Finite Group Databasebeta

← #15836774

Subgroups — #15836774

Order 3328.

1064 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125608C212yesmaximal, complemented
2#13120641C211yesmaximal, characteristic
3#13113861C214yesmaximal, complemented
4#13113793C214yesmaximal, complemented
5#13113408C212yesmaximal, complemented
6C2×#11718117C211yesmaximal, characteristic
7C2×C13×#543C211yesmaximal, characteristic
8C13×#543—24no—
9C13×#543—22no—
10C13×#543C2214yes—
11C2×C13×C22⋊Q8—22no—
12C2×C13×C22⋊Q8—21no—
13C2×C13×C22⋊Q8—22no—
14C2×C13×C22⋊Q8C2212yes—
15C22×C13×C22⋊C4—21no—
16C22×C13×C22⋊C4C2211yescharacteristic
17C2×#88178C2211yescharacteristic
18#11718908—21no—
19#11718889—22no—
20#11718873C2211yescharacteristic
21#11718871—21no—
22C2×C22⋊#1055C2212yes—
23#11718712—24no—
24#11718711C2214yescomplemented
25#11718705C2214yescomplemented
26#11718631—22no—
27#11718610C2212yes—
28#11718609—22no—
29#11718608C2212yes—
30#11718130C2218yescomplemented
31#11718117C2214yes—
32C25×C13D411yescharacteristic
33C22×Q8×C13—21no—
34C22×Q8×C13D412yes—
35C23×C4×C13—21no—
36C23×C4×C13—21no—
37C23×C4×C13D411yescharacteristic
38C23×C4×C13C2311yescharacteristic
39C22×Dic26C2311yescharacteristic
40C13×C22⋊Q8—28no—
41C13×C22⋊Q8—28no—
42C13×C22⋊Q8—24no—
43C13×C22⋊Q8—28no—
44C13×C22⋊Q8—24no—
45C13×C22⋊Q8—22no—
46C13×C22⋊Q8—22no—
47C13×C22⋊Q8—28no—
48C13×C22⋊Q8D414yes—
49C13×C22⋊Q8C2314yes—
50C2×C13×C4⋊C4—21no—
51C2×C13×C4⋊C4—22no—
52C2×C13×C4⋊C4—22no—
53C2×C13×C4⋊C4D412yes—
54C2×C13×C22⋊C4—22no—
55C2×C13×C22⋊C4—24no—
56C2×C13×C22⋊C4—22no—
57C2×C13×C22⋊C4—24no—
58C2×C13×C22⋊C4—22no—
59C2×C13×C22⋊C4—22no—
60C2×C13×C22⋊C4D412yes—
61C2×C13×C22⋊C4C2312yes—
62C2×C22⋊Dic13—21no—
63C2×C22⋊Dic13—21no—
64C2×C22⋊Dic13—22no—
65C2×C22⋊Dic13—22no—
66C2×#6107—24no—
67C2×C4⋊Dic13—21no—
68C2×C4⋊Dic13C2311yescharacteristic
69#88178—22no—
70#88178—22no—
71C2×#6101—21no—
72C2×#6101—22no—
73C22×#1055—22no—
74#88075—24no—
75#88071—22no—
76#88071C2312yes—
77C22⋊#1055—22no—
78C22⋊#1055C2314yes—
79#88050—24no—
80#88047—24no—
81#18287—131nomaximal, Sylow
82C24×C13—21no—
83C24×C13—21no—
84C24×C13—22no—
85C24×C13—81no—
86C24×C13—81no—
87C24×C13Q1614yescomplemented
88C24×C13C2×D412yes—
89C24×C13C2×D411yescharacteristic
90C2×Q8×C13—24no—
91C2×Q8×C13—22no—
92C2×Q8×C13—82no—
93C2×Q8×C13—82no—
94C2×Q8×C13C2×D414yes—
95C22×C4×C13—22no—
96C22×C4×C13—21no—
97C22×C4×C13—22no—
98C22×C4×C13—24no—
99C22×C4×C13—24no—
100C22×C4×C13—21no—
101C22×C4×C13—81no—
102C22×C4×C13—81no—
103C22×C4×C13—82no—
104C22×C4×C13—82no—
105C22×C4×C13C2×D412yes—
106C22×C4×C13C2×D411yescharacteristic
107C22×C4×C13C2×D412yes—
108C22×C4×C13C2×D412yes—
109C22×C4×C13C2411yescharacteristic
110C22×Dic13—21no—
111C22×Dic13—22no—
112C22×Dic13—21no—
113C2×Dic26—22no—
114C2×Dic26—81no—
115C13×C4⋊C4—24no—
116C13×C4⋊C4—28no—
117C13×C4⋊C4—24no—
118C13×C4⋊C4—22no—
119C13×C4⋊C4—24no—
120C13×C4⋊C4C2×D414yes—
121C13×C22⋊C4—28no—
122C13×C22⋊C4—24no—
123C13×C22⋊C4—24no—
124C13×C22⋊C4—24no—
125C13×C22⋊C4—24no—
126C22⋊Dic13—41no—
127C22⋊Dic13—41no—
128C22⋊Dic13—42no—
129C22⋊Dic13—82no—
130C22⋊Dic13—82no—
131#6107—84no—
132C4⋊Dic13—22no—
133C4⋊Dic13—81no—
134#6101—42no—
135#6101—82no—
136C2×#1055—24no—
137C2×#1055—82no—
138C2×#543D1311yescharacteristic
139#3189—132no—
140#2626—131no—
141#1614—134no—
142#1602—134no—
143#1520—132no—
144C2×#323—131no—
145C23×C13—22no—
146C23×C13—24no—
147C23×C13—22no—
148C23×C13—24no—
149C23×C13—41no—
150C23×C13—41no—
151C23×C13—42no—
152C23×C13—42no—
153C23×C13—42no—
154C23×C13—42no—
155C23×C13—81no—
156C23×C13—81no—
157C23×C13—84no—
158C23×C13—84no—
159C23×C13#12011yescharacteristic
160C23×C13#12011yescharacteristic
161C23×C13C2×Q1614yes—
162C23×C13C2×Q1612yes—
163C23×C13#13712yes—
164C23×C13C22×D411yescharacteristic
165Q8×C13—44no—
166Q8×C13—82no—
167Q8×C13—84no—
168C2×C4×C13—24no—
169C2×C4×C13—22no—
170C2×C4×C13—24no—
171C2×C4×C13—42no—
172C2×C4×C13—41no—
173C2×C4×C13—41no—
174C2×C4×C13—42no—
175C2×C4×C13—42no—
176C2×C4×C13—44no—
177C2×C4×C13—44no—
178C2×C4×C13—42no—
179C2×C4×C13—81no—
180C2×C4×C13—84no—
181C2×C4×C13—84no—
182C2×C4×C13#12012yes—
183C2×C4×C13C22×D412yes—
184C2×Dic13—42no—
185C2×Dic13—42no—
186C2×Dic13—42no—
187C2×Dic13—42no—
188C2×Dic13—41no—
189C2×Dic13—41no—
190C2×Dic13—42no—
191Dic26—44no—
192#1055—44no—
193#543—24no—
194#543—22no—
195#543C2×D1314yes—
196C2×C22⋊Q8—22no—
197C2×C22⋊Q8—21no—
198C2×C22⋊Q8—22no—
199C2×C22⋊Q8—131no—
200C2×C22⋊Q8C2×D1312yes—
201C22×C22⋊C4—21no—
202C22×C22⋊C4C2×D1311yescharacteristic
203#484—134no—
204#484—262no—
205#467—264no—
206#451—134no—
207#451—262no—
208C2×C22⋊C8—132no—
209#399—132no—
210#396—262no—
211#393—132no—
212#385—261no—
213#380—131no—
214#379—261no—
215#328—138no—
216#323—134no—
217C22×C13—21no—
218C22×C13—21no—
219C22×C13—21no—
220C22×C13—21no—
221C22×C13—21no—
222C22×C13—21no—
223C22×C13—24no—
224C22×C13—42no—
225C22×C13—42no—
226C22×C13—44no—
227C22×C13—48no—
228C22×C13—42no—
229C22×C13—42no—
230C22×C13—82no—
231C22×C13—84no—
232C22×C13#45112yes—
233C22×C13#45112yes—
234C22×C13C2×#12011yescharacteristic
235C22×C13C22×Q1611yescharacteristic
236C22×C13C2×#13711yescharacteristic
237C4×C13—22no—
238C4×C13—22no—
239C4×C13—42no—
240C4×C13—44no—
241C4×C13—44no—
242Dic13—82no—
243Dic13—81no—
244Dic13—81no—
245C25C22⋊D1311yescharacteristic
246C22×Q8—21no—
247C22×Q8—131no—
248C22×Q8C22⋊D1312yes—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4C22⋊D1311yescharacteristic
252C23×C4C22×D1311yescharacteristic
253C2×Q16—264no—
254C22×C8—262no—
255C22⋊Q8—28no—
256C22⋊Q8—28no—
257C22⋊Q8—24no—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262C22⋊Q8—28no—
263C22⋊Q8—262no—
264C22⋊Q8—262no—
265C22⋊Q8C22⋊D1314yes—
266C22⋊Q8C22×D1314yes—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—131no—
271C2×C4⋊C4—261no—
272C2×C4⋊C4—262no—
273C2×C4⋊C4—261no—
274C2×C4⋊C4C22⋊D1312yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—261no—
282C2×C22⋊C4—261no—
283C2×C22⋊C4—262no—
284C2×C22⋊C4—262no—
285C2×C22⋊C4C22⋊D1312yes—
286C2×C22⋊C4C22×D1312yes—
287#107—264no—
288#103—264no—
289#103—264no—
290C22⋊C8—134no—
291C22⋊C8—262no—
292#95—132no—
293#95—262no—
294C2×C13—21no—
295C2×C13—21no—
296C2×C13—22no—
297C2×C13—22no—
298C2×C13—44no—
299C2×C13C2×#45111yescharacteristic
300C2×C13#301911yescharacteristic
301C2×C13#302611yescharacteristic
302C24—21no—
303C24—22no—
304C24—21no—
305C24—81no—
306C24—81no—
307C24#610714yescomplemented
308C24C2×C22⋊D1312yes—
309C24C2×C22⋊D1311yescharacteristic
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—82no—
313C2×Q8—82no—
314C2×Q8—262no—
315C2×Q8—1041no—
316C2×Q8C2×C22⋊D1314yes—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—262no—
328C22×C4—261no—
329C22×C4—261no—
330C22×C4C2×C22⋊D1312yes—
331C22×C4C2×C22⋊D1311yescharacteristic
332C22×C4C2×C22⋊D1312yes—
333C22×C4C2×C22⋊D1312yes—
334C22×C4C23×D1311yescharacteristic
335Q16—1044no—
336C2×C8—264no—
337C2×C8—1042no—
338C4⋊C4—24no—
339C4⋊C4—28no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—262no—
344C4⋊C4—522no—
345C4⋊C4—1042no—
346C4⋊C4—1041no—
347C4⋊C4C2×C22⋊D1314yes—
348C22⋊C4—28no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—522no—
354C22⋊C4—521no—
355C22⋊C4—521no—
356C22⋊C4—1042no—
357C22⋊C4—1042no—
358C13#1828711yescharacteristic, complemented, Sylow
359C23—22no—
360C23—24no—
361C23—22no—
362C23—24no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—84no—
372C23—84no—
373C23#8819612yes—
374C23C2×#610714yes—
375C23C2×#610712yes—
376C23#8820711yescharacteristic
377C23#8820711yescharacteristic
378C23C22×C22⋊D1311yescharacteristic
379Q8—44no—
380Q8—82no—
381Q8—84no—
382Q8—524no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—81no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—522no—
398C2×C4—522no—
399C2×C4—522no—
400C2×C4—522no—
401C2×C4—522no—
402C2×C4—521no—
403C2×C4—521no—
404C2×C4#8820712yes—
405C2×C4C22×C22⋊D1312yes—
406C8—524no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—24no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—48no—
418C22—42no—
419C22—42no—
420C22—82no—
421C22—84no—
422C22#1171888912yes—
423C22#1171888912yes—
424C22C2×#8819611yescharacteristic
425C22C22×#610711yescharacteristic
426C22C2×#8820711yescharacteristic
427C4—22no—
428C4—22no—
429C4—42no—
430C4—44no—
431C4—44no—
432C4—1042no—
433C4—1041no—
434C4—1041no—
435C2—21no—
436C2—21no—
437C2—22no—
438C2—22no—
439C2—44no—
440C2C2×#1171888911yescharacteristic
441C2#1312419311yescharacteristic
442C2#1312423311yescharacteristic
subgroupG/H
centerC22C2×#88207
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC2×C13×#543C2
socleC22×C13C2×#120
prime radicalC25×C13D4
hypercenterC2×#543D13
hypocenterC13#18287
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #13125608C2semidirect
3. #13113861C2semidirect
4. #13113793C2semidirect
5. #13113408C2semidirect
24. #11718711C22semidirect
25. #11718705C22semidirect
30. #11718130C22semidirect
87. C24×C13Q16semidirect
307. C24#6107semidirect
358. C13#18287semidirect

Maximal subgroups

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

subgroupindex
#131256082
#131206412
#131138612
#131137932
#131134082
C2×#117181172
C2×C13×#5432
#1828713