Finite Group Databasebeta

← #27294593

Subgroups — #27294593

Order 13568.

622 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810392C211yesmaximal, characteristic, complemented
2#20273045C211yesmaximal, characteristic, complemented
3#20272745C212yesmaximal, complemented
4#20272718C212yesmaximal, complemented
5C53×#2406C211yesmaximal, characteristic
6C22×#11719917—21no—
7#16810587—21no—
8#16810392—21no—
9#16810392C2212yes—
10#16810349—21no—
11#16810349C2212yes—
12C22×#41×C53C411yescharacteristic
13C2×#123×C53—21no—
14C22×C4⋊C4×C53C2211yescharacteristic
15C53×#395C411yescharacteristic
16C53×#395C2211yescharacteristic
17C53×#394C411yescharacteristic
18C53×#394C2211yescharacteristic
19C53×#386—21no—
20C53×#385—21no—
21C53×#385—21no—
22C53×#381C411yescharacteristic
23C2×#11719917—24no—
24C2×#11719917—22no—
25C2×#11719917—41no—
26C2×#11719917—42no—
27C2×#11719917—41no—
28C22×#88322—41no—
29C22⋊#88322—22no—
30C22⋊#88322—41no—
31C2×#41×C53—41no—
32C2×#41×C53—41no—
33C2×#41×C53—44no—
34C22×Q8×C53C2×C411yescharacteristic
35C22×D4×C53—21no—
36C22×D4×C53C2×C411yescharacteristic
37C23×C4×C53—21no—
38C23×C4×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53D411yescharacteristic
41#123×C53—44no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—41no—
45C2×C4⋊C4×C53—41no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53C2311yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53C2×C411yescharacteristic
59#95×C53C2×C411yescharacteristic
60C24×C53—21no—
61C24×C53C22⋊C411yescharacteristic
62#41×C53—44no—
63#41×C53—44no—
64#41×C53—44no—
65#41×C53—44no—
66C2×Q8×C53—42no—
67C2×Q8×C53—41no—
68C2×D4×C53—24no—
69C2×D4×C53—41no—
70C2×D4×C53—42no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—42no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—22no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—42no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53—42no—
90C22×C4×C53—42no—
91C22×C4×C53C22⋊C411yescharacteristic
92C22×C4×C53C22⋊C411yescharacteristic
93C22×C4×C53C22⋊C411yescharacteristic
94C22×C4×C53C22×C411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—42no—
102C4⋊C4×C53—42no—
103C22⋊C4×C53—42no—
104C22⋊C4×C53—42no—
105C22⋊C4×C53—42no—
106C22⋊C4×C53—42no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109#11719917—24no—
110#11719917—44no—
111#11719917—42no—
112#11719917—82no—
113C2×#88322—44no—
114C2×#88322—42no—
115C2×#88322—42no—
116C23×C53—22no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—22no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53#9911yescharacteristic
128C23×C53#9911yescharacteristic
129C23×C53C2×C22⋊C411yescharacteristic
130Q8×C53—42no—
131Q8×C53—42no—
132D4×C53—44no—
133D4×C53—42no—
134D4×C53—42no—
135D4×C53—44no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—22no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—48no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—81no—
166C2×C4×C53#10412yes—
167C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C53#13711yescharacteristic
169#88322—44no—
170#88322—82no—
171#11762—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53—42no—
182C22×C53—42no—
183C22×C53—42no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53AΣL(1,𝔽16)12yes—
189C22×C53C2×#9911yescharacteristic
190C22×C53#41711yescharacteristic
191C22×C53C2×#10411yescharacteristic
192C4×C53—21no—
193C4×C53—22no—
194C4×C53—21no—
195C4×C53—41no—
196C4×C53—41no—
197C4×C53—42no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—81no—
201C4×C53—81no—
202#2406D5311yescharacteristic
203C2×#343—531no—
204#1463—531no—
205#1354—532no—
206#1345—532no—
207C2×C53—21no—
208C2×C53—21no—
209C2×C53—22no—
210C2×C53—41no—
211C2×C53—82no—
212C2×C53#150411yescharacteristic
213C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
214C2×C53#213311yescharacteristic
215C22×#41Dic5311yescharacteristic
216C22×M16—1061no—
217C2×#123—21no—
218C22×C4⋊C4C2×D5311yescharacteristic
219#407—1061no—
220#395Dic5311yescharacteristic
221#395C2×D5311yescharacteristic
222#394Dic5311yescharacteristic
223#394C2×D5311yescharacteristic
224#386—21no—
225#385—21no—
226#385—21no—
227#381Dic5311yescharacteristic
228#343—532no—
229#343—1061no—
230#324—532no—
231#324—1061no—
232C53#1176211yescharacteristic, complemented, Sylow
233C2×#41—41no—
234C2×#41—41no—
235C2×#41—44no—
236C22×Q8C2×Dic5311yescharacteristic
237C22×D4—21no—
238C22×D4C2×Dic5311yescharacteristic
239C23×C4—21no—
240C23×C4C2×Dic5311yescharacteristic
241C23×C4C22⋊D5311yescharacteristic
242C23×C4C22⋊D5311yescharacteristic
243C2×M16—1064no—
244C2×M16—1062no—
245C2×M16—2121no—
246C2×M16—2121no—
247C2×M16—2122no—
248C22×C8—2121no—
249#123—44no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4C2×Dic5311yescharacteristic
255C2×C4⋊C4C22⋊D5311yescharacteristic
256C2×C4⋊C4C22⋊D5311yescharacteristic
257C2×C4⋊C4C22×D5311yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C22⋊C8—1062no—
264C22⋊C8—2121no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95C2×Dic5311yescharacteristic
269#95C2×Dic5311yescharacteristic
270C24—21no—
271C24C22⋊Dic5311yescharacteristic
272#41—44no—
273#41—44no—
274#41—44no—
275#41—44no—
276C2×Q8—42no—
277C2×Q8—41no—
278C2×D4—24no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4C22⋊Dic5311yescharacteristic
302C22×C4C22⋊Dic5311yescharacteristic
303C22×C4C22⋊Dic5311yescharacteristic
304C22×C4C22×Dic5311yescharacteristic
305C22×C4C2×C22⋊D5311yescharacteristic
306C22×C4C2×C22⋊D5311yescharacteristic
307M16—1064no—
308M16—4242no—
309M16—2124no—
310M16—2122no—
311C2×C8—2122no—
312C2×C8—2124no—
313C2×C8—2122no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23#1313313211yescharacteristic
338C23#1313313211yescharacteristic
339C23C2×C22⋊Dic5311yescharacteristic
340Q8—42no—
341Q8—42no—
342D4—44no—
343D4—42no—
344D4—42no—
345D4—44no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—48no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4#1313314412yes—
377C2×C4#1313324211yescharacteristic
378C2×C4#1313324711yescharacteristic
379C8—4242no—
380C8—2124no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22#1681041012yes—
398C22C2×#1313313211yescharacteristic
399C22#1681062411yescharacteristic
400C22C2×#1313314411yescharacteristic
401C4—21no—
402C4—22no—
403C4—21no—
404C4—41no—
405C4—41no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4—81no—
410C4—81no—
411C2—21no—
412C2—21no—
413C2—22no—
414C2—41no—
415C2—82no—
416C2#2027321911yescharacteristic
417C2C2×#1681041011yescharacteristic
418C2#2027640411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2406C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2406D53
hypocenterC53#11762
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
1. C2×#16810392C2semidirect
2. #20273045C2semidirect
3. #20272745C2semidirect
4. #20272718C2semidirect
232. C53#11762semidirect

Maximal subgroups

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

subgroupindex
C2×#168103922
#202730452
#202727452
#202727182
C53×#24062
#1176253