Finite Group Databasebeta

← #27282102

Subgroups — #27282102

Order 13568.

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

normal subgroups only

This group has 520 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#20283744C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20273045C212yesmaximal, complemented
4#20272621C212yesmaximal, complemented
5C53×#2579C211yesmaximal, characteristic
6#16810695—24no—
7#16810680—21no—
8#16810587C2212yes—
9#16810587C2212yes—
10C2×C22⋊#88322—21no—
11#16810349—24no—
12#16810349—21no—
13#16810347—21no—
14C2×C4×D4×C53C2211yescharacteristic
15C22×C4⋊C4×C53C2211yescharacteristic
16C53×#398—21no—
17C53×#396—21no—
18C53×#387—21no—
19C53×#386C411yescharacteristic
20C53×#386C411yescharacteristic
21C53×#385—21no—
22C53×#385C411yescharacteristic
23C53×#385C411yescharacteristic
24C53×#380C2211yescharacteristic
25C2×#11719917—22no—
26C2×#11719917—22no—
27C22×#88322—22no—
28C4⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33C22⋊#88322D414yes—
34#13133128—22no—
35C22×D4×C53C2×C411yescharacteristic
36C23×C4×C53C2×C411yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40C4×D4×C53—22no—
41C4×D4×C53—22no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—22no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53D411yescharacteristic
56C2×C42×C53D411yescharacteristic
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62C24×C53C22⋊C411yescharacteristic
63C24×C53C22×C411yescharacteristic
64C2×D4×C53—22no—
65C2×D4×C53—22no—
66C2×D4×C53—41no—
67C22×C4×C53—21no—
68C22×C4×C53—22no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—22no—
75C22×C4×C53—42no—
76C22×C4×C53—81no—
77C22×C4×C53—81no—
78C22×C4×C53—81no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C2×D412yes—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53#4112yes—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—42no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—81no—
92C4⋊C4×C53—81no—
93C22⋊C4×C53—22no—
94C22⋊C4×C53—22no—
95C22⋊C4×C53—42no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—81no—
99C42×C53—21no—
100C42×C53—21no—
101#11719917—82no—
102#11719917—82no—
103C2×#88322—24no—
104C2×#88322—42no—
105C2×#88322—82no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53#10112yes—
115C23×C53C2×C22⋊C411yescharacteristic
116C23×C53C4×D412yes—
117D4×C53—44no—
118C2×C4×C53—24no—
119C2×C4×C53—22no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
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—81no—
145C2×C4×C53—82no—
146C2×C4×C53#12011yescharacteristic
147C2×C4×C53#12112yes—
148C2×C4×C53#12311yescharacteristic
149#88322—44no—
150#88322—82no—
151#9517—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—42no—
159C22×C53—42no—
160C22×C53—42no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—81no—
164C22×C53#38611yescharacteristic
165C22×C53C2×#10111yescharacteristic
166C22×C53#41311yescharacteristic
167C4×C53—21no—
168C4×C53—22no—
169C4×C53—21no—
170C4×C53—42no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—81no—
175C4×C53—81no—
176C4×C53—81no—
177#2973—531no—
178#2579D5311yescharacteristic
179#1465—531no—
180#1463—532no—
181#1305—532no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—42no—
185C2×C53—42no—
186C2×C53#189111yescharacteristic
187C2×C53#190111yescharacteristic
188C2×C53#191311yescharacteristic
189C2×C4×D4C2×D5311yescharacteristic
190C22×C4⋊C4C2×D5311yescharacteristic
191#435—1064no—
192#432—1061no—
193#407—532no—
194#407—532no—
195C2×C22⋊C8—1061no—
196#398—21no—
197#396—21no—
198#387—21no—
199#386Dic5311yescharacteristic
200#386Dic5311yescharacteristic
201#385—21no—
202#385Dic5311yescharacteristic
203#385Dic5311yescharacteristic
204#380C2×D5311yescharacteristic
205#324—1064no—
206#324—1061no—
207#323—1061no—
208C53#951711yescharacteristic, complemented, Sylow
209C22×D4C2×Dic5311yescharacteristic
210C23×C4C2×Dic5311yescharacteristic
211C23×C4C2×Dic5311yescharacteristic
212C23×C4C22⋊D5311yescharacteristic
213C23×C4C22×D5311yescharacteristic
214C2×M16—1062no—
215C2×M16—1062no—
216C22×C8—1062no—
217C4×D4—22no—
218C4×D4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C22⋊D5311yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4C2×Dic5311yescharacteristic
230C2×C22⋊C4C2×Dic5311yescharacteristic
231C2×C22⋊C4C2×Dic5311yescharacteristic
232C2×C22⋊C4C22⋊D5311yescharacteristic
233C2×C42C22⋊D5311yescharacteristic
234C4⋊C8—1062no—
235C22⋊C8—534no—
236C22⋊C8—1062no—
237C22⋊C8—1062no—
238C22⋊C8—1062no—
239C22⋊C8—1062no—
240#97—1062no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246C24C22⋊Dic5311yescharacteristic
247C24C22×Dic5311yescharacteristic
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4C22⋊Dic5311yescharacteristic
264C22×C4C22⋊Dic5311yescharacteristic
265C22×C4C22⋊Dic5311yescharacteristic
266C22×C4D4×D5312yes—
267C22×C4#1171994712yes—
268C22×C4C2×C22⋊D5311yescharacteristic
269C22×C4C2×C22⋊D5311yescharacteristic
270M16—4242no—
271M16—4242no—
272C2×C8—1064no—
273C2×C8—4242no—
274C2×C8—2122no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—81no—
281C4⋊C4—81no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—42no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—81no—
288C42—21no—
289C42—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#1313313612yes—
299C23C2×C22⋊Dic5311yescharacteristic
300C23D4×Dic5312yes—
301D4—44no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4#1313318511yescharacteristic
331C2×C4#1313318912yes—
332C2×C4#1313319811yescharacteristic
333C8—4242no—
334C8—2124no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22#1681050311yescharacteristic
348C22C2×#1313313611yescharacteristic
349C22#1681060711yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—41no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—42no—
364C2#2027513411yescharacteristic
365C2#2027518411yescharacteristic
366C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2579C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#2579D53
hypocenterC53#9517
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2579

Complements

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

NH ≅ G/Ntype
3. #20273045C2semidirect
4. #20272621C2semidirect
208. C53#9517semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202730532
#202730452
#202726212
C53×#25792
#951753