Finite Group Databasebeta

← #27539676

Subgroups — #27539676

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285301C211yesmaximal, characteristic, complemented
2#20285285C211yesmaximal, characteristic
3#20277001C211yesmaximal, characteristic, complemented
4#20276986C211yesmaximal, characteristic, complemented
5#20276961C211yesmaximal, characteristic, complemented
6#20276952C211yesmaximal, characteristic, complemented
7#20276828C212yesmaximal
8#20276823C212yesmaximal, complemented
9C4×Dic424C212yesmaximal, complemented
10C4×#13133162C212yesmaximal, complemented
11C4×M32×C53C211yesmaximal, characteristic, complemented
12#16811180—21no—
13#16811157—21no—
14C2×Dic424C2212yes—
15C2×#13133162C2212yes—
16#16811028C2211yescharacteristic
17#16811025C2211yescharacteristic
18#16811007C2211yescharacteristic
19#16811003C2211yescharacteristic
20#16810978—21no—
21#16810974—21no—
22#16810864—21no—
23#16810864C2211yescharacteristic
24#16810864C2212yes—
25#16810859C2211yescharacteristic
26#16810852—21no—
27C4×Dic212—21no—
28C4×Dic212C2212yes—
29C4×Dic212C2211yescharacteristic
30C4×#11719913—21no—
31C4×D424C2211yescharacteristic
32#16810451C2212yes—
33#16810449C2212yes—
34#16810428C2212yes—
35#16810428C2212yes—
36#16810428C2212yes—
37#16810425C2212yes—
38C2×M32×C53C2212yes—
39C2×C4×C8×C53C2211yescharacteristic
40C53×#346C2211yescharacteristic
41C53×#346C2211yescharacteristic
42C4×C16×C53C2212yes—
43C2×Dic212—21no—
44C2×Dic212C2312yes—
45C2×Dic212C2311yescharacteristic
46C2×#11719913—21no—
47C2×D424C2311yescharacteristic
48#13133217—21no—
49#13133217—21no—
50#13133211—41no—
51#13133209—21no—
52#13133209—41no—
53#13133205—21no—
54#13133200—41no—
55#13133196—21no—
56#13133196—22no—
57#13133196—41no—
58#13133191—21no—
59C4×Dic106—21no—
60C4×Dic106—22no—
61C4×Dic106—41no—
62C4×D212—21no—
63C4×C22⋊D53—41no—
64#13133176—41no—
65#13133174—41no—
66Dic424—24no—
67#13133162—24no—
68#13133151—21no—
69#13133151C2311yescharacteristic
70#13133151C2311yescharacteristic
71#13133151C2312yes—
72#13133149—21no—
73#13133142—21no—
74#13133142—22no—
75#13133142—21no—
76#13133142—22no—
77#13133139—21no—
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×D212—21no—
94C2×C4×D53—41no—
95C2×Dic106—21no—
96C2×Dic106—22no—
97C2×Dic106—41no—
98C2×C8×C53—22no—
99C2×C8×C53—22no—
100C2×C8×C53C2×D411yescharacteristic
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53C2×D411yescharacteristic
103C2×C8×C53C2411yescharacteristic
104C42×C53D811yescharacteristic
105C42×C53D811yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C22⋊Dic53—42no—
109C22⋊Dic53—41no—
110#11719921—41no—
111#11719921—41no—
112#11719921—41no—
113#11719921—41no—
114C4⋊Dic53—21no—
115C4⋊Dic53—21no—
116C4⋊Dic53—22no—
117C4⋊Dic53—41no—
118C4⋊Dic53—41no—
119C4⋊Dic53—41no—
120#11719919—41no—
121#11719919—41no—
122#11719919—41no—
123#11719919—42no—
124#11719919—41no—
125#11719919—42no—
126C4×Dic53—41no—
127C4×Dic53—42no—
128Dic212—22no—
129Dic212—22no—
130Dic212—22no—
131Dic212—42no—
132D424—22no—
133#11719913—42no—
134C16×C53—22no—
135C16×C53#4114yes—
136C23×C53C2×D811yescharacteristic
137C22×D53—41no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—22no—
141C2×C4×C53—41no—
142C2×C4×C53C2×D811yescharacteristic
143C2×C4×C53C2×D811yescharacteristic
144C2×C4×C53C2×D811yescharacteristic
145C2×C4×C53C2×D811yescharacteristic
146C2×C4×C53C2×D811yescharacteristic
147C2×C4×C53C22×D411yescharacteristic
148C22⋊D53—82no—
149C2×Dic53—42no—
150C2×Dic53—41no—
151C2×Dic53—42no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154D212—41no—
155D212—41no—
156C4×D53—82no—
157Dic106—41no—
158Dic106—42no—
159Dic106—41no—
160Dic106—42no—
161Dic106—82no—
162C8×C53—21no—
163C8×C53—21no—
164C8×C53#13512yes—
165C8×C53C2×#4112yes—
166#33738—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53C22×D811yescharacteristic
171C2×D53—81no—
172C2×D53—81no—
173C2×D53—81no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53#51012yes—
179C4×C53#53211yescharacteristic
180C4×C53C2×#13511yescharacteristic
181Dic53—81no—
182Dic53—82no—
183Dic53—81no—
184Dic53—81no—
185Dic53—82no—
186#3170—531no—
187#3168—531no—
188#2255—531no—
189#2251—531no—
190#2245—531no—
191#2243—531no—
192#2220—532no—
193#2219—532no—
194C4×Q32—532no—
195C4×SD32—532no—
196C4×M32D5311yescharacteristic, complemented
197C2×C53—41no—
198C2×C53#316911yescharacteristic
199C2×C53C2×#51011yescharacteristic
200C2×C53#341411yescharacteristic
201D53—161no—
202#533—1061no—
203#529—1061no—
204C2×Q32—532no—
205C2×SD32—532no—
206C2×M32C2×D5312yes—
207#500—531no—
208#499—531no—
209#495—531no—
210#494—531no—
211#488—1061no—
212#487—1061no—
213#467—531no—
214#467—532no—
215#467—1061no—
216#466—531no—
217#465—1061no—
218C4×Q16—531no—
219C4×Q16—532no—
220C4×Q16—1061no—
221C4×SD16—1061no—
222C4×D8—531no—
223C2×C4×C8C2×D5311yescharacteristic
224#367—532no—
225#366—532no—
226#358—532no—
227#358—532no—
228#358—532no—
229#357—532no—
230#346C2×D5311yescharacteristic
231#346C2×D5311yescharacteristic
232C4×C16C2×D5312yes—
233C53#3373811yescharacteristic, complemented, Sylow
234C2×Q16—532no—
235C2×Q16—531no—
236C2×Q16—1061no—
237C2×SD16—1061no—
238C2×D8—531no—
239C22×C8D21211yescharacteristic
240C22×C8C22×D5311yescharacteristic
241C4⋊Q8—1061no—
242C4⋊Q8—1061no—
243#126—2121no—
244#125—1061no—
245#125—2121no—
246#124—1061no—
247#123—2121no—
248C22⋊Q8—1061no—
249C22⋊Q8—1062no—
250C22⋊Q8—2121no—
251#121—1061no—
252C4×Q8—1061no—
253C4×Q8—1062no—
254C4×Q8—2121no—
255C4×D4—1061no—
256C4×D4—2121no—
257#117—2121no—
258#117—2121no—
259C2×C42D21211yescharacteristic
260Q32—1064no—
261SD32—1064no—
262M32—24no—
263C2×C16C22×D5312yes—
264C2×C16C22×D5312yes—
265#107—532no—
266#107—531no—
267#107—531no—
268#107—1061no—
269#106—1061no—
270#103—1061no—
271#103—1061no—
272#103—1062no—
273#103—1062no—
274#102—1061no—
275#102—1061no—
276C4×C8D21211yescharacteristic
277C4×C8D21211yescharacteristic
278C4×C8C22×D5311yescharacteristic
279C4×C8C22×D5311yescharacteristic
280C2×Q8—1061no—
281C2×Q8—1062no—
282C2×Q8—2121no—
283C2×D4—1061no—
284C2×D4—2121no—
285C22×C4—2121no—
286C22×C4D42411yescharacteristic
287C22×C4D42411yescharacteristic
288C22×C4C2×D21211yescharacteristic
289Q16—1062no—
290Q16—1062no—
291Q16—1062no—
292Q16—2122no—
293SD16—2122no—
294D8—1062no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8C2×D21211yescharacteristic
298C2×C8C2×D21211yescharacteristic
299C2×C8C2×D21211yescharacteristic
300C2×C8C23×D5311yescharacteristic
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4—1062no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C4⋊C4—2121no—
307C4⋊C4—2121no—
308C4⋊C4—2121no—
309C4⋊C4—2122no—
310C4⋊C4—2121no—
311C4⋊C4—2122no—
312C4⋊C4—2121no—
313C22⋊C4—2122no—
314C22⋊C4—2121no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C22⋊C4—2121no—
318C22⋊C4—2121no—
319C42—2121no—
320C42—2122no—
321C42D42411yescharacteristic
322C42D42411yescharacteristic
323C42C2×D21211yescharacteristic
324C42C2×D21211yescharacteristic
325C16—22no—
326C16#1171994514yes—
327C23—2121no—
328C23C2×D42411yescharacteristic
329Q8—4242no—
330Q8—2121no—
331Q8—2122no—
332Q8—2122no—
333Q8—2121no—
334D4—4242no—
335D4—2121no—
336D4—2121no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—4242no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4—2121no—
347C2×C4C2×D42411yescharacteristic
348C2×C4C2×D42411yescharacteristic
349C2×C4C2×D42411yescharacteristic
350C2×C4C2×D42411yescharacteristic
351C2×C4C2×D42411yescharacteristic
352C2×C4C22×D21211yescharacteristic
353C8—21no—
354C8—21no—
355C8#1313323912yes—
356C8C2×#1171994512yes—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—4241no—
361C22—4241no—
362C22—4241no—
363C22C22×D42411yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—4241no—
369C4—4241no—
370C4—4242no—
371C4—4241no—
372C4—4242no—
373C4#1681106612yes—
374C4#1681117311yescharacteristic
375C4C2×#1313323911yescharacteristic
376C2—41no—
377C2—8481no—
378C2#2028529411yescharacteristic
379C2C2×#1681106611yescharacteristic
380C2#2028785111yescharacteristic
subgroupG/H
centerC2×C4C2×D424
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC4×M32×C53C2
socleC22×C53C22×D8
prime radical#16810859C22
hypercenterC4×M32D53
hypocenterC53#33738
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. #20285301C2semidirect
3. #20277001C2semidirect
4. #20276986C2semidirect
5. #20276961C2semidirect
6. #20276952C2semidirect
8. #20276823C2semidirect
9. C4×Dic424C2semidirect
10. C4×#13133162C2semidirect
11. C4×M32×C53C2semidirect
196. C4×M32D53semidirect
233. C53#33738semidirect

Maximal subgroups

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

subgroupindex
#202853012
#202852852
#202770012
#202769862
#202769612
#202769522
#202768282
#202768232
C4×Dic4242
C4×#131331622
C4×M32×C532
#3373853