Finite Group Databasebeta

← #27368046

Subgroups — #27368046

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810970C211yesmaximal, characteristic, complemented
2#20284530C211yesmaximal, characteristic, complemented
3#20284195C211yesmaximal, characteristic
4#20274131C214yesmaximal, complemented
5#20274124C214yesmaximal, complemented
6#20273471C211yesmaximal, characteristic, complemented
7#20273395C211yesmaximal, characteristic, complemented
8#20273382C211yesmaximal, characteristic, complemented
9C8×#41×C53C211yesmaximal, characteristic, complemented
10C2×#13133242—21no—
11C2×#13133217C2211yescharacteristic
12C2×#13133214C2211yescharacteristic
13#16810970—21no—
14#16810970—21no—
15#16810970C2214yes—
16#16810918C2212yescomplemented
17#16810913C2212yescomplemented
18#16810908—21no—
19#16810903—21no—
20#16810823—21no—
21#16810823C2212yes—
22#16810818—21no—
23#16810818C2212yes—
24#16810666—21no—
25#16810647C2211yescharacteristic
26#16810647C2211yescharacteristic
27#16810627—21no—
28#16810624—21no—
29C2×#13133139—22no—
30#16810374C2214yescomplemented
31#16810371C2214yescomplemented
32#16810366C2214yes—
33C4×#41×C53C2211yescharacteristic
34C8×Q8×C53C2212yescomplemented
35C8×D4×C53—22no—
36C8×D4×C53C2212yescomplemented
37C53×#431—21no—
38C53×#431C2211yescharacteristic
39C2×C4×C8×C53—21no—
40C2×C4×C8×C53C2211yescharacteristic
41C22×Dic106—21no—
42C22×D212—21no—
43C22×D212—21no—
44C22×D212—21no—
45#13133242—44no—
46C2×#11719924—22no—
47C2×#11719922—22no—
48C2×#11719917—21no—
49C2×#11719917—21no—
50#13133217—21no—
51#13133217—21no—
52#13133217—21no—
53#13133217C2312yes—
54#13133214—21no—
55#13133214—21no—
56#13133214—21no—
57#13133214C2312yes—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53—21no—
60#13133150—22no—
61#13133148—22no—
62C4⋊#88322—21no—
63C4⋊#88322—21no—
64C4⋊#88322C2312yes—
65C4⋊#88322C2312yes—
66#13133141—22no—
67#13133139—24no—
68#13133139—24no—
69#13133138—22no—
70#13133137—22no—
71#13133137—22no—
72C2×#41×C53D411yescharacteristic
73C22×C8×C53—22no—
74C22×C8×C53D412yes—
75C4×Q8×C53D411yescharacteristic
76C4×Q8×C53C2311yescharacteristic
77C4×D4×C53—21no—
78C4×D4×C53—21no—
79C4×D4×C53D411yescharacteristic
80C4×D4×C53C2311yescharacteristic
81#117×C53—21no—
82#117×C53D411yescharacteristic
83C2×C42×C53—21no—
84C2×C42×C53C2311yescharacteristic
85C4⋊C8×C53—22no—
86C4⋊C8×C53C2312yes—
87C22⋊C8×C53—22no—
88C22⋊C8×C53—21no—
89C4×C8×C53—22no—
90C4×C8×C53—21no—
91C4×C8×C53D411yescharacteristic
92C4×C8×C53D411yescharacteristic
93C4×C8×C53C2312yes—
94C23×D53—41no—
95#41×C53—22no—
96#41×C53SD1614yes—
97C2×Q8×C53C2×D411yescharacteristic
98C2×D4×C53—21no—
99C2×D4×C53C2×D411yescharacteristic
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53#4111yescharacteristic
106C22×Dic53—41no—
107C2×D212—22no—
108C2×D212—22no—
109C2×D212—22no—
110C2×D212—44no—
111C2×D212—42no—
112C2×D212—42no—
113C2×D212—41no—
114C2×D212—41no—
115C2×D212—41no—
116C2×D212—82no—
117C2×Dic106—22no—
118C2×Dic106—41no—
119C2×Dic106—82no—
120C2×C8×C53—22no—
121C2×C8×C53—42no—
122C2×C8×C53—44no—
123C2×C8×C53C2×D412yes—
124C2×C8×C53C2×D412yes—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53C2×D411yescharacteristic
128C4⋊C4×C53C2×D411yescharacteristic
129C22⋊C4×C53—21no—
130C22⋊C4×C53—21no—
131C22⋊C4×C53—21no—
132C42×C53—21no—
133C42×C53—21no—
134C42×C53—21no—
135C42×C53C2×D411yescharacteristic
136C42×C53C2×D411yescharacteristic
137C42×C53#4111yescharacteristic
138C42×C53C2411yescharacteristic
139#11719924—44no—
140#11719922—44no—
141C4⋊Dic53—22no—
142C4⋊Dic53—22no—
143C4⋊Dic53—41no—
144C4⋊Dic53—41no—
145#11719917—42no—
146#11719917—42no—
147C2×#88322—22no—
148C2×#88322—22no—
149C23×C53—21no—
150C23×C53#12111yescharacteristic
151C22×D53—42no—
152C22×D53—82no—
153C22×D53—82no—
154C22×D53—82no—
155Q8×C53—21no—
156Q8×C53C2×SD1612yes—
157D4×C53—24no—
158D4×C53—21no—
159D4×C53C2×SD1612yes—
160C2×C4×C53—21no—
161C2×C4×C53—22no—
162C2×C4×C53—24no—
163C2×C4×C53—22no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53C2×SD1612yes—
172C2×C4×C53#13512yes—
173C2×C4×C53C22×D411yescharacteristic
174C2×C4×C53C22×D411yescharacteristic
175C2×C4×C53C2×#4111yescharacteristic
176C2×Dic53—42no—
177C2×Dic53—82no—
178D212—42no—
179D212—42no—
180D212—44no—
181D212—82no—
182D212—82no—
183D212—82no—
184D212—82no—
185Dic106—82no—
186Dic106—82no—
187C8×C53—22no—
188C8×C53—22no—
189C8×C53—42no—
190#88322—42no—
191#88322—42no—
192#19274—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—22no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53#46512yes—
199C22×C53C2×#12111yescharacteristic
200C2×D53—82no—
201C2×D53—82no—
202C2×D53—82no—
203C2×D53—164no—
204C4×C53—21no—
205C4×C53—22no—
206C4×C53—21no—
207C4×C53—41no—
208C4×C53—41no—
209C4×C53#46512yes—
210C4×C53#47212yes—
211C4×C53C22×SD1611yescharacteristic
212C4×C53C2×#13511yescharacteristic
213C4×C53#57512yes—
214Dic53—82no—
215C2×#486—531no—
216#3079—531no—
217#3038—531no—
218C8×#41D5311yescharacteristic, complemented
219#1678—534no—
220#1677—534no—
221#1566—531no—
222#1548—531no—
223#1545—531no—
224C2×C53—22no—
225C2×C53—42no—
226C2×C53#304011yescharacteristic
227C2×C53C2×#46511yescharacteristic
228C2×C53C2×#47211yescharacteristic
229D53—162no—
230C2×AGL(1,ℤ/8)—1061no—
231C2×C4⋊Q8—531no—
232C2×#127—531no—
233C4×#41C2×D5311yescharacteristic
234#486—534no—
235#486—1061no—
236#486—1061no—
237#477—532no—
238#476—532no—
239#475—1061no—
240#474—1061no—
241#460—532no—
242#460—1061no—
243#459—532no—
244#459—1061no—
245C8×Q8C2×D5312yescomplemented
246C8×D4—22no—
247C8×D4C2×D5312yescomplemented
248#431—21no—
249#431C2×D5311yescharacteristic
250#428—1061no—
251#423—531no—
252#423—531no—
253#418—1061no—
254#417—1061no—
255C2×#102—1062no—
256C2×C4×C8—21no—
257C2×C4×C8C2×D5311yescharacteristic
258#334—534no—
259#333—534no—
260#331—534no—
261C53#1927411yescharacteristic, complemented, Sylow
262C2×#41D21211yescharacteristic
263C22×Q8—1061no—
264C22×D4—1061no—
265C22×D4—1061no—
266C22×D4—1061no—
267AGL(1,ℤ/8)—2124no—
268C2×SD16—1062no—
269C2×D8—1062no—
270C2×M16—1061no—
271C2×M16—1061no—
272C22×C8—22no—
273C22×C8C22⋊D5312yes—
274C4⋊Q8—532no—
275C4⋊Q8—1061no—
276C4⋊Q8—1061no—
277C4⋊Q8—1061no—
278#127—532no—
279#127—1061no—
280#127—1061no—
281#127—1061no—
282C4×Q8D21211yescharacteristic
283C4×Q8C22×D5311yescharacteristic
284C4×D4—21no—
285C4×D4—21no—
286C4×D4D21211yescharacteristic
287C4×D4C22×D5311yescharacteristic
288#117—21no—
289#117D21211yescharacteristic
290C2×C4⋊C4—1061no—
291C2×C4⋊C4—1061no—
292C2×C42—21no—
293C2×C42C22×D5311yescharacteristic
294#107—1062no—
295#106—1062no—
296C4⋊C8—22no—
297C4⋊C8—532no—
298C4⋊C8—532no—
299C4⋊C8—1061no—
300C4⋊C8—1061no—
301C4⋊C8C22×D5312yes—
302#103—1062no—
303#102—1062no—
304#102—1064no—
305#102—1064no—
306#102—1062no—
307#102—1062no—
308C22⋊C8—21no—
309C22⋊C8—22no—
310C4×C8—22no—
311C4×C8—21no—
312C4×C8C22⋊D5311yescharacteristic
313C4×C8C22⋊D5311yescharacteristic
314C4×C8C22×D5312yes—
315C24—2121no—
316#41—22no—
317#41#1171991314yes—
318C2×Q8—1062no—
319C2×Q8—4242no—
320C2×Q8—2121no—
321C2×Q8C2×D21211yescharacteristic
322C2×D4—21no—
323C2×D4—1062no—
324C2×D4—1062no—
325C2×D4—1062no—
326C2×D4—4242no—
327C2×D4—2124no—
328C2×D4—2121no—
329C2×D4—2121no—
330C2×D4—2121no—
331C2×D4—2122no—
332C2×D4—2122no—
333C2×D4C2×D21211yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—2121no—
338C22×C4C2×D21211yescharacteristic
339C22×C4#1171994511yescharacteristic
340C22×C4C2×C22⋊D5311yescharacteristic
341SD16—2124no—
342D8—2124no—
343M16—2122no—
344M16—2122no—
345C2×C8—22no—
346C2×C8—42no—
347C2×C8—44no—
348C2×C8—1062no—
349C2×C8—1062no—
350C2×C8C2×C22⋊D5312yes—
351C2×C8C2×C22⋊D5312yes—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—1062no—
355C4⋊C4—1062no—
356C4⋊C4—2121no—
357C4⋊C4—2121no—
358C4⋊C4C2×D21211yescharacteristic
359C4⋊C4C2×D21211yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42C2×D21211yescharacteristic
367C42#1171994511yescharacteristic
368C42C2×C22⋊D5311yescharacteristic
369C42C23×D5311yescharacteristic
370C23—21no—
371C23—4242no—
372C23—4242no—
373C23—4242no—
374C23—2122no—
375C23#1313319111yescharacteristic
376Q8—21no—
377Q8—4242no—
378Q8—4242no—
379Q8C2×#1171991312yes—
380D4—24no—
381D4—21no—
382D4—4242no—
383D4—4242no—
384D4—4242no—
385D4—4242no—
386D4—2122no—
387D4—2122no—
388D4—2124no—
389D4C2×#1171991312yes—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—4242no—
399C2×C4—2122no—
400C2×C4#1313319111yescharacteristic
401C2×C4#1313319111yescharacteristic
402C2×C4#1313319111yescharacteristic
403C2×C4C2×#1171991312yes—
404C2×C4#1313323912yes—
405C2×C4C22×C22⋊D5311yescharacteristic
406C2×C4C22×D21211yescharacteristic
407C2×C4C2×#1171994511yescharacteristic
408C8—22no—
409C8—22no—
410C8—42no—
411C8—2122no—
412C8—2122no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—8484no—
419C22—4242no—
420C22—4242no—
421C22—4242no—
422C22#1681085212yes—
423C22C2×#1313319111yescharacteristic
424C4—21no—
425C4—21no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—4242no—
430C4#1681085212yes—
431C4#1681089312yes—
432C4C22×#1171991311yescharacteristic
433C4C2×#1313323911yescharacteristic
434C4#1681142112yes—
435C2—22no—
436C2—42no—
437C2—8482no—
438C2#2028420911yescharacteristic
439C2C2×#1681085211yescharacteristic
440C2C2×#1681089311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC8×#41×C53C2
socleC22×C53C2×#121
prime radical#20284195C2
hypercenterC8×#41D53
hypocenterC53#19274
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16810970C2semidirect
2. #20284530C2semidirect
4. #20274131C2semidirect
5. #20274124C2semidirect
6. #20273471C2semidirect
7. #20273395C2semidirect
8. #20273382C2semidirect
9. C8×#41×C53C2semidirect
16. #16810918C22semidirect
17. #16810913C22semidirect
30. #16810374C22semidirect
31. #16810371C22semidirect
34. C8×Q8×C53C22semidirect
36. C8×D4×C53C22semidirect
218. C8×#41D53semidirect
245. C8×Q8C2×D53semidirect
247. C8×D4C2×D53semidirect
261. C53#19274semidirect

Maximal subgroups

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

subgroupindex
C2×#168109702
#202845302
#202841952
#202741312
#202741242
#202734712
#202733952
#202733822
C8×#41×C532
#1927453