Finite Group Databasebeta

← #27290816

Subgroups — #27290816

Order 13568.

1506 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133144C214yesmaximal, complemented
2#20274409C211yesmaximal, characteristic, complemented
3C4×#13133124C211yesmaximal, characteristic, complemented
4C2×C53×#532C211yesmaximal, characteristic
5C22×#11719917C2212yes—
6C22×C4×Dic53C2212yes—
7C2×#13133144—28no—
8C2×#13133144—28no—
9C2×#13133144—28no—
10C2×#13133124—22no—
11#16810378—22no—
12C22×#41×C53C2212yes—
13C53×#532—28no—
14C2×C4⋊Q8×C53C411yescharacteristic, complemented
15C2×#127×C53C411yescharacteristic, complemented
16C2×#124×C53C412yescomplemented
17C2×#121×C53—22no—
18C2×C4×D4×C53—22no—
19C22×C42×C53C2211yescharacteristic
20C23×Dic53—22no—
21C2×#11719917—24no—
22C2×#11719917—24no—
23C2×#11719917—22no—
24C2×#11719917D414yescomplemented
25C22×#88322—22no—
26C2×C4×Dic53—22no—
27C2×C4×Dic53—24no—
28C2×C4×Dic53—24no—
29C2×C4×Dic53D414yescomplemented
30#13133144—232no—
31#13133124—28no—
32C2×#41×C53—24no—
33C2×#41×C53—24no—
34C2×#41×C53—24no—
35C2×#41×C53—24no—
36C2×#41×C53—24no—
37C2×#41×C53D418yes—
38C22×Q8×C53C2×C412yes—
39C22×D4×C53—22no—
40C22×D4×C53—22no—
41C22×D4×C53C2×C412yes—
42C23×C4×C53—22no—
43C23×C4×C53D412yes—
44C23×C4×C53C2311yescharacteristic
45C4⋊Q8×C53—24no—
46#127×C53—24no—
47#124×C53—24no—
48#124×C53—24no—
49#121×C53—216no—
50C4×D4×C53—216no—
51C2×C4⋊C4×C53—22no—
52C2×C22⋊C4×C53—24no—
53C2×C42×C53—24no—
54C2×C42×C53C2×C411yescharacteristic
55C2×C42×C53C2×C411yescharacteristic
56C2×C42×C53D412yes—
57C24×C53—22no—
58C24×C53C2×D411yescharacteristic
59#41×C53—28no—
60#41×C53—28no—
61#41×C53—28no—
62#41×C53—28no—
63#41×C53—44no—
64#41×C53—44no—
65#41×C53—44no—
66#41×C53—44no—
67C2×Q8×C53—22no—
68C2×Q8×C53—24no—
69C2×Q8×C53—24no—
70C2×Q8×C53C22⋊C414yes—
71C2×D4×C53—28no—
72C2×D4×C53—28no—
73C2×D4×C53—22no—
74C2×D4×C53—24no—
75C2×D4×C53—24no—
76C2×D4×C53—44no—
77C2×D4×C53—44no—
78C2×D4×C53—42no—
79C2×D4×C53—42no—
80C2×D4×C53—44no—
81C2×D4×C53C22⋊C414yes—
82C22×C4×C53—22no—
83C22×C4×C53—24no—
84C22×C4×C53—24no—
85C22×C4×C53—28no—
86C22×C4×C53—42no—
87C22×C4×C53—44no—
88C22×C4×C53—42no—
89C22×C4×C53—42no—
90C22×C4×C53C22⋊C412yes—
91C22×C4×C53C22⋊C412yes—
92C22×C4×C53C22×C411yescharacteristic
93C22×C4×C53C2×D414yes—
94C22×C4×C53C2×D412yes—
95C22×C4×C53C2×D412yes—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53#4112yes—
98C22×C4×C53#4112yes—
99C22×Dic53—28no—
100C22×Dic53—24no—
101C22×Dic53—44no—
102C22×Dic53—42no—
103C22×Dic53—42no—
104C4⋊C4×C53—28no—
105C22⋊C4×C53—216no—
106C42×C53—22no—
107C42×C53—22no—
108C42×C53—24no—
109C4×Dic53—28no—
110C4×Dic53—44no—
111#11719917—28no—
112#11719917—44no—
113C2×#88322—24no—
114C2×#88322—42no—
115C2×#88322—42no—
116C23×C53—22no—
117C23×C53—22no—
118C23×C53—24no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—44no—
122C23×C53—44no—
123C23×C53C2×C22⋊C411yescharacteristic
124C23×C53#12012yes—
125C23×C53#12112yes—
126C23×C53#12411yescharacteristic
127C23×C53#12711yescharacteristic
128Q8×C53—28no—
129Q8×C53—44no—
130D4×C53—28no—
131D4×C53—416no—
132D4×C53—48no—
133D4×C53—48no—
134D4×C53—44no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—24no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—48no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—48no—
145C2×C4×C53—48no—
146C2×C4×C53#10414yescomplemented
147C2×C4×C53C2×C22⋊C412yes—
148C2×C4×C53C4×D412yes—
149C2×C4×C53C4×D412yes—
150C2×C4×C53#12014yes—
151C2×C4×C53#12012yes—
152C2×C4×C53#12112yes—
153C2×C4×C53#12114yes—
154C2×C4×C53#12114yes—
155C2×C4×C53#12314yes—
156C2×C4×C53#12411yescharacteristic
157C2×C4×C53#12711yescharacteristic
158C2×Dic53—28no—
159C2×Dic53—44no—
160C2×Dic53—48no—
161C2×Dic53—48no—
162C2×Dic53—44no—
163#88322—44no—
164#11011—531nomaximal, Sylow
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—24no—
170C22×C53—44no—
171C22×C53—48no—
172C22×C53—82no—
173C22×C53—82no—
174C22×C53—84no—
175C22×C53#37911yescharacteristic
176C22×C53#39011yescharacteristic
177C22×C53#39011yescharacteristic
178C22×C53#39212yes—
179C22×C53#39412yes—
180C22×C53C2×#10412yes—
181C22×C53#42112yes—
182C4×C53—24no—
183C4×C53—44no—
184C4×C53—42no—
185C4×C53#38614yes—
186C4×C53#39212yes—
187C4×C53#39412yes—
188Dic53—48no—
189C2×#532D5311yescharacteristic
190C22×#104—534no—
191#1728—531no—
192C4×#95—531no—
193C2×C53—22no—
194C2×C53—22no—
195C2×C53—84no—
196C2×C53#143811yescharacteristic
197C2×C53#178912yes—
198C2×C53#196211yescharacteristic
199C2×C53#196311yescharacteristic
200C2×C53#198012yes—
201C22×#41C2×D5312yes—
202C22×M16—532no—
203#532—28no—
204C2×C4⋊Q8Dic5311yescharacteristic, complemented
205C2×#127Dic5311yescharacteristic, complemented
206C2×#124Dic5312yescomplemented
207C2×#121—22no—
208C2×C4×D4—22no—
209C22×C42—532no—
210C22×C42C2×D5311yescharacteristic
211C2×#104—1068no—
212C2×#104—1068no—
213C2×#104—1068no—
214C2×#95—1062no—
215#336—1062no—
216C53#1101111yescharacteristic, complemented, Sylow
217C2×#41—24no—
218C2×#41—24no—
219C2×#41—24no—
220C2×#41—24no—
221C2×#41—24no—
222C2×#41C22⋊D5318yes—
223C22×Q8C2×Dic5312yes—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4C2×Dic5312yes—
227C23×C4—22no—
228C23×C4—1062no—
229C23×C4C22⋊D5312yes—
230C23×C4C22×D5311yescharacteristic
231C2×M16—534no—
232C2×M16—1062no—
233C2×M16—1064no—
234C2×M16—1064no—
235C22×C8—1062no—
236C4⋊Q8—24no—
237#127—24no—
238#124—24no—
239#124—24no—
240#121—216no—
241C4×D4—216no—
242C2×C4⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C42—24no—
245C2×C42—534no—
246C2×C42—1064no—
247C2×C42—1064no—
248C2×C42—1062no—
249C2×C42C2×Dic5311yescharacteristic
250C2×C42C2×Dic5311yescharacteristic
251C2×C42C22⋊D5312yes—
252#104—10632no—
253#95—1068no—
254C24—22no—
255C24C2×C22⋊D5311yescharacteristic
256#41—28no—
257#41—28no—
258#41—28no—
259#41—28no—
260#41—44no—
261#41—44no—
262#41—44no—
263#41—44no—
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8C22⋊Dic5314yes—
268C2×D4—28no—
269C2×D4—28no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—44no—
278C2×D4C22⋊Dic5314yes—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—28no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—1068no—
288C22×C4—1064no—
289C22×C4—2124no—
290C22×C4—2122no—
291C22×C4—2122no—
292C22×C4C22⋊Dic5312yes—
293C22×C4C22⋊Dic5312yes—
294C22×C4D4×D5312yes—
295C22×C4D4×D5312yes—
296C22×C4#1171994712yes—
297C22×C4#1171994712yes—
298C22×C4C22×Dic5311yescharacteristic
299C22×C4C2×C22⋊D5314yes—
300C22×C4C2×C22⋊D5311yescharacteristic
301M16—1068no—
302M16—2124no—
303C2×C8—1064no—
304C2×C8—2122no—
305C2×C8—2122no—
306C4⋊C4—28no—
307C22⋊C4—216no—
308C42—22no—
309C42—22no—
310C42—24no—
311C42—1068no—
312C42—2124no—
313C23—22no—
314C23—22no—
315C23—24no—
316C23—42no—
317C23—42no—
318C23—44no—
319C23—44no—
320C23C2×C22⋊Dic5311yescharacteristic
321C23#1313318612yes—
322C23#1313318812yes—
323C23#1313320411yescharacteristic
324C23#1313321311yescharacteristic
325Q8—28no—
326Q8—44no—
327D4—28no—
328D4—416no—
329D4—48no—
330D4—48no—
331D4—44no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—48no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—48no—
342C2×C4—48no—
343C2×C4—1068no—
344C2×C4—2124no—
345C2×C4—2128no—
346C2×C4—2128no—
347C2×C4—2124no—
348C2×C4#1313314414yescomplemented
349C2×C4C2×C22⋊Dic5312yes—
350C2×C4D4×Dic5312yes—
351C2×C4D4×Dic5312yes—
352C2×C4#1313318514yes—
353C2×C4#1313318612yes—
354C2×C4#1313318812yes—
355C2×C4#1313318914yes—
356C2×C4#1313318914yes—
357C2×C4#1313319814yes—
358C2×C4#1313320411yescharacteristic
359C2×C4#1313321311yescharacteristic
360C8—2124no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—24no—
366C22—44no—
367C22—48no—
368C22—82no—
369C22—82no—
370C22—84no—
371C22#1681047911yescharacteristic
372C22#1681052611yescharacteristic
373C22#1681052611yescharacteristic
374C22#1681053112yes—
375C22#1681053712yes—
376C22C2×#1313314412yes—
377C22#1681064012yes—
378C4—24no—
379C4—44no—
380C4—42no—
381C4—2128no—
382C4#1681050314yes—
383C4#1681053112yes—
384C4#1681053712yes—
385C2—22no—
386C2—22no—
387C2—84no—
388C2#2027295611yescharacteristic
389C2#2027465912yes—
390C2#2027554411yescharacteristic
391C2#2027554811yescharacteristic
392C2#2027563512yes—
subgroupG/H
centerC22×C4C2×C22⋊D53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC2×C53×#532C2
socleC23×C53C2×C22⋊C4
cosocleC23×C4×C53C23
hypercenterC2×#532D53
hypocenterC53#11011
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C22×#13133144C2semidirect
2. #20274409C2semidirect
3. C4×#13133124C2semidirect
14. C2×C4⋊Q8×C53C4semidirect
15. C2×#127×C53C4semidirect
16. C2×#124×C53C4semidirect
24. C2×#11719917D4semidirect
29. C2×C4×Dic53D4semidirect
146. C2×C4×C53#104semidirect
204. C2×C4⋊Q8Dic53semidirect
205. C2×#127Dic53semidirect
206. C2×#124Dic53semidirect
216. C53#11011semidirect
348. C2×C4#13133144semidirect

Maximal subgroups

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

subgroupindex
C22×#131331442
#202744092
C4×#131331242
C2×C53×#5322
#1101153