Finite Group Databasebeta

← #27280592

Subgroups — #27280592

Order 13568.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285640C211yesmaximal, characteristic
2#20283458C211yesmaximal, characteristic
3#20275705C211yesmaximal, characteristic, complemented
4#20273390C211yesmaximal, characteristic
5#20272612C211yesmaximal, characteristic, complemented
6#20272565C211yesmaximal, characteristic, complemented
7C2×C53×#333C211yesmaximal, characteristic, complemented
8C2×#13133223—21no—
9#16811173C2211yescharacteristic
10#16810997—21no—
11#16810973—21no—
12#16810825—22no—
13#16810762C414yes—
14#16810700—21no—
15#16810693—22no—
16#16810688—21no—
17#16810676C2211yescharacteristic
18#16810647C2211yescharacteristic
19#16810589—21no—
20C4×#11719917C2211yescharacteristic
21#16810392—21no—
22#16810392—21no—
23#16810368—21no—
24#16810366—21no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C53×#333—21no—
29C53×#333—21no—
30C2×#11719945—21no—
31C2×#11719924—24no—
32#13133223—42no—
33#13133223—42no—
34C2×#11719917—21no—
35C2×#11719917D411yescharacteristic
36C2×#11719917D411yescharacteristic
37C2×#11719912—22no—
38C2×C8×D53—22no—
39#13133217D411yescharacteristic
40#13133214D411yescharacteristic
41#13133205—21no—
42#13133191—21no—
43#13133191C2×C412yes—
44C4×D212—21no—
45C4×D212D412yes—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49#13133141—24no—
50C22⋊#88322C2×C412yes—
51#13133129—22no—
52#13133129—22no—
53#13133128—21no—
54C4×#88322—21no—
55C4×#88322—21no—
56C22×Q8×C53C2×C412yes—
57C22×C8×C53—21no—
58C22×C8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C42×C53C2311yescharacteristic
66C4⋊C8×C53—21no—
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69C4⋊C8×C53D411yescharacteristic
70C4⋊C8×C53D411yescharacteristic
71C2×Q8×C53—22no—
72C2×Q8×C53—22no—
73C2×Q8×C53—82no—
74C22×C4×C53—22no—
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53#4111yescharacteristic
78C2×C22⋊D53—22no—
79#11719945—41no—
80#11719945—41no—
81#11719945—41no—
82#11719945—41no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212C22⋊C412yes—
86C2×C4×D53—22no—
87C2×Dic106—21no—
88C2×C8×C53—21no—
89C2×C8×C53—21no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53—81no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C42×C53C2×D411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53#4111yescharacteristic
102#11719924—84no—
103#11719921—22no—
104#11719921—22no—
105C4⋊Dic53—21no—
106C4⋊Dic53C22⋊C412yes—
107#11719917—21no—
108#11719917—21no—
109#11719917—21no—
110#11719917—41no—
111#11719917—41no—
112#11719917SD1612yescomplemented
113C2×#88322—21no—
114C2×#88322—22no—
115C2×#88322—21no—
116C2×#88322—41no—
117#11719912—44no—
118C8×D53—44no—
119C23×C53C4×D411yescharacteristic
120C22×D53—22no—
121Q8×C53—44no—
122Q8×C53—82no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—42no—
127C2×C4×C53—82no—
128C2×C4×C53C2×C22⋊C411yescharacteristic
129C2×C4×C53C4×D411yescharacteristic
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53#12111yescharacteristic
132C2×C4×C53#12111yescharacteristic
133C2×C4×C53#12311yescharacteristic
134C2×C4×C53C2×SD1611yescharacteristic
135C2×C4×C53#13511yescharacteristic
136C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C53#13711yescharacteristic
138C22⋊D53—42no—
139C22⋊D53—42no—
140C2×Dic53—22no—
141D212—42no—
142D212—41no—
143D212—41no—
144D212—41no—
145D212—41no—
146C4×D53—42no—
147C4×D53—42no—
148Dic106—41no—
149Dic106—41no—
150C8×C53—41no—
151C8×C53—41no—
152C8×C53—81no—
153#88322—22no—
154#88322—42no—
155#88322—41no—
156#88322—81no—
157#9267—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53#38611yescharacteristic
161C22×C53C4×SD1611yescharacteristic
162C22×C53#44011yescharacteristic
163C2×D53—42no—
164C2×D53—42no—
165C2×D53—42no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—82no—
170C4×C53#45011yescharacteristic
171C4×C53#45211yescharacteristic
172C4×C53#46111yescharacteristic
173C4×C53#46311yescharacteristic
174C4×C53#47111yescharacteristic
175C4×C53#47311yescharacteristic
176Dic53—42no—
177#3206—531no—
178#2935—531no—
179#1997—531no—
180#1547—531no—
181C2×#333D5311yescharacteristic, complemented
182#1301—531no—
183#1283—531no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53#187711yescharacteristic
187C2×C53#190811yescharacteristic
188C2×C53#191411yescharacteristic
189D53—82no—
190C2×#131—1061no—
191#532—531no—
192C2×C4⋊Q8C2×D5311yescharacteristic
193#492—1061no—
194#487—1061no—
195#461—1062no—
196#448—534no—
197#436—1061no—
198#435—1062no—
199C8×D4—1061no—
200#431—531no—
201#423—531no—
202C2×C4⋊C8C2×D5311yescharacteristic
203C2×C4⋊C8C2×D5311yescharacteristic
204#408—1061no—
205C4×M16—531no—
206#343—1061no—
207#343—1061no—
208#333—21no—
209#333—21no—
210#332—1061no—
211#331—1061no—
212C53#926711yescharacteristic, complemented, Sylow
213C2×#41—1061no—
214C22×Q8C4×D5312yes—
215C2×SD16—1064no—
216#131—2122no—
217#131—2122no—
218C2×M16—531no—
219C2×M16—531no—
220C2×M16—1062no—
221C2×M16—1061no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—1062no—
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⋊C4—21no—
238C2×C42C22×D5311yescharacteristic
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—1061no—
243C4⋊C8—1061no—
244C4⋊C8—1061no—
245C4⋊C8C22⋊D5311yescharacteristic
246C4⋊C8C22⋊D5311yescharacteristic
247#103—1064no—
248C22⋊C8—532no—
249C22⋊C8—1062no—
250C22⋊C8—1062no—
251#97—1061no—
252C4×C8—1061no—
253C4×C8—1061no—
254#41—2121no—
255#41—2121no—
256#41—2121no—
257#41—2121no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—82no—
261C2×Q8—1061no—
262C2×D4—532no—
263C2×D4—1062no—
264C2×D4—1061no—
265C2×D4—1061no—
266C22×C4—22no—
267C22×C4—1062no—
268C22×C4C2×C4×D5311yescharacteristic
269C22×C4D4×D5311yescharacteristic
270C22×C4#1171994711yescharacteristic
271SD16—4244no—
272M16—532no—
273M16—1061no—
274M16—1061no—
275M16—1061no—
276M16—2124no—
277M16—2121no—
278M16—2121no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—81no—
285C2×C8—1061no—
286C2×C8—1062no—
287C2×C8—1061no—
288C2×C8—2124no—
289C2×C8—2121no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—532no—
295C4⋊C4—1061no—
296C22⋊C4—1062no—
297C22⋊C4—1062no—
298C42#1171994511yescharacteristic
299C42D4×D5311yescharacteristic
300C42D4×D5311yescharacteristic
301C42C2×C22⋊D5311yescharacteristic
302C23—1062no—
303C23#1313317711yescharacteristic
304Q8—44no—
305Q8—82no—
306Q8—2121no—
307Q8—2121no—
308D4—2122no—
309D4—2122no—
310D4—2122no—
311D4—2121no—
312D4—2121no—
313D4—2121no—
314D4—2121no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—1062no—
321C2×C4—2122no—
322C2×C4—2122no—
323C2×C4C22⋊C4×D5311yescharacteristic
324C2×C4C4×C22⋊D5311yescharacteristic
325C2×C4#1313318511yescharacteristic
326C2×C4#1313318811yescharacteristic
327C2×C4#1313319011yescharacteristic
328C2×C4#1313319911yescharacteristic
329C2×C4SD16×D5311yescharacteristic
330C2×C4#1313324011yescharacteristic
331C2×C4#1313324411yescharacteristic
332C2×C4#1313324911yescharacteristic
333C8—41no—
334C8—41no—
335C8—81no—
336C8—1062no—
337C8—4241no—
338C8—2122no—
339C8—2121no—
340C22—21no—
341C22—21no—
342C22—2122no—
343C22—2122no—
344C22—2122no—
345C22#1681049911yescharacteristic
346C22#1681070811yescharacteristic
347C22#1681072611yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—82no—
352C4—2122no—
353C4#1681077211yescharacteristic
354C4#1681078411yescharacteristic
355C4#1681082711yescharacteristic
356C4#1681084111yescharacteristic
357C4#1681088311yescharacteristic
358C4#1681089511yescharacteristic
359C2—21no—
360C2—21no—
361C2—4242no—
362C2#2027504611yescharacteristic
363C2#2027523311yescharacteristic
364C2#2027525911yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#333C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenterC2×#333D53
hypocenterC53#9267
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
3. #20275705C2semidirect
5. #20272612C2semidirect
6. #20272565C2semidirect
7. C2×C53×#333C2semidirect
112. #11719917SD16semidirect
181. C2×#333D53semidirect
212. C53#9267semidirect

Maximal subgroups

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

subgroupindex
#202856402
#202834582
#202757052
#202733902
#202726122
#202725652
C2×C53×#3332
#926753