Finite Group Databasebeta

← #27357143

Subgroups — #27357143

Order 13568.

1254 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 203 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#20284742C212yesmaximal, complemented
2#20280069C211yesmaximal, characteristic, complemented
3#20273735C214yesmaximal, complemented
4#20273667C214yesmaximal, complemented
5#20273275C212yesmaximal, complemented
6#20273036C211yesmaximal, characteristic, complemented
7C53×#3496C211yesmaximal, characteristic
8C2×C4×Dic106C2211yescharacteristic, complemented
9#16810939C2214yescomplemented
10#16810905—22no—
11#16810820—22no—
12#16810761C2214yescomplemented
13C4×#11719923—24no—
14#16810676C2212yescomplemented
15#16810559C2212yescomplemented
16#16810535C2212yescomplemented
17#16810526—21no—
18#16810519—21no—
19#16810496—22no—
20#16810489C2211yescharacteristic, complemented
21#16810357C2218yescomplemented
22#16810347C2214yescomplemented
23C2×H(5,𝔽2)×C53—21no—
24C53×#547—22no—
25C53×#546—24no—
26D42×C53—22no—
27C53×#537—21no—
28C53×#534C2214yes—
29C53×#532C2212yes—
30C2×#121×C53—22no—
31C2×C4×D4×C53C2211yescharacteristic
32C22×Dic106C2311yescharacteristic
33C2×#11719923—24no—
34C4×Dic106—22no—
35C4×Dic106—22no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53C2311yescharacteristic
38C2×#11719919—21no—
39C2×#11719919—22no—
40C2×C22⋊Dic53—22no—
41C2×C22⋊Dic53—22no—
42C2×C4×Dic53—21no—
43#13133148—24no—
44C4⋊#88322—22no—
45#13133140—24no—
46#13133138—24no—
47C22⋊#88322C2314yes—
48C4×#88322—22no—
49#13133124—21no—
50#13133124—22no—
51#13133124C2312yes—
52H(5,𝔽2)×C53—44no—
53H(5,𝔽2)×C53—44no—
54C2×#41×C53—24no—
55C2×#41×C53—22no—
56C22×D4×C53—24no—
57C22×D4×C53—24no—
58C22×D4×C53—22no—
59C22×D4×C53D411yescharacteristic
60C23×C4×C53D412yes—
61C4⋊Q8×C53D412yes—
62#127×C53D412yes—
63#124×C53—22no—
64#123×C53—28no—
65C22⋊Q8×C53—24no—
66#121×C53—24no—
67#121×C53—24no—
68#121×C53—24no—
69#121×C53—22no—
70#121×C53—24no—
71#121×C53C2314yes—
72#120×C53—28no—
73C4×D4×C53—22no—
74C4×D4×C53—22no—
75C4×D4×C53—22no—
76C4×D4×C53D414yes—
77C2×C4⋊C4×C53—22no—
78C2×C4⋊C4×C53D411yescharacteristic
79C2×C22⋊C4×C53—24no—
80C2×C22⋊C4×C53C2312yes—
81C2×C42×C53C2311yescharacteristic
82C24×C53—24no—
83C24×C53C2×D412yes—
84#41×C53—416no—
85#41×C53—44no—
86#41×C53—44no—
87C2×Q8×C53—22no—
88C2×D4×C53—28no—
89C2×D4×C53—24no—
90C2×D4×C53—24no—
91C2×D4×C53—22no—
92C2×D4×C53—22no—
93C2×D4×C53—22no—
94C2×D4×C53—22no—
95C2×D4×C53—416no—
96C2×D4×C53—48no—
97C2×D4×C53—48no—
98C2×D4×C53—84no—
99C2×D4×C53SD1614yescomplemented
100C2×D4×C53C2×D414yes—
101C22×C4×C53—24no—
102C22×C4×C53—22no—
103C22×C4×C53—24no—
104C22×C4×C53—22no—
105C22×C4×C53—82no—
106C22×C4×C53C2×D412yes—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2411yescharacteristic
110C22×Dic53—21no—
111C22×Dic53—22no—
112C22×Dic53—21no—
113C2×Dic106—22no—
114C2×Dic106—42no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53C2×D414yes—
119C22⋊C4×C53—28no—
120C22⋊C4×C53—24no—
121C22⋊C4×C53—24no—
122C22⋊C4×C53—24no—
123C42×C53C2×D412yes—
124C42×C53C2×D412yes—
125C22⋊Dic53—44no—
126C22⋊Dic53—82no—
127#11719923—48no—
128C4⋊Dic53—22no—
129C4⋊Dic53—42no—
130#11719919—41no—
131#11719919—41no—
132#11719919—82no—
133C4×Dic53—42no—
134C2×#88322—24no—
135C23×C53—22no—
136C23×C53—22no—
137C23×C53—24no—
138C23×C53—42no—
139C23×C53—42no—
140C23×C53—48no—
141C23×C53—84no—
142C23×C53—84no—
143C23×C53C2×SD1614yes—
144C23×C53C22×D411yescharacteristic
145Q8×C53—44no—
146D4×C53—28no—
147D4×C53—44no—
148D4×C53—44no—
149D4×C53—416no—
150D4×C53—44no—
151D4×C53—44no—
152D4×C53—44no—
153D4×C53—84no—
154C2×C4×C53—24no—
155C2×C4×C53—22no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—44no—
159C2×C4×C53—48no—
160C2×C4×C53—48no—
161C2×C4×C53—42no—
162C2×C4×C53—82no—
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12011yescharacteristic
165C2×C4×C53#12012yes—
166C2×C4×C53C2×SD1612yes—
167C2×C4×C53AGL(1,ℤ/8)12yes—
168C2×C4×C53C22×D412yes—
169C2×Dic53—22no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—42no—
174C2×Dic53—42no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—42no—
178Dic106—24no—
179Dic106—42no—
180#88322—44no—
181#18211—531nomaximal, Sylow
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—24no—
185C22×C53—44no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53—44no—
189C22×C53—44no—
190C22×C53—44no—
191C22×C53—82no—
192C22×C53—168no—
193C22×C53C2×#12011yescharacteristic
194C22×C53C22×SD1611yescharacteristic
195C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—22no—
199C4×C53—44no—
200C4×C53—42no—
201C4×C53#45014yes—
202Dic53—42no—
203Dic53—82no—
204Dic53—81no—
205#3496D5311yescharacteristic
206#3103—532no—
207#2580—531no—
208#1613—534no—
209#1601—534no—
210#1517—532no—
211#1460—531no—
212C2×C53—21no—
213C2×C53—21no—
214C2×C53—44no—
215C2×C53—84no—
216C2×C53C2×#45011yescharacteristic
217C2×C53#301911yescharacteristic
218C2×C53#302511yescharacteristic
219C2×H(5,𝔽2)—21no—
220#547—22no—
221#546—24no—
222D42—22no—
223#537—21no—
224#534C2×D5314yes—
225#532C2×D5312yes—
226C2×#121—22no—
227C2×C4×Q8—531no—
228C2×C4×D4C2×D5311yescharacteristic
229#481—534no—
230#475—1062no—
231#460—1062no—
232#448—534no—
233C4×SD16—1064no—
234#431—532no—
235#399—532no—
236#393—532no—
237#390—1061no—
238#389—1061no—
239#385—1062no—
240#382—531no—
241#327—538no—
242#323—534no—
243C53#1821111yescharacteristic, complemented, Sylow
244H(5,𝔽2)—44no—
245H(5,𝔽2)—44no—
246C2×#41—24no—
247C2×#41—22no—
248C22×Q8—531no—
249C22×D4—24no—
250C22×D4—24no—
251C22×D4—22no—
252C22×D4C22⋊D5311yescharacteristic
253C23×C4C22⋊D5312yes—
254C2×SD16—1064no—
255C4⋊Q8C22⋊D5312yes—
256#127C22⋊D5312yes—
257#124—22no—
258#123—28no—
259C22⋊Q8—24no—
260#121—24no—
261#121—24no—
262#121—24no—
263#121—22no—
264#121—24no—
265#121C22×D5314yes—
266#120—28no—
267C4×Q8—1062no—
268C4×Q8—1062no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4C22⋊D5314yes—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—531no—
275C2×C4⋊C4—1062no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4—1061no—
278C2×C4⋊C4C22⋊D5311yescharacteristic
279C2×C22⋊C4—24no—
280C2×C22⋊C4—1062no—
281C2×C22⋊C4—1062no—
282C2×C22⋊C4C22×D5312yes—
283C2×C42—1061no—
284C2×C42C22×D5311yescharacteristic
285#106—1064no—
286C4⋊C8—1062no—
287#103—1064no—
288#102—1064no—
289C22⋊C8—534no—
290C4×C8—1062no—
291#95—532no—
292#95—1061no—
293#95—1062no—
294C24—24no—
295C24C2×C22⋊D5312yes—
296#41—416no—
297#41—44no—
298#41—44no—
299C2×Q8—22no—
300C2×Q8—1062no—
301C2×Q8—2122no—
302C2×D4—28no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—416no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4—84no—
313C2×D4#1171992314yescomplemented
314C2×D4C2×C22⋊D5314yes—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—82no—
320C22×C4—1061no—
321C22×C4—1062no—
322C22×C4—1061no—
323C22×C4C2×C22⋊D5312yes—
324C22×C4C2×C22⋊D5311yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326C22×C4C23×D5311yescharacteristic
327SD16—2128no—
328C2×C8—1064no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—1062no—
333C4⋊C4—4242no—
334C4⋊C4—2121no—
335C4⋊C4—2121no—
336C4⋊C4—2122no—
337C4⋊C4C2×C22⋊D5314yes—
338C22⋊C4—28no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—4242no—
343C22⋊C4—2124no—
344C42—2122no—
345C42C2×C22⋊D5312yes—
346C42C2×C22⋊D5312yes—
347C23—22no—
348C23—22no—
349C23—24no—
350C23—42no—
351C23—42no—
352C23—48no—
353C23—84no—
354C23—84no—
355C23C2×#1171992314yes—
356C23C22×C22⋊D5311yescharacteristic
357Q8—44no—
358Q8—1064no—
359Q8—2122no—
360D4—28no—
361D4—44no—
362D4—44no—
363D4—416no—
364D4—44no—
365D4—44no—
366D4—44no—
367D4—84no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—1062no—
378C2×C4—2121no—
379C2×C4—2122no—
380C2×C4—2122no—
381C2×C4—2121no—
382C2×C4—2121no—
383C2×C4—2122no—
384C2×C4—2121no—
385C2×C4—2121no—
386C2×C4#1313318611yescharacteristic
387C2×C4#1313318611yescharacteristic
388C2×C4#1313318612yes—
389C2×C4C2×#1171992312yes—
390C2×C4#1313324312yes—
391C2×C4C22×C22⋊D5312yes—
392C8—2124no—
393C22—21no—
394C22—21no—
395C22—24no—
396C22—44no—
397C22—42no—
398C22—42no—
399C22—44no—
400C22—44no—
401C22—44no—
402C22—82no—
403C22—168no—
404C22C2×#1313318611yescharacteristic
405C22C22×#1171992311yescharacteristic
406C22C2×#1313324311yescharacteristic
407C4—21no—
408C4—21no—
409C4—22no—
410C4—42no—
411C4—44no—
412C4—4242no—
413C4—4241no—
414C4—2122no—
415C4#1681077414yes—
416C2—21no—
417C2—21no—
418C2—44no—
419C2—84no—
420C2C2×#1681077411yescharacteristic
421C2#2028407211yescharacteristic
422C2#2028410711yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC53×#3496C2
socleC22×C53C2×#120
hypercenter#3496D53
hypocenterC53#18211
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. #20284742C2semidirect
2. #20280069C2semidirect
3. #20273735C2semidirect
4. #20273667C2semidirect
5. #20273275C2semidirect
6. #20273036C2semidirect
8. C2×C4×Dic106C22semidirect
9. #16810939C22semidirect
12. #16810761C22semidirect
14. #16810676C22semidirect
15. #16810559C22semidirect
16. #16810535C22semidirect
20. #16810489C22semidirect
21. #16810357C22semidirect
22. #16810347C22semidirect
99. C2×D4×C53SD16semidirect
243. C53#18211semidirect
313. C2×D4#11719923semidirect

Maximal subgroups

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

subgroupindex
#202847422
#202800692
#202737352
#202736672
#202732752
#202730362
C53×#34962
#1821153