Finite Group Databasebeta

← #27278988

Subgroups — #27278988

Order 13568.

484 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283381C211yesmaximal, characteristic, complemented
2#20273166C211yesmaximal, characteristic, complemented
3#20273114C211yesmaximal, characteristic, complemented
4#20273097C211yesmaximal, characteristic, complemented
5#20272612C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C53×#3035C211yesmaximal, characteristic
8#16810684—22no—
9#16810680—21no—
10#16810676—21no—
11#16810676C2211yescharacteristic
12#16810647C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14#16810590—21no—
15#16810579—21no—
16C2×C4×#88322C2211yescharacteristic
17Q8⋊#88322—21no—
18D4⋊#88322—22no—
19D4⋊#88322—21no—
20D4⋊#88322—21no—
21D4⋊#88322—21no—
22C2×#135×C53—21no—
23C53×#532C2211yescharacteristic
24C4×#41×C53C2211yescharacteristic
25C53×#464—21no—
26C53×#463—21no—
27C53×#462C411yescharacteristic
28C53×#461C411yescharacteristic
29C53×#460C411yescharacteristic
30C53×#459C411yescharacteristic
31C2×C4⋊C8×C53C2211yescharacteristic
32C53×#416—21no—
33C2×#11719917—21no—
34C22×#88322—21no—
35C22×#88322—21no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322D411yescharacteristic
40C4⋊#88322D411yescharacteristic
41C4⋊#88322D412yes—
42C22⋊#88322—22no—
43C22⋊#88322—22no—
44C22⋊#88322—21no—
45#13133128—22no—
46C4×#88322—22no—
47C4×#88322—21no—
48C4×#88322—21no—
49C4×#88322—21no—
50C2×#41×C53—21no—
51C2×#41×C53D411yescharacteristic
52#135×C53—42no—
53#135×C53—42no—
54C2×Q16×C53—21no—
55C2×SD16×C53—21no—
56C2×SD16×C53—21no—
57C2×D8×C53—21no—
58C22×C8×C53—21no—
59C4⋊Q8×C53C2×C411yescharacteristic
60#127×C53C2×C411yescharacteristic
61#124×C53—21no—
62#121×C53—21no—
63#121×C53—21no—
64C4×Q8×C53C2×C411yescharacteristic
65C4×Q8×C53D411yescharacteristic
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69C4×D4×C53—21no—
70C4×D4×C53C2×C411yescharacteristic
71C4×D4×C53D411yescharacteristic
72#117×C53—21no—
73#117×C53D411yescharacteristic
74C2×C42×C53—21no—
75C2×C42×C53C2311yescharacteristic
76C4⋊C8×C53—21no—
77C4⋊C8×C53C2×C411yescharacteristic
78C4⋊C8×C53C2×C411yescharacteristic
79#103×C53—21no—
80#103×C53—21no—
81#102×C53—21no—
82#102×C53—21no—
83#41×C53—21no—
84#41×C53—21no—
85#41×C53—21no—
86#41×C53—21no—
87#41×C53—41no—
88#41×C53—41no—
89#41×C53—41no—
90#41×C53—41no—
91C2×Q8×C53—21no—
92C2×Q8×C53C22⋊C411yescharacteristic
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53—21no—
97C2×D4×C53—41no—
98C2×D4×C53C22⋊C411yescharacteristic
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—41no—
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C22×C4×C53#4111yescharacteristic
106Q16×C53—42no—
107SD16×C53—42no—
108SD16×C53—42no—
109D8×C53—42no—
110C2×C8×C53—21no—
111C2×C8×C53—21no—
112C2×C8×C53—41no—
113C2×C8×C53—41no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53C22⋊C411yescharacteristic
119C4⋊C4×C53C22⋊C411yescharacteristic
120C22⋊C4×C53—21no—
121C22⋊C4×C53—21no—
122C22⋊C4×C53—21no—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C42×C53—21no—
126C42×C53—21no—
127C42×C53—21no—
128C42×C53C22×C411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53C2×D411yescharacteristic
131C42×C53C2×D411yescharacteristic
132#11719917—41no—
133#11719917—41no—
134C2×#88322—22no—
135C2×#88322—21no—
136C2×#88322—22no—
137C2×#88322—21no—
138C2×#88322—41no—
139C2×#88322—41no—
140C2×#88322—41no—
141C2×#88322—41no—
142C2×#88322—41no—
143C23×C53—21no—
144C23×C53—41no—
145C23×C53#12311yescharacteristic
146Q8×C53—21no—
147Q8×C53—21no—
148Q8×C53—41no—
149Q8×C53—41no—
150D4×C53—21no—
151D4×C53—21no—
152D4×C53—42no—
153D4×C53—41no—
154D4×C53—41no—
155D4×C53—41no—
156D4×C53—41no—
157D4×C53—41no—
158D4×C53—41no—
159D4×C53—81no—
160D4×C53—81no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—81no—
173C2×C4×C53—81no—
174C2×C4×C53—81no—
175C2×C4×C53C2×C22⋊C411yescharacteristic
176C2×C4×C53C4×D411yescharacteristic
177C2×C4×C53C4×D411yescharacteristic
178C2×C4×C53#12011yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53#12111yescharacteristic
181C2×C4×C53#13511yescharacteristic
182C2×C4×C53#13511yescharacteristic
183C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
185C8×C53—41no—
186C8×C53—41no—
187#88322—42no—
188#88322—41no—
189#88322—41no—
190#88322—42no—
191#88322—81no—
192#8996—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53—81no—
201C22×C53#38611yescharacteristic
202C22×C53#48211yescharacteristic
203C22×C53#48211yescharacteristic
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—41no—
208C4×C53—41no—
209C4×C53—41no—
210C4×C53—81no—
211C4×C53#41611yescharacteristic
212C4×C53#41811yescharacteristic
213C4×C53#44311yescharacteristic
214C4×C53#44411yescharacteristic
215C4×C53#44911yescharacteristic
216C4×C53#44911yescharacteristic
217#3035D5311yescharacteristic
218#2925—531no—
219#1492—531no—
220#1480—531no—
221#1476—531no—
222#1301—531no—
223#1277—531no—
224C2×C53—21no—
225C2×C53—21no—
226C2×C53—81no—
227C2×C53—161no—
228C2×C53#187511yescharacteristic
229C2×C53#187611yescharacteristic
230C2×C53#188111yescharacteristic
231C2×#135—21no—
232#532C2×D5311yescharacteristic
233C4×#41C2×D5311yescharacteristic
234#464—21no—
235#463—21no—
236#462Dic5311yescharacteristic
237#461Dic5311yescharacteristic
238#460Dic5311yescharacteristic
239#459Dic5311yescharacteristic
240#433—1062no—
241#432—1061no—
242#431—531no—
243#431—1061no—
244#423—531no—
245C2×C4⋊C8—531no—
246C2×C4⋊C8C2×D5311yescharacteristic
247#416—21no—
248#408—1061no—
249#405—1061no—
250C2×C4×C8—531no—
251Q8⋊C8—1061no—
252D4⋊C8—1061no—
253D4⋊C8—1061no—
254D4⋊C8—1062no—
255D4⋊C8—1061no—
256C53#899611yescharacteristic, complemented, Sylow
257C2×#41—21no—
258C2×#41C22⋊D5311yescharacteristic
259#135—42no—
260#135—42no—
261C2×Q16—21no—
262C2×SD16—21no—
263C2×SD16—21no—
264C2×D8—21no—
265C2×M16—1061no—
266C22×C8—21no—
267C22×C8—1061no—
268C22×C8—1061no—
269C4⋊Q8C2×Dic5311yescharacteristic
270#127C2×Dic5311yescharacteristic
271#124—21no—
272#121—21no—
273#121—21no—
274C4×Q8C2×Dic5311yescharacteristic
275C4×Q8C22⋊D5311yescharacteristic
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4C2×Dic5311yescharacteristic
281C4×D4C22⋊D5311yescharacteristic
282#117—21no—
283#117C22⋊D5311yescharacteristic
284C2×C42—21no—
285C2×C42C22×D5311yescharacteristic
286C4⋊C8—21no—
287C4⋊C8—531no—
288C4⋊C8—531no—
289C4⋊C8—532no—
290C4⋊C8—1061no—
291C4⋊C8—1061no—
292C4⋊C8—1061no—
293C4⋊C8C2×Dic5311yescharacteristic
294C4⋊C8C2×Dic5311yescharacteristic
295#103—21no—
296#103—21no—
297#102—21no—
298#102—21no—
299C22⋊C8—1062no—
300C22⋊C8—1062no—
301C22⋊C8—1061no—
302#97—1062no—
303C4×C8—1062no—
304C4×C8—1061no—
305C4×C8—1061no—
306C4×C8—1061no—
307#41—21no—
308#41—21no—
309#41—21no—
310#41—21no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315C2×Q8—21no—
316C2×Q8C22⋊Dic5311yescharacteristic
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4C22⋊Dic5311yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—41no—
327C22×C4#1171994711yescharacteristic
328C22×C4#1171994711yescharacteristic
329C22×C4C2×C22⋊D5311yescharacteristic
330Q16—42no—
331SD16—42no—
332SD16—42no—
333D8—42no—
334M16—2121no—
335M16—2121no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—1062no—
341C2×C8—1061no—
342C2×C8—1062no—
343C2×C8—1061no—
344C2×C8—2121no—
345C2×C8—2121no—
346C2×C8—2121no—
347C2×C8—2121no—
348C2×C8—2121no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4C22⋊Dic5311yescharacteristic
354C4⋊C4C22⋊Dic5311yescharacteristic
355C22⋊C4—21no—
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42D4×D5311yescharacteristic
364C42D4×D5311yescharacteristic
365C42C22×Dic5311yescharacteristic
366C42C2×C22⋊D5311yescharacteristic
367C23—21no—
368C23—41no—
369C23#1313319811yescharacteristic
370Q8—21no—
371Q8—21no—
372Q8—41no—
373Q8—41no—
374D4—21no—
375D4—21no—
376D4—42no—
377D4—41no—
378D4—41no—
379D4—41no—
380D4—41no—
381D4—41no—
382D4—41no—
383D4—81no—
384D4—81no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4C2×C22⋊Dic5311yescharacteristic
400C2×C4D4×Dic5311yescharacteristic
401C2×C4D4×Dic5311yescharacteristic
402C2×C4#1313318511yescharacteristic
403C2×C4#1313318911yescharacteristic
404C2×C4#1313318911yescharacteristic
405C2×C4#1313323711yescharacteristic
406C2×C4#1313323711yescharacteristic
407C2×C4#1313324311yescharacteristic
408C2×C4#1313324311yescharacteristic
409C8—41no—
410C8—41no—
411C8—4241no—
412C8—2122no—
413C8—2121no—
414C8—2121no—
415C8—2122no—
416C22—21no—
417C22—21no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22#1681050311yescharacteristic
425C22#1681094411yescharacteristic
426C22#1681094411yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—81no—
434C4#1681061911yescharacteristic
435C4#1681062811yescharacteristic
436C4#1681074011yescharacteristic
437C4#1681074511yescharacteristic
438C4#1681076611yescharacteristic
439C4#1681076611yescharacteristic
440C2—21no—
441C2—21no—
442C2—81no—
443C2—161no—
444C2#2027503811yescharacteristic
445C2#2027504411yescharacteristic
446C2#2027507911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3035C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3035D53
hypocenterC53#8996
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #20283381C2semidirect
2. #20273166C2semidirect
3. #20273114C2semidirect
4. #20273097C2semidirect
5. #20272612C2semidirect
6. #20272549C2semidirect
256. C53#8996semidirect

Maximal subgroups

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

subgroupindex
#202833812
#202731662
#202731142
#202730972
#202726122
#202725492
C53×#30352
#899653