Finite Group Databasebeta

← #27294995

Subgroups — #27294995

Order 13568.

572 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 91 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#20274795C211yesmaximal, characteristic, complemented
2C2×#16810349C211yesmaximal, characteristic, complemented
3#20272745C212yesmaximal, complemented
4#20272718C212yesmaximal, complemented
5C53×#2406C211yesmaximal, characteristic
6#16810587—21no—
7C2×C22⋊#88322—21no—
8#16810392—21no—
9#16810392C2212yes—
10#16810349—21no—
11#16810349C2212yes—
12C22×#41×C53C411yescharacteristic
13C2×#123×C53—21no—
14C22×C4⋊C4×C53C2211yescharacteristic
15C53×#395C411yescharacteristic
16C53×#395C2211yescharacteristic
17C53×#394C411yescharacteristic
18C53×#394C2211yescharacteristic
19C53×#386—21no—
20C53×#385—21no—
21C53×#385—21no—
22C53×#381C411yescharacteristic
23C2×#11719917—22no—
24C22×#88322—41no—
25C22⋊#88322—22no—
26C22⋊#88322—41no—
27C22⋊#88322—41no—
28C22⋊#88322—41no—
29C2×#41×C53—41no—
30C2×#41×C53—41no—
31C2×#41×C53—44no—
32C22×Q8×C53C2×C411yescharacteristic
33C22×D4×C53—21no—
34C22×D4×C53C2×C411yescharacteristic
35C23×C4×C53—21no—
36C23×C4×C53C2×C411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39#123×C53—44no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—41no—
43C2×C4⋊C4×C53—41no—
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C4⋊C4×C53C2311yescharacteristic
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53—21no—
59C24×C53C22⋊C411yescharacteristic
60#41×C53—44no—
61#41×C53—44no—
62#41×C53—44no—
63#41×C53—44no—
64C2×Q8×C53—42no—
65C2×Q8×C53—41no—
66C2×D4×C53—24no—
67C2×D4×C53—41no—
68C2×D4×C53—42no—
69C2×D4×C53—41no—
70C2×D4×C53—41no—
71C2×D4×C53—42no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—22no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—41no—
83C22×C4×C53—42no—
84C22×C4×C53—42no—
85C22×C4×C53—42no—
86C22×C4×C53—42no—
87C22×C4×C53—42no—
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C22⋊C411yescharacteristic
90C22×C4×C53C22⋊C411yescharacteristic
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—42no—
99C4⋊C4×C53—42no—
100C22⋊C4×C53—42no—
101C22⋊C4×C53—42no—
102C22⋊C4×C53—42no—
103C22⋊C4×C53—42no—
104C22⋊C4×C53—42no—
105#11719917—42no—
106#11719917—42no—
107C2×#88322—42no—
108C2×#88322—42no—
109C2×#88322—82no—
110C23×C53—22no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—22no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53#9911yescharacteristic
122C23×C53#9911yescharacteristic
123C23×C53C2×C22⋊C411yescharacteristic
124Q8×C53—42no—
125Q8×C53—42no—
126D4×C53—44no—
127D4×C53—42no—
128D4×C53—42no—
129D4×C53—44no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—22no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—48no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—81no—
159C2×C4×C53#13512yes—
160#88322—82no—
161#88322—82no—
162#11848—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—42no—
173C22×C53—42no—
174C22×C53—42no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53AΣL(1,𝔽16)12yes—
180C22×C53C2×#9911yescharacteristic
181C22×C53#41611yescharacteristic
182C22×C53#42111yescharacteristic
183C4×C53—22no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—42no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—41no—
190C4×C53—81no—
191C4×C53—81no—
192#2406D5311yescharacteristic
193#1821—531no—
194C2×#324—531no—
195#1354—532no—
196#1345—532no—
197C2×C53—21no—
198C2×C53—21no—
199C2×C53—22no—
200C2×C53—41no—
201C2×C53—82no—
202C2×C53#151611yescharacteristic
203C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
204C2×C53#213311yescharacteristic
205C22×#41Dic5311yescharacteristic
206C2×#123—21no—
207C22×C4⋊C4C2×D5311yescharacteristic
208#407—1061no—
209C2×C22⋊C8—1061no—
210#395Dic5311yescharacteristic
211#395C2×D5311yescharacteristic
212#394Dic5311yescharacteristic
213#394C2×D5311yescharacteristic
214#386—21no—
215#385—21no—
216#385—21no—
217#381Dic5311yescharacteristic
218#343—532no—
219#343—1061no—
220#324—532no—
221#324—1061no—
222C53#1184811yescharacteristic, complemented, Sylow
223C2×#41—41no—
224C2×#41—41no—
225C2×#41—44no—
226C22×Q8C2×Dic5311yescharacteristic
227C22×D4—21no—
228C22×D4C2×Dic5311yescharacteristic
229C23×C4—21no—
230C23×C4C2×Dic5311yescharacteristic
231C23×C4C22⋊D5311yescharacteristic
232C23×C4C22⋊D5311yescharacteristic
233C2×M16—1062no—
234C22×C8—2121no—
235#123—44no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4C2×Dic5311yescharacteristic
241C2×C4⋊C4C22⋊D5311yescharacteristic
242C2×C4⋊C4C22⋊D5311yescharacteristic
243C2×C4⋊C4C22×D5311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C22⋊C8—1062no—
250C22⋊C8—2121no—
251C22⋊C8—2121no—
252C22⋊C8—2121no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95C2×Dic5311yescharacteristic
257#95C2×Dic5311yescharacteristic
258C24—21no—
259C24C22⋊Dic5311yescharacteristic
260#41—44no—
261#41—44no—
262#41—44no—
263#41—44no—
264C2×Q8—42no—
265C2×Q8—41no—
266C2×D4—24no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4C22⋊Dic5311yescharacteristic
289C22×C4C22⋊Dic5311yescharacteristic
290C22×C4C22⋊Dic5311yescharacteristic
291C22×C4C22×Dic5311yescharacteristic
292C22×C4C2×C22⋊D5311yescharacteristic
293C22×C4C2×C22⋊D5311yescharacteristic
294M16—2122no—
295M16—2122no—
296C2×C8—4242no—
297C2×C8—2122no—
298C2×C8—2122no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—22no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23#1313313211yescharacteristic
322C23#1313313211yescharacteristic
323C23C2×C22⋊Dic5311yescharacteristic
324Q8—42no—
325Q8—42no—
326D4—44no—
327D4—42no—
328D4—42no—
329D4—44no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—48no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
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#1313323712yes—
360C8—4242no—
361C8—4242no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#1681041012yes—
379C22C2×#1313313211yescharacteristic
380C22#1681061911yescharacteristic
381C22#1681064011yescharacteristic
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2—41no—
395C2—82no—
396C2#2027327111yescharacteristic
397C2C2×#1681041011yescharacteristic
398C2#2027640411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2406C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2406D53
hypocenterC53#11848
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
1. #20274795C2semidirect
2. C2×#16810349C2semidirect
3. #20272745C2semidirect
4. #20272718C2semidirect
222. C53#11848semidirect

Maximal subgroups

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

subgroupindex
#202747952
C2×#168103492
#202727452
#202727182
C53×#24062
#1184853