Finite Group Databasebeta

← #27357946

Subgroups — #27357946

Order 13568.

1064 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1064 conjugacy classes of subgroups — too many to draw. Only its 187 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×#16811249C211yesmaximal, characteristic
2#20285487C212yesmaximal, complemented
3#20273739C214yesmaximal, complemented
4#20273671C214yesmaximal, complemented
5#20273286C212yesmaximal, complemented
6C2×#16810347C211yesmaximal, characteristic
7C53×#2626C211yesmaximal, characteristic
8#16811249—24no—
9#16811249—22no—
10#16811249C2214yes—
11C2×#13133196—21no—
12C2×#13133196C2212yes—
13C2×#13133194—22no—
14C2×#13133194—22no—
15C22×C22⋊Dic53—21no—
16#16810958C2214yescomplemented
17#16810956—22no—
18#16810862—24no—
19#16810779C2214yescomplemented
20#16810778—22no—
21C2×C22⋊#88322C2212yes—
22#16810359C2218yescomplemented
23#16810347C2214yes—
24C2×C22⋊Q8×C53C2211yescharacteristic
25C22×C22⋊C4×C53C2211yescharacteristic
26C53×#399C2212yes—
27C53×#396—22no—
28C53×#393C2212yes—
29C53×#385—21no—
30C53×#380C2211yescharacteristic
31C53×#379—21no—
32C22×Dic106—21no—
33C22×Dic106D412yes—
34C23×Dic53—21no—
35C23×Dic53—21no—
36C2×#11719925—24no—
37C22×#88322—22no—
38#13133196—24no—
39#13133196—24no—
40#13133196—22no—
41#13133196—22no—
42#13133196D414yes—
43#13133196C2314yes—
44#13133194—28no—
45#13133194—28no—
46#13133194—28no—
47#13133194—28no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53D412yes—
50C2×#11719919—22no—
51C2×#11719919—22no—
52C2×C22⋊Dic53—22no—
53C2×C22⋊Dic53—24no—
54C2×C22⋊Dic53—22no—
55C2×C22⋊Dic53—24no—
56#13133150—24no—
57#13133141—24no—
58#13133140—24no—
59C22⋊#88322—22no—
60C22⋊#88322C2314yes—
61C25×C53D411yescharacteristic
62C22×Q8×C53C2311yescharacteristic
63C23×C4×C53D411yescharacteristic
64C23×C4×C53C2311yescharacteristic
65C22⋊Q8×C53—22no—
66C22⋊Q8×C53—22no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53C2311yescharacteristic
71C2×C22⋊C4×C53—21no—
72C2×C22⋊C4×C53—21no—
73C2×C22⋊C4×C53—22no—
74C2×C22⋊C4×C53—22no—
75C2×C22⋊C4×C53—22no—
76C2×C22⋊C4×C53—22no—
77C2×C22⋊C4×C53D412yes—
78C2×C22⋊C4×C53C2312yes—
79#95×C53—22no—
80#95×C53C2312yes—
81C24×C53—21no—
82C24×C53—21no—
83C24×C53—22no—
84C24×C53—81no—
85C24×C53—81no—
86C24×C53Q1614yescomplemented
87C24×C53C2×D412yes—
88C24×C53C2×D411yescharacteristic
89C2×Q8×C53—22no—
90C2×Q8×C53—81no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—22no—
95C22×C4×C53—21no—
96C22×C4×C53—81no—
97C22×C4×C53—81no—
98C22×C4×C53C2×D412yes—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D412yes—
101C22×C4×C53C2×D412yes—
102C22×C4×C53C2411yescharacteristic
103C22×Dic53—22no—
104C22×Dic53—22no—
105C22×Dic53—24no—
106C22×Dic53—24no—
107C22×Dic53—82no—
108C22×Dic53—82no—
109C2×Dic106—24no—
110C2×Dic106—22no—
111C2×Dic106—82no—
112C2×Dic106—82no—
113C2×Dic106C2×D414yes—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—42no—
116C4⋊C4×C53—81no—
117C4⋊C4×C53—82no—
118C22⋊C4×C53—24no—
119C22⋊C4×C53—24no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—42no—
123C22⋊C4×C53—82no—
124C22⋊C4×C53—82no—
125C22⋊Dic53—28no—
126C22⋊Dic53—24no—
127C22⋊Dic53—24no—
128#11719925—84no—
129C4⋊Dic53—24no—
130C4⋊Dic53—22no—
131C4⋊Dic53C2×D414yes—
132#11719919—28no—
133#11719919—24no—
134#11719919—24no—
135C2×#88322—24no—
136C2×#88322—82no—
137C23×C53—22no—
138C23×C53—24no—
139C23×C53—22no—
140C23×C53—24no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—42no—
144C23×C53—42no—
145C23×C53—42no—
146C23×C53—42no—
147C23×C53—81no—
148C23×C53—81no—
149C23×C53—84no—
150C23×C53—84no—
151C23×C53#12011yescharacteristic
152C23×C53#12011yescharacteristic
153C23×C53C2×Q1614yes—
154C23×C53C2×Q1612yes—
155C23×C53#13712yes—
156C23×C53C22×D411yescharacteristic
157Q8×C53—44no—
158C2×C4×C53—22no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—42no—
172C2×C4×C53—81no—
173C2×C4×C53#12012yes—
174C2×C4×C53C22×D412yes—
175C2×Dic53—24no—
176C2×Dic53—24no—
177C2×Dic53—44no—
178C2×Dic53—44no—
179C2×Dic53—84no—
180C2×Dic53—84no—
181Dic106—44no—
182Dic106—82no—
183Dic106—84no—
184#88322—44no—
185#18287—531nomaximal, Sylow
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—24no—
193C22×C53—42no—
194C22×C53—42no—
195C22×C53—44no—
196C22×C53—48no—
197C22×C53—42no—
198C22×C53—42no—
199C22×C53—82no—
200C22×C53—84no—
201C22×C53#45112yes—
202C22×C53#45112yes—
203C22×C53C2×#12011yescharacteristic
204C22×C53C22×Q1611yescharacteristic
205C22×C53C2×#13711yescharacteristic
206C4×C53—22no—
207C4×C53—22no—
208C4×C53—42no—
209C4×C53—82no—
210C4×C53—81no—
211C4×C53—81no—
212Dic53—44no—
213Dic53—44no—
214C2×#543—531no—
215#3189—532no—
216#2626D5311yescharacteristic
217#1614—534no—
218#1602—534no—
219#1520—532no—
220C2×#323—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—22no—
224C2×C53—22no—
225C2×C53—44no—
226C2×C53C2×#45111yescharacteristic
227C2×C53#301911yescharacteristic
228C2×C53#302611yescharacteristic
229#543—534no—
230#543—1064no—
231#543—1062no—
232C2×C22⋊Q8—532no—
233C2×C22⋊Q8—1062no—
234C2×C22⋊Q8—1062no—
235C2×C22⋊Q8—1061no—
236C2×C22⋊Q8C2×D5311yescharacteristic
237C22×C22⋊C4—1061no—
238C22×C22⋊C4C2×D5311yescharacteristic
239#484—534no—
240#484—1062no—
241#467—1064no—
242#451—534no—
243#451—1062no—
244C2×C22⋊C8—532no—
245#399C2×D5312yes—
246#396—22no—
247#393C2×D5312yes—
248#385—21no—
249#380C2×D5311yescharacteristic
250#379—21no—
251#328—538no—
252#323—534no—
253C53#1828711yescharacteristic, complemented, Sylow
254C25C22⋊D5311yescharacteristic
255C22×Q8—532no—
256C22×Q8—1061no—
257C22×Q8C22×D5311yescharacteristic
258C23×C4—1061no—
259C23×C4—1061no—
260C23×C4C22⋊D5311yescharacteristic
261C23×C4C22×D5311yescharacteristic
262C2×Q16—1064no—
263C22×C8—1062no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C22⋊Q8—534no—
267C22⋊Q8—534no—
268C22⋊Q8—1068no—
269C22⋊Q8—1068no—
270C22⋊Q8—1068no—
271C22⋊Q8—1068no—
272C22⋊Q8—1064no—
273C22⋊Q8—1062no—
274C22⋊Q8—1062no—
275C22⋊Q8—1064no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—532no—
280C2×C4⋊C4—1062no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4—1062no—
283C2×C4⋊C4C22×D5311yescharacteristic
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—1062no—
291C2×C22⋊C4—1064no—
292C2×C22⋊C4—1062no—
293C2×C22⋊C4—1064no—
294C2×C22⋊C4C22⋊D5312yes—
295C2×C22⋊C4C22×D5312yes—
296#107—1064no—
297#103—1064no—
298#103—1064no—
299C22⋊C8—534no—
300C22⋊C8—1062no—
301#95—22no—
302#95C22×D5312yes—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—81no—
307C24—81no—
308C24#1171992514yescomplemented
309C24C2×C22⋊D5312yes—
310C24C2×C22⋊D5311yescharacteristic
311C2×Q8—22no—
312C2×Q8—81no—
313C2×Q8—534no—
314C2×Q8—1064no—
315C2×Q8—1062no—
316C2×Q8—4242no—
317C2×Q8—4242no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—1064no—
326C22×C4—1064no—
327C22×C4—1062no—
328C22×C4—1062no—
329C22×C4—4242no—
330C22×C4—4242no—
331C22×C4D4×D5312yes—
332C22×C4D4×D5312yes—
333C22×C4C2×C22⋊D5312yes—
334C22×C4C2×C22⋊D5311yescharacteristic
335C22×C4C23×D5311yescharacteristic
336Q16—4244no—
337C2×C8—1064no—
338C2×C8—4242no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—81no—
342C4⋊C4—82no—
343C4⋊C4—534no—
344C4⋊C4—1068no—
345C4⋊C4—1064no—
346C4⋊C4—1062no—
347C4⋊C4—1064no—
348C4⋊C4—1064no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—42no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—1068no—
357C22⋊C4—1064no—
358C22⋊C4—1064no—
359C23—24no—
360C23—22no—
361C23—24no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—84no—
372C23—84no—
373C23#1313318511yescharacteristic
374C23#1313318511yescharacteristic
375C23C2×#1171992514yes—
376C23C2×#1171992512yes—
377C23#1313324712yes—
378C23C22×C22⋊D5311yescharacteristic
379Q8—44no—
380Q8—4242no—
381Q8—4244no—
382Q8—2124no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—81no—
398C2×C4—1064no—
399C2×C4—1064no—
400C2×C4—4244no—
401C2×C4—4244no—
402C2×C4—2124no—
403C2×C4—2124no—
404C2×C4#1313318512yes—
405C2×C4C2×D4×D5312yes—
406C8—2124no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—24no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—48no—
418C22—42no—
419C22—42no—
420C22—82no—
421C22—84no—
422C22#1681077812yes—
423C22#1681077812yes—
424C22C2×#1313318511yescharacteristic
425C22C22×#1171992511yescharacteristic
426C22C2×#1313324711yescharacteristic
427C4—22no—
428C4—22no—
429C4—42no—
430C4—82no—
431C4—81no—
432C4—81no—
433C4—2124no—
434C4—2124no—
435C2—21no—
436C2—22no—
437C2—21no—
438C2—22no—
439C2—44no—
440C2C2×#1681077811yescharacteristic
441C2#2028407111yescharacteristic
442C2#2028411111yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2626C2
socleC22×C53C2×#120
prime radicalC25×C53D4
hypercenter#2626D53
hypocenterC53#18287
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
2. #20285487C2semidirect
3. #20273739C2semidirect
4. #20273671C2semidirect
5. #20273286C2semidirect
16. #16810958C22semidirect
19. #16810779C22semidirect
22. #16810359C22semidirect
86. C24×C53Q16semidirect
253. C53#18287semidirect
308. C24#11719925semidirect

Maximal subgroups

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

subgroupindex
C2×#168112492
#202854872
#202737392
#202736712
#202732862
C2×#168103472
C53×#26262
#1828753