Finite Group Databasebeta

← #27477156

Subgroups — #27477156

Order 13568.

1066 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810969C211yesmaximal, characteristic, complemented
2C2×#16810688C211yesmaximal, characteristic, complemented
3#20279710C211yesmaximal, characteristic, complemented
4#20279662C211yesmaximal, characteristic
5#20275895C214yesmaximal, complemented
6#20275666C211yesmaximal, characteristic, complemented
7#20275140C214yesmaximal, complemented
8C4×#13133150C211yesmaximal, characteristic, complemented
9C53×#1982C211yesmaximal, characteristic, complemented
10C22×C8×D53C2212yes—
11C2×#13133217C2211yescharacteristic
12C2×#13133213—21no—
13C2×#13133205—21no—
14C2×#13133188C2214yes—
15C2×C4×D212C2211yescharacteristic
16#16810969—24no—
17#16810688—24no—
18C2×#13133150C2212yes—
19C2×#13133138C2212yes—
20C2×#13133138C2212yes—
21C2×#13133129C2212yes—
22C2×C4×#88322C2211yescharacteristic
23#16810550—22no—
24#16810531—22no—
25#16810529—21no—
26#16810526—21no—
27#16810480C2214yes—
28C4×C4⋊Dic53C2211yescharacteristic
29#16810386C2214yes—
30#16810385C2212yes—
31C2×#127×C53C2211yescharacteristic
32C2×C4⋊C8×C53C2211yescharacteristic
33C2×#102×C53C2214yes—
34C4×C4⋊C4×C53C2211yescharacteristic
35C22×Dic106—21no—
36C22×D212D411yescharacteristic
37C22×C22⋊D53—24no—
38C22×C4×D53C2312yes—
39C22×#88322C2311yescharacteristic
40C22×#88322C2311yescharacteristic
41C2×C8×D53—24no—
42C2×C8×D53—44no—
43#13133217—22no—
44#13133217—22no—
45#13133213—44no—
46#13133205—44no—
47#13133188—28no—
48#13133188—44no—
49C4×D212—22no—
50C4×D212—22no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53D411yescharacteristic
53C2×C4⋊Dic53C2312yes—
54C2×C4⋊Dic53C2312yes—
55C2×C22⋊Dic53—24no—
56C2×#11719921—24no—
57C2×#11719921D412yes—
58C2×C4×Dic53—21no—
59#13133150—24no—
60#13133138—24no—
61#13133138—24no—
62#13133129—24no—
63C4×#88322D414yescomplemented
64#13133124—21no—
65#13133124—24no—
66C22×D4×C53—21no—
67C22×D4×C53—21no—
68C22×D4×C53C2312yes—
69C22×C8×C53C2312yes—
70#127×C53—22no—
71#127×C53—22no—
72C2×C4⋊C4×C53C2312yes—
73C2×C42×C53—21no—
74C2×C42×C53C2311yescharacteristic
75C4⋊C8×C53—22no—
76#102×C53—44no—
77#95×C53—21no—
78C24×C53—22no—
79C23×D53C2×D412yes—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—24no—
83C2×D4×C53—24no—
84C2×D4×C53—42no—
85C2×D4×C53—42no—
86C2×D4×C53—42no—
87C2×D4×C53—42no—
88C2×D4×C53—82no—
89C22×C4×C53—22no—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2411yescharacteristic
93C2×C22⋊D53—48no—
94C2×C22⋊D53—48no—
95C2×C22⋊D53—84no—
96C22×Dic53—22no—
97C22×Dic53—22no—
98C22×Dic53C2×D412yes—
99C2×D212—22no—
100C2×D212—22no—
101C2×D212D814yes—
102C2×C4×D53—22no—
103C2×C4×D53—22no—
104C2×C4×D53—24no—
105C2×C4×D53—24no—
106C2×Dic106—42no—
107C2×Dic106—42no—
108C2×Dic106—42no—
109C2×C8×C53—22no—
110C2×C8×C53—24no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—22no—
113C42×C53—42no—
114C42×C53C2×D412yes—
115C42×C53#4112yes—
116C22⋊Dic53—48no—
117#11719921—24no—
118#11719921—48no—
119C4⋊Dic53—22no—
120C4⋊Dic53—22no—
121C4⋊Dic53—22no—
122C4⋊Dic53—22no—
123C4⋊Dic53—42no—
124C4⋊Dic53#4114yes—
125C4×Dic53—42no—
126C2×#88322—22no—
127C2×#88322—22no—
128C2×#88322C2×D412yes—
129C2×#88322C2×D412yes—
130C8×D53—48no—
131C23×C53—42no—
132C23×C53—44no—
133C23×C53—42no—
134C23×C53—42no—
135C23×C53—44no—
136C23×C53C22×D411yescharacteristic
137C22×D53—22no—
138C22×D53—24no—
139C22×D53—22no—
140C22×D53—24no—
141C22×D53C2×D814yes—
142D4×C53—42no—
143D4×C53—42no—
144D4×C53—44no—
145D4×C53—84no—
146D4×C53—84no—
147C2×C4×C53—22no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53#12112yes—
152C2×C4×C53#12112yes—
153C2×C4×C53C2×D812yes—
154C2×C4×C53#13512yes—
155C2×C4×C53C22×D411yescharacteristic
156C2×C4×C53C2×#4112yes—
157C2×C4×C53C2×#4111yescharacteristic
158C2×C4×C53H(5,𝔽2)12yes—
159C22⋊D53—816no—
160C2×Dic53—22no—
161C2×Dic53—24no—
162C2×Dic53—24no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168D212—28no—
169C4×D53—42no—
170C4×D53—42no—
171C4×D53—44no—
172Dic106—84no—
173C8×C53—42no—
174#88322—22no—
175#88322—22no—
176#28597—531nomaximal, Sylow
177C22×C53—84no—
178C22×C53—82no—
179C22×C53—84no—
180C22×C53—82no—
181C22×C53—82no—
182C22×C53C2×#12111yescharacteristic
183C22×C53#54612yes—
184C22×C53C22×D811yescharacteristic
185C22×C53C2×#13511yescharacteristic
186C22×C53#57712yes—
187C2×D53—28no—
188C2×D53—24no—
189C2×D53—44no—
190C2×D53—42no—
191C2×D53—42no—
192C4×C53—22no—
193C4×C53—82no—
194C4×C53#46614yes—
195C4×C53#53211yescharacteristic
196C4×C53#53511yescharacteristic
197C4×C53#55912yes—
198Dic53—44no—
199Dic53—42no—
200Dic53—82no—
201C2×#486—531no—
202C2×C8×D4—531no—
203#2547—531no—
204#2541—531no—
205#2033—534no—
206#1988—531no—
207#1982D5311yescharacteristic, complemented
208#1893—534no—
209C4×#107—531no—
210C2×C53—162no—
211C2×C53#254411yescharacteristic
212C2×C53C2×#46611yescharacteristic
213C2×C53#305711yescharacteristic
214C2×C53#329412yes—
215C2×C53#330412yes—
216D53—44no—
217C23×C8—532no—
218C2×C4⋊Q8—531no—
219C2×#127—1061no—
220C2×#127C2×D5311yescharacteristic
221C2×#124—1061no—
222C2×#121—534no—
223C2×C4×D4—531no—
224#486—1064no—
225C8×D4—1064no—
226C2×#107—532no—
227C2×C4⋊C8C2×D5311yescharacteristic
228C2×#102—532no—
229C2×#102—532no—
230C2×#102C2×D5314yes—
231C2×C22⋊C8—532no—
232C2×C4×C8—531no—
233#397—1062no—
234#392—1062no—
235#391—1061no—
236#390—1061no—
237#380—534no—
238C4×C4⋊C4—531no—
239C4×C4⋊C4C2×D5311yescharacteristic
240#340—534no—
241#340—532no—
242C53#2859711yescharacteristic, complemented, Sylow
243C22×Q8—1061no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4—531no—
247C22×D4—1064no—
248C22×D4C22×D5312yes—
249C23×C4—532no—
250C22×C8—531no—
251C22×C8—531no—
252C22×C8—1064no—
253C22×C8—2124no—
254C22×C8C22×D5312yes—
255C4⋊Q8—1062no—
256C4⋊Q8—1062no—
257#127—22no—
258#127—22no—
259#127—2124no—
260#124—2124no—
261#121—1068no—
262#121—2124no—
263C4×D4—1062no—
264C4×D4—1062no—
265C2×C4⋊C4—532no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4—532no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4C22×D5312yes—
270C2×C22⋊C4—532no—
271C2×C22⋊C4—1064no—
272C2×C22⋊C4—1064no—
273C2×C42—21no—
274C2×C42—1061no—
275C2×C42C22×D5311yescharacteristic
276#107—1064no—
277C4⋊C8—22no—
278#102—44no—
279#102—1064no—
280#102—1064no—
281C22⋊C8—1064no—
282C4×C8—534no—
283#95—21no—
284#95—1064no—
285#95—1061no—
286C24—22no—
287C24—532no—
288C2×Q8—2122no—
289C2×Q8—2122no—
290C2×Q8—2122no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—82no—
300C2×D4—534no—
301C2×D4—1062no—
302C2×D4—1062no—
303C2×D4—4244no—
304C2×D4—2128no—
305C2×D4—2128no—
306C22×C4—22no—
307C22×C4—532no—
308C22×C4—1062no—
309C22×C4—1062no—
310C22×C4—1064no—
311C22×C4—1062no—
312C22×C4—1064no—
313C22×C4—1062no—
314C22×C4D4×D5311yescharacteristic
315C22×C4D4×D5311yescharacteristic
316C22×C4C23×D5311yescharacteristic
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—532no—
320C2×C8—532no—
321C2×C8—1062no—
322C2×C8—1062no—
323C2×C8—2128no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—534no—
327C4⋊C4—1062no—
328C4⋊C4—1062no—
329C4⋊C4—1062no—
330C4⋊C4—1062no—
331C4⋊C4—2122no—
332C22⋊C4—1064no—
333C22⋊C4—2128no—
334C22⋊C4—2128no—
335C42—42no—
336C42—2122no—
337C42#1171994512yes—
338C42D4×D5312yes—
339C23—44no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—44no—
344C23—534no—
345C23—1064no—
346C23—1062no—
347C23—1062no—
348C23—1064no—
349C23C2×D4×D5311yescharacteristic
350Q8—4244no—
351D4—44no—
352D4—42no—
353D4—42no—
354D4—84no—
355D4—84no—
356D4—1068no—
357D4—42416no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—1064no—
363C2×C4—1064no—
364C2×C4—1062no—
365C2×C4—2122no—
366C2×C4—2122no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4—2124no—
371C2×C4—2122no—
372C2×C4—2122no—
373C2×C4#1313319012yes—
374C2×C4#1313319012yes—
375C2×C4D8×D5312yes—
376C2×C4#1313323812yes—
377C2×C4C2×D4×D5311yescharacteristic
378C2×C4#41×D5312yes—
379C2×C4C2×#1171994511yescharacteristic
380C2×C4#1313326212yes—
381C8—42no—
382C8—1062no—
383C8—1062no—
384C22—84no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—84no—
389C22—1068no—
390C22—1064no—
391C22—2122no—
392C22—2122no—
393C22—2124no—
394C22C2×#1313319011yescharacteristic
395C22#1681127012yes—
396C22C2×D8×D5311yescharacteristic
397C22C2×#1313323811yescharacteristic
398C22#1681143312yes—
399C4—22no—
400C4—82no—
401C4—4242no—
402C4—2124no—
403C4—2122no—
404C4#1681085814yes—
405C4#1681117211yescharacteristic
406C4#1681118611yescharacteristic
407C4#1681135612yes—
408C2—162no—
409C2—2124no—
410C2#2027968911yescharacteristic
411C2C2×#1681085811yescharacteristic
412C2#2028435311yescharacteristic
413C2#2028653112yes—
414C2#2028665712yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1982C2
socleC23×C53C22×D4
prime radical#20279662C2
hypercenter#1982D53
hypocenterC53#28597
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
1. C2×#16810969C2semidirect
2. C2×#16810688C2semidirect
3. #20279710C2semidirect
5. #20275895C2semidirect
6. #20275666C2semidirect
7. #20275140C2semidirect
8. C4×#13133150C2semidirect
9. C53×#1982C2semidirect
63. C4×#88322D4semidirect
207. #1982D53semidirect
242. C53#28597semidirect

Maximal subgroups

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

subgroupindex
C2×#168109692
C2×#168106882
#202797102
#202796622
#202758952
#202756662
#202751402
C4×#131331502
C53×#19822
#2859753