Finite Group Databasebeta

← #27278986

Subgroups — #27278986

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#20284170C211yesmaximal, characteristic
2#20273165C211yesmaximal, characteristic, complemented
3#20273113C211yesmaximal, characteristic, complemented
4#20273097C211yesmaximal, characteristic, complemented
5#20272610C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C53×#2925C211yesmaximal, characteristic, complemented
8C2×#13133237—21no—
9#16811173C2211yescharacteristic
10#16810845—21no—
11#16810838—21no—
12#16810833C411yescharacteristic
13#16810828C411yescharacteristic
14#16810823C411yescharacteristic
15#16810818C411yescharacteristic
16#16810676C2211yescharacteristic
17C2×C4⋊#88322C2211yescharacteristic
18#16810617—21no—
19C2×C4×#88322C2211yescharacteristic
20Q8⋊#88322—21no—
21D4⋊#88322—21no—
22#16810350—22no—
23#16810350—21no—
24#16810350—21no—
25C4×#41×C53C2211yescharacteristic
26C53×#433—22no—
27C53×#432—21no—
28C53×#431—21no—
29C53×#423C2211yescharacteristic
30C2×C4⋊C8×C53C2211yescharacteristic
31C53×#408—21no—
32C53×#405—21no—
33C2×#11719945—21no—
34#13133237—42no—
35#13133237—42no—
36C2×#11719925—21no—
37C2×#11719923—21no—
38C2×#11719924—21no—
39C2×#11719922—21no—
40C22×#88322—21no—
41C22×#88322—21no—
42#13133217C2×C411yescharacteristic
43#13133214C2×C411yescharacteristic
44#13133205—21no—
45#13133191—21no—
46#13133191—21no—
47C4×D212—21no—
48C4×D212—21no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322C2×C411yescharacteristic
52C4⋊#88322C2×C411yescharacteristic
53#13133140—21no—
54#13133140—21no—
55#13133137—21no—
56#13133137—21no—
57C22⋊#88322—21no—
58C4×#88322—21no—
59C4×#88322—21no—
60C4×#88322—21no—
61C2×#41×C53D411yescharacteristic
62C2×M16×C53—21no—
63C22×C8×C53—21no—
64C4×Q8×C53C2×C411yescharacteristic
65C4×Q8×C53D411yescharacteristic
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C4×D4×C53C2×C411yescharacteristic
69C4×D4×C53D411yescharacteristic
70#117×C53—21no—
71#117×C53D411yescharacteristic
72C2×C42×C53—21no—
73C2×C42×C53C2311yescharacteristic
74C4⋊C8×C53—21no—
75C4⋊C8×C53—21no—
76C4⋊C8×C53D411yescharacteristic
77C4⋊C8×C53D411yescharacteristic
78C4⋊C8×C53D412yes—
79C22⋊C8×C53—22no—
80C22⋊C8×C53—22no—
81#97×C53—22no—
82C4×C8×C53—22no—
83#41×C53—21no—
84#41×C53—21no—
85#41×C53—21no—
86#41×C53—21no—
87C2×Q8×C53C22⋊C411yescharacteristic
88C2×D4×C53—21no—
89C2×D4×C53C22⋊C411yescharacteristic
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53#4111yescharacteristic
95C22×C4×C53#4111yescharacteristic
96C2×C22⋊D53—41no—
97#11719945—41no—
98#11719945—41no—
99#11719945—41no—
100#11719945—41no—
101C2×D212—21no—
102C2×D212—21no—
103C2×D212—21no—
104C2×C4×D53—41no—
105C2×Dic106—21no—
106M16×C53—41no—
107M16×C53—41no—
108C2×C8×C53—21no—
109C2×C8×C53—22no—
110C2×C8×C53—21no—
111C2×C8×C53—41no—
112C2×C8×C53—41no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53C22⋊C411yescharacteristic
116C4⋊C4×C53C22⋊C411yescharacteristic
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—21no—
120C42×C53—21no—
121C42×C53—21no—
122C42×C53—21no—
123C42×C53C22×C411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127#11719925—42no—
128#11719924—42no—
129#11719923—42no—
130#11719922—42no—
131#11719921—41no—
132#11719921—41no—
133C4⋊Dic53—21no—
134C4⋊Dic53—21no—
135C2×#88322—22no—
136C2×#88322—21no—
137C2×#88322—21no—
138C2×#88322—41no—
139C2×#88322—41no—
140C2×#88322—41no—
141C2×#88322—41no—
142C2×#88322—41no—
143C23×C53—21no—
144C23×C53#12311yescharacteristic
145C22×D53—41no—
146Q8×C53—21no—
147Q8×C53—21no—
148D4×C53—21no—
149D4×C53—21no—
150D4×C53—41no—
151D4×C53—41no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—81no—
163C2×C4×C53C2×C22⋊C411yescharacteristic
164C2×C4×C53C4×D411yescharacteristic
165C2×C4×C53C4×D411yescharacteristic
166C2×C4×C53#12011yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#13511yescharacteristic
170C2×C4×C53#13511yescharacteristic
171C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
173C22⋊D53—81no—
174C22⋊D53—81no—
175C2×Dic53—41no—
176D212—42no—
177D212—41no—
178D212—41no—
179D212—41no—
180D212—41no—
181C4×D53—81no—
182C4×D53—81no—
183Dic106—41no—
184Dic106—41no—
185C8×C53—41no—
186C8×C53—41no—
187C8×C53—42no—
188#88322—42no—
189#88322—41no—
190#88322—41no—
191#88322—81no—
192#8996—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53#38611yescharacteristic
199C22×C53#48211yescharacteristic
200C22×C53#48211yescharacteristic
201C2×D53—81no—
202C2×D53—81no—
203C2×D53—81no—
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—41no—
208C4×C53—41no—
209C4×C53—41no—
210C4×C53#41611yescharacteristic
211C4×C53#41811yescharacteristic
212C4×C53#44311yescharacteristic
213C4×C53#44411yescharacteristic
214C4×C53#44911yescharacteristic
215C4×C53#44911yescharacteristic
216Dic53—81no—
217#3035—531no—
218#2925D5311yescharacteristic, complemented
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#187511yescharacteristic
228C2×C53#187611yescharacteristic
229C2×C53#188111yescharacteristic
230D53—161no—
231C2×#135—1061no—
232#532—531no—
233C4×#41C2×D5311yescharacteristic
234#464—1061no—
235#463—1061no—
236#462—531no—
237#461—531no—
238#460—531no—
239#459—531no—
240#433—22no—
241#432—21no—
242#431—21no—
243#431—531no—
244#423C2×D5311yescharacteristic
245C2×C4⋊C8—531no—
246C2×C4⋊C8C2×D5311yescharacteristic
247#416—1061no—
248#408—21no—
249#405—21no—
250C2×C4×C8—531no—
251Q8⋊C8—1061no—
252D4⋊C8—1061no—
253D4⋊C8—1061no—
254D4⋊C8—1061no—
255D4⋊C8—1062no—
256C53#899611yescharacteristic, complemented, Sylow
257C2×#41—1061no—
258C2×#41C22⋊D5311yescharacteristic
259#135—2122no—
260#135—2122no—
261C2×Q16—1061no—
262C2×SD16—1061no—
263C2×SD16—1061no—
264C2×D8—1061no—
265C2×M16—21no—
266C22×C8—21no—
267C22×C8—1061no—
268C22×C8—1061no—
269C4⋊Q8—531no—
270#127—531no—
271#124—1061no—
272#121—1061no—
273#121—1061no—
274C4×Q8C4×D5311yescharacteristic
275C4×Q8D21211yescharacteristic
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—1061no—
279C4×D4—1061no—
280C4×D4C4×D5311yescharacteristic
281C4×D4D21211yescharacteristic
282#117—21no—
283#117C22⋊D5311yescharacteristic
284C2×C42—21no—
285C2×C42C22×D5311yescharacteristic
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—531no—
289C4⋊C8—531no—
290C4⋊C8—1061no—
291C4⋊C8—1061no—
292C4⋊C8C22⋊D5311yescharacteristic
293C4⋊C8C22⋊D5311yescharacteristic
294C4⋊C8C22⋊D5312yes—
295#103—1061no—
296#103—1061no—
297#102—1061no—
298#102—1061no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—1061no—
302#97—22no—
303C4×C8—22no—
304C4×C8—1061no—
305C4×C8—1061no—
306C4×C8—1061no—
307#41—21no—
308#41—21no—
309#41—21no—
310#41—21no—
311#41—2121no—
312#41—2121no—
313#41—2121no—
314#41—2121no—
315C2×Q8—1061no—
316C2×Q8#1171992111yescharacteristic
317C2×D4—21no—
318C2×D4—1061no—
319C2×D4—1061no—
320C2×D4—1061no—
321C2×D4—2121no—
322C2×D4#1171992111yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—2121no—
327C22×C4#1171994511yescharacteristic
328C22×C4#1171994511yescharacteristic
329C22×C4C2×C22⋊D5311yescharacteristic
330Q16—2122no—
331SD16—2122no—
332SD16—2122no—
333D8—2122no—
334M16—41no—
335M16—41no—
336C2×C8—21no—
337C2×C8—22no—
338C2×C8—21no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—1062no—
342C2×C8—1061no—
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—1061no—
352C4⋊C4—1061no—
353C4⋊C4#1171992111yescharacteristic
354C4⋊C4#1171992111yescharacteristic
355C22⋊C4—21no—
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—2121no—
359C22⋊C4—2121no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42C2×C4×D5311yescharacteristic
364C42C2×D21211yescharacteristic
365C42C2×C22⋊D5311yescharacteristic
366C42C2×C22⋊D5311yescharacteristic
367C23—21no—
368C23—2121no—
369C23#1313320011yescharacteristic
370Q8—21no—
371Q8—21no—
372Q8—2121no—
373Q8—2121no—
374D4—21no—
375D4—21no—
376D4—41no—
377D4—41no—
378D4—4241no—
379D4—4241no—
380D4—2122no—
381D4—2121no—
382D4—2121no—
383D4—2121no—
384D4—2121no—
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—81no—
396C2×C4—4241no—
397C2×C4—4241no—
398C2×C4—2121no—
399C2×C4C2×#1171992111yescharacteristic
400C2×C4C4×C22⋊D5311yescharacteristic
401C2×C4C4×C22⋊D5311yescharacteristic
402C2×C4#1313318611yescharacteristic
403C2×C4#1313319111yescharacteristic
404C2×C4#1313319111yescharacteristic
405C2×C4#1313323711yescharacteristic
406C2×C4#1313323711yescharacteristic
407C2×C4#1313324211yescharacteristic
408C2×C4#1313324211yescharacteristic
409C8—41no—
410C8—41no—
411C8—42no—
412C8—4241no—
413C8—2122no—
414C8—2121no—
415C8—2121no—
416C22—21no—
417C22—21no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—4241no—
422C22—4241no—
423C22—4241no—
424C22#1681050211yescharacteristic
425C22#1681094711yescharacteristic
426C22#1681094711yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—4241no—
434C4#1681061711yescharacteristic
435C4#1681062711yescharacteristic
436C4#1681073811yescharacteristic
437C4#1681074311yescharacteristic
438C4#1681076811yescharacteristic
439C4#1681076811yescharacteristic
440C2—21no—
441C2—21no—
442C2—81no—
443C2—8481no—
444C2#2027503711yescharacteristic
445C2#2027504211yescharacteristic
446C2#2027507711yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2925C2
socleC22×C53#386
prime radical#20284170C2
cosocleC2×C42×C53C23
hypercenter#2925D53
hypocenterC53#8996
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2925

Complements

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

NH ≅ G/Ntype
2. #20273165C2semidirect
3. #20273113C2semidirect
4. #20273097C2semidirect
5. #20272610C2semidirect
6. #20272549C2semidirect
7. C53×#2925C2semidirect
218. #2925D53semidirect
256. C53#8996semidirect

Maximal subgroups

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

subgroupindex
#202841702
#202731652
#202731132
#202730972
#202726102
#202725492
C53×#29252
#899653