Finite Group Databasebeta

← #27283036

Subgroups — #27283036

Order 13568.

466 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×#11719913C211yesmaximal, characteristic
2#20273111C211yesmaximal, characteristic, complemented
3#20273107C211yesmaximal, characteristic
4#20273038C211yesmaximal, characteristic, complemented
5#20273035C211yesmaximal, characteristic
6#20273007C211yesmaximal, characteristic, complemented
7C53×#1564C211yesmaximal, characteristic, complemented
8C22×#11719913C411yescharacteristic
9C2×C4×Dic106C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11C4×#11719913—21no—
12C4×#11719913—21no—
13C4×#11719913—21no—
14C4×#11719913—21no—
15#16810676C2211yescharacteristic
16#16810676C2211yescharacteristic
17C2×#13133149C411yescharacteristic
18C2×#13133142C411yescharacteristic
19C2×#13133139C411yescharacteristic
20#16810355—22no—
21#16810352—22no—
22#16810348—22no—
23#16810346—22no—
24#16810345—21no—
25#16810345—21no—
26C53×#431C2211yescharacteristic
27C53×#431C2211yescharacteristic
28C2×C4×C8×C53C2211yescharacteristic
29C53×#335—21no—
30C53×#334—21no—
31C22×Dic106C2×C411yescharacteristic
32C22×D212C2×C411yescharacteristic
33C22×C4×D53—21no—
34C2×#11719913—21no—
35C2×#11719913—21no—
36C2×#11719913—22no—
37C2×#11719913—22no—
38C4×Dic106—21no—
39C4×Dic106—21no—
40C4×Dic106—21no—
41C4×Dic106—21no—
42C4×D212—21no—
43C4×D212—21no—
44C4×D212—21no—
45C4×D212—21no—
46C2×C4⋊Dic53C2×C411yescharacteristic
47C2×C4⋊Dic53C2×C411yescharacteristic
48C2×#11719919—21no—
49C2×#11719921—21no—
50C2×C4×Dic53—21no—
51#13133149—21no—
52#13133149—21no—
53C4⋊#88322—21no—
54C4⋊#88322D412yes—
55#13133142—21no—
56#13133142—21no—
57#13133139—21no—
58#13133139—21no—
59C22⋊#88322—21no—
60C22⋊#88322D412yes—
61C4×#88322—21no—
62C4×#88322—21no—
63C22×C8×C53C2×C411yescharacteristic
64C22×C8×C53C2×C411yescharacteristic
65C2×C42×C53C2311yescharacteristic
66C4⋊C8×C53—21no—
67C4⋊C8×C53—21no—
68C22⋊C8×C53—21no—
69C22⋊C8×C53—21no—
70C4×C8×C53—21no—
71C4×C8×C53—21no—
72C4×C8×C53D411yescharacteristic
73C4×C8×C53D411yescharacteristic
74C4×C8×C53D411yescharacteristic
75C4×C8×C53D411yescharacteristic
76C23×D53—21no—
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212M1612yes—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C4×D53—82no—
92C2×Dic106—21no—
93C2×Dic106—21no—
94C2×Dic106—21no—
95C2×Dic106—21no—
96C2×Dic106—21no—
97C2×Dic106M1612yes—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—21no—
101C2×C8×C53—21no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C2×C8×C53C22⋊C411yescharacteristic
105C2×C8×C53C22⋊C411yescharacteristic
106C2×C8×C53C22⋊C411yescharacteristic
107C2×C8×C53C22⋊C411yescharacteristic
108C42×C53C2×D411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53#4111yescharacteristic
112#11719921—41no—
113#11719921—41no—
114C4⋊Dic53—21no—
115C4⋊Dic53—21no—
116C4⋊Dic53—21no—
117C4⋊Dic53—21no—
118#11719919—41no—
119#11719919—41no—
120C4×Dic53—41no—
121C4×Dic53—41no—
122C2×#88322—22no—
123C2×#88322—22no—
124#11719913—24no—
125#11719913—24no—
126C23×C53C4×D411yescharacteristic
127C22×D53—22no—
128C22×D53—41no—
129C22×D53—41no—
130C22×D53—81no—
131C22×D53—81no—
132C2×C4×C53—22no—
133C2×C4×C53—41no—
134C2×C4×C53C2×C22⋊C411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53#12411yescharacteristic
139C2×C4×C53#12711yescharacteristic
140C2×C4×C53C2×M1611yescharacteristic
141C2×C4×C53C2×M1611yescharacteristic
142C2×C4×C53#13111yescharacteristic
143C2×C4×C53#13111yescharacteristic
144C2×Dic53—22no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—81no—
150C2×Dic53—81no—
151D212—22no—
152D212—22no—
153D212—41no—
154D212—41no—
155C4×D53—82no—
156C4×D53—82no—
157Dic106—22no—
158Dic106—22no—
159Dic106—41no—
160Dic106—41no—
161C8×C53—21no—
162C8×C53—21no—
163C8×C53—21no—
164C8×C53—21no—
165C8×C53—81no—
166C8×C53—81no—
167#88322—42no—
168#88322—42no—
169#9672—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53#39011yescharacteristic
173C22×C53#43511yescharacteristic
174C22×C53#43511yescharacteristic
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#40711yescharacteristic
184C4×C53#40811yescharacteristic
185C4×C53#43511yescharacteristic
186C4×C53#43511yescharacteristic
187C4×C53#47112yes—
188Dic53—42no—
189Dic53—81no—
190Dic53—81no—
191C2×C4×SD16—531no—
192#1564D5311yescharacteristic, complemented
193#1479—531no—
194#1478—531no—
195#1461—531no—
196#1460—531no—
197#1452—531no—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53#195511yescharacteristic
201C2×C53#197611yescharacteristic
202C2×C53#197711yescharacteristic
203D53—82no—
204C22×SD16—531no—
205C2×C4×Q8—531no—
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×C8C2×D5311yescharacteristic
219#335—21no—
220#334—21no—
221Q8⋊C8—1062no—
222D4⋊C8—1062no—
223#324—1062no—
224#323—1062no—
225#322—1061no—
226#322—1061no—
227C53#967211yescharacteristic, complemented, Sylow
228C22×Q8—531no—
229C22×D4—531no—
230C23×C4—1061no—
231C2×SD16—1062no—
232C2×SD16—1062no—
233C2×SD16—1061no—
234C2×SD16—1061no—
235C22×C8C4×D5311yescharacteristic
236C22×C8C4×D5311yescharacteristic
237C4×Q8—1061no—
238C4×Q8—1061no—
239C4×Q8—1061no—
240C4×Q8—1061no—
241C4×D4—1061no—
242C4×D4—1061no—
243C4×D4—1061no—
244C4×D4—1061no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—1061no—
248C2×C22⋊C4—1061no—
249C2×C42—1061no—
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—21no—
263C22⋊C8—532no—
264C22⋊C8—1061no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—1061no—
268C4×C8—1061no—
269C4×C8D21211yescharacteristic
270C4×C8D21211yescharacteristic
271C4×C8C22⋊D5311yescharacteristic
272C4×C8C22⋊D5311yescharacteristic
273C24—1061no—
274C2×Q8—532no—
275C2×Q8—1061no—
276C2×Q8—1061no—
277C2×Q8—1061no—
278C2×Q8—1061no—
279C2×Q8—1061no—
280C2×D4—532no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4—1061no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—4242no—
290C22×C4—2121no—
291C22×C4—2121no—
292C22×C4C2×C4×D5311yescharacteristic
293C22×C4D4×D5311yescharacteristic
294C22×C4#1171994911yescharacteristic
295SD16—1064no—
296SD16—1064no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—1062no—
304C2×C8—1062no—
305C2×C8#1171992111yescharacteristic
306C2×C8#1171992111yescharacteristic
307C2×C8#1171992111yescharacteristic
308C2×C8#1171992111yescharacteristic
309C4⋊C4—1061no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C4⋊C4—2121no—
314C4⋊C4—2121no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C42—2121no—
318C42—2121no—
319C42C2×D21211yescharacteristic
320C42D4×D5311yescharacteristic
321C42#1171994911yescharacteristic
322C42C2×C22⋊D5311yescharacteristic
323C23—1062no—
324C23—4241no—
325C23—4241no—
326C23—2121no—
327C23—2121no—
328C23#1313317911yescharacteristic
329Q8—1062no—
330Q8—1062no—
331Q8—2121no—
332Q8—2121no—
333D4—1062no—
334D4—1062no—
335D4—2121no—
336D4—2121no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—1062no—
340C2×C4—4242no—
341C2×C4—4242no—
342C2×C4—4241no—
343C2×C4—4241no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4C2×#1171992111yescharacteristic
349C2×C4#1313317911yescharacteristic
350C2×C4#1313319211yescharacteristic
351C2×C4#1313319211yescharacteristic
352C2×C4#1313320611yescharacteristic
353C2×C4#1313321311yescharacteristic
354C2×C4M16×D5311yescharacteristic
355C2×C4M16×D5311yescharacteristic
356C2×C4#1313322311yescharacteristic
357C2×C4#1313322311yescharacteristic
358C8—21no—
359C8—21no—
360C8—21no—
361C8—21no—
362C8—81no—
363C8—81no—
364C8—2122no—
365C8—2122no—
366C22—21no—
367C22—21no—
368C22—4242no—
369C22—4242no—
370C22—4241no—
371C22—4241no—
372C22—2122no—
373C22#1681052411yescharacteristic
374C22#1681069311yescharacteristic
375C22#1681069311yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—4241no—
380C4—4241no—
381C4—2122no—
382C4#1681058611yescharacteristic
383C4#1681058911yescharacteristic
384C4#1681069311yescharacteristic
385C4#1681069311yescharacteristic
386C4#1681088712yes—
387C2—21no—
388C2—21no—
389C2—4242no—
390C2#2027551211yescharacteristic
391C2#2027561111yescharacteristic
392C2#2027561611yescharacteristic
subgroupG/H
centerC2×C4#13133192
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1564C2
socleC22×C53#390
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1564D53
hypocenterC53#9672
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
2. #20273111C2semidirect
4. #20273038C2semidirect
6. #20273007C2semidirect
7. C53×#1564C2semidirect
192. #1564D53semidirect
227. C53#9672semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199132
#202731112
#202731072
#202730382
#202730352
#202730072
C53×#15642
#967253