Finite Group Databasebeta

← #27281813

Subgroups — #27281813

Order 13568.

512 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 93 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#20279201C211yesmaximal, characteristic
2#20277316C211yesmaximal, characteristic
3#20274793C211yesmaximal, characteristic
4#20273248C211yesmaximal, characteristic
5#20272569C211yesmaximal, characteristic
6#20272567C211yesmaximal, characteristic
7C53×#1511C211yesmaximal, characteristic
8C22×C4⋊Dic53C2211yescharacteristic
9#16810587C2211yescharacteristic
10#16810562—22no—
11#16810530C412yescomplemented
12#16810508—22no—
13#16810506—22no—
14#16810488C412yescomplemented
15#16810486C2211yescharacteristic
16#16810484—21no—
17#16810484—21no—
18#16810481—21no—
19C22⋊C4×Dic53—21no—
20C4×C22⋊Dic53C2211yescharacteristic
21#16810391—22no—
22#16810360—22no—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C53×#407C2211yescharacteristic
25C53×#381C2211yescharacteristic
26C53×#328—22no—
27C23×Dic53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C2×#11719917—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53C2×C412yes—
35C2×C4⋊Dic53D412yes—
36C2×#11719919—22no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C4×Dic53—22no—
40C2×C4×Dic53C2×C412yes—
41C22⋊#88322—21no—
42C22⋊#88322—21no—
43#13133124—22no—
44#13133124—22no—
45#13133124—21no—
46#13133124—21no—
47#13133124—22no—
48#13133124—22no—
49#13133124—22no—
50#13133124—22no—
51#13133124D412yes—
52C22×Q8×C53C2×C411yescharacteristic
53C23×C4×C53C2311yescharacteristic
54C2×M16×C53—21no—
55C22⋊Q8×C53—21no—
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53C2×C411yescharacteristic
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C22⋊C8×C53—21no—
64C22⋊C8×C53—21no—
65#95×C53—21no—
66#95×C53D412yes—
67C24×C53C2×D411yescharacteristic
68C2×Q8×C53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—42no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—22no—
74C22×C4×C53—42no—
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—22no—
86C22×Dic53—22no—
87C22×Dic53—42no—
88C22×Dic53—42no—
89C22×Dic53C22⋊C412yes—
90C22×Dic53C22⋊C412yes—
91M16×C53—42no—
92C2×C8×C53—41no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C22⋊C4×C53—42no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊Dic53—42no—
101C22⋊Dic53—42no—
102C4⋊Dic53—22no—
103C4⋊Dic53—22no—
104C4⋊Dic53—22no—
105C4⋊Dic53—22no—
106#11719919—82no—
107C4×Dic53—42no—
108C4×Dic53—82no—
109#11719917—42no—
110C2×#88322—41no—
111C23×C53—22no—
112C23×C53—22no—
113C23×C53—41no—
114C23×C53C2×C22⋊C411yescharacteristic
115C23×C53#12011yescharacteristic
116C23×C53#12311yescharacteristic
117Q8×C53—81no—
118Q8×C53—81no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53C4×D411yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×Dic53—22no—
137C2×Dic53—22no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144C2×Dic53—42no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—42no—
148C2×Dic53—81no—
149C2×Dic53—81no—
150C8×C53—81no—
151#88322—81no—
152#9460—531nomaximal, Sylow
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—44no—
161C22×C53#38611yescharacteristic
162C22×C53#41012yes—
163C22×C53#45512yes—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—82no—
169C4×C53—81no—
170C4×C53—81no—
171Dic53—42no—
172Dic53—42no—
173Dic53—82no—
174Dic53—82no—
175#2503—531no—
176#2313—531no—
177#1821—531no—
178#1511—531no—
179#1511D5311yescharacteristic
180#1284—531no—
181#1284—531no—
182C2×C53—22no—
183C2×C53—22no—
184C2×C53—41no—
185C2×C53#188511yescharacteristic
186C2×C53#189911yescharacteristic
187C2×C53#191211yescharacteristic
188C2×C22⋊Q8C2×D5311yescharacteristic
189C22×C4⋊C4—531no—
190#407—531no—
191#407C2×D5311yescharacteristic
192#400—1062no—
193#391—532no—
194#387—1062no—
195#387—1062no—
196#382—532no—
197#381—531no—
198#381—1061no—
199#381—1061no—
200#381C2×D5311yescharacteristic
201#380—1061no—
202C4×C22⋊C4—531no—
203C4×C22⋊C4—1061no—
204#343—1062no—
205#328—22no—
206#328—1062no—
207C53#946011yescharacteristic, complemented, Sylow
208C22×Q8C4×D5311yescharacteristic
209C23×C4—531no—
210C23×C4—531no—
211C23×C4C22×D5311yescharacteristic
212C2×M16—21no—
213C2×M16—1061no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—532no—
220C2×C4⋊C4—532no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1062no—
226C2×C4⋊C4C4×D5311yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—1061no—
230C2×C22⋊C4—1061no—
231C2×C42—532no—
232C2×C42—1062no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235C22⋊C8—1061no—
236C22⋊C8—1061no—
237#95—21no—
238#95—532no—
239#95—1062no—
240#95—1062no—
241#95—1062no—
242#95—1062no—
243#95—1062no—
244#95—1062no—
245#95—1061no—
246#95—1061no—
247#95C22⋊D5312yes—
248C24D4×D5311yescharacteristic
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—42no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—532no—
257C22×C4—532no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1062no—
263C22×C4—1062no—
264C22×C4—2122no—
265C22×C4—2122no—
266C22×C4C2×C4×D5311yescharacteristic
267C22×C4#1171994511yescharacteristic
268C22×C4D4×D5311yescharacteristic
269C22×C4D4×D5311yescharacteristic
270C22×C4#1171994711yescharacteristic
271C22×C4C2×C22⋊D5311yescharacteristic
272M16—42no—
273M16—2122no—
274C2×C8—41no—
275C2×C8—2121no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—1062no—
281C4⋊C4—1062no—
282C4⋊C4—1062no—
283C4⋊C4—1062no—
284C4⋊C4—4242no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—2122no—
289C22⋊C4—2122no—
290C42—4242no—
291C42—2122no—
292C23—22no—
293C23—22no—
294C23—41no—
295C23C22⋊C4×D5311yescharacteristic
296C23#1313318511yescharacteristic
297C23#1313319911yescharacteristic
298Q8—81no—
299Q8—81no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—1062no—
314C2×C4—1062no—
315C2×C4—4241no—
316C2×C4—4241no—
317C2×C4—2122no—
318C2×C4—2122no—
319C2×C4—2122no—
320C2×C4—2122no—
321C2×C4—2122no—
322C2×C4—2122no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4#1313317711yescharacteristic
328C2×C4C4×C22⋊D5311yescharacteristic
329C2×C4#1313318811yescharacteristic
330C2×C4#1313319011yescharacteristic
331C8—81no—
332C8—4241no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—44no—
341C22#1681049911yescharacteristic
342C22#1681059512yes—
343C22#1681079912yes—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—82no—
349C4—81no—
350C4—81no—
351C4—4242no—
352C4—4242no—
353C4—2122no—
354C4—2122no—
355C2—22no—
356C2—22no—
357C2—41no—
358C2#2027510011yescharacteristic
359C2#2027517011yescharacteristic
360C2#2027524911yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1511C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1511D53
hypocenterC53#9460
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
11. #16810530C4semidirect
14. #16810488C4semidirect
207. C53#9460semidirect

Maximal subgroups

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

subgroupindex
#202792012
#202773162
#202747932
#202732482
#202725692
#202725672
C53×#15112
#946053