Finite Group Databasebeta

← #28304452

Subgroups — #28304452

Order 13568.

644 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133284C211yesmaximal, characteristic
2#20290571C211yesmaximal, characteristic
3C2×#16811483C212yesmaximal, complemented
4#20289593C212yesmaximal, complemented
5#20283063C211yesmaximal, characteristic
6C23×#88324—21no—
7C2×#13133284—24no—
8C2×#13133284—21no—
9C2×#13133284C2212yes—
10C2×#13133284C2212yes—
11C2×C22⋊#88324—21no—
12#16811489—21no—
13#16811483—24no—
14C2×#13133215C2211yescharacteristic
15C2×#13133196C411yescharacteristic
16C22×C4×Dic53C2211yescharacteristic
17#16810530C411yescharacteristic
18#16810528C411yescharacteristic
19#16810518—21no—
20#16810518—21no—
21#16810492C2211yescharacteristic
22#16810481—21no—
23#16810481—21no—
24C2×C22⋊Q8×C53C411yescharacteristic
25C22×#88324—24no—
26C22×#88324—22no—
27C22×#88324—22no—
28C22×#88324—42no—
29C22×#88324—42no—
30#13133284—24no—
31#13133284—22no—
32#13133284—22no—
33#13133284—22no—
34#13133284D414yes—
35C22⋊#88324—42no—
36C22×Dic106C2×C411yescharacteristic
37C23×Dic53D411yescharacteristic
38C23×Dic53D411yescharacteristic
39#13133215—22no—
40#13133215—22no—
41#13133196—42no—
42C2×C4⋊Dic53C2×C411yescharacteristic
43C2×#11719919—21no—
44C2×#11719919—21no—
45C2×#11719919—21no—
46C2×#11719919—21no—
47C2×#11719919—21no—
48C2×C22⋊Dic53—21no—
49C2×C4×Dic53—22no—
50C2×C4×Dic53—22no—
51C2×C4×Dic53C2×C411yescharacteristic
52C2×C4×Dic53D411yescharacteristic
53C2×C4×Dic53D411yescharacteristic
54C2×C4×Dic53C2311yescharacteristic
55#13133124—21no—
56#13133124—21no—
57#13133124—21no—
58#13133124—21no—
59C22×Q8×C53C2×C411yescharacteristic
60C23×C4×C53C2×C411yescharacteristic
61C22⋊Q8×C53—42no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53C2×C411yescharacteristic
64C2×C22⋊C4×C53—21no—
65C24×C53C22⋊C411yescharacteristic
66C2×Q8×C53—22no—
67C2×Q8×C53—41no—
68C2×Q8×C53—41no—
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—41no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—22no—
78C22×Dic53—21no—
79C22×Dic53—22no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53C22⋊C411yescharacteristic
85C22×Dic53C22⋊C411yescharacteristic
86C22×Dic53SD1612yes—
87C22×Dic53SD1612yes—
88C22×Dic53Q1612yes—
89C22×Dic53Q1612yes—
90C22×Dic53C2×D411yescharacteristic
91C22×Dic53C2×D411yescharacteristic
92C2×Dic106—22no—
93C2×Dic106—41no—
94C2×Dic106—41no—
95C2×#88324—24no—
96C2×#88324—48no—
97C2×#88324—42no—
98C2×#88324—42no—
99C2×#88324—42no—
100C2×#88324—42no—
101C2×#88324—42no—
102C4⋊C4×C53—42no—
103C4⋊C4×C53—41no—
104C22⋊C4×C53—42no—
105C22⋊Dic53—42no—
106C4⋊Dic53—41no—
107#11719919—42no—
108#11719919—41no—
109#11719919—42no—
110#11719919—41no—
111#11719919—42no—
112#11719919—41no—
113#11719919—41no—
114C4×Dic53—22no—
115C4×Dic53—21no—
116C4×Dic53—22no—
117C4×Dic53—21no—
118C4×Dic53C2×D412yes—
119C4×Dic53#4112yes—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53#10012yes—
125C23×C53#10312yes—
126C23×C53#10312yes—
127C23×C53C2×C22⋊C411yescharacteristic
128Q8×C53—82no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—44no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53C4×D412yes—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—24no—
149C2×Dic53—24no—
150C2×Dic53—22no—
151C2×Dic53—22no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53#10312yes—
159C2×Dic53#10312yes—
160C2×Dic53#12011yescharacteristic
161C2×Dic53#12112yes—
162C2×Dic53#12311yescharacteristic
163C2×Dic53C2×SD1611yescharacteristic
164C2×Dic53C2×SD1611yescharacteristic
165C2×Dic53C2×Q1611yescharacteristic
166C2×Dic53C2×Q1611yescharacteristic
167C2×Dic53#13512yes—
168C2×Dic53AGL(1,ℤ/8)11yescharacteristic
169C2×Dic53#13711yescharacteristic
170Dic106—82no—
171#88324—44no—
172#88324—82no—
173#9127—531nomaximal, Sylow
174C22×C53—22no—
175C22×C53—22no—
176C22×C53—22no—
177C22×C53—22no—
178C22×C53—22no—
179C22×C53—22no—
180C22×C53#33314yescomplemented
181C22×C53#38611yescharacteristic
182C22×C53C2×#10011yescharacteristic
183C22×C53#41311yescharacteristic
184C22×C53C2×#10311yescharacteristic
185C22×C53C2×#10311yescharacteristic
186C22×C53#41611yescharacteristic
187C22×C53#41711yescharacteristic
188C4×C53—22no—
189C4×C53—41no—
190C4×C53—81no—
191C4×C53—81no—
192Dic53—22no—
193Dic53—22no—
194Dic53—81no—
195Dic53—81no—
196Dic53#45011yescharacteristic
197Dic53#45111yescharacteristic
198Dic53#46512yes—
199Dic53#46712yes—
200Dic53#48311yescharacteristic
201Dic53#48411yescharacteristic
202C22×C4⋊C8—531no—
203#2857—531no—
204#1797—531no—
205C2×#333—532no—
206#1296—532no—
207C2×C53—24no—
208C2×C53C2×#33311yescharacteristic
209C2×C53#154211yescharacteristic
210C2×C53#155011yescharacteristic
211C2×C53#155111yescharacteristic
212C2×C53#187211yescharacteristic
213C2×C53#189211yescharacteristic
214C2×C53#190011yescharacteristic
215C23×C8—1061no—
216C2×C4⋊Q8—531no—
217C2×C22⋊Q8—531no—
218C2×C22⋊Q8#615711yescharacteristic
219C22×C42—531no—
220C2×C4⋊C8—532no—
221C2×C4⋊C8—532no—
222C2×C4⋊C8—1064no—
223C2×C4⋊C8—1061no—
224C2×C22⋊C8—1061no—
225#391—531no—
226#391—531no—
227#389—1061no—
228#389—1061no—
229#384—531no—
230#380—1061no—
231#380—1061no—
232#336—1061no—
233#333—1064no—
234C53#912711yescharacteristic, complemented, Sylow
235C22×Q8—531no—
236C22×Q8C2×#615711yescharacteristic
237C23×C4—531no—
238C23×C4—531no—
239C23×C4C2×#615711yescharacteristic
240C22×C8—1062no—
241C22×C8—1064no—
242C22×C8—1062no—
243C22×C8—2122no—
244C22×C8—2122no—
245C4⋊Q8—1062no—
246C4⋊Q8—1062no—
247C22⋊Q8—42no—
248C22⋊Q8—2122no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—531no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4C2×#615711yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—1061no—
259C2×C42—531no—
260C2×C42—531no—
261C2×C42—531no—
262C2×C42—531no—
263C2×C42—1062no—
264C2×C42—1062no—
265C4⋊C8—534no—
266C4⋊C8—1062no—
267C4⋊C8—1062no—
268C4⋊C8—1062no—
269C4⋊C8—1064no—
270C22⋊C8—2122no—
271#95—1061no—
272#95—1061no—
273#95—1061no—
274#95—1061no—
275C24C22⋊#615711yescharacteristic
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—1062no—
280C2×Q8—2121no—
281C2×Q8—2121no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—532no—
290C22×C4—532no—
291C22×C4—532no—
292C22×C4—532no—
293C22×C4—531no—
294C22×C4—531no—
295C22×C4—1061no—
296C22×C4—1062no—
297C22×C4—1061no—
298C22×C4—1061no—
299C22×C4—1062no—
300C22×C4—1061no—
301C22×C4—1061no—
302C22×C4—1061no—
303C22×C4C22⋊#615711yescharacteristic
304C22×C4C22×#615711yescharacteristic
305C2×C8—1064no—
306C2×C8—2122no—
307C2×C8—2122no—
308C2×C8—2122no—
309C2×C8—2122no—
310C2×C8—2122no—
311C2×C8—2128no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—2121no—
315C4⋊C4—2122no—
316C4⋊C4—2121no—
317C4⋊C4—2122no—
318C4⋊C4—2121no—
319C4⋊C4—2122no—
320C4⋊C4—2121no—
321C4⋊C4—2121no—
322C22⋊C4—42no—
323C22⋊C4—2122no—
324C42—532no—
325C42—532no—
326C42—1061no—
327C42—1062no—
328C42—1061no—
329C42—1062no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23#1313327712yes—
335C23#1313328012yes—
336C23#1313328012yes—
337C23C2×C22⋊#615711yescharacteristic
338Q8—82no—
339Q8—4242no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—44no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—531no—
353C2×C4—532no—
354C2×C4—531no—
355C2×C4—531no—
356C2×C4—531no—
357C2×C4—531no—
358C2×C4—531no—
359C2×C4—532no—
360C2×C4—531no—
361C2×C4—531no—
362C2×C4—532no—
363C2×C4—532no—
364C2×C4—1062no—
365C2×C4—1062no—
366C2×C4—1062no—
367C2×C4—1062no—
368C2×C4—1064no—
369C2×C4—1064no—
370C2×C4—1061no—
371C2×C4—1061no—
372C2×C4—1061no—
373C2×C4—1061no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2121no—
380C2×C4D4×#615712yes—
381C8—4242no—
382C8—2124no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22#1681148314yescomplemented
390C22#1681157111yescharacteristic
391C22C2×#1313327711yescharacteristic
392C22#1681161011yescharacteristic
393C22C2×#1313328011yescharacteristic
394C22C2×#1313328011yescharacteristic
395C22#1681161311yescharacteristic
396C22#1681161411yescharacteristic
397C4—22no—
398C4—41no—
399C4—81no—
400C4—81no—
401C4—531no—
402C4—531no—
403C4—531no—
404C4—531no—
405C4—532no—
406C4—532no—
407C4—1062no—
408C4—1062no—
409C4—4241no—
410C4—4241no—
411C2—24no—
412C2C2×#1681148311yescharacteristic
413C2#2029021711yescharacteristic
414C2#2029023311yescharacteristic
415C2#2029023411yescharacteristic
416C2#2029070011yescharacteristic
417C2#2029072811yescharacteristic
418C2#2029074011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C22⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C22⋊Q8#6157
hypocenterC53#9127
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16811483C2semidirect
4. #20289593C2semidirect
180. C22×C53#333semidirect
234. C53#9127semidirect
389. C22#16811483semidirect

Maximal subgroups

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

subgroupindex
C22×#131332842
#202905712
C2×#168114832
#202895932
#202830632
#912753