Finite Group Databasebeta

← #27538738

Subgroups — #27538738

Order 13568.

546 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285229C211yesmaximal, characteristic, complemented
2#20276879C212yesmaximal, complemented
3#20276872C212yesmaximal, complemented
4#20276867C211yesmaximal, characteristic, complemented
5#20276862C211yesmaximal, characteristic, complemented
6#20276857C211yesmaximal, characteristic, complemented
7#20276852C211yesmaximal, characteristic, complemented
8#20276497C212yesmaximal, complemented
9C2×C4⋊#11719908C211yesmaximal, characteristic, complemented
10#20276422C212yesmaximal
11C4×#135×C53C211yesmaximal, characteristic
12#16811180—21no—
13C22×#11719908C2212yes—
14#16811028C2211yescharacteristic
15#16811028C2211yescharacteristic
16#16811025—21no—
17#16811025C2212yes—
18#16811021—21no—
19#16810661—21no—
20#16810661C2212yes—
21#16810450C2212yes—
22#16810450C2212yes—
23#16810448C2212yes—
24#16810448C2212yes—
25C4⋊#11719908C2211yescharacteristic
26C4⋊#11719908C2211yescharacteristic
27C4⋊#11719908C2212yes—
28#16810427C2212yes—
29#16810427C2212yes—
30#16810424C2212yes—
31#16810424C2212yes—
32C2×#135×C53C2211yescharacteristic
33C4×#41×C53—21no—
34C4×Q16×C53C2211yescharacteristic
35C4×Q16×C53C2211yescharacteristic
36C4×SD16×C53—21no—
37C4×SD16×C53—21no—
38C4×D8×C53C2211yescharacteristic
39C4×D8×C53C2211yescharacteristic
40C53×#427C2211yescharacteristic
41C53×#416—21no—
42C2×C4×C8×C53C2211yescharacteristic
43#13133217—21no—
44#13133217—21no—
45#13133209—22no—
46#13133196—41no—
47#13133196—41no—
48C4×Dic106—41no—
49C4×Dic106—41no—
50#13133176—22no—
51C2×#11719908—24no—
52C2×#11719908C2312yes—
53C2×#11719908C2312yes—
54#13133151—21no—
55#13133151—21no—
56#13133151C2312yes—
57#13133151C2312yes—
58#13133149—22no—
59#13133149—22no—
60C2×#41×C53—21no—
61#135×C53—22no—
62#135×C53Q814yescomplemented
63C2×Q16×C53C2311yescharacteristic
64C2×SD16×C53—21no—
65C2×D8×C53C2311yescharacteristic
66C22×C8×C53D411yescharacteristic
67C22×C8×C53C2311yescharacteristic
68C4×Q8×C53—21no—
69C4×Q8×C53—21no—
70C4×D4×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53—41no—
73C4×D4×C53—41no—
74#117×C53—21no—
75#117×C53—41no—
76C2×C42×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
89#41×C53—24no—
90#41×C53—42no—
91C2×Q8×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53D811yescharacteristic
98C22×C4×C53D811yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C2×Dic106—41no—
101C2×Dic106—41no—
102Q16×C53—21no—
103Q16×C53C2×Q812yes—
104SD16×C53—24no—
105D8×C53—21no—
106D8×C53C2×Q812yes—
107C2×C8×C53—22no—
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2×Q812yes—
112C2×C8×C53#4112yes—
113C2×C8×C53C2411yescharacteristic
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C42×C53—41no—
122C42×C53—41no—
123C42×C53—41no—
124C42×C53D811yescharacteristic
125C42×C53D811yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53C2×D411yescharacteristic
128C22⋊Dic53—42no—
129C4⋊Dic53—22no—
130C4⋊Dic53—22no—
131C4⋊Dic53—41no—
132C4⋊Dic53—41no—
133#11719919—42no—
134#11719919—42no—
135C4×Dic53—42no—
136#11719908—22no—
137#11719908—22no—
138C23×C53—41no—
139C23×C53C2×D811yescharacteristic
140Q8×C53—22no—
141Q8×C53—41no—
142D4×C53—22no—
143D4×C53—44no—
144D4×C53—41no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—44no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—82no—
152C2×C4×C53—82no—
153C2×C4×C53C22⋊Q812yes—
154C2×C4×C53C2×D811yescharacteristic
155C2×C4×C53C2×D811yescharacteristic
156C2×C4×C53C2×D811yescharacteristic
157C2×C4×C53C2×D811yescharacteristic
158C2×C4×C53C2×D811yescharacteristic
159C2×C4×C53#13712yes—
160C2×C4×C53C22×D411yescharacteristic
161C2×Dic53—42no—
162C2×Dic53—42no—
163Dic106—82no—
164Dic106—82no—
165C8×C53—21no—
166C8×C53—21no—
167C8×C53C22⋊Q812yes—
168C8×C53C22×Q811yescharacteristic
169C8×C53C2×#4111yescharacteristic
170#33626—531nomaximal, Sylow
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—42no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53#47412yes—
177C22×C53C22×D811yescharacteristic
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—42no—
182C4×C53—81no—
183C4×C53—81no—
184C4×C53#47412yes—
185C4×C53#50812yes—
186C4×C53C2×C22⋊Q811yescharacteristic
187C4×C53C2×#13711yescharacteristic
188Dic53—82no—
189Dic53—82no—
190#3160—531no—
191C4×#135D5311yescharacteristic
192#2229—532no—
193#2228—532no—
194#2227—531no—
195#2226—531no—
196#2225—531no—
197#2224—531no—
198#2158—532no—
199C2×C4⋊C16—531no—
200#2138—532no—
201C2×C53—22no—
202C2×C53—82no—
203C2×C53C2×#47411yescharacteristic
204C2×C53C2×#50811yescharacteristic
205C2×C53#341411yescharacteristic
206C2×#135C2×D5311yescharacteristic
207#533—1061no—
208C4×#41—21no—
209C22×C16—532no—
210#500—531no—
211#500—531no—
212#499—532no—
213#499—1061no—
214#498—1061no—
215C4×Q16C2×D5311yescharacteristic
216C4×Q16C2×D5311yescharacteristic
217C4×SD16—21no—
218C4×SD16—21no—
219C4×D8C2×D5311yescharacteristic
220C4×D8C2×D5311yescharacteristic
221#427—532no—
222#427—1061no—
223#427C2×D5311yescharacteristic
224#416—21no—
225C2×C4×C8C2×D5311yescharacteristic
226#367—532no—
227#367—532no—
228#366—532no—
229#366—532no—
230C4⋊C16—531no—
231C4⋊C16—531no—
232C4⋊C16—532no—
233#358—532no—
234#358—532no—
235#357—532no—
236#357—532no—
237C53#3362611yescharacteristic, complemented, Sylow
238C2×#41—21no—
239#135—22no—
240#135Dic10614yescomplemented
241C2×Q16C22×D5311yescharacteristic
242C2×SD16—21no—
243C2×D8C22×D5311yescharacteristic
244C22×C8C22⋊D5311yescharacteristic
245C22×C8C22×D5311yescharacteristic
246C4⋊Q8—1061no—
247C4⋊Q8—1061no—
248#125—1062no—
249C22⋊Q8—2121no—
250C22⋊Q8—2121no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—2121no—
254C4×Q8—2121no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—41no—
258C4×D4—41no—
259#117—21no—
260#117—41no—
261#117—1062no—
262C2×C42—41no—
263C2×C42C22⋊D5311yescharacteristic
264C2×C16—532no—
265C2×C16—532no—
266C2×C16—1064no—
267#107—532no—
268#107—532no—
269#107—1061no—
270#107—1061no—
271#107C22×D5311yescharacteristic
272#107C22×D5311yescharacteristic
273#106—21no—
274#106—1062no—
275#106—1062no—
276#103—21no—
277#103—21no—
278#102—21no—
279#102—21no—
280C4×C8C22⋊D5311yescharacteristic
281C4×C8C22⋊D5311yescharacteristic
282C4×C8C22×D5311yescharacteristic
283C4×C8C22×D5311yescharacteristic
284#41—24no—
285#41—42no—
286C2×Q8—21no—
287C2×Q8—2121no—
288C2×Q8—2121no—
289C2×D4—21no—
290C2×D4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4#1171992211yescharacteristic
295C22×C4#1171992211yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297Q16—21no—
298Q16C2×Dic10612yes—
299SD16—24no—
300D8—21no—
301D8C2×Dic10612yes—
302C2×C8—22no—
303C2×C8C2×Dic10612yes—
304C2×C8#1171994512yes—
305C2×C8C2×C22⋊D5311yescharacteristic
306C2×C8C2×C22⋊D5311yescharacteristic
307C2×C8C2×C22⋊D5311yescharacteristic
308C2×C8C23×D5311yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—1062no—
314C4⋊C4—1062no—
315C4⋊C4—2122no—
316C4⋊C4—2122no—
317C4⋊C4—2121no—
318C4⋊C4—2121no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—2122no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—2122no—
327C42#1171992211yescharacteristic
328C42#1171992211yescharacteristic
329C42C2×C22⋊D5311yescharacteristic
330C42C2×C22⋊D5311yescharacteristic
331C16—1062no—
332C16—1062no—
333C23—41no—
334C23C2×#1171992211yescharacteristic
335Q8—22no—
336Q8—41no—
337Q8—4242no—
338Q8—4242no—
339D4—22no—
340D4—44no—
341D4—41no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—2122no—
351C2×C4—2122no—
352C2×C4#1313319612yes—
353C2×C4C2×#1171992211yescharacteristic
354C2×C4C2×#1171992211yescharacteristic
355C2×C4C2×#1171992211yescharacteristic
356C2×C4C2×#1171992211yescharacteristic
357C2×C4C2×#1171992211yescharacteristic
358C2×C4#1313324712yes—
359C2×C4C22×C22⋊D5311yescharacteristic
360C8—21no—
361C8—21no—
362C8#1313319612yes—
363C8C22×Dic10611yescharacteristic
364C8C2×#1171994511yescharacteristic
365C22—21no—
366C22—21no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22#1681090312yes—
371C22C22×#1171992211yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—4242no—
379C4—4242no—
380C4#1681090312yes—
381C4#1681105312yes—
382C4C2×#1313319611yescharacteristic
383C4C2×#1313324711yescharacteristic
384C2—22no—
385C2—82no—
386C2C2×#1681090311yescharacteristic
387C2C2×#1681105311yescharacteristic
388C2#2028784911yescharacteristic
subgroupG/H
centerC2×C4C2×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC4×#135×C53C2
socleC22×C53C22×D8
prime radicalC2×#135×C53C22
hypercenterC4×#135D53
hypocenterC53#33626
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #20285229C2semidirect
2. #20276879C2semidirect
3. #20276872C2semidirect
4. #20276867C2semidirect
5. #20276862C2semidirect
6. #20276857C2semidirect
7. #20276852C2semidirect
8. #20276497C2semidirect
9. C2×C4⋊#11719908C2semidirect
62. #135×C53Q8semidirect
237. C53#33626semidirect
240. #135Dic106semidirect

Maximal subgroups

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

subgroupindex
#202852292
#202768792
#202768722
#202768672
#202768622
#202768572
#202768522
#202764972
C2×C4⋊#117199082
#202764222
C4×#135×C532
#3362653