Finite Group Databasebeta

← #27293639

Subgroups — #27293639

Order 13568.

492 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 83 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#20278993C211yesmaximal, characteristic
2C2×#16810347C211yesmaximal, characteristic
3#20272664C212yesmaximal, complemented
4#20272644C212yesmaximal, complemented
5C53×#1462C211yesmaximal, characteristic
6C22×C22⋊Dic53C411yescharacteristic
7C2×C22⋊#88322—21no—
8#16810551—21no—
9#16810551—21no—
10#16810496—21no—
11#16810495C411yescharacteristic
12#16810495C2211yescharacteristic
13#16810486—21no—
14#16810484C411yescharacteristic
15#16810484C2211yescharacteristic
16#16810479C411yescharacteristic
17#16810347—21no—
18#16810347C2212yes—
19C22×C22⋊C4×C53C2211yescharacteristic
20C53×#407—21no—
21C53×#323—21no—
22C53×#323C2212yes—
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C22×#88322—41no—
26C2×#11719919—21no—
27C2×#11719919—21no—
28C2×C22⋊Dic53—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—41no—
33C2×C22⋊Dic53—41no—
34C2×C22⋊Dic53C2×C411yescharacteristic
35C2×C22⋊Dic53C2×C411yescharacteristic
36C22⋊#88322—22no—
37C22⋊#88322—41no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C25×C53C2×C411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C2×M16×C53—41no—
48C2×C22⋊C4×C53—41no—
49C2×C22⋊C4×C53—41no—
50C2×C22⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53C2311yescharacteristic
54C22⋊C8×C53—22no—
55C22⋊C8×C53—41no—
56C24×C53—21no—
57C24×C53—21no—
58C24×C53—22no—
59C24×C53—41no—
60C24×C53—42no—
61C24×C53—81no—
62C24×C53C22⋊C411yescharacteristic
63C24×C53C22⋊C411yescharacteristic
64C24×C53C22×C411yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—42no—
70C22×C4×C53—81no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—22no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—81no—
84M16×C53—82no—
85C2×C8×C53—42no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊Dic53—22no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C22⋊Dic53—41no—
94C22⋊Dic53—41no—
95C22⋊Dic53—41no—
96C22⋊Dic53—81no—
97#11719919—81no—
98#11719919—81no—
99C2×#88322—42no—
100C2×#88322—82no—
101C23×C53—22no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—41no—
110C23×C53—42no—
111C23×C53—42no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—42no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—42no—
118C23×C53—81no—
119C23×C53—81no—
120C23×C53—84no—
121C23×C53—84no—
122C23×C53#9911yescharacteristic
123C23×C53#9911yescharacteristic
124C23×C53C2×C22⋊C411yescharacteristic
125C23×C53#13112yes—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—81no—
137C2×C4×C53—81no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—82no—
151C2×Dic53—82no—
152C8×C53—82no—
153#88322—82no—
154#11547—531nomaximal, Sylow
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—22no—
162C22×C53—21no—
163C22×C53—42no—
164C22×C53—42no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—84no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53—84no—
176C22×C53#35212yes—
177C22×C53#40811yescharacteristic
178C22×C53C2×#9911yescharacteristic
179C22×C53#41311yescharacteristic
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—41no—
184Dic53—81no—
185Dic53—81no—
186Dic53—81no—
187Dic53—81no—
188#2483—531no—
189#1462D5311yescharacteristic
190C2×#323—531no—
191#1326—532no—
192#1317—532no—
193C2×C53—21no—
194C2×C53—21no—
195C2×C53—22no—
196C2×C53—41no—
197C2×C53—42no—
198C2×C53—81no—
199C2×C53#146811yescharacteristic
200C2×C53C2×#35211yescharacteristic
201C2×C53#212111yescharacteristic
202C22×C22⋊C4—531no—
203C22×C22⋊C4C2×D5311yescharacteristic
204#407—21no—
205C2×C22⋊C8—1061no—
206#397—1061no—
207#397—1061no—
208#385—531no—
209#385—531no—
210#385—1061no—
211#381—531no—
212#381—531no—
213#381—1061no—
214#379—531no—
215#323—21no—
216#323—532no—
217#323—1061no—
218#323C2×D5312yes—
219C53#1154711yescharacteristic, complemented, Sylow
220C25C4×D5311yescharacteristic
221C23×C4—531no—
222C23×C4—531no—
223C23×C4D21211yescharacteristic
224C23×C4C22⋊D5311yescharacteristic
225C2×M16—41no—
226C22×C8—2121no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—531no—
232C2×C22⋊C4—531no—
233C2×C22⋊C4—1061no—
234C2×C22⋊C4—1061no—
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1061no—
237C2×C22⋊C4—2121no—
238C2×C22⋊C4—2121no—
239C2×C22⋊C4C4×D5311yescharacteristic
240C2×C22⋊C4D21211yescharacteristic
241C2×C22⋊C4C22⋊D5311yescharacteristic
242C2×C22⋊C4C22×D5311yescharacteristic
243C22⋊C8—22no—
244C22⋊C8—41no—
245C22⋊C8—1062no—
246C22⋊C8—2121no—
247#95—1061no—
248#95—1061no—
249#95—1061no—
250#95—1061no—
251#95—1061no—
252#95—1061no—
253C24—21no—
254C24—21no—
255C24—22no—
256C24—41no—
257C24—42no—
258C24—81no—
259C24#1171992111yescharacteristic
260C24#1171992111yescharacteristic
261C24C2×C4×D5311yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—81no—
268C22×C4—1062no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—4241no—
274C22×C4—2121no—
275C22×C4—2121no—
276C22×C4—2121no—
277C22×C4#1171992111yescharacteristic
278C22×C4#1171992111yescharacteristic
279C22×C4C2×D21211yescharacteristic
280C22×C4C2×C22⋊D5311yescharacteristic
281M16—82no—
282C2×C8—42no—
283C2×C8—4242no—
284C2×C8—2122no—
285C4⋊C4—4241no—
286C4⋊C4—4241no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—1062no—
292C22⋊C4—4241no—
293C22⋊C4—2122no—
294C22⋊C4—2122no—
295C22⋊C4—2121no—
296C22⋊C4—2121no—
297C22⋊C4—2121no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—41no—
307C23—42no—
308C23—42no—
309C23—41no—
310C23—41no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—84no—
318C23—84no—
319C23#1313313111yescharacteristic
320C23#1313313111yescharacteristic
321C23C2×#1171992111yescharacteristic
322C23#1313322512yes—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—4242no—
336C2×C4—4242no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4—2121no—
349C8—82no—
350C8—4242no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—81no—
370C22—81no—
371C22—84no—
372C22#1681041212yes—
373C22#1681059111yescharacteristic
374C22C2×#1313313111yescharacteristic
375C22#1681060511yescharacteristic
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—4241no—
381C4—4241no—
382C4—4241no—
383C4—4241no—
384C2—21no—
385C2—21no—
386C2—22no—
387C2—41no—
388C2—42no—
389C2—81no—
390C2#2027306411yescharacteristic
391C2C2×#1681041211yescharacteristic
392C2#2027636211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1462C2
socleC22×C53C2×#99
prime radicalC25×C53C2×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#1462D53
hypocenterC53#11547
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #20272664C2semidirect
4. #20272644C2semidirect
219. C53#11547semidirect

Maximal subgroups

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

subgroupindex
#202789932
C2×#168103472
#202726642
#202726442
C53×#14622
#1154753