Finite Group Databasebeta

← #27279722

Subgroups — #27279722

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 368 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
5C2×C53×#432C211yesmaximal, 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—
20C22×C42×C53C2211yescharacteristic
21C53×#432—24no—
22C2×C4⋊C8×C53—21no—
23C2×C22⋊C8×C53—21no—
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—22no—
51C2×C42×C53—22no—
52C2×C42×C53—22no—
53C2×C42×C53C2×C411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C2×C42×C53C2311yescharacteristic
57C4⋊C8×C53—24no—
58C22⋊C8×C53—24no—
59#97×C53—22no—
60#97×C53D414yes—
61C24×C53C22⋊C411yescharacteristic
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—21no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C2×C8×C53—24no—
83C2×C8×C53—42no—
84C2×C8×C53—42no—
85C42×C53—22no—
86C42×C53—21no—
87C42×C53—22no—
88C42×C53—21no—
89C42×C53C2×D412yes—
90C42×C53#4112yes—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C4⋊Dic53—22no—
94C4⋊Dic53—22no—
95C4⋊Dic53—41no—
96C4⋊Dic53—41no—
97#11719919—42no—
98#11719919—42no—
99#11719919—41no—
100#11719919—42no—
101#11719919—41no—
102C2×#88322—24no—
103C2×#88322—42no—
104C2×#88322—42no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53#10112yes—
112C23×C53C2×C22⋊C411yescharacteristic
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
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—44no—
128C2×C4×C53—44no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53#10412yes—
132C2×C4×C53C4×D412yes—
133C2×C4×C53#12011yescharacteristic
134C2×C4×C53#12112yes—
135C2×C4×C53#12311yescharacteristic
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
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—
148C8×C53—44no—
149#88322—44no—
150#9126—531nomaximal, Sylow
151C22×C53—22no—
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53#38611yescharacteristic
159C22×C53C2×#10111yescharacteristic
160C22×C53#41311yescharacteristic
161C22×C53C2×#10411yescharacteristic
162C22×C53#42111yescharacteristic
163C22×C53#42111yescharacteristic
164C22×C53#42111yescharacteristic
165C4×C53—22no—
166C4×C53—22no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—22no—
170C4×C53—41no—
171C4×C53—41no—
172Dic53—81no—
173Dic53—81no—
174Dic53—81no—
175Dic53—81no—
176C2×#432D5311yescharacteristic
177#2851—531no—
178#1749—531no—
179C2×#330—532no—
180#1295—532no—
181C2×C53—42no—
182C2×C53#152911yescharacteristic
183C2×C53#153111yescharacteristic
184C2×C53#153411yescharacteristic
185C2×C53#153511yescharacteristic
186C2×C53#187111yescharacteristic
187C2×C53#189711yescharacteristic
188C2×C53#190111yescharacteristic
189C2×#125—531no—
190C22×C42C2×D5311yescharacteristic
191#432—24no—
192C2×C4⋊C8—21no—
193C2×C22⋊C8—21no—
194C2×C22⋊C8—1061no—
195C2×#97—532no—
196C2×#97C2×D5312yes—
197#389—531no—
198#389—531no—
199#387—1061no—
200#387—1061no—
201#385—1061no—
202#385—1061no—
203#382—531no—
204#380—531no—
205#380—531no—
206#336—1061no—
207#330—1064no—
208C53#912611yescharacteristic, complemented, Sylow
209C23×C4C4×D5311yescharacteristic
210C23×C4D21211yescharacteristic
211C23×C4C22⋊D5311yescharacteristic
212C22×C8—22no—
213C22×C8—1062no—
214#125—1062no—
215#125—1062no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42C4×D5311yescharacteristic
229C2×C42D21211yescharacteristic
230C2×C42C22⋊D5311yescharacteristic
231C2×C42C22×D5311yescharacteristic
232C4⋊C8—24no—
233C22⋊C8—24no—
234C22⋊C8—1064no—
235#97—22no—
236#97—1062no—
237#97—1062no—
238#97C22⋊D5314yes—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245C24#1171992111yescharacteristic
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4#1171992111yescharacteristic
261C22×C4#1171992111yescharacteristic
262C22×C4#1171992111yescharacteristic
263C22×C4C2×C4×D5311yescharacteristic
264C22×C4C2×D21211yescharacteristic
265C22×C4C2×C22⋊D5311yescharacteristic
266C2×C8—24no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—1064no—
270C2×C8—2122no—
271C2×C8—2122no—
272C4⋊C4—1062no—
273C4⋊C4—1062no—
274C4⋊C4—2121no—
275C4⋊C4—2121no—
276C4⋊C4—2122no—
277C4⋊C4—2122no—
278C4⋊C4—2121no—
279C4⋊C4—2122no—
280C4⋊C4—2121no—
281C22⋊C4—2122no—
282C22⋊C4—2122no—
283C42—22no—
284C42—21no—
285C42—22no—
286C42—21no—
287C42#1171994512yes—
288C42C2×C22⋊D5312yes—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1313313512yes—
296C23C2×#1171992111yescharacteristic
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4#1313314312yes—
328C2×C4C4×C22⋊D5312yes—
329C2×C4#1313318611yescharacteristic
330C2×C4#1313319112yes—
331C2×C4#1313320011yescharacteristic
332C8—44no—
333C8—2124no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22#1681050211yescharacteristic
342C22C2×#1313313511yescharacteristic
343C22#1681060511yescharacteristic
344C22C2×#1313314311yescharacteristic
345C22#1681063911yescharacteristic
346C22#1681064111yescharacteristic
347C22#1681064111yescharacteristic
348C4—22no—
349C4—22no—
350C4—21no—
351C4—21no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—4241no—
356C4—4241no—
357C4—4241no—
358C4—4241no—
359C2—42no—
360C2#2027332311yescharacteristic
361C2#2027333111yescharacteristic
362C2#2027334111yescharacteristic
363C2#2027334411yescharacteristic
364C2#2027501711yescharacteristic
365C2#2027516311yescharacteristic
366C2#2027518311yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#432C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC2×#432D53
hypocenterC53#9126
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
3. C2×#16810364C2semidirect
4. #20272597C2semidirect
208. C53#9126semidirect

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
C2×C53×#4322
#912653