Finite Group Databasebeta

← #27280158

Subgroups — #27280158

Order 13568.

622 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 127 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#20280457C211yesmaximal, characteristic
2#20277333C211yesmaximal, characteristic, complemented
3#20273284C212yesmaximal
4#20272583C212yesmaximal, complemented
5C2×C53×#324C211yesmaximal, characteristic, complemented
6C2×#13133200—21no—
7C2×#13133191C2211yescharacteristic
8C2×#13133176—21no—
9#16810553C412yescomplemented
10#16810545—22no—
11#16810540C412yescomplemented
12#16810522—24no—
13#16810502—21no—
14#16810483—21no—
15#16810483C2211yescharacteristic
16#16810483C2212yes—
17#16810358—24no—
18C22×C4⋊C4×C53C2211yescharacteristic
19C2×C22⋊C8×C53C2212yes—
20C53×#324—22no—
21C22×D212C2×C411yescharacteristic
22C22×C22⋊D53—21no—
23#13133200—42no—
24#13133200—42no—
25#13133191—22no—
26#13133191—22no—
27#13133176—42no—
28#13133176—42no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—22no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—22no—
37C2×#11719921—22no—
38C2×#11719921—22no—
39C2×#11719921—21no—
40C2×C4×Dic53—22no—
41#13133124—22no—
42#13133124—22no—
43#13133124C2×C412yes—
44#13133124D414yes—
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C23×C4×C53C2311yescharacteristic
48C22×C8×C53—22no—
49C2×C4⋊C4×C53—22no—
50C2×C4⋊C4×C53—22no—
51C2×C4⋊C4×C53C2×C412yes—
52C2×C4⋊C4×C53D412yes—
53C22⋊C8×C53—22no—
54C22⋊C8×C53—22no—
55C24×C53C2×D411yescharacteristic
56C23×D53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—81no—
64C22×C4×C53—81no—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22⋊C412yes—
67C22×C4×C53D812yes—
68C22×C4×C53SD1612yes—
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D412yes—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53#4112yes—
73C2×C22⋊D53—41no—
74C2×C22⋊D53—41no—
75C2×C22⋊D53—41no—
76C2×C22⋊D53—41no—
77C2×C22⋊D53—41no—
78C2×C22⋊D53—41no—
79C22×Dic53—21no—
80C22×Dic53—22no—
81C22×Dic53—24no—
82C2×D212—22no—
83C2×D212—81no—
84C2×C8×C53—42no—
85C2×C8×C53—82no—
86C4⋊C4×C53—24no—
87C4⋊C4×C53—24no—
88C22⋊Dic53—42no—
89C22⋊Dic53—41no—
90C22⋊Dic53—41no—
91C22⋊Dic53—42no—
92#11719921—44no—
93#11719921—42no—
94#11719921—82no—
95C4⋊Dic53—22no—
96C4⋊Dic53—44no—
97C4⋊Dic53—82no—
98C4⋊Dic53—82no—
99C4⋊Dic53—81no—
100#11719919—44no—
101C4×Dic53—44no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53#12011yescharacteristic
106C23×C53#12311yescharacteristic
107C23×C53C2×D811yescharacteristic
108C23×C53C2×SD1611yescharacteristic
109C23×C53AGL(1,ℤ/8)11yescharacteristic
110C23×C53#13711yescharacteristic
111C22×D53—42no—
112C22×D53—42no—
113C22×D53—41no—
114C22×D53—41no—
115C22×D53—41no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—41no—
124C2×C4×C53—42no—
125C2×C4×C53—41no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—81no—
130C2×C4×C53#10214yes—
131C2×C4×C53C4×D412yes—
132C2×C4×C53#12112yes—
133C22⋊D53—81no—
134C22⋊D53—81no—
135C22⋊D53—81no—
136C22⋊D53—81no—
137C2×Dic53—44no—
138C2×Dic53—42no—
139C2×Dic53—44no—
140C2×Dic53—42no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—42no—
144C2×Dic53—41no—
145C2×Dic53—44no—
146D212—82no—
147C8×C53—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#38611yescharacteristic
160C22×C53C2×#10211yescharacteristic
161C22×C53#41711yescharacteristic
162C22×C53#44711yescharacteristic
163C22×C53#44811yescharacteristic
164C22×C53#48011yescharacteristic
165C22×C53#48111yescharacteristic
166C2×D53—82no—
167C2×D53—82no—
168C2×D53—81no—
169C2×D53—81no—
170C2×D53—81no—
171C4×C53—24no—
172C4×C53—22no—
173C4×C53—22no—
174C4×C53—42no—
175Dic53—84no—
176Dic53—82no—
177Dic53—81no—
178#2618—531no—
179#2316—531no—
180#1519—532no—
181C2×#324D5311yescharacteristic, complemented
182#1289—532no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53#187411yescharacteristic
190C2×C53#191211yescharacteristic
191C2×C53#191311yescharacteristic
192D53—161no—
193C2×#123—1061no—
194C2×#121—531no—
195C2×#117—1061no—
196C22×C4⋊C4C2×D5311yescharacteristic
197C2×C22⋊C8C2×D5312yes—
198#398—532no—
199#396—1062no—
200#394—532no—
201#389—1064no—
202#386—1061no—
203#380—531no—
204#380—532no—
205#380—1061no—
206#327—1064no—
207#324—22no—
208C53#920411yescharacteristic, complemented, Sylow
209C22×D4—531no—
210C22×D4—1061no—
211C23×C4D21211yescharacteristic
212C23×C4D21211yescharacteristic
213C23×C4C22×D5311yescharacteristic
214C22×C8—22no—
215#123—2122no—
216#123—2122no—
217#121—1062no—
218#121—1062no—
219#117—2122no—
220#117—2122no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—1062no—
225C2×C4⋊C4—1062no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1062no—
228C2×C4⋊C4—1062no—
229C2×C4⋊C4C4×D5312yes—
230C2×C4⋊C4C22⋊D5312yes—
231C2×C22⋊C4—1062no—
232C2×C22⋊C4—1062no—
233C2×C22⋊C4—1061no—
234C2×C22⋊C4—1061no—
235C2×C22⋊C4—1062no—
236C2×C42—1062no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239#95—532no—
240#95—534no—
241#95—1062no—
242#95—1062no—
243C24—1061no—
244C24C2×D21211yescharacteristic
245C2×D4—1062no—
246C2×D4—4241no—
247C2×D4—2121no—
248C2×D4—2121no—
249C2×D4—2121no—
250C2×D4—2121no—
251C2×D4—2121no—
252C2×D4—2121no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—1062no—
262C22×C4—1061no—
263C22×C4—1064no—
264C22×C4#1171991312yes—
265C22×C4D42412yes—
266C22×C4#1171992112yes—
267C22×C4#1171992112yes—
268C22×C4C2×C4×D5311yescharacteristic
269C22×C4D4×D5312yes—
270C22×C4#1171994712yes—
271C22×C4C2×C22⋊D5311yescharacteristic
272C2×C8—42no—
273C2×C8—82no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—1062no—
277C4⋊C4—4242no—
278C4⋊C4—4241no—
279C4⋊C4—4242no—
280C4⋊C4—2124no—
281C4⋊C4—2124no—
282C22⋊C4—4242no—
283C22⋊C4—2122no—
284C22⋊C4—2121no—
285C22⋊C4—2121no—
286C22⋊C4—2122no—
287C22⋊C4—2124no—
288C22⋊C4—2122no—
289C42—2124no—
290C23—41no—
291C23—41no—
292C23—2122no—
293C23—2122no—
294C23—2121no—
295C23—2121no—
296C23—2121no—
297C23C2×#1171992111yescharacteristic
298C23#1313318711yescharacteristic
299C23#1313320111yescharacteristic
300C23C2×D42411yescharacteristic
301C23C2×#1171991311yescharacteristic
302C23#1313324511yescharacteristic
303C23#1313325011yescharacteristic
304D4—4242no—
305D4—4241no—
306D4—4241no—
307D4—4241no—
308D4—4241no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—2122no—
324C2×C4—2124no—
325C2×C4—2124no—
326C2×C4—2122no—
327C2×C4—2121no—
328C2×C4—2124no—
329C2×C4—2122no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4#1313313914yes—
333C2×C4#1313317712yes—
334C2×C4#1313318912yes—
335C8—82no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—4242no—
347C22—4242no—
348C22—4241no—
349C22—4241no—
350C22—4241no—
351C22#1681050011yescharacteristic
352C22C2×#1313313911yescharacteristic
353C22#1681062511yescharacteristic
354C22#1681075911yescharacteristic
355C22#1681076411yescharacteristic
356C22#1681093711yescharacteristic
357C22#1681094211yescharacteristic
358C4—22no—
359C4—22no—
360C4—24no—
361C4—42no—
362C4—4244no—
363C4—4242no—
364C4—4241no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—8481no—
372C2#2027503011yescharacteristic
373C2#2027525011yescharacteristic
374C2#2027525511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#324C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenterC2×#324D53
hypocenterC53#9204
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #20277333C2semidirect
4. #20272583C2semidirect
5. C2×C53×#324C2semidirect
9. #16810553C4semidirect
11. #16810540C4semidirect
181. C2×#324D53semidirect
208. C53#9204semidirect

Maximal subgroups

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

subgroupindex
#202804572
#202773332
#202732842
#202725832
C2×C53×#3242
#920453