Finite Group Databasebeta

← #27281812

Subgroups — #27281812

Order 13568.

904 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277293C211yesmaximal, characteristic
2#20274636C211yesmaximal, characteristic
3#20273248C212yesmaximal, complemented
4#20272568C212yesmaximal, complemented
5C53×#2435C211yesmaximal, characteristic
6C22×C22⋊Dic53—21no—
7C2×#13133134—22no—
8#16810587C2212yes—
9#16810503—24no—
10#16810486C2212yes—
11#16810479—21no—
12#16810360—24no—
13C23×D4×C53C2211yescharacteristic
14C2×#124×C53C412yescomplemented
15C2×C22⋊Q8×C53C2211yescharacteristic
16C53×#394—22no—
17C53×#390C412yescomplemented
18C53×#379—21no—
19C53×#379—21no—
20C4×C22⋊C4×C53C2211yescharacteristic
21C23×Dic53—22no—
22C2×#11719917—22no—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53—44no—
28#13133134—44no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133124—22no—
32#13133124D414yes—
33C25×C53D411yescharacteristic
34C25×C53D411yescharacteristic
35C22×Q8×C53C2×C411yescharacteristic, complemented
36C22×D4×C53—22no—
37C22×D4×C53—22no—
38C22×D4×C53—44no—
39C22×D4×C53C2×C412yes—
40C22×D4×C53D412yes—
41C23×C4×C53C2311yescharacteristic
42#124×C53—44no—
43C22⋊Q8×C53—22no—
44C22⋊Q8×C53—22no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53—22no—
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C42×C53C2×C412yescomplemented
54#95×C53—21no—
55C24×C53—21no—
56C24×C53—24no—
57C24×C53—21no—
58C24×C53—21no—
59C24×C53—21no—
60C24×C53—42no—
61C24×C53—42no—
62C24×C53—42no—
63C24×C53—81no—
64C24×C53—81no—
65C24×C53C22⋊C412yes—
66C24×C53C22⋊C412yes—
67C24×C53C2×D411yescharacteristic
68C2×Q8×C53—22no—
69C2×Q8×C53—42no—
70C2×D4×C53—24no—
71C2×D4×C53—24no—
72C2×D4×C53—24no—
73C2×D4×C53—24no—
74C2×D4×C53—42no—
75C2×D4×C53—42no—
76C2×D4×C53—44no—
77C2×D4×C53—44no—
78C2×D4×C53—48no—
79C22×C4×C53—22no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—42no—
83C22×C4×C53C22×C411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D412yes—
86C22×C4×C53#4112yes—
87C22×Dic53—24no—
88C22×Dic53—44no—
89C22×Dic53—82no—
90C22×Dic53—82no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—42no—
93C22⋊C4×C53—42no—
94C22⋊C4×C53—44no—
95C22⋊C4×C53—44no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—82no—
99C22⋊C4×C53—82no—
100C42×C53—42no—
101C22⋊Dic53—44no—
102C22⋊Dic53—44no—
103C22⋊Dic53—82no—
104C22⋊Dic53—82no—
105#11719917—44no—
106C2×#88322—42no—
107C23×C53—22no—
108C23×C53—22no—
109C23×C53—24no—
110C23×C53—22no—
111C23×C53—22no—
112C23×C53—22no—
113C23×C53—22no—
114C23×C53—22no—
115C23×C53—22no—
116C23×C53—22no—
117C23×C53—41no—
118C23×C53—42no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—42no—
123C23×C53—42no—
124C23×C53—42no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—81no—
129C23×C53—81no—
130C23×C53—81no—
131C23×C53—81no—
132C23×C53—88no—
133C23×C53—88no—
134C23×C53#9914yes—
135C23×C53C2×C22⋊C411yescharacteristic
136C23×C53#12011yescharacteristic
137C23×C53#12311yescharacteristic
138Q8×C53—82no—
139D4×C53—48no—
140D4×C53—48no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53C4×D412yes—
153C2×C4×C53#12112yes—
154C2×Dic53—44no—
155C2×Dic53—44no—
156C2×Dic53—44no—
157C2×Dic53—88no—
158#88322—82no—
159#9459—531nomaximal, Sylow
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—24no—
167C22×C53—21no—
168C22×C53—44no—
169C22×C53—44no—
170C22×C53—44no—
171C22×C53—44no—
172C22×C53—44no—
173C22×C53—42no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53—42no—
178C22×C53—82no—
179C22×C53—82no—
180C22×C53—84no—
181C22×C53—84no—
182C22×C53—84no—
183C22×C53#38611yescharacteristic
184C22×C53C2×#9912yes—
185C22×C53#45512yes—
186C4×C53—22no—
187C4×C53—22no—
188C4×C53—82no—
189C4×C53—81no—
190C4×C53—81no—
191Dic53—84no—
192#2435D5311yescharacteristic
193#2308—531no—
194#1783—531no—
195#1511—532no—
196#1284—532no—
197C2×C53—22no—
198C2×C53—22no—
199C2×C53—41no—
200C2×C53—44no—
201C2×C53—82no—
202C2×C53#188511yescharacteristic
203C2×C53#189811yescharacteristic
204C2×C53#191111yescharacteristic
205C23×D4C2×D5311yescharacteristic
206C2×#124Dic5312yescomplemented
207C2×C22⋊Q8C2×D5311yescharacteristic
208C22×C22⋊C4—1061no—
209C2×#100—1062no—
210#407—532no—
211#394—22no—
212#390Dic5312yescomplemented
213#386—1064no—
214#381—532no—
215#379—21no—
216#379—21no—
217#379—1061no—
218C4×C22⋊C4C2×D5311yescharacteristic
219#328—1064no—
220C53#945911yescharacteristic, complemented, Sylow
221C25C22⋊D5311yescharacteristic
222C25C22⋊D5311yescharacteristic
223C22×Q8C2×Dic5311yescharacteristic, complemented
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4C2×Dic5312yes—
228C22×D4C22⋊D5312yes—
229C23×C4—1062no—
230C23×C4C22×D5311yescharacteristic
231C2×M16—1062no—
232#124—44no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C2×Dic5311yescharacteristic, complemented
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—1062no—
244C2×C22⋊C4—1062no—
245C2×C22⋊C4—1062no—
246C2×C22⋊C4—1062no—
247C2×C22⋊C4—2124no—
248C2×C42C2×Dic5312yescomplemented
249#100—2124no—
250C22⋊C8—1062no—
251C22⋊C8—1062no—
252#95—21no—
253#95—534no—
254#95—1062no—
255C24—21no—
256C24—24no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—81no—
264C24—81no—
265C24C22⋊Dic5312yes—
266C24C22⋊Dic5312yes—
267C24C2×C22⋊D5311yescharacteristic
268C2×Q8—22no—
269C2×Q8—42no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—48no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—42no—
283C22×C4—1064no—
284C22×C4—4242no—
285C22×C4—4242no—
286C22×C4—2124no—
287C22×C4D4×D5312yes—
288C22×C4#1171994712yes—
289C22×C4C22×Dic5311yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291M16—2124no—
292C2×C8—2122no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C22⋊C4—44no—
296C22⋊C4—42no—
297C22⋊C4—44no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—4242no—
303C22⋊C4—4242no—
304C22⋊C4—2124no—
305C22⋊C4—2124no—
306C42—42no—
307C23—22no—
308C23—22no—
309C23—24no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—41no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—88no—
333C23—88no—
334C23#1313313214yes—
335C23C2×C22⋊Dic5311yescharacteristic
336C23#1313318511yescharacteristic
337C23#1313319811yescharacteristic
338Q8—82no—
339D4—48no—
340D4—48no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—4248no—
353C2×C4—2124no—
354C2×C4—2124no—
355C2×C4—2124no—
356C2×C4D4×Dic5312yes—
357C2×C4#1313318912yes—
358C8—4242no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—24no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—44no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—82no—
378C22—82no—
379C22—84no—
380C22—84no—
381C22—84no—
382C22#1681050311yescharacteristic
383C22C2×#1313313212yes—
384C22#1681080012yes—
385C4—22no—
386C4—22no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4—4244no—
391C2—22no—
392C2—22no—
393C2—41no—
394C2—44no—
395C2—82no—
396C2#2027510411yescharacteristic
397C2#2027516911yescharacteristic
398C2#2027524811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2435C2
socleC22×C53#386
prime radicalC23×D4×C53C22
cosocleC23×C4×C53C23
hypercenter#2435D53
hypocenterC53#9459
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2435

Complements

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

NH ≅ G/Ntype
3. #20273248C2semidirect
4. #20272568C2semidirect
14. C2×#124×C53C4semidirect
17. C53×#390C4semidirect
35. C22×Q8×C53C2×C4semidirect
46. C2×C4⋊C4×C53C2×C4semidirect
53. C2×C42×C53C2×C4semidirect
206. C2×#124Dic53semidirect
212. #390Dic53semidirect
220. C53#9459semidirect
223. C22×Q8C2×Dic53semidirect
236. C2×C4⋊C4C2×Dic53semidirect
248. C2×C42C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202772932
#202746362
#202732482
#202725682
C53×#24352
#945953