Finite Group Databasebeta

← #27280777

Subgroups — #27280777

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285630C211yesmaximal, characteristic
2#20275539C211yesmaximal, characteristic, complemented
3#20273384C211yesmaximal, characteristic, complemented
4#20273275C211yesmaximal, characteristic, complemented
5#20272575C211yesmaximal, characteristic, complemented
6#20272546C211yesmaximal, characteristic, complemented
7C53×#2932C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9#16810999C412yescomplemented
10#16810971C412yescomplemented
11#16810818—22no—
12#16810756—22no—
13#16810676C2211yescharacteristic
14#16810676C2211yescharacteristic
15C2×#13133139—21no—
16C2×#13133139—21no—
17#16810489C2211yescharacteristic
18#16810366—22no—
19#16810357—22no—
20C2×#131×C53—21no—
21C53×#532C2211yescharacteristic
22C53×#435—22no—
23C53×#435—22no—
24C53×#423C2211yescharacteristic
25C53×#408—21no—
26C2×#97×C53C2211yescharacteristic
27C22×D212—21no—
28C22×D212—21no—
29C22×D212—21no—
30C2×#11719922—24no—
31#13133214—21no—
32#13133214—21no—
33#13133214—21no—
34#13133214C2×C412yes—
35C2×C4⋊Dic53—21no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38#13133139—42no—
39#13133139—42no—
40#13133137—24no—
41C22⋊#88322—21no—
42C22⋊#88322—21no—
43C4×#88322—21no—
44C4×#88322C2×C412yes—
45#13133124—21no—
46#13133124—21no—
47C2×#41×C53—21no—
48#131×C53—42no—
49#131×C53—42no—
50C2×M16×C53—22no—
51C2×M16×C53—21no—
52C22×C8×C53—22no—
53C22×C8×C53D411yescharacteristic
54C22×C8×C53D411yescharacteristic
55C4⋊Q8×C53C2×C411yescharacteristic
56#127×C53C2×C411yescharacteristic
57#124×C53—21no—
58#121×C53—21no—
59#121×C53D412yes—
60C4×D4×C53—21no—
61C4×D4×C53D412yes—
62C2×C42×C53C2311yescharacteristic
63C4⋊C8×C53—21no—
64C4⋊C8×C53D412yes—
65C22⋊C8×C53—24no—
66#97×C53—21no—
67#97×C53—21no—
68C23×D53—41no—
69#41×C53—41no—
70#41×C53—41no—
71#41×C53—41no—
72#41×C53—41no—
73C2×Q8×C53—21no—
74C2×D4×C53—22no—
75C2×D4×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53C22⋊C412yes—
78C22×C4×C53—22no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C22×Dic53—41no—
83C2×D212—22no—
84C2×D212—22no—
85C2×D212—22no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—82no—
94C2×D212—82no—
95M16×C53—44no—
96M16×C53—41no—
97M16×C53—41no—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—21no—
101C2×C8×C53—22no—
102C2×C8×C53—21no—
103C2×C8×C53—21no—
104C2×C8×C53—44no—
105C2×C8×C53D812yescomplemented
106C4⋊C4×C53—21no—
107C4⋊C4×C53C22⋊C412yes—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—22no—
110C42×C53C22×C411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53#4111yescharacteristic
114#11719922—84no—
115C4⋊Dic53—41no—
116C4⋊Dic53—41no—
117C2×#88322—22no—
118C2×#88322—41no—
119C23×C53—22no—
120C23×C53#12111yescharacteristic
121C22×D53—42no—
122C22×D53—82no—
123C22×D53—82no—
124C22×D53—81no—
125C22×D53—81no—
126Q8×C53—41no—
127Q8×C53—41no—
128D4×C53—42no—
129D4×C53—42no—
130D4×C53—42no—
131D4×C53—41no—
132D4×C53—41no—
133D4×C53—81no—
134C2×C4×C53—22no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53C2×C22⋊C411yescharacteristic
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53C4×D411yescharacteristic
142C2×C4×C53#12011yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12311yescharacteristic
145C2×C4×C53C2×D811yescharacteristic
146C2×C4×C53#13511yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×Dic53—81no—
150C2×Dic53—81no—
151C2×Dic53—81no—
152D212—44no—
153D212—42no—
154D212—42no—
155D212—82no—
156D212—82no—
157D212—82no—
158D212—81no—
159D212—81no—
160C8×C53—22no—
161C8×C53—42no—
162C8×C53—41no—
163#88322—42no—
164#88322—81no—
165#9293—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—42no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53#38611yescharacteristic
172C22×C53#45911yescharacteristic
173C22×C53#46311yescharacteristic
174C2×D53—82no—
175C2×D53—82no—
176C2×D53—82no—
177C2×D53—162no—
178C2×D53—162no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—42no—
184C4×C53C4×D811yescharacteristic
185C4×C53#44211yescharacteristic
186C4×C53#44711yescharacteristic
187C4×C53#44911yescharacteristic
188Dic53—161no—
189#3205—531no—
190#2932D5311yescharacteristic, complemented
191#1961—531no—
192#1545—531no—
193#1517—531no—
194#1286—531no—
195#1275—531no—
196C2×C53—21no—
197C2×C53—21no—
198C2×C53—82no—
199C2×C53#187911yescharacteristic
200C2×C53#190311yescharacteristic
201C2×C53#191011yescharacteristic
202D53—162no—
203C2×#131—21no—
204#532C2×D5311yescharacteristic
205C2×#127—531no—
206#493—532no—
207#486—532no—
208#459—1062no—
209#447—1062no—
210#435—22no—
211#435—22no—
212#431—531no—
213#431—531no—
214#423C2×D5311yescharacteristic
215C2×#102—1061no—
216C2×#102—1061no—
217#408—21no—
218C2×#97C2×D5311yescharacteristic
219#382—531no—
220#331—1062no—
221#327—1062no—
222C53#929311yescharacteristic, complemented, Sylow
223C2×#41—21no—
224C22×D4—1061no—
225C22×D4—1061no—
226C22×D4—1061no—
227C2×D8—1064no—
228#131—42no—
229#131—42no—
230C2×M16—22no—
231C2×M16—21no—
232C22×C8—22no—
233C22×C8C22⋊D5311yescharacteristic
234C22×C8C22⋊D5311yescharacteristic
235C4⋊Q8C4×D5311yescharacteristic
236#127—532no—
237#127—1061no—
238#127—1061no—
239#127—1061no—
240#127C4×D5311yescharacteristic
241#124—21no—
242#121—21no—
243#121D21212yes—
244C4×D4—21no—
245C4×D4C22⋊D5312yes—
246C2×C4⋊C4—1061no—
247C2×C42C22×D5311yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—1061no—
250C4⋊C8—1061no—
251C4⋊C8C22⋊D5312yes—
252#102—1064no—
253#102—2122no—
254#102—2122no—
255C22⋊C8—24no—
256C22⋊C8—1061no—
257C22⋊C8—1061no—
258#97—21no—
259#97—21no—
260C4×C8—532no—
261C4×C8—1061no—
262#95—1061no—
263#95—1061no—
264C24—2121no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×D4—22no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—1062no—
274C2×D4—1062no—
275C2×D4—1062no—
276C2×D4—4242no—
277C2×D4—4242no—
278C2×D4—2121no—
279C2×D4—2121no—
280C2×D4—2121no—
281C2×D4—2121no—
282C2×D4—2121no—
283C2×D4—2121no—
284C2×D4—2121no—
285C2×D4#1171992112yes—
286C22×C4—22no—
287C22×C4—2121no—
288C22×C4C2×D21211yescharacteristic
289C22×C4#1171994511yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291D8—4244no—
292M16—44no—
293M16—41no—
294M16—41no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—44no—
302C2×C8—1062no—
303C2×C8—2121no—
304C2×C8#1171992212yescomplemented
305C4⋊C4—21no—
306C4⋊C4—2121no—
307C4⋊C4—2121no—
308C4⋊C4#1171992112yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C42C2×C4×D5311yescharacteristic
312C42#1171994511yescharacteristic
313C42C2×C22⋊D5311yescharacteristic
314C42C2×C22⋊D5311yescharacteristic
315C23—22no—
316C23—4242no—
317C23—4242no—
318C23—4241no—
319C23—4241no—
320C23—2122no—
321C23#1313319111yescharacteristic
322Q8—41no—
323Q8—41no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—41no—
328D4—41no—
329D4—81no—
330D4—4242no—
331D4—4242no—
332D4—4242no—
333D4—4241no—
334D4—4241no—
335D4—2124no—
336D4—2122no—
337D4—2122no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—4241no—
344C2×C4—4241no—
345C2×C4—4241no—
346C2×C4C2×#1171992111yescharacteristic
347C2×C4C4×C22⋊D5311yescharacteristic
348C2×C4C4×C22⋊D5311yescharacteristic
349C2×C4#1313318611yescharacteristic
350C2×C4#1313319111yescharacteristic
351C2×C4#1313320011yescharacteristic
352C2×C4C2×#1171992211yescharacteristic
353C2×C4#1313323711yescharacteristic
354C2×C4#1313324211yescharacteristic
355C2×C4#1313324311yescharacteristic
356C8—22no—
357C8—42no—
358C8—41no—
359C8—4241no—
360C8—2122no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—8482no—
367C22—8482no—
368C22—4242no—
369C22—4242no—
370C22—4242no—
371C22#1681050211yescharacteristic
372C22#1681081811yescharacteristic
373C22#1681083811yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—8481no—
380C4C4×#1171992211yescharacteristic
381C4#1681073311yescharacteristic
382C4#1681075711yescharacteristic
383C4#1681076811yescharacteristic
384C2—21no—
385C2—21no—
386C2—82no—
387C2—8482no—
388C2#2027506311yescharacteristic
389C2#2027519511yescharacteristic
390C2#2027524211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2932C2
socleC22×C53#386
prime radical#20285630C2
cosocleC2×C42×C53C23
hypercenter#2932D53
hypocenterC53#9293
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2932

Complements

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

NH ≅ G/Ntype
2. #20275539C2semidirect
3. #20273384C2semidirect
4. #20273275C2semidirect
5. #20272575C2semidirect
6. #20272546C2semidirect
7. C53×#2932C2semidirect
9. #16810999C4semidirect
10. #16810971C4semidirect
105. C2×C8×C53D8semidirect
190. #2932D53semidirect
222. C53#9293semidirect
304. C2×C8#11719922semidirect

Maximal subgroups

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

subgroupindex
#202856302
#202755392
#202733842
#202732752
#202725752
#202725462
C53×#29322
#929353