Finite Group Databasebeta

← #27280546

Subgroups — #27280546

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810936C211yesmaximal, characteristic
2#20275551C211yesmaximal, characteristic
3#20274801C211yesmaximal, characteristic, complemented
4#20273282C211yesmaximal, characteristic
5#20272582C211yesmaximal, characteristic, complemented
6#20272563C211yesmaximal, characteristic, complemented
7C53×#2344C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9#16810936—22no—
10#16810936—21no—
11#16810936—21no—
12C2×#13133150C412yescomplemented
13C2×#13133137C412yescomplemented
14C2×#13133131—21no—
15C2×#13133131—21no—
16C2×C22⋊#88322C2211yescharacteristic
17C2×C22⋊#88322C2211yescharacteristic
18#16810483C2211yescharacteristic
19#16810385—22no—
20#16810356—22no—
21C2×#117×C53—21no—
22C22×C4⋊C4×C53C2211yescharacteristic
23C53×#389—21no—
24C53×#387—22no—
25C53×#381—21no—
26C53×#380C2211yescharacteristic
27C4×C4⋊C4×C53—21no—
28C4×C22⋊C4×C53C2211yescharacteristic
29C22×D212C2×C411yescharacteristic
30C22×C22⋊D53—21no—
31C22×#88322—21no—
32C22×#88322C2×C412yes—
33#13133191—21no—
34#13133191—21no—
35#13133191—21no—
36#13133191—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C22⋊Dic53—21no—
40C2×#11719921—21no—
41#13133150—22no—
42#13133150—22no—
43#13133137—22no—
44#13133137—22no—
45#13133131—42no—
46#13133131—42no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51#13133124—21no—
52C23×C4×C53—21no—
53C23×C4×C53C2311yescharacteristic
54#117×C53—42no—
55#117×C53—42no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53C2×C412yes—
62C2×C4⋊C4×C53D412yes—
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53D411yescharacteristic
67C2×C22⋊C4×C53D411yescharacteristic
68C2×C42×C53—22no—
69C2×C42×C53—21no—
70#95×C53—22no—
71#95×C53—22no—
72#95×C53—21no—
73#95×C53D412yes—
74C24×C53C2×D411yescharacteristic
75C23×D53—21no—
76C22×C4×C53—22no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—22no—
81C22×C4×C53—21no—
82C22×C4×C53—22no—
83C22×C4×C53—42no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C2×C22⋊D53—41no—
95C2×C22⋊D53—41no—
96C2×C22⋊D53—41no—
97C2×C22⋊D53—41no—
98C2×C22⋊D53—41no—
99C2×C22⋊D53—41no—
100C22×Dic53—21no—
101C22×Dic53—41no—
102C2×D212—21no—
103C2×D212—21no—
104C2×D212—42no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—42no—
110C4⋊C4×C53—44no—
111C4⋊C4×C53C22⋊C412yes—
112C4⋊C4×C53C22⋊C412yes—
113C22⋊C4×C53—21no—
114C22⋊C4×C53—21no—
115C22⋊C4×C53—21no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C22⋊C4×C53D812yes—
119C42×C53—42no—
120C42×C53—44no—
121C22⋊Dic53—41no—
122C22⋊Dic53—41no—
123#11719921—41no—
124#11719921—41no—
125C4⋊Dic53—21no—
126C4⋊Dic53—21no—
127C4⋊Dic53—42no—
128C2×#88322—22no—
129C2×#88322—42no—
130C2×#88322—42no—
131C2×#88322—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53C4×D411yescharacteristic
135C23×C53#12011yescharacteristic
136C23×C53#12111yescharacteristic
137C23×C53C2×D811yescharacteristic
138C23×C53#13511yescharacteristic
139C23×C53AGL(1,ℤ/8)11yescharacteristic
140C23×C53AGL(1,ℤ/8)11yescharacteristic
141C22×D53—41no—
142C22×D53—41no—
143C22×D53—41no—
144C22×D53—81no—
145C22×D53—81no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—42no—
167C2×C4×C53—41no—
168C2×C4×C53—42no—
169C2×C4×C53—42no—
170C2×C4×C53—41no—
171C2×C4×C53C2×C22⋊C411yescharacteristic
172C2×C4×C53C4×D411yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12311yescharacteristic
175C22⋊D53—81no—
176C22⋊D53—81no—
177C22⋊D53—81no—
178C22⋊D53—81no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—41no—
182C2×Dic53—81no—
183C2×Dic53—81no—
184C2×Dic53—81no—
185D212—42no—
186D212—81no—
187#88322—42no—
188#88322—81no—
189#9260—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53#38611yescharacteristic
201C22×C53C4×D811yescharacteristic
202C22×C53#44211yescharacteristic
203C22×C53#44711yescharacteristic
204C22×C53#44911yescharacteristic
205C22×C53#45911yescharacteristic
206C22×C53#46311yescharacteristic
207C2×D53—81no—
208C2×D53—81no—
209C2×D53—81no—
210C2×D53—161no—
211C2×D53—161no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—42no—
216C4×C53—42no—
217C4×C53—42no—
218C4×C53—82no—
219C4×C53—81no—
220C4×C53#41012yes—
221Dic53—81no—
222Dic53—161no—
223C2×#480—531no—
224#2344D5311yescharacteristic, complemented
225#1964—531no—
226#1823—531no—
227#1519—531no—
228#1289—531no—
229#1282—531no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53—21no—
233C2×C53—21no—
234C2×C53—21no—
235C2×C53—21no—
236C2×C53#187911yescharacteristic
237C2×C53#189911yescharacteristic
238C2×C53#190511yescharacteristic
239D53—161no—
240C2×#121—531no—
241C2×#117—21no—
242C22×C4⋊C4C2×D5311yescharacteristic
243#480—1062no—
244#480—1061no—
245#480—1061no—
246C2×#107—532no—
247C2×#102—532no—
248C2×#99—1061no—
249C2×#99—1061no—
250C2×C22⋊C8—531no—
251C2×C22⋊C8—531no—
252#389—21no—
253#387—22no—
254#381—21no—
255#380—531no—
256#380C2×D5311yescharacteristic
257C4×C4⋊C4—21no—
258C4×C22⋊C4C2×D5311yescharacteristic
259#340—1062no—
260#327—1062no—
261C53#926011yescharacteristic, complemented, Sylow
262C22×D4—531no—
263C22×D4—1061no—
264C23×C4—21no—
265C23×C4C22×D5311yescharacteristic
266C22×C8—532no—
267C22×C8—1061no—
268#121—1061no—
269#121—1061no—
270#121—1061no—
271#121—1061no—
272#117—42no—
273#117—42no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—531no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4C4×D5312yes—
282C2×C4⋊C4D21212yes—
283C2×C4⋊C4C22⋊D5311yescharacteristic
284C2×C4⋊C4C22⋊D5311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—1061no—
287C2×C22⋊C4—1061no—
288C2×C22⋊C4C22⋊D5311yescharacteristic
289C2×C22⋊C4C22⋊D5311yescharacteristic
290C2×C42—22no—
291C2×C42—21no—
292#107—1062no—
293#107—1062no—
294#102—1062no—
295#102—1062no—
296#99—2122no—
297#99—2122no—
298C22⋊C8—1061no—
299C22⋊C8—1061no—
300C22⋊C8—1061no—
301C22⋊C8—1061no—
302#95—22no—
303#95—22no—
304#95—21no—
305#95—1061no—
306#95C22⋊D5312yes—
307C24—1061no—
308C24C2×C22⋊D5311yescharacteristic
309C2×D4—1061no—
310C2×D4—1061no—
311C2×D4—2122no—
312C2×D4—2121no—
313C2×D4—2121no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4—2121no—
317C2×D4—2121no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—1061no—
331C22×C4—2121no—
332C22×C4C2×C4×D5311yescharacteristic
333C22×C4C2×D21211yescharacteristic
334C22×C4#1171994511yescharacteristic
335C22×C4#1171994511yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338C2×C8—1062no—
339C2×C8—2122no—
340C2×C8—2121no—
341C2×C8—2122no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—1061no—
349C4⋊C4—1061no—
350C4⋊C4—2122no—
351C4⋊C4#1171992112yes—
352C4⋊C4#1171992112yes—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—2121no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C22⋊C4—2121no—
362C22⋊C4#1171992212yes—
363C42—44no—
364C42—42no—
365C23—41no—
366C23—41no—
367C23—4241no—
368C23—4241no—
369C23—2121no—
370C23—2121no—
371C23—2121no—
372C23C4×C22⋊D5311yescharacteristic
373C23#1313318611yescharacteristic
374C23#1313319111yescharacteristic
375C23C2×#1171992211yescharacteristic
376C23#1313323711yescharacteristic
377C23#1313324211yescharacteristic
378C23#1313324311yescharacteristic
379D4—4241no—
380D4—4241no—
381D4—4241no—
382D4—4241no—
383D4—4241no—
384D4—2122no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—4241no—
411C2×C4—4241no—
412C2×C4—4241no—
413C2×C4—2121no—
414C2×C4—2121no—
415C2×C4—2121no—
416C2×C4C2×#1171992111yescharacteristic
417C2×C4C4×C22⋊D5311yescharacteristic
418C2×C4#1313319111yescharacteristic
419C2×C4#1313320011yescharacteristic
420C8—4241no—
421C8—2122no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—8481no—
433C22—8481no—
434C22—4241no—
435C22—4241no—
436C22—4241no—
437C22#1681050211yescharacteristic
438C22C4×#1171992211yescharacteristic
439C22#1681073311yescharacteristic
440C22#1681075711yescharacteristic
441C22#1681076811yescharacteristic
442C22#1681081811yescharacteristic
443C22#1681083811yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—82no—
451C4—81no—
452C4—8481no—
453C4—4241no—
454C4#1681059612yes—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—8481no—
462C2#2027506311yescharacteristic
463C2#2027517311yescharacteristic
464C2#2027520911yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2344C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#2344D53
hypocenterC53#9260
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2344

Complements

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

NH ≅ G/Ntype
3. #20274801C2semidirect
5. #20272582C2semidirect
6. #20272563C2semidirect
7. C53×#2344C2semidirect
12. C2×#13133150C4semidirect
13. C2×#13133137C4semidirect
224. #2344D53semidirect
261. C53#9260semidirect

Maximal subgroups

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

subgroupindex
C2×#168109362
#202755512
#202748012
#202732822
#202725822
#202725632
C53×#23442
#926053