Finite Group Databasebeta

← #27279822

Subgroups — #27279822

Order 13568.

1010 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 147 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#20283002C211yesmaximal, characteristic
2#20274625C211yesmaximal, characteristic
3C2×#16810359C212yesmaximal, complemented
4#20272594C212yesmaximal, complemented
5C2×C4×C22⋊C4×C53C211yesmaximal, characteristic
6C2×#13133196C412yescomplemented
7C2×#13133196C2211yescharacteristic
8C2×C22⋊#88322—22no—
9C2×C22⋊#88322C2212yes—
10#16810496—21no—
11#16810496—22no—
12#16810483C412yescomplemented
13#16810483C2211yescharacteristic
14#16810479—21no—
15#16810359—24no—
16C24×C4×C53C2211yescharacteristic
17C22×C22⋊C4×C53—21no—
18C22×C42×C53—21no—
19C4×C22⋊C4×C53—24no—
20C2×#95×C53C2212yes—
21C22×Dic106C2×C411yescharacteristic
22C22×#88322—22no—
23#13133196—22no—
24#13133196—22no—
25#13133196—44no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C412yes—
28C2×#11719919—22no—
29C2×#11719919—21no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C22⋊#88322—22no—
35C22⋊#88322—22no—
36C22⋊#88322—44no—
37#13133124—22no—
38#13133124—21no—
39C25×C53D411yescharacteristic
40C23×C4×C53—22no—
41C23×C4×C53—22no—
42C23×C4×C53—22no—
43C23×C4×C53—44no—
44C23×C4×C53C2×C412yes—
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D412yes—
47C23×C4×C53C2311yescharacteristic
48C2×C22⋊C4×C53—28no—
49C2×C22⋊C4×C53—24no—
50C2×C42×C53—24no—
51C2×C42×C53—44no—
52#95×C53—22no—
53#95×C53D414yes—
54C24×C53—24no—
55C24×C53—21no—
56C24×C53—21no—
57C24×C53—22no—
58C24×C53—22no—
59C24×C53—42no—
60C24×C53C22⋊C412yes—
61C24×C53C2×D411yescharacteristic
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—24no—
65C22×C4×C53—28no—
66C22×C4×C53—24no—
67C22×C4×C53—24no—
68C22×C4×C53—24no—
69C22×C4×C53—24no—
70C22×C4×C53—24no—
71C22×C4×C53—42no—
72C22×C4×C53—42no—
73C22×C4×C53—42no—
74C22×C4×C53—48no—
75C22×C4×C53—48no—
76C22×C4×C53—44no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53C22⋊C412yes—
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D412yes—
83C22×C4×C53#4112yes—
84C22×Dic53—21no—
85C22×Dic53—22no—
86C22×Dic53—21no—
87C2×Dic106—22no—
88C2×Dic106—42no—
89C22⋊C4×C53—216no—
90C42×C53—48no—
91C22⋊Dic53—44no—
92C22⋊Dic53—44no—
93C22⋊Dic53—41no—
94C22⋊Dic53—41no—
95C22⋊Dic53—42no—
96C4⋊Dic53—22no—
97C4⋊Dic53—42no—
98#11719919—44no—
99#11719919—42no—
100C2×#88322—44no—
101C2×#88322—42no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—24no—
105C23×C53—24no—
106C23×C53—22no—
107C23×C53—22no—
108C23×C53—22no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53—22no—
112C23×C53—48no—
113C23×C53—48no—
114C23×C53—42no—
115C23×C53—42no—
116C23×C53—42no—
117C23×C53—42no—
118C23×C53#10414yes—
119C23×C53C2×C22⋊C411yescharacteristic
120C23×C53#12011yescharacteristic
121C23×C53#12311yescharacteristic
122C23×C53#13512yes—
123C23×C53#13712yes—
124C2×C4×C53—24no—
125C2×C4×C53—24no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—28no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—48no—
134C2×C4×C53—48no—
135C2×C4×C53—48no—
136C2×C4×C53—48no—
137C2×C4×C53—44no—
138C2×C4×C53—44no—
139C2×C4×C53#9914yes—
140C2×C4×C53C4×D412yes—
141C2×C4×C53#12112yes—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—42no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—42no—
148C2×Dic53—42no—
149Dic106—82no—
150#88322—82no—
151#9146—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—24no—
157C22×C53—24no—
158C22×C53—24no—
159C22×C53—24no—
160C22×C53—24no—
161C22×C53—48no—
162C22×C53—48no—
163C22×C53—42no—
164C22×C53—42no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53#38611yescharacteristic
168C22×C53C2×#9912yes—
169C22×C53#41611yescharacteristic
170C22×C53#41911yescharacteristic
171C22×C53C2×#10412yes—
172C22×C53#45212yes—
173C22×C53#48512yes—
174C4×C53—22no—
175C4×C53—22no—
176C4×C53—24no—
177C4×C53—48no—
178Dic53—81no—
179Dic53—81no—
180Dic53—82no—
181#2848—531no—
182C2×C4×C22⋊C4D5311yescharacteristic
183#1779—531no—
184C2×#328—532no—
185#1294—532no—
186C2×C53—22no—
187C2×C53—22no—
188C2×C53—44no—
189C2×C53#150112yes—
190C2×C53#150812yes—
191C2×C53#187311yescharacteristic
192C2×C53#189611yescharacteristic
193C2×C53#189811yescharacteristic
194C24×C4C2×D5311yescharacteristic
195C2×C22⋊Q8—531no—
196C2×C22⋊Q8—532no—
197C22×C22⋊C4—21no—
198C22×C42—21no—
199C2×C22⋊C8—532no—
200C2×C22⋊C8—1062no—
201#385—1062no—
202#385—1061no—
203#380—532no—
204#380—531no—
205#379—1061no—
206C4×C22⋊C4—24no—
207C2×#95C2×D5312yes—
208#328—1064no—
209C53#914611yescharacteristic, complemented, Sylow
210C25C22⋊D5311yescharacteristic
211C22×Q8—531no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—44no—
216C23×C4C4×D5312yes—
217C23×C4D21212yes—
218C23×C4C22⋊D5311yescharacteristic
219C23×C4C22×D5311yescharacteristic
220C22×C8—1062no—
221C22⋊Q8—1062no—
222C22⋊Q8—1062no—
223C22⋊Q8—2124no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—532no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1062no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—1061no—
231C2×C22⋊C4—1061no—
232C2×C22⋊C4—1062no—
233C2×C22⋊C4—1062no—
234C2×C42—24no—
235C2×C42—44no—
236C22⋊C8—1062no—
237C22⋊C8—1062no—
238C22⋊C8—2124no—
239#95—22no—
240#95—1061no—
241#95—1062no—
242#95C22⋊D5314yes—
243C24—24no—
244C24—21no—
245C24—21no—
246C24—22no—
247C24—22no—
248C24—42no—
249C24#1171992112yes—
250C24C2×C22⋊D5311yescharacteristic
251C2×Q8—1062no—
252C2×Q8—2122no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—1061no—
271C22×C4—1062no—
272C22×C4—1061no—
273C22×C4#1171992112yes—
274C22×C4C2×C4×D5311yescharacteristic
275C22×C4C2×D21211yescharacteristic
276C22×C4#1171994512yes—
277C22×C4C2×C22⋊D5312yes—
278C2×C8—2124no—
279C2×C8—2122no—
280C4⋊C4—1062no—
281C4⋊C4—2122no—
282C4⋊C4—2124no—
283C4⋊C4—2122no—
284C22⋊C4—216no—
285C22⋊C4—2124no—
286C22⋊C4—2124no—
287C22⋊C4—2121no—
288C22⋊C4—2121no—
289C22⋊C4—2122no—
290C42—48no—
291C23—22no—
292C23—22no—
293C23—24no—
294C23—24no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—48no—
302C23—48no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23#1313314514yes—
308C23C2×#1171992111yescharacteristic
309C23#1313318611yescharacteristic
310C23#1313320011yescharacteristic
311C23#1313323712yes—
312C23#1313324712yes—
313Q8—4242no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—28no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—48no—
324C2×C4—48no—
325C2×C4—44no—
326C2×C4—48no—
327C2×C4—48no—
328C2×C4—44no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4#1313313114yes—
337C2×C4C4×C22⋊D5312yes—
338C2×C4#1313319112yes—
339C8—4242no—
340C22—24no—
341C22—24no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—24no—
347C22—24no—
348C22—24no—
349C22—48no—
350C22—48no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22#1681050211yescharacteristic
356C22C2×#1313313112yes—
357C22#1681061711yescharacteristic
358C22#1681063111yescharacteristic
359C22C2×#1313314512yes—
360C22#1681078512yes—
361C22#1681096412yes—
362C4—22no—
363C4—24no—
364C4—22no—
365C4—48no—
366C4—4242no—
367C4—4241no—
368C4—4241no—
369C2—22no—
370C2—22no—
371C2—44no—
372C2#2027320412yes—
373C2#2027323412yes—
374C2#2027502711yescharacteristic
375C2#2027515811yescharacteristic
376C2#2027516811yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C4×C22⋊C4×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenterC2×C4×C22⋊C4D53
hypocenterC53#9146
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#16810359C2semidirect
4. #20272594C2semidirect
6. C2×#13133196C4semidirect
12. #16810483C4semidirect
209. C53#9146semidirect

Maximal subgroups

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

subgroupindex
#202830022
#202746252
C2×#168103592
#202725942
C2×C4×C22⋊C4×C532
#914653