Finite Group Databasebeta

← #27537508

Subgroups — #27537508

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287849C211yesmaximal, characteristic
2#20283632C211yesmaximal, characteristic, complemented
3#20283631C211yesmaximal, characteristic, complemented
4#20276598C214yesmaximal, complemented
5#20276593C214yesmaximal, complemented
6#20276509C211yesmaximal, characteristic, complemented
7C2×#16810431C211yesmaximal, characteristic, complemented
8#20276311C211yesmaximal, characteristic, complemented
9C2×C53×#359C211yesmaximal, characteristic, complemented
10C2×#13133239C2211yescharacteristic
11#16811390—21no—
12#16811389—21no—
13#16811064C412yescomplemented
14#16811063C412yescomplemented
15#16811059C412yescomplemented
16#16811058C412yescomplemented
17#16811053C2214yes—
18C2×#13133157C2211yescharacteristic
19#16810745—22no—
20#16810744—22no—
21#16810740C2214yescomplemented
22#16810739C2214yescomplemented
23C2×#13133153C2211yescharacteristic
24#16810640—21no—
25#16810639—21no—
26#16810578C2211yescharacteristic
27#16810452C2214yescomplemented
28#16810431C2212yescomplemented
29#16810431C2212yescomplemented
30#16810398C2212yescomplemented
31C16×Dic53C2212yescomplemented
32C2×#135×C53C2211yescharacteristic
33C22×C16×C53C2211yescharacteristic
34C2×#108×C53C2211yescharacteristic
35C53×#359C2212yescomplemented
36C53×#359C2212yescomplemented
37C2×#11719945—21no—
38#13133239—21no—
39#13133239—21no—
40#13133239C2×C412yes—
41#13133239C2312yes—
42C2×Dic212C2×C411yescharacteristic
43C2×#11719913—21no—
44C2×D424C2×C411yescharacteristic
45#13133225—24no—
46#13133224—24no—
47#13133224—42no—
48#13133223—42no—
49C2×#11719917—21no—
50C2×#11719917—42no—
51C2×#11719917D411yescharacteristic
52M16×D53—42no—
53#13133176D411yescharacteristic
54#13133163C2×C414yes—
55#13133161C2×C414yes—
56#13133160C2×C414yes—
57#13133158C2×C414yes—
58#13133157C2×C412yes—
59#13133157C2×C412yes—
60C2×#11719908C2312yes—
61#13133153—21no—
62#13133153C2312yes—
63#13133144—24no—
64#13133143—24no—
65#13133127D412yes—
66C8×Dic53C2312yes—
67C2×#41×C53—21no—
68#135×C53—21no—
69#135×C53—21no—
70#135×C53C2×C412yes—
71#135×C53C2312yes—
72C2×Q16×C53C2×C411yescharacteristic
73C2×SD16×C53—21no—
74C2×D8×C53C2×C411yescharacteristic
75C2×M16×C53—21no—
76C22×C8×C53C2311yescharacteristic
77C2×C16×C53—21no—
78C2×C16×C53—21no—
79C2×C16×C53C2311yescharacteristic
80C2×C16×C53C2311yescharacteristic
81#108×C53—21no—
82#108×C53C2312yes—
83#41×C53—22no—
84#41×C53—22no—
85#41×C53—41no—
86#41×C53—41no—
87C2×Q8×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53C2×D411yescharacteristic
92C2×C22⋊D53—41no—
93#11719945—22no—
94#11719945—22no—
95#11719945—41no—
96#11719945—41no—
97C2×D212—21no—
98C2×C4×D53—41no—
99C2×Dic106—21no—
100Q16×C53—21no—
101Q16×C53C22×C412yes—
102SD16×C53—22no—
103SD16×C53—22no—
104D8×C53—21no—
105D8×C53C22×C412yes—
106M16×C53—22no—
107M16×C53—41no—
108C2×C8×C53—41no—
109C2×C8×C53C22×C411yescharacteristic
110C2×C8×C53C22×C411yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53#4111yescharacteristic
113C2×C8×C53#4111yescharacteristic
114C2×C8×C53C2411yescharacteristic
115C22⋊Dic53—21no—
116C4⋊Dic53—21no—
117C4×Dic53C2×D412yes—
118#11719917—22no—
119#11719917—41no—
120#11719917—44no—
121#11719917D812yes—
122#11719917D812yes—
123C2×#88322—41no—
124C2×#88322—44no—
125C2×#88322C2×D412yes—
126Dic212—21no—
127Dic212C22×C412yes—
128D424—21no—
129D424C22×C412yes—
130#11719913—22no—
131#11719913—22no—
132#11719912—44no—
133C8×D53—44no—
134C16×C53—21no—
135C16×C53—21no—
136#11719908C22×C414yes—
137C23×C53—41no—
138C23×C53#13511yescharacteristic
139C22×D53—41no—
140Q8×C53—22no—
141Q8×C53—41no—
142D4×C53—22no—
143D4×C53—42no—
144D4×C53—42no—
145D4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53C4×D411yescharacteristic
151C2×C4×C53C2×D811yescharacteristic
152C2×C4×C53C2×D811yescharacteristic
153C2×C4×C53#13511yescharacteristic
154C2×C4×C53C22×D411yescharacteristic
155C22⋊D53—42no—
156C22⋊D53—42no—
157C2×Dic53—22no—
158C2×Dic53—41no—
159D212—22no—
160D212—41no—
161C4×D53—42no—
162C4×D53—42no—
163Dic106—22no—
164Dic106—41no—
165C8×C53—42no—
166C8×C53C4×D411yescharacteristic
167C8×C53C4×D411yescharacteristic
168C8×C53C23×C411yescharacteristic
169C8×C53C2×#4111yescharacteristic
170#88322—42no—
171#88322C2×D814yes—
172#33464—531nomaximal, Sylow
173C22×C53—41no—
174C22×C53—42no—
175C22×C53—161no—
176C22×C53C4×D811yescharacteristic
177C22×C53C4×D811yescharacteristic
178C22×C53C2×#13511yescharacteristic
179C2×D53—42no—
180C2×D53—161no—
181C4×C53—42no—
182C4×C53C4×D811yescharacteristic
183C4×C53C4×D811yescharacteristic
184C4×C53C2×C4×D411yescharacteristic
185C4×C53C22×D811yescharacteristic
186Dic53—42no—
187Dic53—42no—
188#3414—531no—
189#2957—531no—
190#2957—531no—
191#2179—534no—
192#2178—534no—
193#2161—531no—
194C2×#359—531no—
195C2×#359D5311yescharacteristic, complemented
196#2108—531no—
197C2×C53—21no—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53—82no—
201C2×C53C2×C4×D811yescharacteristic
202D53—82no—
203C2×#135—531no—
204C2×#135C2×D5311yescharacteristic
205#568—1061no—
206#568—1061no—
207#510—532no—
208#510—532no—
209#509—532no—
210#509—532no—
211#508—534no—
212C2×M32—531no—
213C22×C16C2×D5311yescharacteristic
214#444—1062no—
215#444—1062no—
216#443—534no—
217#443—534no—
218C2×#108—531no—
219C2×#108C2×D5311yescharacteristic
220#421—1061no—
221#421—1061no—
222#405—531no—
223#368—534no—
224#359—532no—
225#359—532no—
226#359C2×D5312yescomplemented
227#359C2×D5312yescomplemented
228#346—532no—
229C4×C16—532no—
230C53#3346411yescharacteristic, complemented, Sylow
231C2×#41—21no—
232C2×#41—1061no—
233#135—21no—
234#135—21no—
235#135—532no—
236#135—532no—
237#135—1061no—
238#135—1061no—
239#135C4×D5312yes—
240#135C22×D5312yes—
241C2×Q16—531no—
242C2×Q16C4×D5311yescharacteristic
243C2×SD16—21no—
244C2×SD16—1061no—
245C2×D8—531no—
246C2×D8C4×D5311yescharacteristic
247#131—1064no—
248#131—1064no—
249#131—2122no—
250#131—2122no—
251C2×M16—21no—
252C2×M16—531no—
253C2×M16—1061no—
254C2×M16—2122no—
255C2×M16—2122no—
256C22×C8C22×D5311yescharacteristic
257#117—531no—
258Q32—534no—
259SD32—534no—
260SD32—534no—
261D16—534no—
262M32—532no—
263M32—532no—
264C2×C16—21no—
265C2×C16—21no—
266C2×C16—532no—
267C2×C16C22×D5311yescharacteristic
268C2×C16C22×D5311yescharacteristic
269#108—21no—
270#108—532no—
271#108—1061no—
272#108C22×D5312yes—
273#104—1064no—
274#104—1064no—
275#97—532no—
276C4×C8—532no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—1062no—
282#41—1062no—
283#41—2121no—
284#41—2121no—
285C2×Q8—21no—
286C2×Q8—1061no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—1061no—
290C2×D4—2121no—
291C22×C4—41no—
292C22×C4—2121no—
293C22×C4D4×D5311yescharacteristic
294Q16—21no—
295Q16—532no—
296Q16—1061no—
297Q16C2×C4×D5312yes—
298SD16—22no—
299SD16—22no—
300SD16—1062no—
301SD16—1062no—
302D8—21no—
303D8—532no—
304D8—1061no—
305D8C2×C4×D5312yes—
306M16—22no—
307M16—41no—
308M16—532no—
309M16—532no—
310M16—1062no—
311M16—2124no—
312M16—2124no—
313M16—2121no—
314C2×C8—41no—
315C2×C8—532no—
316C2×C8—2124no—
317C2×C8—2124no—
318C2×C8—2121no—
319C2×C8C2×C4×D5311yescharacteristic
320C2×C8C2×C4×D5311yescharacteristic
321C2×C8D4×D5311yescharacteristic
322C2×C8#1171994711yescharacteristic
323C2×C8#1171994711yescharacteristic
324C2×C8C23×D5311yescharacteristic
325C4⋊C4—1061no—
326C22⋊C4—1061no—
327C42—532no—
328C16—21no—
329C16—21no—
330C16—534no—
331C23—41no—
332C23—2121no—
333C23#1313323811yescharacteristic
334Q8—22no—
335Q8—41no—
336Q8—1062no—
337Q8—2121no—
338D4—22no—
339D4—42no—
340D4—42no—
341D4—41no—
342D4—1062no—
343D4—2122no—
344D4—2122no—
345D4—2121no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—1062no—
350C2×C4—2121no—
351C2×C4—2122no—
352C2×C4—2122no—
353C2×C4#1313317711yescharacteristic
354C2×C4#1313317711yescharacteristic
355C2×C4D8×D5311yescharacteristic
356C2×C4D8×D5311yescharacteristic
357C2×C4#1313323811yescharacteristic
358C2×C4C2×D4×D5311yescharacteristic
359C8—42no—
360C8—534no—
361C8—2122no—
362C8#1313317711yescharacteristic
363C8#1313317711yescharacteristic
364C8C22×C4×D5311yescharacteristic
365C8C2×#1171994711yescharacteristic
366C22—41no—
367C22—42no—
368C22—161no—
369C22—8481no—
370C22—2122no—
371C22#1681070311yescharacteristic
372C22#1681070311yescharacteristic
373C22C2×#1313323811yescharacteristic
374C4—42no—
375C4—2122no—
376C4—2122no—
377C4#1681070311yescharacteristic
378C4#1681070311yescharacteristic
379C4C2×#1313317711yescharacteristic
380C4C2×D8×D5311yescharacteristic
381C2—21no—
382C2—21no—
383C2—21no—
384C2—82no—
385C2—4242no—
386C2C2×#1681070311yescharacteristic
subgroupG/H
centerC4C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC2×C53×#359C2
socleC2×C53C2×C4×D8
prime radical#20287849C2
cosocleC2×C8×C53C24
hypercenterC2×#359D53
hypocenterC53#33464
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#359

Complements

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

NH ≅ G/Ntype
2. #20283632C2semidirect
3. #20283631C2semidirect
4. #20276598C2semidirect
5. #20276593C2semidirect
6. #20276509C2semidirect
7. C2×#16810431C2semidirect
8. #20276311C2semidirect
9. C2×C53×#359C2semidirect
13. #16811064C4semidirect
14. #16811063C4semidirect
15. #16811059C4semidirect
16. #16811058C4semidirect
21. #16810740C22semidirect
22. #16810739C22semidirect
27. #16810452C22semidirect
28. #16810431C22semidirect
29. #16810431C22semidirect
30. #16810398C22semidirect
31. C16×Dic53C22semidirect
35. C53×#359C22semidirect
36. C53×#359C22semidirect
195. C2×#359D53semidirect
226. #359C2×D53semidirect
227. #359C2×D53semidirect
230. C53#33464semidirect

Maximal subgroups

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

subgroupindex
#202878492
#202836322
#202836312
#202765982
#202765932
#202765092
C2×#168104312
#202763112
C2×C53×#3592
#3346453