Finite Group Databasebeta

← #27274087

Subgroups — #27274087

Order 13568.

850 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810676C211yesmaximal, characteristic
2C2×C4×C4⋊Dic53C211yesmaximal, characteristic
3#20272593C214yesmaximal
4C53×#2285C211yesmaximal, characteristic
5C22×C4⋊Dic53—21no—
6C22×C4×Dic53—21no—
7#16810676—22no—
8#16810676C414yescomplemented
9C2×C4⋊#88322—21no—
10C2×C22⋊#88322C2212yes—
11C2×C4×#88322—21no—
12C4×C4⋊Dic53—22no—
13C4×C4⋊Dic53C414yescomplemented
14C2×#13133124C2212yes—
15C2×#117×C53—21no—
16C22×C42×C53C2211yescharacteristic
17C53×#384C412yescomplemented
18C53×#382—21no—
19C53×#380—21no—
20C53×#380C2212yes—
21C4×C4⋊C4×C53—21no—
22C4×C4⋊C4×C53C412yescomplemented
23C23×Dic53—22no—
24C22×#88322—22no—
25C2×C4⋊Dic53—28no—
26C2×C4⋊Dic53—24no—
27C2×C4×Dic53—24no—
28C2×C4×Dic53—42no—
29C2×C4×Dic53—42no—
30C4⋊#88322—24no—
31C22⋊#88322—22no—
32C22⋊#88322C2×C414yes—
33C4×#88322—24no—
34#13133124—22no—
35#13133124C2×C414yes—
36C23×C4×C53D411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53C2311yescharacteristic
39#117×C53—42no—
40#117×C53—42no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53C2×C414yes—
44C2×C22⋊C4×C53—22no—
45C2×C42×C53—22no—
46C2×C42×C53C2×C412yes—
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53C2×C411yescharacteristic
49C2×C42×C53C2×C412yes—
50C2×C42×C53D412yes—
51C2×C42×C53D411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53Q812yes—
54#95×C53—22no—
55#95×C53—22no—
56C24×C53C2×D411yescharacteristic
57C22×C4×C53—24no—
58C22×C4×C53—24no—
59C22×C4×C53—42no—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C4⋊C412yes—
67C22×C4×C53C4⋊C412yes—
68C22×C4×C53D812yes—
69C22×C4×C53SD1612yes—
70C22×C4×C53SD1612yes—
71C22×C4×C53Q1612yes—
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×Q811yescharacteristic
78C22×Dic53—28no—
79C22×Dic53—24no—
80C22×Dic53—44no—
81C22×Dic53—42no—
82C22×Dic53—42no—
83C4⋊C4×C53—44no—
84C4⋊C4×C53—44no—
85C4⋊C4×C53—44no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C42×C53—44no—
89C42×C53C4212yes—
90C42×C53C4212yes—
91C42×C53C22⋊C412yes—
92C42×C53C22⋊C412yes—
93C42×C53C4⋊C412yes—
94C42×C53C4⋊C412yes—
95C42×C53C4⋊C412yes—
96C42×C53C4⋊C412yes—
97C4⋊Dic53—216no—
98C4×Dic53—48no—
99C2×#88322—24no—
100C2×#88322—44no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53C2×C4⋊C411yescharacteristic
106C23×C53C2×D811yescharacteristic
107C23×C53C2×SD1611yescharacteristic
108C23×C53C2×SD1611yescharacteristic
109C23×C53C2×Q1611yescharacteristic
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—24no—
114C2×C4×C53—24no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—44no—
120C2×C4×C53—44no—
121C2×C4×C53—44no—
122C2×C4×C53#9512yes—
123C2×C4×C53#9512yes—
124C2×C4×C53#9512yes—
125C2×C4×C53#9512yes—
126C2×C4×C53#10214yes—
127C2×C4×C53#10212yes—
128C2×C4×C53#10212yes—
129C2×C4×C53#10314yes—
130C2×C4×C53#10312yes—
131C2×C4×C53#10312yes—
132C2×C4×C53#10414yes—
133C2×C4×C53#10614yes—
134C2×C4×C53#10714yes—
135C2×C4×C53C2×C4211yescharacteristic
136C2×C4×C53C2×C22⋊C411yescharacteristic
137C2×C4×C53C2×C4⋊C411yescharacteristic
138C2×C4×C53C2×C4⋊C411yescharacteristic
139C2×Dic53—28no—
140C2×Dic53—44no—
141C2×Dic53—48no—
142C2×Dic53—48no—
143C2×Dic53—44no—
144#88322—44no—
145#8014—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—22no—
151C22×C53—22no—
152C22×C53—22no—
153C22×C53—22no—
154C22×C53C2×#9511yescharacteristic
155C22×C53#41012yes—
156C22×C53C2×#10211yescharacteristic
157C22×C53C2×#10211yescharacteristic
158C22×C53C2×#10311yescharacteristic
159C22×C53C2×#10311yescharacteristic
160C22×C53C2×#10412yes—
161C22×C53C2×#10611yescharacteristic
162C22×C53C2×#10711yescharacteristic
163C4×C53—22no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—84no—
169C4×C53#33914yes—
170C4×C53#34014yes—
171C4×C53#34014yes—
172Dic53—48no—
173C2×#431—531no—
174#2285D5311yescharacteristic
175C2×C4×C4⋊C4—531no—
176#1294—534no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53#150212yes—
182C2×C53#150312yes—
183C2×C53C2×#33911yescharacteristic
184C2×C53C2×#34011yescharacteristic
185C2×C53#174011yescharacteristic
186C2×#117—21no—
187C22×C4⋊C4—1061no—
188C22×C42—1061no—
189C22×C42C2×D5311yescharacteristic
190#431—534no—
191#431—1062no—
192C2×C4⋊C8—1061no—
193C2×C22⋊C8—532no—
194C2×C4×C8—1061no—
195#384Dic5312yescomplemented
196#382—21no—
197#380—21no—
198#380C2×D5312yes—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—534no—
201C4×C4⋊C4—1062no—
202C4×C4⋊C4Dic5312yescomplemented
203C2×#95—532no—
204C53#801411yescharacteristic, complemented, Sylow
205C23×C4—1062no—
206C23×C4C22⋊D5311yescharacteristic
207C23×C4C22⋊D5311yescharacteristic
208C23×C4C22×D5311yescharacteristic
209C22×C8—1062no—
210#117—42no—
211#117—42no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—1068no—
215C2×C4⋊C4—1064no—
216C2×C4⋊C4C2×Dic5314yes—
217C2×C22⋊C4—22no—
218C2×C42—22no—
219C2×C42—1064no—
220C2×C42—2122no—
221C2×C42—2122no—
222C2×C42Dic10612yes—
223C2×C42C4×D5312yes—
224C2×C42C4×D5312yes—
225C2×C42D21212yes—
226C2×C42C2×Dic5311yescharacteristic
227C2×C42C2×Dic5311yescharacteristic
228C2×C42C22⋊D5311yescharacteristic
229C2×C42C22⋊D5311yescharacteristic
230C4⋊C8—1064no—
231C22⋊C8—534no—
232C22⋊C8—1062no—
233C4×C8—1064no—
234#95—22no—
235#95—22no—
236#95—534no—
237#95—1062no—
238C24C2×C22⋊D5311yescharacteristic
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—42no—
242C22×C4—1068no—
243C22×C4—1064no—
244C22×C4—2124no—
245C22×C4—2122no—
246C22×C4—2122no—
247C22×C4#1171991912yes—
248C22×C4#1171991912yes—
249C22×C4#1171992112yes—
250C22×C4#1171992112yes—
251C22×C4#1171992212yes—
252C22×C4#1171992312yes—
253C22×C4#1171992412yes—
254C22×C4#1171992512yes—
255C22×C4C22⋊Dic5311yescharacteristic
256C22×C4C22⋊Dic5311yescharacteristic
257C22×C4C22⋊Dic5311yescharacteristic
258C22×C4C22⋊Dic5311yescharacteristic
259C22×C4C2×Dic10611yescharacteristic
260C22×C4C2×C4×D5311yescharacteristic
261C22×C4C2×C4×D5311yescharacteristic
262C22×C4C2×D21211yescharacteristic
263C22×C4C22×Dic5311yescharacteristic
264C22×C4C2×C22⋊D5311yescharacteristic
265C2×C8—1064no—
266C2×C8—2124no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—10616no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C42—44no—
274C42—2128no—
275C42C4×Dic5312yes—
276C42C4×Dic5312yes—
277C42#1171991912yes—
278C42#1171991912yes—
279C42C4⋊Dic5312yes—
280C42C4⋊Dic5312yes—
281C42#1171992112yes—
282C42#1171992112yes—
283C23—22no—
284C23—22no—
285C23C2×#1171992111yescharacteristic
286C23C2×C22⋊Dic5311yescharacteristic
287C23C2×#1171991911yescharacteristic
288C23C2×#1171992211yescharacteristic
289C23C2×#1171992411yescharacteristic
290C23C2×#1171992311yescharacteristic
291C23C2×#1171992511yescharacteristic
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—1068no—
305C2×C4—2124no—
306C2×C4—2128no—
307C2×C4—2128no—
308C2×C4—2124no—
309C2×C4#1313312412yes—
310C2×C4#1313312412yes—
311C2×C4#1313312412yes—
312C2×C4#1313312412yes—
313C2×C4#1313313714yes—
314C2×C4#1313313812yes—
315C2×C4#1313313812yes—
316C2×C4#1313314014yes—
317C2×C4#1313314112yes—
318C2×C4#1313314112yes—
319C2×C4#1313314414yes—
320C2×C4#1313314814yes—
321C2×C4#1313315014yes—
322C2×C4C2×C4×Dic5311yescharacteristic
323C2×C4C2×#1171992111yescharacteristic
324C2×C4C2×#1171991911yescharacteristic
325C2×C4C2×C4⋊Dic5311yescharacteristic
326C8—2124no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22C2×#1313312411yescharacteristic
336C22#1681059712yes—
337C22C2×#1313313711yescharacteristic
338C22C2×#1313313811yescharacteristic
339C22C2×#1313314011yescharacteristic
340C22C2×#1313314111yescharacteristic
341C22C2×#1313314412yes—
342C22C2×#1313314811yescharacteristic
343C22C2×#1313315011yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—22no—
349C4—84no—
350C4—2128no—
351C4#1681038214yes—
352C4#1681038514yes—
353C4#1681038514yes—
354C2—21no—
355C2—21no—
356C2—21no—
357C2—21no—
358C2#2027321012yes—
359C2#2027321412yes—
360C2C2×#1681038211yescharacteristic
361C2C2×#1681038511yescharacteristic
362C2#2027444811yescharacteristic
subgroupG/H
centerC22×C4C22⋊Dic53
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#2285C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2285D53
hypocenterC53#8014
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
8. #16810676C4semidirect
13. C4×C4⋊Dic53C4semidirect
17. C53×#384C4semidirect
22. C4×C4⋊C4×C53C4semidirect
195. #384Dic53semidirect
202. C4×C4⋊C4Dic53semidirect
204. C53#8014semidirect

Maximal subgroups

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

subgroupindex
C2×#168106762
C2×C4×C4⋊Dic532
#202725932
C53×#22852
#801453