Finite Group Databasebeta

← #27287875

Subgroups — #27287875

Order 13568.

366 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 93 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#20284520C211yesmaximal, characteristic
2#20273489C211yesmaximal, characteristic
3#20273373C211yesmaximal, characteristic
4#20273278C211yesmaximal, characteristic
5#20273039C211yesmaximal, characteristic
6#20272573C211yesmaximal, characteristic
7C53×#1805C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810930—22no—
10#16810918—21no—
11#16810908—21no—
12#16810676C2211yescharacteristic
13#16810676C2211yescharacteristic
14#16810663—21no—
15#16810647C2211yescharacteristic
16#16810632—21no—
17#16810375—22no—
18#16810371—21no—
19#16810368—21no—
20#16810359—22no—
21#16810349—22no—
22C2×C4×Q8×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×#103×C53—21no—
25C53×#382C2211yescharacteristic
26C53×#340—21no—
27C2×#11719917—21no—
28#13133217D411yescharacteristic
29#13133217D411yescharacteristic
30#13133209—21no—
31#13133196—21no—
32#13133196—21no—
33C4×Dic106—21no—
34C4×Dic106—21no—
35#13133176—21no—
36#13133150—22no—
37#13133148—22no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322D412yes—
42#13133141—22no—
43#13133141—22no—
44C22⋊#88322—21no—
45C22⋊#88322D412yes—
46C4×#88322—21no—
47C4×#88322—21no—
48C22×Q8×C53D411yescharacteristic
49C22×C8×C53—21no—
50C4×Q8×C53—21no—
51C4×Q8×C53—21no—
52C4×Q8×C53—21no—
53C4×Q8×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C42×C53—21no—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61#103×C53—42no—
62#95×C53—21no—
63#95×C53—21no—
64C2×Q8×C53—21no—
65C2×Q8×C53—21no—
66C2×Q8×C53—21no—
67C2×Q8×C53—21no—
68C2×Q8×C53—21no—
69C2×Q8×C53Q1612yes—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—41no—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53#4111yescharacteristic
76C2×Dic106—21no—
77C2×Dic106—21no—
78C2×C8×C53—41no—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C42×C53—41no—
88C42×C53—41no—
89C42×C53C2×D411yescharacteristic
90C42×C53C2×D411yescharacteristic
91C42×C53#4111yescharacteristic
92C42×C53#4111yescharacteristic
93C22⋊Dic53—41no—
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98#11719919—41no—
99#11719919—41no—
100C4×Dic53—41no—
101#11719917—41no—
102#11719917—41no—
103C2×#88322—22no—
104C2×#88322—22no—
105C2×#88322—41no—
106C23×C53#12111yescharacteristic
107Q8×C53—22no—
108Q8×C53—22no—
109Q8×C53—41no—
110Q8×C53—41no—
111C2×C4×C53—22no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—81no—
118C2×C4×C53—81no—
119C2×C4×C53—81no—
120C2×C4×C53—81no—
121C2×C4×C53#12011yescharacteristic
122C2×C4×C53#12111yescharacteristic
123C2×C4×C53#12111yescharacteristic
124C2×C4×C53#12311yescharacteristic
125C2×C4×C53#12311yescharacteristic
126C2×C4×C53#12411yescharacteristic
127C2×C4×C53C2×Q1611yescharacteristic
128C2×C4×C53#13511yescharacteristic
129C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C53#13711yescharacteristic
131C2×Dic53—41no—
132C2×Dic53—41no—
133Dic106—81no—
134Dic106—81no—
135C8×C53—81no—
136#88322—42no—
137#88322—42no—
138#88322—81no—
139#10471—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53#39411yescharacteristic
143C22×C53#46711yescharacteristic
144C22×C53#46811yescharacteristic
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—42no—
150C4×C53—81no—
151C4×C53—161no—
152C4×C53#44911yescharacteristic
153C4×C53#45111yescharacteristic
154C4×C53#48411yescharacteristic
155C4×C53#48511yescharacteristic
156Dic53—81no—
157Dic53—81no—
158#3078—531no—
159#1805D5311yescharacteristic
160#1571—531no—
161#1542—531no—
162#1518—531no—
163#1461—531no—
164#1286—531no—
165C2×C53—21no—
166C2×C53—21no—
167C2×C53#203611yescharacteristic
168C2×C53#204911yescharacteristic
169C2×C53#205311yescharacteristic
170#533—531no—
171C2×C4×Q8C2×D5311yescharacteristic
172#479—1062no—
173#477—1061no—
174#475—1061no—
175#431—531no—
176#431—531no—
177#428—1061no—
178#423—531no—
179C2×C4⋊C8C2×D5311yescharacteristic
180#419—1061no—
181C2×#103—21no—
182#382C2×D5311yescharacteristic
183#340—21no—
184#335—1062no—
185#333—1061no—
186#332—1061no—
187#328—1062no—
188#324—1062no—
189C53#1047111yescharacteristic, complemented, Sylow
190C22×Q8C22⋊D5311yescharacteristic
191C2×M16—1061no—
192C22×C8—21no—
193C4⋊Q8—531no—
194C4⋊Q8—531no—
195#125—1061no—
196C22⋊Q8—1061no—
197C22⋊Q8—1061no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—1061no—
203C4×Q8—1061no—
204#117—1061no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C22⋊D5311yescharacteristic
208C2×C42—21no—
209C2×C42C22×D5311yescharacteristic
210#107—1062no—
211#106—1062no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—532no—
215C4⋊C8—1061no—
216C4⋊C8—1061no—
217C4⋊C8—1061no—
218#103—42no—
219#103—1062no—
220#103—1062no—
221C22⋊C8—532no—
222C22⋊C8—1061no—
223C4×C8—1061no—
224C4×C8—1061no—
225#95—21no—
226#95—21no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—1061no—
233C2×Q8—1061no—
234C2×Q8#1171992512yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4D4×D5311yescharacteristic
239C22×C4#1171994711yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241M16—2121no—
242M16—2121no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—1062no—
247C2×C8—1062no—
248C2×C8—2121no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—1061no—
256C4⋊C4—1061no—
257C4⋊C4—1061no—
258C4⋊C4—1061no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C22⋊C4—2121no—
262C42—41no—
263C42—41no—
264C42—2121no—
265C42#1171994511yescharacteristic
266C42#1171994511yescharacteristic
267C42D4×D5311yescharacteristic
268C42D4×D5311yescharacteristic
269C23#1313318911yescharacteristic
270Q8—22no—
271Q8—22no—
272Q8—41no—
273Q8—41no—
274Q8—4241no—
275Q8—4241no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4#1313318511yescharacteristic
289C2×C4#1313319011yescharacteristic
290C2×C4#1313319011yescharacteristic
291C2×C4#1313319911yescharacteristic
292C2×C4#1313320011yescharacteristic
293C2×C4#1313320311yescharacteristic
294C2×C4C2×#1171992511yescharacteristic
295C2×C4#1313323711yescharacteristic
296C2×C4#1313324311yescharacteristic
297C2×C4#1313324711yescharacteristic
298C8—81no—
299C8—4241no—
300C8—2122no—
301C8—2122no—
302C22—21no—
303C22—21no—
304C22#1681053811yescharacteristic
305C22#1681086211yescharacteristic
306C22#1681086711yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—42no—
312C4—81no—
313C4—161no—
314C4—4241no—
315C4—4241no—
316C4#1681076611yescharacteristic
317C4#1681077811yescharacteristic
318C4#1681095811yescharacteristic
319C4#1681096411yescharacteristic
320C2—21no—
321C2—21no—
322C2#2027591811yescharacteristic
323C2#2027599311yescharacteristic
324C2#2027601911yescharacteristic
subgroupG/H
centerC22#16810538
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1805C2
socleC22×C53#394
prime radicalC23×C53#121
hypercenter#1805D53
hypocenterC53#10471
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
189. C53#10471semidirect

Maximal subgroups

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

subgroupindex
#202845202
#202734892
#202733732
#202732782
#202730392
#202725732
C53×#18052
#1047153