Finite Group Databasebeta

← #27279141

Subgroups — #27279141

Order 13568.

768 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 147 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×#16810548C211yesmaximal, characteristic
2#20274799C212yesmaximal, complemented
3C2×#16810347C211yesmaximal, characteristic
4#20272593C212yesmaximal, complemented
5C53×#2315C211yesmaximal, characteristic
6C22×C4⋊Dic53C2211yescharacteristic
7C22×C22⋊Dic53—21no—
8C2×C22⋊#88322C2212yes—
9#16810548—22no—
10#16810548C414yescomplemented
11C2×#13133124—21no—
12C2×#13133124C2211yescharacteristic
13#16810385—24no—
14#16810347—22no—
15C22×C22⋊C4×C53—21no—
16C22×C22⋊C4×C53C2211yescharacteristic
17C53×#385—24no—
18C53×#380—21no—
19C53×#380C2212yes—
20C23×Dic53—21no—
21C23×Dic53—21no—
22C22×#88322—22no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53C2×C412yescomplemented
27C2×C22⋊Dic53—24no—
28C2×C22⋊Dic53—42no—
29C2×C22⋊Dic53—42no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32#13133124—24no—
33#13133124—21no—
34#13133124C2×C412yescomplemented
35C25×C53D411yescharacteristic
36C23×C4×C53—22no—
37C23×C4×C53D411yescharacteristic
38C23×C4×C53C2311yescharacteristic
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53D414yes—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—42no—
46C2×C22⋊C4×C53—42no—
47C2×C22⋊C4×C53C2×C412yes—
48C2×C22⋊C4×C53D412yes—
49#95×C53—22no—
50#95×C53—22no—
51C24×C53—21no—
52C24×C53—21no—
53C24×C53—21no—
54C24×C53—21no—
55C24×C53—42no—
56C24×C53—42no—
57C24×C53C22⋊C412yes—
58C24×C53D812yes—
59C24×C53SD1612yes—
60C24×C53C2×D411yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—24no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53—44no—
67C22×C4×C53—82no—
68C22×C4×C53—82no—
69C22×C4×C53C22⋊C412yes—
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D412yes—
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53#4112yes—
74C22×Dic53—22no—
75C22×Dic53—22no—
76C22×Dic53—42no—
77C22×Dic53—42no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—41no—
84C22×Dic53—41no—
85C22×Dic53—41no—
86C4⋊C4×C53—44no—
87C4⋊C4×C53—82no—
88C22⋊C4×C53—24no—
89C22⋊C4×C53—24no—
90C22⋊C4×C53—44no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—42no—
93C22⋊Dic53—48no—
94C4⋊Dic53—24no—
95C4⋊Dic53—42no—
96C2×#88322—44no—
97C2×#88322—42no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—42no—
107C23×C53—42no—
108C23×C53—44no—
109C23×C53—44no—
110C23×C53—42no—
111C23×C53—42no—
112C23×C53—82no—
113C23×C53—82no—
114C23×C53—81no—
115C23×C53—81no—
116C23×C53#10214yes—
117C23×C53C2×C22⋊C411yescharacteristic
118C23×C53#12011yescharacteristic
119C23×C53#12311yescharacteristic
120C23×C53C2×D811yescharacteristic
121C23×C53C2×SD1611yescharacteristic
122C23×C53AGL(1,ℤ/8)11yescharacteristic
123C23×C53#13711yescharacteristic
124C2×C4×C53—24no—
125C2×C4×C53—22no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—44no—
132C2×C4×C53—44no—
133C2×C4×C53—44no—
134C2×C4×C53—81no—
135C2×C4×C53—88no—
136C2×C4×C53C4×D412yes—
137C2×C4×C53#12112yes—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144C2×Dic53—82no—
145C2×Dic53—82no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149C2×Dic53—81no—
150#88322—82no—
151#9025—531nomaximal, Sylow
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—24no—
161C22×C53—44no—
162C22×C53—44no—
163C22×C53—44no—
164C22×C53—42no—
165C22×C53—42no—
166C22×C53—82no—
167C22×C53—82no—
168C22×C53—82no—
169C22×C53#38611yescharacteristic
170C22×C53C2×#10211yescharacteristic
171C22×C53#41711yescharacteristic
172C22×C53#44711yescharacteristic
173C22×C53#44811yescharacteristic
174C22×C53#45512yes—
175C22×C53#45812yes—
176C22×C53#48011yescharacteristic
177C22×C53#48111yescharacteristic
178C4×C53—22no—
179C4×C53—22no—
180C4×C53—42no—
181C4×C53—84no—
182Dic53—82no—
183Dic53—82no—
184C2×#397—531no—
185#2315D5311yescharacteristic
186#1823—532no—
187C2×#323—531no—
188#1294—532no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—44no—
194C2×C53#160712yes—
195C2×C53#160812yes—
196C2×C53#187411yescharacteristic
197C2×C53#188511yescharacteristic
198C2×C53#189011yescharacteristic
199C22×C4⋊C4—531no—
200C22×C22⋊C4—21no—
201C22×C22⋊C4—1061no—
202C22×C22⋊C4C2×D5311yescharacteristic
203C2×C22⋊C8—532no—
204#397—534no—
205#397—1062no—
206#385—24no—
207#380—21no—
208#380C2×D5312yes—
209C2×#95—531no—
210C2×#95—1061no—
211#340—1064no—
212#323—1062no—
213C53#902511yescharacteristic, complemented, Sylow
214C25C22⋊D5311yescharacteristic
215C23×C4—22no—
216C23×C4—1061no—
217C23×C4—1061no—
218C23×C4C22⋊D5311yescharacteristic
219C23×C4C22×D5311yescharacteristic
220C22×C8—1062no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—532no—
223C2×C4⋊C4—1062no—
224C2×C4⋊C4—1062no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4C22⋊D5314yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—1064no—
234C2×C22⋊C4—2122no—
235C2×C22⋊C4—2122no—
236C2×C22⋊C4C4×D5312yes—
237C2×C22⋊C4D21212yes—
238C22⋊C8—1062no—
239C22⋊C8—1062no—
240#95—22no—
241#95—22no—
242#95—532no—
243#95—1061no—
244#95—1064no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—42no—
250C24—42no—
251C24#1171992112yes—
252C24#1171992212yes—
253C24#1171992412yes—
254C24C2×C22⋊D5311yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—1062no—
264C22×C4—1062no—
265C22×C4—2122no—
266C22×C4—2122no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4—2121no—
270C22×C4—2121no—
271C22×C4—2121no—
272C22×C4—2121no—
273C22×C4—2121no—
274C22×C4—2121no—
275C22×C4#1171992112yes—
276C22×C4C2×C4×D5311yescharacteristic
277C22×C4C2×D21211yescharacteristic
278C22×C4#1171994512yes—
279C22×C4C2×C22⋊D5312yes—
280C2×C8—2124no—
281C2×C8—2122no—
282C4⋊C4—44no—
283C4⋊C4—82no—
284C4⋊C4—1064no—
285C4⋊C4—2122no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—44no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—2128no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—42no—
301C23—42no—
302C23—44no—
303C23—44no—
304C23—42no—
305C23—42no—
306C23—82no—
307C23—82no—
308C23—81no—
309C23—81no—
310C23#1313313714yes—
311C23C2×#1171992111yescharacteristic
312C23#1313318611yescharacteristic
313C23#1313320011yescharacteristic
314C23C2×#1171992211yescharacteristic
315C23C2×#1171992411yescharacteristic
316C23#1313324311yescharacteristic
317C23#1313324811yescharacteristic
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—81no—
329C2×C4—88no—
330C2×C4—4242no—
331C2×C4—4242no—
332C2×C4—4241no—
333C2×C4—4241no—
334C2×C4—4241no—
335C2×C4—4241no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4C4×C22⋊D5312yes—
343C2×C4#1313319112yes—
344C8—4242no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—24no—
354C22—44no—
355C22—44no—
356C22—44no—
357C22—42no—
358C22—42no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22#1681050211yescharacteristic
363C22C2×#1313313711yescharacteristic
364C22#1681062211yescharacteristic
365C22#1681075711yescharacteristic
366C22#1681076211yescharacteristic
367C22#1681080112yes—
368C22#1681081312yes—
369C22#1681093611yescharacteristic
370C22#1681094111yescharacteristic
371C4—22no—
372C4—22no—
373C4—42no—
374C4—84no—
375C4—4242no—
376C4—4242no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2027370112yes—
383C2#2027370612yes—
384C2#2027503211yescharacteristic
385C2#2027510311yescharacteristic
386C2#2027512811yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2315C2
socleC23×C53#120
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenter#2315D53
hypocenterC53#9025
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
2. #20274799C2semidirect
4. #20272593C2semidirect
10. #16810548C4semidirect
26. C2×C4⋊Dic53C2×C4semidirect
34. #13133124C2×C4semidirect
213. C53#9025semidirect

Maximal subgroups

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

subgroupindex
C2×#168105482
#202747992
C2×#168103472
#202725932
C53×#23152
#902553