Finite Group Databasebeta

← #27367237

Subgroups — #27367237

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285836C211yesmaximal, characteristic
2#20285640C211yesmaximal, characteristic
3#20283428C211yesmaximal, characteristic
4#20274259C214yesmaximal, complemented
5#20274135C214yesmaximal, complemented
6#20273463C211yesmaximal, characteristic, complemented
7#20273400C211yesmaximal, characteristic, complemented
8#20273387C211yesmaximal, characteristic, complemented
9C2×C53×#487C211yesmaximal, characteristic, complemented
10C2×#13133223—21no—
11#16811173C2211yescharacteristic
12#16811026—21no—
13#16810997—21no—
14#16810973—21no—
15#16810971—21no—
16#16810936—22no—
17#16810900C2214yes—
18#16810825C2214yes—
19#16810762—22no—
20#16810702—21no—
21#16810702C2212yes—
22C8×D212—21no—
23C8×D212C2212yes—
24#16810676C2211yescharacteristic
25#16810676C2211yescharacteristic
26#16810647C2211yescharacteristic
27#16810589—21no—
28#16810376C2212yes—
29#16810374C2212yes—
30#16810369C2214yes—
31#16810369C2214yes—
32C2×C4⋊Q8×C53C2211yescharacteristic
33C2×C4⋊Q8×C53C2211yescharacteristic
34C53×#487—21no—
35C53×#487—21no—
36C53×#487C2212yes—
37C53×#487C2212yes—
38C2×#103×C53—21no—
39C2×#103×C53—21no—
40C2×C4×C8×C53C2211yescharacteristic
41C2×#11719945—21no—
42C2×#11719924—24no—
43#13133223—44no—
44C2×#11719917—21no—
45C2×#11719912—22no—
46C2×C8×D53—22no—
47#13133217D411yescharacteristic
48#13133214D411yescharacteristic
49#13133205—21no—
50#13133191—21no—
51#13133191D412yes—
52C4×D212—21no—
53C4×D212C2312yes—
54#13133150—24no—
55C4⋊#88322—21no—
56C4⋊#88322C2312yes—
57C4⋊#88322C2312yes—
58C4⋊#88322C2312yes—
59#13133141—24no—
60#13133137—24no—
61C22⋊#88322—21no—
62C22⋊#88322—21no—
63#13133129—22no—
64#13133129—22no—
65C4×#88322—21no—
66C4×#88322—21no—
67C22×Q8×C53—21no—
68C22×Q8×C53—21no—
69C22×C8×C53D411yescharacteristic
70C22×C8×C53D411yescharacteristic
71C4⋊Q8×C53—21no—
72C4⋊Q8×C53—21no—
73C4⋊Q8×C53—21no—
74C4⋊Q8×C53—21no—
75C4⋊Q8×C53—21no—
76C4⋊Q8×C53—21no—
77C4⋊Q8×C53C2312yes—
78C4⋊Q8×C53C2312yes—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53—21no—
83C2×C42×C53C2311yescharacteristic
84#103×C53—22no—
85#103×C53—22no—
86#103×C53—22no—
87#103×C53—22no—
88C4×C8×C53D411yescharacteristic
89C4×C8×C53D411yescharacteristic
90C4×C8×C53C2311yescharacteristic
91C4×C8×C53C2311yescharacteristic
92C2×Q8×C53—22no—
93C2×Q8×C53—22no—
94C2×Q8×C53—41no—
95C2×Q8×C53—41no—
96C2×Q8×C53—42no—
97C2×Q8×C53—42no—
98C2×Q8×C53—82no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C2×C22⋊D53—22no—
105#11719945—42no—
106#11719945—42no—
107C2×D212—21no—
108C2×D212—21no—
109C2×D212C2×D412yes—
110C2×C4×D53—22no—
111C2×Dic106—21no—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C2×C8×C53C2×D411yescharacteristic
115C2×C8×C53C2×D411yescharacteristic
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53C2×D411yescharacteristic
118C4⋊C4×C53—22no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53—41no—
124C4⋊C4×C53—41no—
125C4⋊C4×C53—41no—
126C42×C53C2×D411yescharacteristic
127C42×C53C2×D411yescharacteristic
128C42×C53#4111yescharacteristic
129C42×C53C2411yescharacteristic
130#11719924—84no—
131#11719921—22no—
132#11719921—22no—
133C4⋊Dic53—21no—
134C4⋊Dic53C2×D412yes—
135#11719917—42no—
136C2×#88322—22no—
137C2×#88322—22no—
138C2×#88322—22no—
139#11719912—44no—
140C8×D53—44no—
141C23×C53#12111yescharacteristic
142C22×D53—22no—
143Q8×C53—42no—
144Q8×C53—42no—
145Q8×C53—44no—
146Q8×C53—82no—
147Q8×C53—82no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—82no—
153C2×C4×C53—82no—
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#13512yes—
158C2×C4×C53AGL(1,ℤ/8)12yes—
159C2×C4×C53C22×D411yescharacteristic
160C2×C4×C53C22×D411yescharacteristic
161C2×C4×C53C2×#4111yescharacteristic
162C22⋊D53—44no—
163C2×Dic53—22no—
164D212—24no—
165D212—42no—
166D212—42no—
167C4×D53—42no—
168C4×D53—42no—
169Dic106—42no—
170C8×C53—21no—
171C8×C53—21no—
172C8×C53—21no—
173C8×C53—21no—
174#88322—42no—
175#88322—42no—
176#88322—42no—
177#19197—531nomaximal, Sylow
178C22×C53—21no—
179C22×C53—21no—
180C22×C53#46312yes—
181C22×C53C2×#12111yescharacteristic
182C2×D53—42no—
183C2×D53—42no—
184C2×D53—42no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—82no—
188C4×C53—82no—
189C4×C53#46312yes—
190C4×C53#47312yes—
191C4×C53C2×#13511yescharacteristic
192C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
193C4×C53#57712yes—
194Dic53—42no—
195#3226—531no—
196#3206—531no—
197C2×#487D5311yescharacteristic, complemented
198#2931—531no—
199#1699—534no—
200#1679—534no—
201#1564—531no—
202#1549—531no—
203#1546—531no—
204C2×C53—22no—
205C2×C53C2×#46311yescharacteristic
206C2×C53#305711yescharacteristic
207C2×C53C2×#47311yescharacteristic
208D53—82no—
209C2×#131—1061no—
210#532—531no—
211C2×C4⋊Q8C2×D5311yescharacteristic
212C2×C4⋊Q8C2×D5311yescharacteristic
213#500—1061no—
214#492—1061no—
215#487—21no—
216#487—21no—
217#487—1061no—
218#487C2×D5312yes—
219#487C2×D5312yes—
220#486—1061no—
221#480—1062no—
222#474—534no—
223#461—534no—
224#448—1062no—
225#436—532no—
226#436—1061no—
227C8×D4—532no—
228C8×D4—1061no—
229#431—531no—
230#431—531no—
231#423—531no—
232C2×#103—21no—
233C2×#103—21no—
234#408—1061no—
235C2×C4×C8C2×D5311yescharacteristic
236#335—532no—
237#334—532no—
238#332—534no—
239#332—534no—
240C53#1919711yescharacteristic, complemented, Sylow
241C2×#41—1061no—
242C22×Q8—21no—
243C22×Q8—21no—
244C2×SD16—1064no—
245#131—2124no—
246C2×M16—1061no—
247C2×M16—1062no—
248C22×C8—1062no—
249C22×C8C22⋊D5311yescharacteristic
250C22×C8C22⋊D5311yescharacteristic
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—531no—
258C4⋊Q8C22×D5312yes—
259C4⋊Q8C22×D5312yes—
260#127—531no—
261#124—1061no—
262#121—532no—
263#121—1061no—
264C4×D4—532no—
265C4×D4—1061no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C42C22×D5311yescharacteristic
271#107—1064no—
272C4⋊C8—532no—
273C4⋊C8—532no—
274C4⋊C8—532no—
275C4⋊C8—1061no—
276#103—22no—
277#103—22no—
278#103—22no—
279#103—22no—
280#103—1064no—
281#102—1064no—
282C22⋊C8—1061no—
283C22⋊C8—1061no—
284C22⋊C8—1062no—
285C22⋊C8—1062no—
286C4×C8—1061no—
287C4×C8—1061no—
288C4×C8C22⋊D5311yescharacteristic
289C4×C8C22⋊D5311yescharacteristic
290C4×C8C22×D5311yescharacteristic
291C4×C8C22×D5311yescharacteristic
292#41—2122no—
293#41—2122no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—82no—
301C2×Q8—1061no—
302C2×D4—532no—
303C2×D4—1062no—
304C2×D4—1061no—
305C2×D4—1061no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—1062no—
309C22×C4D4×D5311yescharacteristic
310C22×C4#1171994711yescharacteristic
311C22×C4C2×C22⋊D5311yescharacteristic
312SD16—4244no—
313M16—2124no—
314M16—2122no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—1062no—
318C2×C8—1062no—
319C2×C8—1062no—
320C2×C8—2124no—
321C2×C8C2×C22⋊D5311yescharacteristic
322C2×C8C2×C22⋊D5311yescharacteristic
323C2×C8C2×C22⋊D5311yescharacteristic
324C2×C8C2×C22⋊D5311yescharacteristic
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—532no—
334C4⋊C4—1061no—
335C22⋊C4—1062no—
336C22⋊C4—1062no—
337C42D4×D5311yescharacteristic
338C42#1171994711yescharacteristic
339C42C2×C22⋊D5311yescharacteristic
340C42C23×D5311yescharacteristic
341C23—1062no—
342C23#1313318811yescharacteristic
343Q8—42no—
344Q8—42no—
345Q8—44no—
346Q8—82no—
347Q8—82no—
348Q8—2122no—
349D4—1064no—
350D4—2124no—
351D4—2122no—
352D4—2122no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—1062no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4#1313318811yescharacteristic
363C2×C4#1313318811yescharacteristic
364C2×C4#1313318811yescharacteristic
365C2×C4#1313323712yes—
366C2×C4#1313324212yes—
367C2×C4C2×D4×D5311yescharacteristic
368C2×C4C22×C22⋊D5311yescharacteristic
369C2×C4C2×#1171994711yescharacteristic
370C8—21no—
371C8—21no—
372C8—21no—
373C8—21no—
374C8—2122no—
375C8—2122no—
376C8—2122no—
377C22—21no—
378C22—21no—
379C22—2122no—
380C22—2122no—
381C22—2122no—
382C22#1681083512yes—
383C22C2×#1313318811yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—82no—
388C4—2122no—
389C4#1681083512yes—
390C4#1681089512yes—
391C4C2×#1313323711yescharacteristic
392C4C2×#1313324211yescharacteristic
393C4#1681142912yes—
394C2—22no—
395C2—4242no—
396C2C2×#1681083511yescharacteristic
397C2#2028435011yescharacteristic
398C2C2×#1681089511yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#487C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterC2×#487D53
hypocenterC53#19197
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #20274259C2semidirect
5. #20274135C2semidirect
6. #20273463C2semidirect
7. #20273400C2semidirect
8. #20273387C2semidirect
9. C2×C53×#487C2semidirect
197. C2×#487D53semidirect
240. C53#19197semidirect

Maximal subgroups

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

subgroupindex
#202858362
#202856402
#202834282
#202742592
#202741352
#202734632
#202734002
#202733872
C2×C53×#4872
#1919753