Finite Group Databasebeta

← #27285343

Subgroups — #27285343

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811019C211yesmaximal, characteristic, complemented
2C2×#16810900C211yesmaximal, characteristic
3#20279710C211yesmaximal, characteristic
4C2×#16810368C211yesmaximal, characteristic, complemented
5C2×#16810350C211yesmaximal, characteristic
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1981C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×#13133205—21no—
10C2×C4×D212C2211yescharacteristic
11#16811019—22no—
12#16811019—22no—
13#16810900—22no—
14#16810900—22no—
15C2×#13133150—21no—
16C2×#13133148—22no—
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133137—21no—
19C2×C4×#88322C2211yescharacteristic
20#16810550—22no—
21#16810529—21no—
22#16810480—22no—
23#16810368—22no—
24#16810350—22no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×#103×C53—22no—
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×Dic106—21no—
30C22×D212D411yescharacteristic
31C22×C4×D53—21no—
32C22×#88322—21no—
33C22×#88322—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217—21no—
37#13133217—21no—
38#13133205—42no—
39#13133205—42no—
40C4×D212—21no—
41C4×D212—21no—
42C4×D212—21no—
43C4×D212—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53D411yescharacteristic
46C2×C4⋊Dic53D412yes—
47C2×#11719921—24no—
48C2×#11719921—21no—
49#13133150—24no—
50#13133148—28no—
51C4⋊#88322—21no—
52C4⋊#88322D412yes—
53#13133137—24no—
54C4×#88322—21no—
55C4×#88322D412yes—
56#13133124—24no—
57C22×Q8×C53—21no—
58C22×C8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53D412yes—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D412yes—
66C2×C42×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70#103×C53—44no—
71#95×C53—21no—
72C23×D53—21no—
73C2×Q8×C53—22no—
74C2×Q8×C53—41no—
75C2×Q8×C53—42no—
76C2×Q8×C53—42no—
77C22×C4×C53—22no—
78C22×C4×C53—41no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×Dic53—21no—
83C22×Dic53—22no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212SD1612yes—
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×C4×D53—42no—
93C2×C4×D53—42no—
94C2×Dic106—41no—
95C2×Dic106—41no—
96C2×Dic106—41no—
97C2×Dic106—41no—
98C2×Dic106—41no—
99C2×Dic106—41no—
100C2×C8×C53—41no—
101C2×C8×C53—42no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C42×C53—41no—
109C42×C53—41no—
110C42×C53C2×D411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53#4111yescharacteristic
114#11719921—48no—
115#11719921—41no—
116#11719921—41no—
117C4⋊Dic53—22no—
118C4⋊Dic53—21no—
119C4⋊Dic53—21no—
120C4⋊Dic53—22no—
121C4⋊Dic53—41no—
122C4⋊Dic53—41no—
123C2×#88322—24no—
124C2×#88322—22no—
125C2×#88322—22no—
126C2×#88322—41no—
127C2×#88322—41no—
128C23×C53#12011yescharacteristic
129C22×D53—22no—
130C22×D53—41no—
131C22×D53—42no—
132C22×D53—41no—
133C22×D53—42no—
134Q8×C53—44no—
135Q8×C53—81no—
136Q8×C53—81no—
137C2×C4×C53—22no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—82no—
143C2×C4×C53#12011yescharacteristic
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12711yescharacteristic
149C2×C4×C53C2×SD1611yescharacteristic
150C2×C4×C53#13512yes—
151C2×C4×C53#13511yescharacteristic
152C2×C4×C53AGL(1,ℤ/8)12yes—
153C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C53#13711yescharacteristic
155C2×Dic53—42no—
156C2×Dic53—41no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160D212—22no—
161D212—22no—
162D212—41no—
163D212—41no—
164C4×D53—82no—
165C4×D53—81no—
166C4×D53—81no—
167Dic106—81no—
168Dic106—81no—
169Dic106—81no—
170Dic106—81no—
171C8×C53—81no—
172#88322—24no—
173#88322—42no—
174#10074—531nomaximal, Sylow
175C22×C53#39211yescharacteristic
176C22×C53#44912yes—
177C22×C53#45011yescharacteristic
178C22×C53#45211yescharacteristic
179C22×C53#45612yes—
180C2×D53—44no—
181C2×D53—42no—
182C2×D53—82no—
183C2×D53—81no—
184C2×D53—81no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—82no—
188C4×C53—82no—
189C4×C53#44811yescharacteristic
190C4×C53#44911yescharacteristic
191C4×C53#46312yes—
192C4×C53#46511yescharacteristic
193C4×C53#46811yescharacteristic
194C4×C53#49511yescharacteristic
195C4×C53#49611yescharacteristic
196Dic53—82no—
197Dic53—81no—
198C2×#498—531no—
199C2×#474—531no—
200#2547—531no—
201#1981D5311yescharacteristic, complemented
202C2×#332—531no—
203C2×D4⋊C8—531no—
204#1279—531no—
205C2×C53#165212yes—
206C2×C53#165412yes—
207C2×C53#200011yescharacteristic
208C2×C53#200511yescharacteristic
209C2×C53#201211yescharacteristic
210D53—82no—
211C2×C4⋊Q8—531no—
212C2×C4⋊Q8C2×D5311yescharacteristic
213C2×#124—1061no—
214C2×C4×D4—531no—
215#498—1062no—
216#498—1062no—
217#474—1062no—
218#474—1062no—
219C2×#107—1061no—
220C2×#106—1062no—
221C2×C4⋊C8—531no—
222C2×C4⋊C8C2×D5311yescharacteristic
223C2×#103—22no—
224C2×#102—1061no—
225C2×C4×C8—531no—
226#397—1062no—
227#391—1061no—
228#380—1062no—
229C4×C4⋊C4C2×D5311yescharacteristic
230#332—1062no—
231D4⋊C8—1062no—
232C53#1007411yescharacteristic, complemented, Sylow
233C22×Q8—21no—
234C22×Q8—1061no—
235C22×D4—531no—
236C23×C4—1061no—
237C22×C8—21no—
238C22×C8—1061no—
239C22×C8—1061no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—1061no—
244C4⋊Q8—1061no—
245C4⋊Q8—1061no—
246C4⋊Q8—1061no—
247C4⋊Q8D21212yes—
248#124—2122no—
249#124—2122no—
250C4×D4—1061no—
251C4×D4—1061no—
252C4×D4—1061no—
253C4×D4—1061no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—531no—
257C2×C4⋊C4—532no—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4C22⋊D5312yes—
260C2×C22⋊C4—1064no—
261C2×C22⋊C4—1061no—
262C2×C42—21no—
263C2×C42C22×D5311yescharacteristic
264#107—1064no—
265#106—1068no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—532no—
269C4⋊C8—1061no—
270#103—44no—
271#102—1064no—
272C4×C8—532no—
273C4×C8—1061no—
274#95—21no—
275#95—1064no—
276C24—1061no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—2121no—
282C2×Q8—2121no—
283C2×Q8—2121no—
284C2×Q8—2121no—
285C2×Q8—2121no—
286C2×Q8—2121no—
287C2×D4—532no—
288C2×D4—1061no—
289C2×D4—1061no—
290C2×D4—1061no—
291C2×D4—1061no—
292C2×D4—1061no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—1061no—
296C22×C4—1062no—
297C22×C4—2121no—
298C22×C4—2121no—
299C22×C4—2122no—
300C22×C4—2122no—
301C22×C4D4×D5311yescharacteristic
302C22×C4D4×D5311yescharacteristic
303C22×C4C2×C22⋊D5311yescharacteristic
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—1064no—
307C2×C8—1062no—
308C2×C8—1062no—
309C2×C8—2121no—
310C2×C8—2121no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—1062no—
318C4⋊C4—1061no—
319C4⋊C4—1061no—
320C4⋊C4—1062no—
321C4⋊C4—2121no—
322C4⋊C4—2121no—
323C22⋊C4—2128no—
324C22⋊C4—2121no—
325C22⋊C4—2121no—
326C42—41no—
327C42—41no—
328C42C2×D21211yescharacteristic
329C42#1171994511yescharacteristic
330C42D4×D5311yescharacteristic
331C42D4×D5311yescharacteristic
332C23—1062no—
333C23—2122no—
334C23—2121no—
335C23—2121no—
336C23—2122no—
337C23#1313318511yescharacteristic
338Q8—44no—
339Q8—81no—
340Q8—81no—
341Q8—4241no—
342Q8—4241no—
343Q8—4241no—
344Q8—4241no—
345D4—1062no—
346D4—1062no—
347D4—2121no—
348D4—2121no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4—4241no—
356C2×C4—4242no—
357C2×C4—4241no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4—2122no—
361C2×C4—2121no—
362C2×C4—2122no—
363C2×C4#1313318711yescharacteristic
364C2×C4#1313318711yescharacteristic
365C2×C4#1313319011yescharacteristic
366C2×C4#1313319011yescharacteristic
367C2×C4#1313319111yescharacteristic
368C2×C4#1313321311yescharacteristic
369C2×C4SD16×D5311yescharacteristic
370C2×C4#1313323712yes—
371C2×C4#1313324011yescharacteristic
372C2×C4#1313324312yes—
373C2×C4#1313324411yescharacteristic
374C2×C4#1313324911yescharacteristic
375C8—81no—
376C8—1064no—
377C8—2122no—
378C22—4241no—
379C22—4242no—
380C22—4241no—
381C22—2122no—
382C22—2124no—
383C22#1681053211yescharacteristic
384C22#1681076612yes—
385C22#1681077211yescharacteristic
386C22#1681078411yescharacteristic
387C22#1681080412yes—
388C4—21no—
389C4—21no—
390C4—82no—
391C4—82no—
392C4—4242no—
393C4—4241no—
394C4#1681076311yescharacteristic
395C4#1681077111yescharacteristic
396C4#1681083812yes—
397C4#1681085511yescharacteristic
398C4#1681087211yescharacteristic
399C4#1681100511yescharacteristic
400C4#1681101011yescharacteristic
401C2—4242no—
402C2#2027398312yes—
403C2#2027399712yes—
404C2#2027571711yescharacteristic
405C2#2027575211yescharacteristic
406C2#2027579111yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1981C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenter#1981D53
hypocenterC53#10074
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
1. C2×#16811019C2semidirect
4. C2×#16810368C2semidirect
6. #20272556C2semidirect
7. C53×#1981C2semidirect
201. #1981D53semidirect
232. C53#10074semidirect

Maximal subgroups

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

subgroupindex
C2×#168110192
C2×#168109002
#202797102
C2×#168103682
C2×#168103502
#202725562
C53×#19812
#1007453