Finite Group Databasebeta

← #27286821

Subgroups — #27286821

Order 13568.

504 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 103 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#20285863C211yesmaximal, characteristic
2#20285856C211yesmaximal, characteristic
3#20283093C211yesmaximal, characteristic
4#20273401C211yesmaximal, characteristic
5#20273388C211yesmaximal, characteristic
6#20272548C211yesmaximal, characteristic
7C53×#1994C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9C2×#13133217C2211yescharacteristic
10#16811026—21no—
11#16811019—21no—
12#16810973—21no—
13#16810973—21no—
14#16810956—22no—
15#16810951—22no—
16#16810918—22no—
17#16810908—22no—
18#16810676C2211yescharacteristic
19#16810676C2211yescharacteristic
20#16810529—22no—
21#16810529—22no—
22#16810489—22no—
23#16810370—22no—
24#16810368—22no—
25C53×#533C2211yescharacteristic
26C53×#423C2211yescharacteristic
27C43×C53C2211yescharacteristic
28C53×#339—22no—
29C22×Dic106—21no—
30C22×Dic106—21no—
31#13133217—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217—21no—
37#13133217—21no—
38#13133217—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53D412yes—
42C2×C4⋊Dic53D412yes—
43#13133150—24no—
44#13133148—24no—
45C4⋊#88322D412yes—
46C4⋊#88322D412yes—
47#13133141—24no—
48#13133141—24no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C4×#88322—21no—
52C4×#88322—21no—
53#13133124—28no—
54C2×M16×C53—21no—
55C4⋊Q8×C53D411yescharacteristic
56C4⋊Q8×C53D411yescharacteristic
57#125×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62#117×C53—21no—
63C2×C42×C53—21no—
64C2×C42×C53—21no—
65C2×C42×C53D412yes—
66C2×C42×C53C2311yescharacteristic
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C22×C4×C53—22no—
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—22no—
76C22×Dic53—22no—
77C2×Dic106—44no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C2×Dic106—42no—
83C2×Dic106—42no—
84M16×C53—42no—
85C2×C8×C53—41no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C42×C53—24no—
94C42×C53—22no—
95C42×C53—22no—
96C42×C53—42no—
97C42×C53—41no—
98C42×C53C2×D411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53#4111yescharacteristic
102C4⋊Dic53—22no—
103C4⋊Dic53—22no—
104C4⋊Dic53—22no—
105C4⋊Dic53—22no—
106C4⋊Dic53—41no—
107C4⋊Dic53—41no—
108C4⋊Dic53—41no—
109C4⋊Dic53—41no—
110C2×#88322—22no—
111C2×#88322—22no—
112C23×C53#12011yescharacteristic
113Q8×C53—42no—
114Q8×C53—42no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—24no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53#12011yescharacteristic
123C2×C4×C53#12011yescharacteristic
124C2×C4×C53#12111yescharacteristic
125C2×C4×C53#12111yescharacteristic
126C2×C4×C53#12111yescharacteristic
127C2×C4×C53#12711yescharacteristic
128C2×C4×C53#13512yes—
129C2×C4×C53#13712yes—
130C2×Dic53—44no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—44no—
134Dic106—82no—
135Dic106—82no—
136Dic106—82no—
137Dic106—81no—
138Dic106—81no—
139C8×C53—81no—
140#88322—42no—
141#88322—42no—
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—81no—
151C4×C53—81no—
152C4×C53#45412yes—
153C4×C53#45612yes—
154C4×C53#46412yes—
155C4×C53#49511yescharacteristic
156C4×C53#49711yescharacteristic
157Dic53—82no—
158Dic53—82no—
159#3230—531no—
160#3229—531no—
161#2862—531no—
162#1994D5311yescharacteristic
163#1549—531no—
164#1546—531no—
165#1276—531no—
166C2×C53—22no—
167C2×C53#200311yescharacteristic
168C2×C53#200511yescharacteristic
169C2×C53#201511yescharacteristic
170#533C2×D5311yescharacteristic
171C2×C4⋊Q8—531no—
172C2×C4⋊Q8—531no—
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#423C2×D5311yescharacteristic
184#391—1062no—
185#391—1062no—
186#382—1062no—
187C43C2×D5311yescharacteristic
188#339—22no—
189#332—1062no—
190#332—1062no—
191C53#1029611yescharacteristic, complemented, Sylow
192C22×Q8—1061no—
193C22×Q8—1061no—
194C2×M16—21no—
195C4⋊Q8—1061no—
196C4⋊Q8—1061no—
197C4⋊Q8—1061no—
198C4⋊Q8—1061no—
199C4⋊Q8—1061no—
200C4⋊Q8—1061no—
201C4⋊Q8—1061no—
202C4⋊Q8—1061no—
203C4⋊Q8D21211yescharacteristic
204C4⋊Q8D21211yescharacteristic
205#125—21no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210#117—21no—
211C2×C4⋊C4—532no—
212C2×C4⋊C4—532no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42C22⋊D5312yes—
218C2×C42C22×D5311yescharacteristic
219#107—1064no—
220#106—1064no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—532no—
224C4⋊C8—532no—
225#103—1064no—
226#103—1064no—
227C22⋊C8—1061no—
228C22⋊C8—1061no—
229C4×C8—1061no—
230C4×C8—1061no—
231#95—1068no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—2124no—
235C2×Q8—2121no—
236C2×Q8—2121no—
237C2×Q8—2121no—
238C2×Q8—2121no—
239C2×Q8—2122no—
240C2×Q8—2122no—
241C22×C4—22no—
242C22×C4—1062no—
243C22×C4—1062no—
244C22×C4D4×D5311yescharacteristic
245C22×C4D4×D5311yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247M16—42no—
248C2×C8—41no—
249C2×C8—1062no—
250C2×C8—1062no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—1062no—
258C4⋊C4—1062no—
259C4⋊C4—1062no—
260C4⋊C4—1062no—
261C4⋊C4—2121no—
262C4⋊C4—2121no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C22⋊C4—41no—
266C42—24no—
267C42—22no—
268C42—22no—
269C42—42no—
270C42—41no—
271C42C2×D21211yescharacteristic
272C42#1171994511yescharacteristic
273C42D4×D5311yescharacteristic
274C42D4×D5311yescharacteristic
275C23#1313318511yescharacteristic
276Q8—42no—
277Q8—42no—
278Q8—4242no—
279Q8—4241no—
280Q8—4242no—
281Q8—4242no—
282Q8—4241no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—2124no—
291C2×C4—2124no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4#1313318711yescharacteristic
295C2×C4#1313318711yescharacteristic
296C2×C4#1313319011yescharacteristic
297C2×C4#1313319011yescharacteristic
298C2×C4#1313319111yescharacteristic
299C2×C4#1313321311yescharacteristic
300C2×C4#1313323712yes—
301C2×C4#1313324812yes—
302C8—81no—
303C8—2122no—
304C8—2122no—
305C22—21no—
306C22—21no—
307C22#1681053211yescharacteristic
308C22#1681078312yes—
309C4—21no—
310C4—21no—
311C4—44no—
312C4—81no—
313C4—81no—
314C4—4242no—
315C4—4242no—
316C4#1681079712yes—
317C4#1681080712yes—
318C4#1681084512yes—
319C4#1681100511yescharacteristic
320C4#1681101511yescharacteristic
321C2—22no—
322C2#2027573611yescharacteristic
323C2#2027575011yescharacteristic
324C2#2027580811yescharacteristic
subgroupG/H
centerC22#16810532
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1994C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#1994D53
hypocenterC53#10296
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1994

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
#202858632
#202858562
#202830932
#202734012
#202733882
#202725482
C53×#19942
#1029653