Finite Group Databasebeta

← #27273123

Subgroups — #27273123

Order 13568.

430 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 85 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#20281732C211yesmaximal, characteristic
2#20276274C211yesmaximal, characteristic, complemented
3#20276125C211yesmaximal, characteristic, complemented
4#20276118C211yesmaximal, characteristic
5#20272540C211yesmaximal, characteristic, complemented
6#20272539C211yesmaximal, characteristic, complemented
7C53×#1270C211yesmaximal, characteristic
8C2×#13133194—21no—
9C2×#13133142—21no—
10C2×#13133138—21no—
11#16810559—21no—
12#16810556—21no—
13#16810546—21no—
14#16810529C2211yescharacteristic
15#16810521C2211yescharacteristic
16#16810503—21no—
17#16810498—21no—
18#16810493C2211yescharacteristic
19#16810386—21no—
20#16810386—21no—
21#16810386—21no—
22#16810385—21no—
23#16810378C2211yescharacteristic
24C53×#390C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Dic106D411yescharacteristic
28C23×Dic53—21no—
29C22×#88322—21no—
30#13133194—42no—
31#13133194—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53D411yescharacteristic
37C2×C4⋊Dic53D411yescharacteristic
38C2×C4⋊Dic53D411yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—21no—
45#13133142—42no—
46#13133138—42no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×D4×C53D411yescharacteristic
52C22×C8×C53—21no—
53C22×C8×C53—21no—
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C42×C53C2311yescharacteristic
58C24×C53—21no—
59C2×D4×C53—21no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53SD1612yes—
63C22×C4×C53—21no—
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—42no—
72C22×Dic53—41no—
73C22×Dic53—81no—
74C22×Dic53—81no—
75C2×Dic106—21no—
76C2×Dic106—41no—
77C2×Dic106—41no—
78C2×Dic106D812yes—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C2×C8×C53—41no—
82C2×C8×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C4⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C42×C53—41no—
91C22⋊Dic53—42no—
92C22⋊Dic53—81no—
93C22⋊Dic53—81no—
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53—42no—
97C4⋊Dic53—41no—
98C4⋊Dic53—41no—
99C4⋊Dic53—42no—
100C4⋊Dic53—41no—
101C4⋊Dic53—41no—
102C4⋊Dic53—41no—
103C4⋊Dic53—41no—
104#11719919—42no—
105#11719919—41no—
106#11719919—41no—
107#11719919—81no—
108C2×#88322—41no—
109C2×#88322—41no—
110C2×#88322—41no—
111C23×C53—22no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53#12011yescharacteristic
119D4×C53—42no—
120D4×C53—81no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53C2×D811yescharacteristic
131C2×C4×C53C2×SD1611yescharacteristic
132C2×C4×C53#13511yescharacteristic
133C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C53#13711yescharacteristic
135C2×C4×C53#13711yescharacteristic
136C2×Dic53—22no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—84no—
149C2×Dic53—81no—
150C2×Dic53—81no—
151C2×Dic53—81no—
152Dic106—42no—
153Dic106—81no—
154C8×C53—81no—
155C8×C53—81no—
156#88322—81no—
157#7787—531nomaximal, Sylow
158C22×C53—42no—
159C22×C53—42no—
160C22×C53—42no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53—81no—
164C22×C53—81no—
165C22×C53#44711yescharacteristic
166C22×C53#44811yescharacteristic
167C22×C53#45211yescharacteristic
168C22×C53#45411yescharacteristic
169C22×C53#45511yescharacteristic
170C22×C53#45511yescharacteristic
171C22×C53#45811yescharacteristic
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—81no—
179Dic53—42no—
180Dic53—81no—
181Dic53—81no—
182Dic53—81no—
183Dic53—161no—
184#2732—531no—
185#2099—531no—
186#2072—531no—
187#2071—531no—
188#1272—531no—
189#1271—531no—
190#1270D5311yescharacteristic
191C2×C53—82no—
192C2×C53#160711yescharacteristic
193C2×C53#160811yescharacteristic
194C2×C53#161511yescharacteristic
195C2×C53#162711yescharacteristic
196C2×C53#164111yescharacteristic
197C2×C53#164311yescharacteristic
198C2×C53#166211yescharacteristic
199C2×C22⋊Q8—1061no—
200C2×#103—1061no—
201C2×#102—1061no—
202#399—1061no—
203#398—1061no—
204#396—1061no—
205#391—531no—
206#390C2×D5311yescharacteristic
207#389—531no—
208#386—1061no—
209#385—1061no—
210#384—531no—
211#340—1061no—
212#340—1061no—
213#340—1061no—
214#340—1061no—
215#336—531no—
216#336C2×D5311yescharacteristic
217#336C2×D5311yescharacteristic
218C53#778711yescharacteristic, complemented, Sylow
219C22×Q8—531no—
220C22×D4D21211yescharacteristic
221C23×C4—1061no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—1061no—
225C22⋊Q8—2122no—
226C22⋊Q8—2122no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—531no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4D21211yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1061no—
243C2×C42C22×D5311yescharacteristic
244#103—2122no—
245#102—2122no—
246#95—1061no—
247#95—1061no—
248#95—1061no—
249#95—1061no—
250C24—21no—
251C2×Q8—532no—
252C2×Q8—1061no—
253C2×Q8—2121no—
254C2×Q8—2121no—
255C2×D4—21no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4#1171991312yes—
259C22×C4—21no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—4241no—
265C22×C4—4241no—
266C22×C4—2122no—
267C22×C4—2121no—
268C22×C4C2×D21211yescharacteristic
269C22×C4D4×D5311yescharacteristic
270C22×C4D4×D5311yescharacteristic
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—2121no—
278C2×C8—2121no—
279C2×C8—2121no—
280C4⋊C4—41no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—4241no—
284C4⋊C4—2122no—
285C4⋊C4—2121no—
286C4⋊C4—2121no—
287C4⋊C4—2122no—
288C4⋊C4—2122no—
289C4⋊C4—2121no—
290C4⋊C4—2121no—
291C4⋊C4—2121no—
292C4⋊C4—2121no—
293C4⋊C4—2121no—
294C4⋊C4—2121no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—4241no—
300C22⋊C4—4241no—
301C22⋊C4—2122no—
302C42—41no—
303C23—22no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23#1313318711yescharacteristic
311Q8—4241no—
312Q8—2122no—
313D4—42no—
314D4—81no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—1062no—
325C2×C4—4241no—
326C2×C4—4241no—
327C2×C4—4241no—
328C2×C4—4244no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4D8×D5311yescharacteristic
341C2×C4C2×#1171991311yescharacteristic
342C2×C4#1313323611yescharacteristic
343C2×C4#1313324411yescharacteristic
344C2×C4#1313324611yescharacteristic
345C2×C4#1313325011yescharacteristic
346C8—81no—
347C8—81no—
348C8—4241no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#1681075811yescharacteristic
357C22#1681076411yescharacteristic
358C22#1681078611yescharacteristic
359C22#1681079711yescharacteristic
360C22#1681080211yescharacteristic
361C22#1681080311yescharacteristic
362C22#1681081411yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—81no—
370C4—8481no—
371C4—4241no—
372C4—4241no—
373C4—4241no—
374C4—2122no—
375C2—82no—
376C2#2027370211yescharacteristic
377C2#2027370811yescharacteristic
378C2#2027374711yescharacteristic
379C2#2027382111yescharacteristic
380C2#2027390911yescharacteristic
381C2#2027392411yescharacteristic
382C2#2027404011yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1270C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#1270D53
hypocenterC53#7787
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #20276274C2semidirect
3. #20276125C2semidirect
5. #20272540C2semidirect
6. #20272539C2semidirect
218. C53#7787semidirect

Maximal subgroups

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

subgroupindex
#202817322
#202762742
#202761252
#202761182
#202725402
#202725392
C53×#12702
#778753