Finite Group Databasebeta

← #27468191

Subgroups — #27468191

Order 13568.

442 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276298C211yesmaximal, characteristic, complemented
2#20276296C211yesmaximal, characteristic, complemented
3#20276170C211yesmaximal, characteristic
4#20276167C211yesmaximal, characteristic
5#20275504C211yesmaximal, characteristic, complemented
6#20275462C211yesmaximal, characteristic, complemented
7#20275385C211yesmaximal, characteristic
8#20275384C211yesmaximal, characteristic
9#20274861C211yesmaximal, characteristic, complemented
10#20274837C211yesmaximal, characteristic, complemented
11#20274768C211yesmaximal, characteristic
12#20274764C211yesmaximal, characteristic
13C4×#13133152C211yesmaximal, characteristic, complemented
14#20274549C211yesmaximal, characteristic
15C53×#1841C211yesmaximal, characteristic, complemented
16C2×#13133225C2211yescharacteristic
17#16810672C2211yescharacteristic
18#16810671C2211yescharacteristic
19C2×#13133152C2211yescharacteristic
20C2×#13133152C2211yescharacteristic
21C2×#13133152C2211yescharacteristic
22#16810659C2211yescharacteristic
23#16810651C2211yescharacteristic
24#16810650C2211yescharacteristic
25#16810641C2211yescharacteristic
26#16810639C2211yescharacteristic
27C2×#13133145C2211yescharacteristic
28#16810617C2211yescharacteristic
29#16810605C2211yescharacteristic
30#16810605C2211yescharacteristic
31#16810579C2211yescharacteristic
32C4×#11719917C2211yescharacteristic
33C2×C4×#88322C2211yescharacteristic
34#16810395C2211yescharacteristic
35#16810395C2211yescharacteristic
36#16810393C2211yescharacteristic
37#16810393C2211yescharacteristic
38#16810387C2211yescharacteristic
39#16810387C2211yescharacteristic
40#16810387C2211yescharacteristic
41#16810383C2211yescharacteristic
42#16810381C2211yescharacteristic
43#16810379C2211yescharacteristic
44C4×M16×C53C2211yescharacteristic
45C53×#344C2211yescharacteristic
46C53×#344C2211yescharacteristic
47C53×#339C2211yescharacteristic
48C53×#339C2211yescharacteristic
49C53×#338C2211yescharacteristic
50C53×H(3,ℤ/4)C2211yescharacteristic
51C2×#11719945C2311yescharacteristic
52#13133225—21no—
53#13133225—21no—
54#13133225—21no—
55#13133225—21no—
56C2×#11719917C2311yescharacteristic
57C2×#11719917C2311yescharacteristic
58C2×#11719917C2311yescharacteristic
59C2×#11719917C2311yescharacteristic
60C2×#11719912—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C2×C8×D53—21no—
64#13133176C2311yescharacteristic
65#13133153—41no—
66#13133152—21no—
67#13133152—21no—
68#13133152—21no—
69#13133152—21no—
70#13133152—21no—
71#13133152—21no—
72#13133152—41no—
73#13133150—21no—
74#13133148—21no—
75C4⋊#88322—41no—
76#13133146—41no—
77#13133145—21no—
78#13133145—21no—
79#13133145—41no—
80#13133143—41no—
81#13133140—21no—
82#13133137—21no—
83#13133135—21no—
84#13133135—21no—
85#13133133—21no—
86#13133133—21no—
87#13133128—21no—
88#13133128—21no—
89C4×#88322—21no—
90C4×#88322—21no—
91C4×#88322—21no—
92C4×#88322—21no—
93C2×M16×C53C2311yescharacteristic
94C2×M16×C53C2311yescharacteristic
95C2×M16×C53C2311yescharacteristic
96C2×M16×C53C2311yescharacteristic
97#117×C53C2311yescharacteristic
98#117×C53C2311yescharacteristic
99C2×C42×C53C2311yescharacteristic
100#97×C53—21no—
101C4×C8×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53C2411yescharacteristic
104C2×C22⋊D53—21no—
105#11719945—21no—
106#11719945—21no—
107#11719945—21no—
108#11719945—21no—
109C2×D212#4111yescharacteristic
110C2×C4×D53—21no—
111C2×Dic106#4111yescharacteristic
112M16×C53—21no—
113M16×C53—21no—
114M16×C53—21no—
115M16×C53—21no—
116M16×C53—21no—
117M16×C53—21no—
118M16×C53—21no—
119M16×C53—21no—
120C2×C8×C53—21no—
121C2×C8×C53—21no—
122C2×C8×C53—21no—
123C2×C8×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C22⋊C4×C53—21no—
127C22⋊C4×C53—21no—
128C42×C53—21no—
129C42×C53—21no—
130C42×C53—21no—
131C42×C53#4111yescharacteristic
132C42×C53#4111yescharacteristic
133C22⋊Dic53—21no—
134C4⋊Dic53—21no—
135C4×Dic53—21no—
136#11719917—21no—
137#11719917—21no—
138#11719917—21no—
139#11719917—21no—
140#11719917—21no—
141#11719917—21no—
142#11719917—21no—
143#11719917—21no—
144C2×#88322—21no—
145C2×#88322—21no—
146C2×#88322—21no—
147C2×#88322—21no—
148C2×#88322—21no—
149C2×#88322—21no—
150C2×#88322—21no—
151C2×#88322#4111yescharacteristic
152C2×#88322#4111yescharacteristic
153C2×#88322#4111yescharacteristic
154C2×#88322#4111yescharacteristic
155C2×#88322#4111yescharacteristic
156C2×#88322#4111yescharacteristic
157#11719912—41no—
158#11719912—41no—
159C8×D53—41no—
160C8×D53—41no—
161C23×C532−1+411yescharacteristic
162C22×D53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—42no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53C2×#4111yescharacteristic
169C2×C4×C53C2×#4111yescharacteristic
170C2×C4×C53C2×#4111yescharacteristic
171C2×C4×C53C2×#4111yescharacteristic
172C2×C4×C53C2×#4111yescharacteristic
173C2×C4×C53H(5,𝔽2)11yescharacteristic
174C22⋊D53—41no—
175C22⋊D53—41no—
176C2×Dic53—21no—
177C2×Dic53—41no—
178D212—21no—
179D212—21no—
180C4×D53—41no—
181C4×D53—41no—
182Dic106—21no—
183Dic106—21no—
184C8×C53—41no—
185C8×C53—41no—
186C8×C53—41no—
187C8×C53—41no—
188#88322—41no—
189#88322—41no—
190#88322—41no—
191#88322—41no—
192#88322—41no—
193#88322#12611yescharacteristic
194#88322#12611yescharacteristic
195#88322#12611yescharacteristic
196#88322#12611yescharacteristic
197#27833—531nomaximal, Sylow
198C22×C53—41no—
199C22×C53#54211yescharacteristic
200C22×C53#55211yescharacteristic
201C22×C53#55211yescharacteristic
202C2×D53—41no—
203C2×D53—81no—
204C4×C53—22no—
205C4×C53—41no—
206C4×C53—81no—
207C4×C53—81no—
208C4×C53C2×#12611yescharacteristic
209C4×C53#52911yescharacteristic
210C4×C53#55311yescharacteristic
211C4×C53#55511yescharacteristic
212Dic53—41no—
213Dic53—81no—
214#2103—531no—
215#2103—531no—
216#2080—531no—
217#2080—531no—
218#1953—531no—
219#1945—531no—
220#1932—531no—
221#1932—531no—
222#1841—531no—
223#1841D5311yescharacteristic, complemented
224#1833—531no—
225#1814—531no—
226#1813—531no—
227C4×#108—531no—
228#1764—531no—
229C2×C53—21no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53#249511yescharacteristic
233D53—81no—
234C2×#131—531no—
235#430—531no—
236#430—531no—
237C2×#108—531no—
238C2×#108—531no—
239C2×#108—531no—
240#427—531no—
241#424—531no—
242#424—531no—
243#421—531no—
244#421—531no—
245C2×#104—531no—
246#416—531no—
247#413—531no—
248#413—531no—
249#405—531no—
250C4×M16—531no—
251C4×M16C2×D5311yescharacteristic
252C2×C4×C8—531no—
253#344—531no—
254#344—531no—
255#344—531no—
256#344—531no—
257#344C2×D5311yescharacteristic
258#344C2×D5311yescharacteristic
259#341—531no—
260#341—531no—
261#341—531no—
262#339—531no—
263#339C2×D5311yescharacteristic
264#339C2×D5311yescharacteristic
265#338—531no—
266#338C2×D5311yescharacteristic
267H(3,ℤ/4)—531no—
268H(3,ℤ/4)C2×D5311yescharacteristic
269C53#2783311yescharacteristic, complemented, Sylow
270C2×#41—531no—
271#131—1061no—
272#131—1061no—
273#131—1061no—
274#131—1061no—
275C2×M16—531no—
276C2×M16—531no—
277C2×M16—531no—
278C2×M16—531no—
279C2×M16—1061no—
280C2×M16C22×D5311yescharacteristic
281C2×M16C22×D5311yescharacteristic
282C2×M16C22×D5311yescharacteristic
283C2×M16C22×D5311yescharacteristic
284C22×C8—531no—
285C22×C8—531no—
286C22×C8—1061no—
287#117—531no—
288#117C22×D5311yescharacteristic
289#117C22×D5311yescharacteristic
290C2×C42C22×D5311yescharacteristic
291#108—1061no—
292#108—1061no—
293#108—1061no—
294#108—1061no—
295#108—1061no—
296#108—1061no—
297#108—2121no—
298#108—2121no—
299#107—1061no—
300#106—1061no—
301C4⋊C8—2121no—
302C4⋊C8—2121no—
303#104—1061no—
304#104—1061no—
305#104—2121no—
306#104—2121no—
307#103—1061no—
308#102—1061no—
309#101—1061no—
310#101—1061no—
311#100—1061no—
312#100—1061no—
313#97—21no—
314#97—1061no—
315#97—1061no—
316C4×C8—21no—
317C4×C8—1061no—
318C4×C8—1061no—
319C4×C8—1061no—
320C4×C8—1061no—
321#41—1061no—
322#41—1061no—
323#41—1061no—
324#41—1061no—
325C2×Q8—531no—
326C2×D4—531no—
327C2×D4—1061no—
328C22×C4—21no—
329C22×C4—1061no—
330C22×C4C23×D5311yescharacteristic
331M16—21no—
332M16—21no—
333M16—21no—
334M16—21no—
335M16—21no—
336M16—21no—
337M16—21no—
338M16—21no—
339M16—1061no—
340M16—1061no—
341M16—1061no—
342M16—1061no—
343M16—1061no—
344M16—1061no—
345M16—1061no—
346M16—1061no—
347M16—2121no—
348M16—2121no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—531no—
354C2×C8—531no—
355C2×C8—531no—
356C2×C8—531no—
357C2×C8—531no—
358C2×C8—531no—
359C2×C8—1061no—
360C2×C8—1061no—
361C2×C8—1061no—
362C2×C8—1061no—
363C2×C8—1061no—
364C2×C8—1061no—
365C2×C8—1061no—
366C2×C8—2121no—
367C2×C8—2121no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—1061no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—1061no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—1061no—
378C42#1171994511yescharacteristic
379C42#1171994511yescharacteristic
380C23—1061no—
381C23#1313326411yescharacteristic
382Q8—1061no—
383Q8—1061no—
384D4—1061no—
385D4—1061no—
386D4—2121no—
387D4—2121no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—1061no—
394C2×C4—2121no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4#41×D5311yescharacteristic
398C2×C4#41×D5311yescharacteristic
399C2×C4#41×D5311yescharacteristic
400C2×C4#41×D5311yescharacteristic
401C2×C4C2×#1171994511yescharacteristic
402C2×C4#1313326311yescharacteristic
403C8—41no—
404C8—41no—
405C8—41no—
406C8—41no—
407C8—531no—
408C8—531no—
409C8—531no—
410C8—531no—
411C8—2121no—
412C8—2121no—
413C8—2121no—
414C8—2121no—
415C8—2121no—
416C22—41no—
417C22—4241no—
418C22—2121no—
419C22#1681123911yescharacteristic
420C22#1681130611yescharacteristic
421C22#1681130911yescharacteristic
422C4—22no—
423C4—41no—
424C4—81no—
425C4—81no—
426C4—4241no—
427C4—2121no—
428C4#126×D5311yescharacteristic
429C4#1681115111yescharacteristic
430C4#1681132411yescharacteristic
431C4#1681133611yescharacteristic
432C2—21no—
433C2—21no—
434C2—21no—
435C2—4241no—
436C2#2027912511yescharacteristic
subgroupG/H
centerC4#16811336
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1841C2
socleC2×C53#2495
prime radicalC2×#11719945C23
hypercenter#1841D53
hypocenterC53#27833
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1841

Complements

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

NH ≅ G/Ntype
1. #20276298C2semidirect
2. #20276296C2semidirect
5. #20275504C2semidirect
6. #20275462C2semidirect
9. #20274861C2semidirect
10. #20274837C2semidirect
13. C4×#13133152C2semidirect
15. C53×#1841C2semidirect
223. #1841D53semidirect
269. C53#27833semidirect

Maximal subgroups

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

subgroupindex
#202762982
#202762962
#202761702
#202761672
#202755042
#202754622
#202753852
#202753842
#202748612
#202748372
#202747682
#202747642
C4×#131331522
#202745492
C53×#18412
#2783353