Finite Group Databasebeta

← #27278700

Subgroups — #27278700

Order 13568.

560 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810754C211yesmaximal, characteristic, complemented
2C2×C4×#11719924C211yesmaximal, characteristic
3#20275508C211yesmaximal, characteristic
4C2×#16810352C211yesmaximal, characteristic
5#20272549C211yesmaximal, characteristic, complemented
6#20272542C211yesmaximal, characteristic, complemented
7C2×C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C22×#11719924C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810754—22no—
11#16810754—22no—
12C4×#11719924—22no—
13C4×#11719924—22no—
14C2×#13133148C411yescharacteristic
15C2×C4⋊#88322—21no—
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133141C411yescharacteristic
18C2×#13133137C411yescharacteristic
19C2×#13133129—21no—
20C2×#13133129—21no—
21C2×#13133128—21no—
22C2×C4×#88322C2211yescharacteristic
23C2×C4×#88322C2211yescharacteristic
24#16810352—22no—
25C2×C4×Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4×C8×C53C2211yescharacteristic
28C53×Q8⋊C8—22no—
29C22×D212C2×C411yescharacteristic
30C22×C4×D53—21no—
31C2×#11719924—22no—
32C2×#11719924—22no—
33C2×#11719924—22no—
34C22×#88322—21no—
35C22×#88322C2×C411yescharacteristic
36C22×#88322C2×C411yescharacteristic
37C22×#88322D411yescharacteristic
38C22×#88322D411yescharacteristic
39C4×D212—21no—
40C4×D212—21no—
41C4×D212—21no—
42C4×D212—21no—
43C2×C4⋊Dic53C2×C411yescharacteristic
44C2×#11719921—21no—
45#13133148—22no—
46C4⋊#88322—24no—
47C4⋊#88322—21no—
48C4⋊#88322D412yes—
49#13133141—22no—
50#13133137—22no—
51#13133129—42no—
52#13133129—42no—
53#13133128—24no—
54C4×#88322—21no—
55C4×#88322—21no—
56C4×#88322—21no—
57C4×#88322—21no—
58C22×Q8×C53C2×C411yescharacteristic
59C22×C8×C53—21no—
60C22×C8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53D412yes—
64C4×Q8×C53D412yes—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53C2×C411yescharacteristic
67C2×C42×C53—21no—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4×C8×C53—21no—
72C4×C8×C53—21no—
73C23×D53—21no—
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53C22⋊C412yes—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53#4111yescharacteristic
85C22×Dic53—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212—21no—
90C2×D212—21no—
91C2×D212M1612yes—
92C2×C4×D53—41no—
93C2×C4×D53—41no—
94C2×C4×D53—41no—
95C2×C4×D53—41no—
96C2×C4×D53—41no—
97C2×C4×D53—41no—
98C2×C8×C53—22no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53C22⋊C412yes—
108C42×C53—22no—
109C42×C53—22no—
110C42×C53C2×D411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53#4111yescharacteristic
114#11719924—28no—
115#11719921—41no—
116#11719921—41no—
117C4⋊Dic53—21no—
118C4⋊Dic53—21no—
119C2×#88322—21no—
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—21no—
125C2×#88322—22no—
126C2×#88322—21no—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322M1612yes—
133C2×#88322SD1612yescomplemented
134C23×C53C4×D411yescharacteristic
135C22×D53—22no—
136C22×D53—41no—
137C22×D53—41no—
138C22×D53—41no—
139C22×D53—41no—
140C22×D53—41no—
141C22×D53—41no—
142Q8×C53—22no—
143Q8×C53—22no—
144Q8×C53—41no—
145Q8×C53—41no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53C2×C22⋊C411yescharacteristic
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53#12011yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12311yescharacteristic
160C2×C4×C53C2×M1611yescharacteristic
161C2×C4×C53C2×M1611yescharacteristic
162C2×C4×C53#13111yescharacteristic
163C2×C4×C53#13111yescharacteristic
164C2×C4×C53C2×SD1611yescharacteristic
165C2×C4×C53#13511yescharacteristic
166C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C53#13711yescharacteristic
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171D212—22no—
172D212—22no—
173D212—41no—
174D212—41no—
175C4×D53—81no—
176C4×D53—81no—
177C4×D53—81no—
178C4×D53—81no—
179C8×C53—42no—
180C8×C53—81no—
181#88322—22no—
182#88322—22no—
183#88322—41no—
184#88322—41no—
185#88322—42no—
186#8952—531nomaximal, Sylow
187C22×C53#38611yescharacteristic
188C22×C53#43511yescharacteristic
189C22×C53#43611yescharacteristic
190C22×C53C4×SD1611yescharacteristic
191C22×C53#44011yescharacteristic
192C22×C53#44311yescharacteristic
193C22×C53#44411yescharacteristic
194C2×D53—42no—
195C2×D53—42no—
196C2×D53—42no—
197C2×D53—81no—
198C2×D53—81no—
199C2×D53—81no—
200C2×D53—81no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—42no—
204C4×C53—42no—
205C4×C53#40711yescharacteristic
206C4×C53#40811yescharacteristic
207C4×C53#43511yescharacteristic
208C4×C53#43611yescharacteristic
209C4×C53#44811yescharacteristic
210C4×C53#44911yescharacteristic
211C4×C53#46011yescharacteristic
212C4×C53#46411yescharacteristic
213Dic53—81no—
214C2×#446—531no—
215C2×C4×SD16—531no—
216#1954—531no—
217C2×Q8⋊C8D5311yescharacteristic, complemented
218C2×D4⋊C8—531no—
219#1277—531no—
220#1273—531no—
221C2×C53#158211yescharacteristic
222C2×C53#158511yescharacteristic
223C2×C53#159011yescharacteristic
224C2×C53#159211yescharacteristic
225C2×C53#187011yescharacteristic
226C2×C53#187611yescharacteristic
227C2×C53#188011yescharacteristic
228D53—82no—
229C22×SD16—531no—
230C2×C4×Q8C2×D5311yescharacteristic
231C2×C4×D4—531no—
232#446—1062no—
233#446—1062no—
234C4×SD16—1062no—
235C4×SD16—1062no—
236C2×#106—531no—
237C2×C4⋊C8—531no—
238C2×C4⋊C8—1061no—
239C2×C4⋊C8C2×D5311yescharacteristic
240C2×#103—531no—
241C2×#102—531no—
242C2×C22⋊C8—1061no—
243C2×C22⋊C8—1061no—
244C2×#97—1061no—
245C2×C4×C8—531no—
246C2×C4×C8—531no—
247C2×C4×C8C2×D5311yescharacteristic
248Q8⋊C8—22no—
249D4⋊C8—1062no—
250C53#895211yescharacteristic, complemented, Sylow
251C22×Q8C4×D5311yescharacteristic
252C22×D4—531no—
253C23×C4—1061no—
254C2×SD16—1062no—
255C2×SD16—1062no—
256C2×SD16—1062no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—531no—
260C22×C8—531no—
261C22×C8—531no—
262C22×C8—531no—
263C22×C8—1061no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8D21212yes—
267C4×Q8C22⋊D5312yes—
268C4×D4—1061no—
269C4×D4—1061no—
270C4×D4—1061no—
271C4×D4—1061no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—531no—
274C2×C4⋊C4C4×D5311yescharacteristic
275C2×C22⋊C4—1061no—
276C2×C42—21no—
277C2×C42C22×D5311yescharacteristic
278#106—1062no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—532no—
282C4⋊C8—1064no—
283C4⋊C8—1061no—
284#103—1062no—
285#102—1062no—
286C22⋊C8—2122no—
287C22⋊C8—2122no—
288#97—1064no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—1061no—
292C4×C8—1061no—
293C4×C8—1061no—
294C4×C8—1061no—
295C24—1061no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8#1171992112yes—
302C2×D4—532no—
303C2×D4—1061no—
304C2×D4—1061no—
305C2×D4—1061no—
306C2×D4—1061no—
307C2×D4—1061no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—1061no—
311C22×C4—2121no—
312C22×C4—2121no—
313C22×C4—2121no—
314C22×C4—2121no—
315C22×C4—2121no—
316C22×C4—2121no—
317C22×C4C2×C4×D5311yescharacteristic
318C22×C4D4×D5311yescharacteristic
319C22×C4#1171994711yescharacteristic
320SD16—1068no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—532no—
328C2×C8—532no—
329C2×C8—1061no—
330C2×C8—1061no—
331C2×C8—1061no—
332C2×C8—1061no—
333C2×C8—1061no—
334C2×C8—1061no—
335C2×C8—1062no—
336C2×C8—1061no—
337C2×C8—1061no—
338C2×C8—1061no—
339C2×C8—1061no—
340C2×C8—2121no—
341C2×C8—2121no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—1061no—
346C4⋊C4—1061no—
347C4⋊C4#1171992112yes—
348C22⋊C4—2121no—
349C22⋊C4—2121no—
350C42—22no—
351C42—22no—
352C42C2×D21211yescharacteristic
353C42D4×D5311yescharacteristic
354C42#1171994711yescharacteristic
355C42C2×C22⋊D5311yescharacteristic
356C23—1062no—
357C23—2121no—
358C23—2121no—
359C23—2121no—
360C23—2121no—
361C23—2121no—
362C23—2121no—
363C23#1313317711yescharacteristic
364Q8—22no—
365Q8—22no—
366Q8—41no—
367Q8—41no—
368D4—1062no—
369D4—1062no—
370D4—2121no—
371D4—2121no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—4241no—
381C2×C4—4241no—
382C2×C4—4241no—
383C2×C4—4241no—
384C2×C4—2121no—
385C2×C4—2121no—
386C2×C4—2121no—
387C2×C4C2×#1171992111yescharacteristic
388C2×C4#1313317711yescharacteristic
389C2×C4#1313318711yescharacteristic
390C2×C4#1313318911yescharacteristic
391C2×C4#1313318911yescharacteristic
392C2×C4#1313320111yescharacteristic
393C2×C4M16×D5311yescharacteristic
394C2×C4M16×D5311yescharacteristic
395C2×C4#1313322311yescharacteristic
396C2×C4#1313322311yescharacteristic
397C2×C4SD16×D5311yescharacteristic
398C2×C4#1313324011yescharacteristic
399C2×C4#1313324411yescharacteristic
400C2×C4#1313324911yescharacteristic
401C8—42no—
402C8—81no—
403C8—1062no—
404C8—1062no—
405C8—2121no—
406C8—2122no—
407C8—2121no—
408C22—4241no—
409C22—4241no—
410C22—4241no—
411C22—4241no—
412C22—2122no—
413C22—2122no—
414C22—2122no—
415C22#1681050011yescharacteristic
416C22#1681069711yescharacteristic
417C22#1681069811yescharacteristic
418C22#1681070811yescharacteristic
419C22#1681072611yescharacteristic
420C22#1681073711yescharacteristic
421C22#1681074211yescharacteristic
422C4—21no—
423C4—21no—
424C4—42no—
425C4—42no—
426C4—4241no—
427C4#1681058611yescharacteristic
428C4#1681058911yescharacteristic
429C4#1681069711yescharacteristic
430C4#1681069811yescharacteristic
431C4#1681076311yescharacteristic
432C4#1681077111yescharacteristic
433C4#1681082111yescharacteristic
434C4#1681084711yescharacteristic
435C2—4242no—
436C2#2027355411yescharacteristic
437C2#2027357511yescharacteristic
438C2#2027360211yescharacteristic
439C2#2027362011yescharacteristic
440C2#2027501011yescharacteristic
441C2#2027504011yescharacteristic
442C2#2027506811yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×Q8⋊C8C2
socleC23×C53C4×D4
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenterC2×Q8⋊C8D53
hypocenterC53#8952
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#16810754C2semidirect
5. #20272549C2semidirect
6. #20272542C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
133. C2×#88322SD16semidirect
217. C2×Q8⋊C8D53semidirect
250. C53#8952semidirect

Maximal subgroups

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

subgroupindex
C2×#168107542
C2×C4×#117199242
#202755082
C2×#168103522
#202725492
#202725422
C2×C53×Q8⋊C82
#895253