Finite Group Databasebeta

← #27282077

Subgroups — #27282077

Order 13568.

558 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 103 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#20283353C211yesmaximal, characteristic, complemented
2#20274798C211yesmaximal, characteristic, complemented
3#20273042C212yesmaximal
4#20272623C212yesmaximal, complemented
5C53×#2482C211yesmaximal, characteristic
6#16810684—24no—
7#16810647—21no—
8#16810587C2212yes—
9#16810587C2212yes—
10C4×#11719917—21no—
11#16810392—21no—
12#16810392—21no—
13#16810347—24no—
14C2×#117×C53C2211yescharacteristic
15C22×C22⋊C4×C53C2211yescharacteristic
16C53×#399—21no—
17C53×#397—21no—
18C53×#388—21no—
19C53×#388—21no—
20C53×#385C411yescharacteristic
21C53×#385C411yescharacteristic
22C53×#381C2211yescharacteristic
23C53×#379C411yescharacteristic
24C53×#379C411yescharacteristic
25C2×#11719917—22no—
26C2×#11719917—22no—
27C4⋊#88322—24no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31C22⋊#88322D414yes—
32#13133128—22no—
33C4×#88322—22no—
34C25×C53C2×C411yescharacteristic
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53C2311yescharacteristic
37#117×C53—22no—
38#117×C53—22no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53C2×C411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C2×C42×C53D411yescharacteristic
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61C24×C53—21no—
62C24×C53—22no—
63C24×C53—42no—
64C24×C53—41no—
65C24×C53—41no—
66C24×C53—81no—
67C24×C53C22×C411yescharacteristic
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—22no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—81no—
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D412yes—
82C22×C4×C53#4112yes—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—81no—
86C4⋊C4×C53—81no—
87C22⋊C4×C53—22no—
88C22⋊C4×C53—22no—
89C22⋊C4×C53—42no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—81no—
97C22⋊C4×C53—81no—
98C42×C53—21no—
99C42×C53—21no—
100C42×C53—21no—
101C42×C53—21no—
102#11719917—42no—
103#11719917—42no—
104#11719917—42no—
105#11719917—42no—
106C2×#88322—24no—
107C2×#88322—42no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—22no—
111C23×C53—24no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—42no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—82no—
122C23×C53—82no—
123C23×C53—84no—
124C23×C53—84no—
125C23×C53C2×C22⋊C411yescharacteristic
126C23×C53C4×D412yes—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
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—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—82no—
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12112yes—
153C2×C4×C53#12311yescharacteristic
154#88322—44no—
155#88322—82no—
156#9512—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—22no—
160C22×C53—22no—
161C22×C53—22no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—42no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53—84no—
170C22×C53—84no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53#38611yescharacteristic
174C22×C53#41011yescharacteristic
175C22×C53#41011yescharacteristic
176C4×C53—21no—
177C4×C53—22no—
178C4×C53—21no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—81no—
184C4×C53—81no—
185C4×C53—81no—
186#2921—531no—
187#2482D5311yescharacteristic
188#1822—531no—
189#1462—532no—
190#1306—532no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—42no—
194C2×C53—82no—
195C2×C53#189111yescharacteristic
196C2×C53#189911yescharacteristic
197C2×C53#191411yescharacteristic
198C2×#117C2×D5311yescharacteristic
199C22×C22⋊C4C2×D5311yescharacteristic
200#433—1064no—
201#423—1061no—
202#407—532no—
203#407—532no—
204C4×M16—1061no—
205#399—21no—
206#397—21no—
207#388—21no—
208#388—21no—
209#385Dic5311yescharacteristic
210#385Dic5311yescharacteristic
211#381C2×D5311yescharacteristic
212#379Dic5311yescharacteristic
213#379Dic5311yescharacteristic
214#343—1061no—
215#343—1061no—
216#323—1064no—
217C53#951211yescharacteristic, complemented, Sylow
218C25C2×Dic5311yescharacteristic
219C23×C4C2×Dic5311yescharacteristic
220C23×C4C22×D5311yescharacteristic
221C2×M16—1062no—
222C2×M16—1062no—
223#117—22no—
224#117—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C22⋊D5311yescharacteristic
228C2×C4⋊C4C22⋊D5311yescharacteristic
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4C2×Dic5311yescharacteristic
238C2×C22⋊C4C2×Dic5311yescharacteristic
239C2×C22⋊C4C2×Dic5311yescharacteristic
240C2×C22⋊C4C2×Dic5311yescharacteristic
241C2×C42C22⋊D5311yescharacteristic
242C2×C42C22⋊D5311yescharacteristic
243C4⋊C8—1064no—
244C22⋊C8—534no—
245C22⋊C8—1062no—
246C22⋊C8—1062no—
247C22⋊C8—1062no—
248#97—1062no—
249C4×C8—1062no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254C24—21no—
255C24—22no—
256C24—42no—
257C24—41no—
258C24—41no—
259C24—81no—
260C24C22×Dic5311yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—81no—
268C22×C4C22⋊Dic5311yescharacteristic
269C22×C4C22⋊Dic5311yescharacteristic
270C22×C4C22⋊Dic5311yescharacteristic
271C22×C4C22⋊Dic5311yescharacteristic
272C22×C4D4×D5312yes—
273C22×C4#1171994712yes—
274C22×C4C2×C22⋊D5311yescharacteristic
275C22×C4C2×C22⋊D5311yescharacteristic
276M16—2122no—
277M16—2122no—
278M16—2122no—
279M16—2122no—
280C2×C8—1064no—
281C2×C8—2122no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C23—21no—
302C23—21no—
303C23—22no—
304C23—24no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—42no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—82no—
315C23—82no—
316C23—84no—
317C23—84no—
318C23C2×C22⋊Dic5311yescharacteristic
319C23D4×Dic5312yes—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—82no—
344C2×C4#1313318511yescharacteristic
345C2×C4#1313318912yes—
346C2×C4#1313319811yescharacteristic
347C8—4242no—
348C8—2124no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—81no—
360C22—81no—
361C22—84no—
362C22—84no—
363C22—81no—
364C22—81no—
365C22#1681050311yescharacteristic
366C22#1681059711yescharacteristic
367C22#1681059711yescharacteristic
368C4—22no—
369C4—21no—
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—82no—
382C2#2027513411yescharacteristic
383C2#2027517411yescharacteristic
384C2#2027526311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2482C2
socleC22×C53#386
prime radicalC25×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#2482D53
hypocenterC53#9512
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
1. #20283353C2semidirect
2. #20274798C2semidirect
4. #20272623C2semidirect
217. C53#9512semidirect

Maximal subgroups

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

subgroupindex
#202833532
#202747982
#202730422
#202726232
C53×#24822
#951253