Finite Group Databasebeta

← #27278638

Subgroups — #27278638

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133196C211yesmaximal, characteristic
2C4×C22⋊#88322C211yesmaximal, characteristic
3C2×#16810353C212yesmaximal, complemented
4#20272554C212yesmaximal, complemented
5C2×C53×#423C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic
7C2×C4×Dic106C2211yescharacteristic
8C2×C22⋊#88322—21no—
9C2×C4×#88322C2212yes—
10#16810529C411yescharacteristic
11#16810521C411yescharacteristic
12#16810507—21no—
13#16810496—21no—
14#16810483C411yescharacteristic
15C4×#11719919—21no—
16C4×C4⋊Dic53C2211yescharacteristic
17C4×C22⋊Dic53—21no—
18#16810378—21no—
19#16810353—24no—
20C22×M16×C53—21no—
21C22×C42×C53C2211yescharacteristic
22C53×#423—24no—
23C2×C4⋊C8×C53—21no—
24C2×C4⋊C8×C53C2212yes—
25C22×Dic106C2×C411yescharacteristic
26C22×#88322—22no—
27#13133196—22no—
28#13133196—22no—
29C4×Dic106—22no—
30C4×Dic106—22no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41C22⋊#88322—24no—
42C4×#88322—22no—
43C4×#88322—22no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C23×C4×C53C2×C411yescharacteristic
50C23×C4×C53D411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C2×M16×C53—24no—
53C2×M16×C53—42no—
54C2×M16×C53—42no—
55C22×C8×C53—22no—
56C2×C42×C53—22no—
57C2×C42×C53—22no—
58C2×C42×C53C2×C411yescharacteristic
59C2×C42×C53D411yescharacteristic
60C2×C42×C53D411yescharacteristic
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—24no—
63C4⋊C8×C53—22no—
64C4⋊C8×C53D414yes—
65C24×C53C22⋊C411yescharacteristic
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—22no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—22no—
76C22×C4×C53—41no—
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53SD1612yes—
81C22×C4×C53Q1612yes—
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C2×Dic106—22no—
90C2×Dic106—22no—
91C2×Dic106—21no—
92C2×Dic106M1612yescomplemented
93M16×C53—48no—
94C2×C8×C53—24no—
95C2×C8×C53—42no—
96C2×C8×C53—42no—
97C42×C53—22no—
98C42×C53—21no—
99C42×C53—22no—
100C42×C53—21no—
101C42×C53C2×D412yes—
102C42×C53#4112yes—
103C22⋊Dic53—24no—
104C22⋊Dic53—42no—
105C4⋊Dic53—21no—
106C4⋊Dic53—22no—
107C4⋊Dic53—22no—
108C4⋊Dic53M1612yescomplemented
109#11719919—24no—
110#11719919—42no—
111#11719919—42no—
112C4×Dic53—41no—
113C4×Dic53—41no—
114C4×Dic53—41no—
115C4×Dic53—41no—
116C2×#88322—24no—
117C2×#88322—42no—
118C2×#88322—42no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53#10312yes—
126C23×C53C2×C22⋊C411yescharacteristic
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—44no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53#10312yes—
144C2×C4×C53#10412yes—
145C2×C4×C53C4×D412yes—
146C2×C4×C53#12011yescharacteristic
147C2×C4×C53#12112yes—
148C2×C4×C53#12311yescharacteristic
149C2×C4×C53C2×M1611yescharacteristic
150C2×C4×C53C2×M1611yescharacteristic
151C2×C4×C53#13112yes—
152C2×C4×C53C2×SD1611yescharacteristic
153C2×C4×C53C2×Q1611yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C53#13711yescharacteristic
156C2×Dic53—22no—
157C2×Dic53—22no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168Dic106—24no—
169Dic106—42no—
170C8×C53—44no—
171#88322—44no—
172#8942—531nomaximal, Sylow
173C22×C53—22no—
174C22×C53—22no—
175C22×C53—22no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53#38611yescharacteristic
181C22×C53#40711yescharacteristic
182C22×C53#40811yescharacteristic
183C22×C53C2×#10311yescharacteristic
184C22×C53#41711yescharacteristic
185C22×C53C2×#10411yescharacteristic
186C22×C53#42111yescharacteristic
187C4×C53—22no—
188C4×C53—22no—
189C4×C53—22no—
190C4×C53—41no—
191C4×C53#43512yes—
192C4×C53#43612yes—
193C4×C53#45011yescharacteristic
194C4×C53#45111yescharacteristic
195C4×C53#48311yescharacteristic
196C4×C53#48411yescharacteristic
197Dic53—42no—
198Dic53—42no—
199Dic53—81no—
200Dic53—81no—
201C2×#423D5311yescharacteristic
202C4×C22⋊Q8—531no—
203C4×C22⋊C8—531no—
204C2×Q8⋊C8—532no—
205#1279—532no—
206C2×C53—42no—
207C2×C53#148711yescharacteristic
208C2×C53#148911yescharacteristic
209C2×C53#149311yescharacteristic
210C2×C53#149511yescharacteristic
211C2×C53#187011yescharacteristic
212C2×C53#187111yescharacteristic
213C2×C53#187211yescharacteristic
214C22×M16—21no—
215C2×C22⋊Q8—531no—
216C2×C4×Q8—531no—
217C22×C42C2×D5311yescharacteristic
218#423—24no—
219C2×C4⋊C8—21no—
220C2×C4⋊C8C2×D5312yes—
221C2×C22⋊C8—1061no—
222C2×C4×C8—532no—
223#391—531no—
224#389—531no—
225#387—1061no—
226#385—1061no—
227#380—531no—
228C4×C4⋊C4—531no—
229C4×C4⋊C4—1061no—
230C4×C22⋊C4—1061no—
231#336—1061no—
232Q8⋊C8—1064no—
233C53#894211yescharacteristic, complemented, Sylow
234C22×Q8—531no—
235C23×C4C4×D5311yescharacteristic
236C23×C4D21211yescharacteristic
237C23×C4C22⋊D5311yescharacteristic
238C2×M16—24no—
239C2×M16—42no—
240C2×M16—42no—
241C22×C8—22no—
242C22×C8—1062no—
243C22⋊Q8—1062no—
244C22⋊Q8—1062no—
245C4×Q8—1062no—
246C4×Q8—1062no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4—531no—
249C2×C4⋊C4—531no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1061no—
253C2×C22⋊C4—1061no—
254C2×C22⋊C4—1061no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—1061no—
258C2×C42—1061no—
259C2×C42C4×D5311yescharacteristic
260C2×C42D21211yescharacteristic
261C2×C42C22⋊D5311yescharacteristic
262C2×C42C22×D5311yescharacteristic
263C4⋊C8—24no—
264C4⋊C8—22no—
265C4⋊C8C22⋊D5314yes—
266C22⋊C8—1064no—
267C4×C8—1062no—
268C4×C8—1062no—
269#95—1061no—
270#95—1061no—
271#95—1061no—
272#95—1061no—
273#95—1061no—
274C24#1171992111yescharacteristic
275C2×Q8—532no—
276C2×Q8—1061no—
277C2×Q8—1062no—
278C2×Q8—1062no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—41no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4#1171992111yescharacteristic
295C22×C4#1171992111yescharacteristic
296C22×C4#1171992111yescharacteristic
297C22×C4#1171992312yes—
298C22×C4#1171992512yes—
299C22×C4C2×C4×D5311yescharacteristic
300C22×C4C2×D21211yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302M16—48no—
303C2×C8—24no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—1064no—
307C2×C8—2122no—
308C2×C8—2122no—
309C4⋊C4—532no—
310C4⋊C4—1064no—
311C4⋊C4—1061no—
312C4⋊C4—1062no—
313C4⋊C4—1062no—
314C4⋊C4—2122no—
315C4⋊C4—2122no—
316C22⋊C4—1064no—
317C22⋊C4—2122no—
318C42—22no—
319C42—21no—
320C42—22no—
321C42—21no—
322C42—2121no—
323C42—2121no—
324C42—2121no—
325C42—2121no—
326C42#1171994512yes—
327C42C2×C22⋊D5312yes—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23#1313314012yes—
335C23C2×#1171992111yescharacteristic
336Q8—1064no—
337Q8—2122no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—1062no—
355C2×C4—1062no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4#1313314012yes—
367C2×C4#1313314312yes—
368C2×C4C4×C22⋊D5312yes—
369C2×C4#1313318611yescharacteristic
370C2×C4#1313319112yes—
371C2×C4#1313320011yescharacteristic
372C2×C4M16×D5311yescharacteristic
373C2×C4M16×D5311yescharacteristic
374C2×C4#1313322512yes—
375C2×C4C2×#1171992311yescharacteristic
376C2×C4C2×#1171992511yescharacteristic
377C2×C4#1313324311yescharacteristic
378C2×C4#1313324811yescharacteristic
379C8—44no—
380C8—2124no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22#1681050211yescharacteristic
389C22#1681058611yescharacteristic
390C22#1681059111yescharacteristic
391C22C2×#1313314011yescharacteristic
392C22#1681062211yescharacteristic
393C22C2×#1313314311yescharacteristic
394C22#1681063911yescharacteristic
395C4—22no—
396C4—22no—
397C4—22no—
398C4—41no—
399C4—4241no—
400C4—4241no—
401C4—2122no—
402C4—2122no—
403C4#1681069212yes—
404C4#1681069912yes—
405C4#1681077411yescharacteristic
406C4#1681077911yescharacteristic
407C4#1681095311yescharacteristic
408C4#1681095811yescharacteristic
409C2—42no—
410C2#2027314311yescharacteristic
411C2#2027315211yescharacteristic
412C2#2027316911yescharacteristic
413C2#2027317811yescharacteristic
414C2#2027501211yescharacteristic
415C2#2027501711yescharacteristic
416C2#2027502211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#423C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC2×#423D53
hypocenterC53#8942
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
3. C2×#16810353C2semidirect
4. #20272554C2semidirect
92. C2×Dic106M16semidirect
108. C4⋊Dic53M16semidirect
233. C53#8942semidirect

Maximal subgroups

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

subgroupindex
C4×#131331962
C4×C22⋊#883222
C2×#168103532
#202725542
C2×C53×#4232
#894253