Finite Group Databasebeta

← #28299956

Subgroups — #28299956

Order 13568.

526 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 98 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#20290571C211yesmaximal, characteristic, complemented
2#20290569C211yesmaximal, characteristic, complemented
3#20289534C212yesmaximal, complemented
4#20289531C212yesmaximal, complemented
5#20283041C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C4⋊#88324—21no—
8C2×C22⋊#88324—21no—
9C2×C22⋊#88324—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133198—21no—
13C22×C4×Dic53C2211yescharacteristic
14#16810526C411yescharacteristic
15#16810526C2211yescharacteristic
16#16810519C411yescharacteristic
17#16810519C2211yescharacteristic
18#16810511—21no—
19#16810511—21no—
20#16810503—21no—
21#16810483C411yescharacteristic
22C2×#121×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26C4⋊#88324—42no—
27C22⋊#88324—42no—
28C22⋊#88324—42no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31#13133198—42no—
32#13133198—42no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C2×C4×Dic53—42no—
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53C2311yescharacteristic
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C22×D4×C53—21no—
51C22×D4×C53C2×C411yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53#121×C53—42no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53—21no—
56C24×C53—21no—
57C24×C53C22⋊C411yescharacteristic
58C2×D4×C53—22no—
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C2×D4×C53—42no—
62C2×D4×C53—42no—
63C2×D4×C53—42no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—41no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53C22⋊C411yescharacteristic
79C22×Dic53C22⋊C411yescharacteristic
80C22×Dic53D812yes—
81C22×Dic53SD1612yes—
82C22×Dic53C2×D411yescharacteristic
83C22×Dic53C2×D411yescharacteristic
84C2×#88324—42no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C2×#88324—42no—
89C2×#88324—42no—
90C4⋊C4×C53—41no—
91C22⋊C4×C53—42no—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C22⋊Dic53—42no—
95C22⋊Dic53—41no—
96C22⋊Dic53—41no—
97C22⋊Dic53—41no—
98C22⋊Dic53—41no—
99C4⋊Dic53—22no—
100C4⋊Dic53—41no—
101#11719919—42no—
102#11719919—41no—
103#11719919—41no—
104C4×Dic53—22no—
105C4×Dic53—22no—
106C4×Dic53—41no—
107C4×Dic53—41no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53#10212yes—
122C23×C53C2×C22⋊C411yescharacteristic
123D4×C53—82no—
124D4×C53—84no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—44no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53#9912yes—
134C2×C4×C53#13112yes—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—42no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—42no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53#10212yes—
155C2×Dic53C2×D811yescharacteristic
156C2×Dic53C2×SD1611yescharacteristic
157C2×Dic53AGL(1,ℤ/8)11yescharacteristic
158C2×Dic53#13711yescharacteristic
159#88324—82no—
160#88324—82no—
161#7714—531nomaximal, Sylow
162C22×C53—22no—
163C22×C53—22no—
164C22×C53—22no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53#40811yescharacteristic
177C22×C53C2×#9911yescharacteristic
178C22×C53#41311yescharacteristic
179C22×C53C2×#10211yescharacteristic
180C22×C53#41711yescharacteristic
181C22×C53#42111yescharacteristic
182C22×C53#42111yescharacteristic
183C4×C53—22no—
184C4×C53—22no—
185C4×C53—81no—
186Dic53—21no—
187Dic53—21no—
188Dic53—21no—
189Dic53—22no—
190Dic53—21no—
191Dic53—41no—
192Dic53—81no—
193Dic53—81no—
194#2854—531no—
195#1797—531no—
196#1797—531no—
197#1271—532no—
198#1270—532no—
199C2×C53—42no—
200C2×C53—161no—
201C2×C53#146811yescharacteristic
202C2×C53#148011yescharacteristic
203C2×C53#149511yescharacteristic
204C2×C53#151311yescharacteristic
205C2×C53#151911yescharacteristic
206C2×C53#153511yescharacteristic
207C2×C53#154711yescharacteristic
208C2×#123—1061no—
209C2×#121#615711yescharacteristic
210C22×C42—531no—
211C2×C4⋊C8—1061no—
212C2×C4⋊C8—1061no—
213C2×C22⋊C8—1061no—
214C2×C22⋊C8—1061no—
215#390—531no—
216#390—531no—
217#389—531no—
218#389—531no—
219#388—1061no—
220#388—1061no—
221#386—1061no—
222#380—531no—
223#336—532no—
224#336—532no—
225C53#771411yescharacteristic, complemented, Sylow
226C22×D4—21no—
227C22×D4C2×#615711yescharacteristic
228C23×C4—531no—
229C23×C4—531no—
230C23×C4C2×#615711yescharacteristic
231C22×C8—1062no—
232C22×C8—1062no—
233#123—2122no—
234#123—2122no—
235#121—42no—
236C2×C4⋊C4—531no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—1061no—
239C2×C4⋊C4—1061no—
240C2×C4⋊C4C2×#615711yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C22⋊C4—1061no—
246C2×C22⋊C4—1061no—
247C2×C42—531no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42—531no—
251C2×C42—2122no—
252C4⋊C8—2122no—
253C4⋊C8—2122no—
254C22⋊C8—2122no—
255C22⋊C8—2122no—
256#95—1061no—
257#95—1061no—
258#95—1061no—
259C24—21no—
260C24C22⋊#615711yescharacteristic
261C2×D4—22no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—532no—
275C22×C4—532no—
276C22×C4—531no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—2121no—
282C22×C4—2121no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4C22⋊#615711yescharacteristic
286C22×C4C22×#615711yescharacteristic
287C2×C8—2122no—
288C2×C8—2122no—
289C2×C8—2122no—
290C2×C8—2122no—
291C2×C8—2122no—
292C2×C8—2122no—
293C4⋊C4—41no—
294C4⋊C4—1062no—
295C4⋊C4—2121no—
296C4⋊C4—2122no—
297C4⋊C4—2121no—
298C4⋊C4—2121no—
299C22⋊C4—42no—
300C22⋊C4—2122no—
301C22⋊C4—2121no—
302C22⋊C4—2121no—
303C22⋊C4—2121no—
304C22⋊C4—2121no—
305C22⋊C4—2122no—
306C22⋊C4—2122no—
307C42—1062no—
308C42—1062no—
309C42—2121no—
310C42—2121no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23#1313327912yes—
325C23C2×C22⋊#615711yescharacteristic
326D4—82no—
327D4—84no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—44no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—532no—
337C2×C4—531no—
338C2×C4—531no—
339C2×C4—531no—
340C2×C4—531no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—2122no—
351C2×C4—2122no—
352C2×C4—2122no—
353C2×C4—2122no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4#1313327512yes—
361C2×C4#1313330412yes—
362C8—4242no—
363C8—4242no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#1681159611yescharacteristic
379C22C2×#1313327511yescharacteristic
380C22#1681161011yescharacteristic
381C22C2×#1313327911yescharacteristic
382C22#1681161411yescharacteristic
383C22#1681162011yescharacteristic
384C22#1681162111yescharacteristic
385C4—22no—
386C4—22no—
387C4—81no—
388C4—1061no—
389C4—1061no—
390C4—1061no—
391C4—1062no—
392C4—1061no—
393C4—4241no—
394C4—4241no—
395C4—2121no—
396C2—42no—
397C2—161no—
398C2#2028998011yescharacteristic
399C2#2029002011yescharacteristic
400C2#2029007011yescharacteristic
401C2#2029013011yescharacteristic
402C2#2029015111yescharacteristic
403C2#2029020011yescharacteristic
404C2#2029022711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#121×C53C4
socleC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#121#6157
hypocenterC53#7714
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #20290571C2semidirect
2. #20290569C2semidirect
3. #20289534C2semidirect
4. #20289531C2semidirect
225. C53#7714semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202905692
#202895342
#202895312
#202830412
#771453