Finite Group Databasebeta

← #15755515

Subgroups — #15755515

Order 3328.

1008 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13114764C211yesmaximal, characteristic, complemented
2#13114748C211yesmaximal, characteristic, complemented
3#13113184C214yesmaximal, complemented
4C13×#3485C211yesmaximal, characteristic
5C2×C13×H(5,𝔽2)C411yescharacteristic
6C22×C13×#41C2211yescharacteristic
7C13×#547—24no—
8C13×#546—24no—
9C13×#536—21no—
10C13×#534—21no—
11C13×#534C412yes—
12C13×#525—21no—
13C2×C13×#123—21no—
14C2×C13×#123C2212yes—
15C13×#518C411yescharacteristic
16C2×#88076—21no—
17C2×#88073—21no—
18#11718868C2212yescomplemented
19C2×C22⋊#1055—21no—
20C2×C22⋊#1055—21no—
21C2×C22⋊#1055C2212yescomplemented
22#11718120—24no—
23#11718117—24no—
24C13×H(5,𝔽2)—42no—
25C13×H(5,𝔽2)—42no—
26C2×C13×#41—21no—
27C2×C13×#41—22no—
28C2×C13×#41—24no—
29C2×C13×#41—42no—
30C2×C13×#41—42no—
31C22×Q8×C13D411yescharacteristic
32C22×D4×C13—21no—
33C22×D4×C13—22no—
34C22×D4×C13C2×C411yescharacteristic
35C22×D4×C13C2×C412yes—
36C22×D4×C13C2×C411yescharacteristic
37C22×D4×C13D411yescharacteristic
38C23×C4×C13D411yescharacteristic
39C23×C4×C13D411yescharacteristic
40C23×C4×C13C2311yescharacteristic
41C13×C4⋊Q8—22no—
42C13×#124—22no—
43C13×#123—24no—
44C13×#123—22no—
45C13×#123—22no—
46C13×#123—22no—
47C13×#123—22no—
48C13×#123—22no—
49C13×#123—24no—
50C13×C22⋊Q8—22no—
51C13×C22⋊Q8—24no—
52C13×#121—22no—
53C13×#121—24no—
54C13×#121—24no—
55C13×#120—22no—
56C13×#120—22no—
57C13×#120—22no—
58C4×D4×C13—24no—
59C4×D4×C13—24no—
60C13×#117—21no—
61C2×C13×C4⋊C4—22no—
62C2×C13×C4⋊C4D414yes—
63C2×C13×C22⋊C4—22no—
64C2×C13×C22⋊C4—21no—
65C2×C13×C22⋊C4C2×C412yes—
66C2×C13×C22⋊C4D414yes—
67C2×#6099—22no—
68C22×#1055—22no—
69#88076—42no—
70#88073—42no—
71C22⋊#1055—24no—
72C22⋊#1055—24no—
73C22⋊#1055—42no—
74C22⋊#1055—42no—
75#8539—131nomaximal, Sylow
76C24×C13—21no—
77C24×C13C22⋊C411yescharacteristic
78C24×C13C22⋊C411yescharacteristic
79C24×C13C22⋊C412yes—
80C24×C13C22×C411yescharacteristic
81C13×#41—42no—
82C13×#41—416no—
83C13×#41—48no—
84C13×#41—42no—
85C2×Q8×C13—21no—
86C2×Q8×C13—22no—
87C2×Q8×C13—22no—
88C2×Q8×C13—41no—
89C2×D4×C13—22no—
90C2×D4×C13—22no—
91C2×D4×C13—22no—
92C2×D4×C13—21no—
93C2×D4×C13—22no—
94C2×D4×C13—22no—
95C2×D4×C13—22no—
96C2×D4×C13—22no—
97C2×D4×C13—24no—
98C2×D4×C13—22no—
99C2×D4×C13—22no—
100C2×D4×C13—41no—
101C2×D4×C13—41no—
102C2×D4×C13—41no—
103C2×D4×C13—48no—
104C2×D4×C13—42no—
105C2×D4×C13—42no—
106C2×D4×C13—42no—
107C2×D4×C13—42no—
108C22×C4×C13—21no—
109C22×C4×C13—22no—
110C22×C4×C13—22no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—41no—
116C22×C4×C13—41no—
117C22×C4×C13—41no—
118C22×C4×C13—41no—
119C22×C4×C13C22⋊C411yescharacteristic
120C22×C4×C13C22⋊C411yescharacteristic
121C22×C4×C13C22⋊C414yes—
122C22×C4×C13C22⋊C412yes—
123C22×C4×C13C2×D411yescharacteristic
124C22×C4×C13C2×D411yescharacteristic
125C22×C4×C13C2×D414yes—
126C13×C4⋊C4—24no—
127C13×C4⋊C4—22no—
128C13×C4⋊C4—24no—
129C13×C4⋊C4—24no—
130C13×C22⋊C4—22no—
131C13×C22⋊C4—24no—
132C13×C22⋊C4—24no—
133C13×C22⋊C4—22no—
134C13×C22⋊C4—22no—
135C13×C22⋊C4—24no—
136C13×C22⋊C4—24no—
137C42×C13—22no—
138#6099—42no—
139#6099—42no—
140C2×#1055—42no—
141C2×#1055—42no—
142C2×#1055—42no—
143C2×#1055—42no—
144#3485D1311yescharacteristic
145#1785—131no—
146#1780—131no—
147#1468—134no—
148C23×C13—21no—
149C23×C13—21no—
150C23×C13—21no—
151C23×C13—21no—
152C23×C13—22no—
153C23×C13—21no—
154C23×C13—21no—
155C23×C13—21no—
156C23×C13—42no—
157C23×C13—42no—
158C23×C13—42no—
159C23×C13—42no—
160C23×C13—42no—
161C23×C13—44no—
162C23×C13—82no—
163C23×C13#9912yes—
164C23×C13C2×C22⋊C412yes—
165C23×C13C2×C22⋊C411yescharacteristic
166Q8×C13—44no—
167Q8×C13—42no—
168D4×C13—28no—
169D4×C13—44no—
170D4×C13—42no—
171D4×C13—44no—
172D4×C13—44no—
173D4×C13—44no—
174D4×C13—42no—
175D4×C13—42no—
176D4×C13—84no—
177C2×C4×C13—21no—
178C2×C4×C13—21no—
179C2×C4×C13—21no—
180C2×C4×C13—21no—
181C2×C4×C13—21no—
182C2×C4×C13—21no—
183C2×C4×C13—24no—
184C2×C4×C13—21no—
185C2×C4×C13—22no—
186C2×C4×C13—21no—
187C2×C4×C13—21no—
188C2×C4×C13—44no—
189C2×C4×C13—44no—
190C2×C4×C13—44no—
191C2×C4×C13—44no—
192C2×C4×C13—42no—
193C2×C4×C13—44no—
194C2×C4×C13—42no—
195C2×C4×C13—82no—
196C2×C4×C13#9912yes—
197C2×C4×C13#12012yes—
198C2×C4×C13#12012yes—
199#1055—82no—
200#1055—82no—
201C2×H(5,𝔽2)Dic1311yescharacteristic
202C22×#41C2×D1311yescharacteristic
203#547—24no—
204#546—24no—
205#536—21no—
206#534—21no—
207#534Dic1312yes—
208#525—21no—
209C2×#123—21no—
210C2×#123C2×D1312yes—
211#518Dic1311yescharacteristic
212C2×#101—261no—
213C2×#100—261no—
214#407—132no—
215C2×C22⋊C8—132no—
216C2×C22⋊C8—261no—
217C2×C22⋊C8—261no—
218#324—264no—
219#323—264no—
220C22×C13—22no—
221C22×C13—22no—
222C22×C13—22no—
223C22×C13—24no—
224C22×C13—42no—
225C22×C13—42no—
226C22×C13—42no—
227C22×C13—41no—
228C22×C13—42no—
229C22×C13—42no—
230C22×C13—44no—
231C22×C13—81no—
232C22×C13—81no—
233C22×C13—81no—
234C22×C13—81no—
235C22×C13—162no—
236C22×C13—162no—
237C22×C13#37911yescharacteristic
238C22×C13C2×#9911yescharacteristic
239C22×C13C2×#9911yescharacteristic
240C4×C13—22no—
241C4×C13—22no—
242C4×C13—22no—
243C4×C13—41no—
244C4×C13—42no—
245C4×C13—44no—
246H(5,𝔽2)—42no—
247H(5,𝔽2)—42no—
248C2×#41—21no—
249C2×#41—22no—
250C2×#41—24no—
251C2×#41—42no—
252C2×#41—42no—
253C22×Q8C22⋊D1311yescharacteristic
254C22×D4—21no—
255C22×D4—22no—
256C22×D4C2×Dic1311yescharacteristic
257C22×D4C2×Dic1312yes—
258C22×D4C2×Dic1311yescharacteristic
259C22×D4C22⋊D1311yescharacteristic
260C23×C4C22⋊D1311yescharacteristic
261C23×C4C22⋊D1311yescharacteristic
262C23×C4C22×D1311yescharacteristic
263C2×M16—262no—
264C22×C8—262no—
265C4⋊Q8—22no—
266#124—22no—
267#123—24no—
268#123—22no—
269#123—22no—
270#123—22no—
271#123—22no—
272#123—22no—
273#123—24no—
274C22⋊Q8—22no—
275C22⋊Q8—24no—
276#121—22no—
277#121—24no—
278#121—24no—
279#120—22no—
280#120—22no—
281#120—22no—
282C4×D4—24no—
283C4×D4—24no—
284#117—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4C22⋊D1314yes—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4C2×Dic1312yes—
290C2×C22⋊C4C22⋊D1314yes—
291#101—522no—
292#100—522no—
293C22⋊C8—264no—
294C22⋊C8—264no—
295C22⋊C8—522no—
296C22⋊C8—522no—
297C2×C13—21no—
298C2×C13—21no—
299C2×C13—42no—
300C2×C13—42no—
301C2×C13—81no—
302C2×C13—82no—
303C2×C13#178111yescharacteristic
304C2×C13#179112yes—
305C24—21no—
306C24C22⋊Dic1311yescharacteristic
307C24C22⋊Dic1311yescharacteristic
308C24C22⋊Dic1312yes—
309C24C22×Dic1311yescharacteristic
310#41—42no—
311#41—416no—
312#41—48no—
313#41—42no—
314C2×Q8—21no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—21no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—48no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4C22⋊Dic1311yescharacteristic
349C22×C4C22⋊Dic1311yescharacteristic
350C22×C4C22⋊Dic1314yes—
351C22×C4C22⋊Dic1312yes—
352C22×C4C2×C22⋊D1311yescharacteristic
353C22×C4C2×C22⋊D1311yescharacteristic
354C22×C4C2×C22⋊D1314yes—
355M16—522no—
356M16—522no—
357C2×C8—522no—
358C2×C8—522no—
359C2×C8—522no—
360C2×C8—522no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C22⋊C4—22no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C42—22no—
373C13#853911yescharacteristic, complemented, Sylow
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—22no—
380C23—21no—
381C23—21no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—44no—
388C23—82no—
389C23#8807412yes—
390C23C2×C22⋊Dic1312yes—
391C23C2×C22⋊Dic1311yescharacteristic
392Q8—44no—
393Q8—42no—
394D4—28no—
395D4—44no—
396D4—42no—
397D4—44no—
398D4—44no—
399D4—44no—
400D4—42no—
401D4—42no—
402D4—84no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—82no—
422C2×C4#8807412yes—
423C2×C4#8820712yes—
424C2×C4#8820712yes—
425C8—1042no—
426C8—1042no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—24no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—41no—
435C22—42no—
436C22—42no—
437C22—44no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—162no—
443C22—162no—
444C22C2×#8807411yescharacteristic
445C22C2×#8807411yescharacteristic
446C22#1171890811yescharacteristic
447C4—22no—
448C4—22no—
449C4—22no—
450C4—41no—
451C4—42no—
452C4—44no—
453C2—21no—
454C2—21no—
455C2—42no—
456C2—42no—
457C2—81no—
458C2—82no—
459C2#1311475111yescharacteristic
460C2#1311478712yes—
subgroupG/H
centerC22#11718908
commutatorC24×C13C22×C4
FrattiniC23×C4C22×D13
Fitting, prime radicalC13×#3485C2
socleC22×C13#379
cosocleC23×C4×C13C23
hypercenter#3485D13
hypocenterC13#8539
LCSG ▷ C24×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3485

Complements

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

NH ≅ G/Ntype
1. #13114764C2semidirect
2. #13114748C2semidirect
3. #13113184C2semidirect
18. #11718868C22semidirect
21. C2×C22⋊#1055C22semidirect
373. C13#8539semidirect

Maximal subgroups

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

subgroupindex
#131147642
#131147482
#131131842
C13×#34852
#853913