Finite Group Databasebeta

← #27280628

Subgroups — #27280628

Order 13568.

428 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285856C211yesmaximal, characteristic
2#20283664C211yesmaximal, characteristic
3#20275558C211yesmaximal, characteristic
4C2×#16810368C211yesmaximal, characteristic
5#20272610C211yesmaximal, characteristic
6#20272565C211yesmaximal, characteristic
7C53×#1550C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9#16811019—21no—
10#16810973—21no—
11#16810951C414yes—
12#16810908—22no—
13#16810754—21no—
14#16810754—22no—
15Q8×#88322—21no—
16#16810676C2211yescharacteristic
17#16810650—21no—
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133135—21no—
20C2×#13133135—21no—
21#16810579—21no—
22C4×#11719917C2211yescharacteristic
23#16810368—21no—
24#16810368—21no—
25C53×#533C2211yescharacteristic
26C53×#423C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C53×#333—21no—
29C53×#332—21no—
30C22×Dic106—21no—
31C2×#11719917D411yescharacteristic
32C2×#11719917D411yescharacteristic
33C22×#88322—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217—21no—
37#13133217—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53C2×C412yes—
40#13133148—24no—
41C4⋊#88322—22no—
42C4⋊#88322—22no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D411yescharacteristic
46C4⋊#88322D411yescharacteristic
47#13133141—24no—
48#13133135—42no—
49#13133135—42no—
50C22⋊#88322C2×C412yes—
51#13133128—22no—
52#13133128—21no—
53C4×#88322—22no—
54C4×#88322—21no—
55C4×#88322—21no—
56C2×M16×C53—21no—
57C22×C8×C53—21no—
58C4⋊Q8×C53D411yescharacteristic
59C4⋊Q8×C53D411yescharacteristic
60#125×C53—21no—
61C22⋊Q8×C53—21no—
62C22⋊Q8×C53C2×C412yes—
63C4×Q8×C53—21no—
64C4×Q8×C53D412yes—
65#117×C53—21no—
66C2×C42×C53C2311yescharacteristic
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53C22⋊C412yes—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53#4111yescharacteristic
76C22×Dic53—22no—
77C2×Dic106—41no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C2×Dic106—41no—
83M16×C53—41no—
84M16×C53—41no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C2×C8×C53—81no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53C22⋊C412yes—
94C22⋊C4×C53—22no—
95C42×C53—22no—
96C42×C53C2×D411yescharacteristic
97C42×C53C2×D411yescharacteristic
98C42×C53C2×D411yescharacteristic
99C42×C53#4111yescharacteristic
100C4⋊Dic53—22no—
101C4⋊Dic53—22no—
102C4⋊Dic53—41no—
103C4⋊Dic53—41no—
104#11719917—21no—
105#11719917—21no—
106#11719917—21no—
107#11719917SD1612yes—
108C2×#88322—21no—
109C2×#88322—22no—
110C2×#88322—21no—
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322—41no—
114C2×#88322—41no—
115C23×C53C4×D411yescharacteristic
116Q8×C53—42no—
117Q8×C53—41no—
118Q8×C53—41no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—41no—
124C2×C4×C53C2×C22⋊C411yescharacteristic
125C2×C4×C53C4×D411yescharacteristic
126C2×C4×C53#12011yescharacteristic
127C2×C4×C53#12111yescharacteristic
128C2×C4×C53#12111yescharacteristic
129C2×C4×C53#12311yescharacteristic
130C2×C4×C53C2×SD1611yescharacteristic
131C2×C4×C53#13511yescharacteristic
132C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C53#13711yescharacteristic
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137Dic106—81no—
138Dic106—81no—
139Dic106—81no—
140Dic106—81no—
141C8×C53—81no—
142C8×C53—81no—
143#88322—22no—
144#88322—42no—
145#88322—41no—
146#88322—41no—
147#88322—41no—
148#9272—531nomaximal, Sylow
149C22×C53—21no—
150C22×C53—21no—
151C22×C53#38611yescharacteristic
152C22×C53C4×SD1611yescharacteristic
153C22×C53#44011yescharacteristic
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—42no—
158C4×C53—42no—
159C4×C53#44811yescharacteristic
160C4×C53#44911yescharacteristic
161C4×C53#46011yescharacteristic
162C4×C53#46411yescharacteristic
163C4×C53#47111yescharacteristic
164C4×C53#47311yescharacteristic
165Dic53—82no—
166#3229—531no—
167#2962—531no—
168#1966—531no—
169#1550D5311yescharacteristic
170C2×#332—531no—
171#1301—531no—
172#1283—531no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53#188011yescharacteristic
176C2×C53#190811yescharacteristic
177C2×C53#191511yescharacteristic
178#533C2×D5311yescharacteristic
179C2×C4⋊Q8—531no—
180#498—1061no—
181#487—1061no—
182#483—534no—
183#475—1062no—
184#446—1061no—
185#446—1062no—
186C8×Q8—1061no—
187#431—531no—
188#424—1061no—
189#423C2×D5311yescharacteristic
190C2×C4⋊C8—531no—
191C2×C4⋊C8C2×D5311yescharacteristic
192C2×#101—1061no—
193C2×#101—1061no—
194#405—1061no—
195C4×M16—531no—
196#333—21no—
197#332—21no—
198#332—1061no—
199#332—1061no—
200C53#927211yescharacteristic, complemented, Sylow
201C22×Q8—1061no—
202C2×M16—21no—
203C2×M16—531no—
204C2×M16—531no—
205C22×C8—21no—
206C22×C8—1061no—
207C4⋊Q8—1061no—
208C4⋊Q8—1061no—
209C4⋊Q8—1061no—
210C4⋊Q8—1061no—
211C4⋊Q8C22⋊D5311yescharacteristic
212C4⋊Q8C22⋊D5311yescharacteristic
213#125—21no—
214C22⋊Q8—21no—
215C22⋊Q8C4×D5312yes—
216C4×Q8—21no—
217C4×Q8D21212yes—
218#117—21no—
219C2×C4⋊C4—532no—
220C2×C4⋊C4—1061no—
221C2×C42C22×D5311yescharacteristic
222#106—1064no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—531no—
228C4⋊C8—531no—
229C4⋊C8—1062no—
230C4⋊C8—1062no—
231C4⋊C8—1061no—
232C4⋊C8—1061no—
233#103—1064no—
234#101—2122no—
235#101—2122no—
236C22⋊C8—532no—
237#97—1062no—
238#97—1061no—
239C4×C8—1062no—
240C4×C8—1061no—
241C4×C8—1061no—
242C2×Q8—21no—
243C2×Q8—2121no—
244C2×Q8—2121no—
245C2×Q8—2121no—
246C2×Q8—2121no—
247C2×Q8—2121no—
248C2×Q8—2121no—
249C2×Q8#1171992112yes—
250C22×C4—1062no—
251C22×C4C2×C4×D5311yescharacteristic
252C22×C4D4×D5311yescharacteristic
253C22×C4#1171994711yescharacteristic
254M16—41no—
255M16—41no—
256M16—532no—
257M16—1061no—
258M16—1061no—
259M16—1061no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—81no—
263C2×C8—1061no—
264C2×C8—1062no—
265C2×C8—1061no—
266C2×C8—1061no—
267C2×C8—1061no—
268C2×C8—2121no—
269C2×C8—2121no—
270C4⋊C4—22no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—1062no—
276C4⋊C4—1062no—
277C4⋊C4—2121no—
278C4⋊C4—2121no—
279C4⋊C4#1171992112yes—
280C22⋊C4—22no—
281C42—22no—
282C42C2×D21211yescharacteristic
283C42D4×D5311yescharacteristic
284C42#1171994711yescharacteristic
285C42C2×C22⋊D5311yescharacteristic
286C23#1313317711yescharacteristic
287Q8—42no—
288Q8—41no—
289Q8—41no—
290Q8—4241no—
291Q8—4241no—
292Q8—4241no—
293Q8—4241no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—2122no—
300C2×C4—2122no—
301C2×C4—2122no—
302C2×C4C2×#1171992111yescharacteristic
303C2×C4#1313317711yescharacteristic
304C2×C4#1313318711yescharacteristic
305C2×C4#1313318911yescharacteristic
306C2×C4#1313318911yescharacteristic
307C2×C4#1313320111yescharacteristic
308C2×C4SD16×D5311yescharacteristic
309C2×C4#1313324011yescharacteristic
310C2×C4#1313324411yescharacteristic
311C2×C4#1313324911yescharacteristic
312C8—81no—
313C8—81no—
314C8—1062no—
315C8—2122no—
316C8—2121no—
317C8—2121no—
318C8—2121no—
319C22—21no—
320C22—21no—
321C22#1681050011yescharacteristic
322C22#1681070811yescharacteristic
323C22#1681072611yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—42no—
328C4—42no—
329C4—4242no—
330C4#1681076311yescharacteristic
331C4#1681077111yescharacteristic
332C4#1681082111yescharacteristic
333C4#1681084711yescharacteristic
334C4#1681088411yescharacteristic
335C4#1681089611yescharacteristic
336C2—21no—
337C2—21no—
338C2#2027506811yescharacteristic
339C2#2027522811yescharacteristic
340C2#2027526511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1550C2
socleC22×C53#386
prime radicalC23×C53C4×D4
cosocleC2×C42×C53C23
hypercenter#1550D53
hypocenterC53#9272
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
200. C53#9272semidirect

Maximal subgroups

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

subgroupindex
#202858562
#202836642
#202755582
C2×#168103682
#202726102
#202725652
C53×#15502
#927253