Finite Group Databasebeta

← #27300852

Subgroups — #27300852

Order 13568.

524 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810413C211yesmaximal, characteristic
2#20275129C211yesmaximal, characteristic
3#20274995C211yesmaximal, characteristic
4#20274924C211yesmaximal, characteristic
5#20272840C211yesmaximal, characteristic
6#20272833C211yesmaximal, characteristic
7C53×#2374C211yesmaximal, characteristic
8#16811098C2211yescharacteristic
9#16810663—21no—
10C2×#13133134C2211yescharacteristic
11C2×#13133132C2211yescharacteristic
12C2×#13133132C2211yescharacteristic
13#16810496—21no—
14#16810486—21no—
15#16810413C414yescomplemented
16#16810389—21no—
17#16810389—21no—
18#16810383—21no—
19C2×#123×C53C2211yescharacteristic
20C53×#390C2211yescharacteristic
21C53×#388—21no—
22C53×#388—21no—
23C53×#387—21no—
24C53×#386C2211yescharacteristic
25C4×C4⋊C4×C53—21no—
26C4×C22⋊C4×C53—21no—
27C4×C22⋊C4×C53—21no—
28C2×#11719917—21no—
29D4×Dic53—41no—
30D4×Dic53—41no—
31#13133176—21no—
32C2×#11719919—41no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37#13133150—41no—
38#13133148—41no—
39#13133134—21no—
40#13133134C2×C412yes—
41#13133132—21no—
42#13133132—21no—
43#13133132—21no—
44#13133132C2×C412yes—
45#13133124—41no—
46#13133124—41no—
47#13133124—41no—
48C22×D4×C53C2311yescharacteristic
49C23×C4×C53D411yescharacteristic
50C23×C4×C53D411yescharacteristic
51#123×C53—21no—
52#123×C53—21no—
53#123×C53C2×C412yes—
54#123×C53Q812yes—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53D411yescharacteristic
64C2×C42×C53—21no—
65C2×C42×C53—21no—
66C2×C42×C53—21no—
67C2×C42×C53D411yescharacteristic
68#95×C53—21no—
69#95×C53—21no—
70#95×C53—21no—
71#95×C53—21no—
72#95×C53—21no—
73C24×C53C2×D411yescharacteristic
74C24×C53C2×D411yescharacteristic
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53C22×C411yescharacteristic
78C2×D4×C53C2×Q811yescharacteristic
79C2×D4×C53#4111yescharacteristic
80C2×D4×C53#4111yescharacteristic
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—42no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53C4⋊C412yescomplemented
93C22×C4×C53C2×D411yescharacteristic
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—41no—
99C22×Dic53—41no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—42no—
103C4⋊C4×C53—41no—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—21no—
107C22⋊C4×C53—42no—
108C22⋊C4×C53—42no—
109C22⋊C4×C53—42no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53C4⋊C412yescomplemented
112C42×C53—42no—
113C42×C53—41no—
114C42×C53—81no—
115C42×C53—81no—
116C22⋊Dic53—22no—
117C22⋊Dic53—41no—
118C22⋊Dic53—41no—
119C22⋊Dic53—41no—
120C22⋊Dic53—41no—
121C22⋊Dic53—41no—
122C22⋊Dic53—41no—
123C4⋊Dic53—21no—
124C4⋊Dic53—21no—
125#11719919—82no—
126C4×Dic53—41no—
127#11719917—22no—
128#11719917—41no—
129C2×#88322—41no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53C2×C4⋊C411yescharacteristic
139C23×C53C4×D411yescharacteristic
140C23×C53#12011yescharacteristic
141C23×C53C22⋊Q811yescharacteristic
142C23×C53#12311yescharacteristic
143D4×C53—41no—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—41no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—81no—
170C2×C4×C53—81no—
171C2×C4×C53C4×D411yescharacteristic
172C2×C4×C53C22⋊Q811yescharacteristic
173C2×Dic53—42no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—81no—
177C2×Dic53—81no—
178C2×Dic53—81no—
179C2×Dic53—81no—
180C2×Dic53—81no—
181C2×Dic53—81no—
182C2×Dic53—81no—
183C2×Dic53—81no—
184C2×Dic53—81no—
185#88322—42no—
186#12959—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—81no—
202C22×C53—81no—
203C22×C53#38511yescharacteristic
204C22×C53B(4,𝔽2)11yescharacteristic
205C22×C53#45811yescharacteristic
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—42no—
209C4×C53—42no—
210C4×C53—42no—
211C4×C53—42no—
212C4×C53—81no—
213C4×C53—81no—
214Dic53—82no—
215Dic53—81no—
216Dic53—81no—
217Dic53—161no—
218#2374D5311yescharacteristic
219C2×#352—531no—
220#1890—531no—
221#1867—531no—
222#1856—531no—
223#1393—531no—
224#1390—531no—
225C2×C53—21no—
226C2×C53—21no—
227C2×C53—41no—
228C2×C53—41no—
229C2×C53—41no—
230C2×C53—41no—
231C2×C53#186711yescharacteristic
232C2×C53#219711yescharacteristic
233C2×C53#219811yescharacteristic
234C2×#123C2×D5311yescharacteristic
235#518—531no—
236#428—1061no—
237C2×#100—531no—
238C2×#99—531no—
239C2×#99—531no—
240#390C2×D5311yescharacteristic
241#388—21no—
242#388—21no—
243#387—21no—
244#386C2×D5311yescharacteristic
245#385—1061no—
246#381—1061no—
247C4×C4⋊C4—21no—
248C4×C22⋊C4—21no—
249C4×C22⋊C4—21no—
250#352—534no—
251#342—1061no—
252#342—1061no—
253#339—1061no—
254C53#1295911yescharacteristic, complemented, Sylow
255C22×D4C22×D5311yescharacteristic
256C23×C4C22⋊D5311yescharacteristic
257C23×C4C22⋊D5311yescharacteristic
258C2×M16—1061no—
259#123—21no—
260#123—21no—
261#123Dic10612yes—
262#123C4×D5312yes—
263C4×D4—2121no—
264C4×D4—2121no—
265#117—1061no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—2121no—
269C2×C4⋊C4C22⋊D5311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—1061no—
275C2×C22⋊C4—1061no—
276C2×C22⋊C4—1061no—
277C2×C22⋊C4—1061no—
278C2×C22⋊C4C22⋊D5311yescharacteristic
279C2×C22⋊C4C22⋊D5311yescharacteristic
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42C22⋊D5311yescharacteristic
284#107—2121no—
285#106—2121no—
286#100—532no—
287#100—1061no—
288#99—532no—
289#99—1061no—
290#99—1061no—
291#99—1061no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—2121no—
298#95—2121no—
299#95—2121no—
300C24C2×C22⋊D5311yescharacteristic
301C24C2×C22⋊D5311yescharacteristic
302C2×D4—41no—
303C2×D4—41no—
304C2×D4C2×Dic10611yescharacteristic
305C2×D4C2×C4×D5311yescharacteristic
306C2×D4#1171994511yescharacteristic
307C2×D4#1171994511yescharacteristic
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—1061no—
320C22×C4—1061no—
321C22×C4—1061no—
322C22×C4—1061no—
323C22×C4—2121no—
324C22×C4—2121no—
325C22×C4#1171991912yescomplemented
326C22×C4C2×C22⋊D5311yescharacteristic
327M16—1062no—
328M16—2121no—
329C2×C8—2121no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—1061no—
335C4⋊C4—1061no—
336C4⋊C4—4242no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—21no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—41no—
344C22⋊C4—1062no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C22⋊C4—2121no—
348C22⋊C4—2121no—
349C22⋊C4—2121no—
350C22⋊C4—2121no—
351C22⋊C4#1171991912yescomplemented
352C42—41no—
353C42—42no—
354C42—81no—
355C42—81no—
356C42—2121no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23C2×#1171991911yescharacteristic
366C23C4×C22⋊D5311yescharacteristic
367C23#1313318611yescharacteristic
368C23#1313319611yescharacteristic
369C23#1313320011yescharacteristic
370D4—41no—
371D4—41no—
372D4—41no—
373D4—41no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—4241no—
399C2×C4—4241no—
400C2×C4—4241no—
401C2×C4—4241no—
402C2×C4—4241no—
403C2×C4—4241no—
404C2×C4—4241no—
405C2×C4—4241no—
406C2×C4—4241no—
407C2×C4—2122no—
408C2×C4—2121no—
409C2×C4—2121no—
410C2×C4C4×C22⋊D5311yescharacteristic
411C2×C4#1313319611yescharacteristic
412C8—2122no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—81no—
428C22—81no—
429C22#1681049611yescharacteristic
430C22#1681081011yescharacteristic
431C22#1681081311yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—42no—
438C4—81no—
439C4—81no—
440C4—8481no—
441C4—4242no—
442C4—4241no—
443C4—4241no—
444C2—21no—
445C2—21no—
446C2—41no—
447C2—41no—
448C2—41no—
449C2—41no—
450C2#2027499611yescharacteristic
451C2#2027670511yescharacteristic
452C2#2027671111yescharacteristic
subgroupG/H
centerC22#16810810
commutatorC2×D4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2374C2
socleC22×C53B(4,𝔽2)
prime radical, cosocleC22×D4×C53C23
hypercenter#2374D53
hypocenterC53#12959
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2374

Complements

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

NH ≅ G/Ntype
15. #16810413C4semidirect
92. C22×C4×C53C4⋊C4semidirect
111. C22⋊C4×C53C4⋊C4semidirect
254. C53#12959semidirect
325. C22×C4#11719919semidirect
351. C22⋊C4#11719919semidirect

Maximal subgroups

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

subgroupindex
C2×#168104132
#202751292
#202749952
#202749242
#202728402
#202728332
C53×#23742
#1295953