Finite Group Databasebeta

← #27357157

Subgroups — #27357157

Order 13568.

974 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284750C212yesmaximal, complemented
2#20280069C211yesmaximal, characteristic
3#20273740C214yesmaximal, complemented
4#20273672C214yesmaximal, complemented
5#20273279C212yesmaximal, complemented
6#20273036C211yesmaximal, characteristic
7C53×#3501C211yesmaximal, characteristic
8C2×C4×Dic106C2211yescharacteristic
9#16810956C2214yes—
10#16810915—22no—
11#16810830—22no—
12#16810778C2214yes—
13C4×#11719925—24no—
14#16810676C2212yes—
15#16810559C2212yes—
16#16810535C2212yes—
17#16810526—21no—
18#16810519—21no—
19#16810496—22no—
20#16810489C2211yescharacteristic
21#16810360C2218yes—
22#16810347C2214yes—
23C53×#554—22no—
24C53×#554—24no—
25D4×Q8×C53—22no—
26C53×#544—21no—
27C53×#543C2214yes—
28C53×#533C2212yes—
29C2×C22⋊Q8×C53—22no—
30C53×#518—21no—
31C2×C4×D4×C53C2211yescharacteristic
32C22×Dic106C2311yescharacteristic
33C2×#11719925—24no—
34C4×Dic106—22no—
35C4×Dic106—22no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53C2311yescharacteristic
38C2×#11719919—21no—
39C2×#11719919—22no—
40C2×C22⋊Dic53—22no—
41C2×C22⋊Dic53—22no—
42C2×C4×Dic53—21no—
43#13133150—24no—
44C4⋊#88322—22no—
45#13133141—24no—
46#13133140—24no—
47C22⋊#88322C2314yes—
48C4×#88322—22no—
49#13133124—21no—
50#13133124—22no—
51#13133124C2312yes—
52C22×Q8×C53—22no—
53C22×D4×C53D411yescharacteristic
54C23×C4×C53D412yes—
55C4⋊Q8×C53—24no—
56C4⋊Q8×C53D412yes—
57C4⋊Q8×C53D412yes—
58#125×C53—24no—
59#125×C53—22no—
60C22⋊Q8×C53—24no—
61C22⋊Q8×C53—24no—
62C22⋊Q8×C53—28no—
63C22⋊Q8×C53—28no—
64C22⋊Q8×C53—22no—
65C22⋊Q8×C53—24no—
66C22⋊Q8×C53C2314yes—
67C4×Q8×C53—22no—
68C4×Q8×C53D414yes—
69C4×D4×C53—24no—
70C4×D4×C53—24no—
71C4×D4×C53—22no—
72C4×D4×C53—22no—
73#117×C53—22no—
74C2×C4⋊C4×C53—22no—
75C2×C4⋊C4×C53—24no—
76C2×C4⋊C4×C53D411yescharacteristic
77C2×C22⋊C4×C53—24no—
78C2×C22⋊C4×C53C2312yes—
79C2×C42×C53C2311yescharacteristic
80C24×C53C2×D412yes—
81C2×Q8×C53—24no—
82C2×Q8×C53—22no—
83C2×Q8×C53—84no—
84C2×Q8×C53C2×D414yes—
85C2×D4×C53—22no—
86C2×D4×C53—22no—
87C2×D4×C53Q1614yescomplemented
88C22×C4×C53—24no—
89C22×C4×C53—24no—
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—82no—
93C22×C4×C53C2×D412yes—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2411yescharacteristic
97C22×Dic53—21no—
98C22×Dic53—22no—
99C22×Dic53—21no—
100C2×Dic106—22no—
101C2×Dic106—42no—
102C4⋊C4×C53—24no—
103C4⋊C4×C53—28no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—24no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53C2×D414yes—
111C22⋊C4×C53—28no—
112C22⋊C4×C53—24no—
113C22⋊C4×C53—24no—
114C42×C53—24no—
115C42×C53C2×D412yes—
116C42×C53C2×D412yes—
117C22⋊Dic53—44no—
118C22⋊Dic53—82no—
119#11719925—48no—
120C4⋊Dic53—22no—
121C4⋊Dic53—42no—
122#11719919—41no—
123#11719919—41no—
124#11719919—82no—
125C4×Dic53—42no—
126C2×#88322—24no—
127C23×C53—22no—
128C23×C53—22no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53C2×Q1614yes—
132C23×C53C22×D411yescharacteristic
133Q8×C53—44no—
134Q8×C53—44no—
135Q8×C53—84no—
136D4×C53—28no—
137C2×C4×C53—22no—
138C2×C4×C53—24no—
139C2×C4×C53—24no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—44no—
143C2×C4×C53—42no—
144C2×C4×C53—82no—
145C2×C4×C53—84no—
146C2×C4×C53—84no—
147C2×C4×C53#12011yescharacteristic
148C2×C4×C53#12011yescharacteristic
149C2×C4×C53#12012yes—
150C2×C4×C53C2×Q1612yes—
151C2×C4×C53#13712yes—
152C2×C4×C53C22×D412yes—
153C2×Dic53—22no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—42no—
162Dic106—24no—
163Dic106—42no—
164#88322—44no—
165#18213—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—24no—
169C22×C53—44no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—82no—
173C22×C53C2×#12011yescharacteristic
174C22×C53C22×Q1611yescharacteristic
175C22×C53C2×#13711yescharacteristic
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—22no—
179C4×C53—42no—
180C4×C53—44no—
181C4×C53—44no—
182C4×C53#45114yes—
183Dic53—42no—
184Dic53—82no—
185Dic53—81no—
186#3501D5311yescharacteristic
187#3104—532no—
188#2580—531no—
189#1614—534no—
190#1602—534no—
191#1518—532no—
192#1460—531no—
193C2×C53—21no—
194C2×C53—21no—
195C2×C53—44no—
196C2×C53C2×#45111yescharacteristic
197C2×C53#301911yescharacteristic
198C2×C53#302511yescharacteristic
199#554—22no—
200#554—24no—
201D4×Q8—22no—
202#544—21no—
203#543C2×D5314yes—
204#533C2×D5312yes—
205C2×C22⋊Q8—22no—
206#518—21no—
207C2×C4×Q8—531no—
208C2×C4×D4C2×D5311yescharacteristic
209#484—534no—
210#477—1062no—
211#462—1062no—
212#451—534no—
213C4×Q16—1064no—
214#431—532no—
215#399—532no—
216#393—532no—
217#390—1061no—
218#389—1061no—
219#385—1062no—
220#382—531no—
221#328—538no—
222#323—534no—
223C53#1821311yescharacteristic, complemented, Sylow
224C22×Q8—22no—
225C22×Q8—531no—
226C22×D4C22⋊D5311yescharacteristic
227C23×C4C22⋊D5312yes—
228C2×Q16—1064no—
229C4⋊Q8—24no—
230C4⋊Q8C22⋊D5312yes—
231C4⋊Q8C22⋊D5312yes—
232#125—24no—
233#125—22no—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236C22⋊Q8—28no—
237C22⋊Q8—22no—
238C22⋊Q8—24no—
239C22⋊Q8—28no—
240C22⋊Q8C22×D5314yes—
241C4×Q8—22no—
242C4×Q8—1062no—
243C4×Q8—1062no—
244C4×Q8C22⋊D5314yes—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—22no—
248C4×D4—22no—
249#117—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4—1062no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4C22⋊D5311yescharacteristic
257C2×C22⋊C4—24no—
258C2×C22⋊C4—1062no—
259C2×C22⋊C4—1062no—
260C2×C22⋊C4C22×D5312yes—
261C2×C42—1061no—
262C2×C42C22×D5311yescharacteristic
263#107—1064no—
264C4⋊C8—1062no—
265#103—1064no—
266#103—1064no—
267C22⋊C8—534no—
268C4×C8—1062no—
269#95—532no—
270#95—1061no—
271#95—1062no—
272C24C2×C22⋊D5312yes—
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—84no—
276C2×Q8—1062no—
277C2×Q8—2122no—
278C2×Q8C2×C22⋊D5314yes—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4#1171992514yescomplemented
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—82no—
287C22×C4—1061no—
288C22×C4—1062no—
289C22×C4—1061no—
290C22×C4C2×C22⋊D5312yes—
291C22×C4C2×C22⋊D5311yescharacteristic
292C22×C4C2×C22⋊D5311yescharacteristic
293C22×C4C23×D5311yescharacteristic
294Q16—2128no—
295C2×C8—1064no—
296C4⋊C4—24no—
297C4⋊C4—28no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—1062no—
305C4⋊C4—4242no—
306C4⋊C4—2121no—
307C4⋊C4—2121no—
308C4⋊C4—2122no—
309C4⋊C4C2×C22⋊D5314yes—
310C22⋊C4—28no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—4242no—
314C22⋊C4—2124no—
315C42—24no—
316C42—2122no—
317C42C2×C22⋊D5312yes—
318C42C2×C22⋊D5312yes—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23C2×#1171992514yes—
324C23C22×C22⋊D5311yescharacteristic
325Q8—44no—
326Q8—44no—
327Q8—84no—
328Q8—1064no—
329Q8—2122no—
330D4—28no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—82no—
339C2×C4—84no—
340C2×C4—84no—
341C2×C4—1062no—
342C2×C4—2121no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2122no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4#1313318611yescharacteristic
351C2×C4#1313318611yescharacteristic
352C2×C4#1313318612yes—
353C2×C4C2×#1171992512yes—
354C2×C4#1313324812yes—
355C2×C4C22×C22⋊D5312yes—
356C8—2124no—
357C22—21no—
358C22—21no—
359C22—24no—
360C22—44no—
361C22—42no—
362C22—42no—
363C22—82no—
364C22C2×#1313318611yescharacteristic
365C22C22×#1171992511yescharacteristic
366C22C2×#1313324811yescharacteristic
367C4—21no—
368C4—21no—
369C4—22no—
370C4—42no—
371C4—44no—
372C4—44no—
373C4—4242no—
374C4—4241no—
375C4—2122no—
376C4#1681077914yes—
377C2—21no—
378C2—21no—
379C2—44no—
380C2C2×#1681077911yescharacteristic
381C2#2028407211yescharacteristic
382C2#2028410711yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3501C2
socleC22×C53C2×#120
prime radicalC22×D4×C53D4
hypercenter#3501D53
hypocenterC53#18213
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #20284750C2semidirect
3. #20273740C2semidirect
4. #20273672C2semidirect
5. #20273279C2semidirect
87. C2×D4×C53Q16semidirect
223. C53#18213semidirect
281. C2×D4#11719925semidirect

Maximal subgroups

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

subgroupindex
#202847502
#202800692
#202737402
#202736722
#202732792
#202730362
C53×#35012
#1821353