Finite Group Databasebeta

← #27284603

Subgroups — #27284603

Order 13568.

568 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 109 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#20285487C211yesmaximal, characteristic
2C2×#16810958C211yesmaximal, characteristic
3#20275700C211yesmaximal, characteristic
4#20274805C211yesmaximal, characteristic
5#20273287C211yesmaximal, characteristic
6#20272563C211yesmaximal, characteristic
7C53×#2502C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810958—22no—
10#16810958—21no—
11#16810958—22no—
12#16810956—21no—
13#16810862—22no—
14#16810779—21no—
15#16810778—21no—
16C2×#13133150—21no—
17C2×#13133140—21no—
18C2×C22⋊#88322C2211yescharacteristic
19C2×C22⋊#88322C2211yescharacteristic
20#16810483C2211yescharacteristic
21#16810389—22no—
22#16810385—22no—
23#16810360—22no—
24C2×C22⋊Q8×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#398—22no—
27C53×#391—21no—
28C53×#385—22no—
29C53×#384—21no—
30C4×C22⋊C4×C53C2211yescharacteristic
31C22×Dic106D411yescharacteristic
32C2×#11719925—22no—
33C22×#88322—21no—
34C22×#88322D412yes—
35#13133196—21no—
36#13133196—21no—
37#13133196—21no—
38#13133196—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53D411yescharacteristic
41C2×#11719919—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—21no—
44#13133150—22no—
45#13133150—24no—
46#13133141—22no—
47#13133140—22no—
48#13133140—24no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322D412yes—
53#13133124—21no—
54C22×Q8×C53D411yescharacteristic
55C23×C4×C53—21no—
56C23×C4×C53C2311yescharacteristic
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53—21no—
61C2×C4⋊C4×C53—24no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53D412yes—
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53D412yes—
72C2×C42×C53—21no—
73#95×C53—24no—
74#95×C53—21no—
75C24×C53C2×D411yescharacteristic
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—42no—
79C22×C4×C53—22no—
80C22×C4×C53—21no—
81C22×C4×C53—22no—
82C22×C4×C53—21no—
83C22×C4×C53—42no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—82no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×Dic53—21no—
94C22×Dic53—21no—
95C22×Dic53—41no—
96C2×Dic106—21no—
97C2×Dic106—21no—
98C2×Dic106—42no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—84no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53Q1614yescomplemented
112C42×C53—42no—
113C22⋊Dic53—41no—
114C22⋊Dic53—41no—
115C22⋊Dic53—42no—
116#11719925—82no—
117C4⋊Dic53—21no—
118C4⋊Dic53—21no—
119C4⋊Dic53—42no—
120#11719919—41no—
121#11719919—41no—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—42no—
125C2×#88322—42no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53#12011yescharacteristic
129C23×C53#12111yescharacteristic
130C23×C53#12711yescharacteristic
131C23×C53C2×Q1612yes—
132C23×C53#13712yes—
133Q8×C53—42no—
134Q8×C53—81no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—24no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—44no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—82no—
151C2×C4×C53—82no—
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12011yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—81no—
159C2×Dic53—81no—
160C2×Dic53—81no—
161C2×Dic53—81no—
162C2×Dic53—81no—
163Dic106—42no—
164Dic106—42no—
165#88322—42no—
166#88322—42no—
167#9959—531nomaximal, Sylow
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53#39211yescharacteristic
177C22×C53#45112yes—
178C22×C53#46212yes—
179C22×C53#49411yescharacteristic
180C22×C53#49711yescharacteristic
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—44no—
185C4×C53—81no—
186C4×C53—82no—
187C4×C53—81no—
188C4×C53#45812yes—
189Dic53—81no—
190Dic53—81no—
191Dic53—161no—
192#3189—531no—
193C2×#484—531no—
194#2502D5311yescharacteristic
195#1996—531no—
196#1824—531no—
197#1520—531no—
198#1282—531no—
199C2×C53—21no—
200C2×C53—22no—
201C2×C53—22no—
202C2×C53—21no—
203C2×C53#200811yescharacteristic
204C2×C53#201611yescharacteristic
205C2×C53#202211yescharacteristic
206C2×C22⋊Q8—531no—
207C2×C22⋊Q8C2×D5311yescharacteristic
208C22×C4⋊C4C2×D5311yescharacteristic
209#484—1062no—
210#484—1061no—
211#484—1062no—
212#484—1061no—
213#467—1062no—
214#451—1061no—
215#451—1061no—
216C2×#107—1061no—
217C2×#103—1061no—
218C2×C22⋊C8—531no—
219C2×C22⋊C8—531no—
220#398—22no—
221#391—21no—
222#385—22no—
223#384—21no—
224#380—531no—
225C4×C22⋊C4C2×D5311yescharacteristic
226#342—1062no—
227#340—1062no—
228#328—1062no—
229C53#995911yescharacteristic, complemented, Sylow
230C22×Q8—531no—
231C22×Q8C22⋊D5311yescharacteristic
232C23×C4—21no—
233C23×C4C22×D5311yescharacteristic
234C2×Q16—1062no—
235C22×C8—532no—
236C22×C8—1061no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—1061no—
242C22⋊Q8—1061no—
243C22⋊Q8—1061no—
244C22⋊Q8—1061no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4C22⋊D5312yes—
256C2×C4⋊C4C22⋊D5311yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—1061no—
259C2×C22⋊C4—1061no—
260C2×C22⋊C4C22⋊D5312yes—
261C2×C42—21no—
262#107—1064no—
263#107—1062no—
264#103—1064no—
265#103—1062no—
266#103—1062no—
267C22⋊C8—532no—
268C22⋊C8—1061no—
269C22⋊C8—1061no—
270C22⋊C8—1061no—
271#95—24no—
272#95—21no—
273#95—1061no—
274C24C2×C22⋊D5311yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—42no—
278C2×Q8—1061no—
279C2×Q8—1061no—
280C2×Q8—2122no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—82no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—2121no—
292C22×C4D4×D5311yescharacteristic
293C22×C4D4×D5311yescharacteristic
294C22×C4D4×D5311yescharacteristic
295C22×C4#1171994711yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297C22×C4C2×C22⋊D5311yescharacteristic
298Q16—4242no—
299C2×C8—1062no—
300C2×C8—1062no—
301C2×C8—2122no—
302C2×C8—2122no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—84no—
312C4⋊C4—1061no—
313C4⋊C4—1061no—
314C4⋊C4—2122no—
315C4⋊C4—2121no—
316C4⋊C4—2121no—
317C22⋊C4—22no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—2122no—
321C22⋊C4—2121no—
322C22⋊C4—2121no—
323C22⋊C4#1171992514yescomplemented
324C42—42no—
325C23—41no—
326C23—41no—
327C23#1313318611yescharacteristic
328C23#1313318811yescharacteristic
329C23#1313321311yescharacteristic
330C23C2×#1171992512yes—
331C23#1313324812yes—
332Q8—42no—
333Q8—81no—
334Q8—2122no—
335Q8—2122no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—4241no—
354C2×C4—4241no—
355C2×C4—4241no—
356C2×C4—4241no—
357C2×C4—4241no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4#1313318511yescharacteristic
361C2×C4#1313318511yescharacteristic
362C2×C4#1313318911yescharacteristic
363C2×C4#1313318911yescharacteristic
364C8—2122no—
365C8—2122no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22#1681053111yescharacteristic
375C22#1681077912yes—
376C22#1681083012yes—
377C22#1681100211yescharacteristic
378C22#1681101511yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—44no—
383C4—82no—
384C4—81no—
385C4—81no—
386C4—8481no—
387C4—4241no—
388C4—4241no—
389C4#1681081212yes—
390C2—22no—
391C2—21no—
392C2—22no—
393C2—21no—
394C2#2027576811yescharacteristic
395C2#2027581511yescharacteristic
396C2#2027584311yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2502C2
socleC22×C53#392
prime radicalC24×C53C2×D4
hypercenter#2502D53
hypocenterC53#9959
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
111. C22⋊C4×C53Q16semidirect
229. C53#9959semidirect
323. C22⋊C4#11719925semidirect

Maximal subgroups

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

subgroupindex
#202854872
C2×#168109582
#202757002
#202748052
#202732872
#202725632
C53×#25022
#995953