Finite Group Databasebeta

← #28320256

Subgroups — #28320256

Order 13568.

866 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811638C211yesmaximal, characteristic
2C22×#13133288C211yesmaximal, characteristic
3#20291358C211yesmaximal, characteristic
4#20291280C211yesmaximal, characteristic
5#20290831C214yesmaximal, complemented
6#20290829C214yesmaximal, complemented
7#20290409C211yesmaximal, characteristic
8#20290407C211yesmaximal, characteristic
9#424×D53C211yesmaximal, characteristic
10C22×#11719936C2211yescharacteristic
11C22×#11719936C2211yescharacteristic
12C2×D53⋊M16C2211yescharacteristic
13C2×D53⋊M16C2211yescharacteristic
14#16811638—24no—
15C2×#13133288—24no—
16C2×#13133288C2212yes—
17C2×#13133288C2214yes—
18C2×#13133287C2212yes—
19#16811630—22no—
20#16811629—22no—
21#16811628C2212yes—
22#16811627C2212yes—
23#16811588—22no—
24#16811587—22no—
25C4×#11719936C2214yes—
26#16811499C2212yes—
27#16811499C2214yes—
28#16811499C2212yes—
29C2×M16×D53C2211yescharacteristic
30C22×C8×D53C2211yescharacteristic
31#117×D53C2211yescharacteristic
32#16810651C411yescharacteristic
33#16810650C411yescharacteristic
34#16810649C411yescharacteristic
35#16810648C414yes—
36C4⋊C8×D53C2214yes—
37C53×#424C411yescharacteristic
38C2×#11719936—21no—
39C2×#11719936—22no—
40C2×#11719936—22no—
41C2×#11719936—22no—
42C2×#11719936—21no—
43C2×#11719936D414yes—
44C2×#11719936C2311yescharacteristic
45C2×#11719936C2312yes—
46C2×#11719936C2311yescharacteristic
47C2×#11719936C2312yes—
48C2×#11719936C2312yes—
49D53⋊M16—22no—
50D53⋊M16—22no—
51D53⋊M16—22no—
52D53⋊M16Q814yes—
53C2×#11719940—21no—
54C2×#11719940—21no—
55C2×D53⋊C8—21no—
56C2×D53⋊C8—21no—
57C22×#88324—21no—
58C22×#88324—21no—
59#13133288—28no—
60#13133288—24no—
61#13133287—24no—
62#13133284—24no—
63C4⋊#88324—24no—
64#13133272—24no—
65C4×#88324—24no—
66C22×C4×D53C2311yescharacteristic
67C2×#11719917C2×C411yescharacteristic
68C2×#11719912C2×C412yes—
69M16×D53—22no—
70M16×D53—22no—
71C22×#88322C2×C411yescharacteristic
72C2×C8×D53—22no—
73C2×C8×D53—22no—
74C2×C8×D53C2×C412yes—
75C2×C8×D53C2312yes—
76C2×C8×D53C2312yes—
77#13133176C2×C411yescharacteristic
78#13133175D412yes—
79#13133174C2×C412yes—
80#13133173D412yes—
81C4⋊C4×D53Q812yes—
82C22⋊C4×D53Q812yes—
83C42×D53C2312yes—
84C4⋊#88322C2×C414yes—
85#13133146C2×C414yes—
86#13133146C2×C414yes—
87C2×M16×C53C2×C411yescharacteristic
88C22×C8×C53C2×C411yescharacteristic
89#117×C53C2×C411yescharacteristic
90C4⋊C8×C53C2×C414yes—
91C23×D53C2×Q811yescharacteristic
92C22×C4×C53C22×C411yescharacteristic
93C22×Dic53C2×D411yescharacteristic
94C2×C4×D53C22×C411yescharacteristic
95C2×C4×D53C2×D412yes—
96C2×C4×D53C2×D411yescharacteristic
97C2×C4×D53C2×Q814yes—
98C2×C4×D53C2×Q812yes—
99C2×C4×D53C2×Q811yescharacteristic
100C2×C4×D53#4112yes—
101C2×C4×D53#4112yes—
102C2×C4×D53C2411yescharacteristic
103#11719940—24no—
104#11719940—42no—
105C2×#88324—24no—
106C2×#88324—22no—
107C2×#88324—22no—
108C2×#88324—21no—
109C2×#88324—22no—
110C2×#88324—21no—
111C2×#88324—42no—
112#11719936—24no—
113#11719936—22no—
114#11719936—22no—
115#11719936—21no—
116#11719936—21no—
117#11719936—22no—
118#11719936—21no—
119#11719936—21no—
120#11719936—22no—
121#11719936—42no—
122#11719936C2×D414yes—
123#11719936C2×Q814yes—
124D53⋊C8—24no—
125D53⋊C8—42no—
126M16×C53—22no—
127C2×C8×C53—22no—
128C2×C8×C53C22×C412yes—
129C2×C8×C53C22×C412yes—
130C4⋊C4×C53C4⋊C412yes—
131C22⋊C4×C53C4⋊C412yes—
132C42×C53C22×C412yes—
133C22⋊Dic53C4⋊C412yes—
134#11719921C4⋊C414yes—
135C4⋊Dic53C4⋊C412yes—
136#11719919C4⋊C414yes—
137C4×Dic53C22×C412yes—
138C4×Dic53C2×D414yes—
139#11719917—22no—
140C2×#88322—22no—
141C2×#88322C22×C412yes—
142C2×#88322C22×C412yes—
143#11719912—24no—
144C8×D53—24no—
145C8×D53—22no—
146C8×D53—22no—
147C8×D53—22no—
148C8×D53—22no—
149C23×C53C2×C4⋊C411yescharacteristic
150C22×D53—42no—
151C22×D53C2×C4⋊C411yescharacteristic
152C22×D53#12512yes—
153C22×D53C4⋊Q812yes—
154C22×D53C22×Q811yescharacteristic
155C2×C4×C53C2×C4⋊C414yes—
156C2×C4×C53C2×C4⋊C411yescharacteristic
157C2×C4×C53C4×D412yes—
158C2×C4×C53C4×Q812yes—
159C2×C4×C53C23×C411yescharacteristic
160C2×Dic53C2×C4⋊C414yes—
161C2×Dic53C2×C4⋊C411yescharacteristic
162C2×Dic53#12112yes—
163C2×Dic53C22⋊Q812yes—
164C2×Dic53C22×D411yescharacteristic
165C4×D53—44no—
166C4×D53C4×D412yes—
167C4×D53C4×Q812yes—
168C4×D53#12112yes—
169C4×D53C22⋊Q812yes—
170C4×D53#12512yes—
171C4×D53C4⋊Q812yes—
172C4×D53C22×D411yescharacteristic
173C4×D53C22×Q811yescharacteristic
174C4×D53C2×#4111yescharacteristic
175C4×D53C2×#4111yescharacteristic
176#88324—24no—
177#88324—24no—
178#88324—42no—
179#88324—41no—
180#88324—41no—
181C8×C53—22no—
182C8×C53—22no—
183#88322—22no—
184#88322—22no—
185#21178—531nomaximal, Sylow
186C22×C53—22no—
187C22×C53#38912yes—
188C22×C53C22×C4⋊C411yescharacteristic
189C2×D53—22no—
190C2×D53—22no—
191C2×D53—42no—
192C2×D53—42no—
193C2×D53#38912yes—
194C2×D53C2×#12511yescharacteristic
195C2×D53C2×C4⋊Q811yescharacteristic
196C4×C53—24no—
197C4×C53#38912yes—
198C4×C53C2×C4×D411yescharacteristic
199C4×C53C2×C4×Q811yescharacteristic
200Dic53—24no—
201Dic53#38912yes—
202Dic53C2×#12111yescharacteristic
203Dic53C2×C22⋊Q811yescharacteristic
204C2×#430—531no—
205C22×#108—531no—
206#2905—531no—
207C2×#424—531no—
208#2873—531no—
209#1950—534no—
210#1950—534no—
211#1738—531no—
212C2×#341—531no—
213C2×C53—22no—
214C2×C53C2×#38911yescharacteristic
215C2×C53#291612yes—
216D53—21no—
217D53—21no—
218D53—22no—
219C22×M16—531no—
220C22×M16—531no—
221C22×M16—531no—
222C22×M16—531no—
223C22×M16—531no—
224C23×C8—531no—
225C2×#117—531no—
226#430—1064no—
227C2×#108—534no—
228C2×#108—532no—
229C2×#108—532no—
230C2×#108—1064no—
231#424—531no—
232#424—531no—
233#424—531no—
234#424—534no—
235#424—1062no—
236#424—1062no—
237#424#615711yescharacteristic
238#423—532no—
239#423—532no—
240C2×C4⋊C8—534no—
241#405—1062no—
242#405—1062no—
243C4×M16—534no—
244#341—532no—
245#341—532no—
246#341—534no—
247C53#2117811yescharacteristic, complemented, Sylow
248C23×C4—531no—
249C2×M16—531no—
250C2×M16—532no—
251C2×M16—532no—
252C2×M16—532no—
253C2×M16—534no—
254C2×M16—534no—
255C2×M16—532no—
256C2×M16—531no—
257C2×M16—531no—
258C2×M16—1062no—
259C2×M16—1062no—
260C2×M16—1061no—
261C2×M16—1061no—
262C2×M16—1062no—
263C2×M16—1062no—
264C2×M16—1061no—
265C2×M16—1061no—
266C2×M16—1062no—
267C2×M16—1062no—
268C2×M16—1062no—
269C2×M16—1062no—
270C2×M16C2×#615711yescharacteristic
271C22×C8—531no—
272C22×C8—532no—
273C22×C8—532no—
274C22×C8—532no—
275C22×C8—1062no—
276C22×C8—1062no—
277C22×C8—1061no—
278C22×C8—1061no—
279C22×C8—1061no—
280C22×C8—1061no—
281C22×C8C2×#615711yescharacteristic
282#117—531no—
283#117—532no—
284#117—532no—
285#117—532no—
286#117C2×#615711yescharacteristic
287C2×C4⋊C4—532no—
288C2×C22⋊C4—532no—
289C2×C42—532no—
290#108—1068no—
291#108—1064no—
292#108—1064no—
293C4⋊C8—534no—
294C4⋊C8—534no—
295C4⋊C8—534no—
296C4⋊C8—1064no—
297C4⋊C8—1064no—
298C4⋊C8C2×#615714yes—
299#97—1064no—
300C4×C8—1064no—
301C24—531no—
302C22×C4—531no—
303C22×C4—531no—
304C22×C4—531no—
305C22×C4—532no—
306C22×C4—532no—
307C22×C4—532no—
308C22×C4—532no—
309C22×C4—531no—
310C22×C4—531no—
311C22×C4—534no—
312C22×C4C22×#615711yescharacteristic
313M16—22no—
314M16—534no—
315M16—534no—
316M16—1064no—
317M16—1064no—
318M16—1062no—
319M16—1062no—
320M16—1064no—
321M16—1062no—
322M16—1061no—
323M16—1061no—
324M16—1062no—
325M16—1061no—
326M16—1061no—
327M16—1062no—
328M16—2122no—
329M16—2122no—
330C2×C8—22no—
331C2×C8—532no—
332C2×C8—532no—
333C2×C8—1062no—
334C2×C8—1062no—
335C2×C8—1064no—
336C2×C8—1062no—
337C2×C8—1062no—
338C2×C8—1062no—
339C2×C8—1062no—
340C2×C8—1064no—
341C2×C8—1064no—
342C2×C8—1062no—
343C2×C8—1061no—
344C2×C8—1062no—
345C2×C8—1061no—
346C2×C8—2122no—
347C2×C8—2122no—
348C2×C8C22×#615712yes—
349C2×C8C22×#615712yes—
350C4⋊C4—534no—
351C4⋊C4—532no—
352C4⋊C4C4⋊#615712yes—
353C22⋊C4—532no—
354C22⋊C4—534no—
355C22⋊C4C4⋊#615712yes—
356C42—534no—
357C42—532no—
358C42C22×#615712yes—
359C23—531no—
360C23—531no—
361C23—532no—
362C23—532no—
363C23—2122no—
364C23C2×C4⋊#615711yescharacteristic
365C2×C4—532no—
366C2×C4—532no—
367C2×C4—531no—
368C2×C4—531no—
369C2×C4—532no—
370C2×C4—532no—
371C2×C4—532no—
372C2×C4—532no—
373C2×C4—532no—
374C2×C4—531no—
375C2×C4—531no—
376C2×C4—532no—
377C2×C4—531no—
378C2×C4—531no—
379C2×C4—534no—
380C2×C4—2124no—
381C2×C4C2×C4⋊#615714yes—
382C2×C4C2×C4⋊#615711yescharacteristic
383C2×C4D4×#615712yes—
384C2×C4Q8×#615712yes—
385C2×C4C23×#615711yescharacteristic
386C8—22no—
387C8—22no—
388C8—1064no—
389C8—1064no—
390C8—1062no—
391C8—1062no—
392C8—2122no—
393C8—2121no—
394C8—2121no—
395C22—22no—
396C22—531no—
397C22—531no—
398C22—532no—
399C22—1062no—
400C22—1062no—
401C22—2122no—
402C22—2122no—
403C22#1681157412yes—
404C22C22×C4⋊#615711yescharacteristic
405C4—24no—
406C4—531no—
407C4—531no—
408C4—532no—
409C4—1064no—
410C4#1681157412yes—
411C4C2×D4×#615711yescharacteristic
412C4C2×Q8×#615711yescharacteristic
413C2—22no—
414C2—1061no—
415C2—1061no—
416C2—1062no—
417C2C2×#1681157411yescharacteristic
418C2#2029137712yes—
subgroupG/H
centerC22C22×C4⋊#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#424C4
socleC22×C53C22×C4⋊C4
prime radicalC23×D53C2×Q8
cosocleC2×C4×D53C24
hypercenter#424#6157
hypocenterC53#21178
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
5. #20290831C2semidirect
6. #20290829C2semidirect
247. C53#21178semidirect

Maximal subgroups

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

subgroupindex
C2×#168116382
C22×#131332882
#202913582
#202912802
#202908312
#202908292
#202904092
#202904072
#424×D532
#2117853