Finite Group Databasebeta

← #27279122

Subgroups — #27279122

Order 13568.

812 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277329C211yesmaximal, characteristic
2#20274805C212yesmaximal, complemented
3C2×#16810347C211yesmaximal, characteristic
4#20272594C212yesmaximal, complemented
5C2×C53×#399C211yesmaximal, characteristic
6C22×C22⋊Dic53—21no—
7C2×C22⋊#88322C2212yes—
8#16810498—24no—
9#16810483—21no—
10#16810483C2212yes—
11#16810385—24no—
12#16810347—22no—
13C22×C4⋊C4×C53—21no—
14C22×C4⋊C4×C53C2211yescharacteristic
15C22×C22⋊C4×C53—21no—
16C22×C22⋊C4×C53C2211yescharacteristic
17C53×#399—22no—
18C53×#399C414yescomplemented
19C2×#95×C53C2211yescharacteristic
20C23×Dic53—22no—
21C22×#88322—22no—
22C2×C4⋊Dic53—22no—
23C2×C4⋊Dic53D414yes—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53—42no—
27C2×C22⋊Dic53—42no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30#13133124—22no—
31#13133124—22no—
32C25×C53D411yescharacteristic
33C23×C4×C53—21no—
34C23×C4×C53—21no—
35C23×C4×C53D411yescharacteristic
36C23×C4×C53C2311yescharacteristic
37C2×C4⋊C4×C53—24no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—44no—
42C2×C4⋊C4×C53C2×C412yescomplemented
43C2×C22⋊C4×C53—24no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—44no—
47C2×C22⋊C4×C53C2×C412yes—
48C2×C22⋊C4×C53D412yes—
49#95×C53—21no—
50#95×C53C2×C412yescomplemented
51C24×C53—21no—
52C24×C53—21no—
53C24×C53—21no—
54C24×C53—21no—
55C24×C53—42no—
56C24×C53—42no—
57C24×C53C22⋊C412yes—
58C24×C53SD1612yes—
59C24×C53Q1612yes—
60C24×C53C2×D411yescharacteristic
61C22×C4×C53—24no—
62C22×C4×C53—22no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—22no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53—42no—
71C22×C4×C53—42no—
72C22×C4×C53—42no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53C22⋊C412yes—
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D412yes—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4112yes—
80C22×Dic53—24no—
81C22×Dic53—44no—
82C22×Dic53—82no—
83C22×Dic53—82no—
84C4⋊C4×C53—24no—
85C4⋊C4×C53—48no—
86C4⋊C4×C53—42no—
87C22⋊C4×C53—24no—
88C22⋊C4×C53—24no—
89C22⋊C4×C53—48no—
90C22⋊Dic53—44no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C4⋊Dic53—44no—
94C4⋊Dic53—82no—
95C2×#88322—44no—
96C2×#88322—42no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—22no—
105C23×C53—42no—
106C23×C53—42no—
107C23×C53—44no—
108C23×C53—44no—
109C23×C53—42no—
110C23×C53—42no—
111C23×C53—82no—
112C23×C53—82no—
113C23×C53—81no—
114C23×C53—81no—
115C23×C53#10314yes—
116C23×C53C2×C22⋊C411yescharacteristic
117C23×C53#12011yescharacteristic
118C23×C53#12311yescharacteristic
119C23×C53C2×SD1611yescharacteristic
120C23×C53C2×Q1611yescharacteristic
121C23×C53AGL(1,ℤ/8)11yescharacteristic
122C23×C53#13711yescharacteristic
123C2×C4×C53—24no—
124C2×C4×C53—24no—
125C2×C4×C53—22no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—44no—
130C2×C4×C53—44no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—81no—
139C2×C4×C53—82no—
140C2×C4×C53—82no—
141C2×C4×C53C4×D412yes—
142C2×C4×C53#12112yes—
143C2×Dic53—44no—
144C2×Dic53—44no—
145C2×Dic53—44no—
146C2×Dic53—88no—
147#88322—82no—
148#9021—531nomaximal, Sylow
149C22×C53—22no—
150C22×C53—22no—
151C22×C53—22no—
152C22×C53—22no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—24no—
158C22×C53—44no—
159C22×C53—44no—
160C22×C53—44no—
161C22×C53—42no—
162C22×C53—42no—
163C22×C53—82no—
164C22×C53—82no—
165C22×C53—82no—
166C22×C53#38611yescharacteristic
167C22×C53C2×#10311yescharacteristic
168C22×C53#41711yescharacteristic
169C22×C53#45011yescharacteristic
170C22×C53#45111yescharacteristic
171C22×C53#45312yes—
172C22×C53#45812yes—
173C22×C53#48311yescharacteristic
174C22×C53#48411yescharacteristic
175C4×C53—22no—
176C4×C53—22no—
177C4×C53—44no—
178C4×C53—42no—
179C4×C53—82no—
180Dic53—84no—
181C2×#399D5311yescharacteristic
182#2315—531no—
183#1824—532no—
184C2×#323—531no—
185#1294—532no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—21no—
190C2×C53—44no—
191C2×C53#161312yes—
192C2×C53#161412yes—
193C2×C53#187211yescharacteristic
194C2×C53#188711yescharacteristic
195C2×C53#189011yescharacteristic
196C22×C4⋊C4—21no—
197C22×C4⋊C4C2×D5311yescharacteristic
198C22×C22⋊C4—21no—
199C22×C22⋊C4—1061no—
200C22×C22⋊C4C2×D5311yescharacteristic
201C2×C22⋊C8—532no—
202#399—22no—
203#399Dic5314yescomplemented
204#385—1064no—
205#380—532no—
206#380—1061no—
207C2×#95C2×D5311yescharacteristic
208#340—1064no—
209#323—1062no—
210C53#902111yescharacteristic, complemented, Sylow
211C25C22⋊D5311yescharacteristic
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—1062no—
215C23×C4C22⋊D5311yescharacteristic
216C23×C4C22×D5311yescharacteristic
217C22×C8—1062no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—534no—
224C2×C4⋊C4—1062no—
225C2×C4⋊C4C2×Dic5312yescomplemented
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—1062no—
231C2×C22⋊C4—1062no—
232C2×C22⋊C4—2122no—
233C2×C22⋊C4—2122no—
234C2×C22⋊C4C2×Dic5312yes—
235C2×C22⋊C4C22⋊D5312yes—
236C22⋊C8—1062no—
237C22⋊C8—1062no—
238#95—21no—
239#95—1062no—
240#95—1062no—
241#95C2×Dic5312yescomplemented
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—42no—
247C24—42no—
248C24#1171992412yes—
249C24#1171992512yes—
250C24C22⋊Dic5312yes—
251C24C2×C22⋊D5311yescharacteristic
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—1064no—
267C22×C4—4242no—
268C22×C4—4242no—
269C22×C4—2124no—
270C22×C4C22⋊Dic5312yes—
271C22×C4D4×D5312yes—
272C22×C4#1171994712yes—
273C22×C4C22×Dic5311yescharacteristic
274C22×C4C2×C22⋊D5311yescharacteristic
275C2×C8—2124no—
276C2×C8—2122no—
277C4⋊C4—24no—
278C4⋊C4—48no—
279C4⋊C4—42no—
280C4⋊C4—4242no—
281C4⋊C4—2124no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—48no—
285C22⋊C4—2124no—
286C22⋊C4—2122no—
287C22⋊C4—2122no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—42no—
297C23—44no—
298C23—44no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—82no—
303C23—82no—
304C23—81no—
305C23—81no—
306C23#1313314114yes—
307C23C2×C22⋊Dic5311yescharacteristic
308C23#1313318511yescharacteristic
309C23#1313319811yescharacteristic
310C23C2×#1171992411yescharacteristic
311C23C2×#1171992511yescharacteristic
312C23#1313324211yescharacteristic
313C23#1313324711yescharacteristic
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—4248no—
333C2×C4—2124no—
334C2×C4—2124no—
335C2×C4—2124no—
336C2×C4D4×Dic5312yes—
337C2×C4#1313318912yes—
338C8—4242no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—24no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22#1681050311yescharacteristic
357C22C2×#1313314111yescharacteristic
358C22#1681062411yescharacteristic
359C22#1681077311yescharacteristic
360C22#1681077811yescharacteristic
361C22#1681079012yes—
362C22#1681081212yes—
363C22#1681095111yescharacteristic
364C22#1681095611yescharacteristic
365C4—22no—
366C4—22no—
367C4—44no—
368C4—42no—
369C4—82no—
370C4—4244no—
371C2—21no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—44no—
376C2#2027373412yes—
377C2#2027373912yes—
378C2#2027502311yescharacteristic
379C2#2027511411yescharacteristic
380C2#2027512911yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#399C2
socleC23×C53#120
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenterC2×#399D53
hypocenterC53#9021
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. #20274805C2semidirect
4. #20272594C2semidirect
18. C53×#399C4semidirect
42. C2×C4⋊C4×C53C2×C4semidirect
50. #95×C53C2×C4semidirect
203. #399Dic53semidirect
210. C53#9021semidirect
225. C2×C4⋊C4C2×Dic53semidirect
241. #95C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202773292
#202748052
C2×#168103472
#202725942
C2×C53×#3992
#902153