Finite Group Databasebeta

← #27281851

Subgroups — #27281851

Order 13568.

738 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 135 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#20278080C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic
3#20274651C212yesmaximal, complemented
4#20272583C212yesmaximal, complemented
5C53×#2972C211yesmaximal, characteristic
6C23×Dic106C2211yescharacteristic
7C2×#13133196—21no—
8C2×#13133142—24no—
9C2×#13133132—21no—
10#16810543C412yescomplemented
11#16810535C412yescomplemented
12#16810529—21no—
13#16810503—21no—
14#16810495—21no—
15#16810483C2212yes—
16#16810483C2211yescharacteristic
17#16810390—21no—
18C2×C4×D4×C53C2211yescharacteristic
19C8×D4×C53—24no—
20C53×#431—21no—
21C2×C22⋊C8×C53—21no—
22C2×C22⋊C8×C53C2212yes—
23C22×Dic106—22no—
24C22×Dic106—21no—
25C22×Dic106—22no—
26C22×Dic106—44no—
27C22×Dic106C2×C412yes—
28C23×Dic53—21no—
29#13133196—44no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—22no—
32C2×#11719919—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36#13133142—48no—
37#13133132—42no—
38#13133124—22no—
39#13133124—22no—
40#13133124—22no—
41#13133124C2×C412yes—
42C22×D4×C53D411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C23×C4×C53C2311yescharacteristic
45C22×C8×C53—22no—
46C22×C8×C53D414yes—
47C4×D4×C53—22no—
48C4×D4×C53—22no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C4⋊C8×C53—22no—
54C22⋊C8×C53—22no—
55C22⋊C8×C53—22no—
56C22⋊C8×C53—22no—
57C4×C8×C53—22no—
58C24×C53C22⋊C411yescharacteristic
59C24×C53C2×D411yescharacteristic
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53SD1612yes—
63C2×D4×C53Q1612yes—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D412yes—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53#4112yes—
75C22×Dic53—22no—
76C22×Dic53—22no—
77C22×Dic53—44no—
78C22×Dic53—42no—
79C22×Dic53—42no—
80C2×Dic106—28no—
81C2×Dic106—24no—
82C2×Dic106—24no—
83C2×Dic106—44no—
84C2×Dic106—48no—
85C2×Dic106—44no—
86C2×Dic106—42no—
87C2×Dic106—41no—
88C2×Dic106—41no—
89C2×C8×C53—24no—
90C2×C8×C53—44no—
91C2×C8×C53—82no—
92C4⋊C4×C53—22no—
93C22⋊C4×C53—22no—
94C22⋊C4×C53—22no—
95C42×C53—21no—
96C42×C53—21no—
97C22⋊Dic53—44no—
98C22⋊Dic53—42no—
99C22⋊Dic53—42no—
100C22⋊Dic53—42no—
101C4⋊Dic53—42no—
102C4⋊Dic53—44no—
103#11719919—44no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53#10312yescomplemented
111C23×C53C2×C22⋊C411yescharacteristic
112C23×C53#12112yes—
113C23×C53C2×SD1611yescharacteristic
114C23×C53C2×Q1611yescharacteristic
115C23×C53#13512yes—
116D4×C53—24no—
117D4×C53—24no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—24no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—81no—
130C2×C4×C53#10312yescomplemented
131C2×C4×C53C4×D412yes—
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53#12311yescharacteristic
134C2×Dic53—24no—
135C2×Dic53—48no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—82no—
142C2×Dic53—82no—
143C2×Dic53—82no—
144Dic106—28no—
145Dic106—44no—
146Dic106—44no—
147Dic106—44no—
148C8×C53—44no—
149#9467—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—22no—
152C22×C53—21no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—42no—
158C22×C53—42no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—81no—
162C22×C53#38611yescharacteristic
163C22×C53C2×#10311yescharacteristic
164C22×C53#41611yescharacteristic
165C22×C53#46512yes—
166C22×C53#46712yes—
167C4×C53—22no—
168C4×C53—21no—
169C4×C53—22no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53B(4,𝔽2)12yes—
173Dic53—44no—
174Dic53—82no—
175Dic53—162no—
176#2972D5311yescharacteristic
177#2409—531no—
178#1799—531no—
179#1787—532no—
180#1289—532no—
181C2×C53—21no—
182C2×C53—22no—
183C2×C53—21no—
184C2×C53—22no—
185C2×C53—42no—
186C2×C53#188911yescharacteristic
187C2×C53#189211yescharacteristic
188C2×C53#190911yescharacteristic
189C23×Q8—531no—
190C2×C22⋊Q8—1061no—
191C2×C4×D4C2×D5311yescharacteristic
192C8×D4—24no—
193#431—21no—
194C2×#103—1064no—
195C2×#99—1061no—
196C2×C22⋊C8—21no—
197C2×C22⋊C8C2×D5312yes—
198#395—532no—
199#393—532no—
200#391—1061no—
201#386—1061no—
202#385—1061no—
203#380—531no—
204#380—532no—
205#342—1061no—
206C53#946711yescharacteristic, complemented, Sylow
207C22×Q8—532no—
208C22×Q8—1061no—
209C22×Q8—1062no—
210C22×Q8—1062no—
211C22×Q8—2124no—
212C22×D4D21211yescharacteristic
213C23×C4—1061no—
214C23×C4C22⋊D5311yescharacteristic
215C23×C4C22×D5311yescharacteristic
216C22×C8—22no—
217C22×C8C22⋊D5314yes—
218C22⋊Q8—2124no—
219C4×D4—22no—
220C4×D4—22no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4—1062no—
224C2×C4⋊C4C4×D5311yescharacteristic
225C2×C22⋊C4—1061no—
226C2×C22⋊C4—1062no—
227C2×C22⋊C4—1062no—
228C2×C22⋊C4C4×D5311yescharacteristic
229C2×C22⋊C4D21211yescharacteristic
230C2×C42C22⋊D5311yescharacteristic
231C4⋊C8—22no—
232#103—2128no—
233#99—2122no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C4×C8—22no—
238#95—532no—
239#95—1062no—
240#95—1062no—
241#95—1062no—
242C24#1171992111yescharacteristic
243C24C2×D21211yescharacteristic
244C2×Q8—1064no—
245C2×Q8—1068no—
246C2×Q8—1064no—
247C2×Q8—2124no—
248C2×Q8—2122no—
249C2×Q8—2124no—
250C2×Q8—2128no—
251C2×Q8—2121no—
252C2×Q8—2121no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4#1171991312yes—
256C2×D4Dic21212yes—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—1062no—
262C22×C4—1062no—
263C22×C4—2124no—
264C22×C4—2122no—
265C22×C4—2122no—
266C22×C4#1171992111yescharacteristic
267C22×C4#1171992111yescharacteristic
268C22×C4#1171992111yescharacteristic
269C22×C4C2×C4×D5311yescharacteristic
270C22×C4#1171994512yes—
271C22×C4C2×C22⋊D5312yes—
272C22×C4C2×C22⋊D5311yescharacteristic
273C2×C8—24no—
274C2×C8—44no—
275C2×C8—82no—
276C4⋊C4—22no—
277C4⋊C4—2124no—
278C4⋊C4—2122no—
279C4⋊C4—2124no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—2124no—
283C22⋊C4—2122no—
284C22⋊C4—2122no—
285C22⋊C4—2122no—
286C42—21no—
287C42—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#1313314212yescomplemented
295C23C2×#1171992111yescharacteristic
296C23#1313319112yes—
297C23C2×#1171991311yescharacteristic
298C23C2×Dic21211yescharacteristic
299C23#1313323912yes—
300Q8—1068no—
301Q8—2124no—
302Q8—2124no—
303Q8—2124no—
304D4—24no—
305D4—24no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—1064no—
319C2×C4—4242no—
320C2×C4—4242no—
321C2×C4—4242no—
322C2×C4—2122no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2128no—
327C2×C4—2122no—
328C2×C4#1313314212yescomplemented
329C2×C4C4×C22⋊D5312yes—
330C2×C4#1313318611yescharacteristic
331C2×C4#1313320011yescharacteristic
332C8—44no—
333C22—21no—
334C22—22no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—81no—
345C22#1681050211yescharacteristic
346C22C2×#1313314211yescharacteristic
347C22#1681062011yescharacteristic
348C22#1681085212yes—
349C22#1681086412yes—
350C4—21no—
351C4—22no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—8482no—
356C4—4242no—
357C4—2124no—
358C4#1681081012yes—
359C2—22no—
360C2—21no—
361C2—21no—
362C2—22no—
363C2—42no—
364C2#2027512311yescharacteristic
365C2#2027513811yescharacteristic
366C2#2027523711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2972C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#2972D53
hypocenterC53#9467
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
3. #20274651C2semidirect
4. #20272583C2semidirect
10. #16810543C4semidirect
11. #16810535C4semidirect
110. C23×C53#103semidirect
130. C2×C4×C53#103semidirect
206. C53#9467semidirect
294. C23#13133142semidirect
328. C2×C4#13133142semidirect

Maximal subgroups

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

subgroupindex
#202780802
#202747032
#202746512
#202725832
C53×#29722
#946753