Finite Group Databasebeta

← #27278437

Subgroups — #27278437

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1M16×D212C211yesmaximal, characteristic
2#20275490C211yesmaximal, characteristic, complemented
3#20273471C211yesmaximal, characteristic, complemented
4#20273141C211yesmaximal, characteristic
5C2×#16810352C211yesmaximal, characteristic
6#20272611C211yesmaximal, characteristic, complemented
7C53×#1278C211yesmaximal, characteristic, complemented
8C2×M16×D53—21no—
9C2×C4×D212C2211yescharacteristic
10#16810702—21no—
11#16810696—22no—
12C8×D212—21no—
13C2×#13133152—21no—
14#16810647C2211yescharacteristic
15#16810647C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17#16810586—21no—
18#16810387—21no—
19#16810374—22no—
20#16810352—21no—
21#16810352—21no—
22#16810352C411yescharacteristic
23#16810352C411yescharacteristic
24#16810352C411yescharacteristic
25#16810352C411yescharacteristic
26C53×#431C2211yescharacteristic
27C4×M16×C53C2211yescharacteristic
28C2×C4×C8×C53C2211yescharacteristic
29C22×D212Q811yescharacteristic
30C22×C4×D53—21no—
31C2×#11719917—21no—
32C2×#11719917—21no—
33C2×#11719912—22no—
34M16×D53—42no—
35M16×D53—42no—
36C22×#88322—21no—
37C2×C8×D53—22no—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212C2×C411yescharacteristic
41C4×D212C2×C411yescharacteristic
42C4×D212D411yescharacteristic
43C4×D212D411yescharacteristic
44C2×C4⋊Dic53Q811yescharacteristic
45C2×#11719921—21no—
46#13133152—42no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322C2×C411yescharacteristic
52C4⋊#88322C2×C411yescharacteristic
53C4⋊#88322D412yes—
54#13133129—22no—
55#13133129—22no—
56C2×M16×C53D411yescharacteristic
57C2×M16×C53D411yescharacteristic
58C22×C8×C53—21no—
59C2×C42×C53C2311yescharacteristic
60C4⋊C8×C53—21no—
61C22⋊C8×C53—21no—
62#97×C53—21no—
63C4×C8×C53—21no—
64C4×C8×C53—21no—
65C4×C8×C53—21no—
66C4×C8×C53C2×C411yescharacteristic
67C4×C8×C53C2×C411yescharacteristic
68C23×D53—21no—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×Q811yescharacteristic
71C22×C4×C53#4111yescharacteristic
72C22×Dic53—21no—
73C2×D212—22no—
74C2×D212—21no—
75C2×D212—41no—
76C2×D212C4⋊C411yescharacteristic
77C2×D212C4⋊C411yescharacteristic
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86M16×C53—21no—
87M16×C53—21no—
88M16×C53—21no—
89M16×C53SD1612yes—
90C2×C8×C53—21no—
91C2×C8×C53—21no—
92C2×C8×C53—21no—
93C2×C8×C53—21no—
94C2×C8×C53—42no—
95C2×C8×C53—41no—
96C42×C53C22×C411yescharacteristic
97C42×C53C2×D411yescharacteristic
98C42×C53C2×D411yescharacteristic
99C42×C53#4111yescharacteristic
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104C4⋊Dic53—21no—
105C4⋊Dic53C4⋊C411yescharacteristic
106C4⋊Dic53C4⋊C411yescharacteristic
107#11719917—41no—
108#11719917—41no—
109#11719917—41no—
110#11719917—41no—
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322—22no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117#11719912—44no—
118C8×D53—44no—
119C23×C53C22⋊Q811yescharacteristic
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—42no—
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—81no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—41no—
129C2×C4×C53C2×C4⋊C411yescharacteristic
130C2×C4×C53C4×D411yescharacteristic
131C2×C4×C53C4×D411yescharacteristic
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53C22⋊Q811yescharacteristic
134C2×C4×C53#12311yescharacteristic
135C2×C4×C53C2×SD1611yescharacteristic
136C2×C4×C53#13511yescharacteristic
137C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143D212—21no—
144D212—21no—
145D212—21no—
146D212—42no—
147D212#10812yescomplemented
148C4×D53—41no—
149C4×D53—41no—
150C4×D53—41no—
151C4×D53—41no—
152C4×D53—41no—
153C4×D53—41no—
154C4×D53—41no—
155C4×D53—41no—
156C8×C53—22no—
157C8×C53—22no—
158C8×C53—41no—
159C8×C53—41no—
160C8×C53—81no—
161#88322—41no—
162#88322—41no—
163#88322—42no—
164#88322—81no—
165#8908—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53#38511yescharacteristic
169C22×C53#47611yescharacteristic
170C22×C53#47811yescharacteristic
171C2×D53—22no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—82no—
178C2×D53—82no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53C2×#10811yescharacteristic
183C4×C53#43011yescharacteristic
184C4×C53C4×SD1611yescharacteristic
185C4×C53#44211yescharacteristic
186C4×C53#44911yescharacteristic
187C4×C53#45011yescharacteristic
188Dic53—41no—
189Dic53—41no—
190D4×M16—531no—
191#1950—531no—
192#1566—531no—
193#1486—531no—
194C2×D4⋊C8—531no—
195#1301—531no—
196#1278D5311yescharacteristic, complemented
197C2×C53—21no—
198C2×C53—21no—
199C2×C53#185511yescharacteristic
200C2×C53#186111yescharacteristic
201C2×C53#186411yescharacteristic
202D53—42no—
203D53—81no—
204C22×M16—1061no—
205C2×C4×D4—531no—
206#436—1061no—
207#435—1062no—
208C8×D4—1061no—
209#431C2×D5311yescharacteristic
210C2×#108—1061no—
211#423—531no—
212#423—531no—
213C2×C4⋊C8—531no—
214#407—1061no—
215C4×M16C2×D5311yescharacteristic
216C2×C4×C8C2×D5311yescharacteristic
217#341—1061no—
218#334—1062no—
219D4⋊C8—531no—
220D4⋊C8—531no—
221D4⋊C8—531no—
222D4⋊C8—531no—
223D4⋊C8—1061no—
224D4⋊C8—1061no—
225C53#890811yescharacteristic, complemented, Sylow
226C22×D4—531no—
227C23×C4—1061no—
228C2×M16—1062no—
229C2×M16—1061no—
230C2×M16—1061no—
231C2×M16—2122no—
232C2×M16—2122no—
233C2×M16D21211yescharacteristic
234C2×M16D21211yescharacteristic
235C22×C8—21no—
236C22×C8—1062no—
237C22×C8—1061no—
238C4×D4—531no—
239C4×D4—531no—
240C4×D4—531no—
241C4×D4—531no—
242C4×D4—1061no—
243C4×D4—1061no—
244C2×C4⋊C4—531no—
245C2×C22⋊C4—1061no—
246C2×C42C22×D5311yescharacteristic
247#108—2122no—
248C4⋊C8—21no—
249C4⋊C8—531no—
250C4⋊C8—531no—
251C4⋊C8—532no—
252C4⋊C8—1061no—
253C4⋊C8—1061no—
254C4⋊C8—1061no—
255C4⋊C8—1061no—
256C22⋊C8—21no—
257C22⋊C8—1062no—
258C22⋊C8—1062no—
259#97—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8C4×D5311yescharacteristic
264C4×C8C4×D5311yescharacteristic
265C24—1061no—
266C2×D4—531no—
267C2×D4—531no—
268C2×D4—1062no—
269C2×D4—1061no—
270C2×D4—2121no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—2121no—
277C22×C4—2121no—
278C22×C4—2121no—
279C22×C4—2121no—
280C22×C4C2×D21211yescharacteristic
281C22×C4#1171994711yescharacteristic
282C22×C4Q8×D5311yescharacteristic
283M16—21no—
284M16—21no—
285M16—21no—
286M16—2124no—
287M16—2121no—
288M16—2121no—
289M16—2121no—
290M16—2121no—
291M16#1171991312yes—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—1061no—
299C2×C8—1061no—
300C2×C8—1062no—
301C2×C8—2124no—
302C2×C8—2121no—
303C2×C8—2121no—
304C2×C8—2121no—
305C4⋊C4—531no—
306C4⋊C4—531no—
307C4⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C42C2×C4×D5311yescharacteristic
313C42#1171994511yescharacteristic
314C42D4×D5311yescharacteristic
315C42D4×D5311yescharacteristic
316C23—1061no—
317C23—1061no—
318C23—4241no—
319C23—2122no—
320C23—2121no—
321C23—2121no—
322C23#1313319711yescharacteristic
323D4—532no—
324D4—1061no—
325D4—1061no—
326D4—1061no—
327D4—2122no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4C4⋊C4×D5311yescharacteristic
344C2×C4#1313317711yescharacteristic
345C2×C4C4×D21211yescharacteristic
346C2×C4#1313318711yescharacteristic
347C2×C4#1313319511yescharacteristic
348C2×C4#1313319911yescharacteristic
349C2×C4C2×#1171991311yescharacteristic
350C2×C4#1313323911yescharacteristic
351C2×C4#1313324511yescharacteristic
352C2×C4#1313324511yescharacteristic
353C8—22no—
354C8—22no—
355C8—41no—
356C8—41no—
357C8—81no—
358C8—4241no—
359C8—2122no—
360C8—2121no—
361C8—2121no—
362C22—21no—
363C22—21no—
364C22—1062no—
365C22—4242no—
366C22—4242no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22—2121no—
372C22#1681049711yescharacteristic
373C22#1681091411yescharacteristic
374C22#1681092411yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—2121no—
379C4—2121no—
380C4#108×D5311yescharacteristic
381C4#1681067011yescharacteristic
382C4C4×#1171991311yescharacteristic
383C4#1681073611yescharacteristic
384C4#1681077011yescharacteristic
385C4#1681077611yescharacteristic
386C2—21no—
387C2—21no—
388C2—4241no—
389C2—2122no—
390C2#2027492111yescharacteristic
391C2#2027495611yescharacteristic
392C2#2027497911yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1278C2
socleC22×C53#385
prime radicalC22×D212Q8
cosocleC2×C42×C53C23
hypercenter#1278D53
hypocenterC53#8908
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
2. #20275490C2semidirect
3. #20273471C2semidirect
6. #20272611C2semidirect
7. C53×#1278C2semidirect
147. D212#108semidirect
196. #1278D53semidirect
225. C53#8908semidirect

Maximal subgroups

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

subgroupindex
M16×D2122
#202754902
#202734712
#202731412
C2×#168103522
#202726112
C53×#12782
#890853