Finite Group Databasebeta

← #27294800

Subgroups — #27294800

Order 13568.

680 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 95 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#20274789C211yesmaximal, characteristic
2#20273042C211yesmaximal, characteristic
3#20272736C214yesmaximal, complemented
4C2×C53×#393C211yesmaximal, characteristic
5#16810587—21no—
6#16810479—21no—
7#16810390—21no—
8#16810390C2212yes—
9#16810347—21no—
10#16810347C2212yes—
11C23×Q8×C53C411yescharacteristic, complemented
12C22×C22⋊C4×C53—21no—
13C22×C22⋊C4×C53C2211yescharacteristic
14C53×#393—21no—
15C53×#393—21no—
16C53×#393C412yescomplemented
17C53×#393C2212yes—
18C2×#95×C53—21no—
19C2×#95×C53C411yescharacteristic, complemented
20C2×#11719917—41no—
21C2×C22⋊Dic53—22no—
22C2×C22⋊Dic53—41no—
23C2×C22⋊Dic53—41no—
24C22⋊#88322—22no—
25C22⋊#88322—41no—
26C25×C53D411yescharacteristic
27C22×Q8×C53—21no—
28C22×Q8×C53—44no—
29C22×Q8×C53—41no—
30C22×Q8×C53—41no—
31C22×Q8×C53C2×C412yes—
32C23×C4×C53—21no—
33C23×C4×C53—21no—
34C23×C4×C53C2×C411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53—41no—
39C2×C22⋊C4×C53—41no—
40C2×C22⋊C4×C53—44no—
41C2×C22⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53C2311yescharacteristic
45#95×C53—22no—
46#95×C53—21no—
47#95×C53—22no—
48#95×C53C2×C412yes—
49C24×C53—22no—
50C24×C53—21no—
51C24×C53—21no—
52C24×C53—41no—
53C24×C53—42no—
54C24×C53—42no—
55C24×C53C22⋊C411yescharacteristic
56C24×C53C22⋊C411yescharacteristic
57C24×C53C2×D411yescharacteristic
58C2×Q8×C53—44no—
59C2×Q8×C53—42no—
60C2×Q8×C53—44no—
61C2×Q8×C53—42no—
62C2×Q8×C53—48no—
63C2×Q8×C53—44no—
64C2×Q8×C53—44no—
65C22×C4×C53—24no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—22no—
70C22×C4×C53—21no—
71C22×C4×C53—22no—
72C22×C4×C53—42no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—42no—
76C22×C4×C53—42no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—42no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—44no—
90C22⋊C4×C53—44no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C22⋊Dic53—82no—
94C22⋊Dic53—82no—
95#11719917—82no—
96C2×#88322—42no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—42no—
107C23×C53—42no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—48no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—81no—
116C23×C53—81no—
117C23×C53—84no—
118C23×C53#9911yescharacteristic
119C23×C53#9911yescharacteristic
120C23×C53C2×C22⋊C411yescharacteristic
121C23×C53#13712yes—
122Q8×C53—48no—
123Q8×C53—44no—
124Q8×C53—44no—
125C2×C4×C53—24no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—42no—
131C2×C4×C53—44no—
132C2×C4×C53—42no—
133C2×C4×C53—44no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—81no—
145C2×C4×C53—84no—
146C2×Dic53—82no—
147C2×Dic53—82no—
148C2×Dic53—82no—
149#88322—82no—
150#11808—531nomaximal, Sylow
151C22×C53—24no—
152C22×C53—24no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—22no—
160C22×C53—42no—
161C22×C53—42no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—81no—
168C22×C53—84no—
169C22×C53—84no—
170C22×C53—81no—
171C22×C53AΣL(1,𝔽16)12yes—
172C22×C53#35412yes—
173C22×C53C2×#9911yescharacteristic
174C22×C53#41911yescharacteristic
175C22×C53#42111yescharacteristic
176C4×C53—44no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—82no—
182Dic53—162no—
183C2×#393D5311yescharacteristic
184#1819—531no—
185#1462—531no—
186#1351—534no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—22no—
190C2×C53—22no—
191C2×C53—42no—
192C2×C53—81no—
193C2×C53#151111yescharacteristic
194C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
195C2×C53C2×#35411yescharacteristic
196C23×Q8Dic5311yescharacteristic, complemented
197C22×C22⋊C4—21no—
198C22×C22⋊C4C2×D5311yescharacteristic
199#407—1061no—
200#393—21no—
201#393—21no—
202#393Dic5312yescomplemented
203#393C2×D5312yes—
204#379—1061no—
205C2×#95—21no—
206C2×#95Dic5311yescharacteristic, complemented
207#342—532no—
208#342—1061no—
209#323—532no—
210#323—1061no—
211C53#1180811yescharacteristic, complemented, Sylow
212C25C22⋊D5311yescharacteristic
213C22×Q8—21no—
214C22×Q8—44no—
215C22×Q8—41no—
216C22×Q8—41no—
217C22×Q8C2×Dic5312yes—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4C2×Dic5311yescharacteristic
221C23×C4C22⋊D5311yescharacteristic
222C2×M16—2121no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—1062no—
229C2×C22⋊C4—2121no—
230C2×C22⋊C4—2121no—
231C2×C22⋊C4C2×Dic5311yescharacteristic
232C2×C22⋊C4C22⋊D5311yescharacteristic
233C2×C22⋊C4C22⋊D5311yescharacteristic
234C2×C22⋊C4C22×D5311yescharacteristic
235C22⋊C8—1062no—
236C22⋊C8—2121no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95C2×Dic5312yes—
241C24—22no—
242C24—21no—
243C24—21no—
244C24—41no—
245C24—42no—
246C24—42no—
247C24C22⋊Dic5311yescharacteristic
248C24C22⋊Dic5311yescharacteristic
249C24C2×C22⋊D5311yescharacteristic
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—48no—
255C2×Q8—44no—
256C2×Q8—44no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—2122no—
273C22×C4C22⋊Dic5311yescharacteristic
274C22×C4C22⋊Dic5311yescharacteristic
275C22×C4C22×Dic5311yescharacteristic
276C22×C4C2×C22⋊D5311yescharacteristic
277M16—4242no—
278C2×C8—2122no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—4242no—
286C22⋊C4—4242no—
287C22⋊C4—2122no—
288C22⋊C4—2122no—
289C23—21no—
290C23—21no—
291C23—22no—
292C23—22no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—42no—
299C23—42no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—48no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—81no—
309C23—84no—
310C23#1313313211yescharacteristic
311C23#1313313211yescharacteristic
312C23C2×C22⋊Dic5311yescharacteristic
313C23#1313324812yes—
314Q8—44no—
315Q8—44no—
316Q8—48no—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—81no—
337C2×C4—84no—
338C2×C4—4242no—
339C2×C4—4242no—
340C2×C4—4242no—
341C8—4242no—
342C22—24no—
343C22—24no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22—84no—
360C22—84no—
361C22—81no—
362C22#1681041012yes—
363C22#1681041812yes—
364C22C2×#1313313211yescharacteristic
365C22#1681063211yescharacteristic
366C22#1681064011yescharacteristic
367C4—44no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—82no—
373C4—8482no—
374C2—21no—
375C2—21no—
376C2—22no—
377C2—22no—
378C2—42no—
379C2—81no—
380C2#2027324811yescharacteristic
381C2C2×#1681041011yescharacteristic
382C2C2×#1681041811yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC2×C53×#393C2
socleC22×C53C2×#99
prime radicalC25×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenterC2×#393D53
hypocenterC53#11808
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
3. #20272736C2semidirect
11. C23×Q8×C53C4semidirect
16. C53×#393C4semidirect
19. C2×#95×C53C4semidirect
196. C23×Q8Dic53semidirect
202. #393Dic53semidirect
206. C2×#95Dic53semidirect
211. C53#11808semidirect

Maximal subgroups

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

subgroupindex
#202747892
#202730422
#202727362
C2×C53×#3932
#1180853