Finite Group Databasebeta

← #27286865

Subgroups — #27286865

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810995C211yesmaximal, characteristic, complemented
2C2×#16810969C211yesmaximal, characteristic, complemented
3#20275676C211yesmaximal, characteristic, complemented
4#20275522C211yesmaximal, characteristic, complemented
5#20275522C211yesmaximal, characteristic, complemented
6#20272542C211yesmaximal, characteristic, complemented
7C2×C53×#493C211yesmaximal, characteristic
8C22×#11719923—22no—
9C2×#13133217C2211yescharacteristic
10#16810995—22no—
11#16810995—22no—
12#16810969—22no—
13#16810969—22no—
14C2×#13133138—22no—
15C2×#13133138—22no—
16C2×#13133138—22no—
17C2×C4×#88322C2211yescharacteristic
18C2×C4×#88322C2211yescharacteristic
19#16810493C2211yescharacteristic
20#16810386—22no—
21C22×D8×C53—22no—
22C2×#127×C53C2211yescharacteristic
23C2×#127×C53C2211yescharacteristic
24C53×#493—22no—
25C53×#493—22no—
26C2×C4×C8×C53C2211yescharacteristic
27C22×Dic106—21no—
28C2×#11719923—28no—
29C2×#11719923—44no—
30C2×#11719923—44no—
31C22×#88322D412yes—
32C22×#88322D412yes—
33#13133217—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217D412yes—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—21no—
42#13133138—28no—
43#13133138—44no—
44#13133138—44no—
45C4×#88322—21no—
46C4×#88322—21no—
47C4×#88322—21no—
48C4×#88322—21no—
49C22×D4×C53—21no—
50C22×D4×C53—21no—
51C22×D4×C53—21no—
52C22×D4×C53—21no—
53C22×D4×C53—21no—
54C22×D4×C53—21no—
55C2×D8×C53—28no—
56C2×D8×C53—44no—
57C2×D8×C53—44no—
58C22×C8×C53D412yes—
59#127×C53—21no—
60#127×C53—21no—
61#127×C53—21no—
62#127×C53—21no—
63#127×C53—21no—
64#127×C53—21no—
65#127×C53D412yes—
66#127×C53D412yes—
67C2×C42×C53C2311yescharacteristic
68C4×C8×C53—21no—
69C4×C8×C53—21no—
70C24×C53—41no—
71C24×C53—41no—
72C2×D4×C53—22no—
73C2×D4×C53—22no—
74C2×D4×C53—22no—
75C2×D4×C53—22no—
76C2×D4×C53—22no—
77C2×D4×C53—22no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—42no—
85C2×D4×C53—42no—
86C2×D4×C53—42no—
87C2×D4×C53—42no—
88C2×D4×C53—42no—
89C2×D4×C53—42no—
90C2×D4×C53—42no—
91C2×D4×C53—42no—
92C2×D4×C53—42no—
93C2×D4×C53—42no—
94C2×D4×C53—42no—
95C2×D4×C53—42no—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×Dic53—41no—
100C22×Dic53—41no—
101C2×Dic106—22no—
102C2×Dic106—42no—
103C2×Dic106—42no—
104C2×Dic106—41no—
105D8×C53—416no—
106C2×C8×C53—22no—
107C2×C8×C53—22no—
108C2×C8×C53SD1614yescomplemented
109C42×C53C2×D411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53#4111yescharacteristic
113#11719923—416no—
114C4⋊Dic53—22no—
115C4⋊Dic53—22no—
116C4⋊Dic53—41no—
117C4⋊Dic53—41no—
118C4⋊Dic53—42no—
119C4⋊Dic53—42no—
120C4⋊Dic53—42no—
121C2×#88322—22no—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—22no—
125C2×#88322—22no—
126C2×#88322D814yescomplemented
127C23×C53—42no—
128C23×C53—42no—
129C23×C53—81no—
130C23×C53—81no—
131C23×C53—81no—
132C23×C53—81no—
133C23×C53—81no—
134C23×C53—81no—
135C23×C53—81no—
136C23×C53—81no—
137C23×C53—81no—
138C23×C53—81no—
139C23×C53—81no—
140C23×C53—81no—
141C23×C53#12711yescharacteristic
142D4×C53—44no—
143D4×C53—44no—
144D4×C53—44no—
145D4×C53—44no—
146D4×C53—44no—
147D4×C53—44no—
148D4×C53—82no—
149D4×C53—82no—
150D4×C53—82no—
151D4×C53—82no—
152D4×C53—82no—
153D4×C53—82no—
154C2×C4×C53#12011yescharacteristic
155C2×C4×C53#12011yescharacteristic
156C2×C4×C53#12011yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12111yescharacteristic
160C2×C4×C53C2×D812yes—
161C2×C4×C53C2×SD1612yes—
162C2×C4×C53#13512yes—
163C2×C4×C53AGL(1,ℤ/8)12yes—
164C2×C4×C53AGL(1,ℤ/8)12yes—
165C2×C4×C53AGL(1,ℤ/8)12yes—
166C2×Dic53—42no—
167C2×Dic53—81no—
168C2×Dic53—81no—
169C2×Dic53—81no—
170C2×Dic53—81no—
171C2×Dic53—81no—
172Dic106—44no—
173Dic106—82no—
174C8×C53—24no—
175#88322—24no—
176#88322—42no—
177#10305—531nomaximal, Sylow
178C22×C53—82no—
179C22×C53—82no—
180C22×C53—82no—
181C22×C53—82no—
182C22×C53—82no—
183C22×C53—82no—
184C22×C53—161no—
185C22×C53—161no—
186C22×C53—161no—
187C22×C53—161no—
188C22×C53—161no—
189C22×C53—161no—
190C22×C53—161no—
191C22×C53—161no—
192C22×C53#39211yescharacteristic
193C22×C53#49211yescharacteristic
194C22×C53#49311yescharacteristic
195C22×C53#49511yescharacteristic
196C22×C53#49611yescharacteristic
197C22×C53#49611yescharacteristic
198C22×C53#49611yescharacteristic
199C4×C53#44712yes—
200C4×C53#44912yes—
201C4×C53#45012yes—
202C4×C53#45912yes—
203C4×C53#46012yes—
204C4×C53#46312yes—
205Dic53—82no—
206Dic53—161no—
207C2×#493D5311yescharacteristic
208C2×#492—531no—
209C2×#486—531no—
210#1990—531no—
211#1957—531no—
212#1957—531no—
213#1273—531no—
214C2×C53—162no—
215C2×C53—162no—
216C2×C53#170611yescharacteristic
217C2×C53#171511yescharacteristic
218C2×C53#171711yescharacteristic
219C2×C53#172111yescharacteristic
220C2×C53#201111yescharacteristic
221C2×C53#201211yescharacteristic
222C2×C53#201411yescharacteristic
223C22×SD16—1062no—
224C22×D8—22no—
225C2×C4⋊Q8—531no—
226C2×#127C2×D5311yescharacteristic
227C2×#127C2×D5311yescharacteristic
228#493—22no—
229#493—22no—
230#492—1062no—
231#492—1062no—
232#486—1062no—
233#486—1062no—
234C2×#102—1062no—
235C2×#102—1062no—
236C2×#102—1062no—
237C2×C4×C8—531no—
238C2×C4×C8—531no—
239C2×C4×C8C2×D5311yescharacteristic
240#384—531no—
241#340—1062no—
242C53#1030511yescharacteristic, complemented, Sylow
243C22×Q8—1061no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—21no—
250C2×SD16—1068no—
251C2×SD16—2124no—
252C2×SD16—2124no—
253C2×D8—28no—
254C2×D8—44no—
255C2×D8—44no—
256C22×C8—532no—
257C22×C8—532no—
258C22×C8C22⋊D5312yes—
259C4⋊Q8—532no—
260C4⋊Q8—1061no—
261C4⋊Q8—1061no—
262C4⋊Q8—1061no—
263#127—21no—
264#127—21no—
265#127—21no—
266#127—21no—
267#127—21no—
268#127—21no—
269#127C22⋊D5312yes—
270#127C22⋊D5312yes—
271C2×C4⋊C4—1061no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4—1061no—
274C2×C4⋊C4—1061no—
275C2×C4⋊C4—1061no—
276C2×C42C22×D5311yescharacteristic
277#102—1068no—
278#102—2124no—
279#102—2124no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—1061no—
283C4×C8—1061no—
284C4×C8—1061no—
285C4×C8—1061no—
286C24—41no—
287C24—41no—
288C2×Q8—1062no—
289C2×Q8—2122no—
290C2×Q8—2122no—
291C2×Q8—2121no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C22×C4—2121no—
317C22×C4—2121no—
318C22×C4D4×D5311yescharacteristic
319C22×C4D4×D5311yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321SD16—21216no—
322D8—416no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—534no—
326C2×C8—1062no—
327C2×C8—1062no—
328C2×C8—1062no—
329C2×C8—1062no—
330C2×C8—1062no—
331C2×C8#1171992314yescomplemented
332C4⋊C4—1062no—
333C4⋊C4—1062no—
334C4⋊C4—2122no—
335C4⋊C4—2122no—
336C4⋊C4—2122no—
337C4⋊C4—2121no—
338C4⋊C4—2121no—
339C42D4×D5311yescharacteristic
340C42#1171994711yescharacteristic
341C42C2×C22⋊D5311yescharacteristic
342C42C2×C22⋊D5311yescharacteristic
343C23—42no—
344C23—42no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23#1313321311yescharacteristic
358Q8—4242no—
359Q8—2124no—
360D4—44no—
361D4—44no—
362D4—44no—
363D4—44no—
364D4—44no—
365D4—44no—
366D4—82no—
367D4—82no—
368D4—82no—
369D4—82no—
370D4—82no—
371D4—82no—
372C2×C4—4241no—
373C2×C4—4241no—
374C2×C4—4241no—
375C2×C4—4241no—
376C2×C4—4241no—
377C2×C4—2122no—
378C2×C4#1313318511yescharacteristic
379C2×C4#1313318511yescharacteristic
380C2×C4#1313318611yescharacteristic
381C2×C4#1313318811yescharacteristic
382C2×C4#1313318911yescharacteristic
383C2×C4#1313318911yescharacteristic
384C2×C4D8×D5312yes—
385C2×C4C2×#1171992312yes—
386C2×C4#1313323612yes—
387C2×C4#1313324112yes—
388C2×C4#1313324112yes—
389C2×C4#1313324312yes—
390C8—24no—
391C8—1064no—
392C8—2122no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22#1681053111yescharacteristic
408C22#1681099511yescharacteristic
409C22#1681099811yescharacteristic
410C22#1681100611yescharacteristic
411C22#1681101011yescharacteristic
412C22#1681101111yescharacteristic
413C22#1681101111yescharacteristic
414C4—8481no—
415C4—4242no—
416C4#1681075512yes—
417C4#1681076512yes—
418C4#1681077412yes—
419C4#1681081612yes—
420C4#1681082012yes—
421C4#1681083612yes—
422C2—162no—
423C2—162no—
424C2#2027430411yescharacteristic
425C2#2027434111yescharacteristic
426C2#2027435311yescharacteristic
427C2#2027438011yescharacteristic
428C2#2027578411yescharacteristic
429C2#2027579011yescharacteristic
430C2#2027580311yescharacteristic
subgroupG/H
centerC23#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#493C2
socleC23×C53#127
hypercenterC2×#493D53
hypocenterC53#10305
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#493

Complements

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

NH ≅ G/Ntype
1. C2×#16810995C2semidirect
2. C2×#16810969C2semidirect
3. #20275676C2semidirect
4. #20275522C2semidirect
5. #20275522C2semidirect
6. #20272542C2semidirect
108. C2×C8×C53SD16semidirect
126. C2×#88322D8semidirect
242. C53#10305semidirect
331. C2×C8#11719923semidirect

Maximal subgroups

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

subgroupindex
C2×#168109952
C2×#168109692
#202756762
#202755222
#202725422
C2×C53×#4932
#1030553