Finite Group Databasebeta

← #27302334

Subgroups — #27302334

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285139C211yesmaximal, characteristic
2C2×#16810431C211yesmaximal, characteristic, complemented
3C2×#16810430C211yesmaximal, characteristic, complemented
4C2×#16810429C211yesmaximal, characteristic, complemented
5C2×#16810398C211yesmaximal, characteristic, complemented
6#20274885C211yesmaximal, characteristic, complemented
7C2×C53×#359C211yesmaximal, characteristic, complemented
8C2×#13133239C2211yescharacteristic
9C2×#13133237—21no—
10#16811169—21no—
11C22×#11719908C2211yescharacteristic
12#16811008C411yescharacteristic
13#16811006C411yescharacteristic
14#16811004—21no—
15#16811001C411yescharacteristic
16#16810998C411yescharacteristic
17#16810994—21no—
18C2×#13133153C2211yescharacteristic
19C2×#13133152—21no—
20#16810646—21no—
21C2×C8×Dic53C2211yescharacteristic
22#16810431—22no—
23#16810430—22no—
24#16810429—22no—
25#16810398—21no—
26#16810398—21no—
27C2×#135×C53C2211yescharacteristic
28C22×C16×C53C2211yescharacteristic
29C2×#108×C53C2211yescharacteristic
30C53×#359—22no—
31C2×#11719945—21no—
32#13133239—21no—
33#13133239—21no—
34#13133239—21no—
35#13133239—21no—
36#13133237—42no—
37#13133237—42no—
38C2×Dic212C2×C411yescharacteristic
39C2×#11719925—21no—
40C2×#11719913—21no—
41C2×#11719923—21no—
42C2×#11719924—21no—
43C2×D424C2×C411yescharacteristic
44C2×#11719922—21no—
45C2×#11719917—21no—
46C22×#88322D411yescharacteristic
47#13133215—21no—
48#13133213—21no—
49#13133206—21no—
50#13133204—21no—
51#13133189—42no—
52C4×C22⋊D53—42no—
53C2×C4×Dic53D411yescharacteristic
54C2×#11719908—21no—
55C2×#11719908—21no—
56C2×#11719908D412yescomplemented
57#13133153—21no—
58#13133153—21no—
59#13133152—42no—
60#13133146—42no—
61C8×Dic53C2×C411yescharacteristic
62C8×Dic53C2×C411yescharacteristic
63C8×Dic53D411yescharacteristic
64C8×Dic53D411yescharacteristic
65C2×#41×C53—21no—
66#135×C53—21no—
67#135×C53—21no—
68#135×C53—21no—
69#135×C53—21no—
70C2×Q16×C53C2×C411yescharacteristic
71C2×SD16×C53—21no—
72C2×D8×C53C2×C411yescharacteristic
73C2×M16×C53—21no—
74C22×C8×C53C2311yescharacteristic
75C2×C16×C53—21no—
76C2×C16×C53—21no—
77C2×C16×C53D412yescomplemented
78#108×C53—21no—
79#108×C53—21no—
80#41×C53—41no—
81#41×C53—41no—
82#41×C53—41no—
83#41×C53—41no—
84C2×Q8×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53C2×D411yescharacteristic
89C2×C22⋊D53—41no—
90C2×C22⋊D53—42no—
91C22×Dic53SD1612yes—
92#11719945—41no—
93#11719945—41no—
94#11719945—41no—
95#11719945—41no—
96C2×D212—21no—
97C2×C4×D53—41no—
98C2×Dic106—21no—
99Q16×C53—21no—
100Q16×C53—21no—
101SD16×C53—41no—
102SD16×C53—41no—
103D8×C53—21no—
104D8×C53—21no—
105M16×C53—41no—
106M16×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53C22×C411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53#4111yescharacteristic
113C2×C8×C53#4111yescharacteristic
114C22⋊Dic53—42no—
115#11719925—42no—
116#11719924—42no—
117#11719923—42no—
118#11719922—42no—
119#11719921—42no—
120#11719919—42no—
121C4×Dic53C22⋊C411yescharacteristic
122C4×Dic53C22⋊C411yescharacteristic
123C4×Dic53D812yes—
124#11719917—41no—
125#11719917—41no—
126C2×#88322—41no—
127C2×#88322C22⋊C411yescharacteristic
128C2×#88322C22⋊C411yescharacteristic
129C2×#88322D812yes—
130C2×#88322SD1612yes—
131Dic212—21no—
132Dic212—21no—
133D424—21no—
134D424—21no—
135#11719913—41no—
136#11719913—41no—
137C16×C53—22no—
138#11719908—22no—
139C23×C53—41no—
140C23×C53C2×SD1611yescharacteristic
141C22×D53—41no—
142Q8×C53—41no—
143Q8×C53—41no—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—81no—
147D4×C53—81no—
148C2×C4×C53—41no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53C2×C22⋊C411yescharacteristic
152C2×C4×C53#12411yescharacteristic
153C2×C4×C53#12711yescharacteristic
154C2×C4×C53C2×D811yescharacteristic
155C2×C4×C53C2×D811yescharacteristic
156C2×C4×C53C2×SD1611yescharacteristic
157C22⋊D53—84no—
158C22⋊D53—81no—
159C22⋊D53—81no—
160C2×Dic53—42no—
161C2×Dic53—41no—
162C2×Dic53#10212yes—
163C2×Dic53#10212yes—
164D212—41no—
165D212—41no—
166C4×D53—81no—
167C4×D53—81no—
168Dic106—41no—
169Dic106—41no—
170C8×C53—81no—
171C8×C53C4×D411yescharacteristic
172C8×C53C4×D411yescharacteristic
173C8×C53#12111yescharacteristic
174C8×C53#12111yescharacteristic
175#88322—81no—
176#88322#10212yes—
177#88322#10212yes—
178#13209—531nomaximal, Sylow
179C22×C53—22no—
180C22×C53—82no—
181C22×C53—81no—
182C22×C53C2×#10211yescharacteristic
183C22×C53#48611yescharacteristic
184C22×C53#49211yescharacteristic
185C2×D53—82no—
186C2×D53—81no—
187C4×C53—81no—
188C4×C53#39011yescharacteristic
189C4×C53C2×#10211yescharacteristic
190C4×C53#48611yescharacteristic
191C4×C53#49311yescharacteristic
192Dic53—22no—
193Dic53—22no—
194Dic53—81no—
195#3147—531no—
196C2×#359—531no—
197C2×#359—531no—
198C2×#359—531no—
199C2×#359D5311yescharacteristic, complemented
200C2×#346—531no—
201#1848—531no—
202C2×C53—21no—
203C2×C53—21no—
204C2×C53—161no—
205C2×C53#195711yescharacteristic
206C2×C53#214312yes—
207D53—161no—
208C2×#135—531no—
209C2×#135—1061no—
210C2×#135C2×D5311yescharacteristic
211#532—1061no—
212C22×C16—531no—
213C22×C16C2×D5311yescharacteristic
214#495—531no—
215#495—531no—
216#495—1061no—
217#494—531no—
218#493—531no—
219#492—1061no—
220C2×#108—531no—
221C2×#108—1061no—
222C2×#108C2×D5311yescharacteristic
223#423—1061no—
224C2×C4×C8—531no—
225#359—22no—
226#359—1062no—
227#359—1062no—
228#359—1062no—
229#346—1061no—
230#346—1061no—
231C53#1320911yescharacteristic, complemented, Sylow
232C2×#41—21no—
233C2×#41—1061no—
234#135—21no—
235#135—21no—
236#135—21no—
237#135—21no—
238#135—1061no—
239#135—1061no—
240#135—1061no—
241#135—1061no—
242#135—2122no—
243#135—2122no—
244C2×Q16—531no—
245C2×Q16—1061no—
246C2×Q16C4×D5311yescharacteristic
247C2×SD16—21no—
248C2×SD16—1061no—
249C2×SD16—1061no—
250C2×SD16—1061no—
251C2×D8—531no—
252C2×D8—1061no—
253C2×D8C4×D5311yescharacteristic
254C2×M16—21no—
255C2×M16—1061no—
256C22×C8—531no—
257C22×C8C22×D5311yescharacteristic
258C4⋊Q8—1061no—
259#127—1061no—
260#124—1061no—
261#124—1061no—
262#121—2122no—
263C4×D4—2122no—
264C2×C42—531no—
265C2×C16—21no—
266C2×C16—21no—
267C2×C16—532no—
268C2×C16—1061no—
269C2×C16—1061no—
270C2×C16C22⋊D5312yescomplemented
271#108—21no—
272#108—21no—
273#108—1061no—
274#108—1061no—
275#108—2122no—
276C4⋊C8—2122no—
277C4×C8—531no—
278C4×C8—531no—
279C4×C8—531no—
280C4×C8—531no—
281#41—41no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—2121no—
286#41—2121no—
287#41—2121no—
288#41—2121no—
289C2×Q8—21no—
290C2×Q8—1061no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—1061no—
294C2×D4—2122no—
295C2×D4—2121no—
296C22×C4—41no—
297C22×C4—532no—
298C22×C4—2121no—
299C22×C4D4×D5311yescharacteristic
300Q16—21no—
301Q16—21no—
302Q16—1061no—
303Q16—1061no—
304Q16—2122no—
305SD16—41no—
306SD16—41no—
307SD16—2122no—
308SD16—2122no—
309SD16—2121no—
310SD16—2121no—
311D8—21no—
312D8—21no—
313D8—1061no—
314D8—1061no—
315D8—2122no—
316M16—41no—
317M16—41no—
318M16—2121no—
319M16—2121no—
320C2×C8—41no—
321C2×C8—531no—
322C2×C8—531no—
323C2×C8—532no—
324C2×C8—532no—
325C2×C8—2121no—
326C2×C8C2×C4×D5311yescharacteristic
327C2×C8#1171994511yescharacteristic
328C2×C8D4×D5311yescharacteristic
329C2×C8D4×D5311yescharacteristic
330C2×C8#1171994711yescharacteristic
331C2×C8C2×C22⋊D5311yescharacteristic
332C4⋊C4—2122no—
333C22⋊C4—2122no—
334C22⋊C4—2122no—
335C42—531no—
336C42—531no—
337C42—532no—
338C16—22no—
339C16—1062no—
340C23—41no—
341C23—2121no—
342C23SD16×D5311yescharacteristic
343Q8—41no—
344Q8—41no—
345Q8—2121no—
346Q8—2121no—
347D4—41no—
348D4—41no—
349D4—81no—
350D4—81no—
351D4—4244no—
352D4—4241no—
353D4—4241no—
354D4—2121no—
355D4—2121no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—532no—
360C2×C4—532no—
361C2×C4—4241no—
362C2×C4—4241no—
363C2×C4—2122no—
364C2×C4—2121no—
365C2×C4C22⋊C4×D5311yescharacteristic
366C2×C4#1313320311yescharacteristic
367C2×C4#1313321311yescharacteristic
368C2×C4D8×D5311yescharacteristic
369C2×C4D8×D5311yescharacteristic
370C2×C4SD16×D5311yescharacteristic
371C8—81no—
372C8—532no—
373C8—532no—
374C8—4241no—
375C8#1313317711yescharacteristic
376C8C4×C22⋊D5311yescharacteristic
377C8#1313318911yescharacteristic
378C8#1313319011yescharacteristic
379C22—22no—
380C22—81no—
381C22—82no—
382C22—4241no—
383C22—4242no—
384C22#102×D5311yescharacteristic
385C22#1681096811yescharacteristic
386C22#1681099411yescharacteristic
387C4—81no—
388C4—1062no—
389C4—1062no—
390C4—4241no—
391C4#1681052311yescharacteristic
392C4#102×D5311yescharacteristic
393C4#1681096811yescharacteristic
394C4#1681099811yescharacteristic
395C2—21no—
396C2—21no—
397C2—161no—
398C2—8481no—
399C2#2027551911yescharacteristic
400C2#2027644112yes—
subgroupG/H
centerC2×C4D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#359C2
socleC22×C53C2×#102
prime radical#20285139C2
cosocleC22×C8×C53C23
hypercenterC2×#359D53
hypocenterC53#13209
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#359

Complements

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

NH ≅ G/Ntype
2. C2×#16810431C2semidirect
3. C2×#16810430C2semidirect
4. C2×#16810429C2semidirect
5. C2×#16810398C2semidirect
6. #20274885C2semidirect
7. C2×C53×#359C2semidirect
56. C2×#11719908D4semidirect
77. C2×C16×C53D4semidirect
199. C2×#359D53semidirect
231. C53#13209semidirect
270. C2×C16C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202851392
C2×#168104312
C2×#168104302
C2×#168104292
C2×#168103982
#202748852
C2×C53×#3592
#1320953