Finite Group Databasebeta

← #27285355

Subgroups — #27285355

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810900C211yesmaximal, characteristic
2#20279669C211yesmaximal, characteristic
3#20275639C211yesmaximal, characteristic, complemented
4C2×#16810369C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic
6#20272555C211yesmaximal, characteristic, complemented
7C2×C53×#500C211yesmaximal, characteristic, complemented
8C2×#13133193—22no—
9C2×C4×D212C2211yescharacteristic
10#16810900—22no—
11#16810900—22no—
12C2×#13133150—21no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133141—22no—
16C2×#13133137—21no—
17#16810558—22no—
18#16810524—21no—
19#16810492—21no—
20C4×C4⋊Dic53C2211yescharacteristic
21#16810369—22no—
22#16810352—22no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×C4⋊Q8×C53C2211yescharacteristic
25C53×#500—22no—
26C53×#500—22no—
27C2×#107×C53—22no—
28C2×C4×C8×C53C2211yescharacteristic
29C22×D212D411yescharacteristic
30C22×C4×D53—21no—
31C22×#88322—21no—
32C22×#88322—21no—
33#13133193—44no—
34#13133193—44no—
35C4×D212—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C2×C4⋊Dic53D411yescharacteristic
40C2×C4⋊Dic53D412yes—
41C2×#11719919—24no—
42C2×#11719921—24no—
43C2×#11719921—21no—
44C2×C4×Dic53—21no—
45#13133150—24no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322D412yes—
50#13133141—44no—
51#13133137—24no—
52#13133124—21no—
53C22×Q8×C53—21no—
54C22×Q8×C53—21no—
55C22×C8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53D412yes—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53D412yes—
68C2×C42×C53C2311yescharacteristic
69#107×C53—28no—
70C4×C8×C53—21no—
71C4×C8×C53D412yes—
72C23×D53—21no—
73C2×Q8×C53—22no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53—42no—
78C2×Q8×C53—42no—
79C2×Q8×C53—42no—
80C2×Q8×C53—42no—
81C22×C4×C53—22no—
82C22×C4×C53—41no—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×Dic53—22no—
87C22×Dic53—21no—
88C2×D212—21no—
89C2×D212—21no—
90C2×D212—21no—
91C2×D212—21no—
92C2×D212—21no—
93C2×D212Q1612yes—
94C2×C4×D53—41no—
95C2×C4×D53—41no—
96C2×C4×D53—42no—
97C2×C4×D53—42no—
98C2×C8×C53—24no—
99C2×C8×C53—22no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C42×C53C2×D411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53#4111yescharacteristic
112#11719921—48no—
113#11719921—41no—
114#11719921—41no—
115C4⋊Dic53—22no—
116C4⋊Dic53—21no—
117C4⋊Dic53—21no—
118C4⋊Dic53D814yes—
119#11719919—48no—
120C4×Dic53—41no—
121C4×Dic53—41no—
122C2×#88322—22no—
123C2×#88322—41no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—42no—
127C23×C53#12011yescharacteristic
128C22×D53—22no—
129C22×D53—41no—
130C22×D53—42no—
131C22×D53—41no—
132C22×D53—42no—
133Q8×C53—44no—
134Q8×C53—82no—
135Q8×C53—81no—
136Q8×C53—81no—
137Q8×C53—81no—
138Q8×C53—81no—
139C2×C4×C53—22no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—44no—
143C2×C4×C53—82no—
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12011yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12711yescharacteristic
150C2×C4×C53C2×D812yes—
151C2×C4×C53C2×Q1611yescharacteristic
152C2×C4×C53#13511yescharacteristic
153C2×C4×C53AGL(1,ℤ/8)12yes—
154C2×C4×C53#13711yescharacteristic
155C2×C4×C53#13711yescharacteristic
156C2×Dic53—24no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—41no—
160C2×Dic53—42no—
161D212—22no—
162D212—22no—
163D212—41no—
164D212—41no—
165C4×D53—82no—
166C4×D53—81no—
167C4×D53—81no—
168C8×C53—24no—
169#88322—42no—
170#88322—81no—
171#10076—531nomaximal, Sylow
172C22×C53#39211yescharacteristic
173C22×C53#44712yes—
174C22×C53#45111yescharacteristic
175C22×C53#45211yescharacteristic
176C22×C53#45612yes—
177C2×D53—44no—
178C2×D53—42no—
179C2×D53—82no—
180C2×D53—81no—
181C2×D53—81no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—82no—
185C4×C53—82no—
186C4×C53#45111yescharacteristic
187C4×C53#45211yescharacteristic
188C4×C53#45912yes—
189C4×C53#46711yescharacteristic
190C4×C53#47011yescharacteristic
191C4×C53#49311yescharacteristic
192C4×C53#49611yescharacteristic
193Dic53—44no—
194Dic53—81no—
195C2×#500D5311yescharacteristic, complemented
196C2×#474—531no—
197#2542—531no—
198#1981—531no—
199C2×#332—531no—
200C2×D4⋊C8—531no—
201#1279—531no—
202C2×C53#164712yes—
203C2×C53#164912yes—
204C2×C53#200111yescharacteristic
205C2×C53#200511yescharacteristic
206C2×C53#201111yescharacteristic
207D53—82no—
208C2×C4⋊Q8C2×D5311yescharacteristic
209C2×C4⋊Q8C2×D5311yescharacteristic
210C2×C22⋊Q8—1062no—
211C2×C4×D4—531no—
212#500—22no—
213#500—22no—
214#474—1062no—
215#474—1062no—
216C2×#107—22no—
217C2×#107—1061no—
218C2×C4⋊C8—531no—
219C2×C4⋊C8—531no—
220C2×#103—1062no—
221C2×#102—1061no—
222C2×C4×C8C2×D5311yescharacteristic
223#399—1062no—
224#390—1061no—
225#384—1061no—
226C4×C4⋊C4—531no—
227#332—1062no—
228D4⋊C8—1062no—
229C53#1007611yescharacteristic, complemented, Sylow
230C22×Q8—21no—
231C22×Q8—21no—
232C22×D4—531no—
233C23×C4—1061no—
234C22×C8—21no—
235C22×C8—1061no—
236C22×C8—1061no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C22⋊D5312yes—
245C22⋊Q8—2124no—
246C22⋊Q8—2124no—
247C4×D4—1061no—
248C4×D4—1061no—
249C4×D4—1061no—
250C4×D4—1061no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—532no—
255C2×C4⋊C4—531no—
256C2×C4⋊C4—1064no—
257C2×C4⋊C4C22⋊D5312yes—
258C2×C22⋊C4—1064no—
259C2×C22⋊C4—1061no—
260C2×C42—1061no—
261C2×C42C22×D5311yescharacteristic
262#107—28no—
263#107—1064no—
264C4⋊C8—532no—
265C4⋊C8—1061no—
266C4⋊C8—1061no—
267C4⋊C8—1061no—
268#103—2124no—
269#102—1064no—
270C4×C8—21no—
271C4×C8C22⋊D5312yes—
272#95—1061no—
273C24—1061no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×D4—532no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C22×C4—22no—
289C22×C4—41no—
290C22×C4—1062no—
291C22×C4—1061no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4—2122no—
295C22×C4—2122no—
296C22×C4D4×D5311yescharacteristic
297C22×C4D4×D5311yescharacteristic
298C22×C4C2×C22⋊D5311yescharacteristic
299C2×C8—24no—
300C2×C8—22no—
301C2×C8—1062no—
302C2×C8—2121no—
303C2×C8—2122no—
304C2×C8—2121no—
305C2×C8—2121no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—534no—
315C4⋊C4—1062no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4—2128no—
319C22⋊C4—2128no—
320C22⋊C4—2121no—
321C22⋊C4—2121no—
322C42—2121no—
323C42—2121no—
324C42D4×D5311yescharacteristic
325C42#1171994711yescharacteristic
326C42C2×C22⋊D5311yescharacteristic
327C42C2×C22⋊D5311yescharacteristic
328C23—1062no—
329C23—2122no—
330C23—2121no—
331C23—2121no—
332C23—2122no—
333C23#1313318511yescharacteristic
334Q8—44no—
335Q8—81no—
336Q8—81no—
337Q8—82no—
338Q8—81no—
339Q8—81no—
340D4—1062no—
341D4—1062no—
342D4—2121no—
343D4—2121no—
344C2×C4—22no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—1064no—
350C2×C4—4241no—
351C2×C4—4242no—
352C2×C4—4241no—
353C2×C4—2122no—
354C2×C4—2121no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4#1313318511yescharacteristic
358C2×C4#1313318611yescharacteristic
359C2×C4#1313318811yescharacteristic
360C2×C4#1313318911yescharacteristic
361C2×C4#1313318911yescharacteristic
362C2×C4#1313321311yescharacteristic
363C2×C4D8×D5312yes—
364C2×C4Q16×D5311yescharacteristic
365C2×C4#1313324011yescharacteristic
366C2×C4#1313324112yes—
367C2×C4#1313324911yescharacteristic
368C2×C4#1313324911yescharacteristic
369C8—24no—
370C8—4241no—
371C8—2122no—
372C22—4242no—
373C22—4241no—
374C22—4241no—
375C22—2122no—
376C22—2124no—
377C22#1681053111yescharacteristic
378C22#1681075512yes—
379C22#1681077711yescharacteristic
380C22#1681078411yescharacteristic
381C22#1681080512yes—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—82no—
386C4—4241no—
387C4—2124no—
388C4#1681077711yescharacteristic
389C4#1681078411yescharacteristic
390C4#1681081612yes—
391C4#1681086111yescharacteristic
392C4#1681087811yescharacteristic
393C4#1681099811yescharacteristic
394C4#1681101111yescharacteristic
395C2—4242no—
396C2#2027394712yes—
397C2#2027396112yes—
398C2#2027572211yescharacteristic
399C2#2027575111yescharacteristic
400C2#2027578411yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#500C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenterC2×#500D53
hypocenterC53#10076
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
3. #20275639C2semidirect
4. C2×#16810369C2semidirect
6. #20272555C2semidirect
7. C2×C53×#500C2semidirect
195. C2×#500D53semidirect
229. C53#10076semidirect

Maximal subgroups

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

subgroupindex
C2×#168109002
#202796692
#202756392
C2×#168103692
C2×#168103522
#202725552
C2×C53×#5002
#1007653