Finite Group Databasebeta

← #27279697

Subgroups — #27279697

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283025C211yesmaximal, characteristic
2#20274477C211yesmaximal, characteristic
3C2×#16810364C212yesmaximal, complemented
4#20272597C212yesmaximal, complemented
5C22×#97×C53C211yesmaximal, characteristic
6C2×#13133209C2211yescharacteristic
7C2×C22⋊#88322—21no—
8C2×#13133128C2212yes—
9#16810521C411yescharacteristic
10#16810521C411yescharacteristic
11#16810507—21no—
12#16810507—21no—
13#16810496—21no—
14#16810496—21no—
15#16810489C2211yescharacteristic
16#16810483C411yescharacteristic
17#16810483C411yescharacteristic
18#16810378—21no—
19#16810364—24no—
20C23×C8×C53—21no—
21C22×C42×C53C2211yescharacteristic
22C2×#97×C53—21no—
23C2×#97×C53—24no—
24C2×#97×C53C2212yes—
25C22×#88322—22no—
26#13133209—22no—
27#13133209—22no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C22⋊#88322—24no—
39#13133128—22no—
40#13133128—22no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47C23×C4×C53C2×C411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C23×C4×C53D411yescharacteristic
50C22×C8×C53—24no—
51C22×C8×C53—24no—
52C22×C8×C53—24no—
53C22×C8×C53—22no—
54C2×C42×C53—22no—
55C2×C42×C53—22no—
56C2×C42×C53C2×C411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59C2×C42×C53C2311yescharacteristic
60#97×C53—24no—
61#97×C53—22no—
62#97×C53D414yes—
63C24×C53C22⋊C411yescharacteristic
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C2×C8×C53—216no—
85C2×C8×C53—24no—
86C2×C8×C53—24no—
87C2×C8×C53—24no—
88C42×C53—22no—
89C42×C53—21no—
90C42×C53—22no—
91C42×C53—21no—
92C42×C53C2×D412yes—
93C42×C53#4112yes—
94C22⋊Dic53—42no—
95C22⋊Dic53—42no—
96C4⋊Dic53—22no—
97C4⋊Dic53—22no—
98C4⋊Dic53—41no—
99C4⋊Dic53—41no—
100#11719919—42no—
101#11719919—42no—
102#11719919—41no—
103#11719919—42no—
104#11719919—41no—
105C2×#88322—24no—
106C2×#88322—42no—
107C2×#88322—42no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53#10112yes—
113C23×C53#10412yes—
114C23×C53#10412yes—
115C23×C53C2×C22⋊C411yescharacteristic
116C2×C4×C53—24no—
117C2×C4×C53—24no—
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—21no—
132C2×C4×C53—44no—
133C2×C4×C53—44no—
134C2×C4×C53#10412yes—
135C2×C4×C53C4×D412yes—
136C2×C4×C53#12011yescharacteristic
137C2×C4×C53#12112yes—
138C2×C4×C53#12311yescharacteristic
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C8×C53—28no—
152#88322—44no—
153#9121—531nomaximal, Sylow
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—22no—
160C22×C53#33014yescomplemented
161C22×C53#38611yescharacteristic
162C22×C53C2×#10111yescharacteristic
163C22×C53#41311yescharacteristic
164C22×C53C2×#10411yescharacteristic
165C22×C53C2×#10411yescharacteristic
166C22×C53C2×#10411yescharacteristic
167C22×C53#42111yescharacteristic
168C4×C53—22no—
169C4×C53—21no—
170C4×C53—22no—
171C4×C53—21no—
172C4×C53—22no—
173C4×C53—41no—
174C4×C53—41no—
175Dic53—81no—
176Dic53—81no—
177Dic53—81no—
178Dic53—81no—
179C22×#97D5311yescharacteristic
180#2851—531no—
181#1749—531no—
182C2×#330—532no—
183#1295—532no—
184C2×C53—24no—
185C2×C53C2×#33011yescharacteristic
186C2×C53#152411yescharacteristic
187C2×C53#153111yescharacteristic
188C2×C53#153211yescharacteristic
189C2×C53#187111yescharacteristic
190C2×C53#189611yescharacteristic
191C2×C53#190111yescharacteristic
192C23×C8—21no—
193C2×#125—531no—
194C22×C42C2×D5311yescharacteristic
195C2×C22⋊C8—1061no—
196C2×#97—21no—
197C2×#97—24no—
198C2×#97—532no—
199C2×#97C2×D5312yes—
200#389—531no—
201#389—531no—
202#387—1061no—
203#387—1061no—
204#385—1061no—
205#385—1061no—
206#382—531no—
207#380—531no—
208#380—531no—
209#336—1061no—
210#330—1064no—
211C53#912111yescharacteristic, complemented, Sylow
212C23×C4C4×D5311yescharacteristic
213C23×C4D21211yescharacteristic
214C23×C4C22⋊D5311yescharacteristic
215C22×C8—24no—
216C22×C8—24no—
217C22×C8—24no—
218C22×C8—22no—
219C22×C8—1062no—
220#125—1062no—
221#125—1062no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—531no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1061no—
230C2×C22⋊C4—1061no—
231C2×C22⋊C4—1061no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42C4×D5311yescharacteristic
235C2×C42D21211yescharacteristic
236C2×C42C22⋊D5311yescharacteristic
237C2×C42C22×D5311yescharacteristic
238C22⋊C8—1064no—
239#97—24no—
240#97—22no—
241#97—1062no—
242#97—1062no—
243#97C22⋊D5314yes—
244#95—1061no—
245#95—1061no—
246#95—1061no—
247#95—1061no—
248#95—1061no—
249#95—1061no—
250C24#1171992111yescharacteristic
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4#1171992111yescharacteristic
266C22×C4#1171992111yescharacteristic
267C22×C4#1171992111yescharacteristic
268C22×C4C2×C4×D5311yescharacteristic
269C22×C4C2×D21211yescharacteristic
270C22×C4C2×C22⋊D5311yescharacteristic
271C2×C8—216no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—1064no—
276C2×C8—2122no—
277C2×C8—2122no—
278C4⋊C4—1062no—
279C4⋊C4—1062no—
280C4⋊C4—2121no—
281C4⋊C4—2121no—
282C4⋊C4—2122no—
283C4⋊C4—2122no—
284C4⋊C4—2121no—
285C4⋊C4—2122no—
286C4⋊C4—2121no—
287C22⋊C4—2122no—
288C22⋊C4—2122no—
289C42—22no—
290C42—21no—
291C42—22no—
292C42—21no—
293C42#1171994512yes—
294C42C2×C22⋊D5312yes—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#1313313512yes—
300C23#1313314312yes—
301C23#1313314512yes—
302C23C2×#1171992111yescharacteristic
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4#1313314512yes—
334C2×C4C4×C22⋊D5312yes—
335C2×C4#1313318611yescharacteristic
336C2×C4#1313319112yes—
337C2×C4#1313320011yescharacteristic
338C8—28no—
339C8—2124no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22#1681036414yescomplemented
347C22#1681050211yescharacteristic
348C22C2×#1313313511yescharacteristic
349C22#1681060511yescharacteristic
350C22C2×#1313314311yescharacteristic
351C22C2×#1313314511yescharacteristic
352C22C2×#1313314511yescharacteristic
353C22#1681063911yescharacteristic
354C4—22no—
355C4—21no—
356C4—22no—
357C4—21no—
358C4—22no—
359C4—41no—
360C4—41no—
361C4—4241no—
362C4—4241no—
363C4—4241no—
364C4—4241no—
365C2—24no—
366C2C2×#1681036411yescharacteristic
367C2#2027330111yescharacteristic
368C2#2027332911yescharacteristic
369C2#2027333311yescharacteristic
370C2#2027501711yescharacteristic
371C2#2027515811yescharacteristic
372C2#2027518311yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC22×#97×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC22×#97D53
hypocenterC53#9121
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#97

Complements

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

NH ≅ G/Ntype
3. C2×#16810364C2semidirect
4. #20272597C2semidirect
160. C22×C53#330semidirect
211. C53#9121semidirect
346. C22#16810364semidirect

Maximal subgroups

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

subgroupindex
#202830252
#202744772
C2×#168103642
#202725972
C22×#97×C532
#912153