Finite Group Databasebeta

← #27274613

Subgroups — #27274613

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810507C211yesmaximal, characteristic
2C4×C22⋊#88322C211yesmaximal, characteristic
3C2×#16810348C212yesmaximal
4#20272557C212yesmaximal
5C53×#1749C211yesmaximal, characteristic
6C22×#11719919C411yescharacteristic
7C22×#11719919C2211yescharacteristic
8C2×C22⋊#88322C2212yes—
9C2×C4×#88322—21no—
10#16810507—21no—
11#16810507—21no—
12#16810507—21no—
13#16810507—21no—
14C2×#13133124C411yescharacteristic
15C2×#13133124C411yescharacteristic
16C2×#13133124C411yescharacteristic
17C2×#13133124C2211yescharacteristic
18#16810378—21no—
19#16810348C418yescomplemented
20C22×C42×C53C2211yescharacteristic
21C2×C22⋊C8×C53C2212yes—
22C2×#97×C53—21no—
23C53×#336—21no—
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27C23×Dic53C2×C411yescharacteristic
28C22×#88322—22no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—22no—
34C2×#11719919—42no—
35C2×#11719919C2×C414yes—
36C22⋊#88322—22no—
37C22⋊#88322C2×C414yes—
38C4×#88322—42no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47C23×C4×C53C2×C411yescharacteristic
48C23×C4×C53C2×C411yescharacteristic
49C23×C4×C53C2311yescharacteristic
50C22×C8×C53—22no—
51C2×C42×C53—42no—
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C22⋊C8×C53—22no—
57C22⋊C8×C53C2×C414yes—
58#97×C53—42no—
59C24×C53C22×C411yescharacteristic
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C412yes—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53#4112yes—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—22no—
81C22×Dic53—22no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—22no—
85C22×Dic53—22no—
86C22×Dic53—41no—
87C22×Dic53—41no—
88C22×Dic53—41no—
89C22×Dic53—41no—
90C22×Dic53—41no—
91C22×Dic53—41no—
92C22×Dic53C4212yes—
93C22×Dic53C4212yes—
94C2×C8×C53—24no—
95C2×C8×C53—42no—
96C2×C8×C53—42no—
97C42×C53—22no—
98C42×C53—22no—
99C42×C53—41no—
100C42×C53—41no—
101#11719919—24no—
102#11719919—24no—
103#11719919—42no—
104#11719919—42no—
105C2×#88322—24no—
106C2×#88322—42no—
107C2×#88322—42no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53C2×C4211yescharacteristic
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53#11711yescharacteristic
115C23×C53#13112yes—
116C23×C53#13112yes—
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—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—44no—
134C2×C4×C53—44no—
135C2×C4×C53#9912yes—
136C2×C4×C53#10112yes—
137C2×C4×C53C4×D412yes—
138C2×C4×C53C4×D412yes—
139C2×Dic53—22no—
140C2×Dic53—22no—
141C2×Dic53—24no—
142C2×Dic53—42no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—42no—
148C2×Dic53—42no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—82no—
156C2×Dic53—82no—
157C8×C53—44no—
158#88322—44no—
159#8154—531nomaximal, Sylow
160C22×C53—22no—
161C22×C53—22no—
162C22×C53—22no—
163C22×C53—22no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53C4×C22⋊C411yescharacteristic
169C22×C53#40512yes—
170C22×C53#40811yescharacteristic
171C22×C53#40811yescharacteristic
172C22×C53C2×#9911yescharacteristic
173C22×C53#41011yescharacteristic
174C22×C53C2×#10111yescharacteristic
175C22×C53#41311yescharacteristic
176C22×C53#43312yes—
177C4×C53—22no—
178C4×C53—22no—
179C4×C53—22no—
180C4×C53—22no—
181C4×C53—41no—
182C4×C53—41no—
183Dic53—42no—
184Dic53—42no—
185Dic53—81no—
186Dic53—81no—
187C2×#387—531no—
188#1749D5311yescharacteristic
189C4×C22⋊C8—531no—
190C2×#324—532no—
191#1280—532no—
192C2×C53—22no—
193C2×C53—22no—
194C2×C53#146811yescharacteristic
195C2×C53#147011yescharacteristic
196C2×C53#147211yescharacteristic
197C2×C53#147311yescharacteristic
198C2×C53#175011yescharacteristic
199C2×C53C4×#9911yescharacteristic
200C2×C53C4×#10111yescharacteristic
201C22×C4⋊C4—531no—
202C22×C4⋊C4—531no—
203C22×C42C2×D5311yescharacteristic
204C2×C22⋊C8—532no—
205C2×C22⋊C8C2×D5312yes—
206C2×#97—21no—
207C2×C4×C8—1061no—
208#387—1061no—
209#387—1061no—
210#387—1061no—
211#387—1061no—
212C2×#95—531no—
213C2×#95—531no—
214C2×#95—531no—
215C2×#95—531no—
216#336—21no—
217#336—1061no—
218#324—538no—
219C53#815411yescharacteristic, complemented, Sylow
220C23×C4—531no—
221C23×C4—531no—
222C23×C4—531no—
223C23×C4—531no—
224C23×C4C4×D5311yescharacteristic
225C23×C4C4×D5311yescharacteristic
226C23×C4C22×D5311yescharacteristic
227C22×C8—22no—
228C22×C8—1062no—
229C2×C4⋊C4—534no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1062no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—2122no—
236C2×C42—42no—
237C2×C42D21211yescharacteristic
238C2×C42D21211yescharacteristic
239C2×C42C22⋊D5311yescharacteristic
240C2×C42C22⋊D5311yescharacteristic
241C22⋊C8—22no—
242C22⋊C8—534no—
243C22⋊C8—1062no—
244C22⋊C8C4×D5314yes—
245#97—42no—
246C4×C8—2122no—
247#95—1061no—
248#95—1061no—
249#95—1061no—
250#95—1061no—
251#95—1061no—
252#95—1061no—
253#95—1061no—
254#95—1061no—
255C24C2×C4×D5311yescharacteristic
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—532no—
265C22×C4—532no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1062no—
271C22×C4—1062no—
272C22×C4—1062no—
273C22×C4—1062no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—2121no—
277C22×C4—2121no—
278C22×C4—2121no—
279C22×C4—2121no—
280C22×C4—2121no—
281C22×C4—2121no—
282C22×C4#1171992111yescharacteristic
283C22×C4#1171992111yescharacteristic
284C22×C4#1171992111yescharacteristic
285C22×C4#1171992111yescharacteristic
286C22×C4C2×C4×D5312yes—
287C22×C4C2×D21211yescharacteristic
288C22×C4#1171994512yes—
289C22×C4C2×C22⋊D5311yescharacteristic
290C2×C8—24no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—1064no—
294C2×C8—2122no—
295C2×C8—2122no—
296C4⋊C4—1064no—
297C4⋊C4—1064no—
298C4⋊C4—2122no—
299C4⋊C4—2122no—
300C42—22no—
301C42—22no—
302C42—41no—
303C42—41no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23C42×D5311yescharacteristic
309C23C2×#1171992111yescharacteristic
310C23#1313317411yescharacteristic
311C23#1313322512yes—
312C23#1313322512yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—1064no—
332C2×C4—1062no—
333C2×C4—1062no—
334C2×C4—4242no—
335C2×C4—4242no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2122no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2122no—
348C2×C4—2122no—
349C2×C4#1313313112yes—
350C2×C4#1313313512yes—
351C2×C4C4×C22⋊D5312yes—
352C2×C4C4×D21212yes—
353C8—44no—
354C8—2124no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22C4×#1171992111yescharacteristic
364C22#1681058012yes—
365C22#1681059111yescharacteristic
366C22#1681059111yescharacteristic
367C22C2×#1313313111yescharacteristic
368C22#1681059611yescharacteristic
369C22C2×#1313313511yescharacteristic
370C22#1681060511yescharacteristic
371C22#1681068512yes—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—41no—
377C4—41no—
378C4—4241no—
379C4—4241no—
380C4—2122no—
381C4—2122no—
382C2—22no—
383C2—22no—
384C2#2027306411yescharacteristic
385C2#2027307211yescharacteristic
386C2#2027308011yescharacteristic
387C2#2027308511yescharacteristic
388C2#2027448411yescharacteristic
389C2C4×#1313313111yescharacteristic
390C2C4×#1313313511yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC23×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1749C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC23×C4×C53C23
hypercenter#1749D53
hypocenterC53#8154
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
19. #16810348C4semidirect
219. C53#8154semidirect

Maximal subgroups

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

subgroupindex
C2×#168105072
C4×C22⋊#883222
C2×#168103482
#202725572
C53×#17492
#815453