Finite Group Databasebeta

← #27280612

Subgroups — #27280612

Order 13568.

480 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285843C211yesmaximal, characteristic
2#20283461C211yesmaximal, characteristic
3#20273393C211yesmaximal, characteristic
4C2×#16810369C211yesmaximal, characteristic, complemented
5#20272610C211yesmaximal, characteristic, complemented
6#20272566C211yesmaximal, characteristic, complemented
7C53×#1997C211yesmaximal, characteristic, complemented
8C2×#13133225—21no—
9#16811173C2211yescharacteristic
10#16811019—21no—
11#16810971—21no—
12#16810941C414yes—
13#16810910—22no—
14#16810702—21no—
15#16810696—22no—
16C8×D212—21no—
17#16810676C2211yescharacteristic
18C2×C4⋊#88322C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20#16810591—21no—
21#16810370—21no—
22#16810369—21no—
23#16810369—21no—
24#16810366—21no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C53×#423C2211yescharacteristic
27C4×M16×C53C2211yescharacteristic
28C53×#343—21no—
29C53×#343—21no—
30C2×#11719945—21no—
31#13133225—42no—
32#13133225—42no—
33C2×#11719912—22no—
34C22×#88322—21no—
35C22×#88322—21no—
36C2×C8×D53—22no—
37#13133217D411yescharacteristic
38#13133214D411yescharacteristic
39#13133205—21no—
40#13133191—21no—
41#13133191C2×C412yes—
42C4×D212—21no—
43C4×D212D412yes—
44#13133148—24no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322D411yescharacteristic
50C4⋊#88322D411yescharacteristic
51#13133137—24no—
52C22⋊#88322C2×C412yes—
53#13133129—22no—
54#13133129—22no—
55C4×#88322—21no—
56C22×Q8×C53—21no—
57C2×M16×C53—21no—
58C2×M16×C53D411yescharacteristic
59C2×M16×C53D411yescharacteristic
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53C2×C412yes—
66C2×C42×C53C2311yescharacteristic
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69#97×C53—21no—
70C4×C8×C53—21no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C2×Q8×C53—81no—
74C2×Q8×C53—81no—
75C22×C4×C53—22no—
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C2×C22⋊D53—22no—
80#11719945—41no—
81#11719945—41no—
82#11719945—41no—
83#11719945—41no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212C22⋊C412yes—
87C2×C4×D53—22no—
88C2×Dic106—21no—
89M16×C53—21no—
90M16×C53—21no—
91M16×C53—21no—
92M16×C53—41no—
93M16×C53—41no—
94M16×C53SD1612yes—
95C2×C8×C53—21no—
96C2×C8×C53—21no—
97C2×C8×C53—41no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53#4111yescharacteristic
106#11719921—22no—
107#11719921—22no—
108C4⋊Dic53—21no—
109C4⋊Dic53C22⋊C412yes—
110C2×#88322—22no—
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322—41no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—81no—
117#11719912—44no—
118C8×D53—44no—
119C23×C53C4×D411yescharacteristic
120C22×D53—22no—
121Q8×C53—82no—
122Q8×C53—81no—
123Q8×C53—81no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×C4×C53—82no—
129C2×C4×C53C2×C22⋊C411yescharacteristic
130C2×C4×C53C4×D411yescharacteristic
131C2×C4×C53#12011yescharacteristic
132C2×C4×C53#12111yescharacteristic
133C2×C4×C53#12111yescharacteristic
134C2×C4×C53#12311yescharacteristic
135C2×C4×C53C2×SD1611yescharacteristic
136C2×C4×C53#13511yescharacteristic
137C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C53#13711yescharacteristic
139C22⋊D53—42no—
140C22⋊D53—42no—
141C2×Dic53—22no—
142D212—42no—
143D212—41no—
144D212—41no—
145D212—41no—
146D212—41no—
147C4×D53—42no—
148C4×D53—42no—
149Dic106—41no—
150Dic106—41no—
151C8×C53—22no—
152C8×C53—41no—
153C8×C53—81no—
154#88322—42no—
155#88322—41no—
156#88322—41no—
157#88322—81no—
158#9269—531nomaximal, Sylow
159C22×C53—21no—
160C22×C53—21no—
161C22×C53#38611yescharacteristic
162C22×C53C4×SD1611yescharacteristic
163C22×C53#44011yescharacteristic
164C2×D53—42no—
165C2×D53—42no—
166C2×D53—42no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—82no—
171C4×C53#45011yescharacteristic
172C4×C53#45211yescharacteristic
173C4×C53#46111yescharacteristic
174C4×C53#46311yescharacteristic
175C4×C53#47111yescharacteristic
176C4×C53#47211yescharacteristic
177Dic53—42no—
178#3227—531no—
179#2935—531no—
180#1997D5311yescharacteristic, complemented
181#1547—531no—
182C2×#332—531no—
183#1301—531no—
184#1283—531no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53#187711yescharacteristic
188C2×C53#190711yescharacteristic
189C2×C53#191511yescharacteristic
190D53—82no—
191C2×#131—1061no—
192#532—531no—
193C2×C4⋊Q8C2×D5311yescharacteristic
194#498—1061no—
195#486—1061no—
196#481—534no—
197#476—1062no—
198#436—1061no—
199#435—1062no—
200C8×D4—1061no—
201#431—531no—
202#423C2×D5311yescharacteristic
203C2×C4⋊C8—531no—
204C2×C4⋊C8—531no—
205#408—1061no—
206C4×M16C2×D5311yescharacteristic
207#343—21no—
208#343—21no—
209#332—1061no—
210#332—1061no—
211#332—1061no—
212#331—1061no—
213C53#926911yescharacteristic, complemented, Sylow
214C2×#41—1061no—
215C22×Q8—21no—
216#131—2122no—
217#131—2122no—
218C2×M16—21no—
219C2×M16—1062no—
220C2×M16C22⋊D5311yescharacteristic
221C2×M16C22⋊D5311yescharacteristic
222C22×C8—1062no—
223C22×C8—1061no—
224C22×C8—1061no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—531no—
230#127—531no—
231#124—1061no—
232#121—532no—
233#121—1061no—
234C4×D4—532no—
235C4×D4—1061no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C4×D5312yes—
238C2×C42C22×D5311yescharacteristic
239#106—1064no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—531no—
243C4⋊C8—531no—
244C4⋊C8—1061no—
245C4⋊C8—1061no—
246C4⋊C8—1061no—
247C4⋊C8—1061no—
248#102—1064no—
249C22⋊C8—532no—
250C22⋊C8—1062no—
251C22⋊C8—1062no—
252#97—21no—
253C4×C8—21no—
254C4×C8—1061no—
255#41—2121no—
256#41—2121no—
257#41—2121no—
258#41—2121no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—81no—
262C2×Q8—81no—
263C2×Q8—1061no—
264C2×D4—532no—
265C2×D4—1062no—
266C2×D4—1061no—
267C2×D4—1061no—
268C22×C4—22no—
269C22×C4—1062no—
270C22×C4C2×C4×D5311yescharacteristic
271C22×C4#1171994511yescharacteristic
272C22×C4C2×C22⋊D5311yescharacteristic
273M16—21no—
274M16—21no—
275M16—21no—
276M16—41no—
277M16—41no—
278M16—2124no—
279M16#1171992412yes—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—1062no—
284C2×C8—1061no—
285C2×C8—1061no—
286C2×C8—4241no—
287C2×C8—2124no—
288C2×C8—2121no—
289C2×C8—2121no—
290C2×C8—2121no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—532no—
296C4⋊C4—1061no—
297C22⋊C4—1062no—
298C22⋊C4—1062no—
299C42D4×D5311yescharacteristic
300C42D4×D5311yescharacteristic
301C42D4×D5311yescharacteristic
302C42#1171994711yescharacteristic
303C23—1062no—
304C23C4×C22⋊D5311yescharacteristic
305Q8—82no—
306Q8—81no—
307Q8—81no—
308Q8—2121no—
309Q8—2121no—
310D4—2122no—
311D4—2122no—
312D4—2122no—
313D4—2121no—
314D4—2121no—
315D4—2121no—
316D4—2121no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—82no—
322C2×C4—1062no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4C22⋊C4×D5311yescharacteristic
326C2×C4#1313317711yescharacteristic
327C2×C4#1313318511yescharacteristic
328C2×C4#1313318811yescharacteristic
329C2×C4#1313319011yescharacteristic
330C2×C4#1313319911yescharacteristic
331C2×C4C2×#1171992411yescharacteristic
332C2×C4#1313323711yescharacteristic
333C2×C4#1313324211yescharacteristic
334C2×C4#1313324811yescharacteristic
335C8—22no—
336C8—41no—
337C8—81no—
338C8—4241no—
339C8—2122no—
340C8—2121no—
341C8—2121no—
342C22—21no—
343C22—21no—
344C22—2122no—
345C22—2122no—
346C22—2122no—
347C22#1681049911yescharacteristic
348C22C4×#1171992411yescharacteristic
349C22#1681072111yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—82no—
354C4—2122no—
355C4#1681077311yescharacteristic
356C4#1681078311yescharacteristic
357C4#1681082511yescharacteristic
358C4#1681083511yescharacteristic
359C4#1681088511yescharacteristic
360C4#1681089211yescharacteristic
361C2—21no—
362C2—21no—
363C2—4242no—
364C2#2027505211yescharacteristic
365C2#2027522011yescharacteristic
366C2#2027526411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1997C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#1997D53
hypocenterC53#9269
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
4. C2×#16810369C2semidirect
5. #20272610C2semidirect
6. #20272566C2semidirect
7. C53×#1997C2semidirect
180. #1997D53semidirect
213. C53#9269semidirect

Maximal subgroups

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

subgroupindex
#202858432
#202834612
#202733932
C2×#168103692
#202726102
#202725662
C53×#19972
#926953