Finite Group Databasebeta

← #27281274

Subgroups — #27281274

Order 13568.

732 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283975C211yesmaximal, characteristic
2#20275699C211yesmaximal, characteristic, complemented
3#20274659C211yesmaximal, characteristic
4#20273234C211yesmaximal, characteristic, complemented
5#20272621C211yesmaximal, characteristic, complemented
6#20272601C211yesmaximal, characteristic, complemented
7C53×#2341C211yesmaximal, characteristic
8C2×#13133247C412yescomplemented
9C2×#13133196C2211yescharacteristic
10#16810785—22no—
11#16810779—21no—
12#16810774—21no—
13C2×#13133144—22no—
14#16810624C412yescomplemented
15#16810587C2211yescharacteristic
16#16810587C2211yescharacteristic
17C4×C22⋊Dic53C2211yescharacteristic
18#16810389—22no—
19#16810359—21no—
20#16810359—21no—
21C22×#41×C53C2211yescharacteristic
22C2×C4×Q8×C53—21no—
23C53×#390—21no—
24C53×#390—22no—
25C53×#380—21no—
26C53×#380C2211yescharacteristic
27C4×C22⋊C4×C53—21no—
28C4×C22⋊C4×C53C2211yescharacteristic
29C22×Dic106C2×C411yescharacteristic
30#13133247—44no—
31C2×#11719925—22no—
32C2×#11719923—22no—
33C2×#11719917—21no—
34C2×#11719917C2×C412yes—
35#13133196—21no—
36#13133196—21no—
37#13133196—21no—
38#13133196—21no—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C4×Dic53—21no—
44#13133144—44no—
45#13133141—22no—
46#13133138—22no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51#13133124—21no—
52C2×#41×C53—21no—
53C2×#41×C53—21no—
54C2×#41×C53—44no—
55C2×#41×C53C2×C412yes—
56C2×#41×C53D412yes—
57C22×Q8×C53D411yescharacteristic
58C22×D4×C53—21no—
59C22×D4×C53D411yescharacteristic
60C23×C4×C53—21no—
61C23×C4×C53C2311yescharacteristic
62C4×Q8×C53—44no—
63C2×C4⋊C4×C53—22no—
64C2×C4⋊C4×C53—21no—
65C2×C22⋊C4×C53—22no—
66C2×C22⋊C4×C53—22no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53D412yes—
69C2×C42×C53—22no—
70C2×C42×C53—21no—
71#95×C53—21no—
72#95×C53D411yescharacteristic
73#95×C53D411yescharacteristic
74C24×C53—21no—
75C24×C53C2×D411yescharacteristic
76#41×C53—24no—
77#41×C53—24no—
78#41×C53—42no—
79#41×C53—42no—
80#41×C53—44no—
81#41×C53—44no—
82C2×Q8×C53—24no—
83C2×Q8×C53—21no—
84C2×Q8×C53C22⋊C412yes—
85C2×D4×C53—22no—
86C2×D4×C53—22no—
87C2×D4×C53—21no—
88C2×D4×C53—42no—
89C2×D4×C53—44no—
90C2×D4×C53C22⋊C412yes—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—21no—
97C22×C4×C53—42no—
98C22×C4×C53—44no—
99C22×C4×C53C22×C411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C22×Dic53—21no—
106C22×Dic53—21no—
107C22×Dic53—41no—
108C2×Dic106—21no—
109C2×Dic106—21no—
110C2×Dic106—42no—
111C4⋊C4×C53—44no—
112C4⋊C4×C53—42no—
113C22⋊C4×C53—42no—
114C22⋊C4×C53—44no—
115C22⋊C4×C53—42no—
116C22⋊C4×C53—82no—
117C42×C53—44no—
118C42×C53—42no—
119C22⋊Dic53—22no—
120C22⋊Dic53—41no—
121C22⋊Dic53—41no—
122C22⋊Dic53—41no—
123#11719925—44no—
124#11719923—44no—
125C4⋊Dic53—21no—
126C4⋊Dic53—21no—
127#11719919—41no—
128#11719919—41no—
129C4×Dic53—42no—
130#11719917—42no—
131#11719917—42no—
132C2×#88322—22no—
133C2×#88322—41no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—22no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—42no—
140C23×C53—42no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53C4×D411yescharacteristic
144C23×C53#12011yescharacteristic
145C23×C53#12111yescharacteristic
146Q8×C53—24no—
147Q8×C53—22no—
148Q8×C53#9914yes—
149D4×C53—22no—
150D4×C53—22no—
151D4×C53—44no—
152D4×C53—44no—
153D4×C53—42no—
154C2×C4×C53—24no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—22no—
158C2×C4×C53—22no—
159C2×C4×C53—22no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—44no—
164C2×C4×C53—44no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—42no—
168C2×C4×C53—41no—
169C2×C4×C53—42no—
170C2×C4×C53—41no—
171C2×C4×C53—42no—
172C2×C4×C53—41no—
173C2×C4×C53C2×C22⋊C411yescharacteristic
174C2×C4×C53C4×D411yescharacteristic
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53#12311yescharacteristic
177C2×Dic53—42no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—81no—
181C2×Dic53—81no—
182C2×Dic53—81no—
183C2×Dic53—81no—
184Dic106—42no—
185Dic106—81no—
186#88322—42no—
187#88322—81no—
188#9364—531nomaximal, Sylow
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—42no—
198C22×C53—42no—
199C22×C53—42no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53—82no—
203C22×C53—82no—
204C22×C53#38611yescharacteristic
205C22×C53#45511yescharacteristic
206C22×C53#45611yescharacteristic
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—24no—
210C4×C53—21no—
211C4×C53—42no—
212C4×C53—82no—
213C4×C53—81no—
214C4×C53C2×#9912yes—
215Dic53—82no—
216Dic53—81no—
217Dic53—81no—
218#3007—531no—
219#2341D5311yescharacteristic
220#1996—531no—
221#1789—531no—
222#1508—531no—
223#1305—531no—
224#1297—531no—
225C2×C53—21no—
226C2×C53—21no—
227C2×C53—41no—
228C2×C53—41no—
229C2×C53—82no—
230C2×C53#188211yescharacteristic
231C2×C53#189811yescharacteristic
232C2×C53#190611yescharacteristic
233C22×#41C2×D5311yescharacteristic
234C2×#137—532no—
235C2×C22⋊Q8—531no—
236C2×C4×Q8—21no—
237#452—1062no—
238#451—1061no—
239#450—1061no—
240C2×#104—1062no—
241#417—532no—
242#407—531no—
243#407—531no—
244#390—21no—
245#390—22no—
246#380—21no—
247#380C2×D5311yescharacteristic
248C4×C22⋊C4—21no—
249C4×C22⋊C4—531no—
250C4×C22⋊C4C2×D5311yescharacteristic
251#342—1062no—
252#328—1061no—
253#328—1061no—
254C53#936411yescharacteristic, complemented, Sylow
255C2×#41—21no—
256C2×#41—21no—
257C2×#41—44no—
258C2×#41C4×D5312yes—
259C2×#41D21212yes—
260C22×Q8—531no—
261C22×Q8C22⋊D5311yescharacteristic
262C22×D4—21no—
263C22×D4C22⋊D5311yescharacteristic
264C23×C4—21no—
265C23×C4C22×D5311yescharacteristic
266#137—2124no—
267C2×Q16—1062no—
268C2×SD16—1062no—
269C2×M16—532no—
270C2×M16—1061no—
271C22⋊Q8—1061no—
272C22⋊Q8—1061no—
273C22⋊Q8—1061no—
274C22⋊Q8—1061no—
275C4×Q8—44no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4—1061no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—1061no—
284C2×C22⋊C4—1061no—
285C2×C22⋊C4C22⋊D5312yes—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—1061no—
289#104—2124no—
290#103—1062no—
291#102—1062no—
292C22⋊C8—1061no—
293C22⋊C8—1061no—
294C22⋊C8—1061no—
295C22⋊C8—1061no—
296#95—21no—
297#95—1061no—
298#95C22⋊D5311yescharacteristic
299#95C22⋊D5311yescharacteristic
300C24—21no—
301C24C2×C22⋊D5311yescharacteristic
302#41—24no—
303#41—24no—
304#41—42no—
305#41—42no—
306#41—44no—
307#41—44no—
308C2×Q8—24no—
309C2×Q8—21no—
310C2×Q8—1061no—
311C2×Q8—1061no—
312C2×Q8—2122no—
313C2×Q8#1171992112yes—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—21no—
317C2×D4—42no—
318C2×D4—44no—
319C2×D4#1171992112yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—42no—
327C22×C4—44no—
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
337Q16—2124no—
338SD16—2124no—
339M16—2122no—
340M16—2122no—
341C2×C8—1062no—
342C2×C8—2121no—
343C4⋊C4—44no—
344C4⋊C4—42no—
345C4⋊C4—1061no—
346C4⋊C4—1061no—
347C4⋊C4—2121no—
348C4⋊C4—2121no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—42no—
352C22⋊C4—82no—
353C22⋊C4—1062no—
354C22⋊C4—2121no—
355C22⋊C4—2121no—
356C22⋊C4—2121no—
357C42—44no—
358C42—42no—
359C42—2122no—
360C23—21no—
361C23—21no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23C4×C22⋊D5311yescharacteristic
370C23#1313318611yescharacteristic
371C23#1313319111yescharacteristic
372Q8—24no—
373Q8—22no—
374Q8—4241no—
375Q8—2122no—
376Q8#1313313114yes—
377D4—22no—
378D4—22no—
379D4—44no—
380D4—44no—
381D4—42no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—4241no—
402C2×C4—4241no—
403C2×C4—4241no—
404C2×C4—4241no—
405C2×C4—2122no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4C2×#1171992111yescharacteristic
409C2×C4C4×C22⋊D5311yescharacteristic
410C2×C4#1313319111yescharacteristic
411C2×C4#1313320011yescharacteristic
412C8—4241no—
413C8—2122no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—81no—
426C22—81no—
427C22—82no—
428C22—82no—
429C22#1681050211yescharacteristic
430C22#1681080111yescharacteristic
431C22#1681080611yescharacteristic
432C4—21no—
433C4—24no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—82no—
438C4—81no—
439C4—4242no—
440C4—4241no—
441C4—4241no—
442C4C2×#1313313112yes—
443C2—21no—
444C2—21no—
445C2—41no—
446C2—41no—
447C2—82no—
448C2#2027508411yescharacteristic
449C2#2027516811yescharacteristic
450C2#2027521611yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2341C2
socleC22×C53#386
prime radicalC22×#41×C53C22
cosocleC23×C4×C53C23
hypercenter#2341D53
hypocenterC53#9364
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2341

Complements

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

NH ≅ G/Ntype
2. #20275699C2semidirect
4. #20273234C2semidirect
5. #20272621C2semidirect
6. #20272601C2semidirect
8. C2×#13133247C4semidirect
14. #16810624C4semidirect
254. C53#9364semidirect

Maximal subgroups

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

subgroupindex
#202839752
#202756992
#202746592
#202732342
#202726212
#202726012
C53×#23412
#936453