Finite Group Databasebeta

← #27281744

Subgroups — #27281744

Order 13568.

516 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 95 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#20283353C211yesmaximal, characteristic, complemented
2#20279173C211yesmaximal, characteristic
3#20273243C212yesmaximal
4#20272567C212yesmaximal, complemented
5C53×#1820C211yesmaximal, characteristic, complemented
6C2×#13133191C2211yescharacteristic
7#16810684—24no—
8#16810647—21no—
9#16810587C2212yes—
10C4×#11719917—21no—
11#16810538—21no—
12#16810538—21no—
13#16810524C411yescharacteristic, complemented
14#16810524C411yescharacteristic, complemented
15#16810514—21no—
16#16810514—21no—
17#16810492C411yescharacteristic, complemented
18#16810488C411yescharacteristic, complemented
19C4×C22⋊Dic53C2211yescharacteristic
20#16810356—24no—
21C2×#117×C53C2211yescharacteristic
22C53×#381C2212yes—
23C53×#342—21no—
24C53×#342—21no—
25C22×D212C2×C411yescharacteristic
26C22×C22⋊D53—21no—
27C2×#11719917—22no—
28#13133191—22no—
29#13133191—22no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53C2×C411yescharacteristic
41C4⋊#88322—24no—
42C22⋊#88322—22no—
43C22⋊#88322D414yes—
44#13133128—22no—
45C4×#88322—22no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C23×C4×C53C2311yescharacteristic
50#117×C53—22no—
51#117×C53—22no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53—22no—
55C2×C22⋊C4×C53D411yescharacteristic
56C2×C22⋊C4×C53D411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59#95×C53—22no—
60#95×C53—22no—
61C24×C53C2×D411yescharacteristic
62C23×D53—21no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53—42no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D412yes—
73C22×C4×C53#4112yes—
74C2×C22⋊D53—42no—
75C2×C22⋊D53—81no—
76C2×C22⋊D53—81no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C2×D212—22no—
81C2×D212—41no—
82C2×D212—41no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—22no—
85C22⋊C4×C53—22no—
86C22⋊C4×C53—22no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—42no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—21no—
93C22⋊Dic53—42no—
94#11719921—42no—
95#11719921—42no—
96#11719921—41no—
97#11719921—41no—
98#11719921—81no—
99#11719921—81no—
100C4⋊Dic53—22no—
101#11719919—81no—
102#11719919—81no—
103C4×Dic53—41no—
104C4×Dic53—41no—
105#11719917—42no—
106#11719917—42no—
107C2×#88322—24no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53C2×C22⋊C411yescharacteristic
113C23×C53#12112yes—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53—41no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—41no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—82no—
136C2×C4×C53—82no—
137C2×C4×C53—82no—
138C2×C4×C53C4×D412yes—
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12311yescharacteristic
141C22⋊D53—84no—
142C2×Dic53—22no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151D212—82no—
152#88322—44no—
153#9446—531nomaximal, Sylow
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—81no—
160C22×C53—81no—
161C22×C53#38611yescharacteristic
162C22×C53#41011yescharacteristic
163C22×C53#41011yescharacteristic
164C2×D53—81no—
165C2×D53—81no—
166C2×D53—81no—
167C2×D53—81no—
168C2×D53—81no—
169C2×D53—81no—
170C2×D53—81no—
171C4×C53—21no—
172C4×C53—22no—
173C4×C53—21no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53—162no—
179Dic53—42no—
180Dic53—81no—
181Dic53—81no—
182#2921—531no—
183#2500—531no—
184#1820D5311yescharacteristic, complemented
185#1510—532no—
186#1284—532no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—42no—
190C2×C53#188411yescharacteristic
191C2×C53#189911yescharacteristic
192C2×C53#191011yescharacteristic
193D53—161no—
194C2×#121—531no—
195C2×#117C2×D5311yescharacteristic
196#433—1064no—
197#423—1061no—
198#407—532no—
199C4×M16—1061no—
200#394—1061no—
201#394—1061no—
202#390—531no—
203#390—531no—
204#388—1061no—
205#388—1061no—
206#384—531no—
207#382—531no—
208#381C2×D5312yes—
209C4×C22⋊C4—531no—
210#342—21no—
211#342—21no—
212#327—1064no—
213C53#944611yescharacteristic, complemented, Sylow
214C22×D4—531no—
215C22×D4—1061no—
216C23×C4C22×D5311yescharacteristic
217C2×M16—1062no—
218#121—1062no—
219#121—1062no—
220#117—22no—
221#117—22no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4C4×D5311yescharacteristic
226C2×C4⋊C4C4×D5311yescharacteristic
227C2×C22⋊C4—22no—
228C2×C22⋊C4—1061no—
229C2×C22⋊C4—1061no—
230C2×C22⋊C4—1061no—
231C2×C22⋊C4—1061no—
232C2×C22⋊C4—1061no—
233C2×C22⋊C4—1061no—
234C2×C22⋊C4C22⋊D5311yescharacteristic
235C2×C22⋊C4C22⋊D5311yescharacteristic
236C2×C42—531no—
237C2×C42—531no—
238C2×C42D21211yescharacteristic
239C2×C42D21211yescharacteristic
240C4⋊C8—1064no—
241C22⋊C8—534no—
242C22⋊C8—1062no—
243#97—1062no—
244C4×C8—1062no—
245#95—22no—
246#95—22no—
247#95—1061no—
248#95—1061no—
249#95—1061no—
250C24—1061no—
251C24C2×C22⋊D5311yescharacteristic
252C2×D4—1062no—
253C2×D4—4241no—
254C2×D4—4241no—
255C2×D4—2121no—
256C2×D4—2121no—
257C2×D4—2122no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4#1171992111yescharacteristic
265C22×C4#1171992111yescharacteristic
266C22×C4#1171992111yescharacteristic
267C22×C4#1171992111yescharacteristic
268C22×C4C2×C4×D5311yescharacteristic
269C22×C4C2×D21211yescharacteristic
270C22×C4D4×D5312yes—
271C22×C4#1171994712yes—
272M16—2122no—
273M16—2122no—
274C2×C8—1064no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—1062no—
278C4⋊C4—4241no—
279C4⋊C4—4241no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—4241no—
285C22⋊C4—4241no—
286C22⋊C4—2122no—
287C22⋊C4—2122no—
288C22⋊C4—2122no—
289C22⋊C4—2121no—
290C22⋊C4—2121no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—2121no—
296C42—2121no—
297C23—21no—
298C23—21no—
299C23—41no—
300C23—41no—
301C23—2121no—
302C23—2121no—
303C23—2121no—
304C23—2121no—
305C23—2121no—
306C23—2121no—
307C23—2121no—
308C23C2×#1171992111yescharacteristic
309C23#1313318912yes—
310D4—4242no—
311D4—4244no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—1062no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4#1313317712yes—
339C2×C4#1313318711yescharacteristic
340C2×C4#1313320111yescharacteristic
341C8—2124no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—81no—
348C22—81no—
349C22—4241no—
350C22—4241no—
351C22—4241no—
352C22—4241no—
353C22—4241no—
354C22—4241no—
355C22—4241no—
356C22#1681050011yescharacteristic
357C22#1681059611yescharacteristic
358C22#1681059611yescharacteristic
359C4—21no—
360C4—22no—
361C4—21no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—162no—
367C4—4241no—
368C4—4241no—
369C4—2122no—
370C2—21no—
371C2—21no—
372C2—42no—
373C2—8481no—
374C2#2027509611yescharacteristic
375C2#2027517111yescharacteristic
376C2#2027524011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1820C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#1820D53
hypocenterC53#9446
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
1. #20283353C2semidirect
4. #20272567C2semidirect
5. C53×#1820C2semidirect
13. #16810524C4semidirect
14. #16810524C4semidirect
17. #16810492C4semidirect
18. #16810488C4semidirect
184. #1820D53semidirect
213. C53#9446semidirect

Maximal subgroups

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

subgroupindex
#202833532
#202791732
#202732432
#202725672
C53×#18202
#944653