Finite Group Databasebeta

← #27301488

Subgroups — #27301488

Order 13568.

484 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M32×D53C211yesmaximal, characteristic
2C2×#16810865C211yesmaximal, characteristic
3C2×#16810440C211yesmaximal, characteristic, complemented
4C2×#16810438C211yesmaximal, characteristic, complemented
5#20274750C211yesmaximal, characteristic
6#20272823C211yesmaximal, characteristic, complemented
7C53×#1387C211yesmaximal, characteristic, complemented
8C22×Dic212C2211yescharacteristic
9C22×C8×D53C2211yescharacteristic
10C2×#13133197—21no—
11C2×#13133157C2211yescharacteristic
12C2×#13133156—21no—
13M32×D53—22no—
14M32×D53—22no—
15C2×C16×D53—21no—
16#16810865—21no—
17#16810865—21no—
18#16810865C412yes—
19#16810865C412yes—
20C2×#13133153C2211yescharacteristic
21C2×#13133140—21no—
22#16810586—21no—
23#16810440—22no—
24#16810438—22no—
25#16810387—21no—
26C2×M32×C53C2211yescharacteristic
27C2×#108×C53C2211yescharacteristic
28C2×#107×C53C2211yescharacteristic
29C22×Dic106—21no—
30C22×C4×D53D411yescharacteristic
31C2×Dic212—21no—
32C2×Dic212—21no—
33C2×Dic212—21no—
34C2×Dic212—21no—
35C2×Dic212—21no—
36C2×Dic212C2×C412yes—
37C2×#11719917—21no—
38C22×#88322D411yescharacteristic
39C2×C8×D53—21no—
40C2×C8×D53—21no—
41C2×C8×D53—21no—
42C2×C8×D53—21no—
43C2×C8×D53C2×C411yescharacteristic
44C2×C8×D53C2×C411yescharacteristic
45C2×C8×D53D411yescharacteristic
46C2×C8×D53D411yescharacteristic
47#13133197—22no—
48#13133197—22no—
49#13133197—41no—
50#13133197—41no—
51C2×C4⋊Dic53—41no—
52C2×#11719921—41no—
53#13133157—21no—
54#13133157D412yes—
55#13133156—24no—
56C2×#11719908—21no—
57C16×D53—24no—
58#13133153—21no—
59#13133153—21no—
60#13133140—22no—
61#13133140—22no—
62#13133129—41no—
63#13133129—41no—
64C2×M16×C53—21no—
65C22×C8×C53C2311yescharacteristic
66C2×C4⋊C4×C53—21no—
67M32×C53—21no—
68M32×C53D412yes—
69C2×C16×C53—21no—
70#108×C53—21no—
71#108×C53—21no—
72#107×C53—21no—
73#107×C53C2×C412yes—
74C23×D53SD1611yescharacteristic
75C22×C4×C53—41no—
76C22×C4×C53C2×D411yescharacteristic
77C22×Dic53—41no—
78C22×Dic53SD1611yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53C22⋊C411yescharacteristic
84C2×C4×D53C22⋊C411yescharacteristic
85C2×C4×D53D811yescharacteristic
86C2×C4×D53D811yescharacteristic
87C2×Dic106—22no—
88C2×Dic106—41no—
89C2×Dic106—41no—
90C2×Dic106—41no—
91C2×Dic106—41no—
92C2×Dic106—41no—
93M16×C53—41no—
94M16×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53C22×C411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53#4111yescharacteristic
101C2×C8×C53#4111yescharacteristic
102C4⋊C4×C53—22no—
103C4⋊C4×C53—41no—
104#11719921—42no—
105#11719921—42no—
106C4⋊Dic53—42no—
107C4⋊Dic53—42no—
108#11719917—41no—
109#11719917—41no—
110C2×#88322—21no—
111C2×#88322—21no—
112C2×#88322—41no—
113C2×#88322C22⋊C411yescharacteristic
114C2×#88322C22⋊C411yescharacteristic
115Dic212—22no—
116Dic212—22no—
117Dic212—41no—
118Dic212—41no—
119C8×D53—21no—
120C8×D53—21no—
121C8×D53—21no—
122C8×D53—21no—
123C8×D53—21no—
124C8×D53—21no—
125C8×D53—21no—
126C8×D53—21no—
127C16×C53—22no—
128#11719908—22no—
129C23×C53C2×SD1611yescharacteristic
130C22×D53—21no—
131C22×D53—21no—
132C22×D53—22no—
133C22×D53—41no—
134C22×D53#10211yescharacteristic
135C22×D53#10211yescharacteristic
136C2×C4×C53—42no—
137C2×C4×C53—82no—
138C2×C4×C53C2×C22⋊C411yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53C2×D811yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53#13511yescharacteristic
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—42no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149C2×Dic53#10211yescharacteristic
150C2×Dic53#10211yescharacteristic
151C4×D53—21no—
152C4×D53—21no—
153C4×D53—21no—
154C4×D53—21no—
155C4×D53—21no—
156C4×D53—21no—
157C4×D53—21no—
158C4×D53—21no—
159Dic106—42no—
160Dic106—42no—
161Dic106—81no—
162Dic106—81no—
163C8×C53—81no—
164C8×C53C4×D411yescharacteristic
165C8×C53C4×D411yescharacteristic
166C8×C53#12011yescharacteristic
167C8×C53#12311yescharacteristic
168#88322—21no—
169#88322—21no—
170#88322—81no—
171#13066—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53C2×#10211yescharacteristic
175C22×C53#46511yescharacteristic
176C22×C53#46511yescharacteristic
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—21no—
181C2×D53—21no—
182C2×D53—42no—
183C2×D53—42no—
184C2×D53#36212yescomplemented
185C4×C53—82no—
186C4×C53#38611yescharacteristic
187C4×C53#41611yescharacteristic
188C4×C53#46611yescharacteristic
189C4×C53#46611yescharacteristic
190Dic53—21no—
191Dic53—21no—
192Dic53—82no—
193C22×M32—531no—
194C2×#467—531no—
195C2×#362—531no—
196C2×#362—531no—
197#1810—531no—
198#1387—531no—
199#1387D5311yescharacteristic, complemented
200C2×C53—21no—
201C2×C53—21no—
202C2×C53#189311yescharacteristic
203C2×C53C2×#36211yescharacteristic
204C2×C53#214811yescharacteristic
205D53—22no—
206D53—41no—
207C22×Q16—531no—
208C23×C8—531no—
209C2×C22⋊Q8—1061no—
210C2×M32—531no—
211C2×M32—1061no—
212C2×M32—1062no—
213C2×M32—1062no—
214C2×M32C2×D5311yescharacteristic
215C22×C16—1061no—
216#467—532no—
217#467—532no—
218#467—1061no—
219#467—1061no—
220C2×#108—531no—
221C2×#108C2×D5311yescharacteristic
222C2×#107C2×D5311yescharacteristic
223C2×#103—1061no—
224#407—1061no—
225#362—1062no—
226#362—1062no—
227#341—1061no—
228C53#1306611yescharacteristic, complemented, Sylow
229C22×Q8—1061no—
230C23×C4—531no—
231C2×Q16—532no—
232C2×Q16—1061no—
233C2×Q16—1061no—
234C2×Q16—1061no—
235C2×Q16—1061no—
236C2×Q16—1061no—
237C2×M16—21no—
238C2×M16—1061no—
239C22×C8—531no—
240C22×C8—531no—
241C22×C8—531no—
242C22×C8—531no—
243C22×C8—531no—
244C22×C8—1061no—
245C22×C8—1061no—
246C22×C8—1061no—
247C22×C8—1061no—
248C22×C8C22×D5311yescharacteristic
249C22⋊Q8—1062no—
250C22⋊Q8—1062no—
251C22⋊Q8—2121no—
252C22⋊Q8—2121no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—2121no—
255C2×C22⋊C4—2121no—
256M32—21no—
257M32—532no—
258M32—1064no—
259M32—1061no—
260M32D21212yes—
261C2×C16—21no—
262C2×C16—1064no—
263C2×C16—1061no—
264#108—21no—
265#108—21no—
266#108—1061no—
267#108—1061no—
268#107—21no—
269#107C4×D5312yes—
270#103—1062no—
271#103—1062no—
272C22⋊C8—2121no—
273C22⋊C8—2121no—
274C24—531no—
275C2×Q8—1062no—
276C2×Q8—2121no—
277C2×Q8—2121no—
278C2×Q8—2121no—
279C2×Q8—2121no—
280C2×Q8—2121no—
281C22×C4—41no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—531no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—2121no—
292C22×C4D4×D5311yescharacteristic
293Q16—1062no—
294Q16—1062no—
295Q16—2121no—
296Q16—2121no—
297M16—41no—
298M16—41no—
299M16—2121no—
300M16—2121no—
301C2×C8—41no—
302C2×C8—531no—
303C2×C8—531no—
304C2×C8—1061no—
305C2×C8—1061no—
306C2×C8—1061no—
307C2×C8—1061no—
308C2×C8—1061no—
309C2×C8—1061no—
310C2×C8—1061no—
311C2×C8—1061no—
312C2×C8—1061no—
313C2×C8—1061no—
314C2×C8—2121no—
315C2×C8C2×C4×D5311yescharacteristic
316C2×C8C2×D21211yescharacteristic
317C2×C8#1171994511yescharacteristic
318C2×C8D4×D5311yescharacteristic
319C2×C8D4×D5311yescharacteristic
320C2×C8#1171994911yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—2122no—
324C4⋊C4—2122no—
325C22⋊C4—2122no—
326C22⋊C4—2122no—
327C16—22no—
328C16—1062no—
329C23—531no—
330C23—531no—
331C23—1062no—
332C23—1061no—
333C23—1061no—
334C23—2121no—
335C23SD16×D5311yescharacteristic
336Q8—4241no—
337Q8—4241no—
338Q8—2122no—
339Q8—2122no—
340C2×C4—42no—
341C2×C4—82no—
342C2×C4—531no—
343C2×C4—531no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—4241no—
355C2×C4—4241no—
356C2×C4—2122no—
357C2×C4C22⋊C4×D5311yescharacteristic
358C2×C4#1313319011yescharacteristic
359C2×C4#1313319211yescharacteristic
360C2×C4D8×D5311yescharacteristic
361C2×C4#1313323811yescharacteristic
362C2×C4#1313324011yescharacteristic
363C8—81no—
364C8—1061no—
365C8—1061no—
366C8—4241no—
367C8#1313317911yescharacteristic
368C8C4×D21211yescharacteristic
369C8#1313318711yescharacteristic
370C8#1313320211yescharacteristic
371C22—21no—
372C22—21no—
373C22—532no—
374C22—1061no—
375C22—1061no—
376C22—1061no—
377C22—1061no—
378C22—1061no—
379C22—2122no—
380C22—2122no—
381C22#102×D5311yescharacteristic
382C22#1681085311yescharacteristic
383C22#1681085511yescharacteristic
384C4—82no—
385C4—1061no—
386C4—1061no—
387C4—4242no—
388C4#1681050111yescharacteristic
389C4#1681061811yescharacteristic
390C4#1681085811yescharacteristic
391C4#1681086011yescharacteristic
392C2—21no—
393C2—21no—
394C2—1062no—
395C2—2121no—
396C2#2027514211yescharacteristic
397C2#362×D5311yescharacteristic
398C2#2027646111yescharacteristic
subgroupG/H
centerC22#102×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1387C2
socleC22×C53C2×#102
prime radicalC23×D53SD16
cosocleC22×C8×C53C23
hypercenter#1387D53
hypocenterC53#13066
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1387

Complements

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

NH ≅ G/Ntype
3. C2×#16810440C2semidirect
4. C2×#16810438C2semidirect
6. #20272823C2semidirect
7. C53×#1387C2semidirect
184. C2×D53#362semidirect
199. #1387D53semidirect
228. C53#13066semidirect

Maximal subgroups

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

subgroupindex
C2×M32×D532
C2×#168108652
C2×#168104402
C2×#168104382
#202747502
#202728232
C53×#13872
#1306653