Finite Group Databasebeta

← #27283012

Subgroups — #27283012

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719924C211yesmaximal, characteristic
2#20273461C211yesmaximal, characteristic, complemented
3#20273110C211yesmaximal, characteristic, complemented
4#20273107C211yesmaximal, characteristic
5#20273035C211yesmaximal, characteristic
6#20272999C211yesmaximal, characteristic, complemented
7C53×#1461C211yesmaximal, characteristic, complemented
8C22×#11719924C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10C4×#11719924—21no—
11C4×#11719924—21no—
12C4×#11719924—21no—
13C4×#11719924—21no—
14#16810676C2211yescharacteristic
15#16810676C2211yescharacteristic
16C2×#13133148C411yescharacteristic
17C2×#13133141C411yescharacteristic
18C2×#13133137C411yescharacteristic
19C2×C4×#88322C2211yescharacteristic
20#16810375—21no—
21#16810373—21no—
22Q8⋊#88322—22no—
23#16810352—22no—
24#16810346—22no—
25#16810345—21no—
26C8×#88322—21no—
27C2×C4×Q8×C53C2211yescharacteristic
28C53×#431C2211yescharacteristic
29C53×#431C2211yescharacteristic
30C53×#324—22no—
31C22×D212C2×C411yescharacteristic
32C22×C4×D53—21no—
33C2×#11719924—21no—
34C2×#11719924—21no—
35C2×#11719924—22no—
36C2×#11719924C814yescomplemented
37C22×#88322C2×C411yescharacteristic
38C22×#88322C2×C411yescharacteristic
39C4×D212—21no—
40C4×D212—21no—
41C4×D212—21no—
42C4×D212—21no—
43C2×C4⋊Dic53C2×C411yescharacteristic
44C2×#11719921—21no—
45#13133148—21no—
46#13133148—21no—
47C4⋊#88322—21no—
48C4⋊#88322D412yes—
49#13133141—21no—
50#13133141—21no—
51#13133137—21no—
52#13133137—21no—
53C22⋊#88322—21no—
54C22⋊#88322—21no—
55C4×#88322—21no—
56C4×#88322—21no—
57C4×#88322D411yescharacteristic
58C4×#88322D411yescharacteristic
59C4×#88322D411yescharacteristic
60C4×#88322D411yescharacteristic
61C22×Q8×C53C2×C411yescharacteristic
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×Q8×C53—21no—
65C4×Q8×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53C2×C411yescharacteristic
68C2×C42×C53—21no—
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72C22⋊C8×C53—21no—
73C22⋊C8×C53D412yes—
74C4×C8×C53—21no—
75C4×C8×C53—21no—
76C23×D53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53C2×C812yes—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53C22×C411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C22×Dic53—21no—
89C2×D212—21no—
90C2×D212—21no—
91C2×D212—21no—
92C2×D212—21no—
93C2×D212—21no—
94C2×D212C2×C812yes—
95C2×C4×D53—41no—
96C2×C4×D53—41no—
97C2×C4×D53—82no—
98C2×C8×C53—22no—
99C2×C8×C53—41no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C42×C53—41no—
105C42×C53—41no—
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53C2×D411yescharacteristic
109C42×C53#4111yescharacteristic
110#11719924—24no—
111#11719924—24no—
112#11719921—41no—
113#11719921—41no—
114C4⋊Dic53—21no—
115C4⋊Dic53—21no—
116C2×#88322—21no—
117C2×#88322—21no—
118C2×#88322—22no—
119C2×#88322—41no—
120C2×#88322C22⋊C411yescharacteristic
121C2×#88322C22⋊C411yescharacteristic
122C2×#88322C22⋊C411yescharacteristic
123C2×#88322C22⋊C411yescharacteristic
124C2×#88322C2×C812yes—
125C2×#88322M1612yes—
126C23×C53C4×D411yescharacteristic
127C22×D53—22no—
128C22×D53—41no—
129C22×D53—41no—
130C22×D53—81no—
131C22×D53—81no—
132Q8×C53—22no—
133Q8×C53—22no—
134Q8×C53—41no—
135Q8×C53—41no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—81no—
143C2×C4×C53C2×C22⋊C411yescharacteristic
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12411yescharacteristic
148C2×C4×C53#12711yescharacteristic
149C2×C4×C53C22×C811yescharacteristic
150C2×C4×C53C2×M1611yescharacteristic
151C2×C4×C53#13111yescharacteristic
152C2×C4×C53#13111yescharacteristic
153C2×Dic53—41no—
154C2×Dic53—81no—
155D212—22no—
156D212—22no—
157D212—41no—
158D212—41no—
159C4×D53—82no—
160C4×D53—82no—
161C8×C53—42no—
162C8×C53—81no—
163#88322—21no—
164#88322—21no—
165#88322—42no—
166#88322—81no—
167#88322C22⋊C812yes—
168#88322C22⋊C812yes—
169#9667—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53#39011yescharacteristic
173C22×C53C8×D411yescharacteristic
174C22×C53#43611yescharacteristic
175C2×D53—42no—
176C2×D53—82no—
177C2×D53—81no—
178C2×D53—81no—
179C2×D53—82no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—42no—
184C4×C53—81no—
185C4×C53C2×C22⋊C811yescharacteristic
186C4×C53#40811yescharacteristic
187C4×C53C8×D411yescharacteristic
188C4×C53#43611yescharacteristic
189C4×C53#47112yes—
190Dic53—81no—
191C2×C4×SD16—531no—
192#1564—531no—
193#1479—531no—
194#1478—531no—
195#1461D5311yescharacteristic, complemented
196#1460—531no—
197#1450—531no—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53#195411yescharacteristic
201C2×C53#197611yescharacteristic
202C2×C53#197711yescharacteristic
203D53—82no—
204C22×SD16—531no—
205C2×C4×Q8C2×D5311yescharacteristic
206C2×C4×D4—531no—
207C4×SD16—1061no—
208C4×SD16—1061no—
209C4×SD16—1061no—
210C4×SD16—1061no—
211#431—531no—
212#431—531no—
213#431C2×D5311yescharacteristic
214#431C2×D5311yescharacteristic
215C2×#106—531no—
216C2×#103—531no—
217C2×#102—531no—
218C2×C4×C8—531no—
219#335—1061no—
220#334—1061no—
221Q8⋊C8—1062no—
222D4⋊C8—1062no—
223#324—22no—
224#323—1062no—
225#322—1061no—
226C82—1061no—
227C53#966711yescharacteristic, complemented, Sylow
228C22×Q8C4×D5311yescharacteristic
229C22×D4—531no—
230C23×C4—1061no—
231C2×SD16—534no—
232C2×SD16—1062no—
233C2×SD16—1061no—
234C2×SD16—1061no—
235C22×C8—531no—
236C22×C8—531no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×D4—1061no—
242C4×D4—1061no—
243C4×D4—1061no—
244C4×D4—1061no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4C4×D5311yescharacteristic
248C2×C22⋊C4—1061no—
249C2×C42—21no—
250C2×C42C22×D5311yescharacteristic
251#106—1061no—
252#106—1061no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—532no—
256C4⋊C8—1061no—
257#103—1061no—
258#103—1061no—
259#102—1061no—
260#102—1061no—
261C22⋊C8—21no—
262C22⋊C8—1061no—
263C22⋊C8—1061no—
264C22⋊C8C22⋊D5312yes—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—531no—
268C4×C8—531no—
269C4×C8—531no—
270C4×C8—531no—
271C4×C8—1061no—
272C4×C8—1061no—
273C24—1061no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8C8×D5312yes—
280C2×D4—532no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4—1061no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—1061no—
289C22×C4—4242no—
290C22×C4—2121no—
291C22×C4—2121no—
292C22×C4C2×C4×D5311yescharacteristic
293C22×C4#1171994511yescharacteristic
294C22×C4C2×C22⋊D5311yescharacteristic
295SD16—1064no—
296SD16—1064no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—531no—
300C2×C8—531no—
301C2×C8—532no—
302C2×C8—532no—
303C2×C8—531no—
304C2×C8—531no—
305C2×C8—1061no—
306C2×C8—1061no—
307C2×C8—1062no—
308C2×C8—2121no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—1061no—
314C4⋊C4—1061no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C42—41no—
318C42—41no—
319C42D4×D5311yescharacteristic
320C42D4×D5311yescharacteristic
321C42D4×D5311yescharacteristic
322C42#1171994711yescharacteristic
323C23—1062no—
324C23—4241no—
325C23—4241no—
326C23—2121no—
327C23—2121no—
328C23C4×C22⋊D5311yescharacteristic
329Q8—22no—
330Q8—22no—
331Q8—41no—
332Q8—41no—
333D4—1062no—
334D4—1062no—
335D4—2121no—
336D4—2121no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—4242no—
345C2×C4—4242no—
346C2×C4—4241no—
347C2×C4—2121no—
348C2×C4C22⋊C4×D5311yescharacteristic
349C2×C4#1313317711yescharacteristic
350C2×C4#1313318911yescharacteristic
351C2×C4#1313319011yescharacteristic
352C2×C4#1313320311yescharacteristic
353C2×C4#1313321311yescharacteristic
354C2×C4C2×C8×D5311yescharacteristic
355C2×C4M16×D5311yescharacteristic
356C2×C4#1313322311yescharacteristic
357C2×C4#1313322511yescharacteristic
358C8—42no—
359C8—81no—
360C8—532no—
361C8—532no—
362C8—1061no—
363C8—1061no—
364C8—4241no—
365C8—2122no—
366C22—21no—
367C22—21no—
368C22—4242no—
369C22—4242no—
370C22—4241no—
371C22—4241no—
372C22—2122no—
373C22#1681052311yescharacteristic
374C22C8×C22⋊D5311yescharacteristic
375C22#1681069911yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—81no—
381C4—4241no—
382C4C22⋊C8×D5311yescharacteristic
383C4#1681058811yescharacteristic
384C4#1681068611yescharacteristic
385C4#1681069811yescharacteristic
386C4#1681088912yes—
387C2—21no—
388C2—21no—
389C2—4242no—
390C2#2027550711yescharacteristic
391C2#2027561011yescharacteristic
392C2#2027561911yescharacteristic
subgroupG/H
centerC2×C4#13133190
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1461C2
socleC22×C53#390
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1461D53
hypocenterC53#9667
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #20273461C2semidirect
3. #20273110C2semidirect
6. #20272999C2semidirect
7. C53×#1461C2semidirect
36. C2×#11719924C8semidirect
195. #1461D53semidirect
227. C53#9667semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199242
#202734612
#202731102
#202731072
#202730352
#202729992
C53×#14612
#966753