Finite Group Databasebeta

← #15765342

Subgroups — #15765342

Order 3328.

556 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 105 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#13125980C211yesmaximal, characteristic, complemented
2#13124633C211yesmaximal, characteristic
3#13113508C211yesmaximal, characteristic, complemented
4#13113334C211yesmaximal, characteristic, complemented
5#13113227C211yesmaximal, characteristic
6#13112739C211yesmaximal, characteristic, complemented
7C13×#2547C211yesmaximal, characteristic
8C2×C13×C4⋊Q8C2211yescharacteristic
9C2×C13×#124—21no—
10C2×C4×D4×C13C2211yescharacteristic
11#11719175C2211yescharacteristic
12C13×#397—22no—
13C13×#391—21no—
14C13×#380—22no—
15C4×C13×C4⋊C4C2211yescharacteristic
16#11718876—21no—
17#11718755—22no—
18#11718754—21no—
19#11718740—21no—
20#11718670—22no—
21#11718669—21no—
22#11718668—21no—
23#11718632—22no—
24#11718625—21no—
25#11718586C2211yescharacteristic
26#11718586C2211yescharacteristic
27#11718585C2211yescharacteristic
28#11718139—22no—
29D4⋊#1055—22no—
30#11718044—22no—
31C22×Q8×C13—21no—
32C22×D4×C13D411yescharacteristic
33C23×C4×C13—21no—
34C13×C4⋊Q8—21no—
35C13×C4⋊Q8—21no—
36C13×C4⋊Q8—21no—
37C13×C4⋊Q8—21no—
38C13×#124—42no—
39C13×#124—42no—
40C4×D4×C13—21no—
41C4×D4×C13—21no—
42C4×D4×C13—21no—
43C4×D4×C13—21no—
44C2×C13×C4⋊C4—21no—
45C2×C13×C4⋊C4—21no—
46C2×C13×C4⋊C4D412yes—
47C2×C13×C4⋊C4D411yescharacteristic
48C2×C13×C22⋊C4—24no—
49C2×C13×C22⋊C4—21no—
50C2×C42×C13—21no—
51C2×C42×C13C2311yescharacteristic
52#88180—21no—
53#88178—21no—
54#88178—21no—
55C2×#6099—21no—
56#88124—21no—
57#88123D411yescharacteristic
58#88123D411yescharacteristic
59C4×Dic26—21no—
60C4×Dic26—21no—
61C13×#95—24no—
62C13×#95D412yes—
63#88072—22no—
64#88072—22no—
65C22⋊#1055—21no—
66C22⋊#1055—21no—
67#88050—24no—
68#88048—22no—
69#88048—24no—
70#88047—22no—
71C4⋊#1055—21no—
72C4⋊#1055—21no—
73C4⋊#1055D412yes—
74C4⋊#1055D412yes—
75C4×#1055—21no—
76C4×#1055—21no—
77#10247—131nomaximal, Sylow
78C24×C13—21no—
79C2×Q8×C13—41no—
80C2×Q8×C13—41no—
81C2×Q8×C13—41no—
82C2×Q8×C13—41no—
83C2×Q8×C13—41no—
84C2×Q8×C13—41no—
85C2×D4×C13—21no—
86C2×D4×C13—21no—
87C2×D4×C13—21no—
88C2×D4×C13—21no—
89C2×D4×C13—21no—
90C2×D4×C13SD1612yes—
91C22×C4×C13—21no—
92C22×C4×C13—22no—
93C22×C4×C13—22no—
94C22×C4×C13—44no—
95C22×C4×C13—41no—
96C22×C4×C13—41no—
97C22×C4×C13C2×D411yescharacteristic
98C22×C4×C13C2×D411yescharacteristic
99C22×C4×C13C2×D411yescharacteristic
100C2×Dic26—21no—
101C2×Dic26—21no—
102C13×C4⋊C4—21no—
103C13×C4⋊C4—22no—
104C13×C4⋊C4—21no—
105C13×C4⋊C4—22no—
106C13×C4⋊C4—42no—
107C13×C4⋊C4—41no—
108C13×C4⋊C4—41no—
109C13×C22⋊C4—48no—
110C13×C22⋊C4—41no—
111C13×C22⋊C4—41no—
112C42×C13—41no—
113C42×C13—41no—
114C42×C13C2×D411yescharacteristic
115C42×C13C2×D411yescharacteristic
116C42×C13C2×D411yescharacteristic
117C42×C13#4111yescharacteristic
118C22⋊Dic13—41no—
119C4⋊Dic13—21no—
120C4⋊Dic13—21no—
121C4⋊Dic13—21no—
122C4⋊Dic13—21no—
123#6101—41no—
124#6101—41no—
125C4×Dic13—41no—
126#6099—41no—
127#6099—41no—
128C2×#1055—22no—
129C2×#1055—22no—
130C2×#1055—41no—
131#3229—131no—
132#3077—131no—
133#2547D1311yescharacteristic
134#1546—131no—
135#1503—131no—
136#1478—131no—
137#1303—131no—
138C23×C13—22no—
139C23×C13—42no—
140C23×C13—42no—
141C23×C13—41no—
142C23×C13—41no—
143C23×C13#12011yescharacteristic
144Q8×C13—81no—
145Q8×C13—81no—
146Q8×C13—81no—
147Q8×C13—81no—
148D4×C13—22no—
149D4×C13—22no—
150D4×C13—41no—
151D4×C13—41no—
152C2×C4×C13—22no—
153C2×C4×C13—41no—
154C2×C4×C13—42no—
155C2×C4×C13—42no—
156C2×C4×C13—42no—
157C2×C4×C13—41no—
158C2×C4×C13—42no—
159C2×C4×C13—42no—
160C2×C4×C13—42no—
161C2×C4×C13—42no—
162C2×C4×C13—82no—
163C2×C4×C13—82no—
164C2×C4×C13#12011yescharacteristic
165C2×C4×C13#12011yescharacteristic
166C2×C4×C13#12111yescharacteristic
167C2×C4×C13#12111yescharacteristic
168C2×C4×C13#12111yescharacteristic
169C2×C4×C13#12711yescharacteristic
170C2×C4×C13C2×SD1611yescharacteristic
171C2×C4×C13#13511yescharacteristic
172C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C13#13711yescharacteristic
174C2×Dic13—41no—
175C2×Dic13—41no—
176Dic26—42no—
177Dic26—81no—
178#1055—42no—
179#1055—42no—
180#1055—81no—
181#533—131no—
182C2×C4⋊Q8C2×D1311yescharacteristic
183C2×#124—21no—
184C2×C4×D4C2×D1311yescharacteristic
185#498—261no—
186#487—261no—
187#483—262no—
188#478—262no—
189#476—261no—
190#475—262no—
191#474—261no—
192#431—131no—
193#431—131no—
194#428—261no—
195#423—131no—
196#418—261no—
197#397—22no—
198#391—21no—
199#380—22no—
200C4×C4⋊C4C2×D1311yescharacteristic
201#332—262no—
202#328—262no—
203D4⋊C8—262no—
204C22×C13—21no—
205C22×C13—21no—
206C22×C13—44no—
207C22×C13—42no—
208C22×C13—82no—
209C22×C13—81no—
210C22×C13—81no—
211C22×C13#39211yescharacteristic
212C22×C13#45011yescharacteristic
213C22×C13#45211yescharacteristic
214C4×C13—21no—
215C4×C13—21no—
216C4×C13—21no—
217C4×C13—82no—
218C4×C13—82no—
219C4×C13—81no—
220C4×C13#44811yescharacteristic
221C4×C13#44911yescharacteristic
222C4×C13#45412yes—
223C4×C13#46511yescharacteristic
224C4×C13#46811yescharacteristic
225Dic13—81no—
226Dic13—81no—
227C22×Q8—21no—
228C22×D4C22⋊D1311yescharacteristic
229C23×C4—21no—
230C2×M16—261no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—131no—
236C4⋊Q8—131no—
237#125—261no—
238#124—42no—
239#124—42no—
240C22⋊Q8—261no—
241C22⋊Q8—261no—
242C4×Q8—261no—
243C4×Q8—261no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248#117—261no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C22⋊D1311yescharacteristic
252C2×C4⋊C4C22⋊D1312yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—21no—
255C2×C42—21no—
256C2×C42C22×D1311yescharacteristic
257#107—262no—
258#106—262no—
259#106—264no—
260C4⋊C8—132no—
261C4⋊C8—132no—
262C4⋊C8—261no—
263C4⋊C8—261no—
264#103—264no—
265#102—262no—
266#102—262no—
267C22⋊C8—261no—
268C22⋊C8—261no—
269C4×C8—261no—
270C4×C8—261no—
271#95—24no—
272#95C22⋊D1312yes—
273C2×C13—21no—
274C2×C13—21no—
275C2×C13—82no—
276C2×C13#200011yescharacteristic
277C2×C13#200311yescharacteristic
278C2×C13#200911yescharacteristic
279C24—21no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—261no—
287C2×Q8—261no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4#610512yes—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—44no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4C2×C22⋊D1311yescharacteristic
301C22×C4C2×C22⋊D1311yescharacteristic
302C22×C4C2×C22⋊D1311yescharacteristic
303M16—521no—
304M16—521no—
305C2×C8—262no—
306C2×C8—262no—
307C2×C8—521no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—261no—
316C4⋊C4—261no—
317C4⋊C4—261no—
318C4⋊C4—261no—
319C4⋊C4—521no—
320C4⋊C4—521no—
321C22⋊C4—48no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—521no—
325C42—41no—
326C42—41no—
327C42—521no—
328C42D4×D1311yescharacteristic
329C42D4×D1311yescharacteristic
330C42D4×D1311yescharacteristic
331C42#612911yescharacteristic
332C13#1024711yescharacteristic, complemented, Sylow
333C23—22no—
334C23—42no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23#8820711yescharacteristic
339Q8—81no—
340Q8—81no—
341Q8—81no—
342Q8—81no—
343Q8—522no—
344Q8—1041no—
345D4—22no—
346D4—22no—
347D4—41no—
348D4—41no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—521no—
362C2×C4—521no—
363C2×C4#8818611yescharacteristic
364C2×C4C2×#610511yescharacteristic
365C2×C4#8819111yescharacteristic
366C2×C4#8819111yescharacteristic
367C2×C4#8819211yescharacteristic
368C2×C4#8819311yescharacteristic
369C2×C4#8819311yescharacteristic
370C2×C4#8819411yescharacteristic
371C2×C4#8820411yescharacteristic
372C2×C4#8820511yescharacteristic
373C8—522no—
374C8—522no—
375C8—1041no—
376C22—21no—
377C22—21no—
378C22—44no—
379C22—42no—
380C22—82no—
381C22—81no—
382C22—81no—
383C22#1171887911yescharacteristic
384C22#1171888411yescharacteristic
385C22#1171890011yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—82no—
390C4—82no—
391C4—81no—
392C4—1041no—
393C4—1041no—
394C4#1171869811yescharacteristic
395C4#1171870111yescharacteristic
396C4#1171870211yescharacteristic
397C4#1171870311yescharacteristic
398C4#1171872312yes—
399C2—21no—
400C2—21no—
401C2—82no—
402C2#1311583911yescharacteristic
403C2#1311585811yescharacteristic
404C2#1311589611yescharacteristic
subgroupG/H
centerC22#11718884
commutator, cosocleC2×C42×C13C23
FrattiniC2×C42C22×D13
FittingC13×#2547C2
socleC22×C13#392
prime radicalC22×D4×C13D4
hypercenter#2547D13
hypocenterC13#10247
LCSG ▷ C2×C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
1. #13125980C2semidirect
3. #13113508C2semidirect
4. #13113334C2semidirect
6. #13112739C2semidirect
332. C13#10247semidirect

Maximal subgroups

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

subgroupindex
#131259802
#131246332
#131135082
#131133342
#131132272
#131127392
C13×#25472
#1024713