Finite Group Databasebeta

← #27281024

Subgroups — #27281024

Order 13568.

444 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277366C211yesmaximal, characteristic
2#20275660C211yesmaximal, characteristic
3#20273227C211yesmaximal, characteristic
4#20273162C211yesmaximal, characteristic
5#20272607C211yesmaximal, characteristic
6#20272572C211yesmaximal, characteristic
7C53×#3113C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9C4×#11719917C2211yescharacteristic
10#16810530—21no—
11#16810522—21no—
12#16810509—22no—
13#16810491—21no—
14#16810489—21no—
15C4×C4⋊Dic53C2211yescharacteristic
16#16810469C2211yescharacteristic
17#16810382—22no—
18#16810360—21no—
19#16810360—21no—
20Q8⋊#88322—22no—
21C53×#533C2211yescharacteristic
22C2×C4×Q8×C53C2211yescharacteristic
23C53×#483—21no—
24C53×#475C412yescomplemented
25C53×#462C412yescomplemented
26C53×#451—21no—
27C53×#441—22no—
28C53×#432C2211yescharacteristic
29C2×#11719917—21no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53D412yes—
34C4⋊#88322—21no—
35C22⋊#88322—21no—
36#13133128—21no—
37#13133128—21no—
38C4×#88322—21no—
39#13133124—22no—
40#13133124—22no—
41#13133124—22no—
42#13133124—21no—
43#13133124—21no—
44#13133124D411yescharacteristic
45#13133124D411yescharacteristic
46C22×Q8×C53D411yescharacteristic
47C2×Q16×C53—22no—
48C4⋊Q8×C53C2×C411yescharacteristic
49C4⋊Q8×C53C2×C411yescharacteristic
50#125×C53—21no—
51C22⋊Q8×C53—21no—
52C22⋊Q8×C53—21no—
53C4×Q8×C53—21no—
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53C2×C412yes—
58C4×Q8×C53D412yes—
59#117×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C42×C53—21no—
63C2×C42×C53C2311yescharacteristic
64#106×C53—22no—
65C4⋊C8×C53C2×C412yes—
66#103×C53—22no—
67#103×C53—22no—
68C22⋊C8×C53—21no—
69#97×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53—42no—
74C2×Q8×C53C22⋊C412yes—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—22no—
82C22×Dic53—21no—
83Q16×C53—44no—
84C2×C8×C53—22no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53C22⋊C412yes—
95C22⋊C4×C53—41no—
96C42×C53—22no—
97C42×C53—22no—
98C42×C53—41no—
99C42×C53C22×C411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53#4111yescharacteristic
102C42×C53#4111yescharacteristic
103C4⋊Dic53—42no—
104C4⋊Dic53—82no—
105C4×Dic53—42no—
106C4×Dic53—42no—
107#11719917—22no—
108#11719917—41no—
109C2×#88322—41no—
110C2×#88322—41no—
111C23×C53#12011yescharacteristic
112Q8×C53—22no—
113Q8×C53—22no—
114Q8×C53—42no—
115Q8×C53—42no—
116Q8×C53—81no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—41no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53C2×C22⋊C411yescharacteristic
129C2×C4×C53C4×D411yescharacteristic
130C2×C4×C53C4×D411yescharacteristic
131C2×C4×C53#12111yescharacteristic
132C2×C4×C53#12111yescharacteristic
133C2×C4×C53#12311yescharacteristic
134C2×Dic53—41no—
135C2×Dic53—42no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C8×C53—42no—
144#88322—42no—
145#88322—81no—
146#9328—531nomaximal, Sylow
147C22×C53—21no—
148C22×C53—21no—
149C22×C53#38611yescharacteristic
150C22×C53#45511yescharacteristic
151C22×C53#45611yescharacteristic
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—42no—
158C4×C53—42no—
159C4×C53—81no—
160C4×C53—81no—
161C4×C53#42112yes—
162Dic53—81no—
163Dic53—82no—
164Dic53—81no—
165#3113D5311yescharacteristic
166#2320—531no—
167#1986—531no—
168#1506—531no—
169#1491—531no—
170#1300—531no—
171#1285—531no—
172C2×C53—21no—
173C2×C53—21no—
174C2×C53#188211yescharacteristic
175C2×C53#189711yescharacteristic
176C2×C53#190511yescharacteristic
177#533C2×D5311yescharacteristic
178C2×C4×Q8C2×D5311yescharacteristic
179#483—21no—
180#475Dic5312yescomplemented
181#462Dic5312yescomplemented
182#451—21no—
183#441—22no—
184#432—531no—
185#432C2×D5311yescharacteristic
186C4×M16—531no—
187#391—1061no—
188#389—1061no—
189#387—1062no—
190#383—1061no—
191#382—1061no—
192C4×C4⋊C4—531no—
193#376—531no—
194#339—1062no—
195#328—1061no—
196#328—1061no—
197Q8⋊C8—1062no—
198C53#932811yescharacteristic, complemented, Sylow
199C22×Q8C22⋊D5311yescharacteristic
200C2×Q16—22no—
201C2×M16—1061no—
202C4⋊Q8C2×Dic5311yescharacteristic
203C4⋊Q8C2×Dic5311yescharacteristic
204#125—21no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8C2×Dic5312yes—
212C4×Q8C22⋊D5312yes—
213#117—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—1062no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4C22⋊D5311yescharacteristic
218C2×C42—21no—
219C2×C42—532no—
220C2×C42—1061no—
221C2×C42C22×D5311yescharacteristic
222#106—22no—
223C4⋊C8—1061no—
224C4⋊C8C2×Dic5312yes—
225#103—22no—
226#103—22no—
227C22⋊C8—21no—
228C22⋊C8—1061no—
229#97—21no—
230#97—1061no—
231#97—1061no—
232C4×C8—1061no—
233#95—531no—
234#95—531no—
235#95—1062no—
236#95—1062no—
237#95—1062no—
238#95—1061no—
239#95—1061no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8C22⋊Dic5312yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4—1062no—
250C22×C4D4×D5311yescharacteristic
251C22×C4D4×D5311yescharacteristic
252C22×C4C2×C22⋊D5311yescharacteristic
253Q16—44no—
254M16—1062no—
255M16—2121no—
256C2×C8—22no—
257C2×C8—2121no—
258C2×C8—2121no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—4242no—
269C4⋊C4—2122no—
270C4⋊C4C22⋊Dic5312yes—
271C22⋊C4—41no—
272C42—22no—
273C42—22no—
274C42—41no—
275C42—2122no—
276C42—2122no—
277C42#1171994711yescharacteristic
278C42#1171994711yescharacteristic
279C42C22×Dic5311yescharacteristic
280C42C2×C22⋊D5311yescharacteristic
281C23#1313318511yescharacteristic
282Q8—22no—
283Q8—22no—
284Q8—42no—
285Q8—42no—
286Q8—81no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2122no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2122no—
304C2×C4—2122no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4C2×C22⋊Dic5311yescharacteristic
308C2×C4D4×Dic5311yescharacteristic
309C2×C4D4×Dic5311yescharacteristic
310C2×C4#1313318911yescharacteristic
311C2×C4#1313318911yescharacteristic
312C2×C4#1313319811yescharacteristic
313C8—42no—
314C8—4241no—
315C8—2122no—
316C22—21no—
317C22—21no—
318C22#1681050311yescharacteristic
319C22#1681079911yescharacteristic
320C22#1681080411yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—42no—
328C4—81no—
329C4—81no—
330C4—4241no—
331C4—4242no—
332C4—4241no—
333C4#1681064012yes—
334C2—21no—
335C2—21no—
336C2#2027508611yescharacteristic
337C2#2027516411yescharacteristic
338C2#2027521111yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3113C2
socleC22×C53#386
prime radicalC23×C53#120
cosocleC2×C42×C53C23
hypercenter#3113D53
hypocenterC53#9328
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
24. C53×#475C4semidirect
25. C53×#462C4semidirect
180. #475Dic53semidirect
181. #462Dic53semidirect
198. C53#9328semidirect

Maximal subgroups

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

subgroupindex
#202773662
#202756602
#202732272
#202731622
#202726072
#202725722
C53×#31132
#932853