Finite Group Databasebeta

← #27302261

Subgroups — #27302261

Order 13568.

446 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283425C211yesmaximal, characteristic
2C2×#16810441C211yesmaximal, characteristic, complemented
3C2×#16810408C211yesmaximal, characteristic, complemented
4C2×#16810407C211yesmaximal, characteristic, complemented
5C2×#16810406C211yesmaximal, characteristic, complemented
6M32×Dic53C211yesmaximal, characteristic, complemented
7C2×C53×#350C211yesmaximal, characteristic, complemented
8C2×#13133223C2211yescharacteristic
9#16811169C411yescharacteristic
10C2×#13133157C2211yescharacteristic
11C22×#11719908C2211yescharacteristic
12#16810699—21no—
13#16810699—21no—
14C8×C22⋊D53—21no—
15C8×C22⋊D53—21no—
16#16810646C411yescharacteristic
17#16810591C411yescharacteristic
18#16810590C411yescharacteristic
19C2×C8×Dic53C2211yescharacteristic
20#16810441—21no—
21#16810441—21no—
22#16810408—22no—
23#16810407—22no—
24#16810406—22no—
25#16810398—21no—
26C16×Dic53—21no—
27C2×#131×C53C2211yescharacteristic
28C2×M32×C53C2211yescharacteristic
29C22×C16×C53C2211yescharacteristic
30C53×#350—22no—
31C2×#11719945C2×C411yescharacteristic
32#13133223—21no—
33#13133223—21no—
34#13133223—21no—
35#13133223—21no—
36C2×#11719917C2×C411yescharacteristic
37C2×#11719912—21no—
38C22×#88322C2×C411yescharacteristic
39C2×C8×D53—21no—
40#13133215C811yescharacteristic
41#13133213C811yescharacteristic
42#13133206C811yescharacteristic
43#13133204C811yescharacteristic
44#13133189—21no—
45#13133189—21no—
46C4×C22⋊D53—21no—
47C4×C22⋊D53—21no—
48C2×C4×Dic53C2×C411yescharacteristic
49#13133157—21no—
50#13133157D412yescomplemented
51C2×#11719908—21no—
52C2×#11719908—21no—
53C2×#11719908—21no—
54C2×#11719908—21no—
55#13133146—21no—
56#13133146—21no—
57C22⋊#88322—21no—
58C22⋊#88322—21no—
59#13133129—21no—
60#13133129—21no—
61C8×Dic53D411yescharacteristic
62C8×Dic53D411yescharacteristic
63C8×Dic53D411yescharacteristic
64C8×Dic53D411yescharacteristic
65C2×#41×C53C2×C411yescharacteristic
66#131×C53—21no—
67#131×C53—21no—
68#131×C53—21no—
69#131×C53—21no—
70C2×M16×C53—21no—
71C2×M16×C53C2×C411yescharacteristic
72C22×C8×C53—21no—
73C22×C8×C53C2311yescharacteristic
74M32×C53—21no—
75M32×C53D412yescomplemented
76C2×C16×C53—21no—
77C2×C16×C53—21no—
78C2×C16×C53—21no—
79C2×C16×C53—21no—
80#41×C53—21no—
81#41×C53—21no—
82#41×C53—21no—
83#41×C53—21no—
84C2×Q8×C53C2×C811yescharacteristic
85C2×D4×C53—21no—
86C2×D4×C53C2×C811yescharacteristic
87C22×C4×C53—21no—
88C22×C4×C53C22×C411yescharacteristic
89C2×C22⋊D53—21no—
90C2×C22⋊D53—21no—
91C2×C22⋊D53—21no—
92C22×Dic53M1611yescharacteristic
93C22×Dic53M1611yescharacteristic
94#11719945—21no—
95#11719945—21no—
96#11719945—21no—
97#11719945—21no—
98C2×D212C2×C811yescharacteristic
99C2×C4×D53—21no—
100C2×Dic106C2×C811yescharacteristic
101M16×C53—21no—
102M16×C53—21no—
103M16×C53—41no—
104M16×C53—41no—
105C2×C8×C53—21no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53#4111yescharacteristic
113C2×C8×C53#4111yescharacteristic
114C22⋊Dic53—21no—
115C22⋊Dic53—21no—
116#11719921—21no—
117#11719921—21no—
118#11719919—21no—
119#11719919—21no—
120C4×Dic53C22⋊C411yescharacteristic
121C4×Dic53C22⋊C411yescharacteristic
122C4×Dic53C2×C811yescharacteristic
123C4×Dic53C2×C811yescharacteristic
124#11719917—21no—
125#11719917—21no—
126C2×#88322—21no—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322C22⋊C411yescharacteristic
130C2×#88322C22⋊C411yescharacteristic
131#11719912—41no—
132#11719912—41no—
133C8×D53—41no—
134C8×D53—41no—
135C16×C53—22no—
136C16×C53—41no—
137#11719908—22no—
138#11719908—41no—
139C23×C53—21no—
140C23×C53C2×M1611yescharacteristic
141C22×D53—21no—
142Q8×C53—21no—
143Q8×C53—21no—
144D4×C53—21no—
145D4×C53—21no—
146D4×C53—41no—
147D4×C53—41no—
148C2×C4×C53—21no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53C2×C22⋊C411yescharacteristic
152C2×C4×C53C4×D411yescharacteristic
153C2×C4×C53C4×D411yescharacteristic
154C2×C4×C53C22×C811yescharacteristic
155C2×C4×C53#13111yescharacteristic
156C2×C4×C53#13111yescharacteristic
157C22⋊D53—42no—
158C22⋊D53—41no—
159C22⋊D53—41no—
160C22⋊D53—42no—
161C2×Dic53—22no—
162C2×Dic53—21no—
163C2×Dic53—41no—
164C2×Dic53C22⋊C811yescharacteristic
165C2×Dic53C22⋊C811yescharacteristic
166C2×Dic53C22⋊C811yescharacteristic
167C2×Dic53C22⋊C811yescharacteristic
168D212—21no—
169D212—21no—
170C4×D53—41no—
171C4×D53—41no—
172Dic106—21no—
173Dic106—21no—
174C8×C53—41no—
175C8×C53#12111yescharacteristic
176C8×C53#12111yescharacteristic
177C8×C53#12411yescharacteristic
178C8×C53#12711yescharacteristic
179#88322—21no—
180#88322—21no—
181#88322—41no—
182#13193—531nomaximal, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53C2×C22⋊C811yescharacteristic
189C22×C53#43611yescharacteristic
190C22×C53#43611yescharacteristic
191C2×D53—41no—
192C2×D53—41no—
193C2×D53—41no—
194C4×C53—41no—
195C4×C53#39011yescharacteristic
196C4×C53#40811yescharacteristic
197C4×C53C8×D411yescharacteristic
198C4×C53C8×D411yescharacteristic
199Dic53—21no—
200Dic53—21no—
201Dic53—21no—
202Dic53—41no—
203Dic53#35012yescomplemented
204#2931—531no—
205C2×C4⋊C16—531no—
206C2×#350—531no—
207C2×#350—531no—
208C2×#350—531no—
209C2×#350D5311yescharacteristic, complemented
210C4×M32—531no—
211C2×C53—21no—
212C2×C53—21no—
213C2×C53—81no—
214C2×C53#195411yescharacteristic
215C2×C53C2×#35011yescharacteristic
216C2×C53#211711yescharacteristic
217D53—81no—
218C2×#131—531no—
219C2×#131C2×D5311yescharacteristic
220#532—531no—
221C2×M32—531no—
222C2×M32C2×D5311yescharacteristic
223C22×C16—531no—
224C22×C16C2×D5311yescharacteristic
225#436—1061no—
226#436—1061no—
227C8×D4—1061no—
228C8×D4—1061no—
229#423—531no—
230#408—531no—
231#408—531no—
232C2×C4×C8—531no—
233C4⋊C16—1061no—
234C4⋊C16—1061no—
235#350—22no—
236#350—1062no—
237#350—1062no—
238#350—1062no—
239#346—1061no—
240C4×C16—1061no—
241C53#1319311yescharacteristic, complemented, Sylow
242C2×#41—531no—
243C2×#41C4×D5311yescharacteristic
244#131—21no—
245#131—21no—
246#131—21no—
247#131—21no—
248#131—1061no—
249#131—1061no—
250#131—1061no—
251#131—1061no—
252C2×M16—21no—
253C2×M16—531no—
254C2×M16—1061no—
255C2×M16C4×D5311yescharacteristic
256C22×C8—21no—
257C22×C8—531no—
258C22×C8—1061no—
259C22×C8C22×D5311yescharacteristic
260C4⋊Q8—531no—
261#127—531no—
262#124—531no—
263#124—531no—
264#121—1061no—
265#121—1061no—
266C4×D4—1061no—
267C4×D4—1061no—
268C2×C42—531no—
269M32—21no—
270M32—532no—
271M32—1061no—
272M32C22⋊D5312yescomplemented
273C2×C16—21no—
274C2×C16—21no—
275C2×C16—21no—
276C2×C16—21no—
277C2×C16—1061no—
278C2×C16—1061no—
279C2×C16—1061no—
280C2×C16—1061no—
281C4⋊C8—1061no—
282C4⋊C8—1061no—
283C22⋊C8—1061no—
284C22⋊C8—1061no—
285C22⋊C8—1061no—
286C22⋊C8—1061no—
287C4×C8—531no—
288C4×C8—531no—
289C4×C8—531no—
290C4×C8—531no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295#41—1061no—
296#41—1061no—
297#41—1061no—
298#41—1061no—
299C2×Q8—531no—
300C2×Q8C8×D5311yescharacteristic
301C2×D4—21no—
302C2×D4—531no—
303C2×D4—1061no—
304C2×D4—1061no—
305C2×D4—1061no—
306C2×D4C8×D5311yescharacteristic
307C22×C4—21no—
308C22×C4—531no—
309C22×C4—531no—
310C22×C4—1061no—
311C22×C4C2×C4×D5311yescharacteristic
312M16—21no—
313M16—21no—
314M16—41no—
315M16—41no—
316M16—1061no—
317M16—1061no—
318M16—2121no—
319M16—2121no—
320C2×C8—21no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—531no—
324C2×C8—531no—
325C2×C8—1061no—
326C2×C8—1061no—
327C2×C8—1061no—
328C2×C8—2121no—
329C2×C8—2121no—
330C2×C8#1171994511yescharacteristic
331C2×C8D4×D5311yescharacteristic
332C2×C8D4×D5311yescharacteristic
333C2×C8D4×D5311yescharacteristic
334C2×C8#1171994711yescharacteristic
335C2×C8C2×C22⋊D5311yescharacteristic
336C4⋊C4—1061no—
337C4⋊C4—1061no—
338C22⋊C4—1061no—
339C22⋊C4—1061no—
340C22⋊C4—1061no—
341C22⋊C4—1061no—
342C42—531no—
343C42—531no—
344C42—531no—
345C42—531no—
346C16—22no—
347C16—41no—
348C16—1062no—
349C16—2121no—
350C23—21no—
351C23—1061no—
352C23M16×D5311yescharacteristic
353Q8—21no—
354Q8—21no—
355Q8—1061no—
356Q8—1061no—
357D4—21no—
358D4—21no—
359D4—41no—
360D4—41no—
361D4—1061no—
362D4—1061no—
363D4—2122no—
364D4—2122no—
365D4—2121no—
366D4—2121no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—531no—
371C2×C4—531no—
372C2×C4—531no—
373C2×C4—531no—
374C2×C4—1062no—
375C2×C4—1061no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4C22⋊C4×D5311yescharacteristic
380C2×C4#1313317711yescharacteristic
381C2×C4C4×C22⋊D5311yescharacteristic
382C2×C4C2×C8×D5311yescharacteristic
383C2×C4#1313322311yescharacteristic
384C2×C4#1313322511yescharacteristic
385C8—41no—
386C8—1061no—
387C8—1061no—
388C8—2121no—
389C8#1313318911yescharacteristic
390C8#1313319011yescharacteristic
391C8#1313320311yescharacteristic
392C8#1313321311yescharacteristic
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—2121no—
399C22—2121no—
400C22—2121no—
401C22C22⋊C8×D5311yescharacteristic
402C22#1681069811yescharacteristic
403C22#1681069911yescharacteristic
404C4—41no—
405C4—532no—
406C4—1061no—
407C4—1061no—
408C4—1061no—
409C4—2121no—
410C4#1681052311yescharacteristic
411C4#1681058811yescharacteristic
412C4#1681068611yescharacteristic
413C4C8×C22⋊D5311yescharacteristic
414C2—21no—
415C2—21no—
416C2—81no—
417C2—4241no—
418C2#2027550711yescharacteristic
419C2#350×D5311yescharacteristic
420C2#2027634111yescharacteristic
subgroupG/H
centerC2×C8D4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC2×C53×#350C2
socleC22×C53C2×C22⋊C8
prime radical#16811169C4
cosocleC22×C8×C53C23
hypercenterC2×#350D53
hypocenterC53#13193
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
2. C2×#16810441C2semidirect
3. C2×#16810408C2semidirect
4. C2×#16810407C2semidirect
5. C2×#16810406C2semidirect
6. M32×Dic53C2semidirect
7. C2×C53×#350C2semidirect
50. #13133157D4semidirect
75. M32×C53D4semidirect
203. Dic53#350semidirect
209. C2×#350D53semidirect
241. C53#13193semidirect
272. M32C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202834252
C2×#168104412
C2×#168104082
C2×#168104072
C2×#168104062
M32×Dic532
C2×C53×#3502
#1319353