Finite Group Databasebeta

← #27400295

Subgroups — #27400295

Order 13568.

1062 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811346C211yesmaximal, characteristic
2#20283450C212yesmaximal
3#20275703C212yesmaximal, complemented
4#20275180C214yesmaximal, complemented
5#20274797C212yesmaximal, complemented
6C4×#13133136C211yesmaximal, characteristic, complemented
7#20274452C212yesmaximal, complemented
8C53×#1801C211yesmaximal, characteristic, complemented
9C2×M16×D53C2214yes—
10#16811346—22no—
11#16811346—24no—
12#16811346—22no—
13C2×#13133217C2211yescharacteristic
14C2×#13133212C414yes—
15C2×#13133205C412yes—
16C2×#13133193C2212yes—
17C2×C4×Dic106C412yes—
18C2×C4×D212C2211yescharacteristic
19C2×#13133175C2212yes—
20#16810700—22no—
21#16810693—24no—
22#16810688—22no—
23C2×#13133136C2212yes—
24#16810586C2214yes—
25C4×#11719917C2212yes—
26#16810392C2211yescharacteristic
27#16810392C2211yescharacteristic
28#16810391C2214yes—
29#16810391C2214yes—
30C2×C4×Q8×C53C2211yescharacteristic
31C53×#423C2212yes—
32C2×#101×C53C2212yes—
33C53×#343C2212yes—
34C22×Dic106C2×C412yes—
35C22×D212C2×C411yescharacteristic
36C22×C4×D53C2312yes—
37C2×#11719917C2312yes—
38C2×#11719917C2312yes—
39C2×#11719912—24no—
40M16×D53—28no—
41M16×D53—44no—
42C2×C8×D53—24no—
43#13133217—22no—
44#13133217—22no—
45#13133212—28no—
46#13133212—28no—
47#13133205—24no—
48#13133205—24no—
49#13133193—24no—
50#13133193—24no—
51C4×Dic106—24no—
52C4×Dic106—22no—
53C4×Dic106—22no—
54C4×D212D412yes—
55C4×D212D414yes—
56C4×D212D412yes—
57#13133175—24no—
58#13133175—24no—
59C2×C4⋊Dic53C2×C412yes—
60C2×C4⋊Dic53C2×C411yescharacteristic
61C2×C4⋊Dic53C2312yes—
62C2×#11719919C2×C414yes—
63C2×#11719921C2×C414yes—
64C2×#11719921C2×C414yes—
65C2×#11719921C2×C412yes—
66C2×C4×Dic53C2×C414yes—
67#13133136—24no—
68#13133129—24no—
69#13133129—24no—
70#13133128—22no—
71C4×#88322—22no—
72C22×Q8×C53C2311yescharacteristic
73C2×M16×C53C2314yes—
74C4×Q8×C53—22no—
75C4×Q8×C53—22no—
76C2×C4⋊C4×C53C2×C412yes—
77C2×C4⋊C4×C53C2311yescharacteristic
78C2×C42×C53C2×C412yes—
79C2×C42×C53C2311yescharacteristic
80C4⋊C8×C53—22no—
81C4⋊C8×C53—22no—
82#101×C53—42no—
83C23×D53C22×C412yes—
84C2×Q8×C53—22no—
85C2×Q8×C53—22no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C22×C4×C53C22×C412yes—
89C22×C4×C53C22×C411yescharacteristic
90C22×C4×C53C22×C412yes—
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C2411yescharacteristic
93C22×Dic53C22×C412yes—
94C22×Dic53C22×C414yes—
95C2×D212—42no—
96C2×D212C22⋊C412yes—
97C2×D212C22⋊C412yes—
98C2×C4×D53—24no—
99C2×C4×D53—22no—
100C2×C4×D53—22no—
101C2×C4×D53C2×D414yes—
102C2×C4×D53C2×D414yes—
103C2×Dic106—22no—
104C2×Dic106—22no—
105C2×Dic106—44no—
106M16×C53—24no—
107M16×C53—24no—
108C2×C8×C53—24no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53—22no—
111C42×C53—22no—
112C42×C53—22no—
113C42×C53C2×D412yes—
114C42×C53C2×D412yes—
115#11719921—28no—
116#11719921—28no—
117#11719921C22⋊C414yes—
118#11719921C22⋊C414yes—
119C4⋊Dic53—24no—
120C4⋊Dic53—24no—
121C4⋊Dic53C22⋊C412yes—
122C4⋊Dic53C22⋊C411yescharacteristic
123C4⋊Dic53C22⋊C411yescharacteristic
124#11719919—24no—
125#11719919—24no—
126C4×Dic53—24no—
127C4×Dic53—24no—
128#11719917—22no—
129#11719917—22no—
130#11719917—22no—
131#11719917#4114yes—
132C2×#88322—22no—
133C2×#88322—22no—
134#11719912—48no—
135C8×D53—48no—
136C23×C53C23×C411yescharacteristic
137C22×D53—22no—
138C22×D53—22no—
139C22×D53—42no—
140C22×D53—42no—
141C22×D53C2×C22⋊C414yes—
142Q8×C53—42no—
143Q8×C53—42no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53C2×C22⋊C411yescharacteristic
152C2×C4×C53C2×C22⋊C412yes—
153C2×C4×C53C2×C22⋊C411yescharacteristic
154C2×C4×C53C22×D412yes—
155C2×C4×C53C2×#4112yes—
156C2×C4×C53H(5,𝔽2)11yescharacteristic
157C2×C4×C532−1+411yescharacteristic
158C2×Dic53—22no—
159C2×Dic53—22no—
160C2×Dic53—24no—
161C2×Dic53—44no—
162C2×Dic53C2×C22⋊C412yes—
163C2×Dic53C2×C22⋊C412yes—
164D212—24no—
165D212—42no—
166C4×D53—24no—
167C4×D53—24no—
168C4×D53—24no—
169C4×D53—24no—
170Dic106—44no—
171Dic106—44no—
172C8×C53—44no—
173#88322—24no—
174#88322—42no—
175#22484—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53C22×C22⋊C411yescharacteristic
179C22×C53C4×#4111yescharacteristic
180C22×C53#52011yescharacteristic
181C2×D53—24no—
182C2×D53—24no—
183C2×D53—24no—
184C2×D53—82no—
185C2×D53—82no—
186C4×C53—21no—
187C4×C53—22no—
188C4×C53—42no—
189C4×C53—42no—
190C4×C53#41312yes—
191C4×C53#54612yes—
192C4×C53#54712yes—
193Dic53—22no—
194Dic53—22no—
195Dic53—44no—
196C2×#556—531no—
197D4×M16—532no—
198#1997—532no—
199#1901—534no—
200#1822—532no—
201#1801D5311yescharacteristic, complemented
202C4×#101—531no—
203#1742—532no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53#233911yescharacteristic
207C2×C53C2×#41311yescharacteristic
208C2×C53#288911yescharacteristic
209D53—44no—
210C22×M16—534no—
211#556—1062no—
212#556—1064no—
213#556—1062no—
214C2×C4⋊Q8—531no—
215C2×#126—534no—
216C2×#124—532no—
217C2×C22⋊Q8—532no—
218C2×C4×Q8—532no—
219C2×C4×Q8C2×D5311yescharacteristic
220C2×C4×D4—531no—
221C2×#117—532no—
222#436—1062no—
223#435—1064no—
224C8×D4—1062no—
225#423C2×D5312yes—
226C2×#101—532no—
227C2×#101C2×D5312yes—
228#407—534no—
229C4×M16—532no—
230#343—534no—
231#343—534no—
232#343—531no—
233#343—531no—
234#343C2×D5312yes—
235C53#2248411yescharacteristic, complemented, Sylow
236C22×Q8—532no—
237C22×Q8C22×D5311yescharacteristic
238C22×D4—531no—
239C23×C4—532no—
240C2×M16—532no—
241C2×M16—532no—
242C2×M16—1064no—
243C2×M16—1068no—
244C2×M16—2124no—
245C2×M16C22×D5314yes—
246C22×C8—1064no—
247C4⋊Q8—1062no—
248C4⋊Q8—1062no—
249#126—1068no—
250#126—1068no—
251#124—1064no—
252#124—1064no—
253C22⋊Q8—1064no—
254C22⋊Q8—1064no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—1064no—
258C4×Q8—1062no—
259C4×Q8—1062no—
260C4×D4—532no—
261C4×D4—534no—
262C4×D4—532no—
263#117—1064no—
264#117—1064no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—534no—
267C2×C4⋊C4—532no—
268C2×C4⋊C4—532no—
269C2×C4⋊C4C4×D5312yes—
270C2×C4⋊C4C22×D5311yescharacteristic
271C2×C22⋊C4—532no—
272C2×C22⋊C4—534no—
273C2×C22⋊C4—534no—
274C2×C42—534no—
275C2×C42C4×D5312yes—
276C2×C42C22×D5311yescharacteristic
277C4⋊C8—22no—
278C4⋊C8—22no—
279#101—42no—
280#101—1064no—
281C22⋊C8—1064no—
282C22⋊C8—1064no—
283#97—1062no—
284C4×C8—1062no—
285C24—532no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—1062no—
291C2×Q8—1062no—
292C2×Q8—2124no—
293C2×D4—532no—
294C2×D4—532no—
295C2×D4—2122no—
296C22×C4—532no—
297C22×C4—534no—
298C22×C4—534no—
299C22×C4—534no—
300C22×C4—1064no—
301C22×C4—1062no—
302C22×C4—1062no—
303C22×C4C2×C4×D5312yes—
304C22×C4C2×C4×D5311yescharacteristic
305C22×C4C2×C4×D5312yes—
306C22×C4C2×C4×D5311yescharacteristic
307C22×C4C23×D5311yescharacteristic
308M16—24no—
309M16—24no—
310M16—534no—
311M16—1062no—
312M16—1062no—
313M16—1062no—
314M16—2128no—
315C2×C8—24no—
316C2×C8—1062no—
317C2×C8—1062no—
318C2×C8—2128no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—532no—
322C4⋊C4—531no—
323C4⋊C4—531no—
324C4⋊C4—1064no—
325C4⋊C4—1064no—
326C4⋊C4—1064no—
327C4⋊C4—1064no—
328C22⋊C4—534no—
329C22⋊C4—534no—
330C22⋊C4—1068no—
331C22⋊C4—1068no—
332C42—22no—
333C42—22no—
334C42—1064no—
335C42—1064no—
336C42D4×D5312yes—
337C42D4×D5312yes—
338C23—534no—
339C23—1062no—
340C23—1062no—
341C23—2122no—
342C23—2122no—
343C23C22×C4×D5311yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—2124no—
347Q8—2124no—
348D4—1064no—
349D4—2122no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—532no—
358C2×C4—532no—
359C2×C4—1062no—
360C2×C4—1062no—
361C2×C4—1064no—
362C2×C4—1064no—
363C2×C4—1064no—
364C2×C4—1064no—
365C2×C4—1064no—
366C2×C4—2124no—
367C2×C4C22⋊C4×D5311yescharacteristic
368C2×C4C22⋊C4×D5312yes—
369C2×C4C22⋊C4×D5311yescharacteristic
370C2×C4C2×D4×D5312yes—
371C2×C4#41×D5312yes—
372C2×C4#1313326211yescharacteristic
373C2×C4#1313326511yescharacteristic
374C8—44no—
375C8—1064no—
376C8—2122no—
377C22—21no—
378C22—21no—
379C22—1064no—
380C22—1064no—
381C22—1064no—
382C22—4242no—
383C22—4242no—
384C22C2×C22⋊C4×D5311yescharacteristic
385C22#1681109111yescharacteristic
386C22#1681110411yescharacteristic
387C4—21no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—1062no—
392C4—1062no—
393C4—2124no—
394C4#1681060612yes—
395C4#1681127012yes—
396C4#1681127512yes—
397C2—21no—
398C2—21no—
399C2—2124no—
400C2#2027754311yescharacteristic
401C2C2×#1681060611yescharacteristic
402C2#2028319611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1801C2
socleC22×C53C22×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC22×C4×C53C24
hypercenter#1801D53
hypocenterC53#22484
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
3. #20275703C2semidirect
4. #20275180C2semidirect
5. #20274797C2semidirect
6. C4×#13133136C2semidirect
7. #20274452C2semidirect
8. C53×#1801C2semidirect
201. #1801D53semidirect
235. C53#22484semidirect

Maximal subgroups

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

subgroupindex
C2×#168113462
#202834502
#202757032
#202751802
#202747972
C4×#131331362
#202744522
C53×#18012
#2248453