Finite Group Databasebeta

← #15758898

Subgroups — #15758898

Order 3328.

580 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 107 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#13123877C211yesmaximal, characteristic, complemented
2C4×#88074C211yesmaximal, characteristic, complemented
3#13113352C212yesmaximal, complemented
4#13112688C212yesmaximal, complemented
5C13×#2635C211yesmaximal, characteristic
6C2×C13×#121C2211yescharacteristic
7C2×C4×D4×C13C2211yescharacteristic
8C13×#400C411yescharacteristic, complemented
9C13×#399C411yescharacteristic, complemented
10C13×#394—21no—
11C13×#394C411yescharacteristic, complemented
12C13×#392—21no—
13C13×#392C411yescharacteristic, complemented
14C13×#388—21no—
15C13×#386—21no—
16C13×#381C2211yescharacteristic
17C2×#88074—21no—
18C4×C22⋊Dic13C2212yes—
19#11718868—21no—
20#11718868C2212yes—
21#11718673—24no—
22#11718586—21no—
23#11718129—24no—
24#11718118—21no—
25C22×D4×C13—21no—
26C22×D4×C13—21no—
27C22×D4×C13C2×C411yescharacteristic
28C22×D4×C13D411yescharacteristic
29C23×C4×C13D411yescharacteristic
30C23×C4×C13C2311yescharacteristic
31C13×#121—22no—
32C13×#121—22no—
33C4×D4×C13—22no—
34C4×D4×C13—22no—
35C2×C13×C4⋊C4—21no—
36C2×C13×C4⋊C4C2×C411yescharacteristic
37C2×C13×C4⋊C4C2×C411yescharacteristic
38C2×C13×C22⋊C4—21no—
39C2×C13×C22⋊C4—21no—
40C2×C13×C22⋊C4—21no—
41C2×C13×C22⋊C4—21no—
42C2×C13×C22⋊C4—21no—
43C2×C13×C22⋊C4—21no—
44C2×C13×C22⋊C4C2×C411yescharacteristic
45C2×C13×C22⋊C4D411yescharacteristic
46C2×C42×C13D411yescharacteristic
47C2×C22⋊Dic13—22no—
48C2×C4×Dic13—22no—
49C2×#6099—22no—
50C2×#6099—22no—
51C13×#95—21no—
52C13×#95—21no—
53C13×#95—21no—
54C13×#95C2×C411yescharacteristic
55C13×#95C2×C411yescharacteristic
56#88074—24no—
57#88071—22no—
58C22⋊#1055—22no—
59C22⋊#1055—22no—
60C22⋊#1055D414yescomplemented
61C4⋊#1055—22no—
62C4×#1055—22no—
63#9186—131nomaximal, Sylow
64C24×C13—21no—
65C24×C13C22⋊C411yescharacteristic
66C24×C13C2×D411yescharacteristic
67C2×D4×C13—22no—
68C2×D4×C13—22no—
69C2×D4×C13—22no—
70C2×D4×C13—41no—
71C2×D4×C13—42no—
72C2×D4×C13—42no—
73C2×D4×C13—81no—
74C2×D4×C13—81no—
75C2×D4×C13—81no—
76C2×D4×C13—81no—
77C2×D4×C13—81no—
78C22×C4×C13—21no—
79C22×C4×C13—21no—
80C22×C4×C13—21no—
81C22×C4×C13—21no—
82C22×C4×C13—21no—
83C22×C4×C13—81no—
84C22×C4×C13—81no—
85C22×C4×C13C22⋊C411yescharacteristic
86C22×C4×C13C22⋊C411yescharacteristic
87C22×C4×C13C22⋊C411yescharacteristic
88C22×C4×C13C22×C411yescharacteristic
89C22×C4×C13C2×D412yes—
90C22×C4×C13C2×D411yescharacteristic
91C22×C4×C13#4112yes—
92C22×Dic13—42no—
93C13×C4⋊C4—22no—
94C13×C4⋊C4—22no—
95C13×C4⋊C4—81no—
96C13×C22⋊C4—22no—
97C13×C22⋊C4—22no—
98C13×C22⋊C4—42no—
99C13×C22⋊C4—41no—
100C13×C22⋊C4—41no—
101C13×C22⋊C4—81no—
102C13×C22⋊C4—81no—
103C13×C22⋊C4—81no—
104C13×C22⋊C4—81no—
105C42×C13—21no—
106C42×C13—21no—
107C22⋊Dic13—24no—
108C22⋊Dic13—42no—
109C4×Dic13—82no—
110#6099—82no—
111#6099—82no—
112C2×#1055—24no—
113#2975—131no—
114#2635D1311yescharacteristic
115C4×#99—131no—
116#1507—132no—
117#1284—132no—
118C23×C13—21no—
119C23×C13—21no—
120C23×C13—21no—
121C23×C13—21no—
122C23×C13—21no—
123C23×C13—41no—
124C23×C13—41no—
125C23×C13—41no—
126C23×C13—41no—
127C23×C13—41no—
128C23×C13—41no—
129C23×C13—41no—
130C23×C13—41no—
131C23×C13—41no—
132C23×C13—41no—
133C23×C13—41no—
134C23×C13C2×C22⋊C411yescharacteristic
135C23×C13#12112yes—
136D4×C13—44no—
137D4×C13—82no—
138D4×C13—84no—
139D4×C13—84no—
140C2×C4×C13—21no—
141C2×C4×C13—21no—
142C2×C4×C13—21no—
143C2×C4×C13—21no—
144C2×C4×C13—42no—
145C2×C4×C13—42no—
146C2×C4×C13—41no—
147C2×C4×C13—41no—
148C2×C4×C13—42no—
149C2×C4×C13—41no—
150C2×C4×C13—41no—
151C2×C4×C13—41no—
152C2×C4×C13—41no—
153C2×C4×C13—41no—
154C2×C4×C13—41no—
155C2×C4×C13—41no—
156C2×C4×C13—41no—
157C2×C4×C13—41no—
158C2×C4×C13—41no—
159C2×C4×C13—41no—
160C2×C4×C13—81no—
161C2×C4×C13#10412yes—
162C2×C4×C13C4×D412yes—
163C2×C4×C13#12011yescharacteristic
164C2×C4×C13#12311yescharacteristic
165C2×Dic13—44no—
166C2×Dic13—82no—
167C2×Dic13—82no—
168#1055—44no—
169C2×#121C2×D1311yescharacteristic
170C2×C4×D4C2×D1311yescharacteristic
171#436—264no—
172#431—261no—
173C2×#99—261no—
174#407—132no—
175#407—261no—
176#400Dic1311yescharacteristic, complemented
177#399Dic1311yescharacteristic, complemented
178#394—21no—
179#394Dic1311yescharacteristic, complemented
180#392—21no—
181#392Dic1311yescharacteristic, complemented
182#388—21no—
183#386—21no—
184#381C2×D1311yescharacteristic
185C4×C22⋊C4—132no—
186#342—261no—
187#327—264no—
188C22×C13—21no—
189C22×C13—21no—
190C22×C13—22no—
191C22×C13—22no—
192C22×C13—22no—
193C22×C13—42no—
194C22×C13—41no—
195C22×C13—41no—
196C22×C13—41no—
197C22×C13—81no—
198C22×C13—81no—
199C22×C13—81no—
200C22×C13—81no—
201C22×C13—81no—
202C22×C13—81no—
203C22×C13—81no—
204C22×C13—81no—
205C22×C13—81no—
206C22×C13—81no—
207C22×C13#38611yescharacteristic
208C22×C13C2×#10411yescharacteristic
209C22×C13#42111yescharacteristic
210C4×C13—22no—
211C4×C13—21no—
212C4×C13—21no—
213C4×C13—22no—
214C4×C13—41no—
215C4×C13—41no—
216C4×C13—81no—
217C4×C13—81no—
218C4×C13—81no—
219Dic13—84no—
220C22×D4—21no—
221C22×D4—21no—
222C22×D4C2×Dic1311yescharacteristic
223C22×D4C22⋊D1311yescharacteristic
224C23×C4C22⋊D1311yescharacteristic
225C23×C4C22×D1311yescharacteristic
226C2×M16—262no—
227C2×M16—262no—
228#121—22no—
229#121—22no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4C2×Dic1311yescharacteristic
234C2×C4⋊C4C2×Dic1311yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—262no—
242C2×C22⋊C4C2×Dic1311yescharacteristic
243C2×C22⋊C4C22⋊D1311yescharacteristic
244C2×C42—262no—
245C2×C42C22⋊D1311yescharacteristic
246C4⋊C8—262no—
247#99—264no—
248C22⋊C8—134no—
249C22⋊C8—262no—
250C22⋊C8—262no—
251C4×C8—262no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—262no—
256#95C2×Dic1311yescharacteristic
257#95C2×Dic1311yescharacteristic
258C2×C13—21no—
259C2×C13—21no—
260C2×C13—42no—
261C2×C13—81no—
262C2×C13—161no—
263C2×C13#187111yescharacteristic
264C2×C13#191011yescharacteristic
265C2×C13#191311yescharacteristic
266C24—21no—
267C24C22⋊Dic1311yescharacteristic
268C24C2×C22⋊D1311yescharacteristic
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—81no—
276C2×D4—81no—
277C2×D4—81no—
278C2×D4—81no—
279C2×D4—81no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—522no—
288C22×C4C22⋊Dic1311yescharacteristic
289C22×C4C22⋊Dic1311yescharacteristic
290C22×C4C22⋊Dic1311yescharacteristic
291C22×C4D4×D1312yes—
292C22×C4#612912yes—
293C22×C4C22×Dic1311yescharacteristic
294C22×C4C2×C22⋊D1311yescharacteristic
295M16—1042no—
296M16—1042no—
297C2×C8—264no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—81no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—81no—
310C22⋊C4—264no—
311C22⋊C4—522no—
312C42—21no—
313C42—21no—
314C42—1042no—
315C13#918611yescharacteristic, complemented, Sylow
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23#8819312yes—
333C23C2×C22⋊Dic1311yescharacteristic
334D4—44no—
335D4—82no—
336D4—84no—
337D4—84no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—524no—
360C2×C4—1042no—
361C2×C4—1042no—
362C2×C4#8807712yes—
363C2×C4D4×Dic1312yes—
364C2×C4#8818911yescharacteristic
365C2×C4#8819111yescharacteristic
366C8—524no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#1171888111yescharacteristic
387C22C2×#8807711yescharacteristic
388C22#1171889711yescharacteristic
389C4—22no—
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—1044no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—81no—
403C2—161no—
404C2#1311513911yescharacteristic
405C2#1311536411yescharacteristic
406C2#1311537911yescharacteristic
subgroupG/H
centerC22#11718881
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
Fitting, prime radicalC13×#2635C2
socleC22×C13#386
cosocleC23×C4×C13C23
hypercenter#2635D13
hypocenterC13#9186
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2635

Complements

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

NH ≅ G/Ntype
1. #13123877C2semidirect
2. C4×#88074C2semidirect
3. #13113352C2semidirect
4. #13112688C2semidirect
8. C13×#400C4semidirect
9. C13×#399C4semidirect
11. C13×#394C4semidirect
13. C13×#392C4semidirect
60. C22⋊#1055D4semidirect
176. #400Dic13semidirect
177. #399Dic13semidirect
179. #394Dic13semidirect
181. #392Dic13semidirect
315. C13#9186semidirect

Maximal subgroups

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

subgroupindex
#131238772
C4×#880742
#131133522
#131126882
C13×#26352
#918613