Finite Group Databasebeta

← #27280583

Subgroups — #27280583

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283711C211yesmaximal, characteristic, complemented
2#20273335C211yesmaximal, characteristic, complemented
3C2×#16810363C211yesmaximal, characteristic, complemented
4#20273067C211yesmaximal, characteristic, complemented
5#20272615C211yesmaximal, characteristic, complemented
6#20272559C211yesmaximal, characteristic, complemented
7C53×#3211C211yesmaximal, characteristic
8#16810754—21no—
9#16810695—21no—
10#16810695—21no—
11D4×#88322—21no—
12#16810684—21no—
13#16810680C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810579—21no—
16C2×#13133128C2211yescharacteristic
17C2×#13133128C2211yescharacteristic
18#16810365—21no—
19#16810363—21no—
20#16810363—21no—
21#16810363—21no—
22#16810347—21no—
23#16810347—21no—
24C53×#529C2211yescharacteristic
25C2×#124×C53C2211yescharacteristic
26C53×#495—21no—
27C53×#488—21no—
28C53×#464—21no—
29C53×#463—21no—
30C53×#451C411yescharacteristic
31C53×#450C411yescharacteristic
32C53×#448C411yescharacteristic
33C53×#447C411yescharacteristic
34C53×#431C2211yescharacteristic
35C2×#11719917—21no—
36C22×#88322—21no—
37C22×#88322—21no—
38C22×#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46C22⋊#88322D411yescharacteristic
47C22⋊#88322D411yescharacteristic
48#13133128—21no—
49#13133128—21no—
50#13133128—21no—
51#13133128—21no—
52#13133128—21no—
53#13133128D411yescharacteristic
54#13133128D411yescharacteristic
55C4×#88322—21no—
56C4×#88322—21no—
57C22×Q8×C53C2×C411yescharacteristic
58C22×D4×C53C2×C411yescharacteristic
59C2×Q16×C53—21no—
60C2×SD16×C53—21no—
61C2×SD16×C53—21no—
62C2×D8×C53—21no—
63#126×C53—21no—
64#125×C53D411yescharacteristic
65#124×C53—21no—
66#124×C53—21no—
67#124×C53—21no—
68#124×C53—21no—
69#124×C53D411yescharacteristic
70#123×C53—21no—
71C22⋊Q8×C53C2×C411yescharacteristic
72#121×C53C2×C411yescharacteristic
73C4×Q8×C53D411yescharacteristic
74C4×D4×C53—21no—
75C4×D4×C53D411yescharacteristic
76#117×C53—21no—
77C2×C22⋊C4×C53—21no—
78C2×C22⋊C4×C53—21no—
79C2×C42×C53C2311yescharacteristic
80C4⋊C8×C53—21no—
81#103×C53—21no—
82#103×C53—21no—
83#102×C53—21no—
84#102×C53—21no—
85C22⋊C8×C53C2×C411yescharacteristic
86C22⋊C8×C53C2×C411yescharacteristic
87C4×C8×C53—21no—
88C24×C53—21no—
89C2×Q8×C53—21no—
90C2×Q8×C53—21no—
91C2×Q8×C53—81no—
92C2×Q8×C53C22⋊C411yescharacteristic
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53—81no—
97C2×D4×C53C22⋊C411yescharacteristic
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53C22×C411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53#4111yescharacteristic
103Q16×C53—81no—
104SD16×C53—81no—
105SD16×C53—81no—
106D8×C53—81no—
107C2×C8×C53—21no—
108C2×C8×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53C22⋊C411yescharacteristic
114C4⋊C4×C53C22⋊C411yescharacteristic
115C22⋊C4×C53—21no—
116C22⋊C4×C53—21no—
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—21no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C22⋊C4×C53—41no—
124C42×C53—21no—
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53C2×D411yescharacteristic
128C42×C53#4111yescharacteristic
129#11719917—42no—
130C2×#88322—21no—
131C2×#88322—22no—
132C2×#88322—21no—
133C2×#88322—21no—
134C2×#88322—21no—
135C2×#88322—42no—
136C2×#88322—42no—
137C2×#88322—41no—
138C2×#88322—41no—
139C2×#88322—41no—
140C23×C53—21no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53—81no—
145C23×C53—81no—
146C23×C53C4×D411yescharacteristic
147Q8×C53—41no—
148Q8×C53—41no—
149Q8×C53—41no—
150Q8×C53—81no—
151D4×C53—41no—
152D4×C53—41no—
153D4×C53—41no—
154D4×C53—41no—
155D4×C53—41no—
156D4×C53—81no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—81no—
168C2×C4×C53C2×C22⋊C411yescharacteristic
169C2×C4×C53C4×D411yescharacteristic
170C2×C4×C53#12011yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12311yescharacteristic
174C8×C53—41no—
175C8×C53—41no—
176#88322—22no—
177#88322—42no—
178#88322—41no—
179#88322—41no—
180#88322—41no—
181#88322—81no—
182#9265—531nomaximal, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—81no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53—161no—
192C22×C53—161no—
193C22×C53#38611yescharacteristic
194C22×C53#44311yescharacteristic
195C22×C53#44411yescharacteristic
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—41no—
203C4×C53—41no—
204C4×C53—41no—
205C4×C53—81no—
206#3211D5311yescharacteristic
207#2968—531no—
208#1532—531no—
209C2×#329—531no—
210#1469—531no—
211#1302—531no—
212#1281—531no—
213C2×C53—21no—
214C2×C53—21no—
215C2×C53—81no—
216C2×C53—161no—
217C2×C53#187611yescharacteristic
218C2×C53#190611yescharacteristic
219C2×C53#191411yescharacteristic
220#529C2×D5311yescharacteristic
221C2×#124C2×D5311yescharacteristic
222#495—21no—
223#488—21no—
224#464—21no—
225#463—21no—
226#451Dic5311yescharacteristic
227#450Dic5311yescharacteristic
228#448Dic5311yescharacteristic
229#447Dic5311yescharacteristic
230#446—1061no—
231#435—1061no—
232#435—1061no—
233C8×D4—1061no—
234#433—1061no—
235#432—531no—
236#431—531no—
237#431C2×D5311yescharacteristic
238#405—1061no—
239C2×#97—531no—
240C2×#97—531no—
241#330—1061no—
242#329—1061no—
243#329—1061no—
244#329—1061no—
245#323—1061no—
246#323—1061no—
247C53#926511yescharacteristic, complemented, Sylow
248C22×Q8C2×Dic5311yescharacteristic
249C22×D4C2×Dic5311yescharacteristic
250C2×Q16—21no—
251C2×SD16—21no—
252C2×SD16—21no—
253C2×D8—21no—
254C2×M16—1061no—
255C22×C8—1061no—
256C22×C8—1061no—
257C22×C8—1061no—
258#126—21no—
259#125C22⋊D5311yescharacteristic
260#124—21no—
261#124—21no—
262#124—21no—
263#124—21no—
264#124C22⋊D5311yescharacteristic
265#123—21no—
266C22⋊Q8C2×Dic5311yescharacteristic
267#121C2×Dic5311yescharacteristic
268C4×Q8C22⋊D5311yescharacteristic
269C4×D4—21no—
270C4×D4C22⋊D5311yescharacteristic
271#117—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42C22×D5311yescharacteristic
275C4⋊C8—21no—
276C4⋊C8—1061no—
277C4⋊C8—1061no—
278C4⋊C8—1061no—
279C4⋊C8—1061no—
280#103—21no—
281#103—21no—
282#102—21no—
283#102—21no—
284C22⋊C8—531no—
285C22⋊C8—531no—
286C22⋊C8—1061no—
287C22⋊C8—1061no—
288C22⋊C8—1061no—
289C22⋊C8C2×Dic5311yescharacteristic
290C22⋊C8C2×Dic5311yescharacteristic
291#97—531no—
292#97—531no—
293#97—1061no—
294#97—1061no—
295#97—1061no—
296#97—1061no—
297#97—1061no—
298C4×C8—21no—
299C4×C8—1061no—
300C4×C8—1061no—
301C24—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—81no—
305C2×Q8C22⋊Dic5311yescharacteristic
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—81no—
310C2×D4C22⋊Dic5311yescharacteristic
311C22×C4—21no—
312C22×C4—21no—
313C22×C4D4×D5311yescharacteristic
314C22×C4#1171994711yescharacteristic
315C22×C4C22×Dic5311yescharacteristic
316Q16—81no—
317SD16—81no—
318SD16—81no—
319D8—81no—
320M16—2122no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—1061no—
324C2×C8—1062no—
325C2×C8—1061no—
326C2×C8—1061no—
327C2×C8—1061no—
328C2×C8—2122no—
329C2×C8—2122no—
330C2×C8—2121no—
331C2×C8—2121no—
332C2×C8—2121no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4C22⋊Dic5311yescharacteristic
338C4⋊C4C22⋊Dic5311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C42—21no—
349C42D4×D5311yescharacteristic
350C42#1171994711yescharacteristic
351C42C2×C22⋊D5311yescharacteristic
352C42C2×C22⋊D5311yescharacteristic
353C23—21no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—81no—
358C23—81no—
359C23D4×Dic5311yescharacteristic
360Q8—41no—
361Q8—41no—
362Q8—41no—
363Q8—81no—
364D4—41no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—81no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4C2×C22⋊Dic5311yescharacteristic
382C2×C4D4×Dic5311yescharacteristic
383C2×C4#1313318511yescharacteristic
384C2×C4#1313318911yescharacteristic
385C2×C4#1313318911yescharacteristic
386C2×C4#1313319811yescharacteristic
387C8—41no—
388C8—41no—
389C8—1062no—
390C8—4241no—
391C8—2121no—
392C8—2121no—
393C8—2122no—
394C8—2121no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—161no—
404C22—161no—
405C22#1681050311yescharacteristic
406C22#1681074011yescharacteristic
407C22#1681074511yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—81no—
418C2—21no—
419C2—21no—
420C2—81no—
421C2—161no—
422C2#2027504411yescharacteristic
423C2#2027521811yescharacteristic
424C2#2027526311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3211C2
socleC22×C53#386
prime radicalC53×#447C4
cosocleC2×C42×C53C23
hypercenter#3211D53
hypocenterC53#9265
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3211

Complements

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

NH ≅ G/Ntype
1. #20283711C2semidirect
2. #20273335C2semidirect
3. C2×#16810363C2semidirect
4. #20273067C2semidirect
5. #20272615C2semidirect
6. #20272559C2semidirect
247. C53#9265semidirect

Maximal subgroups

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

subgroupindex
#202837112
#202733352
C2×#168103632
#202730672
#202726152
#202725592
C53×#32112
#926553