Finite Group Databasebeta

← #27280779

Subgroups — #27280779

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#20283433C211yesmaximal, characteristic, complemented
2#20275540C211yesmaximal, characteristic, complemented
3#20273383C211yesmaximal, characteristic, complemented
4#20273275C211yesmaximal, characteristic, complemented
5#20272574C211yesmaximal, characteristic, complemented
6#20272545C211yesmaximal, characteristic, complemented
7C53×#3205C211yesmaximal, characteristic
8C2×#13133224—21no—
9#16810695—22no—
10#16810695—22no—
11#16810676C2211yescharacteristic
12#16810647C2211yescharacteristic
13C2×#13133138—21no—
14C2×#13133138—21no—
15#16810590—21no—
16C2×#13133128C2211yescharacteristic
17#16810489C2211yescharacteristic
18#16810367—22no—
19#16810357—22no—
20C53×#532C2211yescharacteristic
21C2×#127×C53C2211yescharacteristic
22C53×#493C412yescomplemented
23C53×#486C412yescomplemented
24C53×#459—22no—
25C53×#447—22no—
26C53×#431C2211yescharacteristic
27#13133224—42no—
28#13133224—42no—
29C2×#11719917—22no—
30C2×#11719917—21no—
31C22×#88322—22no—
32C22×#88322D411yescharacteristic
33C22×#88322D411yescharacteristic
34C2×C4⋊Dic53—21no—
35C4⋊#88322—21no—
36C4⋊#88322—21no—
37C4⋊#88322D412yes—
38#13133138—42no—
39#13133138—42no—
40C22⋊#88322—24no—
41C22⋊#88322—21no—
42#13133128—21no—
43#13133128—21no—
44C4×#88322—21no—
45#13133124—21no—
46#13133124—21no—
47C2×#41×C53—21no—
48C22×D4×C53—21no—
49C22×D4×C53—21no—
50C22×D4×C53—21no—
51C2×D8×C53—24no—
52C4⋊Q8×C53C2×C411yescharacteristic
53#127×C53—21no—
54#127×C53—21no—
55#127×C53—21no—
56#127×C53C2×C412yes—
57#127×C53C2×C411yescharacteristic
58#124×C53—21no—
59#121×C53—21no—
60#121×C53D412yes—
61C4×D4×C53—21no—
62C4×D4×C53D412yes—
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53—21no—
65#102×C53—24no—
66C22⋊C8×C53—21no—
67C4×C8×C53C2×C412yes—
68C24×C53—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—22no—
76C2×D4×C53—21no—
77C2×D4×C53—22no—
78C2×D4×C53—21no—
79C2×D4×C53—22no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—82no—
88C2×D4×C53—82no—
89C2×D4×C53C22⋊C412yes—
90C22×C4×C53—22no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C22×Dic53—41no—
95D8×C53—84no—
96C2×C8×C53—22no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53C22⋊C412yes—
99C22⋊C4×C53—22no—
100C22⋊C4×C53—22no—
101C42×C53C22×C411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53#4111yescharacteristic
105C4⋊Dic53—41no—
106C4⋊Dic53—41no—
107#11719917—44no—
108#11719917—41no—
109#11719917—41no—
110C2×#88322—21no—
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322—22no—
114C2×#88322—21no—
115C2×#88322—21no—
116C2×#88322—44no—
117C2×#88322—41no—
118C2×#88322D812yescomplemented
119C23×C53—22no—
120C23×C53—42no—
121C23×C53—82no—
122C23×C53—82no—
123C23×C53—81no—
124C23×C53—81no—
125C23×C53#12111yescharacteristic
126Q8×C53—41no—
127Q8×C53—41no—
128D4×C53—42no—
129D4×C53—44no—
130D4×C53—42no—
131D4×C53—42no—
132D4×C53—41no—
133D4×C53—42no—
134D4×C53—41no—
135D4×C53—42no—
136D4×C53—81no—
137D4×C53—82no—
138D4×C53—82no—
139D4×C53—82no—
140D4×C53—81no—
141D4×C53—81no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53C2×C22⋊C411yescharacteristic
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12311yescharacteristic
153C2×C4×C53C2×D811yescharacteristic
154C2×C4×C53#13511yescharacteristic
155C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C2×Dic53—81no—
158C2×Dic53—81no—
159C2×Dic53—81no—
160C8×C53—42no—
161#88322—22no—
162#88322—42no—
163#88322—41no—
164#88322—81no—
165#9293—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—42no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53—82no—
172C22×C53—82no—
173C22×C53—82no—
174C22×C53—162no—
175C22×C53—162no—
176C22×C53#38611yescharacteristic
177C22×C53#45911yescharacteristic
178C22×C53#46311yescharacteristic
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#3205D5311yescharacteristic
190#2932—531no—
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—162no—
200C2×C53#187911yescharacteristic
201C2×C53#190311yescharacteristic
202C2×C53#191011yescharacteristic
203C2×#131—1061no—
204#532C2×D5311yescharacteristic
205C2×#127C2×D5311yescharacteristic
206#493Dic5312yescomplemented
207#486Dic5312yescomplemented
208#459—22no—
209#447—22no—
210#435—1062no—
211#435—1062no—
212#431—531no—
213#431C2×D5311yescharacteristic
214#423—531no—
215C2×#102—1061no—
216C2×#102—1061no—
217#408—1061no—
218C2×#97—531no—
219#382—531no—
220#331—1062no—
221#327—1062no—
222C53#929311yescharacteristic, complemented, Sylow
223C2×#41—21no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—21no—
227C2×D8—24no—
228#131—2122no—
229#131—2122no—
230C2×M16—1062no—
231C2×M16—1061no—
232C22×C8—531no—
233C22×C8—531no—
234C22×C8—1062no—
235C4⋊Q8C2×Dic5311yescharacteristic
236#127—21no—
237#127—21no—
238#127—21no—
239#127C2×Dic5312yes—
240#127C2×Dic5311yescharacteristic
241#124—21no—
242#121—21no—
243#121C22⋊D5312yes—
244C4×D4—21no—
245C4×D4C22⋊D5312yes—
246C2×C4⋊C4—1061no—
247C2×C42C22×D5311yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—532no—
250C4⋊C8—1061no—
251C4⋊C8—1061no—
252#102—24no—
253#102—2122no—
254#102—2122no—
255C22⋊C8—21no—
256C22⋊C8—1064no—
257C22⋊C8—1061no—
258#97—1061no—
259#97—1061no—
260C4×C8—1061no—
261C4×C8C2×Dic5312yes—
262#95—1061no—
263#95—1061no—
264C24—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—21no—
275C2×D4—22no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4C22⋊Dic5312yes—
286C22×C4—22no—
287C22×C4—2121no—
288C22×C4D4×D5311yescharacteristic
289C22×C4#1171994711yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291D8—84no—
292M16—2124no—
293M16—2121no—
294M16—2121no—
295C2×C8—22no—
296C2×C8—532no—
297C2×C8—1061no—
298C2×C8—1061no—
299C2×C8—1061no—
300C2×C8—1062no—
301C2×C8—1061no—
302C2×C8—1061no—
303C2×C8—2124no—
304C2×C8—2121no—
305C4⋊C4—21no—
306C4⋊C4—2121no—
307C4⋊C4—2121no—
308C4⋊C4C22⋊Dic5312yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C42D4×D5311yescharacteristic
312C42#1171994711yescharacteristic
313C42C22×Dic5311yescharacteristic
314C42C2×C22⋊D5311yescharacteristic
315C23—22no—
316C23—42no—
317C23—82no—
318C23—82no—
319C23—81no—
320C23—81no—
321C23#1313318911yescharacteristic
322Q8—41no—
323Q8—41no—
324D4—42no—
325D4—44no—
326D4—42no—
327D4—42no—
328D4—41no—
329D4—42no—
330D4—41no—
331D4—42no—
332D4—81no—
333D4—82no—
334D4—82no—
335D4—82no—
336D4—81no—
337D4—81no—
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×C22⋊Dic5311yescharacteristic
347C2×C4D4×Dic5311yescharacteristic
348C2×C4D4×Dic5311yescharacteristic
349C2×C4#1313318511yescharacteristic
350C2×C4#1313318911yescharacteristic
351C2×C4#1313319811yescharacteristic
352C2×C4D8×D5311yescharacteristic
353C2×C4#1313323611yescharacteristic
354C2×C4#1313324111yescharacteristic
355C2×C4#1313324111yescharacteristic
356C8—42no—
357C8—1062no—
358C8—4241no—
359C8—2122no—
360C8—2121no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—162no—
370C22—162no—
371C22#1681050311yescharacteristic
372C22#1681081611yescharacteristic
373C22#1681083611yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—8481no—
380C4D8×Dic5311yescharacteristic
381C4#1681073511yescharacteristic
382C4#1681075511yescharacteristic
383C4#1681076511yescharacteristic
384C2—21no—
385C2—21no—
386C2—82no—
387C2—162no—
388C2#2027506511yescharacteristic
389C2#2027519711yescharacteristic
390C2#2027524311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3205C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3205D53
hypocenterC53#9293
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #20283433C2semidirect
2. #20275540C2semidirect
3. #20273383C2semidirect
4. #20273275C2semidirect
5. #20272574C2semidirect
6. #20272545C2semidirect
22. C53×#493C4semidirect
23. C53×#486C4semidirect
118. C2×#88322D8semidirect
206. #493Dic53semidirect
207. #486Dic53semidirect
222. C53#9293semidirect

Maximal subgroups

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

subgroupindex
#202834332
#202755402
#202733832
#202732752
#202725742
#202725452
C53×#32052
#929353