Finite Group Databasebeta

← #27278695

Subgroups — #27278695

Order 13568.

636 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719922C211yesmaximal, characteristic
2#20275509C211yesmaximal, characteristic, complemented
3C2×D4⋊#88322C211yesmaximal, characteristic, complemented
4C2×#16810350C211yesmaximal, characteristic, complemented
5#20272549C211yesmaximal, characteristic, complemented
6#20272542C211yesmaximal, characteristic, complemented
7C2×C53×#436C211yesmaximal, characteristic, complemented
8C22×#11719922C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10C4×#11719922—22no—
11C4×#11719922—22no—
12C2×#13133150C411yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133138C411yescharacteristic
15C2×#13133137C411yescharacteristic
16C2×#13133129—21no—
17C2×#13133129—21no—
18C2×C4×#88322C2211yescharacteristic
19C2×C4×#88322C2211yescharacteristic
20D4⋊#88322—22no—
21#16810350—22no—
22C22×M16×C53—21no—
23C2×C4×D4×C53C2211yescharacteristic
24C53×#436—22no—
25C53×#436—22no—
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C22⋊C8×C53—21no—
28C2×C4×C8×C53C2211yescharacteristic
29C22×D212C2×C411yescharacteristic
30C22×C4×D53—21no—
31C2×#11719922—22no—
32C2×#11719922—22no—
33C2×#11719922—22no—
34C22×#88322—21no—
35C22×#88322—21no—
36C22×#88322C2×C411yescharacteristic
37C22×#88322C2×C411yescharacteristic
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C4×D212—21no—
42C2×C4⋊Dic53C2×C411yescharacteristic
43C2×#11719921—21no—
44#13133150—22no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47#13133138—22no—
48#13133137—22no—
49#13133129—42no—
50#13133129—42no—
51C4×#88322—21no—
52C4×#88322—21no—
53C4×#88322—21no—
54C4×#88322—21no—
55C22×D4×C53C2×C411yescharacteristic
56C23×C4×C53—21no—
57C2×M16×C53—24no—
58C2×M16×C53—42no—
59C2×M16×C53—42no—
60C22×C8×C53—21no—
61C22×C8×C53D411yescharacteristic
62C22×C8×C53D411yescharacteristic
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53D412yes—
66C4×D4×C53D412yes—
67C2×C4⋊C4×C53C2×C411yescharacteristic
68C2×C22⋊C4×C53—21no—
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—21no—
71C4⋊C8×C53D412yes—
72C22⋊C8×C53—24no—
73C4×C8×C53—21no—
74C4×C8×C53—21no—
75C24×C53—21no—
76C23×D53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53C22⋊C412yes—
83C22×C4×C53—21no—
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×Dic53—21no—
94C2×D212—21no—
95C2×D212—21no—
96C2×D212—21no—
97C2×D212—21no—
98C2×D212—21no—
99C2×D212M1612yescomplemented
100C2×C4×D53—41no—
101C2×C4×D53—41no—
102C2×C4×D53—41no—
103C2×C4×D53—41no—
104C2×C4×D53—41no—
105C2×C4×D53—41no—
106M16×C53—48no—
107C2×C8×C53—21no—
108C2×C8×C53—21no—
109C2×C8×C53—21no—
110C2×C8×C53—22no—
111C2×C8×C53—21no—
112C2×C8×C53—21no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C2×C8×C53D812yescomplemented
116C4⋊C4×C53—21no—
117C4⋊C4×C53C22⋊C412yes—
118C22⋊C4×C53—22no—
119C22⋊C4×C53—22no—
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53#4111yescharacteristic
124#11719922—28no—
125#11719921—41no—
126#11719921—41no—
127C4⋊Dic53—21no—
128C4⋊Dic53—21no—
129C2×#88322—21no—
130C2×#88322—21no—
131C2×#88322—22no—
132C2×#88322—21no—
133C2×#88322—21no—
134C2×#88322—21no—
135C2×#88322—41no—
136C2×#88322—41no—
137C2×#88322—41no—
138C2×#88322—41no—
139C2×#88322—41no—
140C2×#88322M1612yescomplemented
141C23×C53—22no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53C4×D411yescharacteristic
149C22×D53—22no—
150C22×D53—41no—
151C22×D53—41no—
152C22×D53—41no—
153C22×D53—41no—
154C22×D53—41no—
155C22×D53—41no—
156D4×C53—22no—
157D4×C53—22no—
158D4×C53—41no—
159D4×C53—41no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—22no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—42no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
169C2×C4×C53C2×C22⋊C411yescharacteristic
170C2×C4×C53C4×D411yescharacteristic
171C2×C4×C53#12011yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12311yescharacteristic
175C2×C4×C53C2×M1611yescharacteristic
176C2×C4×C53C2×M1611yescharacteristic
177C2×C4×C53#13111yescharacteristic
178C2×C4×C53#13111yescharacteristic
179C2×C4×C53C2×D811yescharacteristic
180C2×C4×C53#13511yescharacteristic
181C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
183C2×Dic53—41no—
184C2×Dic53—41no—
185C2×Dic53—41no—
186D212—22no—
187D212—22no—
188D212—41no—
189D212—41no—
190C4×D53—81no—
191C4×D53—81no—
192C4×D53—81no—
193C4×D53—81no—
194C8×C53—22no—
195C8×C53—42no—
196C8×C53—41no—
197#88322—22no—
198#88322—42no—
199#88322—41no—
200#88322—81no—
201#8951—531nomaximal, Sylow
202C22×C53—42no—
203C22×C53—42no—
204C22×C53—42no—
205C22×C53—81no—
206C22×C53—81no—
207C22×C53—81no—
208C22×C53—81no—
209C22×C53#38611yescharacteristic
210C22×C53#43511yescharacteristic
211C22×C53#43611yescharacteristic
212C22×C53C4×D811yescharacteristic
213C22×C53#44211yescharacteristic
214C22×C53#44311yescharacteristic
215C22×C53#44411yescharacteristic
216C2×D53—42no—
217C2×D53—42no—
218C2×D53—42no—
219C2×D53—81no—
220C2×D53—81no—
221C2×D53—81no—
222C2×D53—81no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—42no—
226C4×C53#40711yescharacteristic
227C4×C53#40811yescharacteristic
228C4×C53#43511yescharacteristic
229C4×C53#43611yescharacteristic
230C4×C53#44711yescharacteristic
231C4×C53#44911yescharacteristic
232C4×C53#45911yescharacteristic
233C4×C53#46311yescharacteristic
234Dic53—81no—
235C2×C4×D8—531no—
236C2×#436D5311yescharacteristic, complemented
237#1954—531no—
238C2×D4⋊C8—531no—
239C2×D4⋊C8—531no—
240#1277—531no—
241#1273—531no—
242C2×C53—82no—
243C2×C53#158111yescharacteristic
244C2×C53#158611yescharacteristic
245C2×C53#158911yescharacteristic
246C2×C53#159411yescharacteristic
247C2×C53#187011yescharacteristic
248C2×C53#187611yescharacteristic
249C2×C53#187911yescharacteristic
250D53—82no—
251C22×D8—531no—
252C22×M16—21no—
253C2×C4×D4—531no—
254C2×C4×D4C2×D5311yescharacteristic
255C4×D8—1062no—
256C4×D8—1062no—
257#436—22no—
258#436—22no—
259C2×#107—531no—
260C2×C4⋊C8—531no—
261C2×C4⋊C8C2×D5311yescharacteristic
262C2×#102—531no—
263C2×#102—531no—
264C2×C22⋊C8—21no—
265C2×C22⋊C8—1061no—
266C2×C22⋊C8—1061no—
267C2×C4×C8—531no—
268C2×C4×C8—531no—
269C2×C4×C8C2×D5311yescharacteristic
270D4⋊C8—1062no—
271D4⋊C8—1062no—
272C53#895111yescharacteristic, complemented, Sylow
273C22×D4—531no—
274C22×D4C4×D5311yescharacteristic
275C23×C4—21no—
276C23×C4—1061no—
277C2×D8—1062no—
278C2×D8—1062no—
279C2×D8—1062no—
280C2×M16—24no—
281C2×M16—42no—
282C2×M16—42no—
283C22×C8—21no—
284C22×C8—531no—
285C22×C8—531no—
286C22×C8—1061no—
287C22×C8—1061no—
288C22×C8C22⋊D5311yescharacteristic
289C22×C8C22⋊D5311yescharacteristic
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—1061no—
293C4×D4—1061no—
294C4×D4—1061no—
295C4×D4—1061no—
296C4×D4D21212yes—
297C4×D4C22⋊D5312yes—
298C2×C4⋊C4—531no—
299C2×C4⋊C4C4×D5311yescharacteristic
300C2×C22⋊C4—21no—
301C2×C22⋊C4—1061no—
302C2×C42C22×D5311yescharacteristic
303#107—1062no—
304C4⋊C8—21no—
305C4⋊C8—1061no—
306C4⋊C8—1061no—
307C4⋊C8C22⋊D5312yes—
308#102—1062no—
309#102—1062no—
310C22⋊C8—24no—
311C22⋊C8—2122no—
312C22⋊C8—2122no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—1061no—
316C4×C8—1061no—
317C4×C8—1061no—
318C4×C8—1061no—
319C24—21no—
320C24—1061no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—532no—
327C2×D4—1061no—
328C2×D4—1061no—
329C2×D4—1061no—
330C2×D4—1061no—
331C2×D4—1061no—
332C2×D4#1171992112yes—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—1061no—
341C22×C4—2121no—
342C22×C4—2121no—
343C22×C4—2121no—
344C22×C4—2121no—
345C22×C4—2121no—
346C22×C4—2121no—
347C22×C4C2×C4×D5311yescharacteristic
348C22×C4#1171994511yescharacteristic
349C22×C4C2×C22⋊D5311yescharacteristic
350D8—1068no—
351M16—48no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—22no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—532no—
361C2×C8—1061no—
362C2×C8—1061no—
363C2×C8—1062no—
364C2×C8—1061no—
365C2×C8—1061no—
366C2×C8—1061no—
367C2×C8—2121no—
368C2×C8—2121no—
369C2×C8—2121no—
370C2×C8—2121no—
371C2×C8—2121no—
372C2×C8#1171992212yescomplemented
373C4⋊C4—21no—
374C4⋊C4—1061no—
375C4⋊C4—1061no—
376C4⋊C4#1171992112yes—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—2121no—
380C22⋊C4—2121no—
381C42C2×D21211yescharacteristic
382C42#1171994511yescharacteristic
383C42C2×C22⋊D5311yescharacteristic
384C42C2×C22⋊D5311yescharacteristic
385C23—22no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—1062no—
393C23—2121no—
394C23—2121no—
395C23—2121no—
396C23—2121no—
397C23—2121no—
398C23—2121no—
399C23C4×C22⋊D5311yescharacteristic
400D4—22no—
401D4—22no—
402D4—41no—
403D4—41no—
404D4—1062no—
405D4—1062no—
406D4—2121no—
407D4—2121no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—4241no—
418C2×C4—4241no—
419C2×C4—4241no—
420C2×C4—4241no—
421C2×C4—2121no—
422C2×C4—2121no—
423C2×C4—2121no—
424C2×C4C2×#1171992111yescharacteristic
425C2×C4C4×C22⋊D5311yescharacteristic
426C2×C4#1313318611yescharacteristic
427C2×C4#1313319111yescharacteristic
428C2×C4#1313319111yescharacteristic
429C2×C4#1313320011yescharacteristic
430C2×C4M16×D5311yescharacteristic
431C2×C4M16×D5311yescharacteristic
432C2×C4#1313322511yescharacteristic
433C2×C4#1313322511yescharacteristic
434C2×C4C2×#1171992211yescharacteristic
435C2×C4#1313323711yescharacteristic
436C2×C4#1313324211yescharacteristic
437C2×C4#1313324311yescharacteristic
438C8—22no—
439C8—42no—
440C8—41no—
441C8—1062no—
442C8—4241no—
443C8—2122no—
444C8—2121no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—4241no—
453C22—4241no—
454C22—4241no—
455C22—4241no—
456C22—2122no—
457C22—2122no—
458C22—2122no—
459C22#1681050211yescharacteristic
460C22#1681069211yescharacteristic
461C22#1681069911yescharacteristic
462C22C4×#1171992211yescharacteristic
463C22#1681073311yescharacteristic
464C22#1681073811yescharacteristic
465C22#1681074311yescharacteristic
466C4—21no—
467C4—21no—
468C4—42no—
469C4—4241no—
470C4#1681058611yescharacteristic
471C4#1681059111yescharacteristic
472C4#1681069211yescharacteristic
473C4#1681069911yescharacteristic
474C4#1681075711yescharacteristic
475C4#1681076811yescharacteristic
476C4#1681081811yescharacteristic
477C4#1681083811yescharacteristic
478C2—82no—
479C2—4242no—
480C2#2027354211yescharacteristic
481C2#2027357711yescharacteristic
482C2#2027359811yescharacteristic
483C2#2027362711yescharacteristic
484C2#2027501211yescharacteristic
485C2#2027504211yescharacteristic
486C2#2027506311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#436C2
socleC23×C53C4×D4
prime radicalC22×#11719922C4
cosocleC2×C42×C53C23
hypercenterC2×#436D53
hypocenterC53#8951
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
2. #20275509C2semidirect
3. C2×D4⋊#88322C2semidirect
4. C2×#16810350C2semidirect
5. #20272549C2semidirect
6. #20272542C2semidirect
7. C2×C53×#436C2semidirect
99. C2×D212M16semidirect
115. C2×C8×C53D8semidirect
140. C2×#88322M16semidirect
236. C2×#436D53semidirect
272. C53#8951semidirect
372. C2×C8#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199222
#202755092
C2×D4⋊#883222
C2×#168103502
#202725492
#202725422
C2×C53×#4362
#895153