
28C43×C53C2211yescharacteristic
29C2×#11719925—24no—
30C2×#11719917—21no—
31#13133217D411yescharacteristic
32#13133217D411yescharacteristic
33#13133209—21no—
34#13133196—21no—
35#13133196—21no—
36C4×Dic106—21no—
37C4×Dic106—21no—
38#13133176—21no—
39#13133150—24no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322D412yes—
43C4⋊#88322D412yes—
44#13133141—24no—
45#13133140—24no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C4×#88322—21no—
49C4×#88322—21no—
50C22×Q8×C53—21no—
51C22×Q8×C53D412yes—
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—
60C4⋊Q8×C53—48no—
61C2×C4⋊C4×C53—28no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D412yes—
66C2×C42×C53—21no—
67C2×C42×C53—21no—
68C2×C42×C53D412yes—
69C2×C42×C53C2311yescharacteristic
70C2×Q8×C53—22no—
71C2×Q8×C53—22no—
72C2×Q8×C53—44no—
73C2×Q8×C53—44no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
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—416no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C42×C53—24no—
94C42×C53—24no—
95C42×C53—22no—
96C42×C53Q1614yescomplemented
97C42×C53C2×D411yescharacteristic
98C42×C53C2×D411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53#4111yescharacteristic
101C22⋊Dic53—41no—
102#11719925—84no—
103C4⋊Dic53—21no—
104C4⋊Dic53—21no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107#11719919—41no—
108#11719919—41no—
109C4×Dic53—41no—
110#11719917—42no—
111C2×#88322—22no—
112C2×#88322—22no—
113C2×#88322—41no—
114C23×C53#12011yescharacteristic
115Q8×C53—44no—
116Q8×C53—82no—
117Q8×C53—82no—
118Q8×C53—82no—
119C2×C4×C53—24no—
120C2×C4×C53—24no—
121C2×C4×C53—24no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—44no—
125C2×C4×C53—44no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53#12011yescharacteristic
129C2×C4×C53#12011yescharacteristic
130C2×C4×C53#12111yescharacteristic
131C2×C4×C53#12111yescharacteristic
132C2×C4×C53#12111yescharacteristic
133C2×C4×C53#12711yescharacteristic
134C2×C4×C53C2×Q1612yes—
135C2×C4×C53#13712yes—
136C2×Dic53—41no—
137C2×Dic53—41no—
138Dic106—42no—
139Dic106—42no—
140#88322—42no—
141#88322—42no—
142#88322—81no—
143#10293—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53#39211yescharacteristic
147C22×C53#45112yes—
148C4×C53—28no—
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—82no—
152C4×C53—82no—
153C4×C53#45312yes—
154C4×C53#45612yes—
155C4×C53#46212yes—
156C4×C53#49411yescharacteristic
157C4×C53#49711yescharacteristic
158Dic53—81no—
159Dic53—81no—
160#3230—531no—
161#3209—531no—
162#2868D5311yescharacteristic
163#1994—531no—
164#1549—531no—
165#1548—531no—
166#1276—531no—
167C2×C53—22no—
168C2×C53#200211yescharacteristic
169C2×C53#200511yescharacteristic
170C2×C53#201611yescharacteristic
171#533—531no—
172C2×C4⋊Q8—22no—
173C2×C4⋊Q8C2×D5311yescharacteristic
174C2×C4⋊Q8C2×D5311yescharacteristic
175#500—1061no—
176#494—1061no—
177#487—1061no—
178#487—1061no—
179#484—1062no—
180#477—1062no—
181#462—1062no—
182#451—1062no—
183#431—531no—
184#431—531no—
185#423—531no—
186#384—22no—
187#384—22no—
188C43C2×D5311yescharacteristic
189#339—1062no—
190#333—1062no—
191#332—1062no—
192C53#1029311yescharacteristic, complemented, Sylow
193C22×Q8—21no—
194C22×Q8C22⋊D5312yes—
195C2×Q16—1064no—
196C2×M16—1061no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—48no—
206C4⋊Q8—531no—
207C4⋊Q8—531no—
208#125—1061no—
209C22⋊Q8—1061no—
210C22⋊Q8—1061no—
211C4×Q8—1061no—
212C4×Q8—1061no—
213#117—1061no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C22⋊D5312yes—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42C22⋊D5312yes—
222C2×C42C22×D5311yescharacteristic
223#107—1064no—
224C4⋊C8—532no—
225C4⋊C8—532no—
226C4⋊C8—1061no—
227C4⋊C8—1061no—
228#103—1064no—
229#103—1064no—
230C22⋊C8—1061no—
231C22⋊C8—1061no—
232C4×C8—1061no—
233C4×C8—1061no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—1061no—
241C2×Q8—1061no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4C2×C22⋊D5311yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247C22×C4C2×C22⋊D5311yescharacteristic
248Q16—4244no—
249M16—2122no—
250C2×C8—1062no—
251C2×C8—1062no—
252C2×C8—2121no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—416no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—1061no—
263C4⋊C4—1061no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C22⋊C4—2121no—
269C42—24no—
270C42—24no—
271C42—22no—
272C42—2121no—
273C42#1171992514yescomplemented
274C42D4×D5311yescharacteristic
275C42D4×D5311yescharacteristic
276C42D4×D5311yescharacteristic
277C42#1171994711yescharacteristic
278C23#1313318611yescharacteristic
279Q8—44no—
280Q8—82no—
281Q8—82no—
282Q8—82no—
283Q8—2122no—
284Q8—2122no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4#1313318511yescharacteristic
297C2×C4#1313318511yescharacteristic
298C2×C4#1313318811yescharacteristic
299C2×C4#1313318911yescharacteristic
300C2×C4#1313318911yescharacteristic
301C2×C4#1313321311yescharacteristic
302C2×C4C2×#1171992512yes—
303C2×C4#1313324812yes—
304C8—4241no—
305C8—2122no—
306C8—2122no—
307C22—21no—
308C22—21no—
309C22#1681053111yescharacteristic
310C22#1681077912yes—
311C4—28no—
312C4—21no—
313C4—21no—
314C4—82no—
315C4—82no—
316C4—4241no—
317C4—4241no—
318C4#1681079012yes—
319C4#1681080512yes—
320C4#1681083012yes—
321C4#1681100211yescharacteristic
322C4#1681101511yescharacteristic
323C2—22no—
324C2#2027573011yescharacteristic
325C2#2027575111yescharacteristic
326C2#2027581511yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2868C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#2868D53
hypocenterC53#10293
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2868

Complements

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

NH ≅ G/Ntype
96. C42×C53Q16semidirect
192. C53#10293semidirect
273. C42#11719925semidirect

Maximal subgroups

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

subgroupindex
#202858662
#202856632
#202756922
#202734002
#202733962
#202725482
C53×#28682
#1029353