Finite Group Databasebeta

← #27272548

Subgroups — #27272548

Order 13568.

568 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 109 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#20283053C211yesmaximal, characteristic
2#20274407C211yesmaximal, characteristic
3#336×D53C211yesmaximal, characteristic
4#20272540C211yesmaximal, characteristic, complemented
5#20272540C211yesmaximal, characteristic, complemented
6#20272538C211yesmaximal, characteristic, complemented
7C53×#1271C211yesmaximal, characteristic, complemented
8C22×#11719912—21no—
9C22×C8×D53—21no—
10C2×#13133197C411yescharacteristic
11C2×C42×D53C2211yescharacteristic
12#16810529C2211yescharacteristic
13#16810528C411yescharacteristic
14#16810519C411yescharacteristic
15#16810510—21no—
16#16810509—21no—
17#16810494—21no—
18#16810485—21no—
19#16810480C411yescharacteristic
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22#16810377—21no—
23#16810377—21no—
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Dic106C2×C411yescharacteristic
28C22×C4×D53C2×C411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C2×#11719912—24no—
32C2×#11719912—42no—
33C2×#11719912—42no—
34C22×#88322—21no—
35C22×#88322—21no—
36C2×C8×D53—24no—
37C2×C8×D53—42no—
38C2×C8×D53—42no—
39#13133197—42no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×#11719919—21no—
43C2×#11719921—21no—
44C2×#11719921—21no—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C42×D53—42no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55C22×C8×C53—21no—
56C22×C8×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53C2×C411yescharacteristic
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C42×C53C2311yescharacteristic
61#95×C53—21no—
62C23×D53C22⋊C411yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C22⋊C411yescharacteristic
73C2×C4×D53—22no—
74C2×C4×D53—22no—
75C2×C4×D53—22no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—41no—
83C2×C4×D53SD1612yes—
84C2×C4×D53Q1612yes—
85C2×Dic106—21no—
86C2×Dic106—21no—
87C2×Dic106—41no—
88C2×Dic106—41no—
89C2×C8×C53—22no—
90C2×C8×C53—22no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C42×C53—41no—
101#11719921—42no—
102#11719921—42no—
103C4⋊Dic53—42no—
104C4⋊Dic53—41no—
105#11719919—42no—
106C4×Dic53—41no—
107C4×Dic53—41no—
108C4×Dic53—41no—
109C2×#88322—22no—
110C2×#88322—22no—
111C2×#88322—41no—
112C2×#88322—41no—
113C2×#88322—41no—
114C2×#88322—41no—
115#11719912—48no—
116C8×D53—48no—
117C23×C53C2×C22⋊C411yescharacteristic
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53#9912yes—
123C22×D53#10312yes—
124C22×D53#10412yes—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
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#13111yescharacteristic
138C2×C4×C53#13111yescharacteristic
139C2×C4×C53C2×SD1611yescharacteristic
140C2×C4×C53C2×Q1611yescharacteristic
141C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C53#13711yescharacteristic
143C2×Dic53—21no—
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#10312yes—
157C2×Dic53#10412yes—
158C4×D53—24no—
159C4×D53—24no—
160C4×D53—24no—
161C4×D53—24no—
162C4×D53—42no—
163C4×D53—42no—
164Dic106—81no—
165Dic106—81no—
166C8×C53—42no—
167C8×C53—42no—
168#88322—42no—
169#88322—42no—
170#7681—531nomaximal, Sylow
171C22×C53#40811yescharacteristic
172C22×C53C2×#9911yescharacteristic
173C22×C53#41311yescharacteristic
174C22×C53C2×#10311yescharacteristic
175C22×C53#41711yescharacteristic
176C22×C53C2×#10411yescharacteristic
177C22×C53C2×#10411yescharacteristic
178C2×D53—22no—
179C2×D53—22no—
180C2×D53—22no—
181C2×D53—22no—
182C2×D53—22no—
183C2×D53—22no—
184C2×D53#32814yes—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—81no—
192C4×C53—81no—
193Dic53—22no—
194Dic53—22no—
195Dic53—41no—
196Dic53—81no—
197Dic53—81no—
198#2855—531no—
199#1728—531no—
200C2×#336—531no—
201#1272—531no—
202#1272—531no—
203#1271—531no—
204#1271D5311yescharacteristic, complemented
205C2×C53#146811yescharacteristic
206C2×C53#148311yescharacteristic
207C2×C53#149311yescharacteristic
208C2×C53C2×#32811yescharacteristic
209C2×C53#150411yescharacteristic
210C2×C53#152411yescharacteristic
211C2×C53#155011yescharacteristic
212D53—24no—
213C22×M16—1061no—
214C23×C8—1061no—
215C2×C22⋊Q8—531no—
216C22×C42—531no—
217#391—531no—
218#391—531no—
219#389—531no—
220#389C2×D5311yescharacteristic
221#387—1061no—
222#387—1061no—
223#385—1061no—
224#381—1061no—
225#380—531no—
226#336—531no—
227#336—531no—
228#336—1061no—
229#336—1061no—
230#336C2×D5311yescharacteristic
231#336C2×D5311yescharacteristic
232C53#768111yescharacteristic, complemented, Sylow
233C22×Q8—531no—
234C23×C4—531no—
235C23×C4—531no—
236C23×C4—531no—
237C2×M16—1064no—
238C2×M16—2122no—
239C2×M16—2122no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—1064no—
243C22×C8—1061no—
244C22×C8—1061no—
245C22×C8—2122no—
246C22×C8—2122no—
247C22⋊Q8—2122no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—531no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4C4×D5311yescharacteristic
253C2×C4⋊C4C4×D5311yescharacteristic
254C2×C22⋊C4—1061no—
255C2×C22⋊C4—1061no—
256C2×C42—531no—
257C2×C42—531no—
258C2×C42—531no—
259C2×C42—2122no—
260C2×C42C22×D5311yescharacteristic
261#95—21no—
262#95—1061no—
263#95—1061no—
264#95—1061no—
265#95—1061no—
266#95—1061no—
267#95—1061no—
268C24—531no—
269C2×Q8—1061no—
270C2×Q8—1061no—
271C2×Q8—2121no—
272C2×Q8—2121no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4—532no—
278C22×C4—532no—
279C22×C4—531no—
280C22×C4—1062no—
281C22×C4—1062no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—1062no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—2121no—
292C22×C4C2×C4×D5311yescharacteristic
293C22×C4D4×D5311yescharacteristic
294C22×C4D4×D5311yescharacteristic
295M16—2128no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—1062no—
303C2×C8—1062no—
304C2×C8—2128no—
305C2×C8—2121no—
306C2×C8—2121no—
307C2×C8—2121no—
308C2×C8—2121no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—2121no—
315C4⋊C4—2122no—
316C4⋊C4—2122no—
317C22⋊C4—2122no—
318C22⋊C4—2122no—
319C42—41no—
320C42—2121no—
321C42—2121no—
322C42—2121no—
323C23—532no—
324C23—532no—
325C23—532no—
326C23—1061no—
327C23—1061no—
328C23—1061no—
329C23—1061no—
330C23C22⋊C4×D5311yescharacteristic
331Q8—4241no—
332Q8—4241no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—532no—
346C2×C4—532no—
347C2×C4—1064no—
348C2×C4—1064no—
349C2×C4—1064no—
350C2×C4—1064no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4#1313322311yescharacteristic
367C2×C4#1313322511yescharacteristic
368C2×C4SD16×D5311yescharacteristic
369C2×C4Q16×D5311yescharacteristic
370C2×C4#1313324111yescharacteristic
371C2×C4#1313324611yescharacteristic
372C8—42no—
373C8—42no—
374C8—2122no—
375C8—2122no—
376C22—534no—
377C22—1062no—
378C22—1062no—
379C22—1062no—
380C22—1062no—
381C22—1062no—
382C22—1062no—
383C22#1681058811yescharacteristic
384C22#99×D5311yescharacteristic
385C22#1681060611yescharacteristic
386C22#103×D5311yescharacteristic
387C22#1681062111yescharacteristic
388C22#104×D5311yescharacteristic
389C22#104×D5311yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—81no—
397C4—81no—
398C4—1062no—
399C4—1062no—
400C4—4241no—
401C4—4241no—
402C4—2121no—
403C2—1064no—
404C2#2027306211yescharacteristic
405C2#2027312511yescharacteristic
406C2#2027316811yescharacteristic
407C2#328×D5311yescharacteristic
408C2#2027321611yescharacteristic
409C2#2027330211yescharacteristic
410C2#2027340411yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1271D53
hypocenterC53#7681
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #20272540C2semidirect
5. #20272540C2semidirect
6. #20272538C2semidirect
7. C53×#1271C2semidirect
204. #1271D53semidirect
232. C53#7681semidirect

Maximal subgroups

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

subgroupindex
#202830532
#202744072
#336×D532
#202725402
#202725382
C53×#12712
#768153