Finite Group Databasebeta

← #27277067

Subgroups — #27277067

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283410C211yesmaximal, characteristic
2#20274751C211yesmaximal, characteristic, complemented
3#20274742C211yesmaximal, characteristic, complemented
4#20273258C211yesmaximal, characteristic
5#20272622C211yesmaximal, characteristic, complemented
6#20272621C211yesmaximal, characteristic, complemented
7C53×#1513C211yesmaximal, characteristic
8C2×#13133224C411yescharacteristic, complemented
9C23×#88322C2211yescharacteristic
10#16810695—21no—
11#16810695—21no—
12D4×#88322—21no—
13D4×#88322—21no—
14C2×#13133152—21no—
15C2×#13133150—21no—
16#16810650C411yescharacteristic, complemented
17#16810590C411yescharacteristic, complemented
18#16810587C2211yescharacteristic
19#16810587C2211yescharacteristic
20#16810579C411yescharacteristic, complemented
21#16810483C2211yescharacteristic
22#16810387—21no—
23#16810385—21no—
24#16810360—21no—
25#16810357—21no—
26C53×#525C2211yescharacteristic
27C53×#407C2211yescharacteristic
28C53×#380C2211yescharacteristic
29C53×#328—21no—
30C53×#327—21no—
31#13133224—42no—
32C2×#11719917—21no—
33C2×#11719917—21no—
34C2×#11719917C2×C411yescharacteristic
35C2×#11719917C2×C411yescharacteristic
36C22×#88322—21no—
37C22×#88322—42no—
38C22×#88322C2×C411yescharacteristic
39C22×#88322C2×C411yescharacteristic
40C22×#88322D411yescharacteristic
41C22×#88322D411yescharacteristic
42C22×#88322D411yescharacteristic
43C22×#88322D411yescharacteristic
44C2×C4⋊Dic53—21no—
45C2×C22⋊Dic53—21no—
46#13133152—42no—
47#13133150—42no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54C22⋊#88322—21no—
55C22⋊#88322—21no—
56#13133128—21no—
57C4×#88322—21no—
58#13133124—21no—
59C2×#41×C53C2×C411yescharacteristic
60C23×C4×C53C2311yescharacteristic
61C2×M16×C53—21no—
62#123×C53—21no—
63C22⋊Q8×C53D411yescharacteristic
64C22⋊Q8×C53Q811yescharacteristic
65#121×C53D411yescharacteristic
66#121×C53Q811yescharacteristic
67#120×C53—21no—
68C4×D4×C53—21no—
69C4×D4×C53—21no—
70#117×C53C2×C411yescharacteristic
71C2×C4⋊C4×C53—21no—
72C2×C22⋊C4×C53—21no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53—21no—
75#95×C53—21no—
76C24×C53#4111yescharacteristic
77#41×C53—41no—
78#41×C53—41no—
79C2×Q8×C53C4⋊C411yescharacteristic
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53C4⋊C411yescharacteristic
84C22×C4×C53—21no—
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×C53C2×Q811yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C22×Dic53—41no—
95M16×C53—41no—
96M16×C53—41no—
97C2×C8×C53—41no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53C4⋊C411yescharacteristic
103C4⋊C4×C53C4⋊C411yescharacteristic
104C22⋊C4×C53—21no—
105C22⋊C4×C53—21no—
106C22⋊C4×C53—21no—
107C22⋊C4×C53—21no—
108C22⋊C4×C53—21no—
109C22⋊C4×C53—81no—
110C42×C53—21no—
111C22⋊Dic53—81no—
112C4⋊Dic53—41no—
113C4⋊Dic53—41no—
114#11719917—42no—
115#11719917—41no—
116#11719917—41no—
117#11719917—41no—
118#11719917—41no—
119C2×#88322—22no—
120C2×#88322—22no—
121C2×#88322—22no—
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—42no—
128C2×#88322—41no—
129C2×#88322C22⋊C411yescharacteristic
130C2×#88322C22⋊C411yescharacteristic
131C2×#88322C22⋊C411yescharacteristic
132C2×#88322C22⋊C411yescharacteristic
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53#11711yescharacteristic
139C23×C53#12111yescharacteristic
140C23×C53C22⋊Q811yescharacteristic
141Q8×C53—41no—
142D4×C53—41no—
143D4×C53—42no—
144D4×C53—42no—
145D4×C53—41no—
146D4×C53—41no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—81no—
162C2×C4×C53—81no—
163C2×C4×C53—81no—
164C2×C4×C53C2×C22⋊C411yescharacteristic
165C2×C4×C53C2×C4⋊C411yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53C22⋊Q811yescharacteristic
168C2×Dic53—81no—
169C2×Dic53—81no—
170C2×Dic53—81no—
171C8×C53—81no—
172#88322—21no—
173#88322—21no—
174#88322—21no—
175#88322—41no—
176#88322—41no—
177#88322—81no—
178#88322#9912yes—
179#8627—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—42no—
189C22×C53—42no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53#38011yescharacteristic
194C22×C53#47211yescharacteristic
195C22×C53#47311yescharacteristic
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53—41no—
202C4×C53—161no—
203C4×C53C2×#9911yescharacteristic
204C4×C53#41011yescharacteristic
205Dic53—161no—
206#2929—531no—
207#1810—531no—
208#1808—531no—
209#1513—531no—
210#1513D5311yescharacteristic
211#1305—531no—
212#1305—531no—
213C2×C53—21no—
214C2×C53—21no—
215C2×C53—41no—
216C2×C53—41no—
217C2×C53—81no—
218C2×C53#179911yescharacteristic
219C2×C53#181411yescharacteristic
220C2×C53#181611yescharacteristic
221C2×#131—531no—
222C23×C8—531no—
223#525C2×D5311yescharacteristic
224#435—1061no—
225#435—1061no—
226C8×D4—1061no—
227C8×D4—1061no—
228C2×#108—1061no—
229C2×#107—1061no—
230#424—531no—
231#408—531no—
232#407—531no—
233#407—531no—
234#407C2×D5311yescharacteristic
235#405—531no—
236#380—531no—
237#380C2×D5311yescharacteristic
238#341—1061no—
239#340—1061no—
240#328—21no—
241#328—1061no—
242#327—21no—
243#327—1061no—
244C53#862711yescharacteristic, complemented, Sylow
245C2×#41C4×D5311yescharacteristic
246C23×C4C22×D5311yescharacteristic
247#131—2122no—
248C2×M16—21no—
249C2×M16—531no—
250C2×M16—531no—
251C2×M16—1061no—
252C2×M16—1061no—
253C22×C8—531no—
254C22×C8—531no—
255C22×C8—531no—
256C22×C8—531no—
257C22×C8—531no—
258C22×C8—531no—
259C22×C8—1061no—
260C22×C8—2122no—
261#123—21no—
262C22⋊Q8Dic10611yescharacteristic
263C22⋊Q8C22⋊D5311yescharacteristic
264#121Dic10611yescharacteristic
265#121C22⋊D5311yescharacteristic
266#120—21no—
267C4×D4—21no—
268C4×D4—21no—
269#117C4×D5311yescharacteristic
270C2×C4⋊C4—21no—
271C2×C4⋊C4—1061no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—1061no—
274#108—2122no—
275#107—2122no—
276C4⋊C8—1061no—
277C4⋊C8—1061no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—1061no—
281C22⋊C8—1061no—
282C22⋊C8—1061no—
283C22⋊C8—1061no—
284C22⋊C8—1061no—
285C22⋊C8—1061no—
286#97—1061no—
287C4×C8—1061no—
288#95—21no—
289#95—1061no—
290C24#1171994711yescharacteristic
291#41—41no—
292#41—41no—
293C2×Q8#1171991911yescharacteristic
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4#1171991911yescharacteristic
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—2121no—
303C22×C4C2×Dic10611yescharacteristic
304C22×C4C2×C4×D5311yescharacteristic
305C22×C4D4×D5311yescharacteristic
306C22×C4D4×D5311yescharacteristic
307C22×C4#1171994711yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309M16—41no—
310M16—41no—
311M16—2122no—
312M16—2121no—
313M16—2121no—
314M16—2121no—
315M16—2121no—
316C2×C8—41no—
317C2×C8—531no—
318C2×C8—531no—
319C2×C8—531no—
320C2×C8—531no—
321C2×C8—1062no—
322C2×C8—1062no—
323C2×C8—1062no—
324C2×C8—1061no—
325C2×C8—1061no—
326C2×C8—2122no—
327C2×C8—2121no—
328C2×C8—2121no—
329C2×C8—2121no—
330C2×C8—2121no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—2121no—
336C4⋊C4—2121no—
337C4⋊C4#1171991911yescharacteristic
338C4⋊C4#1171991911yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—81no—
345C22⋊C4—4241no—
346C42—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23#1313317311yescharacteristic
353C23#1313318911yescharacteristic
354C23#1313319411yescharacteristic
355Q8—41no—
356D4—41no—
357D4—42no—
358D4—42no—
359D4—41no—
360D4—41no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—4241no—
379C2×C4—4241no—
380C2×C4—4241no—
381C2×C4C22⋊C4×D5311yescharacteristic
382C2×C4C2×#1171991911yescharacteristic
383C2×C4#1313318911yescharacteristic
384C2×C4#1313319411yescharacteristic
385C8—81no—
386C8—532no—
387C8—1061no—
388C8—1061no—
389C8—1061no—
390C8—4241no—
391C8—2121no—
392C8—2121no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22#1681048111yescharacteristic
407C22#1681089111yescharacteristic
408C22#1681089611yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—161no—
416C4—8481no—
417C4#99×D5311yescharacteristic
418C4#1681059511yescharacteristic
419C2—21no—
420C2—21no—
421C2—41no—
422C2—41no—
423C2—81no—
424C2#2027470111yescharacteristic
425C2#2027476711yescharacteristic
426C2#2027477711yescharacteristic
subgroupG/H
centerC22#16810481
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1513C2
socleC22×C53#380
prime radicalC53×#525C22
cosocleC23×C4×C53C23
hypercenter#1513D53
hypocenterC53#8627
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1513

Complements

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

NH ≅ G/Ntype
2. #20274751C2semidirect
3. #20274742C2semidirect
5. #20272622C2semidirect
6. #20272621C2semidirect
8. C2×#13133224C4semidirect
16. #16810650C4semidirect
17. #16810590C4semidirect
20. #16810579C4semidirect
244. C53#8627semidirect

Maximal subgroups

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

subgroupindex
#202834102
#202747512
#202747422
#202732582
#202726222
#202726212
C53×#15132
#862753