Finite Group Databasebeta

← #27539424

Subgroups — #27539424

Order 13568.

602 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#20285254C211yesmaximal, characteristic, complemented
2#20276956C212yesmaximal, complemented
3#20276951C212yesmaximal, complemented
4#20276946C212yesmaximal, complemented
5#20276938C212yesmaximal, complemented
6#20276431C211yesmaximal, characteristic, complemented
7#20276427C211yesmaximal, characteristic, complemented
8#20276422C211yesmaximal, characteristic, complemented
9C2×#16810424C211yesmaximal, characteristic, complemented
10#20276312C211yesmaximal, characteristic, complemented
11C53×#3155C211yesmaximal, characteristic
12#16811251—21no—
13C2×#13133157C2211yescharacteristic
14C22×#11719908C2211yescharacteristic
15#16811032—22no—
16#16811028C2212yescomplemented
17#16811025C2212yescomplemented
18#16810665—21no—
19#16810661C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21#16810427C2211yescharacteristic, complemented
22#16810427C2211yescharacteristic, complemented
23#16810427C2212yescomplemented
24#16810424C2212yes—
25#16810424C2212yes—
26#16810424C2212yes—
27#16810424C2212yes—
28#16810424C2211yescharacteristic, complemented
29#16810424C2211yescharacteristic, complemented
30#16810424C2212yescomplemented
31#16810399C2212yescomplemented
32C4×#11719908C2212yescomplemented
33C2×AGL(1,ℤ/8)×C53—21no—
34C2×#135×C53C2211yescharacteristic
35C22×D8×C53C2211yescharacteristic
36C53×#535—21no—
37C53×#496—22no—
38C53×#495C2212yes—
39C53×#493C2212yes—
40C53×#405C2211yescharacteristic
41#13133217—22no—
42#13133216—41no—
43#13133209—22no—
44#13133208—41no—
45#13133194—42no—
46#13133176—21no—
47C2×C4⋊Dic53—21no—
48#13133157—22no—
49C2×#11719908—22no—
50C2×#11719908C2311yescharacteristic
51C2×#11719908C2311yescharacteristic
52C2×#11719908C2312yes—
53#13133151—22no—
54#13133151—21no—
55#13133151C2311yescharacteristic
56#13133151C2312yes—
57#13133151C2311yescharacteristic
58#13133149—22no—
59#13133149—21no—
60C2×#41×C53—21no—
61C22×D4×C53—21no—
62AGL(1,ℤ/8)×C53—44no—
63#135×C53—22no—
64#135×C53—22no—
65C2×Q16×C53C2311yescharacteristic
66C2×SD16×C53—22no—
67C2×SD16×C53—21no—
68C2×D8×C53—22no—
69C2×D8×C53—21no—
70C2×D8×C53—22no—
71C2×D8×C53—22no—
72C2×D8×C53C2311yescharacteristic
73C2×D8×C53C2312yes—
74C2×M16×C53D411yescharacteristic
75C22×C8×C53C2311yescharacteristic
76#127×C53—22no—
77#124×C53—22no—
78#121×C53—41no—
79#121×C53—41no—
80#120×C53—42no—
81#117×C53D411yescharacteristic
82#97×C53D412yes—
83C4×C8×C53C2312yes—
84C24×C53—41no—
85#41×C53—42no—
86#41×C53—42no—
87C2×Q8×C53—21no—
88C2×D4×C53—22no—
89C2×D4×C53—21no—
90C2×D4×C53—41no—
91C2×D4×C53—42no—
92C2×D4×C53—41no—
93C2×D4×C53—42no—
94C2×D4×C53—42no—
95C2×D4×C53—42no—
96C22×C4×C53—41no—
97C22×C4×C53C2×D411yescharacteristic
98C22×Dic53—41no—
99C2×Dic106—42no—
100Q16×C53—21no—
101Q16×C53—21no—
102SD16×C53—42no—
103SD16×C53—44no—
104D8×C53—22no—
105D8×C53—22no—
106D8×C53—21no—
107D8×C53—21no—
108D8×C53—42no—
109D8×C53—44no—
110M16×C53SD1614yes—
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53C2×D412yes—
113C2×C8×C53#4112yes—
114C2×C8×C53#4112yes—
115C2×C8×C53C2411yescharacteristic
116C4⋊C4×C53D811yescharacteristic
117C4⋊C4×C53D811yescharacteristic
118C22⋊C4×C53—42no—
119C22⋊C4×C53—42no—
120C22⋊C4×C53D812yes—
121C42×C53C2×D412yes—
122C22⋊Dic53—41no—
123C4⋊Dic53—21no—
124C4⋊Dic53—22no—
125C4⋊Dic53—21no—
126C4⋊Dic53—41no—
127C4⋊Dic53—42no—
128#11719919—42no—
129#11719919—42no—
130C4×Dic53—41no—
131#11719908—21no—
132#11719908—21no—
133#11719908—22no—
134C23×C53—41no—
135C23×C53—42no—
136C23×C53—82no—
137C23×C53—161no—
138C23×C53—161no—
139C23×C53C2×D811yescharacteristic
140Q8×C53—41no—
141Q8×C53—41no—
142D4×C53—42no—
143D4×C53—41no—
144D4×C53—42no—
145D4×C53—41no—
146D4×C53—82no—
147D4×C53—82no—
148D4×C53—162no—
149D4×C53—162no—
150C2×C4×C53—41no—
151C2×C4×C53—82no—
152C2×C4×C53#12412yes—
153C2×C4×C53C2×D812yes—
154C2×C4×C53C2×D812yes—
155C2×C4×C53C2×D811yescharacteristic
156C2×C4×C53C2×SD1612yes—
157C2×C4×C53C22×D411yescharacteristic
158C2×Dic53—41no—
159C2×Dic53—42no—
160C2×Dic53—41no—
161C2×Dic53—161no—
162Dic106—162no—
163C8×C53#12412yes—
164C8×C53C2×SD1614yes—
165C8×C53C2×#4111yescharacteristic
166C8×C53C2×#4111yescharacteristic
167#33702—531nomaximal, Sylow
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—81no—
171C22×C53—82no—
172C22×C53—81no—
173C22×C53—82no—
174C22×C53—81no—
175C22×C53—82no—
176C22×C53—164no—
177C22×C53#48612yes—
178C22×C53C22×D811yescharacteristic
179C4×C53—22no—
180C4×C53—22no—
181C4×C53—81no—
182C4×C53#48612yes—
183C4×C53C2×#12411yescharacteristic
184C4×C53C22×SD1611yescharacteristic
185Dic53—81no—
186Dic53—82no—
187Dic53—81no—
188#3164—531no—
189#3155D5311yescharacteristic
190#2244—532no—
191#2243—532no—
192#2242—532no—
193#2240—532no—
194#2140—531no—
195#2139—531no—
196#2138—531no—
197C2×#357—531no—
198#2108—531no—
199C2×C53—22no—
200C2×C53—161no—
201C2×C53—162no—
202C2×C53C2×#48611yescharacteristic
203C2×C53#341511yescharacteristic
204C2×C53#341611yescharacteristic
205C2×AGL(1,ℤ/8)—21no—
206C2×#135C2×D5311yescharacteristic
207C22×D8C2×D5311yescharacteristic
208#544—1061no—
209#535—21no—
210C2×M32—531no—
211C22×C16—531no—
212#501—1062no—
213#500—532no—
214#499—532no—
215#496—22no—
216#495C2×D5312yes—
217#493C2×D5312yes—
218#428—1061no—
219#427—531no—
220C2×#107—531no—
221#405C2×D5311yescharacteristic
222#358—531no—
223#358—531no—
224#358—532no—
225#357—532no—
226#357—532no—
227#357—531no—
228#357—531no—
229#357—532no—
230#357—532no—
231#357—532no—
232#346—532no—
233C4×C16—532no—
234C53#3370211yescharacteristic, complemented, Sylow
235C2×#41—21no—
236C22×D4—21no—
237AGL(1,ℤ/8)—44no—
238#135—22no—
239#135—22no—
240C2×Q16C22×D5311yescharacteristic
241C2×SD16—21no—
242C2×SD16—22no—
243C2×D8—22no—
244C2×D8—22no—
245C2×D8—21no—
246C2×D8—22no—
247C2×D8C22×D5312yes—
248C2×D8C22×D5311yescharacteristic
249C2×M16C22⋊D5311yescharacteristic
250C22×C8C22×D5311yescharacteristic
251C4⋊Q8—1062no—
252C4⋊Q8—2121no—
253#127—22no—
254#125—1062no—
255#125—2121no—
256#124—22no—
257C22⋊Q8—2122no—
258#121—41no—
259#121—41no—
260#120—42no—
261#117—1061no—
262#117C22⋊D5311yescharacteristic
263C2×C4⋊C4—1061no—
264M32—1062no—
265C2×C16—531no—
266C2×C16—532no—
267C2×C16—531no—
268C2×C16—1062no—
269#107—532no—
270#107—531no—
271#107—531no—
272#107—1061no—
273#107—1062no—
274#106—1061no—
275#106—1062no—
276#97C22⋊D5312yes—
277C4×C8C22×D5312yes—
278C24—41no—
279#41—42no—
280#41—42no—
281C2×Q8—21no—
282C2×Q8—2122no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C22×C4—41no—
292C22×C4—2121no—
293C22×C4C2×C22⋊D5311yescharacteristic
294Q16—21no—
295Q16—21no—
296SD16—42no—
297SD16—44no—
298D8—22no—
299D8—22no—
300D8—21no—
301D8—21no—
302D8—42no—
303D8—44no—
304M16#1171992314yes—
305C2×C8#1171994712yes—
306C2×C8#1171994712yes—
307C2×C8C2×C22⋊D5311yescharacteristic
308C2×C8C2×C22⋊D5312yes—
309C2×C8C23×D5311yescharacteristic
310C4⋊C4—1061no—
311C4⋊C4—1062no—
312C4⋊C4—1061no—
313C4⋊C4—2121no—
314C4⋊C4—2122no—
315C4⋊C4—2122no—
316C4⋊C4—2122no—
317C4⋊C4#1171992211yescharacteristic
318C4⋊C4#1171992211yescharacteristic
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—2121no—
322C22⋊C4#1171992212yes—
323C42—2121no—
324C42C2×C22⋊D5312yes—
325C16—1061no—
326C16—1061no—
327C16—1062no—
328C23—42no—
329C23—41no—
330C23—82no—
331C23—161no—
332C23—161no—
333C23C2×#1171992211yescharacteristic
334Q8—41no—
335Q8—41no—
336Q8—8482no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—42no—
341D4—82no—
342D4—82no—
343D4—162no—
344D4—162no—
345C2×C4—41no—
346C2×C4—82no—
347C2×C4—8481no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2122no—
351C2×C4#1313320412yes—
352C2×C4C2×#1171992212yes—
353C2×C4C2×#1171992212yes—
354C2×C4C2×#1171992211yescharacteristic
355C2×C4C2×#1171992312yes—
356C2×C4C22×C22⋊D5311yescharacteristic
357C8#1313320412yes—
358C8C2×#1171992314yes—
359C8C2×#1171994711yescharacteristic
360C8C2×#1171994711yescharacteristic
361C22—21no—
362C22—21no—
363C22—81no—
364C22—82no—
365C22—81no—
366C22—82no—
367C22—81no—
368C22—82no—
369C22—164no—
370C22#1681096912yes—
371C22C22×#1171992211yescharacteristic
372C4—22no—
373C4—22no—
374C4—81no—
375C4—4242no—
376C4—4241no—
377C4—4241no—
378C4#1681096912yes—
379C4C2×#1313320411yescharacteristic
380C4C22×#1171992311yescharacteristic
381C2—22no—
382C2—162no—
383C2—161no—
384C2C2×#1681096911yescharacteristic
385C2#2028785411yescharacteristic
386C2#2028786211yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalC53×#3155C2
socleC22×C53C22×D8
hypercenter#3155D53
hypocenterC53#33702
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3155

Complements

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

NH ≅ G/Ntype
1. #20285254C2semidirect
2. #20276956C2semidirect
3. #20276951C2semidirect
4. #20276946C2semidirect
5. #20276938C2semidirect
6. #20276431C2semidirect
7. #20276427C2semidirect
8. #20276422C2semidirect
9. C2×#16810424C2semidirect
10. #20276312C2semidirect
16. #16811028C22semidirect
17. #16811025C22semidirect
21. #16810427C22semidirect
22. #16810427C22semidirect
23. #16810427C22semidirect
28. #16810424C22semidirect
29. #16810424C22semidirect
30. #16810424C22semidirect
31. #16810399C22semidirect
32. C4×#11719908C22semidirect
234. C53#33702semidirect

Maximal subgroups

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

subgroupindex
#202852542
#202769562
#202769512
#202769462
#202769382
#202764312
#202764272
#202764222
C2×#168104242
#202763122
C53×#31552
#3370253