Finite Group Databasebeta

← #28303343

Subgroups — #28303343

Order 13568.

588 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 108 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#20290607C211yesmaximal, characteristic
2#20289891C211yesmaximal, characteristic
3#20289588C212yesmaximal
4#20289588C212yesmaximal
5#20277119C211yesmaximal, characteristic
6C2×#13133274C2212yes—
7C2×(C2×#6157)⋊C4—21no—
8C2×(C2×#6157)⋊C4—21no—
9C2×(C2×#6157)⋊C4C2212yes—
10#16811489—21no—
11#16811489—21no—
12C2×#13133174—21no—
13C2×C42×D53C2211yescharacteristic
14#16810492C411yescharacteristic, complemented
15#16810492C411yescharacteristic, complemented
16#16810487—21no—
17#16810482—21no—
18#16810480C2211yescharacteristic
19#16810480C2211yescharacteristic
20C4×#11719919—21no—
21C4×C4⋊Dic53C411yescharacteristic, complemented
22C4×C4⋊C4×C53C411yescharacteristic, complemented
23C23×#6157—22no—
24C22×#88324—22no—
25#13133274—24no—
26(C2×#6157)⋊C4—24no—
27(C2×#6157)⋊C4—24no—
28(C2×#6157)⋊C4—42no—
29C22×C4×D53D411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×C4×D53C2311yescharacteristic
32#13133174—44no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C42×D53—24no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C42×C53—21no—
52C2×C42×C53C2×C411yescharacteristic
53#95×C53—21no—
54C23×D53C2×D411yescharacteristic
55C22×#6157—24no—
56C22×#6157—42no—
57C22×#6157—42no—
58C22×#6157—42no—
59C22×#6157—42no—
60C22×#6157—42no—
61C22×#6157—42no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C2×D411yescharacteristic
72C2×C4×D53—22no—
73C2×C4×D53—22no—
74C2×C4×D53—22no—
75C2×C4×D53—22no—
76C2×C4×D53—41no—
77C2×C4×D53—41no—
78C2×C4×D53—41no—
79C2×C4×D53SD1612yes—
80C2×C4×D53Q1612yes—
81C2×C4×D53#4114yes—
82C2×#88324—42no—
83C2×#88324—42no—
84C2×#88324—42no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C42×C53—22no—
88C42×C53—42no—
89#11719921—42no—
90#11719921—42no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93#11719919—42no—
94#11719919—42no—
95#11719919—42no—
96#11719919—42no—
97C4×Dic53—22no—
98C4×Dic53—22no—
99C4×Dic53—22no—
100C4×Dic53—42no—
101C23×C53C2×C22⋊C411yescharacteristic
102C22×D53—21no—
103C22×D53—21no—
104C22×D53—21no—
105C22×D53—21no—
106C22×D53#12312yes—
107C22×D53C2×SD1611yescharacteristic
108C22×D53C2×Q1611yescharacteristic
109C22×D53AGL(1,ℤ/8)12yes—
110C2×#6157—44no—
111C2×#6157—44no—
112C2×#6157—44no—
113C2×#6157—82no—
114C2×#6157—82no—
115C2×#6157—82no—
116C2×#6157—82no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53#10312yes—
131C2×C4×C53#11712yes—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53#10312yes—
145C2×Dic53#10412yes—
146C2×Dic53#12312yes—
147C4×D53—24no—
148C4×D53—24no—
149C4×D53—44no—
150C4×D53—44no—
151#88324—82no—
152#8717—531nomaximal, Sylow
153C22×C53#38111yescharacteristic
154C22×C53#41011yescharacteristic
155C22×C53#41011yescharacteristic
156C22×C53C2×#10311yescharacteristic
157C22×C53#41811yescharacteristic
158C22×C53C2×#10411yescharacteristic
159C22×C53#42111yescharacteristic
160C2×D53—22no—
161C2×D53—22no—
162C2×D53—21no—
163C2×D53—21no—
164C2×D53—22no—
165C2×D53—22no—
166C2×D53—21no—
167C2×D53—21no—
168C2×D53#48112yes—
169C2×D53#48412yes—
170#6157—84no—
171C4×C53—22no—
172C4×C53—22no—
173C4×C53—41no—
174C4×C53—81no—
175C4×C53—81no—
176Dic53—22no—
177Dic53—22no—
178Dic53—22no—
179Dic53—81no—
180Dic53—81no—
181#2285—531no—
182#1817—531no—
183#1434—531no—
184#1294—532no—
185#1294—532no—
186C2×C53#150311yescharacteristic
187C2×C53#150511yescharacteristic
188C2×C53#151411yescharacteristic
189C2×C53#151811yescharacteristic
190C2×C53#182011yescharacteristic
191C2×C53#182411yescharacteristic
192C2×C53#182611yescharacteristic
193D53—22no—
194D53—22no—
195C2×#117—1061no—
196C22×C42—531no—
197C2×C22⋊C8—532no—
198#384—531no—
199#384—531no—
200#382—1061no—
201#380—531no—
202#380—531no—
203#380—1061no—
204C4×C4⋊C4—531no—
205C4×C4⋊C4—1061no—
206C4×C4⋊C4#615711yescharacteristic, complemented
207C2×#95—532no—
208C2×#95—1061no—
209C2×#95—1061no—
210#336—1061no—
211#336—1061no—
212C53#871711yescharacteristic, complemented, Sylow
213C23×C4—531no—
214C23×C4—531no—
215C23×C4—531no—
216C23×C4—1062no—
217C22×C8—1062no—
218#117—2124no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4C2×#615711yescharacteristic
226C2×C4⋊C4C2×#615711yescharacteristic
227C2×C22⋊C4—1061no—
228C2×C22⋊C4—1061no—
229C2×C42—21no—
230C2×C42—531no—
231C2×C42—531no—
232C2×C42—531no—
233C2×C42—1064no—
234C2×C42—1061no—
235C2×C42C2×#615711yescharacteristic
236C22⋊C8—1064no—
237#95—21no—
238#95—1061no—
239#95—1061no—
240#95—1064no—
241#95—1064no—
242#95—1061no—
243#95—2122no—
244C24—531no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—532no—
251C22×C4—532no—
252C22×C4—534no—
253C22×C4—1062no—
254C22×C4—1062no—
255C22×C4—1062no—
256C22×C4—1062no—
257C22×C4—1064no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—2121no—
261C22×C4—2121no—
262C22×C4—2121no—
263C22×C4—2122no—
264C22×C4—2122no—
265C22×C4—2122no—
266C22×C4—2122no—
267C22×C4—2122no—
268C22×C4—2122no—
269C22×C4C22⋊#615711yescharacteristic
270C22×C4C22⋊#615711yescharacteristic
271C22×C4C22×#615711yescharacteristic
272C2×C8—2122no—
273C2×C8—2122no—
274C2×C8—2122no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—2121no—
278C4⋊C4—2121no—
279C4⋊C4—2122no—
280C4⋊C4—2122no—
281C4⋊C4—2122no—
282C4⋊C4—2122no—
283C22⋊C4—2122no—
284C22⋊C4—2122no—
285C42—22no—
286C42—42no—
287C42—1062no—
288C42—1062no—
289C42—1062no—
290C42—2122no—
291C23—532no—
292C23—531no—
293C23—531no—
294C23—532no—
295C23—1061no—
296C23—1061no—
297C23—1061no—
298C23—1061no—
299C23C2×C22⋊#615711yescharacteristic
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—532no—
314C2×C4—532no—
315C2×C4—532no—
316C2×C4—1064no—
317C2×C4—1064no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—4242no—
325C2×C4—4242no—
326C2×C4—4242no—
327C2×C4—4242no—
328C2×C4—2124no—
329C2×C4—2124no—
330C2×C4—2124no—
331C2×C4—2124no—
332C2×C4—2124no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4#1313328012yes—
340C2×C4#1313329612yes—
341C8—4242no—
342C22—532no—
343C22—532no—
344C22—1061no—
345C22—1061no—
346C22—1062no—
347C22—1062no—
348C22—1062no—
349C22—1062no—
350C22—1061no—
351C22—1061no—
352C22#1681156611yescharacteristic
353C22#1681160211yescharacteristic
354C22#1681160211yescharacteristic
355C22C2×#1313328011yescharacteristic
356C22#1681161511yescharacteristic
357C22D53⋊#10411yescharacteristic
358C22#1681162011yescharacteristic
359C4—22no—
360C4—22no—
361C4—41no—
362C4—81no—
363C4—81no—
364C4—1062no—
365C4—1062no—
366C4—1062no—
367C4—4244no—
368C4—4241no—
369C4—4241no—
370C2—1062no—
371C2—1062no—
372C2#2029009711yescharacteristic
373C2#2029010511yescharacteristic
374C2#2029013411yescharacteristic
375C2#2029014811yescharacteristic
376C2#2029061211yescharacteristic
377C2#2029061911yescharacteristic
378C2#2029062311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#8717
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
14. #16810492C4semidirect
15. #16810492C4semidirect
21. C4×C4⋊Dic53C4semidirect
22. C4×C4⋊C4×C53C4semidirect
206. C4×C4⋊C4#6157semidirect
212. C53#8717semidirect

Maximal subgroups

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

subgroupindex
#202906072
#202898912
#202895882
#202771192
#871753