Finite Group Databasebeta

← #27277627

Subgroups — #27277627

Order 13568.

438 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 87 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#20277332C211yesmaximal, characteristic
2#20273044C211yesmaximal, characteristic
3#20273044C211yesmaximal, characteristic
4#20272622C211yesmaximal, characteristic
5#20272621C211yesmaximal, characteristic
6#20272620C211yesmaximal, characteristic
7C53×#1305C211yesmaximal, characteristic
8C2×#13133173C411yescharacteristic
9C2×#13133173C411yescharacteristic
10C22×#11719919C2211yescharacteristic
11#16810587C2211yescharacteristic
12#16810587C2211yescharacteristic
13#16810520—21no—
14#16810520—21no—
15#16810518—21no—
16#16810518—21no—
17#16810483C2211yescharacteristic
18#16810481C411yescharacteristic
19#16810481C411yescharacteristic
20#16810348—21no—
21#16810348—21no—
22#16810348—21no—
23#16810348—21no—
24C53×#407C2211yescharacteristic
25C53×#407C2211yescharacteristic
26C53×#380C2211yescharacteristic
27C23×Dic53C2×C411yescharacteristic
28C23×Dic53C2×C411yescharacteristic
29C2×#11719917—21no—
30C2×#11719917—21no—
31#13133173—42no—
32#13133173—42no—
33C2×C4⋊Dic53—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919D411yescharacteristic
43C2×#11719919D411yescharacteristic
44C2×#11719919D411yescharacteristic
45C2×#11719919D411yescharacteristic
46C2×C22⋊Dic53C2×C411yescharacteristic
47C2×C22⋊Dic53C2×C411yescharacteristic
48C2×C4×Dic53—21no—
49C2×C4×Dic53—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54#13133124—21no—
55#13133124—21no—
56#13133124—21no—
57C23×C4×C53C2311yescharacteristic
58C2×M16×C53—21no—
59C2×M16×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C22⋊C4×C53C2×C411yescharacteristic
62C2×C22⋊C4×C53C2×C411yescharacteristic
63C22⋊C8×C53—21no—
64C22⋊C8×C53—21no—
65C22⋊C8×C53—21no—
66C22⋊C8×C53—21no—
67#95×C53—21no—
68C24×C53C22×C411yescharacteristic
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—81no—
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53#4111yescharacteristic
75C22×C4×C53#4111yescharacteristic
76C22×C4×C53#4111yescharacteristic
77C22×C4×C53#4111yescharacteristic
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—22no—
81C22×Dic53—22no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—81no—
87C22×Dic53C22⋊C411yescharacteristic
88C22×Dic53C22⋊C411yescharacteristic
89C22×Dic53C22⋊C411yescharacteristic
90C22×Dic53C22⋊C411yescharacteristic
91M16×C53—81no—
92M16×C53—81no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C4⋊C4×C53—81no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C22⋊Dic53—41no—
99C22⋊Dic53—41no—
100C4⋊Dic53—81no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919—21no—
107#11719919—21no—
108#11719919—21no—
109#11719919—42no—
110#11719919—42no—
111#11719919—81no—
112#11719919—81no—
113C4×Dic53—42no—
114C4×Dic53—42no—
115#11719917—81no—
116#11719917—81no—
117C2×#88322—41no—
118C2×#88322—41no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53C2×C22⋊C411yescharacteristic
124C23×C53#11711yescharacteristic
125C23×C53#11711yescharacteristic
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
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#12311yescharacteristic
139C2×C4×C53#12311yescharacteristic
140C2×C4×C53#12311yescharacteristic
141C2×C4×C53#12311yescharacteristic
142C2×Dic53—24no—
143C2×Dic53—24no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—81no—
159C2×Dic53—81no—
160C2×Dic53#10012yes—
161C2×Dic53#10112yes—
162C8×C53—81no—
163C8×C53—81no—
164#88322—81no—
165#88322—81no—
166#8757—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53#38111yescharacteristic
178C22×C53C2×#10011yescharacteristic
179C22×C53C2×#10111yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—81no—
185C4×C53—81no—
186Dic53—22no—
187Dic53—22no—
188Dic53—41no—
189Dic53—41no—
190Dic53—81no—
191Dic53—81no—
192#2316—531no—
193#1463—531no—
194#1463—531no—
195#1305—531no—
196#1305—531no—
197#1305—531no—
198#1305D5311yescharacteristic
199C2×C53—21no—
200C2×C53—21no—
201C2×C53—41no—
202C2×C53—41no—
203C2×C53#182111yescharacteristic
204C2×C53#182811yescharacteristic
205C2×C53#183011yescharacteristic
206C2×#117—531no—
207C2×#117—531no—
208C22×C4⋊C4—531no—
209#407—531no—
210#407—531no—
211#407C2×D5311yescharacteristic
212#407C2×D5311yescharacteristic
213#389—1061no—
214#389—1061no—
215#389—1061no—
216#389—1061no—
217#380—531no—
218#380—531no—
219#380—531no—
220#380C2×D5311yescharacteristic
221#324—1061no—
222#324—1061no—
223#324—1061no—
224#324—1061no—
225C53#875711yescharacteristic, complemented, Sylow
226C23×C4—531no—
227C23×C4—531no—
228C23×C4C22×D5311yescharacteristic
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—1061no—
232C2×M16—1061no—
233#117—2122no—
234#117—2122no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—531no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C22⋊C4—531no—
250C2×C22⋊C4—531no—
251C2×C22⋊C4C4×D5311yescharacteristic
252C2×C22⋊C4C4×D5311yescharacteristic
253C2×C42—1061no—
254C2×C42—1061no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259C22⋊C8—1061no—
260C22⋊C8—1061no—
261C22⋊C8—1061no—
262C22⋊C8—1061no—
263#95—21no—
264#95—1061no—
265#95—1061no—
266#95—1061no—
267C24C2×C4×D5311yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—81no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—4241no—
284C22×C4#1171994511yescharacteristic
285C22×C4#1171994511yescharacteristic
286C22×C4D4×D5311yescharacteristic
287C22×C4D4×D5311yescharacteristic
288C22×C4#1171994711yescharacteristic
289C22×C4#1171994911yescharacteristic
290M16—81no—
291M16—81no—
292M16—4241no—
293M16—4241no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—2121no—
297C2×C8—2121no—
298C4⋊C4—81no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4—1061no—
307C4⋊C4—4241no—
308C4⋊C4—4241no—
309C4⋊C4—4241no—
310C4⋊C4—2122no—
311C4⋊C4—2122no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—2121no—
315C22⋊C4—2121no—
316C42—2122no—
317C42—2122no—
318C23—21no—
319C23—21no—
320C23—41no—
321C23—41no—
322C23C22⋊C4×D5311yescharacteristic
323C23#1313317411yescharacteristic
324C23#1313317511yescharacteristic
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—532no—
338C2×C4—532no—
339C2×C4—1064no—
340C2×C4—1064no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—4241no—
348C2×C4—4241no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4#1313319911yescharacteristic
358C2×C4#1313319911yescharacteristic
359C2×C4#1313320211yescharacteristic
360C2×C4#1313320211yescharacteristic
361C8—81no—
362C8—81no—
363C8—4241no—
364C8—4241no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—42no—
374C22—42no—
375C22#1681048511yescharacteristic
376C22#100×D5311yescharacteristic
377C22#101×D5311yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—81no—
383C4—81no—
384C4—1062no—
385C4—1062no—
386C4—4241no—
387C4—4241no—
388C4—2121no—
389C4—2121no—
390C2—21no—
391C2—21no—
392C2—41no—
393C2—41no—
394C2#2027479411yescharacteristic
395C2#2027482211yescharacteristic
396C2#2027482811yescharacteristic
subgroupG/H
centerC22#16810485
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1305C2
socleC22×C53#381
cosocleC23×C4×C53C23
hypercenter#1305D53
hypocenterC53#8757
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
225. C53#8757semidirect

Maximal subgroups

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

subgroupindex
#202773322
#202730442
#202726222
#202726212
#202726202
C53×#13052
#875753