Finite Group Databasebeta

← #27286819

Subgroups — #27286819

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#20285863C211yesmaximal, characteristic
2#20283092C211yesmaximal, characteristic
3#20275693C211yesmaximal, characteristic
4#20273399C211yesmaximal, characteristic
5#20273387C211yesmaximal, characteristic
6#20272547C211yesmaximal, characteristic
7C53×#3229C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9#16811026—21no—
10#16810973—21no—
11#16810956—22no—
12#16810918—22no—
13#16810676C2211yescharacteristic
14#16810647C2211yescharacteristic
15#16810530—22no—
16#16810528—22no—
17#16810488—22no—
18C42×Dic53C2211yescharacteristic
19#16810382—22no—
20#16810370—22no—
21#16810369—22no—
22C53×#533C2211yescharacteristic
23C2×C4⋊Q8×C53C2211yescharacteristic
24C53×#498—21no—
25C53×#487—21no—
26C53×#483—22no—
27C53×#475—22no—
28C53×#431C2211yescharacteristic
29C22×Dic106—21no—
30C2×#11719917—21no—
31#13133217—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53D412yes—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53D412yes—
40#13133150—24no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322D412yes—
44#13133141—24no—
45C22⋊#88322—21no—
46C4×#88322—21no—
47#13133124—28no—
48C22×Q8×C53—21no—
49C4⋊Q8×C53—21no—
50C4⋊Q8×C53—21no—
51C4⋊Q8×C53—21no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53D411yescharacteristic
54C4⋊Q8×C53D411yescharacteristic
55#125×C53—21no—
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58C4×Q8×C53—21no—
59C4×Q8×C53—21no—
60#117×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53D412yes—
63C2×C42×C53C2311yescharacteristic
64#106×C53—24no—
65C4⋊C8×C53D412yes—
66#103×C53—24no—
67C22⋊C8×C53—21no—
68C4×C8×C53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C2×Q8×C53—42no—
74C2×Q8×C53—42no—
75C22×C4×C53—22no—
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×Dic53—22no—
80C22×Dic53—22no—
81C2×Dic106—44no—
82C2×Dic106—41no—
83C2×Dic106—41no—
84C2×C8×C53—22no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C42×C53—41no—
97C42×C53C2×D411yescharacteristic
98C42×C53C2×D411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53#4111yescharacteristic
101C4⋊Dic53—22no—
102C4⋊Dic53—22no—
103C4⋊Dic53—41no—
104C4⋊Dic53—41no—
105C4×Dic53—24no—
106C4×Dic53—22no—
107C4×Dic53—22no—
108C4×Dic53—42no—
109#11719917—42no—
110C2×#88322—22no—
111C2×#88322—41no—
112C23×C53#12011yescharacteristic
113Q8×C53—42no—
114Q8×C53—42no—
115Q8×C53—82no—
116Q8×C53—81no—
117Q8×C53—81no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—42no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—44no—
124C2×C4×C53#12011yescharacteristic
125C2×C4×C53#12011yescharacteristic
126C2×C4×C53#12111yescharacteristic
127C2×C4×C53#12111yescharacteristic
128C2×C4×C53#12111yescharacteristic
129C2×C4×C53#12711yescharacteristic
130C2×C4×C53#13512yes—
131C2×C4×C53#13712yes—
132C2×Dic53—24no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—44no—
136C2×Dic53—42no—
137Dic106—82no—
138Dic106—82no—
139C8×C53—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—81no—
150C4×C53—81no—
151C4×C53—82no—
152C4×C53#45412yes—
153C4×C53#45612yes—
154C4×C53#46412yes—
155C4×C53#49511yescharacteristic
156C4×C53#49711yescharacteristic
157Dic53—44no—
158Dic53—82no—
159#3230—531no—
160#3229D5311yescharacteristic
161#2862—531no—
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#533C2×D5311yescharacteristic
171C2×C4⋊Q8—531no—
172C2×C4⋊Q8C2×D5311yescharacteristic
173#500—1061no—
174#498—21no—
175#487—21no—
176#487—1061no—
177#484—1062no—
178#483—22no—
179#477—1062no—
180#475—22no—
181#431—531no—
182#431C2×D5311yescharacteristic
183#423—531no—
184#391—1062no—
185#391—1062no—
186#382—1062no—
187C43—531no—
188#339—1062no—
189#332—1062no—
190#332—1062no—
191C53#1029611yescharacteristic, complemented, Sylow
192C22×Q8—21no—
193C22×Q8—1061no—
194C2×M16—1061no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—1061no—
200C4⋊Q8—1061no—
201C4⋊Q8—1061no—
202C4⋊Q8—1061no—
203C4⋊Q8C22⋊D5311yescharacteristic
204C4⋊Q8C22⋊D5311yescharacteristic
205#125—21no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210#117—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—532no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4C22⋊D5312yes—
215C2×C42—532no—
216C2×C42—1061no—
217C2×C42—1061no—
218C2×C42C22×D5311yescharacteristic
219#107—1064no—
220#106—24no—
221C4⋊C8—532no—
222C4⋊C8—1061no—
223C4⋊C8—1061no—
224C4⋊C8C22⋊D5312yes—
225#103—24no—
226#103—1064no—
227C22⋊C8—21no—
228C22⋊C8—1061no—
229C4×C8—21no—
230C4×C8—1061no—
231#95—1068no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—2124no—
239C2×Q8—2121no—
240C2×Q8—2121no—
241C22×C4—22no—
242C22×C4—1062no—
243C22×C4—1062no—
244C22×C4D4×D5311yescharacteristic
245C22×C4D4×D5311yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247M16—2122no—
248C2×C8—22no—
249C2×C8—1062no—
250C2×C8—2121no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—1062no—
262C4⋊C4—1062no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C22⋊C4—41no—
266C42—41no—
267C42—1064no—
268C42—1062no—
269C42—1062no—
270C42—2122no—
271C42D4×D5311yescharacteristic
272C42#1171994711yescharacteristic
273C42C2×C22⋊D5311yescharacteristic
274C42C2×C22⋊D5311yescharacteristic
275C23#1313318511yescharacteristic
276Q8—42no—
277Q8—42no—
278Q8—81no—
279Q8—82no—
280Q8—81no—
281Q8—4242no—
282Q8—4242no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—1064no—
290C2×C4—2124no—
291C2×C4—2122no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4#1313318511yescharacteristic
295C2×C4#1313318611yescharacteristic
296C2×C4#1313318811yescharacteristic
297C2×C4#1313318911yescharacteristic
298C2×C4#1313318911yescharacteristic
299C2×C4#1313321311yescharacteristic
300C2×C4#1313323812yes—
301C2×C4#1313324612yes—
302C8—42no—
303C8—4241no—
304C8—2122no—
305C22—21no—
306C22—21no—
307C22#1681053111yescharacteristic
308C22#1681078212yes—
309C4—21no—
310C4—21no—
311C4—81no—
312C4—81no—
313C4—82no—
314C4—4242no—
315C4—2124no—
316C4#1681079612yes—
317C4#1681080412yes—
318C4#1681084312yes—
319C4#1681100411yescharacteristic
320C4#1681101411yescharacteristic
321C2—22no—
322C2#2027573511yescharacteristic
323C2#2027574811yescharacteristic
324C2#2027580711yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3229C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#3229D53
hypocenterC53#10296
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

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
#202830922
#202756932
#202733992
#202733872
#202725472
C53×#32292
#1029653