Finite Group Databasebeta

← #27284559

Subgroups — #27284559

Order 13568.

594 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16810934C211yesmaximal, characteristic
2#20279201C211yesmaximal, characteristic, complemented
3#20275552C211yesmaximal, characteristic
4#20274802C211yesmaximal, characteristic, complemented
5#20273287C211yesmaximal, characteristic, complemented
6#20272562C211yesmaximal, characteristic, complemented
7C53×#3187C211yesmaximal, characteristic
8C22×C4⋊Dic53C2211yescharacteristic
9#16810934—22no—
10#16810934—22no—
11C2×#13133150—21no—
12C2×#13133138—21no—
13C2×#13133132—22no—
14C2×C22⋊#88322C2211yescharacteristic
15#16810562—22no—
16#16810530—21no—
17#16810488—21no—
18#16810484—22no—
19#16810483C2211yescharacteristic
20C4×C22⋊Dic53C2211yescharacteristic
21#16810386—22no—
22#16810360—22no—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C2×#121×C53C2211yescharacteristic
25C53×#483—21no—
26C53×#481—21no—
27C53×#465—22no—
28C53×#450—21no—
29C53×#448—21no—
30C2×C22⋊C8×C53C2211yescharacteristic
31C23×Dic53—21no—
32C22×#88322—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53D412yes—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53D412yes—
42C2×C4×Dic53—21no—
43#13133150—24no—
44#13133138—24no—
45#13133132—44no—
46C22⋊#88322—21no—
47C22⋊#88322D412yes—
48#13133124—24no—
49#13133124—24no—
50#13133124—21no—
51#13133124—21no—
52C22×Q8×C53D411yescharacteristic
53C22×D4×C53—21no—
54C22×D4×C53D411yescharacteristic
55C23×C4×C53C2311yescharacteristic
56C2×SD16×C53—22no—
57C22×C8×C53D412yes—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53—21no—
62#121×C53—21no—
63#121×C53—21no—
64#121×C53—21no—
65#121×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C4⋊C4×C53D411yescharacteristic
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71#106×C53—22no—
72#103×C53—22no—
73#102×C53—22no—
74C22⋊C8×C53—21no—
75C22⋊C8×C53—21no—
76C24×C53—21no—
77C24×C53C2×D411yescharacteristic
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—42no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—42no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—42no—
87C2×D4×C53—42no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—42no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53#4111yescharacteristic
98C22×Dic53—22no—
99C22×Dic53—22no—
100C22×Dic53—41no—
101C22×Dic53—41no—
102C22×Dic53—42no—
103C22×Dic53—42no—
104C22×Dic53—41no—
105SD16×C53—82no—
106C2×C8×C53—22no—
107C2×C8×C53—42no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C22⋊Dic53—22no—
119C22⋊Dic53—22no—
120C22⋊Dic53—42no—
121C4⋊Dic53—22no—
122C4⋊Dic53—22no—
123C4⋊Dic53—41no—
124C4⋊Dic53—41no—
125C4⋊Dic53—42no—
126C4×Dic53—42no—
127C2×#88322—22no—
128C2×#88322—42no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—42no—
133C23×C53—81no—
134C23×C53—81no—
135C23×C53#12011yescharacteristic
136C23×C53#12111yescharacteristic
137C23×C53#12711yescharacteristic
138C23×C53#13512yes—
139C23×C53AGL(1,ℤ/8)12yes—
140Q8×C53—42no—
141Q8×C53—81no—
142D4×C53—42no—
143D4×C53—42no—
144D4×C53—82no—
145D4×C53—81no—
146D4×C53—81no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—44no—
167C2×Dic53—42no—
168C2×Dic53—82no—
169C2×Dic53—81no—
170C2×Dic53—81no—
171C2×Dic53—81no—
172C2×Dic53—81no—
173C2×Dic53—81no—
174C8×C53—42no—
175#88322—42no—
176#9953—531nomaximal, Sylow
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—42no—
184C22×C53—42no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53—162no—
189C22×C53#39211yescharacteristic
190C22×C53#44912yes—
191C22×C53#46312yes—
192C22×C53#49511yescharacteristic
193C22×C53#49611yescharacteristic
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—81no—
198C4×C53—81no—
199C4×C53—81no—
200C4×C53#45812yes—
201Dic53—82no—
202Dic53—82no—
203Dic53—161no—
204#3187D5311yescharacteristic
205C2×#480—531no—
206#2503—531no—
207#1964—531no—
208#1823—531no—
209#1520—531no—
210#1282—531no—
211C2×C53—22no—
212C2×C53—21no—
213C2×C53—22no—
214C2×C53—21no—
215C2×C53—161no—
216C2×C53#200911yescharacteristic
217C2×C53#201211yescharacteristic
218C2×C53#202211yescharacteristic
219C2×C22⋊Q8C2×D5311yescharacteristic
220C2×#121C2×D5311yescharacteristic
221C22×C4⋊C4—531no—
222#483—21no—
223#481—21no—
224#480—1062no—
225#480—1062no—
226#465—22no—
227#450—21no—
228#448—21no—
229C2×#107—1061no—
230C2×#102—1061no—
231C2×#99—1062no—
232C2×C22⋊C8—531no—
233C2×C22⋊C8C2×D5311yescharacteristic
234#400—1062no—
235#391—1061no—
236#382—1061no—
237#381—1062no—
238#380—531no—
239C4×C22⋊C4—531no—
240#340—1062no—
241#328—1062no—
242C53#995311yescharacteristic, complemented, Sylow
243C22×Q8C22⋊D5311yescharacteristic
244C22×D4—21no—
245C22×D4C22⋊D5311yescharacteristic
246C23×C4—1061no—
247C23×C4C22×D5311yescharacteristic
248C2×SD16—22no—
249C22×C8—1061no—
250C22×C8C22⋊D5312yes—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255#121—21no—
256#121—21no—
257#121—21no—
258#121—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—532no—
261C2×C4⋊C4—1061no—
262C2×C4⋊C4—1061no—
263C2×C4⋊C4—1061no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4—1061no—
266C2×C4⋊C4—1061no—
267C2×C4⋊C4C22⋊D5311yescharacteristic
268C2×C4⋊C4C22⋊D5311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—532no—
272C2×C22⋊C4—1061no—
273C2×C42—1061no—
274#107—1064no—
275#106—22no—
276#103—22no—
277#102—22no—
278#102—1064no—
279#99—2124no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—532no—
283C22⋊C8—1061no—
284#95—1064no—
285#95—1064no—
286#95—1061no—
287#95—1061no—
288C24—21no—
289C24D4×D5311yescharacteristic
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—42no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—1062no—
305C22×C4—1062no—
306C22×C4—2121no—
307C22×C4—2121no—
308C22×C4—2121no—
309C22×C4—2122no—
310C22×C4—2122no—
311C22×C4D4×D5311yescharacteristic
312C22×C4D4×D5311yescharacteristic
313C22×C4#1171994711yescharacteristic
314C22×C4C2×C22⋊D5311yescharacteristic
315C22×C4C2×C22⋊D5311yescharacteristic
316C22×C4C2×C22⋊D5311yescharacteristic
317SD16—82no—
318C2×C8—22no—
319C2×C8—42no—
320C2×C8—1062no—
321C2×C8—2122no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—1062no—
329C4⋊C4—1062no—
330C4⋊C4—2122no—
331C4⋊C4—2121no—
332C4⋊C4—2121no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—1062no—
338C22⋊C4—1062no—
339C22⋊C4—2122no—
340C42—2122no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—81no—
346C23—81no—
347C23#1313318511yescharacteristic
348C23#1313318911yescharacteristic
349C23#1313321311yescharacteristic
350C23#1313323812yes—
351C23#1313324412yes—
352Q8—42no—
353Q8—81no—
354D4—42no—
355D4—42no—
356D4—81no—
357D4—82no—
358D4—81no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—4241no—
372C2×C4—4241no—
373C2×C4—4241no—
374C2×C4—4241no—
375C2×C4—4242no—
376C2×C4—4241no—
377C2×C4—2122no—
378C2×C4—2122no—
379C2×C4—2122no—
380C2×C4—2124no—
381C2×C4—2122no—
382C2×C4#1313318511yescharacteristic
383C2×C4#1313318611yescharacteristic
384C2×C4#1313318811yescharacteristic
385C2×C4#1313318911yescharacteristic
386C8—42no—
387C8—2122no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—162no—
400C22#1681053111yescharacteristic
401C22#1681076712yes—
402C22#1681084012yes—
403C22#1681100411yescharacteristic
404C22#1681100911yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—81no—
409C4—81no—
410C4—81no—
411C4—8481no—
412C4—4242no—
413C4—4242no—
414C4#1681081312yes—
415C2—21no—
416C2—22no—
417C2—22no—
418C2—21no—
419C2—161no—
420C2#2027577211yescharacteristic
421C2#2027578911yescharacteristic
422C2#2027584411yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#3187C2
socleC22×C53#392
prime radicalC2×#121×C53C22
hypercenter#3187D53
hypocenterC53#9953
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
2. #20279201C2semidirect
4. #20274802C2semidirect
5. #20273287C2semidirect
6. #20272562C2semidirect
242. C53#9953semidirect

Maximal subgroups

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

subgroupindex
C2×#168109342
#202792012
#202755522
#202748022
#202732872
#202725622
C53×#31872
#995353