Finite Group Databasebeta

← #27279322

Subgroups — #27279322

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283664C211yesmaximal, characteristic, complemented
2#20275635C211yesmaximal, characteristic
3#20273214C211yesmaximal, characteristic, complemented
4#20273114C211yesmaximal, characteristic
5#20272555C211yesmaximal, characteristic, complemented
6#20272552C211yesmaximal, characteristic, complemented
7C53×#3167C211yesmaximal, characteristic
8#16810754—21no—
9#16810754—22no—
10Q8×#88322—21no—
11#16810676C2211yescharacteristic
12#16810650—21no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133144—21no—
15C2×#13133144—21no—
16#16810579—21no—
17C4×#11719917C2211yescharacteristic
18C4×C4⋊Dic53C2211yescharacteristic
19#16810360—22no—
20D4⋊#88322—21no—
21D4⋊#88322—21no—
22C53×#533C2211yescharacteristic
23C53×#532C2211yescharacteristic
24C53×#498C411yescharacteristic, complemented
25C53×#492C411yescharacteristic, complemented
26C53×#487C411yescharacteristic, complemented
27C53×#486C411yescharacteristic, complemented
28C53×#465—22no—
29C4×SD16×C53—22no—
30C2×C4×C8×C53C2211yescharacteristic
31C2×#11719917D411yescharacteristic
32C2×#11719917D411yescharacteristic
33C22×#88322—21no—
34C2×C4⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C4⋊#88322—22no—
37C4⋊#88322—22no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322D411yescharacteristic
41C4⋊#88322D411yescharacteristic
42#13133144—42no—
43#13133144—42no—
44C22⋊#88322—21no—
45#13133128—22no—
46#13133128—21no—
47C4×#88322—22no—
48C4×#88322—21no—
49C4×#88322—21no—
50#13133124—21no—
51C2×#41×C53—21no—
52C2×SD16×C53—22no—
53C22×C8×C53—21no—
54C4⋊Q8×C53C2×C411yescharacteristic
55C4⋊Q8×C53C2×C411yescharacteristic
56C4⋊Q8×C53C2×C411yescharacteristic
57#127×C53C2×C411yescharacteristic
58#125×C53—21no—
59#124×C53—21no—
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53D412yes—
62#121×C53—21no—
63#121×C53—21no—
64C4×Q8×C53—21no—
65C4×Q8×C53D412yes—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68#117×C53—21no—
69C2×C42×C53C2311yescharacteristic
70#106×C53—22no—
71#103×C53—22no—
72#102×C53—22no—
73C4×C8×C53—21no—
74C4×C8×C53C2×C411yescharacteristic
75C4×C8×C53C2×C411yescharacteristic
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—41no—
79#41×C53—41no—
80C2×Q8×C53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53C22⋊C412yes—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53#4111yescharacteristic
91C22×Dic53—41no—
92SD16×C53—44no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53—42no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53C22⋊C412yes—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C42×C53—22no—
108C42×C53C22×C411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53#4111yescharacteristic
112C4⋊Dic53—22no—
113C4⋊Dic53—41no—
114C4×Dic53—41no—
115C4×Dic53—41no—
116#11719917—21no—
117#11719917—21no—
118#11719917—21no—
119#11719917SD1612yescomplemented
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—41no—
127C23×C53—41no—
128C23×C53#12111yescharacteristic
129Q8×C53—42no—
130Q8×C53—41no—
131Q8×C53—41no—
132Q8×C53—41no—
133Q8×C53—41no—
134D4×C53—42no—
135D4×C53—41no—
136D4×C53—41no—
137D4×C53—41no—
138D4×C53—41no—
139D4×C53—81no—
140D4×C53—81no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—81no—
148C2×C4×C53—81no—
149C2×C4×C53C2×C22⋊C411yescharacteristic
150C2×C4×C53C4×D411yescharacteristic
151C2×C4×C53C4×D411yescharacteristic
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12311yescharacteristic
155C2×C4×C53C2×SD1611yescharacteristic
156C2×C4×C53#13511yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53#13711yescharacteristic
159C2×Dic53—42no—
160C2×Dic53—81no—
161C2×Dic53—81no—
162C8×C53—22no—
163C8×C53—41no—
164#88322—22no—
165#88322—41no—
166#88322—41no—
167#88322—41no—
168#88322—81no—
169#9054—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53#38611yescharacteristic
176C22×C53#46011yescharacteristic
177C22×C53#46411yescharacteristic
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—42no—
182C4×C53—42no—
183C4×C53—81no—
184C4×C53C4×SD1611yescharacteristic
185C4×C53#44011yescharacteristic
186C4×C53#44311yescharacteristic
187C4×C53#44411yescharacteristic
188C4×C53#44811yescharacteristic
189C4×C53#44911yescharacteristic
190Dic53—82no—
191#3167D5311yescharacteristic
192#2962—531no—
193#1980—531no—
194#1503—531no—
195#1480—531no—
196#1279—531no—
197#1278—531no—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53—161no—
201C2×C53#187611yescharacteristic
202C2×C53#188011yescharacteristic
203C2×C53#188411yescharacteristic
204#533C2×D5311yescharacteristic
205#532C2×D5311yescharacteristic
206#498Dic5311yescharacteristic, complemented
207#492Dic5311yescharacteristic, complemented
208#487Dic5311yescharacteristic, complemented
209#486Dic5311yescharacteristic, complemented
210#465—22no—
211#446—1061no—
212#446—1062no—
213C8×Q8—1061no—
214C4×SD16—22no—
215#431—531no—
216#424—1061no—
217C2×C4⋊C8—531no—
218C2×#104—1061no—
219C2×#104—1061no—
220#405—1061no—
221C4×M16—531no—
222C2×C4×C8C2×D5311yescharacteristic
223C4×C4⋊C4—531no—
224#328—1062no—
225D4⋊C8—1061no—
226D4⋊C8—1061no—
227C53#905411yescharacteristic, complemented, Sylow
228C2×#41—21no—
229C2×SD16—22no—
230C2×M16—531no—
231C2×M16—531no—
232C22×C8—21no—
233C22×C8—1061no—
234C4⋊Q8C2×Dic5311yescharacteristic
235C4⋊Q8C2×Dic5311yescharacteristic
236C4⋊Q8C2×Dic5311yescharacteristic
237#127C2×Dic5311yescharacteristic
238#125—21no—
239#124—21no—
240C22⋊Q8—21no—
241C22⋊Q8C22⋊D5312yes—
242#121—21no—
243#121—21no—
244C4×Q8—21no—
245C4×Q8C22⋊D5312yes—
246C4×D4—21no—
247C4×D4—21no—
248#117—21no—
249C2×C4⋊C4—1061no—
250C2×C42—1061no—
251C2×C42C22×D5311yescharacteristic
252#106—22no—
253C4⋊C8—531no—
254C4⋊C8—531no—
255C4⋊C8—1062no—
256C4⋊C8—1062no—
257C4⋊C8—1061no—
258C4⋊C8—1061no—
259#104—2122no—
260#104—2122no—
261#103—22no—
262#102—22no—
263C22⋊C8—1061no—
264#97—1062no—
265#97—1061no—
266C4×C8—21no—
267C4×C8—1062no—
268C4×C8—1061no—
269C4×C8—1061no—
270C4×C8C2×Dic5311yescharacteristic
271C4×C8C2×Dic5311yescharacteristic
272#95—1061no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8C22⋊Dic5312yes—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—41no—
285C22×C4—2121no—
286C22×C4D4×D5311yescharacteristic
287C22×C4#1171994711yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289SD16—44no—
290M16—532no—
291M16—1061no—
292M16—1061no—
293M16—1061no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—42no—
297C2×C8—1061no—
298C2×C8—1061no—
299C2×C8—1061no—
300C2×C8—1061no—
301C2×C8—2121no—
302C2×C8—2121no—
303C2×C8—2121no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—1062no—
312C4⋊C4—2121no—
313C4⋊C4C22⋊Dic5312yes—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—22no—
318C42—2121no—
319C42—2121no—
320C42D4×D5311yescharacteristic
321C42#1171994711yescharacteristic
322C42C22×Dic5311yescharacteristic
323C42C2×C22⋊D5311yescharacteristic
324C23—41no—
325C23#1313318911yescharacteristic
326Q8—42no—
327Q8—41no—
328Q8—41no—
329Q8—41no—
330Q8—41no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—4241no—
347C2×C4—4241no—
348C2×C4—2122no—
349C2×C4C2×C22⋊Dic5311yescharacteristic
350C2×C4D4×Dic5311yescharacteristic
351C2×C4D4×Dic5311yescharacteristic
352C2×C4#1313318511yescharacteristic
353C2×C4#1313318911yescharacteristic
354C2×C4#1313319811yescharacteristic
355C2×C4SD16×D5311yescharacteristic
356C2×C4#1313323811yescharacteristic
357C2×C4#1313324411yescharacteristic
358C2×C4#1313324611yescharacteristic
359C8—22no—
360C8—41no—
361C8—1062no—
362C8—4241no—
363C8—2121no—
364C8—2121no—
365C8—2121no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#1681050311yescharacteristic
372C22#1681082111yescharacteristic
373C22#1681084311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—81no—
380C4—4242no—
381C4SD16×Dic5311yescharacteristic
382C4#1681072411yescharacteristic
383C4#1681074011yescharacteristic
384C4#1681074511yescharacteristic
385C4#1681076011yescharacteristic
386C4#1681076711yescharacteristic
387C2—21no—
388C2—21no—
389C2—161no—
390C2#2027504411yescharacteristic
391C2#2027507211yescharacteristic
392C2#2027509911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3167C2
socleC22×C53#386
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#3167D53
hypocenterC53#9054
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #20283664C2semidirect
3. #20273214C2semidirect
5. #20272555C2semidirect
6. #20272552C2semidirect
24. C53×#498C4semidirect
25. C53×#492C4semidirect
26. C53×#487C4semidirect
27. C53×#486C4semidirect
119. #11719917SD16semidirect
206. #498Dic53semidirect
207. #492Dic53semidirect
208. #487Dic53semidirect
209. #486Dic53semidirect
227. C53#9054semidirect

Maximal subgroups

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

subgroupindex
#202836642
#202756352
#202732142
#202731142
#202725552
#202725522
C53×#31672
#905453