Finite Group Databasebeta

← #27280161

Subgroups — #27280161

Order 13568.

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

normal subgroups only

This group has 622 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#20277333C211yesmaximal, characteristic, complemented
2#20273283C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic, complemented
4#20272582C212yesmaximal, complemented
5C53×#2618C211yesmaximal, characteristic
6C2×#13133176—21no—
7C2×C22⋊#88322C2212yes—
8#16810522—24no—
9#16810483—21no—
10#16810483C2212yes—
11#16810357—24no—
12#16810349—22no—
13C2×#123×C53—21no—
14C2×#121×C53C2211yescharacteristic
15C22×C4⋊C4×C53C2211yescharacteristic
16C53×#398C412yescomplemented
17C53×#396—22no—
18C53×#394C412yescomplemented
19C53×#386—21no—
20C53×#380C2211yescharacteristic
21C22×#88322—22no—
22#13133176—42no—
23#13133176—42no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53—22no—
27C2×C22⋊Dic53—22no—
28C2×C4×Dic53—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133124—22no—
32#13133124D414yes—
33C22×D4×C53—21no—
34C22×D4×C53C2×C411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C23×C4×C53D411yescharacteristic
37C23×C4×C53C2311yescharacteristic
38#123×C53—42no—
39#123×C53—42no—
40#121×C53—22no—
41#121×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—22no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C412yes—
48C2×C4⋊C4×C53D412yes—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53—22no—
52C2×C22⋊C4×C53—21no—
53#95×C53—22no—
54#95×C53C2×C412yes—
55C24×C53—21no—
56C24×C53C2×D411yescharacteristic
57C2×D4×C53—22no—
58C2×D4×C53—41no—
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53—81no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—22no—
73C22×C4×C53—81no—
74C22×C4×C53—81no—
75C22×C4×C53C22⋊C412yes—
76C22×C4×C53C22⋊C412yes—
77C22×C4×C53D812yes—
78C22×C4×C53SD1612yes—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D412yes—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53#4112yes—
83C22×Dic53—24no—
84C4⋊C4×C53—24no—
85C4⋊C4×C53—24no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—44no—
88C4⋊C4×C53—81no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—44no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—82no—
94C22⋊Dic53—42no—
95C22⋊Dic53—42no—
96C4⋊Dic53—44no—
97C4⋊Dic53—82no—
98C4⋊Dic53—82no—
99C4×Dic53—44no—
100C2×#88322—42no—
101C2×#88322—82no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53C2×C22⋊C411yescharacteristic
110C23×C53#12011yescharacteristic
111C23×C53#12311yescharacteristic
112C23×C53C2×D811yescharacteristic
113C23×C53C2×SD1611yescharacteristic
114C23×C53AGL(1,ℤ/8)11yescharacteristic
115C23×C53#13711yescharacteristic
116D4×C53—82no—
117D4×C53—81no—
118D4×C53—81no—
119D4×C53—81no—
120D4×C53—81no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—81no—
141C2×C4×C53#10214yes—
142C2×C4×C53C4×D412yes—
143C2×C4×C53#12112yes—
144C2×Dic53—44no—
145C2×Dic53—44no—
146C2×Dic53—44no—
147#88322—82no—
148#9204—531nomaximal, Sylow
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—82no—
160C22×C53—82no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53—81no—
164C22×C53#38611yescharacteristic
165C22×C53C2×#10211yescharacteristic
166C22×C53#41711yescharacteristic
167C22×C53#44711yescharacteristic
168C22×C53#44811yescharacteristic
169C22×C53#48011yescharacteristic
170C22×C53#48111yescharacteristic
171C4×C53—24no—
172C4×C53—22no—
173C4×C53—22no—
174C4×C53—42no—
175C4×C53—82no—
176C4×C53—81no—
177Dic53—84no—
178#2618D5311yescharacteristic
179#2316—531no—
180#1519—532no—
181C2×#324—531no—
182#1289—532no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—161no—
190C2×C53#187411yescharacteristic
191C2×C53#191211yescharacteristic
192C2×C53#191311yescharacteristic
193C2×#123—21no—
194C2×#121C2×D5311yescharacteristic
195C2×#117—1061no—
196C22×C4⋊C4C2×D5311yescharacteristic
197C2×C22⋊C8—532no—
198#398Dic5312yescomplemented
199#396—22no—
200#394Dic5312yescomplemented
201#389—1064no—
202#386—21no—
203#380—532no—
204#380—1061no—
205#380C2×D5311yescharacteristic
206#327—1064no—
207#324—1062no—
208C53#920411yescharacteristic, complemented, Sylow
209C22×D4—21no—
210C22×D4C2×Dic5311yescharacteristic
211C23×C4C22⋊D5311yescharacteristic
212C23×C4C22⋊D5311yescharacteristic
213C23×C4C22×D5311yescharacteristic
214C22×C8—1062no—
215#123—42no—
216#123—42no—
217#121—22no—
218#121—22no—
219#117—2122no—
220#117—2122no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1062no—
226C2×C4⋊C4—1062no—
227C2×C4⋊C4—1062no—
228C2×C4⋊C4C2×Dic5311yescharacteristic
229C2×C4⋊C4C2×Dic5312yes—
230C2×C4⋊C4C22⋊D5312yes—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—1062no—
236C2×C42—1062no—
237C22⋊C8—1062no—
238C22⋊C8—1062no—
239#95—22no—
240#95—534no—
241#95—1062no—
242#95C2×Dic5312yes—
243C24—21no—
244C24C2×C22⋊D5311yescharacteristic
245C2×D4—22no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—81no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—1064no—
264C22×C4#1171992212yes—
265C22×C4#1171992312yes—
266C22×C4C22⋊Dic5312yes—
267C22×C4C22⋊Dic5312yes—
268C22×C4D4×D5312yes—
269C22×C4#1171994712yes—
270C22×C4C22×Dic5311yescharacteristic
271C22×C4C2×C22⋊D5311yescharacteristic
272C2×C8—4242no—
273C2×C8—2122no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—44no—
278C4⋊C4—81no—
279C4⋊C4—4242no—
280C4⋊C4—4242no—
281C4⋊C4—2124no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—82no—
287C22⋊C4—2122no—
288C22⋊C4—2122no—
289C42—2124no—
290C23—41no—
291C23—41no—
292C23—42no—
293C23—42no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊Dic5311yescharacteristic
298C23#1313318511yescharacteristic
299C23#1313319811yescharacteristic
300C23C2×#1171992211yescharacteristic
301C23C2×#1171992311yescharacteristic
302C23#1313324211yescharacteristic
303C23#1313324711yescharacteristic
304D4—82no—
305D4—81no—
306D4—81no—
307D4—81no—
308D4—81no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—2124no—
330C2×C4—2124no—
331C2×C4—2124no—
332C2×C4#1313313814yes—
333C2×C4D4×Dic5312yes—
334C2×C4#1313318912yes—
335C8—4242no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—82no—
347C22—82no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22#1681050311yescharacteristic
352C22C2×#1313313811yescharacteristic
353C22#1681062411yescharacteristic
354C22#1681075611yescharacteristic
355C22#1681076111yescharacteristic
356C22#1681093411yescharacteristic
357C22#1681093911yescharacteristic
358C4—22no—
359C4—22no—
360C4—24no—
361C4—42no—
362C4—82no—
363C4—81no—
364C4—4244no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—161no—
372C2#2027503311yescharacteristic
373C2#2027525311yescharacteristic
374C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2618C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#2618D53
hypocenterC53#9204
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
1. #20277333C2semidirect
3. C2×#16810349C2semidirect
4. #20272582C2semidirect
16. C53×#398C4semidirect
18. C53×#394C4semidirect
198. #398Dic53semidirect
200. #394Dic53semidirect
208. C53#9204semidirect

Maximal subgroups

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

subgroupindex
#202773332
#202732832
C2×#168103492
#202725822
C53×#26182
#920453