Finite Group Databasebeta

← #27279987

Subgroups — #27279987

Order 13568.

562 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283283C211yesmaximal, characteristic, complemented
2#20273383C211yesmaximal, characteristic, complemented
3C2×#16810367C211yesmaximal, characteristic, complemented
4#20273102C211yesmaximal, characteristic, complemented
5#20272610C211yesmaximal, characteristic, complemented
6#20272589C211yesmaximal, characteristic, complemented
7C53×#3038C211yesmaximal, characteristic
8C2×#13133224—21no—
9#16810676C2211yescharacteristic
10#16810650—22no—
11#16810650—21no—
12#16810647—21no—
13#16810647C2211yescharacteristic
14C2×C4⋊#88322—21no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17#16810367—21no—
18#16810367—21no—
19#16810367—22no—
20Q8⋊#88322—21no—
21D4⋊#88322—21no—
22C2×AGL(1,ℤ/8)×C53—21no—
23C2×#127×C53C2211yescharacteristic
24C4×#41×C53C2211yescharacteristic
25C53×#460—21no—
26C53×#460C412yes—
27C53×#459—21no—
28C53×#459C412yes—
29C53×#423C2211yescharacteristic
30C53×#418—21no—
31#13133224—42no—
32#13133224—42no—
33C2×#11719917—22no—
34C2×#11719917—21no—
35C22×#88322—22no—
36C22×#88322—21no—
37C22×#88322—21no—
38C4⋊#88322—22no—
39C4⋊#88322—22no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D411yescharacteristic
46C4⋊#88322D411yescharacteristic
47C4⋊#88322D412yes—
48C22⋊#88322—21no—
49C4×#88322—21no—
50C2×#41×C53D411yescharacteristic
51C22×D4×C53—21no—
52C22×D4×C53—21no—
53C22×D4×C53—21no—
54AGL(1,ℤ/8)×C53—42no—
55AGL(1,ℤ/8)×C53—42no—
56C2×SD16×C53—22no—
57C2×D8×C53—22no—
58C2×M16×C53—21no—
59#127×C53—21no—
60#127×C53—21no—
61#127×C53—21no—
62#127×C53C2×C412yes—
63C4×Q8×C53C2×C411yescharacteristic
64C4×Q8×C53D411yescharacteristic
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53C2×C411yescharacteristic
68C4×D4×C53D411yescharacteristic
69#117×C53—21no—
70#117×C53D411yescharacteristic
71C2×C42×C53—21no—
72C2×C42×C53C2311yescharacteristic
73C4⋊C8×C53—21no—
74C4⋊C8×C53C2×C412yes—
75#102×C53—22no—
76#102×C53—22no—
77C24×C53—41no—
78#41×C53—21no—
79#41×C53—21no—
80#41×C53D812yes—
81#41×C53SD1612yes—
82C2×Q8×C53C22⋊C411yescharacteristic
83C2×D4×C53—21no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—22no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—81no—
95C2×D4×C53—81no—
96C2×D4×C53—82no—
97C2×D4×C53C22⋊C411yescharacteristic
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C22×C4×C53#4111yescharacteristic
104SD16×C53—44no—
105D8×C53—44no—
106M16×C53—41no—
107M16×C53—41no—
108C2×C8×C53—22no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53C22⋊C411yescharacteristic
112C4⋊C4×C53C22⋊C411yescharacteristic
113C22⋊C4×C53—21no—
114C22⋊C4×C53—21no—
115C22⋊C4×C53—21no—
116C42×C53—21no—
117C42×C53—21no—
118C42×C53—21no—
119C42×C53C22×C411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123#11719917—44no—
124#11719917—41no—
125#11719917—41no—
126C2×#88322—21no—
127C2×#88322—22no—
128C2×#88322—21no—
129C2×#88322—44no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—81no—
135C23×C53—21no—
136C23×C53—42no—
137C23×C53—82no—
138C23×C53—82no—
139C23×C53—81no—
140C23×C53—81no—
141C23×C53#12311yescharacteristic
142Q8×C53—21no—
143Q8×C53#10212yes—
144D4×C53—22no—
145D4×C53—22no—
146D4×C53—21no—
147D4×C53—42no—
148D4×C53—42no—
149D4×C53—82no—
150D4×C53—82no—
151D4×C53—82no—
152D4×C53—81no—
153D4×C53—81no—
154D4×C53—81no—
155D4×C53—81no—
156D4×C53#10212yes—
157C2×C4×C53—21no—
158C2×C4×C53—22no—
159C2×C4×C53—22no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—81no—
168C2×C4×C53C2×C22⋊C411yescharacteristic
169C2×C4×C53C4×D411yescharacteristic
170C2×C4×C53C4×D411yescharacteristic
171C2×C4×C53#12011yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53C2×D811yescharacteristic
175C2×C4×C53C2×SD1611yescharacteristic
176C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C53#13711yescharacteristic
178C8×C53—42no—
179#88322—41no—
180#88322—42no—
181#88322—41no—
182#88322—81no—
183#88322—81no—
184#9173—531nomaximal, Sylow
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—22no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—82no—
191C22×C53—82no—
192C22×C53—82no—
193C22×C53—162no—
194C22×C53—162no—
195C22×C53#38611yescharacteristic
196C22×C53#48011yescharacteristic
197C22×C53#48111yescharacteristic
198C4×C53—22no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—41no—
203C4×C53—41no—
204C4×C53C2×#10211yescharacteristic
205C4×C53#41711yescharacteristic
206C4×C53#44711yescharacteristic
207C4×C53#44811yescharacteristic
208C4×C53#47111yescharacteristic
209C4×C53#47211yescharacteristic
210#3038D5311yescharacteristic
211#2906—531no—
212#1545—531no—
213C2×#331—531no—
214#1477—531no—
215#1301—531no—
216#1292—531no—
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—42no—
220C2×C53—162no—
221C2×C53#187411yescharacteristic
222C2×C53#190311yescharacteristic
223C2×C53#190711yescharacteristic
224C2×AGL(1,ℤ/8)—21no—
225C2×#131—1061no—
226C2×#127C2×D5311yescharacteristic
227C4×#41C2×D5311yescharacteristic
228#460—21no—
229#460Dic5312yes—
230#459—21no—
231#459Dic5312yes—
232#431—531no—
233#424—1062no—
234#424—1061no—
235#423—531no—
236#423—1061no—
237#423C2×D5311yescharacteristic
238C2×C4⋊C8—531no—
239C2×C4⋊C8—531no—
240C2×C4⋊C8—1061no—
241#418—21no—
242#331—1061no—
243#331—1061no—
244#331—1062no—
245Q8⋊C8—1061no—
246D4⋊C8—1061no—
247C53#917311yescharacteristic, complemented, Sylow
248C2×#41C22⋊D5311yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252AGL(1,ℤ/8)—42no—
253AGL(1,ℤ/8)—42no—
254C2×SD16—22no—
255C2×D8—22no—
256#131—2122no—
257#131—2122no—
258C2×M16—21no—
259C2×M16—1062no—
260C2×M16—1061no—
261C22×C8—1062no—
262C22×C8—1061no—
263C22×C8—1061no—
264#127—21no—
265#127—21no—
266#127—21no—
267#127C2×Dic5312yes—
268C4×Q8C2×Dic5311yescharacteristic
269C4×Q8C22⋊D5311yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C2×Dic5311yescharacteristic
273C4×D4C22⋊D5311yescharacteristic
274#117—21no—
275#117C22⋊D5311yescharacteristic
276C2×C42—21no—
277C2×C42C22×D5311yescharacteristic
278C4⋊C8—21no—
279C4⋊C8—531no—
280C4⋊C8—531no—
281C4⋊C8—532no—
282C4⋊C8—1062no—
283C4⋊C8—1062no—
284C4⋊C8—1061no—
285C4⋊C8—1061no—
286C4⋊C8—1061no—
287C4⋊C8—1061no—
288C4⋊C8—1061no—
289C4⋊C8C2×Dic5312yes—
290#102—22no—
291#102—22no—
292C22⋊C8—1061no—
293C4×C8—1061no—
294C24—41no—
295#41—21no—
296#41—21no—
297#41#1171992212yes—
298#41#1171992312yes—
299C2×Q8C22⋊Dic5311yescharacteristic
300C2×D4—21no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—81no—
312C2×D4—81no—
313C2×D4—82no—
314C2×D4C22⋊Dic5311yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4#1171994711yescharacteristic
319C22×C4#1171994711yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321SD16—44no—
322D8—44no—
323M16—41no—
324M16—41no—
325M16—2124no—
326M16—2121no—
327M16—2121no—
328C2×C8—22no—
329C2×C8—1061no—
330C2×C8—1061no—
331C2×C8—1062no—
332C2×C8—4241no—
333C2×C8—2121no—
334C2×C8—2121no—
335C2×C8—2124no—
336C2×C8—2121no—
337C2×C8—2121no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4C22⋊Dic5311yescharacteristic
341C4⋊C4C22⋊Dic5311yescharacteristic
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42D4×D5311yescharacteristic
349C42D4×D5311yescharacteristic
350C42C22×Dic5311yescharacteristic
351C42C2×C22⋊D5311yescharacteristic
352C23—21no—
353C23—42no—
354C23—82no—
355C23—82no—
356C23—81no—
357C23—81no—
358C23#1313319811yescharacteristic
359Q8—21no—
360Q8#1313313812yes—
361D4—22no—
362D4—22no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—82no—
367D4—82no—
368D4—82no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4#1313313812yes—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—81no—
385C2×C4C2×C22⋊Dic5311yescharacteristic
386C2×C4D4×Dic5311yescharacteristic
387C2×C4D4×Dic5311yescharacteristic
388C2×C4#1313318511yescharacteristic
389C2×C4#1313318911yescharacteristic
390C2×C4#1313318911yescharacteristic
391C2×C4C2×#1171992211yescharacteristic
392C2×C4C2×#1171992311yescharacteristic
393C2×C4#1313324211yescharacteristic
394C2×C4#1313324711yescharacteristic
395C8—42no—
396C8—4241no—
397C8—4241no—
398C8—2121no—
399C8—2122no—
400C8—2121no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—162no—
410C22—162no—
411C22#1681050311yescharacteristic
412C22#1681093411yescharacteristic
413C22#1681093911yescharacteristic
414C4—21no—
415C4—22no—
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4C2×#1313313811yescharacteristic
421C4#1681062411yescharacteristic
422C4#1681075611yescharacteristic
423C4#1681076111yescharacteristic
424C4#1681088411yescharacteristic
425C4#1681089111yescharacteristic
426C2—21no—
427C2—21no—
428C2—42no—
429C2—162no—
430C2#2027503311yescharacteristic
431C2#2027519711yescharacteristic
432C2#2027522511yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3038C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3038D53
hypocenterC53#9173
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #20283283C2semidirect
2. #20273383C2semidirect
3. C2×#16810367C2semidirect
4. #20273102C2semidirect
5. #20272610C2semidirect
6. #20272589C2semidirect
247. C53#9173semidirect

Maximal subgroups

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

subgroupindex
#202832832
#202733832
C2×#168103672
#202731022
#202726102
#202725892
C53×#30382
#917353