Finite Group Databasebeta

← #27280219

Subgroups — #27280219

Order 13568.

450 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283359C211yesmaximal, characteristic, complemented
2#20273307C211yesmaximal, characteristic, complemented
3#20273166C211yesmaximal, characteristic, complemented
4#20273154C211yesmaximal, characteristic, complemented
5#20272606C211yesmaximal, characteristic, complemented
6#20272551C211yesmaximal, characteristic, complemented
7C53×#3106C211yesmaximal, characteristic
8#16810684—22no—
9#16810676—22no—
10#16810650—21no—
11#16810647C2211yescharacteristic
12#16810590—21no—
13C4×#11719917C2211yescharacteristic
14C4×#11719917C2211yescharacteristic
15C2×C4×#88322C2211yescharacteristic
16#16810363—22no—
17Q8⋊#88322—21no—
18D4⋊#88322—22no—
19D4⋊#88322—21no—
20C53×#532C2211yescharacteristic
21C4×#41×C53C2211yescharacteristic
22C53×#482—21no—
23C53×#477C411yescharacteristic
24C53×#476C411yescharacteristic
25C53×#460C411yescharacteristic
26C53×#459C411yescharacteristic
27C53×#449—21no—
28C4×SD16×C53—21no—
29C4×D8×C53—21no—
30C53×#431C2211yescharacteristic
31C2×#11719917—21no—
32C2×#11719917—21no—
33C2×#11719917—21no—
34C22×#88322—21no—
35C4⋊#88322—24no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38C22⋊#88322—24no—
39#13133128—21no—
40#13133128D412yes—
41C4×#88322—21no—
42C4×#88322—21no—
43C4×#88322—21no—
44C4×#88322D412yes—
45C2×#41×C53—21no—
46C2×#41×C53D411yescharacteristic
47C2×SD16×C53—21no—
48C2×D8×C53—21no—
49C4⋊Q8×C53C2×C411yescharacteristic
50#127×C53C2×C411yescharacteristic
51#124×C53—21no—
52#121×C53—21no—
53#121×C53—21no—
54C4×Q8×C53C2×C411yescharacteristic
55C4×Q8×C53D411yescharacteristic
56C4×D4×C53—21no—
57C4×D4×C53—21no—
58C4×D4×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53C2×C411yescharacteristic
61C4×D4×C53D411yescharacteristic
62#117×C53—21no—
63#117×C53D411yescharacteristic
64C2×C42×C53—21no—
65C2×C42×C53C2311yescharacteristic
66#107×C53—21no—
67#106×C53—21no—
68C4⋊C8×C53C2×C411yescharacteristic
69C4⋊C8×C53C2×C411yescharacteristic
70#103×C53—21no—
71#102×C53—21no—
72#102×C53—21no—
73#102×C53—21no—
74C22⋊C8×C53—21no—
75C4×C8×C53—21no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—81no—
79#41×C53—81no—
80C2×Q8×C53—21no—
81C2×Q8×C53C22⋊C411yescharacteristic
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—41no—
87C2×D4×C53C22⋊C411yescharacteristic
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—41no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C22×C4×C53#4111yescharacteristic
95SD16×C53—42no—
96D8×C53—42no—
97C2×C8×C53—21no—
98C2×C8×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53C22⋊C411yescharacteristic
104C4⋊C4×C53C22⋊C411yescharacteristic
105C22⋊C4×C53—21no—
106C22⋊C4×C53—21no—
107C22⋊C4×C53—21no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—41no—
110C42×C53—21no—
111C42×C53—21no—
112C42×C53—21no—
113C42×C53C22×C411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117#11719917—22no—
118#11719917—41no—
119#11719917—41no—
120#11719917—41no—
121#11719917—81no—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C23×C53—21no—
129C23×C53—41no—
130C23×C53#12311yescharacteristic
131Q8×C53—21no—
132Q8×C53—21no—
133Q8×C53—81no—
134D4×C53—21no—
135D4×C53—21no—
136D4×C53—42no—
137D4×C53—41no—
138D4×C53—41no—
139D4×C53—41no—
140D4×C53—41no—
141D4×C53—81no—
142D4×C53—82no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—42no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53C2×C22⋊C411yescharacteristic
161C2×C4×C53C4×D411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C8×C53—41no—
167C8×C53—41no—
168#88322—42no—
169#88322—42no—
170#88322—42no—
171#88322—81no—
172#9213—531nomaximal, Sylow
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53#38611yescharacteristic
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—81no—
190C4×C53#42111yescharacteristic
191C4×C53#42111yescharacteristic
192C4×C53#44311yescharacteristic
193C4×C53#44411yescharacteristic
194C4×C53#47111yescharacteristic
195C4×C53#47211yescharacteristic
196#3106D5311yescharacteristic
197#2922—531no—
198#1525—531no—
199#1492—531no—
200#1489—531no—
201#1299—531no—
202#1278—531no—
203C2×C53—41no—
204C2×C53—81no—
205C2×C53—161no—
206C2×C53#187611yescharacteristic
207C2×C53#189711yescharacteristic
208C2×C53#190711yescharacteristic
209#532C2×D5311yescharacteristic
210C4×#41C2×D5311yescharacteristic
211#482—21no—
212#477Dic5311yescharacteristic
213#476Dic5311yescharacteristic
214#460Dic5311yescharacteristic
215#459Dic5311yescharacteristic
216#449—21no—
217C4×SD16—21no—
218C4×D8—21no—
219#433—1062no—
220#431—1062no—
221#431C2×D5311yescharacteristic
222#424—1061no—
223#423—531no—
224#408—1061no—
225C4×M16—531no—
226C4×M16—531no—
227C2×C4×C8—531no—
228#329—1062no—
229Q8⋊C8—1061no—
230D4⋊C8—1062no—
231D4⋊C8—1061no—
232C53#921311yescharacteristic, complemented, Sylow
233C2×#41—21no—
234C2×#41C22⋊D5311yescharacteristic
235C2×SD16—21no—
236C2×D8—21no—
237C2×M16—1061no—
238C2×M16—1061no—
239C2×M16—1061no—
240C22×C8—1061no—
241C4⋊Q8C2×Dic5311yescharacteristic
242#127C2×Dic5311yescharacteristic
243#124—21no—
244#121—21no—
245#121—21no—
246C4×Q8C2×Dic5311yescharacteristic
247C4×Q8C22⋊D5311yescharacteristic
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4C2×Dic5311yescharacteristic
253C4×D4C22⋊D5311yescharacteristic
254#117—21no—
255#117C22⋊D5311yescharacteristic
256C2×C42—21no—
257C2×C42C22×D5311yescharacteristic
258#107—21no—
259#106—21no—
260C4⋊C8—1064no—
261C4⋊C8—1061no—
262C4⋊C8—1061no—
263C4⋊C8C2×Dic5311yescharacteristic
264C4⋊C8C2×Dic5311yescharacteristic
265#103—21no—
266#102—21no—
267#102—21no—
268#102—21no—
269C22⋊C8—21no—
270C22⋊C8—1064no—
271#97—532no—
272#97—1061no—
273C4×C8—21no—
274C4×C8—532no—
275C4×C8—1061no—
276C4×C8—1061no—
277C4×C8—1061no—
278#41—41no—
279#41—41no—
280#41—81no—
281#41—81no—
282C2×Q8—21no—
283C2×Q8C22⋊Dic5311yescharacteristic
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4C22⋊Dic5311yescharacteristic
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4#1171994711yescharacteristic
295C22×C4#1171994711yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297SD16—42no—
298D8—42no—
299M16—1062no—
300M16—4241no—
301M16—2121no—
302M16—2121no—
303M16—2121no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—1062no—
307C2×C8—1062no—
308C2×C8—2121no—
309C2×C8—2121no—
310C2×C8—2121no—
311C2×C8—2121no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4C22⋊Dic5311yescharacteristic
317C4⋊C4C22⋊Dic5311yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42D4×D5311yescharacteristic
327C42D4×D5311yescharacteristic
328C42C22×Dic5311yescharacteristic
329C42C2×C22⋊D5311yescharacteristic
330C23—21no—
331C23—41no—
332C23#1313319811yescharacteristic
333Q8—21no—
334Q8—21no—
335Q8—81no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—81no—
344D4—82no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4C2×C22⋊Dic5311yescharacteristic
363C2×C4D4×Dic5311yescharacteristic
364C2×C4D4×Dic5311yescharacteristic
365C2×C4#1313318511yescharacteristic
366C2×C4#1313318911yescharacteristic
367C2×C4#1313318911yescharacteristic
368C8—41no—
369C8—41no—
370C8—4241no—
371C8—2122no—
372C8—2122no—
373C8—2122no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22#1681050311yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—81no—
391C4#1681064011yescharacteristic
392C4#1681064011yescharacteristic
393C4#1681074011yescharacteristic
394C4#1681074511yescharacteristic
395C4#1681088411yescharacteristic
396C4#1681089111yescharacteristic
397C2—41no—
398C2—81no—
399C2—161no—
400C2#2027504411yescharacteristic
401C2#2027516411yescharacteristic
402C2#2027522511yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3106C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3106D53
hypocenterC53#9213
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. #20283359C2semidirect
2. #20273307C2semidirect
3. #20273166C2semidirect
4. #20273154C2semidirect
5. #20272606C2semidirect
6. #20272551C2semidirect
232. C53#9213semidirect

Maximal subgroups

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

subgroupindex
#202833592
#202733072
#202731662
#202731542
#202726062
#202725512
C53×#31062
#921353