Finite Group Databasebeta

← #27367247

Subgroups — #27367247

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285637C211yesmaximal, characteristic, complemented
2C2×#16810969C211yesmaximal, characteristic, complemented
3#20283427C211yesmaximal, characteristic, complemented
4#20274274C214yesmaximal, complemented
5#20274150C214yesmaximal, complemented
6#20273461C211yesmaximal, characteristic, complemented
7#20273399C211yesmaximal, characteristic, complemented
8#20273383C211yesmaximal, characteristic, complemented
9C53×#3205C211yesmaximal, characteristic
10C2×#13133224—21no—
11C2×#13133217C2211yescharacteristic
12#16810995—21no—
13#16810975—21no—
14#16810969—21no—
15#16810969—21no—
16#16810969C2212yes—
17#16810969C2212yes—
18#16810828C2214yescomplemented
19#16810761—22no—
20#16810701—21no—
21#16810701C2212yescomplemented
22D4×#88322—21no—
23D4×#88322C2212yescomplemented
24#16810676C2211yescharacteristic
25#16810647C2211yescharacteristic
26C2×#13133138—21no—
27C2×#13133138—21no—
28#16810590—21no—
29C2×C4×#88322C2211yescharacteristic
30#16810375C2212yescomplemented
31#16810373C2212yescomplemented
32#16810370C2214yescomplemented
33#16810367C2214yes—
34C53×#532C2211yescharacteristic
35C2×#127×C53C2211yescharacteristic
36C53×#493—21no—
37C53×#486—21no—
38C53×#459C2214yes—
39C53×#447—22no—
40C53×#431C2211yescharacteristic
41C22×Dic106—21no—
42C2×#11719923—24no—
43#13133224—44no—
44C2×#11719917—22no—
45C2×#11719917—21no—
46C22×#88322—22no—
47C22×#88322D411yescharacteristic
48C22×#88322D411yescharacteristic
49#13133217—21no—
50#13133217—21no—
51#13133217—21no—
52#13133217C2312yes—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—21no—
55C4⋊#88322—21no—
56C4⋊#88322C2312yes—
57C4⋊#88322C2312yes—
58#13133140—24no—
59#13133138—22no—
60#13133138—22no—
61#13133138—22no—
62#13133138—22no—
63C22⋊#88322—22no—
64C22⋊#88322—22no—
65C22⋊#88322—21no—
66C4×#88322—21no—
67C4×#88322D411yescharacteristic
68C4×#88322D411yescharacteristic
69C4×#88322C2311yescharacteristic
70C4×#88322C2311yescharacteristic
71C2×#41×C53—21no—
72C22×D4×C53—21no—
73C22×D4×C53—21no—
74C22×D4×C53—21no—
75C2×D8×C53—24no—
76C4⋊Q8×C53D411yescharacteristic
77#127×C53—21no—
78#127×C53—21no—
79#127×C53—21no—
80#127×C53D411yescharacteristic
81#127×C53C2312yes—
82#124×C53—21no—
83#121×C53—21no—
84#121×C53D412yes—
85C4×D4×C53—21no—
86C4×D4×C53C2312yes—
87C2×C42×C53C2311yescharacteristic
88C4⋊C8×C53C2312yes—
89#102×C53—24no—
90C22⋊C8×C53—21no—
91C4×C8×C53—21no—
92C24×C53—41no—
93#41×C53—42no—
94#41×C53—42no—
95C2×Q8×C53—21no—
96C2×D4×C53—22no—
97C2×D4×C53—21no—
98C2×D4×C53—21no—
99C2×D4×C53—22no—
100C2×D4×C53—22no—
101C2×D4×C53—22no—
102C2×D4×C53—41no—
103C2×D4×C53—41no—
104C2×D4×C53—41no—
105C2×D4×C53—42no—
106C2×D4×C53—42no—
107C2×D4×C53—82no—
108C2×D4×C53—82no—
109C2×D4×C53C2×D412yes—
110C22×C4×C53—22no—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×Dic53—41no—
115C2×Dic106—22no—
116C2×Dic106—41no—
117C2×Dic106—82no—
118D8×C53—84no—
119C2×C8×C53—22no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53C2×D412yes—
122C22⋊C4×C53—22no—
123C22⋊C4×C53—22no—
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53#4111yescharacteristic
127C42×C53C2411yescharacteristic
128#11719923—84no—
129C4⋊Dic53—22no—
130C4⋊Dic53—22no—
131C4⋊Dic53—41no—
132C4⋊Dic53—41no—
133#11719917—44no—
134#11719917—42no—
135C2×#88322—22no—
136C2×#88322—22no—
137C2×#88322—22no—
138C2×#88322—22no—
139C2×#88322—44no—
140C2×#88322C2×D411yescharacteristic
141C2×#88322C2×D411yescharacteristic
142C2×#88322C2×D411yescharacteristic
143C2×#88322C2×D411yescharacteristic
144C23×C53—22no—
145C23×C53—42no—
146C23×C53—82no—
147C23×C53—82no—
148C23×C53—82no—
149C23×C53#12111yescharacteristic
150Q8×C53—42no—
151D4×C53—24no—
152D4×C53—44no—
153D4×C53—42no—
154D4×C53—42no—
155D4×C53—44no—
156D4×C53—42no—
157D4×C53—42no—
158D4×C53—82no—
159D4×C53—82no—
160D4×C53—82no—
161D4×C53—82no—
162C2×C4×C53—22no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—42no—
166C2×C4×C53—42no—
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#13512yes—
171C2×C4×C53AGL(1,ℤ/8)12yes—
172C2×C4×C53C22×D411yescharacteristic
173C2×C4×C53C22×D411yescharacteristic
174C2×C4×C53C2×#4111yescharacteristic
175C2×Dic53—42no—
176C2×Dic53—82no—
177Dic106—44no—
178Dic106—82no—
179C8×C53—42no—
180#88322—21no—
181#88322—21no—
182#88322—21no—
183#88322—21no—
184#88322—42no—
185#88322—42no—
186#19198—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—42no—
190C22×C53—42no—
191C22×C53—42no—
192C22×C53—82no—
193C22×C53—82no—
194C22×C53—82no—
195C22×C53—164no—
196C22×C53#46312yes—
197C22×C53C2×#12111yescharacteristic
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—42no—
201C4×C53#46312yes—
202C4×C53#47112yes—
203C4×C53C2×#13511yescharacteristic
204C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
205C4×C53#57612yes—
206Dic53—82no—
207#3206—531no—
208#3205D5311yescharacteristic
209C2×#486—531no—
210#2931—531no—
211#1701—534no—
212#1681—534no—
213#1564—531no—
214#1549—531no—
215#1545—531no—
216C2×C53—22no—
217C2×C53—82no—
218C2×C53—162no—
219C2×C53C2×#46311yescharacteristic
220C2×C53#305811yescharacteristic
221C2×C53C2×#47111yescharacteristic
222C2×#131—1061no—
223#532C2×D5311yescharacteristic
224C2×C4⋊Q8—531no—
225C2×#127C2×D5311yescharacteristic
226#493—21no—
227#492—1061no—
228#487—1061no—
229#486—21no—
230#486—532no—
231#486—532no—
232#486—1061no—
233#486—1061no—
234#461—534no—
235#459C2×D5314yes—
236#448—1062no—
237#447—22no—
238#436—532no—
239#436—1061no—
240C8×D4—532no—
241C8×D4—1061no—
242#431—531no—
243#431C2×D5311yescharacteristic
244#423—531no—
245C2×#102—1061no—
246C2×#102—1061no—
247#408—1061no—
248C2×C4×C8—531no—
249#335—532no—
250#334—532no—
251#332—534no—
252#331—534no—
253C53#1919811yescharacteristic, complemented, Sylow
254C2×#41—21no—
255C22×Q8—1061no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—21no—
259C2×SD16—1064no—
260C2×D8—24no—
261#131—2124no—
262C2×M16—1062no—
263C2×M16—1061no—
264C22×C8—531no—
265C22×C8—531no—
266C22×C8—1062no—
267C4⋊Q8—532no—
268C4⋊Q8—1061no—
269C4⋊Q8—1061no—
270C4⋊Q8—1061no—
271C4⋊Q8C22⋊D5311yescharacteristic
272#127—21no—
273#127—21no—
274#127—21no—
275#127C22⋊D5311yescharacteristic
276#127C22×D5312yes—
277#124—21no—
278#121—21no—
279#121C22⋊D5312yes—
280C4×D4—21no—
281C4×D4C22×D5312yes—
282C2×C4⋊C4—1061no—
283C2×C4⋊C4—1061no—
284C2×C42C22×D5311yescharacteristic
285C4⋊C8—532no—
286C4⋊C8—532no—
287C4⋊C8—1061no—
288C4⋊C8C22×D5312yes—
289#103—1064no—
290#102—24no—
291#102—1062no—
292#102—1062no—
293#102—1062no—
294#102—1062no—
295C22⋊C8—21no—
296C22⋊C8—1062no—
297C22⋊C8—1062no—
298C22⋊C8—1061no—
299C4×C8—21no—
300C4×C8—531no—
301C4×C8—531no—
302C4×C8—531no—
303C4×C8—531no—
304C4×C8—1061no—
305C24—41no—
306#41—42no—
307#41—42no—
308C2×Q8—21no—
309C2×Q8—1062no—
310C2×Q8—4242no—
311C2×Q8—2121no—
312C2×D4—22no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—82no—
324C2×D4—82no—
325C2×D4C2×C22⋊D5312yes—
326C22×C4—22no—
327C22×C4—2121no—
328C22×C4D4×D5311yescharacteristic
329C22×C4#1171994711yescharacteristic
330C22×C4C2×C22⋊D5311yescharacteristic
331SD16—4244no—
332D8—84no—
333M16—2124no—
334M16—2122no—
335C2×C8—22no—
336C2×C8—531no—
337C2×C8—531no—
338C2×C8—531no—
339C2×C8—531no—
340C2×C8—1062no—
341C2×C8—1062no—
342C2×C8—1062no—
343C2×C8—1062no—
344C2×C8—2124no—
345C4⋊C4—21no—
346C4⋊C4—1062no—
347C4⋊C4—1062no—
348C4⋊C4—2121no—
349C4⋊C4—2121no—
350C4⋊C4C2×C22⋊D5312yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C42D4×D5311yescharacteristic
354C42#1171994711yescharacteristic
355C42C2×C22⋊D5311yescharacteristic
356C42C23×D5311yescharacteristic
357C23—22no—
358C23—42no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23#1313318911yescharacteristic
363Q8—42no—
364Q8—4242no—
365Q8—2124no—
366D4—24no—
367D4—44no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—42no—
372D4—44no—
373D4—82no—
374D4—82no—
375D4—82no—
376D4—82no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—4242no—
383C2×C4—2122no—
384C2×C4#1313318911yescharacteristic
385C2×C4#1313318911yescharacteristic
386C2×C4#1313318911yescharacteristic
387C2×C4#1313323612yes—
388C2×C4#1313324112yes—
389C2×C4C2×D4×D5311yescharacteristic
390C2×C4C22×C22⋊D5311yescharacteristic
391C2×C4C2×#1171994711yescharacteristic
392C8—42no—
393C8—1061no—
394C8—1061no—
395C8—1061no—
396C8—1061no—
397C8—2122no—
398C8—2122no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—164no—
408C22#1681083612yes—
409C22C2×#1313318911yescharacteristic
410C4—21no—
411C4—21no—
412C4—42no—
413C4—4242no—
414C4#1681083612yes—
415C4#1681088412yes—
416C4C2×#1313323611yescharacteristic
417C4C2×#1313324111yescharacteristic
418C4#1681142612yes—
419C2—22no—
420C2—82no—
421C2—162no—
422C2C2×#1681083611yescharacteristic
423C2#2028436011yescharacteristic
424C2C2×#1681088411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3205C2
socleC22×C53C2×#121
hypercenter#3205D53
hypocenterC53#19198
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #20285637C2semidirect
2. C2×#16810969C2semidirect
3. #20283427C2semidirect
4. #20274274C2semidirect
5. #20274150C2semidirect
6. #20273461C2semidirect
7. #20273399C2semidirect
8. #20273383C2semidirect
18. #16810828C22semidirect
21. #16810701C22semidirect
23. D4×#88322C22semidirect
30. #16810375C22semidirect
31. #16810373C22semidirect
32. #16810370C22semidirect
253. C53#19198semidirect

Maximal subgroups

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

subgroupindex
#202856372
C2×#168109692
#202834272
#202742742
#202741502
#202734612
#202733992
#202733832
C53×#32052
#1919853