Finite Group Databasebeta

← #27539665

Subgroups — #27539665

Order 13568.

554 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285294C211yesmaximal, characteristic, complemented
2#20285285C211yesmaximal, characteristic, complemented
3#20277001C211yesmaximal, characteristic, complemented
4#20276981C211yesmaximal, characteristic, complemented
5#20276957C211yesmaximal, characteristic, complemented
6#20276952C211yesmaximal, characteristic, complemented
7#20276828C212yesmaximal, complemented
8#20276818C212yesmaximal, complemented
9C4×#13133162C212yesmaximal, complemented
10C4×D848C212yesmaximal, complemented
11C4×M32×C53C211yesmaximal, characteristic, complemented
12#16811173—21no—
13#16811157—21no—
14C2×#13133162C2212yes—
15C2×D848C2212yes—
16#16811028C2211yescharacteristic
17#16811025C2211yescharacteristic
18#16811007C2211yescharacteristic
19#16811000C2211yescharacteristic
20#16810978—21no—
21#16810970—21no—
22#16810864C2211yescharacteristic
23#16810859—21no—
24#16810859C2211yescharacteristic
25#16810859C2212yes—
26#16810852—21no—
27C4×Dic212C2211yescharacteristic
28C4×#11719913—21no—
29C4×D424—21no—
30C4×D424C2211yescharacteristic
31C4×D424C2212yes—
32#16810451C2212yes—
33#16810449C2212yes—
34#16810428C2212yes—
35#16810425C2212yes—
36#16810425C2212yes—
37#16810425C2212yes—
38C2×M32×C53C2212yes—
39C2×C4×C8×C53C2211yescharacteristic
40C53×#346C2211yescharacteristic
41C53×#346C2211yescharacteristic
42C4×C16×C53C2212yes—
43C2×#11719945—41no—
44C2×Dic212C2311yescharacteristic
45C2×#11719913—21no—
46C2×D424—21no—
47C2×D424C2311yescharacteristic
48C2×D424C2312yes—
49#13133217—21no—
50#13133214—21no—
51#13133211—41no—
52#13133209—21no—
53#13133205—21no—
54#13133205—41no—
55#13133200—41no—
56#13133196—21no—
57#13133191—21no—
58#13133191—22no—
59#13133191—41no—
60C4×Dic106—21no—
61C4×D212—21no—
62C4×D212—22no—
63C4×D212—41no—
64C4×C22⋊D53—41no—
65#13133174—41no—
66#13133162—24no—
67D848—24no—
68#13133151—21no—
69#13133151C2311yescharacteristic
70#13133151C2311yescharacteristic
71#13133151C2312yes—
72#13133149—21no—
73#13133142—21no—
74#13133142—21no—
75#13133139—21no—
76#13133139—22no—
77#13133139—22no—
78#13133139—21no—
79C22×C8×C53D411yescharacteristic
80C22×C8×C53C2311yescharacteristic
81C2×C42×C53D411yescharacteristic
82M32×C53—24no—
83C2×C16×C53C2312yes—
84C2×C16×C53C2312yes—
85C4×C8×C53D411yescharacteristic
86C4×C8×C53D411yescharacteristic
87C4×C8×C53C2311yescharacteristic
88C4×C8×C53C2311yescharacteristic
89C22×C4×C53D811yescharacteristic
90C22×C4×C53D811yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C2×C22⋊D53—41no—
93C2×C22⋊D53—42no—
94#11719945—84no—
95C2×D212—21no—
96C2×D212—22no—
97C2×D212—41no—
98C2×D212—41no—
99C2×C4×D53—41no—
100C2×C4×D53—42no—
101C2×Dic106—21no—
102C2×Dic106—41no—
103C2×C8×C53—22no—
104C2×C8×C53—22no—
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53C2×D411yescharacteristic
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2411yescharacteristic
109C42×C53D811yescharacteristic
110C42×C53D811yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C22⋊Dic53—41no—
114#11719921—41no—
115#11719921—41no—
116#11719921—41no—
117#11719921—42no—
118#11719921—41no—
119#11719921—42no—
120C4⋊Dic53—22no—
121C4⋊Dic53—21no—
122C4⋊Dic53—21no—
123C4⋊Dic53—41no—
124#11719919—41no—
125#11719919—41no—
126#11719919—41no—
127#11719919—41no—
128C4×Dic53—41no—
129Dic212—22no—
130D424—22no—
131D424—22no—
132D424—22no—
133D424—42no—
134#11719913—42no—
135C16×C53—22no—
136C16×C53#4114yescomplemented
137C23×C53C2×D811yescharacteristic
138C22×D53—42no—
139C22×D53—41no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53C2×D811yescharacteristic
145C2×C4×C53C2×D811yescharacteristic
146C2×C4×C53C2×D811yescharacteristic
147C2×C4×C53C2×D811yescharacteristic
148C2×C4×C53C2×D811yescharacteristic
149C2×C4×C53C22×D411yescharacteristic
150C22⋊D53—84no—
151C22⋊D53—82no—
152C2×Dic53—41no—
153C2×Dic53—42no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156D212—41no—
157D212—42no—
158D212—41no—
159D212—42no—
160D212—82no—
161D212—82no—
162C4×D53—82no—
163C4×D53—84no—
164Dic106—41no—
165Dic106—41no—
166Dic106—82no—
167C8×C53—21no—
168C8×C53—21no—
169C8×C53#13512yes—
170C8×C53C2×#4112yes—
171#33737—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53C22×D811yescharacteristic
176C2×D53—82no—
177C2×D53—81no—
178C2×D53—82no—
179C2×D53—81no—
180C2×D53—81no—
181C2×D53—82no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53#50912yes—
187C4×C53#53211yescharacteristic
188C4×C53C2×#13511yescharacteristic
189Dic53—82no—
190Dic53—81no—
191Dic53—81no—
192Dic53—81no—
193#3169—531no—
194#3168—531no—
195#2255—531no—
196#2250—531no—
197#2244—531no—
198#2243—531no—
199#2220—532no—
200#2218—532no—
201C4×SD32—532no—
202C4×D16—532no—
203C4×M32D5311yescharacteristic, complemented
204C2×C53—41no—
205C2×C53#316911yescharacteristic
206C2×C53C2×#50911yescharacteristic
207C2×C53#341411yescharacteristic
208D53—162no—
209D53—161no—
210#532—1061no—
211#529—1061no—
212C2×SD32—532no—
213C2×D16—532no—
214C2×M32C2×D5312yes—
215#500—531no—
216#499—531no—
217#495—531no—
218#493—531no—
219#488—1061no—
220#486—1061no—
221#467—531no—
222#466—531no—
223#466—532no—
224#466—1061no—
225#465—1061no—
226C4×Q16—531no—
227C4×SD16—1061no—
228C4×D8—531no—
229C4×D8—532no—
230C4×D8—1061no—
231C2×C4×C8C2×D5311yescharacteristic
232#367—532no—
233#366—532no—
234#358—532no—
235#357—532no—
236#357—532no—
237#357—532no—
238#346C2×D5311yescharacteristic
239#346C2×D5311yescharacteristic
240C4×C16C2×D5312yes—
241C53#3373711yescharacteristic, complemented, Sylow
242C2×#41—2121no—
243C2×Q16—531no—
244C2×SD16—1061no—
245C2×D8—532no—
246C2×D8—531no—
247C2×D8—1061no—
248C22×C8D21211yescharacteristic
249C22×C8C22×D5311yescharacteristic
250C4⋊Q8—1061no—
251#127—1061no—
252#126—2121no—
253#125—1061no—
254#124—1061no—
255#124—2121no—
256#123—2121no—
257C22⋊Q8—1061no—
258#121—1061no—
259#121—1062no—
260#121—2121no—
261C4×Q8—1061no—
262C4×D4—1061no—
263C4×D4—1062no—
264C4×D4—2121no—
265C4×D4—2121no—
266#117—2121no—
267C2×C42D21211yescharacteristic
268SD32—1064no—
269D16—1064no—
270M32—24no—
271C2×C16C22×D5312yes—
272C2×C16C22×D5312yes—
273#107—531no—
274#107—531no—
275#107—532no—
276#107—1061no—
277#106—1061no—
278#103—1061no—
279#103—1061no—
280#102—1061no—
281#102—1062no—
282#102—1062no—
283#102—1061no—
284C4×C8D21211yescharacteristic
285C4×C8D21211yescharacteristic
286C4×C8C22×D5311yescharacteristic
287C4×C8C22×D5311yescharacteristic
288#41—4244no—
289C2×Q8—1061no—
290C2×Q8—2121no—
291C2×D4—1061no—
292C2×D4—1062no—
293C2×D4—2121no—
294C2×D4—2122no—
295C2×D4—2121no—
296C2×D4—2121no—
297C22×C4—2121no—
298C22×C4—2122no—
299C22×C4D42411yescharacteristic
300C22×C4D42411yescharacteristic
301C22×C4C2×D21211yescharacteristic
302Q16—1062no—
303SD16—2122no—
304D8—1062no—
305D8—1062no—
306D8—1062no—
307D8—2122no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8C2×D21211yescharacteristic
311C2×C8C2×D21211yescharacteristic
312C2×C8C2×D21211yescharacteristic
313C2×C8C23×D5311yescharacteristic
314C4⋊C4—1062no—
315C4⋊C4—1061no—
316C4⋊C4—1061no—
317C4⋊C4—2121no—
318C4⋊C4—2121no—
319C4⋊C4—2121no—
320C4⋊C4—2121no—
321C4⋊C4—2121no—
322C22⋊C4—2121no—
323C22⋊C4—2121no—
324C22⋊C4—2121no—
325C22⋊C4—2122no—
326C22⋊C4—2121no—
327C22⋊C4—2122no—
328C22⋊C4—2121no—
329C42—2121no—
330C42D42411yescharacteristic
331C42D42411yescharacteristic
332C42C2×D21211yescharacteristic
333C42C2×D21211yescharacteristic
334C16—22no—
335C16#1171994514yescomplemented
336C23—2121no—
337C23—2122no—
338C23C2×D42411yescharacteristic
339Q8—4242no—
340Q8—2121no—
341Q8—2121no—
342D4—4242no—
343D4—4242no—
344D4—4242no—
345D4—4244no—
346D4—2122no—
347D4—2121no—
348D4—2121no—
349D4—2122no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—41no—
354C2×C4—4242no—
355C2×C4—4244no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2122no—
360C2×C4C2×D42411yescharacteristic
361C2×C4C2×D42411yescharacteristic
362C2×C4C2×D42411yescharacteristic
363C2×C4C2×D42411yescharacteristic
364C2×C4C2×D42411yescharacteristic
365C2×C4C22×D21211yescharacteristic
366C8—21no—
367C8—21no—
368C8#1313323912yes—
369C8C2×#1171994512yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—4242no—
374C22—4241no—
375C22—4242no—
376C22—4241no—
377C22—4242no—
378C22—4241no—
379C22C22×D42411yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—4241no—
385C4—4242no—
386C4—4241no—
387C4—4241no—
388C4#1681106112yes—
389C4#1681117311yescharacteristic
390C4C2×#1313323911yescharacteristic
391C2—41no—
392C2—8482no—
393C2—8481no—
394C2#2028529411yescharacteristic
395C2C2×#1681106111yescharacteristic
396C2#2028785111yescharacteristic
subgroupG/H
centerC2×C4C2×D424
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC4×M32×C53C2
socleC22×C53C22×D8
hypercenterC4×M32D53
hypocenterC53#33737
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×M32

Complements

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

NH ≅ G/Ntype
1. #20285294C2semidirect
2. #20285285C2semidirect
3. #20277001C2semidirect
4. #20276981C2semidirect
5. #20276957C2semidirect
6. #20276952C2semidirect
7. #20276828C2semidirect
8. #20276818C2semidirect
9. C4×#13133162C2semidirect
10. C4×D848C2semidirect
11. C4×M32×C53C2semidirect
136. C16×C53#41semidirect
203. C4×M32D53semidirect
241. C53#33737semidirect
335. C16#11719945semidirect

Maximal subgroups

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

subgroupindex
#202852942
#202852852
#202770012
#202769812
#202769572
#202769522
#202768282
#202768182
C4×#131331622
C4×D8482
C4×M32×C532
#3373753