Finite Group Databasebeta

← #27281113

Subgroups — #27281113

Order 13568.

534 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#20285521C211yesmaximal, characteristic
2#20275699C211yesmaximal, characteristic, complemented
3#20273247C211yesmaximal, characteristic, complemented
4#20273231C211yesmaximal, characteristic
5#20272621C211yesmaximal, characteristic, complemented
6#20272567C211yesmaximal, characteristic, complemented
7C53×#2377C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810958—21no—
10#16810934C412yescomplemented
11#16810867—22no—
12#16810774—21no—
13#16810761C412yescomplemented
14#16810587C2211yescharacteristic
15#16810587C2211yescharacteristic
16C4×C22⋊Dic53C2211yescharacteristic
17#16810389—21no—
18#16810389—21no—
19#16810359—21no—
20#16810359—21no—
21#16810357—22no—
22C2×#121×C53C2211yescharacteristic
23C53×#390—21no—
24C53×#388—21no—
25C53×#387—21no—
26C53×#386—21no—
27C53×#386—21no—
28C53×#385—21no—
29C53×#381C2211yescharacteristic
30C53×#380C2211yescharacteristic
31C22×Dic106C2×C411yescharacteristic, complemented
32C2×#11719923—22no—
33C2×#11719917—21no—
34C2×#11719917—21no—
35#13133196—21no—
36#13133196—21no—
37#13133196—21no—
38#13133196—21no—
39C2×C4⋊Dic53C2×C411yescharacteristic, complemented
40C2×#11719919—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C4×Dic53—21no—
44#13133150—22no—
45#13133140—22no—
46#13133138—22no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322C2×C412yescomplemented
51#13133124—21no—
52C22×D4×C53—21no—
53C22×D4×C53D411yescharacteristic
54C23×C4×C53—21no—
55C23×C4×C53C2311yescharacteristic
56#121×C53—21no—
57#121×C53—21no—
58#121×C53C2×C412yes—
59#121×C53D412yes—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53D411yescharacteristic
69C2×C22⋊C4×C53D411yescharacteristic
70C2×C42×C53—21no—
71#95×C53—21no—
72#95×C53—21no—
73#95×C53—21no—
74#95×C53—21no—
75#95×C53D411yescharacteristic
76#95×C53D411yescharacteristic
77C24×C53—21no—
78C24×C53C2×D411yescharacteristic
79C2×D4×C53—22no—
80C2×D4×C53—22no—
81C2×D4×C53—21no—
82C2×D4×C53—81no—
83C2×D4×C53—81no—
84C2×D4×C53—81no—
85C2×D4×C53C22⋊C412yes—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—81no—
92C22×C4×C53—81no—
93C22×C4×C53—81no—
94C22×C4×C53—81no—
95C22×C4×C53C22×C411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53#4111yescharacteristic
100C22×C4×C53#4111yescharacteristic
101C22×Dic53—21no—
102C22×Dic53—21no—
103C22×Dic53—41no—
104C2×Dic106—21no—
105C2×Dic106—21no—
106C2×Dic106—81no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—81no—
109C4⋊C4×C53—81no—
110C4⋊C4×C53C22⋊C412yes—
111C22⋊C4×C53—22no—
112C22⋊C4×C53—22no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—81no—
116C22⋊C4×C53—81no—
117C22⋊C4×C53—81no—
118C22⋊C4×C53—81no—
119C42×C53—81no—
120C22⋊Dic53—22no—
121C22⋊Dic53—41no—
122C22⋊Dic53—41no—
123C22⋊Dic53—41no—
124#11719923—82no—
125C4⋊Dic53—21no—
126C4⋊Dic53—21no—
127#11719919—41no—
128#11719919—41no—
129C4×Dic53—81no—
130#11719917—81no—
131#11719917—81no—
132C2×#88322—22no—
133C2×#88322—41no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—22no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53C4×D411yescharacteristic
146C23×C53#12011yescharacteristic
147C23×C53#12111yescharacteristic
148D4×C53—42no—
149D4×C53—81no—
150D4×C53—82no—
151D4×C53—82no—
152C2×C4×C53—22no—
153C2×C4×C53—41no—
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—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—84no—
174C2×C4×C53C2×C22⋊C411yescharacteristic
175C2×C4×C53C4×D411yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12311yescharacteristic
178C2×Dic53—42no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—81no—
182C2×Dic53—81no—
183C2×Dic53—81no—
184C2×Dic53—81no—
185Dic106—42no—
186Dic106—81no—
187#88322—42no—
188#88322—81no—
189#9341—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—42no—
199C22×C53—42no—
200C22×C53—42no—
201C22×C53—81no—
202C22×C53—81no—
203C22×C53—81no—
204C22×C53—81no—
205C22×C53—81no—
206C22×C53—81no—
207C22×C53#38611yescharacteristic
208C22×C53#45511yescharacteristic
209C22×C53#45611yescharacteristic
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—42no—
215C4×C53—81no—
216C4×C53—81no—
217C4×C53—81no—
218C4×C53—81no—
219Dic53—82no—
220Dic53—81no—
221Dic53—81no—
222#3193—531no—
223#2377D5311yescharacteristic
224#1996—531no—
225#1511—531no—
226#1507—531no—
227#1305—531no—
228#1284—531no—
229C2×C53—21no—
230C2×C53—21no—
231C2×C53—41no—
232C2×C53—41no—
233C2×C53—82no—
234C2×C53#188211yescharacteristic
235C2×C53#190411yescharacteristic
236C2×C53#191311yescharacteristic
237C2×C22⋊Q8—531no—
238C2×#121C2×D5311yescharacteristic
239#484—1061no—
240#480—532no—
241#468—1062no—
242#450—1061no—
243#448—532no—
244#407—531no—
245#407—531no—
246#390—21no—
247#388—21no—
248#387—21no—
249#386—21no—
250#386—21no—
251#385—21no—
252#381C2×D5311yescharacteristic
253#380C2×D5311yescharacteristic
254C4×C22⋊C4—531no—
255#342—1061no—
256#342—1061no—
257#328—1061no—
258#328—1061no—
259#327—1062no—
260C53#934111yescharacteristic, complemented, Sylow
261C22×Q8—531no—
262C22×D4—21no—
263C22×D4C22⋊D5311yescharacteristic
264C23×C4—21no—
265C23×C4C22×D5311yescharacteristic
266C2×SD16—1062no—
267C2×M16—1061no—
268C2×M16—1061no—
269C22⋊Q8—1061no—
270C22⋊Q8—1061no—
271C22⋊Q8—1061no—
272C22⋊Q8—1061no—
273#121—21no—
274#121—21no—
275#121C4×D5312yes—
276#121D21212yes—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—531no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4C22⋊D5311yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—1061no—
288C2×C22⋊C4—1061no—
289C2×C22⋊C4C22⋊D5311yescharacteristic
290C2×C22⋊C4C22⋊D5311yescharacteristic
291C2×C42—21no—
292C2×C42—1061no—
293#107—1062no—
294#103—1062no—
295#102—1062no—
296C22⋊C8—532no—
297C22⋊C8—1061no—
298C22⋊C8—1061no—
299C22⋊C8—1061no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—1061no—
305#95C22⋊D5311yescharacteristic
306#95C22⋊D5311yescharacteristic
307C24—21no—
308C24C2×C22⋊D5311yescharacteristic
309C2×Q8—1061no—
310C2×Q8—1061no—
311C2×Q8—4241no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—21no—
315C2×D4—81no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4#1171992112yes—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—1061no—
329C22×C4—1061no—
330C22×C4—2121no—
331C22×C4C2×C4×D5311yescharacteristic
332C22×C4C2×D21211yescharacteristic
333C22×C4#1171994511yescharacteristic
334C22×C4#1171994511yescharacteristic
335C22×C4C2×C22⋊D5311yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337SD16—4242no—
338M16—4241no—
339M16—4241no—
340C2×C8—1062no—
341C2×C8—2121no—
342C4⋊C4—21no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C4⋊C4—1061no—
346C4⋊C4—1061no—
347C4⋊C4—2121no—
348C4⋊C4—2121no—
349C4⋊C4#1171992112yes—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—81no—
355C22⋊C4—81no—
356C22⋊C4—81no—
357C22⋊C4—81no—
358C22⋊C4—1062no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C22⋊C4—2121no—
362C42—81no—
363C42—4241no—
364C23—21no—
365C23—21no—
366C23—22no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23C4×C22⋊D5311yescharacteristic
376C23#1313318611yescharacteristic
377C23#1313319111yescharacteristic
378Q8—4241no—
379Q8—2122no—
380D4—42no—
381D4—81no—
382D4—82no—
383D4—82no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—84no—
406C2×C4—4241no—
407C2×C4—4241no—
408C2×C4—4241no—
409C2×C4—4241no—
410C2×C4—2122no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4C2×#1171992111yescharacteristic
414C2×C4C4×C22⋊D5311yescharacteristic
415C2×C4#1313319111yescharacteristic
416C2×C4#1313320011yescharacteristic
417C8—4241no—
418C8—2122no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22#1681050211yescharacteristic
437C22#1681080111yescharacteristic
438C22#1681080611yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—42no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—4242no—
449C4—4241no—
450C4—4241no—
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—82no—
456C2#2027508411yescharacteristic
457C2#2027520211yescharacteristic
458C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2377C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#2377D53
hypocenterC53#9341
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2377

Complements

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

NH ≅ G/Ntype
2. #20275699C2semidirect
3. #20273247C2semidirect
5. #20272621C2semidirect
6. #20272567C2semidirect
10. #16810934C4semidirect
13. #16810761C4semidirect
31. C22×Dic106C2×C4semidirect
39. C2×C4⋊Dic53C2×C4semidirect
50. C22⋊#88322C2×C4semidirect
260. C53#9341semidirect

Maximal subgroups

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

subgroupindex
#202855212
#202756992
#202732472
#202732312
#202726212
#202725672
C53×#23772
#934153