Finite Group Databasebeta

← #27368069

Subgroups — #27368069

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284524C211yesmaximal, characteristic
2#20284524C211yesmaximal, characteristic
3C8×#11719945C211yesmaximal, characteristic
4#20274170C214yesmaximal, complemented
5#20274163C214yesmaximal, complemented
6#20273474C211yesmaximal, characteristic, complemented
7#20273400C211yesmaximal, characteristic, complemented
8#20273396C211yesmaximal, characteristic, complemented
9C2×C53×#487C211yesmaximal, characteristic, complemented
10C4×#11719945C2211yescharacteristic
11#16810915—21no—
12#16810915C2212yes—
13#16810910—21no—
14#16810910C2212yes—
15#16810905—21no—
16#16810905C2212yes—
17#16810900—21no—
18#16810900C2212yes—
19C8×Dic106C2212yes—
20C8×D212C2212yes—
21C8×C22⋊D53—22no—
22#16810676C2211yescharacteristic
23#16810674—21no—
24#16810663—21no—
25#16810663—21no—
26#16810647C2211yescharacteristic
27#16810647C2211yescharacteristic
28#16810622—21no—
29#16810622—21no—
30C4×C8×D53—21no—
31#16810376C2214yes—
32#16810372C2214yes—
33#16810369C2214yes—
34C2×C4⋊Q8×C53C2211yescharacteristic
35C2×C4⋊Q8×C53C2211yescharacteristic
36C53×#487—21no—
37C53×#487—21no—
38C53×#487C2214yes—
39C2×#103×C53—22no—
40C2×C4×C8×C53C2211yescharacteristic
41C2×#11719945D411yescharacteristic
42C2×#11719917—21no—
43C2×#11719917—21no—
44C2×C8×D53—22no—
45C4×Dic106D411yescharacteristic
46C4×Dic106C2311yescharacteristic
47C4×D212D411yescharacteristic
48C4×D212C2311yescharacteristic
49C4×C22⋊D53—21no—
50C4×C22⋊D53—21no—
51#13133176D411yescharacteristic
52#13133174—21no—
53C42×D53—21no—
54#13133150—22no—
55#13133150—22no—
56#13133148—22no—
57#13133148—22no—
58C4⋊#88322—21no—
59C4⋊#88322—21no—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62C4⋊#88322C2312yes—
63#13133146—22no—
64#13133140—22no—
65#13133140—22no—
66#13133137—22no—
67#13133137—22no—
68C22⋊#88322—21no—
69#13133129—22no—
70C4×#88322—21no—
71C8×Dic53—22no—
72C22×Q8×C53—21no—
73C22×Q8×C53—21no—
74C22×C8×C53D412yes—
75C4⋊Q8×C53—21no—
76C4⋊Q8×C53—21no—
77C4⋊Q8×C53—21no—
78C4⋊Q8×C53—21no—
79C4⋊Q8×C53—21no—
80C4⋊Q8×C53—21no—
81C4⋊Q8×C53C2312yes—
82C4⋊Q8×C53C2312yes—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C42×C53C2311yescharacteristic
88#103×C53—24no—
89#103×C53—24no—
90C4×C8×C53D411yescharacteristic
91C4×C8×C53D411yescharacteristic
92C4×C8×C53C2312yes—
93C2×Q8×C53—22no—
94C2×Q8×C53—22no—
95C2×Q8×C53—44no—
96C2×Q8×C53—41no—
97C2×Q8×C53—41no—
98C2×Q8×C53—82no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C2×C22⋊D53—21no—
105#11719945—22no—
106#11719945Q1614yes—
107C2×D212C2×D411yescharacteristic
108C2×C4×D53—21no—
109C2×C4×D53—21no—
110C2×C4×D53—21no—
111C2×Dic106C2×D411yescharacteristic
112C2×C8×C53—42no—
113C2×C8×C53C2×D412yes—
114C2×C8×C53C2×D412yes—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C42×C53C2×D411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53#4111yescharacteristic
126C42×C53C2411yescharacteristic
127C22⋊Dic53—21no—
128#11719921—21no—
129#11719921—21no—
130C4⋊Dic53C2×D411yescharacteristic
131C4⋊Dic53C2×D411yescharacteristic
132#11719919—21no—
133#11719919—21no—
134C4×Dic53—21no—
135C4×Dic53—21no—
136C4×Dic53—21no—
137#11719917—42no—
138#11719917—42no—
139C2×#88322—22no—
140C2×#88322—22no—
141C2×#88322—22no—
142C8×D53—44no—
143C23×C53#12111yescharacteristic
144C22×D53—21no—
145Q8×C53—44no—
146Q8×C53—82no—
147Q8×C53—82no—
148Q8×C53—82no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—82no—
154C2×C4×C53—82no—
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53C2×Q1612yes—
159C2×C4×C53#13512yes—
160C2×C4×C53C22×D411yescharacteristic
161C2×C4×C53C22×D411yescharacteristic
162C2×C4×C53C2×#4111yescharacteristic
163C22⋊D53—24no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167D212—21no—
168D212C2×Q1612yes—
169C4×D53—24no—
170C4×D53—42no—
171C4×D53—42no—
172Dic106—21no—
173Dic106C2×Q1612yes—
174C8×C53—22no—
175C8×C53—22no—
176#88322—42no—
177#88322—42no—
178#88322—42no—
179#19277—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53#46712yes—
183C22×C53C2×#12111yescharacteristic
184C2×D53—22no—
185C2×D53—41no—
186C2×D53—41no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—82no—
190C4×C53—82no—
191C4×C53#46712yes—
192C4×C53#47112yes—
193C4×C53C22×Q1611yescharacteristic
194C4×C53C2×#13511yescharacteristic
195C4×C53#57512yes—
196Dic53—22no—
197Dic53—41no—
198Dic53—41no—
199C2×#487D5311yescharacteristic, complemented
200#3079—531no—
201#3079—531no—
202C8×#41—531no—
203#1684—534no—
204#1683—534no—
205#1567—531no—
206#1549—531no—
207#1548—531no—
208C2×C53—22no—
209C2×C53#304011yescharacteristic
210C2×C53C2×#46711yescharacteristic
211C2×C53C2×#47111yescharacteristic
212D53—42no—
213C2×C4⋊Q8C2×D5311yescharacteristic
214C2×C4⋊Q8C2×D5311yescharacteristic
215C4×#41—531no—
216#487—21no—
217#487—21no—
218#487C2×D5314yes—
219#477—532no—
220#477—1061no—
221#476—532no—
222#476—1061no—
223#475—532no—
224#475—1061no—
225#474—532no—
226#474—1061no—
227C8×Q8—532no—
228C8×D4—532no—
229C8×D4—1062no—
230#431—531no—
231#431—1061no—
232#428—1061no—
233#428—1061no—
234#423—531no—
235#423—531no—
236#417—1061no—
237#417—1061no—
238C2×#103—22no—
239C2×C4×C8—1061no—
240C2×C4×C8C2×D5311yescharacteristic
241#335—534no—
242#333—534no—
243#332—534no—
244C53#1927711yescharacteristic, complemented, Sylow
245C2×#41—531no—
246C22×Q8—21no—
247C22×Q8—21no—
248C2×M16—1061no—
249C2×M16—1061no—
250C22×C8—1062no—
251C22×C8C22⋊D5312yes—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8C22×D5312yes—
259C4⋊Q8C22×D5312yes—
260C4×Q8—531no—
261C4×Q8—531no—
262C4×D4—531no—
263C4×D4—531no—
264C4×D4—1061no—
265C4×D4—1061no—
266#117—531no—
267#117—1061no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C42—1061no—
273C2×C42C22×D5311yescharacteristic
274#107—1062no—
275#107—1062no—
276#106—1062no—
277#106—1062no—
278C4⋊C8—532no—
279C4⋊C8—532no—
280C4⋊C8—532no—
281C4⋊C8—1062no—
282C4⋊C8—1061no—
283C4⋊C8—1061no—
284#103—24no—
285#103—24no—
286#103—1062no—
287#103—1062no—
288#102—1062no—
289#102—1062no—
290C22⋊C8—1061no—
291C22⋊C8—1062no—
292C4×C8—1062no—
293C4×C8—1061no—
294C4×C8C22⋊D5311yescharacteristic
295C4×C8C22⋊D5311yescharacteristic
296C4×C8C22×D5312yes—
297#41—534no—
298#41—1062no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—44no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—82no—
305C2×Q8—531no—
306C2×D4—531no—
307C2×D4—1061no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—1061no—
311C22×C4—1061no—
312C22×C4—1061no—
313C22×C4D4×D5311yescharacteristic
314C22×C4#1171994711yescharacteristic
315C22×C4C2×C22⋊D5311yescharacteristic
316M16—2122no—
317M16—2122no—
318C2×C8—42no—
319C2×C8—1062no—
320C2×C8—1062no—
321C2×C8—1062no—
322C2×C8—2124no—
323C2×C8C2×C22⋊D5312yes—
324C2×C8C2×C22⋊D5312yes—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—531no—
334C4⋊C4—531no—
335C4⋊C4—1061no—
336C4⋊C4—1061no—
337C22⋊C4—1061no—
338C22⋊C4—1061no—
339C22⋊C4—1061no—
340C42—1061no—
341C42—1061no—
342C42—1061no—
343C42D4×D5311yescharacteristic
344C42#1171994711yescharacteristic
345C42C2×C22⋊D5311yescharacteristic
346C42C23×D5311yescharacteristic
347C23—1061no—
348C23#1313318811yescharacteristic
349Q8—44no—
350Q8—82no—
351Q8—82no—
352Q8—82no—
353Q8—532no—
354Q8—1061no—
355D4—532no—
356D4—1064no—
357D4—1061no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—1061no—
365C2×C4—1064no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4#1313318811yescharacteristic
371C2×C4#1313318811yescharacteristic
372C2×C4#1313318811yescharacteristic
373C2×C4Q16×D5312yes—
374C2×C4#1313324012yes—
375C2×C4C2×D4×D5311yescharacteristic
376C2×C4C22×C22⋊D5311yescharacteristic
377C2×C4C2×#1171994711yescharacteristic
378C8—22no—
379C8—22no—
380C8—2122no—
381C8—2122no—
382C8—2122no—
383C22—21no—
384C22—21no—
385C22—1062no—
386C22—2121no—
387C22—2121no—
388C22#1681086112yes—
389C22C2×#1313318811yescharacteristic
390C4—21no—
391C4—21no—
392C4—82no—
393C4—82no—
394C4—1062no—
395C4—2121no—
396C4—2121no—
397C4#1681086112yes—
398C4#1681088312yes—
399C4C2×Q16×D5311yescharacteristic
400C4C2×#1313324011yescharacteristic
401C4#1681142112yes—
402C2—22no—
403C2—2122no—
404C2#2028420511yescharacteristic
405C2C2×#1681086111yescharacteristic
406C2C2×#1681088311yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#487C2
socleC22×C53C2×#121
prime radicalC2×#11719945D4
hypercenterC2×#487D53
hypocenterC53#19277
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #20274170C2semidirect
5. #20274163C2semidirect
6. #20273474C2semidirect
7. #20273400C2semidirect
8. #20273396C2semidirect
9. C2×C53×#487C2semidirect
199. C2×#487D53semidirect
244. C53#19277semidirect

Maximal subgroups

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

subgroupindex
#202845242
C8×#117199452
#202741702
#202741632
#202734742
#202734002
#202733962
C2×C53×#4872
#1927753