Finite Group Databasebeta

← #27539161

Subgroups — #27539161

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133239C211yesmaximal, characteristic, complemented
2#20276903C212yesmaximal, complemented
3#20276854C211yesmaximal, characteristic, complemented
4#20276849C211yesmaximal, characteristic, complemented
5#20276752C211yesmaximal, characteristic, complemented
6#20276747C211yesmaximal, characteristic, complemented
7#20276636C212yesmaximal, complemented
8#20276589C212yesmaximal, complemented
9#20276579C212yesmaximal, complemented
10#20276518C211yesmaximal, characteristic, complemented
11C53×#3169C211yesmaximal, characteristic, complemented
12C2×#13133239C2211yescharacteristic
13C4×#11719945—21no—
14C2×#13133161C2212yes—
15C2×#13133158C2212yes—
16C4×Dic212C2211yescharacteristic
17C4×Dic212C2211yescharacteristic
18C4×#11719913—21no—
19C4×#11719913—21no—
20C4×D424C2211yescharacteristic
21C4×D424C2211yescharacteristic
22#16810661C2211yescharacteristic
23#16810620—21no—
24#16810450C2212yes—
25#16810448C2212yes—
26C4⋊#11719908C2211yescharacteristic
27C4⋊#11719908C2211yescharacteristic
28#16810426C2212yes—
29#16810424C2212yes—
30#16810424C2212yes—
31#16810423C2212yes—
32C22⋊#11719908C2212yes—
33#16810399C2212yes—
34C53×#532—21no—
35C53×#500C2211yescharacteristic
36C53×#493C2211yescharacteristic
37C53×#486—21no—
38C53×#466—21no—
39C53×#466C2212yes—
40C4×D8×C53—21no—
41C4×D8×C53C2212yes—
42C2×C4×C8×C53C2211yescharacteristic
43C2×#11719945—21no—
44#13133239—22no—
45#13133239—22no—
46C2×Dic212C2311yescharacteristic
47C2×#11719913—21no—
48C2×D424C2311yescharacteristic
49C4×Dic106—21no—
50C4×Dic106—21no—
51C4×D212—21no—
52C4×D212—21no—
53C4×C22⋊D53—41no—
54C4×C22⋊D53—41no—
55#13133176—21no—
56#13133174—41no—
57C42×D53—41no—
58#13133161—24no—
59#13133158—24no—
60C2×#11719908C2312yes—
61C2×#11719908C2312yes—
62#13133151C2311yescharacteristic
63#13133151C2311yescharacteristic
64#13133149—21no—
65#13133142—21no—
66#13133142—21no—
67#13133139—21no—
68#13133139—21no—
69C2×#41×C53—41no—
70C2×D8×C53—21no—
71C2×D8×C53C2312yes—
72C22×C8×C53D411yescharacteristic
73C22×C8×C53C2311yescharacteristic
74C4⋊Q8×C53—21no—
75#127×C53—21no—
76#124×C53—41no—
77#121×C53—22no—
78#121×C53—41no—
79C4×D4×C53—22no—
80C4×D4×C53—41no—
81C2×C42×C53D411yescharacteristic
82#107×C53—21no—
83#107×C53C2312yes—
84#102×C53—22no—
85#102×C53—22no—
86C4×C8×C53D411yescharacteristic
87C4×C8×C53D411yescharacteristic
88C4×C8×C53C2311yescharacteristic
89C4×C8×C53C2311yescharacteristic
90#41×C53—84no—
91C2×Q8×C53—41no—
92C2×D4×C53—22no—
93C2×D4×C53—42no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C22×C4×C53—42no—
97C22×C4×C53D811yescharacteristic
98C22×C4×C53D811yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C2×C22⋊D53—41no—
101#11719945—24no—
102#11719945—42no—
103C2×D212—21no—
104C2×C4×D53—41no—
105C2×C4×D53—41no—
106C2×C4×D53—41no—
107C2×Dic106—21no—
108D8×C53—22no—
109D8×C53—22no—
110D8×C53—42no—
111C2×C8×C53—21no—
112C2×C8×C53—21no—
113C2×C8×C53—41no—
114C2×C8×C53C2×D411yescharacteristic
115C2×C8×C53C2×D411yescharacteristic
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53C2411yescharacteristic
118C4⋊C4×C53—22no—
119C4⋊C4×C53—41no—
120C22⋊C4×C53—42no—
121C22⋊C4×C53—42no—
122C42×C53D811yescharacteristic
123C42×C53D811yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C22⋊Dic53—41no—
127#11719921—41no—
128#11719921—41no—
129C4⋊Dic53—21no—
130C4⋊Dic53—21no—
131#11719919—41no—
132#11719919—41no—
133C4×Dic53—41no—
134C4×Dic53—41no—
135C4×Dic53—41no—
136Dic212—21no—
137Dic212#4112yescomplemented
138D424—21no—
139D424#4112yescomplemented
140#11719913—24no—
141#11719908—22no—
142#11719908—22no—
143C23×C53—42no—
144C23×C53C2×D811yescharacteristic
145C22×D53—41no—
146Q8×C53—82no—
147D4×C53—42no—
148D4×C53—42no—
149D4×C53—82no—
150D4×C53—84no—
151D4×C53—82no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—22no—
155C2×C4×C53—41no—
156C2×C4×C53—42no—
157C2×C4×C53—84no—
158C2×C4×C53C2×D811yescharacteristic
159C2×C4×C53C2×D811yescharacteristic
160C2×C4×C53C2×D811yescharacteristic
161C2×C4×C53C2×D811yescharacteristic
162C2×C4×C53C2×D811yescharacteristic
163C2×C4×C53C22×D411yescharacteristic
164C22⋊D53—44no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168D212—22no—
169D212—41no—
170C4×D53—44no—
171C4×D53—82no—
172C4×D53—82no—
173Dic106—22no—
174Dic106—41no—
175C8×C53—21no—
176C8×C53—21no—
177C8×C53—21no—
178C8×C53C2×#4111yescharacteristic
179C8×C53C2×#4111yescharacteristic
180#33675—531nomaximal, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—82no—
185C22×C53—82no—
186C22×C53—82no—
187C22×C53C22×D811yescharacteristic
188C2×D53—42no—
189C2×D53—81no—
190C2×D53—81no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—82no—
196C4×C53#50912yes—
197C4×C53#52511yescharacteristic
198C4×C53#57511yescharacteristic
199Dic53—42no—
200Dic53—81no—
201Dic53—81no—
202#3169D5311yescharacteristic, complemented
203C4×#135—531no—
204#2234—532no—
205#2225—531no—
206#2224—531no—
207#2207—531no—
208#2206—531no—
209#2186—532no—
210#2177—532no—
211#2175—532no—
212#2163—531no—
213C2×C53—41no—
214C2×C53—162no—
215C2×C53#310011yescharacteristic
216C2×C53C2×#50911yescharacteristic
217C2×C53#341411yescharacteristic
218D53—82no—
219C2×#135—531no—
220#532—21no—
221C4×#41—1061no—
222C2×SD32—532no—
223C2×D16—532no—
224#500C2×D5311yescharacteristic
225#493C2×D5311yescharacteristic
226#486—21no—
227#466—21no—
228#466C2×D5312yes—
229C4×Q16—531no—
230C4×Q16—531no—
231C4×SD16—1061no—
232C4×SD16—1061no—
233C4×D8—21no—
234C4×D8—531no—
235C4×D8—531no—
236C4×D8C2×D5312yes—
237#427—531no—
238#416—1061no—
239C2×C4×C8C2×D5311yescharacteristic
240#367—532no—
241#366—532no—
242C4⋊C16—531no—
243C4⋊C16—531no—
244#358—532no—
245#357—532no—
246#357—532no—
247#357—532no—
248C22⋊C16—532no—
249#346—532no—
250C53#3367511yescharacteristic, complemented, Sylow
251C2×#41—41no—
252C2×#41—1061no—
253#135—1062no—
254#135—1062no—
255C2×Q16—531no—
256C2×SD16—1061no—
257C2×D8—21no—
258C2×D8—531no—
259C2×D8C22×D5312yes—
260C22×C8C22⋊D5311yescharacteristic
261C22×C8C22×D5311yescharacteristic
262C4⋊Q8—21no—
263#127—21no—
264#124—41no—
265#121—22no—
266#121—41no—
267C4×Q8—1061no—
268C4×Q8—1061no—
269C4×D4—22no—
270C4×D4—41no—
271C4×D4—1061no—
272C4×D4—1061no—
273C4×D4—2121no—
274C4×D4—2121no—
275#117—1061no—
276#117—2121no—
277C2×C42—2121no—
278C2×C42C22⋊D5311yescharacteristic
279SD32—1064no—
280D16—1064no—
281C2×C16—532no—
282C2×C16—532no—
283#107—21no—
284#107—531no—
285#107—531no—
286#107C22×D5312yes—
287#106—1061no—
288#103—1061no—
289#103—1061no—
290#102—22no—
291#102—22no—
292#102—1061no—
293#102—1061no—
294C4×C8C22⋊D5311yescharacteristic
295C4×C8C22⋊D5311yescharacteristic
296C4×C8C22×D5311yescharacteristic
297C4×C8C22×D5311yescharacteristic
298#41—84no—
299#41—1064no—
300#41—2122no—
301C2×Q8—41no—
302C2×Q8—1061no—
303C2×D4—22no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—1061no—
308C2×D4—2121no—
309C22×C4—42no—
310C22×C4—2121no—
311C22×C4—2121no—
312C22×C4—2121no—
313C22×C4#1171992211yescharacteristic
314C22×C4#1171992211yescharacteristic
315C22×C4C2×C22⋊D5311yescharacteristic
316Q16—532no—
317Q16—1061no—
318SD16—1064no—
319D8—22no—
320D8—22no—
321D8—42no—
322D8—532no—
323D8—1061no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—41no—
327C2×C8C2×C22⋊D5311yescharacteristic
328C2×C8C2×C22⋊D5311yescharacteristic
329C2×C8C2×C22⋊D5311yescharacteristic
330C2×C8C23×D5311yescharacteristic
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—1061no—
334C4⋊C4—1061no—
335C4⋊C4—2121no—
336C4⋊C4—2121no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—2121no—
340C22⋊C4—2121no—
341C22⋊C4—2121no—
342C42—2121no—
343C42—2121no—
344C42—2121no—
345C42#1171992211yescharacteristic
346C42#1171992211yescharacteristic
347C42C2×C22⋊D5311yescharacteristic
348C42C2×C22⋊D5311yescharacteristic
349C16—1062no—
350C16—1062no—
351C23—42no—
352C23—2121no—
353C23C2×#1171992211yescharacteristic
354Q8—82no—
355Q8—1062no—
356Q8—2121no—
357D4—42no—
358D4—42no—
359D4—82no—
360D4—84no—
361D4—82no—
362D4—1062no—
363D4—2124no—
364D4—2121no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—84no—
371C2×C4—4242no—
372C2×C4—4242no—
373C2×C4—2124no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4C2×#1171992211yescharacteristic
378C2×C4C2×#1171992211yescharacteristic
379C2×C4C2×#1171992211yescharacteristic
380C2×C4C2×#1171992211yescharacteristic
381C2×C4C2×#1171992211yescharacteristic
382C2×C4C22×C22⋊D5311yescharacteristic
383C8—21no—
384C8—21no—
385C8—21no—
386C8#41×D5311yescharacteristic
387C8#41×D5311yescharacteristic
388C22—21no—
389C22—21no—
390C22—21no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22—4241no—
395C22—4241no—
396C22—2122no—
397C22C22×#1171992211yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—82no—
403C4—4241no—
404C4—4241no—
405C4—2122no—
406C4#1681105912yes—
407C4#1681113011yescharacteristic
408C4#1681142111yescharacteristic
409C2—41no—
410C2—162no—
411C2—4242no—
412C2#2028471111yescharacteristic
413C2C2×#1681105911yescharacteristic
414C2#2028784911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3169C2
socleC22×C53C22×D8
hypercenter#3169D53
hypocenterC53#33675
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. C4×#13133239C2semidirect
2. #20276903C2semidirect
3. #20276854C2semidirect
4. #20276849C2semidirect
5. #20276752C2semidirect
6. #20276747C2semidirect
7. #20276636C2semidirect
8. #20276589C2semidirect
9. #20276579C2semidirect
10. #20276518C2semidirect
11. C53×#3169C2semidirect
137. Dic212#41semidirect
139. D424#41semidirect
202. #3169D53semidirect
250. C53#33675semidirect

Maximal subgroups

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

subgroupindex
C4×#131332392
#202769032
#202768542
#202768492
#202767522
#202767472
#202766362
#202765892
#202765792
#202765182
C53×#31692
#3367553