Finite Group Databasebeta

← #27280833

Subgroups — #27280833

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#20275547C211yesmaximal, characteristic
3#20273396C211yesmaximal, characteristic, complemented
4#20273274C211yesmaximal, characteristic
5#20272573C211yesmaximal, characteristic, complemented
6#20272544C211yesmaximal, characteristic, complemented
7C53×#2331C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16810997C412yescomplemented
10#16810973C412yescomplemented
11#16810825—22no—
12#16810762—22no—
13#16810676C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810647C2211yescharacteristic
16C2×#13133145—21no—
17C2×#13133145—21no—
18#16810372—22no—
19#16810356—22no—
20C2×C4⋊Q8×C53C2211yescharacteristic
21C53×#391—22no—
22C53×#389—21no—
23C53×#387—22no—
24C53×#382C2211yescharacteristic
25C4×C4⋊C4×C53—21no—
26C53×#376C2211yescharacteristic
27C2×#11719945—21no—
28C2×#11719924—24no—
29C2×#11719917—21no—
30#13133217C2×C411yescharacteristic
31#13133214C2×C411yescharacteristic
32#13133205—21no—
33#13133191—21no—
34#13133191—21no—
35C4×D212—21no—
36C4×D212—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41#13133145—42no—
42#13133145—42no—
43#13133141—24no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46C4×#88322—21no—
47C4×#88322C2×C412yes—
48C22×Q8×C53D412yes—
49C4⋊Q8×C53—21no—
50C4⋊Q8×C53—21no—
51C4⋊Q8×C53C2×C412yes—
52C4⋊Q8×C53D412yes—
53C2×C4⋊C4×C53—22no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C42×C53—22no—
58C2×C42×C53D411yescharacteristic
59C2×C42×C53D411yescharacteristic
60C2×C42×C53C2311yescharacteristic
61#95×C53—22no—
62#95×C53—22no—
63#95×C53—21no—
64#95×C53—21no—
65#95×C53—21no—
66#95×C53D412yes—
67C2×Q8×C53—22no—
68C2×Q8×C53—82no—
69C2×Q8×C53C22⋊C414yes—
70C22×C4×C53—21no—
71C22×C4×C53—22no—
72C22×C4×C53—21no—
73C22×C4×C53—22no—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C2×C22⋊D53—41no—
78#11719945—41no—
79#11719945—41no—
80#11719945—41no—
81#11719945—41no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×C4×D53—41no—
86C2×Dic106—21no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—82no—
94C42×C53—21no—
95C42×C53—21no—
96C42×C53—21no—
97C42×C53—82no—
98C42×C53SD1612yescomplemented
99C42×C53C22×C411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53#4111yescharacteristic
102C42×C53#4111yescharacteristic
103#11719924—84no—
104#11719921—41no—
105#11719921—41no—
106C4⋊Dic53—21no—
107C4⋊Dic53—21no—
108#11719917—41no—
109#11719917—41no—
110C2×#88322—22no—
111C2×#88322—41no—
112C2×#88322—41no—
113C23×C53#12011yescharacteristic
114C22×D53—41no—
115Q8×C53—44no—
116Q8×C53—82no—
117C2×C4×C53—24no—
118C2×C4×C53—22no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—42no—
131C2×C4×C53C2×C22⋊C411yescharacteristic
132C2×C4×C53C4×D411yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#12311yescharacteristic
137C2×C4×C53C2×SD1611yescharacteristic
138C2×C4×C53#13511yescharacteristic
139C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C53#13711yescharacteristic
141C22⋊D53—81no—
142C22⋊D53—81no—
143C2×Dic53—41no—
144D212—42no—
145D212—41no—
146D212—41no—
147D212—81no—
148C4×D53—81no—
149C4×D53—81no—
150Dic106—41no—
151Dic106—41no—
152#88322—42no—
153#88322—81no—
154#88322—81no—
155#9301—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53#38611yescharacteristic
159C22×C53#44811yescharacteristic
160C22×C53#44911yescharacteristic
161C2×D53—81no—
162C2×D53—81no—
163C2×D53—81no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—42no—
169C4×C53—44no—
170C4×C53—82no—
171C4×C53—81no—
172C4×C53C4×SD1611yescharacteristic
173C4×C53#44011yescharacteristic
174C4×C53#46011yescharacteristic
175C4×C53#46411yescharacteristic
176Dic53—81no—
177#3206—531no—
178#2331D5311yescharacteristic, complemented
179#1963—531no—
180#1548—531no—
181#1517—531no—
182#1286—531no—
183#1274—531no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53#188011yescharacteristic
187C2×C53#190511yescharacteristic
188C2×C53#191111yescharacteristic
189D53—161no—
190#532—531no—
191C2×C4⋊Q8C2×D5311yescharacteristic
192#492—532no—
193#487—532no—
194#461—1062no—
195#448—1062no—
196#431—531no—
197#431—531no—
198#423—531no—
199C2×#104—1061no—
200C2×#104—1061no—
201#391—22no—
202#389—21no—
203#387—22no—
204#382C2×D5311yescharacteristic
205C4×C4⋊C4—21no—
206#376C2×D5311yescharacteristic
207#333—1062no—
208#327—1062no—
209C53#930111yescharacteristic, complemented, Sylow
210C2×#41—1061no—
211C22×Q8C22⋊D5312yes—
212C2×SD16—1064no—
213C2×M16—1061no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—531no—
217C4⋊Q8C4×D5312yes—
218C4⋊Q8D21212yes—
219#127—531no—
220#124—1061no—
221#121—1061no—
222#121—1061no—
223C4×D4—1061no—
224C4×D4—1061no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C42—22no—
230C2×C42C22⋊D5311yescharacteristic
231C2×C42C22⋊D5311yescharacteristic
232C2×C42C22×D5311yescharacteristic
233C4⋊C8—1061no—
234C4⋊C8—1061no—
235C4⋊C8—1061no—
236C4⋊C8—1061no—
237#104—2122no—
238#104—2122no—
239#103—1064no—
240C22⋊C8—1061no—
241C22⋊C8—1061no—
242C4×C8—532no—
243C4×C8—1061no—
244#95—22no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95C22⋊D5312yes—
250#41—2121no—
251#41—2121no—
252#41—2121no—
253#41—2121no—
254C2×Q8—22no—
255C2×Q8—82no—
256C2×Q8—1061no—
257C2×Q8#1171992114yes—
258C2×D4—1061no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—2121no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—2121no—
267C22×C4C2×C22⋊D5311yescharacteristic
268C22×C4C2×C22⋊D5311yescharacteristic
269C22×C4C2×C22⋊D5311yescharacteristic
270SD16—4244no—
271M16—2121no—
272M16—2121no—
273C2×C8—1062no—
274C2×C8—2121no—
275C2×C8—2121no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—82no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C22⋊C4—2121no—
286C22⋊C4—2121no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—82no—
291C42#1171992412yescomplemented
292C42C2×C4×D5311yescharacteristic
293C42C2×D21211yescharacteristic
294C42#1171994511yescharacteristic
295C42#1171994511yescharacteristic
296C23—2121no—
297C23#1313318611yescharacteristic
298Q8—44no—
299Q8—82no—
300Q8—2121no—
301Q8—2121no—
302D4—4241no—
303D4—4241no—
304D4—4241no—
305D4—2122no—
306D4—2121no—
307D4—2121no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—4241no—
323C2×C4—4241no—
324C2×C4—2121no—
325C2×C4C2×#1171992111yescharacteristic
326C2×C4C4×C22⋊D5311yescharacteristic
327C2×C4C4×C22⋊D5311yescharacteristic
328C2×C4#1313319111yescharacteristic
329C2×C4#1313319111yescharacteristic
330C2×C4#1313320011yescharacteristic
331C2×C4C2×#1171992411yescharacteristic
332C2×C4#1313323711yescharacteristic
333C2×C4#1313324211yescharacteristic
334C2×C4#1313324811yescharacteristic
335C8—4241no—
336C8—4241no—
337C8—2122no—
338C22—21no—
339C22—21no—
340C22—4241no—
341C22—4241no—
342C22—4241no—
343C22#1681050211yescharacteristic
344C22#1681076211yescharacteristic
345C22#1681076811yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—42no—
351C4—44no—
352C4—82no—
353C4—81no—
354C4—4241no—
355C4C4×#1171992411yescharacteristic
356C4#1681072111yescharacteristic
357C4#1681082311yescharacteristic
358C4#1681084511yescharacteristic
359C2—21no—
360C2—21no—
361C2—8481no—
362C2#2027507011yescharacteristic
363C2#2027520911yescharacteristic
364C2#2027524711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2331C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#2331D53
hypocenterC53#9301
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #20273396C2semidirect
5. #20272573C2semidirect
6. #20272544C2semidirect
7. C53×#2331C2semidirect
9. #16810997C4semidirect
10. #16810973C4semidirect
98. C42×C53SD16semidirect
178. #2331D53semidirect
209. C53#9301semidirect
291. C42#11719924semidirect

Maximal subgroups

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

subgroupindex
#202856402
#202755472
#202733962
#202732742
#202725732
#202725442
C53×#23312
#930153