Finite Group Databasebeta

← #27534014

Subgroups — #27534014

Order 13568.

974 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810913C211yesmaximal, characteristic
2#20283092C211yesmaximal, characteristic, complemented
3C4×D4×Dic53C211yesmaximal, characteristic
4#20275662C211yesmaximal, characteristic, complemented
5#20275639C211yesmaximal, characteristic, complemented
6#20275352C214yesmaximal, complemented
7#20275051C214yesmaximal
8#20274831C211yesmaximal, characteristic, complemented
9C2×C53×#461C211yesmaximal, characteristic
10C2×#13133217C2211yescharacteristic
11C2×D4×Dic53C2212yes—
12C22×C4×Dic53—21no—
13#16810913—24no—
14C2×#13133148C2212yes—
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133141C2214yes—
17C2×#13133138C2212yes—
18#16810530C2214yes—
19#16810528—22no—
20#16810488—21no—
21#16810488—21no—
22C4⋊C4×Dic53C2212yes—
23C4×C4⋊Dic53C2211yescharacteristic
24C22⋊C4×Dic53—22no—
25C4×C22⋊Dic53—21no—
26C42×Dic53C2211yescharacteristic
27#16810386C2214yes—
28#16810377C2214yes—
29C22×SD16×C53C2212yes—
30C2×C4⋊Q8×C53C2211yescharacteristic
31C2×C4×D4×C53C2211yescharacteristic
32C53×#461—24no—
33C2×C4⋊C8×C53C2211yescharacteristic
34C2×#103×C53C2212yes—
35C22×Dic106—21no—
36C23×Dic53—22no—
37C22×#88322C2312yes—
38#13133217—22no—
39#13133217—22no—
40D4×Dic53—24no—
41D4×Dic53—24no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53C2312yes—
44C2×C4⋊Dic53C2312yes—
45C2×C22⋊Dic53—22no—
46C2×C4×Dic53—21no—
47C2×C4×Dic53—22no—
48C2×C4×Dic53—44no—
49C2×C4×Dic53D411yescharacteristic
50C2×C4×Dic53D412yes—
51C2×C4×Dic53D411yescharacteristic
52C2×C4×Dic53C2312yes—
53#13133148—24no—
54C4⋊#88322—22no—
55#13133141—28no—
56#13133138—24no—
57#13133124—21no—
58#13133124—24no—
59#13133124—24no—
60#13133124—22no—
61C22×Q8×C53C2312yes—
62C22×D4×C53C2311yescharacteristic
63C23×C4×C53—21no—
64C2×SD16×C53—24no—
65C2×SD16×C53—28no—
66C22×C8×C53C2312yes—
67C4⋊Q8×C53—22no—
68C4⋊Q8×C53—22no—
69C4×D4×C53—22no—
70C4×D4×C53D414yes—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53C2311yescharacteristic
74C2×C22⋊C4×C53—21no—
75C2×C42×C53C2311yescharacteristic
76C4⋊C8×C53—22no—
77#103×C53—24no—
78C24×C53—21no—
79C2×Q8×C53—22no—
80C2×Q8×C53—22no—
81C2×Q8×C53—24no—
82C2×Q8×C53—24no—
83C2×D4×C53—21no—
84C2×D4×C53—22no—
85C2×D4×C53C2×D412yes—
86C2×D4×C53#4114yes—
87C22×C4×C53—21no—
88C22×C4×C53—24no—
89C22×C4×C53—22no—
90C22×C4×C53—42no—
91C22×C4×C53—42no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2411yescharacteristic
95C22×Dic53—28no—
96C22×Dic53—22no—
97C22×Dic53—22no—
98C22×Dic53—44no—
99C22×Dic53—44no—
100C22×Dic53C2×D412yes—
101C22×Dic53C2×D412yes—
102C2×Dic106—44no—
103C2×Dic106—42no—
104SD16×C53—48no—
105C2×C8×C53—22no—
106C2×C8×C53—24no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—42no—
109C4⋊C4×C53—42no—
110C4⋊C4×C53C2×D412yes—
111C22⋊C4×C53—24no—
112C42×C53C2×D412yes—
113C42×C53#4112yes—
114C22⋊Dic53—28no—
115C4⋊Dic53—22no—
116C4⋊Dic53—22no—
117C4⋊Dic53—22no—
118C4⋊Dic53—42no—
119C4⋊Dic53#4114yes—
120C4×Dic53—22no—
121C4×Dic53—22no—
122C4×Dic53—22no—
123C4×Dic53—42no—
124C4×Dic53—42no—
125C4×Dic53—44no—
126C4×Dic53SD1614yescomplemented
127C2×#88322—22no—
128C2×#88322—24no—
129C23×C53—22no—
130C23×C53—24no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—42no—
134C23×C53C22×D411yescharacteristic
135Q8×C53—42no—
136Q8×C53—44no—
137Q8×C53—42no—
138D4×C53—24no—
139D4×C53—22no—
140D4×C53#12414yes—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—48no—
146C2×C4×C53—42no—
147C2×C4×C53—44no—
148C2×C4×C53#12112yes—
149C2×C4×C53#12112yes—
150C2×C4×C53C2×SD1612yes—
151C2×C4×C53#13712yes—
152C2×C4×C53C22×D411yescharacteristic
153C2×C4×C53C2×#4112yes—
154C2×C4×C53C2×#4112yes—
155C2×C4×C53C2×#4111yescharacteristic
156C2×Dic53—22no—
157C2×Dic53—22no—
158C2×Dic53—24no—
159C2×Dic53—22no—
160C2×Dic53—22no—
161C2×Dic53—22no—
162C2×Dic53—416no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—44no—
167C2×Dic53C2×SD1614yes—
168Dic106—84no—
169C8×C53—42no—
170#88322—42no—
171#33026—531nomaximal, Sylow
172C22×C53—24no—
173C22×C53—44no—
174C22×C53—44no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53C2×#12111yescharacteristic
178C22×C53#52512yes—
179C22×C53C22×SD1611yescharacteristic
180C22×C53C2×#13711yescharacteristic
181C22×C53#57512yes—
182C4×C53—22no—
183C4×C53—42no—
184C4×C53—82no—
185C4×C53#46114yes—
186C4×C53C2×#12411yescharacteristic
187C4×C53#52912yes—
188C4×C53#53211yescharacteristic
189Dic53—24no—
190Dic53—44no—
191Dic53—42no—
192Dic53—82no—
193C2×#476—531no—
194C2×#461D5311yescharacteristic
195#2862—531no—
196C42×D4—531no—
197#1987—531no—
198#1981—531no—
199#1927—534no—
200#1877—534no—
201#1831—531no—
202C2×C53—44no—
203C2×C53#285011yescharacteristic
204C2×C53C2×#46111yescharacteristic
205C2×C53#304011yescharacteristic
206C2×C53#310112yes—
207C2×C53#311612yes—
208C22×SD16C2×D5312yes—
209C2×C4⋊Q8—531no—
210C2×C4⋊Q8C2×D5311yescharacteristic
211C2×C4×D4—532no—
212C2×C4×D4C2×D5311yescharacteristic
213C22×C42—1061no—
214#476—1064no—
215#461—24no—
216C2×#106—532no—
217C2×C4⋊C8—531no—
218C2×C4⋊C8C2×D5311yescharacteristic
219C2×#103—534no—
220C2×#103C2×D5312yes—
221C2×#102—532no—
222#391—534no—
223#391—1062no—
224#382—1061no—
225#382—1061no—
226C4×C4⋊C4—532no—
227C4×C4⋊C4—531no—
228C4×C22⋊C4—1062no—
229C4×C22⋊C4—1061no—
230C43—531no—
231#340—534no—
232#336—534no—
233C53#3302611yescharacteristic, complemented, Sylow
234C22×Q8—1061no—
235C22×Q8C22×D5312yes—
236C22×D4C22×D5311yescharacteristic
237C23×C4—21no—
238C23×C4—1062no—
239C2×SD16—28no—
240C2×SD16—24no—
241C22×C8—532no—
242C22×C8C22×D5312yes—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—1062no—
246C4⋊Q8—1062no—
247C4×D4—22no—
248C4×D4—1064no—
249C4×D4—1064no—
250C4×D4C22⋊D5314yes—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—532no—
254C2×C4⋊C4—532no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4C22×D5311yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—1062no—
259C2×C42—531no—
260C2×C42—531no—
261C2×C42—532no—
262C2×C42—532no—
263C2×C42—1062no—
264C2×C42—1061no—
265C2×C42—2124no—
266C2×C42C22×D5311yescharacteristic
267#106—1064no—
268C4⋊C8—22no—
269C4⋊C8—1062no—
270#103—24no—
271#103—1068no—
272#102—1064no—
273#95—1062no—
274#95—1064no—
275#95—1064no—
276#95—1061no—
277C24—21no—
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—24no—
282C2×Q8—2124no—
283C2×Q8—2122no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4#1171994714yes—
287C2×D4C2×C22⋊D5312yes—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—532no—
294C22×C4—532no—
295C22×C4—1068no—
296C22×C4—1062no—
297C22×C4—1062no—
298C22×C4—2124no—
299C22×C4—2124no—
300C22×C4D4×D5311yescharacteristic
301C22×C4D4×D5311yescharacteristic
302C22×C4C23×D5311yescharacteristic
303SD16—48no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—1062no—
307C2×C8—1064no—
308C4⋊C4—21no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—534no—
312C4⋊C4—1062no—
313C4⋊C4—1062no—
314C4⋊C4—1062no—
315C4⋊C4—2122no—
316C4⋊C4C2×C22⋊D5312yes—
317C22⋊C4—24no—
318C22⋊C4—1068no—
319C42—534no—
320C42—1062no—
321C42—1062no—
322C42—1062no—
323C42—2122no—
324C42—2124no—
325C42—2122no—
326C42#1171994712yes—
327C42C2×C22⋊D5312yes—
328C23—22no—
329C23—24no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23C2×D4×D5311yescharacteristic
334Q8—42no—
335Q8—44no—
336Q8—42no—
337Q8—4244no—
338D4—24no—
339D4—22no—
340D4#1313320414yes—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—48no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—534no—
349C2×C4—1062no—
350C2×C4—1062no—
351C2×C4—1062no—
352C2×C4—1064no—
353C2×C4—1062no—
354C2×C4—1062no—
355C2×C4—21216no—
356C2×C4—2122no—
357C2×C4—2124no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4#1313318912yes—
361C2×C4#1313318912yes—
362C2×C4SD16×D5312yes—
363C2×C4#1313324612yes—
364C2×C4C22×C22⋊D5311yescharacteristic
365C2×C4#41×D5312yes—
366C2×C4C2×#1171994712yes—
367C2×C4C2×#1171994711yescharacteristic
368C8—42no—
369C8—2122no—
370C22—24no—
371C22—44no—
372C22—42no—
373C22—44no—
374C22—42no—
375C22C2×#1313318911yescharacteristic
376C22#1681113312yes—
377C22C2×SD16×D5311yescharacteristic
378C22C2×#1313324611yescharacteristic
379C22#1681142212yes—
380C4—22no—
381C4—42no—
382C4—82no—
383C4—1064no—
384C4—4242no—
385C4—2124no—
386C4—2122no—
387C4#1681082614yes—
388C4C2×#1313320411yescharacteristic
389C4#1681115412yes—
390C4#1681116911yescharacteristic
391C2—44no—
392C2#2028301411yescharacteristic
393C2C2×#1681082611yescharacteristic
394C2#2028421111yescharacteristic
395C2#2028472212yes—
396C2#2028487412yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#461C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenterC2×#461D53
hypocenterC53#33026
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
2. #20283092C2semidirect
4. #20275662C2semidirect
5. #20275639C2semidirect
6. #20275352C2semidirect
8. #20274831C2semidirect
126. C4×Dic53SD16semidirect
233. C53#33026semidirect

Maximal subgroups

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

subgroupindex
C2×#168109132
#202830922
C4×D4×Dic532
#202756622
#202756392
#202753522
#202750512
#202748312
C2×C53×#4612
#3302653