Finite Group Databasebeta

← #28320509

Subgroups — #28320509

Order 13568.

1198 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811626C211yesmaximal, characteristic
2C22×C4×#88324C211yesmaximal, characteristic
3#20290886C214yesmaximal, complemented
4#20290885C212yesmaximal, complemented
5#20290884C212yesmaximal, complemented
6#20290381C214yesmaximal
7C2×#16811175C211yesmaximal, characteristic
8C22×#11719940C2212yes—
9C23×#88324C2212yes—
10#16811626—24no—
11C2×#13133284C2212yes—
12C2×C4⋊#88324C2212yes—
13C2×C4×#88324—24no—
14C2×C4×#88324C2212yes—
15C2×C4×#88324C2212yes—
16#16811489C2218yes—
17#16811489C2218yes—
18#16811175—24no—
19#16811175—24no—
20C2×#13133216C411yescharacteristic
21C2×#13133215C411yescharacteristic
22C2×#13133207C412yes—
23C2×#13133196C411yescharacteristic
24C2×#13133194C412yes—
25C2×Q8×Dic53C2211yescharacteristic
26C2×#13133182C2211yescharacteristic
27C2×#13133182C2212yes—
28C2×#13133173C2212yes—
29C22×C4×Dic53C2211yescharacteristic
30C2×C22⋊Q8×C53C411yescharacteristic
31C2×#11719940—24no—
32C2×#11719940—28no—
33C22×#88324—24no—
34C22×#88324—24no—
35C22×#88324Q818yescomplemented
36C22×#88324C2314yes—
37C22×#88324C2314yes—
38#13133284—24no—
39C4⋊#88324—24no—
40C4×#88324—24no—
41C4×#88324—24no—
42C22×Dic106C2×C412yes—
43C22×Dic106C2×C411yescharacteristic
44C23×Dic53C2×C411yescharacteristic
45C23×Dic53C2311yescharacteristic
46#13133216—24no—
47#13133215—24no—
48#13133207—28no—
49#13133196—24no—
50#13133194—28no—
51Q8×Dic53—22no—
52Q8×Dic53—22no—
53#13133182—22no—
54#13133182—22no—
55#13133182—24no—
56#13133182—24no—
57#13133173—24no—
58#13133173—24no—
59C2×C4⋊Dic53C2×C412yes—
60C2×C4⋊Dic53C2×C411yescharacteristic
61C2×#11719919C2×C412yes—
62C2×#11719919C2×C412yes—
63C2×#11719919C2×C412yes—
64C2×#11719919C2×C412yes—
65C2×#11719919C2×C412yes—
66C2×C22⋊Dic53C2×C412yes—
67C2×C4×Dic53C2×C411yescharacteristic
68C2×C4×Dic53C2×C411yescharacteristic
69C2×C4×Dic53D414yes—
70C2×C4×Dic53D414yes—
71C2×C4×Dic53C2312yes—
72C2×C4×Dic53C2312yes—
73C2×C4×Dic53C2311yescharacteristic
74C2×C4×Dic53C2311yescharacteristic
75C22×Q8×C53C2×C411yescharacteristic
76C23×C4×C53C2×C411yescharacteristic
77C22⋊Q8×C53—24no—
78C2×C4⋊C4×C53C2×C412yes—
79C2×C4⋊C4×C53C2×C411yescharacteristic
80C2×C22⋊C4×C53C2×C412yes—
81C24×C53C22×C411yescharacteristic
82C2×Q8×C53—22no—
83C2×Q8×C53—24no—
84C22×C4×C53—24no—
85C22×C4×C53C22⋊C414yes—
86C22×C4×C53C22×C412yes—
87C22×C4×C53C22×C412yes—
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C22×C411yescharacteristic
90C22×Dic53—24no—
91C22×Dic53C22⋊C414yes—
92C22×Dic53C22×C412yes—
93C22×Dic53C22×C412yes—
94C22×Dic53C22×C411yescharacteristic
95C22×Dic53C22×C411yescharacteristic
96C22×Dic53C22×C411yescharacteristic
97C22×Dic53C2×D412yes—
98C22×Dic53C2×D412yes—
99C22×Dic53C2×Q814yes—
100C22×Dic53#4114yes—
101C22×Dic53C2411yescharacteristic
102C2×Dic106—22no—
103C2×Dic106—24no—
104C2×Dic106—24no—
105C2×Dic106—28no—
106#11719940—48no—
107C2×#88324—216no—
108C2×#88324—24no—
109C2×#88324—24no—
110C2×#88324—24no—
111C2×#88324—28no—
112C2×#88324C2×Q818yes—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—22no—
115C22⋊C4×C53—24no—
116C22⋊Dic53—24no—
117C4⋊Dic53—22no—
118C4⋊Dic53—24no—
119#11719919—24no—
120#11719919—24no—
121#11719919—24no—
122#11719919—24no—
123#11719919—24no—
124C4×Dic53—22no—
125C4×Dic53—22no—
126C4×Dic53—22no—
127C4×Dic53—22no—
128C4×Dic53C22⋊C414yes—
129C4×Dic53C22⋊C414yes—
130C4×Dic53C2×D412yes—
131C4×Dic53C2×D412yes—
132C4×Dic53C2×D412yes—
133C4×Dic53C2×D412yes—
134C23×C53—22no—
135C23×C53—22no—
136C23×C53C2×C22⋊C412yes—
137C23×C53C4×Q814yes—
138C23×C53C23×C411yescharacteristic
139Q8×C53—44no—
140C2×C4×C53—22no—
141C2×C4×C53—28no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—24no—
145C2×C4×C53—24no—
146C2×C4×C53C2×C22⋊C412yes—
147C2×C4×C53C2×C22⋊C412yes—
148C2×C4×C53#13114yes—
149C2×Dic53—22no—
150C2×Dic53—28no—
151C2×Dic53—22no—
152C2×Dic53—22no—
153C2×Dic53—24no—
154C2×Dic53—24no—
155C2×Dic53C2×C22⋊C412yes—
156C2×Dic53C2×C22⋊C412yes—
157C2×Dic53C2×C22⋊C412yes—
158C2×Dic53C4×Q814yes—
159C2×Dic53C22⋊Q818yes—
160C2×Dic53#12414yes—
161C2×Dic53C4⋊Q814yes—
162C2×Dic53#13114yes—
163C2×Dic53C22×D411yescharacteristic
164C2×Dic53C22×D411yescharacteristic
165C2×Dic53C22×Q812yes—
166C2×Dic53C2×#4112yes—
167Dic106—44no—
168Dic106—48no—
169#88324—28no—
170#88324—44no—
171#21300—531nomaximal, Sylow
172C22×C53—22no—
173C22×C53—22no—
174C22×C53—28no—
175C22×C53#39114yes—
176C22×C53C22×C22⋊C411yescharacteristic
177C22×C53C2×C4×Q812yes—
178C22×C53C2×#13112yes—
179C22×C53C2×#13112yes—
180C4×C53—22no—
181C4×C53—42no—
182C4×C53—42no—
183C4×C53#40814yes—
184Dic53—22no—
185Dic53—42no—
186Dic53—42no—
187Dic53#39114yes—
188Dic53#40814yes—
189Dic53C2×C22⋊Q812yes—
190Dic53C2×#12411yescharacteristic
191Dic53C2×C4⋊Q811yescharacteristic
192C2×#533—531no—
193C2×#423—531no—
194C22×C4×C8—531no—
195#1984—532no—
196#1984—532no—
197#1984—534no—
198#1729—534no—
199C2×C53—24no—
200C2×C53C2×#39111yescharacteristic
201C2×C53C2×#40811yescharacteristic
202C2×C53C2×#40811yescharacteristic
203C2×C53#296014yes—
204C22×M16—532no—
205C23×C8—532no—
206#533—1064no—
207#533—1064no—
208C2×C4⋊Q8—531no—
209C2×C4⋊Q8—531no—
210C2×#125—532no—
211C2×C22⋊Q8—531no—
212C2×C22⋊Q8—532no—
213C2×C22⋊Q8#615711yescharacteristic
214C2×C4×Q8—531no—
215C2×C4×Q8—531no—
216C2×C4×Q8—532no—
217C2×#117—532no—
218C22×C42—531no—
219#423—1064no—
220C2×C4⋊C8—532no—
221C2×C4⋊C8—532no—
222C2×C4×C8—532no—
223C2×C4×C8—532no—
224C2×C4×C8—1064no—
225#336—538no—
226#336—538no—
227C53#2130011yescharacteristic, complemented, Sylow
228C22×Q8—532no—
229C22×Q8—531no—
230C22×Q8C2×#615711yescharacteristic
231C23×C4—531no—
232C23×C4—531no—
233C23×C4C2×#615711yescharacteristic
234C2×M16—1064no—
235C2×M16—1068no—
236C22×C8—534no—
237C22×C8—538no—
238C22×C8—534no—
239C22×C8—1064no—
240C22×C8—1064no—
241C4⋊Q8—1064no—
242C4⋊Q8—1064no—
243#125—1068no—
244C22⋊Q8—24no—
245C22⋊Q8—1068no—
246C22⋊Q8—1064no—
247C4×Q8—1064no—
248C4×Q8—1062no—
249C4×Q8—1064no—
250C4×Q8—1062no—
251C4×Q8—1062no—
252C4×Q8—1062no—
253#117—1064no—
254#117—1064no—
255C2×C4⋊C4—532no—
256C2×C4⋊C4—532no—
257C2×C4⋊C4—532no—
258C2×C4⋊C4—532no—
259C2×C4⋊C4—532no—
260C2×C4⋊C4—532no—
261C2×C4⋊C4—531no—
262C2×C4⋊C4C2×#615712yes—
263C2×C4⋊C4C2×#615711yescharacteristic
264C2×C22⋊C4—532no—
265C2×C22⋊C4C2×#615712yes—
266C2×C42—534no—
267C2×C42—534no—
268C2×C42—531no—
269C2×C42—531no—
270C2×C42—531no—
271C2×C42—531no—
272C2×C42—532no—
273C2×C42—532no—
274C4⋊C8—1064no—
275C4⋊C8—1064no—
276C4×C8—1064no—
277C4×C8—1064no—
278C24C22×#615711yescharacteristic
279C2×Q8—22no—
280C2×Q8—24no—
281C2×Q8—1064no—
282C2×Q8—1068no—
283C2×Q8—1062no—
284C2×Q8—1064no—
285C22×C4—24no—
286C22×C4—532no—
287C22×C4—532no—
288C22×C4—534no—
289C22×C4—534no—
290C22×C4—531no—
291C22×C4—531no—
292C22×C4—531no—
293C22×C4—534no—
294C22×C4—531no—
295C22×C4—532no—
296C22×C4—532no—
297C22×C4—1064no—
298C22×C4C22⋊#615714yes—
299C22×C4C22×#615712yes—
300C22×C4C22×#615712yes—
301C22×C4C22×#615711yescharacteristic
302C22×C4C22×#615711yescharacteristic
303M16—2128no—
304C2×C8—538no—
305C2×C8—1064no—
306C2×C8—10616no—
307C2×C8—1064no—
308C2×C8—1064no—
309C2×C8—1068no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—1064no—
313C4⋊C4—1064no—
314C4⋊C4—1064no—
315C4⋊C4—1064no—
316C4⋊C4—1064no—
317C4⋊C4—1064no—
318C4⋊C4—1062no—
319C22⋊C4—24no—
320C22⋊C4—1064no—
321C42—532no—
322C42—532no—
323C42—532no—
324C42—532no—
325C42—534no—
326C42—534no—
327C42—1062no—
328C42—1062no—
329C42—1062no—
330C42—1062no—
331C23—22no—
332C23—22no—
333C23C2×C22⋊#615712yes—
334C23Q8×#615714yes—
335C23C23×#615711yescharacteristic
336Q8—44no—
337Q8—2128no—
338Q8—2124no—
339C2×C4—22no—
340C2×C4—28no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—538no—
346C2×C4—534no—
347C2×C4—534no—
348C2×C4—532no—
349C2×C4—532no—
350C2×C4—531no—
351C2×C4—531no—
352C2×C4—534no—
353C2×C4—532no—
354C2×C4—532no—
355C2×C4—534no—
356C2×C4—532no—
357C2×C4—1068no—
358C2×C4—1062no—
359C2×C4—1062no—
360C2×C4—1062no—
361C2×C4—1064no—
362C2×C4—1064no—
363C2×C4C2×C22⋊#615712yes—
364C2×C4C2×C22⋊#615712yes—
365C2×C4#1313330414yes—
366C8—1068no—
367C8—2124no—
368C22—28no—
369C22—22no—
370C22—22no—
371C22(C2×#6157)⋊Q814yes—
372C22C22×C22⋊#615711yescharacteristic
373C22C2×Q8×#615712yes—
374C22C2×#1313330412yes—
375C22D53⋊#13112yes—
376C4—22no—
377C4—42no—
378C4—42no—
379C4—532no—
380C4—531no—
381C4—531no—
382C4—534no—
383C4—534no—
384C4—1062no—
385C4—2122no—
386C4—2122no—
387C4#1681159614yes—
388C2—24no—
389C2C2×(C2×#6157)⋊Q811yescharacteristic
390C2C2×#1681159611yescharacteristic
391C2D53⋊#40811yescharacteristic
392C2#2029150314yes—
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC2×C22⋊Q8×C53C4
prime radicalC24×C53C22×C4
cosocleC22×Dic53C24
hypercenterC2×C22⋊Q8#6157
hypocenterC53#21300
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20290886C2semidirect
4. #20290885C2semidirect
5. #20290884C2semidirect
35. C22×#88324Q8semidirect
227. C53#21300semidirect

Maximal subgroups

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

subgroupindex
C2×#168116262
C22×C4×#883242
#202908862
#202908852
#202908842
#202903812
C2×#168111752
#2130053