Finite Group Databasebeta

← #27539677

Subgroups — #27539677

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#20277002C211yesmaximal, characteristic, complemented
3#20276987C211yesmaximal, characteristic, complemented
4#20276962C211yesmaximal, characteristic, complemented
5#20276951C211yesmaximal, characteristic, complemented
6#20276826C212yesmaximal
7#20276821C212yesmaximal, complemented
8C4×#13133163C212yesmaximal, complemented
9C4×#13133161C212yesmaximal, complemented
10C4×#13133157C211yesmaximal, characteristic, complemented
11C53×#3168C211yesmaximal, characteristic
12#16811180—21no—
13C2×#13133163C2212yes—
14C2×#13133161C2212yes—
15C2×#13133157C2212yes—
16#16811028C2211yescharacteristic
17#16811003C2211yescharacteristic
18#16810974—21no—
19#16810864—21no—
20#16810864C2212yes—
21C4×Dic212—21no—
22C4×Dic212C2212yes—
23#16810450C2212yes—
24#16810448C2212yes—
25#16810427C2212yes—
26#16810426C2212yes—
27#16810426C2212yes—
28#16810424C2212yes—
29#16810399C2211yescharacteristic
30#16810399C2211yescharacteristic
31C4×#11719908C2212yes—
32C53×#529—21no—
33C53×#499C2211yescharacteristic
34C53×#495C2211yescharacteristic
35C53×#488—21no—
36C53×#467C2211yescharacteristic
37C53×#466C2211yescharacteristic
38C53×#465—21no—
39C4×Q16×C53C2211yescharacteristic
40C4×SD16×C53—21no—
41C4×D8×C53C2211yescharacteristic
42C2×C4×C8×C53C2211yescharacteristic
43C2×Dic212—21no—
44C2×Dic212C2312yes—
45#13133217—21no—
46#13133217—21no—
47#13133209—41no—
48#13133196—22no—
49#13133196—41no—
50C4×Dic106—22no—
51C4×Dic106—41no—
52#13133176—41no—
53#13133163—24no—
54#13133161—24no—
55#13133157—24no—
56C2×#11719908C2312yes—
57C2×#11719908C2312yes—
58#13133151—21no—
59#13133151C2312yes—
60#13133142—22no—
61#13133142—22no—
62C2×Q16×C53C2311yescharacteristic
63C2×SD16×C53—21no—
64C2×D8×C53C2311yescharacteristic
65C22×C8×C53D411yescharacteristic
66C22×C8×C53C2311yescharacteristic
67#126×C53—41no—
68#125×C53—21no—
69#124×C53—21no—
70#123×C53—41no—
71C22⋊Q8×C53—21no—
72#121×C53—21no—
73C4×Q8×C53—21no—
74C4×D4×C53—21no—
75C4×D4×C53—41no—
76#117×C53—41no—
77C2×C42×C53D411yescharacteristic
78#107×C53C2311yescharacteristic
79#107×C53C2311yescharacteristic
80#106×C53—21no—
81#103×C53—21no—
82#103×C53—21no—
83#102×C53—21no—
84#102×C53—21no—
85C4×C8×C53D411yescharacteristic
86C4×C8×C53D411yescharacteristic
87C4×C8×C53C2311yescharacteristic
88C4×C8×C53C2311yescharacteristic
89C2×Q8×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53D811yescharacteristic
94C22×C4×C53D811yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C2×Dic106—22no—
97C2×Dic106—41no—
98Q16×C53—22no—
99SD16×C53—42no—
100D8×C53—22no—
101C2×C8×C53—22no—
102C2×C8×C53—22no—
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53C2411yescharacteristic
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C42×C53—41no—
119C42×C53D811yescharacteristic
120C42×C53D811yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C22⋊Dic53—42no—
124C4⋊Dic53—22no—
125C4⋊Dic53—41no—
126C4⋊Dic53—41no—
127C4⋊Dic53—41no—
128#11719919—42no—
129#11719919—42no—
130C4×Dic53—42no—
131Dic212—22no—
132Dic212—22no—
133Dic212—42no—
134#11719908—22no—
135#11719908#4114yes—
136C23×C53—41no—
137C23×C53C2×D811yescharacteristic
138Q8×C53—41no—
139Q8×C53—41no—
140D4×C53—41no—
141D4×C53—41no—
142D4×C53—82no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—82no—
151C2×C4×C53C2×D811yescharacteristic
152C2×C4×C53C2×D811yescharacteristic
153C2×C4×C53C2×D811yescharacteristic
154C2×C4×C53C2×D811yescharacteristic
155C2×C4×C53C2×D811yescharacteristic
156C2×C4×C53C22×D411yescharacteristic
157C2×Dic53—42no—
158C2×Dic53—42no—
159Dic106—42no—
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×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53C22×D811yescharacteristic
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—81no—
179C4×C53—81no—
180C4×C53—81no—
181C4×C53#51012yes—
182C4×C53#53211yescharacteristic
183C4×C53C2×#13511yescharacteristic
184Dic53—82no—
185Dic53—82no—
186#3170—531no—
187#3168D5311yescharacteristic
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×M32—531no—
197C2×C53—41no—
198C2×C53—161no—
199C2×C53#316911yescharacteristic
200C2×C53C2×#51011yescharacteristic
201C2×C53#341411yescharacteristic
202#533—1061no—
203#529—21no—
204C2×Q32—532no—
205C2×SD32—532no—
206C2×M32—532no—
207#500—531no—
208#499C2×D5311yescharacteristic
209#495C2×D5311yescharacteristic
210#494—531no—
211#488—21no—
212#487—1061no—
213#467—532no—
214#467—1061no—
215#467C2×D5311yescharacteristic
216#466C2×D5311yescharacteristic
217#465—21no—
218C4×Q16—532no—
219C4×Q16—1061no—
220C4×Q16C2×D5311yescharacteristic
221C4×SD16—21no—
222C4×D8C2×D5311yescharacteristic
223C2×C4×C8C2×D5311yescharacteristic
224#367—532no—
225#366—532no—
226#358—532no—
227#358—532no—
228#358—532no—
229#357—532no—
230#346—531no—
231#346—531no—
232C4×C16—532no—
233C53#3373811yescharacteristic, complemented, Sylow
234C2×Q16—532no—
235C2×Q16—1061no—
236C2×Q16C22×D5311yescharacteristic
237C2×SD16—21no—
238C2×D8C22×D5311yescharacteristic
239C22×C8C22⋊D5311yescharacteristic
240C22×C8C22×D5311yescharacteristic
241C4⋊Q8—1061no—
242C4⋊Q8—1061no—
243#126—41no—
244#125—21no—
245#125—2121no—
246#124—21no—
247#123—41no—
248C22⋊Q8—21no—
249C22⋊Q8—1062no—
250C22⋊Q8—2121no—
251#121—21no—
252C4×Q8—21no—
253C4×Q8—1062no—
254C4×Q8—2121no—
255C4×D4—21no—
256C4×D4—41no—
257#117—41no—
258#117—2121no—
259C2×C42C22⋊D5311yescharacteristic
260Q32—1064no—
261SD32—1064no—
262M32—1064no—
263C2×C16—532no—
264C2×C16—532no—
265#107—532no—
266#107—1061no—
267#107C22×D5311yescharacteristic
268#107C22×D5311yescharacteristic
269#106—21no—
270#103—21no—
271#103—21no—
272#103—1062no—
273#103—1062no—
274#102—21no—
275#102—21no—
276C4×C8C22⋊D5311yescharacteristic
277C4×C8C22⋊D5311yescharacteristic
278C4×C8C22×D5311yescharacteristic
279C4×C8C22×D5311yescharacteristic
280C2×Q8—21no—
281C2×Q8—1062no—
282C2×Q8—2121no—
283C2×D4—21no—
284C2×D4—41no—
285C22×C4—41no—
286C22×C4#1171992211yescharacteristic
287C22×C4#1171992211yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289Q16—22no—
290Q16—1062no—
291Q16—1062no—
292Q16—2122no—
293SD16—42no—
294D8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8C2×C22⋊D5311yescharacteristic
298C2×C8C2×C22⋊D5311yescharacteristic
299C2×C8C2×C22⋊D5311yescharacteristic
300C2×C8C23×D5311yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—1062no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C4⋊C4—2122no—
311C4⋊C4—2122no—
312C4⋊C4—2121no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—2122no—
319C42—41no—
320C42—2122no—
321C42#1171992211yescharacteristic
322C42#1171992211yescharacteristic
323C42C2×C22⋊D5311yescharacteristic
324C42C2×C22⋊D5311yescharacteristic
325C16—534no—
326C16—1062no—
327C23—41no—
328C23C2×#1171992211yescharacteristic
329Q8—41no—
330Q8—41no—
331Q8—4242no—
332Q8—2122no—
333Q8—2122no—
334D4—41no—
335D4—41no—
336D4—82no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—82no—
345C2×C4—2122no—
346C2×C4—2122no—
347C2×C4C2×#1171992211yescharacteristic
348C2×C4C2×#1171992211yescharacteristic
349C2×C4C2×#1171992211yescharacteristic
350C2×C4C2×#1171992211yescharacteristic
351C2×C4C2×#1171992211yescharacteristic
352C2×C4C22×C22⋊D5311yescharacteristic
353C8—21no—
354C8—21no—
355C8#1313323712yes—
356C8#41×D5312yes—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22C22×#1171992211yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—4242no—
372C4—4242no—
373C4#1681106312yes—
374C4#1681116911yescharacteristic
375C4C2×#1313323711yescharacteristic
376C2—41no—
377C2—161no—
378C2#2028529011yescharacteristic
379C2C2×#1681106311yescharacteristic
380C2#2028784911yescharacteristic
subgroupG/H
centerC2×C4C2×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3168C2
socleC22×C53C22×D8
prime radicalC53×#466C22
hypercenter#3168D53
hypocenterC53#33738
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #20285301C2semidirect
2. #20277002C2semidirect
3. #20276987C2semidirect
4. #20276962C2semidirect
5. #20276951C2semidirect
7. #20276821C2semidirect
8. C4×#13133163C2semidirect
9. C4×#13133161C2semidirect
10. C4×#13133157C2semidirect
233. C53#33738semidirect

Maximal subgroups

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

subgroupindex
#202853012
#202770022
#202769872
#202769622
#202769512
#202768262
#202768212
C4×#131331632
C4×#131331612
C4×#131331572
C53×#31682
#3373853