Finite Group Databasebeta

← #27278270

Subgroups — #27278270

Order 13568.

546 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
1#20277666C211yesmaximal, characteristic
2#20275476C211yesmaximal, characteristic, complemented
3#20274712C211yesmaximal, characteristic
4C2×#16810352C211yesmaximal, characteristic
5#20272598C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C53×#1845C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9C2×#13133175—21no—
10C2×#13133148—21no—
11C2×C4⋊#88322C2211yescharacteristic
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133137—21no—
14#16810524—21no—
15#16810513—22no—
16#16810480—21no—
17C4⋊C4×Dic53—21no—
18C4×C4⋊Dic53C2211yescharacteristic
19#16810385—21no—
20#16810385—21no—
21#16810352C414yescomplemented
22C2×#107×C53—22no—
23C2×C4×C8×C53C2211yescharacteristic
24C53×#384C2211yescharacteristic
25C53×#384C2211yescharacteristic
26C22×D212D411yescharacteristic
27C22×C4×D53—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30C4×D212—21no—
31C4×D212—21no—
32C4×D212C2×C412yes—
33C4×D212Q812yes—
34#13133175—42no—
35#13133175—42no—
36C2×C4⋊Dic53D411yescharacteristic
37C2×C4⋊Dic53D411yescharacteristic
38C2×C4⋊Dic53D411yescharacteristic
39C2×#11719921—22no—
40C2×#11719921—22no—
41C2×#11719921—21no—
42C2×C4×Dic53—22no—
43C2×C4×Dic53—21no—
44#13133148—42no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322C2×C412yes—
49#13133137—42no—
50#13133124—22no—
51#13133124—21no—
52C22×C8×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53D412yes—
59C2×C42×C53C2311yescharacteristic
60#107×C53—44no—
61C4×C8×C53—21no—
62C4×C8×C53C2×C412yes—
63C23×D53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—41no—
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212C4⋊C412yes—
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C8×C53—22no—
85C2×C8×C53—22no—
86C2×C8×C53—22no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—42no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C42×C53C22×C411yescharacteristic
98C42×C53C2×Q811yescharacteristic
99C42×C53#4111yescharacteristic
100C42×C53#4111yescharacteristic
101#11719921—22no—
102#11719921—22no—
103#11719921—44no—
104#11719921—82no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109C4⋊Dic53C4⋊C412yes—
110C4⋊Dic53D812yes—
111C4×Dic53—44no—
112C4×Dic53—41no—
113C4×Dic53—41no—
114C2×#88322—22no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C2×#88322—41no—
119C2×#88322—41no—
120C23×C53#12011yescharacteristic
121C22×D53—22no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—41no—
126C22×D53—41no—
127C22×D53—41no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53C2×C4⋊C411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53C22⋊Q811yescharacteristic
140C2×C4×C53C22⋊Q811yescharacteristic
141C2×C4×C53#12311yescharacteristic
142C2×C4×C53C2×D811yescharacteristic
143C2×C4×C53#13511yescharacteristic
144C2×C4×C53#13511yescharacteristic
145C2×C4×C53#13511yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×C4×C53#13711yescharacteristic
150C2×Dic53—22no—
151C2×Dic53—22no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159D212—22no—
160D212—22no—
161D212—41no—
162D212—41no—
163C4×D53—42no—
164C4×D53—42no—
165C4×D53—42no—
166C4×D53—42no—
167C8×C53—24no—
168#88322—42no—
169#88322—81no—
170#8885—531nomaximal, Sylow
171C22×C53#38511yescharacteristic
172C22×C53#44711yescharacteristic
173C22×C53#44911yescharacteristic
174C22×C53#45211yescharacteristic
175C22×C53#45211yescharacteristic
176C22×C53#45311yescharacteristic
177C22×C53#45611yescharacteristic
178C2×D53—42no—
179C2×D53—42no—
180C2×D53—42no—
181C2×D53—81no—
182C2×D53—81no—
183C2×D53—81no—
184C2×D53—81no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—82no—
188C4×C53—161no—
189C4×C53#42711yescharacteristic
190C4×C53#42811yescharacteristic
191C4×C53C4×D811yescharacteristic
192C4×C53#44011yescharacteristic
193C4×C53#47411yescharacteristic
194C4×C53#47811yescharacteristic
195C4×C53#48211yescharacteristic
196C4×C53#48511yescharacteristic
197Dic53—42no—
198Dic53—42no—
199Dic53—81no—
200#2354—531no—
201#1948—531no—
202#1845D5311yescharacteristic, complemented
203#1802—531no—
204C2×D4⋊C8—531no—
205#1296—531no—
206#1279—531no—
207C2×C53#162111yescharacteristic
208C2×C53#163411yescharacteristic
209C2×C53#164711yescharacteristic
210C2×C53#165011yescharacteristic
211C2×C53#185411yescharacteristic
212C2×C53#185911yescharacteristic
213C2×C53#186211yescharacteristic
214D53—82no—
215C2×C4×D4—531no—
216C2×#117—1061no—
217C2×#107—22no—
218C2×#106—1061no—
219C2×C4⋊C8—531no—
220C2×C4⋊C8—531no—
221C2×#102—1061no—
222C2×C4×C8C2×D5311yescharacteristic
223#390—1061no—
224#388—1062no—
225#384C2×D5311yescharacteristic
226#384C2×D5311yescharacteristic
227#380—1061no—
228C4×C4⋊C4—531no—
229C4×C4⋊C4—1061no—
230#340—1061no—
231#340—1061no—
232D4⋊C8—534no—
233C53#888511yescharacteristic, complemented, Sylow
234C22×D4—531no—
235C23×C4—1061no—
236C22×C8—21no—
237C22×C8—1061no—
238C22×C8—1061no—
239C4×D4—532no—
240C4×D4—532no—
241C4×D4—1061no—
242C4×D4—1061no—
243#117—2122no—
244#117—2122no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—531no—
251C2×C4⋊C4—531no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4C22⋊D5312yes—
254C2×C22⋊C4—1061no—
255C2×C22⋊C4—1062no—
256C2×C22⋊C4—1062no—
257C2×C42—1062no—
258C2×C42—1061no—
259C2×C42C22×D5311yescharacteristic
260#107—44no—
261#106—2122no—
262C4⋊C8—532no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265C4⋊C8—1061no—
266#102—2122no—
267C4×C8—21no—
268C4×C8C4×D5312yes—
269#95—1062no—
270#95—1061no—
271C24—1061no—
272C2×D4—532no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—1061no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—1061no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4—2121no—
288C22×C4—2121no—
289C22×C4D4×D5311yescharacteristic
290C22×C4D4×D5311yescharacteristic
291C22×C4C2×C22⋊D5311yescharacteristic
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—1062no—
296C2×C8—2121no—
297C2×C8—2121no—
298C2×C8—2121no—
299C2×C8—2121no—
300C2×C8—2121no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—532no—
312C4⋊C4—532no—
313C4⋊C4—1061no—
314C4⋊C4—1061no—
315C4⋊C4—1061no—
316C4⋊C4—1061no—
317C22⋊C4—1062no—
318C22⋊C4—1062no—
319C22⋊C4—4242no—
320C22⋊C4—2124no—
321C42—2124no—
322C42—2121no—
323C42—2121no—
324C42C2×C4×D5311yescharacteristic
325C42#1171994511yescharacteristic
326C42Q8×D5311yescharacteristic
327C42#1171994911yescharacteristic
328C23—1062no—
329C23—2121no—
330C23—2121no—
331C23—2121no—
332C23—2121no—
333C23—2121no—
334C23—2121no—
335C23#1313318511yescharacteristic
336D4—1062no—
337D4—1062no—
338D4—2121no—
339D4—2121no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—1062no—
349C2×C4—1062no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4—2122no—
361C2×C4C4⋊C4×D5311yescharacteristic
362C2×C4C4×C22⋊D5311yescharacteristic
363C2×C4#1313317911yescharacteristic
364C2×C4#1313319311yescharacteristic
365C2×C4#1313319511yescharacteristic
366C2×C4#1313320211yescharacteristic
367C2×C4D8×D5311yescharacteristic
368C2×C4#1313323611yescharacteristic
369C2×C4#1313324011yescharacteristic
370C2×C4#1313324011yescharacteristic
371C2×C4#1313324111yescharacteristic
372C2×C4#1313324111yescharacteristic
373C2×C4#1313324911yescharacteristic
374C2×C4#1313324911yescharacteristic
375C8—24no—
376C8—4241no—
377C8—2122no—
378C22—4241no—
379C22—4241no—
380C22—4241no—
381C22—4241no—
382C22—2122no—
383C22—2122no—
384C22—2122no—
385C22#1681049411yescharacteristic
386C22#1681075511yescharacteristic
387C22#1681076511yescharacteristic
388C22#1681078411yescharacteristic
389C22#1681078411yescharacteristic
390C22#1681079011yescharacteristic
391C22#1681080511yescharacteristic
392C4—21no—
393C4—21no—
394C4—82no—
395C4—161no—
396C4—4241no—
397C4—2122no—
398C4—2122no—
399C4#1681066011yescharacteristic
400C4#1681066411yescharacteristic
401C4#1681070511yescharacteristic
402C4#1681072211yescharacteristic
403C4#1681090211yescharacteristic
404C4#1681092611yescharacteristic
405C4#1681095011yescharacteristic
406C4#1681096611yescharacteristic
407C2—4242no—
408C2#2027377711yescharacteristic
409C2#2027385811yescharacteristic
410C2#2027394711yescharacteristic
411C2#2027396811yescharacteristic
412C2#2027491111yescharacteristic
413C2#2027494011yescharacteristic
414C2#2027495911yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1845C2
socleC23×C53#120
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1845D53
hypocenterC53#8885
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1845

Complements

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

NH ≅ G/Ntype
2. #20275476C2semidirect
5. #20272598C2semidirect
6. #20272555C2semidirect
7. C53×#1845C2semidirect
21. #16810352C4semidirect
202. #1845D53semidirect
233. C53#8885semidirect

Maximal subgroups

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

subgroupindex
#202776662
#202754762
#202747122
C2×#168103522
#202725982
#202725552
C53×#18452
#888553