Finite Group Databasebeta

← #27282998

Subgroups — #27282998

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719922C211yesmaximal, characteristic
2#20275509C211yesmaximal, characteristic, complemented
3C2×#16810373C211yesmaximal, characteristic, complemented
4C2×D4⋊#88322C211yesmaximal, characteristic, complemented
5C2×#16810350C211yesmaximal, characteristic, complemented
6C2×#16810345C211yesmaximal, characteristic, complemented
7C53×#1954C211yesmaximal, characteristic, complemented
8C22×#11719922C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10C4×#11719922—22no—
11C4×#11719922—22no—
12C2×#13133150C411yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133138C411yescharacteristic
15C2×#13133137C411yescharacteristic
16C2×#13133129—22no—
17C2×C4×#88322C2211yescharacteristic
18C2×C4×#88322C2211yescharacteristic
19#16810373—22no—
20D4⋊#88322—22no—
21#16810350—22no—
22#16810345—22no—
23C2×C4×D4×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C22⋊C8×C53—22no—
26C2×C4×C8×C53C2211yescharacteristic
27C22×D212C2×C411yescharacteristic
28C22×C4×D53—21no—
29C2×#11719922—22no—
30C2×#11719922—22no—
31C2×#11719922—22no—
32C22×#88322—21no—
33C22×#88322—21no—
34C22×#88322C2×C411yescharacteristic
35C22×#88322C2×C411yescharacteristic
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×#11719921—21no—
42#13133150—22no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45#13133138—22no—
46#13133137—22no—
47#13133129—44no—
48C4×#88322—21no—
49C4×#88322D412yes—
50C4×#88322D412yes—
51C4×#88322D412yes—
52C22×D4×C53C2×C411yescharacteristic
53C23×C4×C53—21no—
54C22×C8×C53—21no—
55C22×C8×C53D412yes—
56C4×D4×C53—21no—
57C4×D4×C53—21no—
58C4×D4×C53—21no—
59C4×D4×C53—21no—
60C2×C4⋊C4×C53C2×C411yescharacteristic
61C2×C22⋊C4×C53—21no—
62C2×C42×C53C2311yescharacteristic
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65C22⋊C8×C53—44no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68C24×C53—21no—
69C23×D53—21no—
70C2×D4×C53—21no—
71C2×D4×C53—21no—
72C2×D4×C53—21no—
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53M1612yes—
76C22×C4×C53—21no—
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53#4111yescharacteristic
84C22×Dic53—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212—21no—
90C2×D212M1612yes—
91C2×C4×D53—41no—
92C2×C4×D53—41no—
93C2×C4×D53—42no—
94C2×C4×D53—42no—
95C2×C8×C53—22no—
96C2×C8×C53—22no—
97C2×C8×C53—22no—
98C2×C8×C53—41no—
99C2×C8×C53—42no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C42×C53C2×D411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53#4111yescharacteristic
108#11719922—28no—
109#11719921—41no—
110#11719921—41no—
111C4⋊Dic53—21no—
112C4⋊Dic53—21no—
113C2×#88322—24no—
114C2×#88322—22no—
115C2×#88322—21no—
116C2×#88322—21no—
117C2×#88322—21no—
118C2×#88322—21no—
119C2×#88322—41no—
120C2×#88322—42no—
121C2×#88322C22⋊C412yes—
122C2×#88322C22⋊C412yes—
123C23×C53—22no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53C4×D411yescharacteristic
129C22×D53—22no—
130C22×D53—41no—
131C22×D53—41no—
132C22×D53—42no—
133C22×D53—42no—
134D4×C53—22no—
135D4×C53—22no—
136D4×C53—41no—
137D4×C53—41no—
138C2×C4×C53—22no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—82no—
142C2×C4×C53—81no—
143C2×C4×C53—81no—
144C2×C4×C53C2×C22⋊C411yescharacteristic
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12411yescharacteristic
149C2×C4×C53#12711yescharacteristic
150C2×C4×C53C2×M1611yescharacteristic
151C2×C4×C53C2×M1611yescharacteristic
152C2×C4×C53#13111yescharacteristic
153C2×C4×C53#13111yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)12yes—
155C2×C4×C53AGL(1,ℤ/8)12yes—
156C2×Dic53—41no—
157C2×Dic53—42no—
158D212—22no—
159D212—22no—
160D212—41no—
161D212—41no—
162C4×D53—82no—
163C4×D53—81no—
164C4×D53—81no—
165C8×C53—42no—
166C8×C53—81no—
167#88322—24no—
168#88322—22no—
169#88322—22no—
170#88322—81no—
171#9665—531nomaximal, Sylow
172C22×C53—44no—
173C22×C53—42no—
174C22×C53—82no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53#39011yescharacteristic
178C22×C53#43511yescharacteristic
179C22×C53#43511yescharacteristic
180C22×C53#44212yes—
181C22×C53#44312yes—
182C2×D53—44no—
183C2×D53—42no—
184C2×D53—82no—
185C2×D53—81no—
186C2×D53—81no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—81no—
190C4×C53#40711yescharacteristic
191C4×C53#40811yescharacteristic
192C4×C53#43511yescharacteristic
193C4×C53#43511yescharacteristic
194C4×C53#46912yes—
195C4×C53#49611yescharacteristic
196C4×C53#49611yescharacteristic
197Dic53—81no—
198C2×C4×D8—531no—
199#1954—531no—
200#1954D5311yescharacteristic, complemented
201C2×#334—531no—
202C2×D4⋊C8—531no—
203C2×D4⋊C8—531no—
204C2×#322—531no—
205C2×C53—82no—
206C2×C53#158612yes—
207C2×C53#158612yes—
208C2×C53#195511yescharacteristic
209C2×C53#197211yescharacteristic
210C2×C53#197411yescharacteristic
211D53—82no—
212C22×D8—531no—
213C2×C4×D4—531no—
214C2×C4×D4C2×D5311yescharacteristic
215C4×D8—1062no—
216C4×D8—1062no—
217C2×#107—531no—
218C2×C4⋊C8—531no—
219C2×C4⋊C8C2×D5311yescharacteristic
220C2×#102—531no—
221C2×#102—531no—
222C2×C22⋊C8—22no—
223C2×C22⋊C8—1062no—
224C2×C4×C8—531no—
225C2×C4×C8—531no—
226C2×C4×C8C2×D5311yescharacteristic
227#334—1062no—
228D4⋊C8—1062no—
229D4⋊C8—1062no—
230#322—1062no—
231C53#966511yescharacteristic, complemented, Sylow
232C22×D4—531no—
233C22×D4C4×D5311yescharacteristic
234C23×C4—21no—
235C23×C4—1061no—
236C2×D8—1062no—
237C2×D8—1062no—
238C2×D8—1062no—
239C22×C8—21no—
240C22×C8—531no—
241C22×C8—531no—
242C22×C8—1061no—
243C22×C8—1061no—
244C22×C8C22⋊D5312yes—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—1061no—
250C4×D4—1061no—
251C4×D4—1061no—
252C4×D4—1061no—
253C2×C4⋊C4—531no—
254C2×C4⋊C4C4×D5311yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—1061no—
257C2×C42C22×D5311yescharacteristic
258#107—1062no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—1061no—
262C4⋊C8—1061no—
263#102—1062no—
264#102—1062no—
265C22⋊C8—44no—
266C22⋊C8—2124no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—532no—
270C4×C8—532no—
271C4×C8—532no—
272C4×C8—1061no—
273C24—21no—
274C24—1061no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—532no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4#1171991212yes—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—1061no—
293C22×C4—2121no—
294C22×C4—2121no—
295C22×C4—2122no—
296C22×C4—2122no—
297C22×C4C2×C4×D5311yescharacteristic
298C22×C4#1171994511yescharacteristic
299C22×C4C2×C22⋊D5311yescharacteristic
300D8—1068no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—532no—
307C2×C8—532no—
308C2×C8—1064no—
309C2×C8—1062no—
310C2×C8—1061no—
311C2×C8—1061no—
312C2×C8—1061no—
313C2×C8—1061no—
314C2×C8—2121no—
315C2×C8—2122no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—1061no—
319C4⋊C4—1061no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—2121no—
323C22⋊C4—2121no—
324C42D4×D5311yescharacteristic
325C42D4×D5311yescharacteristic
326C42D4×D5311yescharacteristic
327C42#1171994711yescharacteristic
328C23—22no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—1062no—
334C23—2121no—
335C23—2121no—
336C23—2122no—
337C23—2122no—
338C23C4×C22⋊D5311yescharacteristic
339D4—22no—
340D4—22no—
341D4—41no—
342D4—41no—
343D4—1062no—
344D4—1062no—
345D4—2121no—
346D4—2121no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—82no—
353C2×C4—4241no—
354C2×C4—4241no—
355C2×C4—4242no—
356C2×C4—2121no—
357C2×C4—2122no—
358C2×C4C22⋊C4×D5311yescharacteristic
359C2×C4#1313317711yescharacteristic
360C2×C4#1313318911yescharacteristic
361C2×C4#1313319011yescharacteristic
362C2×C4#1313320311yescharacteristic
363C2×C4#1313321311yescharacteristic
364C2×C4M16×D5311yescharacteristic
365C2×C4C2×#1171991211yescharacteristic
366C2×C4#1313322311yescharacteristic
367C2×C4#1313322511yescharacteristic
368C2×C4#1313324212yes—
369C2×C4#1313324312yes—
370C8—42no—
371C8—81no—
372C8—1064no—
373C8—1062no—
374C8—1062no—
375C8—4241no—
376C22—42no—
377C22—44no—
378C22—81no—
379C22—81no—
380C22—82no—
381C22—4241no—
382C22—4241no—
383C22—4242no—
384C22—2122no—
385C22—2124no—
386C22#1681052311yescharacteristic
387C22#1681069211yescharacteristic
388C22#1681069411yescharacteristic
389C22#1681073312yes—
390C22#1681073812yes—
391C4—21no—
392C4—21no—
393C4—81no—
394C4—4241no—
395C4#1681058511yescharacteristic
396C4#1681058811yescharacteristic
397C4#1681069111yescharacteristic
398C4#1681069711yescharacteristic
399C4#1681087412yes—
400C4#1681101011yescharacteristic
401C4#1681101311yescharacteristic
402C2—82no—
403C2—4242no—
404C2#2027357712yes—
405C2#2027357912yes—
406C2#2027551111yescharacteristic
407C2#2027559011yescharacteristic
408C2#2027560211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1954C2
socleC23×C53C4×D4
prime radicalC22×#11719922C4
cosocleC2×C42×C53C23
hypercenter#1954D53
hypocenterC53#9665
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1954

Complements

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

NH ≅ G/Ntype
2. #20275509C2semidirect
3. C2×#16810373C2semidirect
4. C2×D4⋊#88322C2semidirect
5. C2×#16810350C2semidirect
6. C2×#16810345C2semidirect
7. C53×#1954C2semidirect
200. #1954D53semidirect
231. C53#9665semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199222
#202755092
C2×#168103732
C2×D4⋊#883222
C2×#168103502
C2×#168103452
C53×#19542
#966553