Finite Group Databasebeta

← #27282039

Subgroups — #27282039

Order 13568.

684 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 103 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#20283744C211yesmaximal, characteristic
2#20278080C211yesmaximal, characteristic
3#20274638C212yesmaximal, complemented
4#20272622C212yesmaximal, complemented
5C53×#1465C211yesmaximal, characteristic
6C23×Dic106C2211yescharacteristic
7C2×#13133196—21no—
8#16810695—24no—
9#16810680—21no—
10C2×#13133135—24no—
11#16810587C2212yes—
12C2×C22⋊#88322—21no—
13#16810543—21no—
14#16810535—21no—
15#16810529—21no—
16#16810503C412yes—
17#16810495C412yes—
18#16810483C2211yescharacteristic
19C2×C4×D4×C53C2211yescharacteristic
20C53×#407C2212yes—
21C53×#324—21no—
22C53×#323—21no—
23C22×Dic106—21no—
24C22×Dic106—22no—
25C22×Dic106—21no—
26C22×Dic106—22no—
27C22×Dic106—44no—
28C23×Dic53C2×C412yes—
29C2×#11719917D414yes—
30C22×#88322—22no—
31#13133196—44no—
32C2×C4⋊Dic53—21no—
33C2×#11719919—22no—
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53C2×C412yes—
36C4⋊#88322—22no—
37#13133135—48no—
38C22⋊#88322—22no—
39C22⋊#88322—22no—
40C22⋊#88322—22no—
41#13133128—22no—
42#13133124—22no—
43#13133124—22no—
44#13133124—21no—
45C22×D4×C53C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C23×C4×C53C2311yescharacteristic
48C2×M16×C53—22no—
49C4×D4×C53—22no—
50C4×D4×C53—22no—
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C22⋊C8×C53—22no—
56C22⋊C8×C53—22no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C22×C411yescharacteristic
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—41no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C2×D412yes—
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53#4112yes—
73C22×Dic53—22no—
74C22×Dic53—24no—
75C22×Dic53—22no—
76C22×Dic53—44no—
77C2×Dic106—28no—
78C2×Dic106—24no—
79C2×Dic106—48no—
80C2×Dic106—44no—
81C2×Dic106—42no—
82C2×Dic106—44no—
83C2×Dic106—41no—
84C2×Dic106—42no—
85C2×Dic106—41no—
86M16×C53—44no—
87C2×C8×C53—42no—
88C4⋊C4×C53—22no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53—22no—
91C42×C53—21no—
92C42×C53—21no—
93C22⋊Dic53—44no—
94C22⋊Dic53—42no—
95C4⋊Dic53—42no—
96#11719919—44no—
97#11719917—44no—
98C2×#88322—24no—
99C2×#88322—82no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53#10112yescomplemented
109C23×C53C2×C22⋊C411yescharacteristic
110C23×C53C4×D412yes—
111D4×C53—44no—
112C2×C4×C53—22no—
113C2×C4×C53—24no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—41no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—81no—
125C2×C4×C53#12011yescharacteristic
126C2×C4×C53#12112yes—
127C2×C4×C53#12311yescharacteristic
128C2×Dic53—28no—
129C2×Dic53—24no—
130C2×Dic53—42no—
131C2×Dic53—44no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136Dic106—48no—
137Dic106—44no—
138Dic106—44no—
139C8×C53—82no—
140#88322—44no—
141#9505—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—81no—
154C22×C53#38611yescharacteristic
155C22×C53C2×#10111yescharacteristic
156C22×C53#41311yescharacteristic
157C4×C53—22no—
158C4×C53—21no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53B(4,𝔽2)12yes—
163Dic53—44no—
164Dic53—82no—
165#2973—531no—
166#2409—531no—
167#1784—532no—
168#1465D5311yescharacteristic
169#1305—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—42no—
173C2×C53—42no—
174C2×C53#188911yescharacteristic
175C2×C53#190111yescharacteristic
176C2×C53#191511yescharacteristic
177C23×Q8—531no—
178C2×C22⋊Q8—1061no—
179C2×C4×D4C2×D5311yescharacteristic
180#435—1064no—
181#432—1061no—
182C2×#101—1064no—
183#407—532no—
184#407C2×D5312yes—
185C2×C22⋊C8—1061no—
186#395—1061no—
187#393—1061no—
188#391—1061no—
189#386—532no—
190#385—532no—
191#380—531no—
192#324—21no—
193#323—21no—
194C53#950511yescharacteristic, complemented, Sylow
195C22×Q8—1061no—
196C22×Q8—1062no—
197C22×Q8—1062no—
198C22×Q8—1061no—
199C22×Q8—2124no—
200C22×D4C4×D5311yescharacteristic
201C23×C4—532no—
202C23×C4D21211yescharacteristic
203C23×C4C22×D5311yescharacteristic
204C2×M16—22no—
205C2×M16—534no—
206C22×C8—1062no—
207C22⋊Q8—2124no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—1062no—
212C2×C4⋊C4C22⋊D5311yescharacteristic
213C2×C22⋊C4—532no—
214C2×C22⋊C4—1062no—
215C2×C22⋊C4C4×D5311yescharacteristic
216C2×C22⋊C4C22⋊D5311yescharacteristic
217C2×C42D21211yescharacteristic
218C4⋊C8—1062no—
219#101—2128no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—1062no—
223C22⋊C8—1062no—
224C22⋊C8—1062no—
225#97—1062no—
226#95—1061no—
227#95—1062no—
228#95—1062no—
229C24#1171992111yescharacteristic
230C24C2×C4×D5311yescharacteristic
231C2×Q8—1068no—
232C2×Q8—1064no—
233C2×Q8—2122no—
234C2×Q8—2124no—
235C2×Q8—2128no—
236C2×Q8—2124no—
237C2×Q8—2121no—
238C2×Q8—2122no—
239C2×Q8—2121no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—1062no—
248C22×C4—1064no—
249C22×C4—1062no—
250C22×C4—2124no—
251C22×C4#1171992111yescharacteristic
252C22×C4#1171992111yescharacteristic
253C22×C4#1171992111yescharacteristic
254C22×C4C2×D21211yescharacteristic
255C22×C4D4×D5312yes—
256C22×C4#1171994712yes—
257C22×C4C2×C22⋊D5311yescharacteristic
258M16—44no—
259M16—2124no—
260C2×C8—42no—
261C2×C8—1064no—
262C2×C8—4242no—
263C4⋊C4—22no—
264C4⋊C4—2122no—
265C4⋊C4—2124no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—2124no—
269C22⋊C4—2122no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#1313313512yescomplemented
281C23C2×#1171992111yescharacteristic
282C23#1313317712yes—
283Q8—2128no—
284Q8—2124no—
285Q8—2124no—
286D4—44no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—1064no—
301C2×C4—1068no—
302C2×C4—2122no—
303C2×C4—2122no—
304C2×C4—2122no—
305C2×C4—2122no—
306C2×C4—2124no—
307C2×C4—2122no—
308C2×C4#1313318711yescharacteristic
309C2×C4#1313318912yes—
310C2×C4#1313320111yescharacteristic
311C8—82no—
312C8—2124no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#1681050011yescharacteristic
326C22C2×#1313313511yescharacteristic
327C22#1681060511yescharacteristic
328C4—22no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—4242no—
334C4—2124no—
335C4#1681081112yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—42no—
340C2#2027512111yescharacteristic
341C2#2027518111yescharacteristic
342C2#2027526511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1465C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#1465D53
hypocenterC53#9505
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #20274638C2semidirect
4. #20272622C2semidirect
108. C23×C53#101semidirect
194. C53#9505semidirect
280. C23#13133135semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202780802
#202746382
#202726222
C53×#14652
#950553