Finite Group Databasebeta

← #27279528

Subgroups — #27279528

Order 13568.

504 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283755C211yesmaximal, characteristic
2C2×#16810721C211yesmaximal, characteristic
3#20273107C211yesmaximal, characteristic
4#20273039C211yesmaximal, characteristic, complemented
5#20272559C211yesmaximal, characteristic, complemented
6#20272545C211yesmaximal, characteristic, complemented
7C53×#1479C211yesmaximal, characteristic, complemented
8C22×#11719924C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810721—22no—
11#16810721—22no—
12#16810702—22no—
13#16810700—22no—
14#16810676C2211yescharacteristic
15#16810676C2211yescharacteristic
16C2×#13133150C411yescharacteristic
17C2×#13133141C411yescharacteristic
18C2×#13133137C411yescharacteristic
19#16810585—22no—
20C2×#13133128C2211yescharacteristic
21#16810352—22no—
22#16810349—22no—
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#431C2211yescharacteristic
25C53×#431C2211yescharacteristic
26C53×Q8⋊C8—22no—
27C22×D212C2×C411yescharacteristic
28C22×C4×D53—21no—
29C2×#11719924—22no—
30C2×#11719924—22no—
31C2×#11719924—22no—
32C2×#11719912—24no—
33C22×#88322C2×C411yescharacteristic
34C22×#88322C2×C411yescharacteristic
35C4×D212—21no—
36C4×D212—21no—
37C4×D212D412yes—
38C4×D212D412yes—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719921—21no—
41#13133150—22no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44#13133141—22no—
45#13133137—22no—
46C22⋊#88322—21no—
47C22⋊#88322D412yes—
48#13133129—24no—
49#13133128—21no—
50#13133128—21no—
51C4×#88322—21no—
52C4×#88322—21no—
53C22×Q8×C53C2×C411yescharacteristic
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C42×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C22⋊C8×C53—21no—
65C22⋊C8×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53D412yes—
68C23×D53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53M1612yescomplemented
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×Dic53—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212C22⋊C412yes—
87C2×C4×D53—22no—
88C2×C4×D53—22no—
89C2×C4×D53—82no—
90C2×C8×C53—22no—
91C2×C8×C53—41no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C42×C53—41no—
97C42×C53—41no—
98C42×C53C2×D411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53#4111yescharacteristic
102#11719924—28no—
103#11719921—22no—
104#11719921—22no—
105C4⋊Dic53—21no—
106C4⋊Dic53C22⋊C412yes—
107C2×#88322—21no—
108C2×#88322—21no—
109C2×#88322—21no—
110C2×#88322—22no—
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322—41no—
114C2×#88322M1612yescomplemented
115#11719912—84no—
116C23×C53C4×D411yescharacteristic
117C22×D53—22no—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—81no—
121C22×D53—81no—
122Q8×C53—22no—
123Q8×C53—22no—
124Q8×C53—41no—
125Q8×C53—41no—
126C2×C4×C53—22no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—81no—
133C2×C4×C53C2×C22⋊C411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53#12011yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53#12311yescharacteristic
139C2×C4×C53C2×M1611yescharacteristic
140C2×C4×C53C2×M1611yescharacteristic
141C2×C4×C53#13111yescharacteristic
142C2×C4×C53#13111yescharacteristic
143C2×Dic53—22no—
144C2×Dic53—81no—
145D212—22no—
146D212—22no—
147D212—41no—
148D212—41no—
149C4×D53—44no—
150C4×D53—82no—
151C8×C53—42no—
152C8×C53—81no—
153#88322—22no—
154#88322—42no—
155#88322—41no—
156#88322—81no—
157#9091—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53#38611yescharacteristic
161C22×C53#43511yescharacteristic
162C22×C53#43611yescharacteristic
163C2×D53—42no—
164C2×D53—42no—
165C2×D53—42no—
166C2×D53—82no—
167C2×D53—82no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—42no—
173C4×C53—81no—
174C4×C53#40711yescharacteristic
175C4×C53#40811yescharacteristic
176C4×C53#43511yescharacteristic
177C4×C53#43611yescharacteristic
178Dic53—42no—
179#2975—531no—
180C2×#440—531no—
181#1479D5311yescharacteristic, complemented
182#1478—531no—
183#1461—531no—
184#1281—531no—
185#1275—531no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53#187011yescharacteristic
189C2×C53#190211yescharacteristic
190C2×C53#190411yescharacteristic
191D53—82no—
192C22×SD16—531no—
193C2×C4×Q8C2×D5311yescharacteristic
194C2×C4×D4—531no—
195#440—1062no—
196#440—1062no—
197#436—1062no—
198#436—1062no—
199#431—531no—
200#431—531no—
201#431C2×D5311yescharacteristic
202#431C2×D5311yescharacteristic
203C2×#107—531no—
204C2×#103—531no—
205C2×#102—531no—
206#407—1062no—
207C2×#97—531no—
208Q8⋊C8—22no—
209D4⋊C8—1062no—
210#324—1062no—
211C53#909111yescharacteristic, complemented, Sylow
212C22×Q8C4×D5311yescharacteristic
213C22×D4—531no—
214C23×C4—1061no—
215C2×SD16—1062no—
216C2×SD16—1062no—
217C2×SD16—1062no—
218C2×M16—1064no—
219C22×C8—531no—
220C22×C8—531no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×D4—532no—
226C4×D4—532no—
227C4×D4—1061no—
228C4×D4—1061no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—531no—
231C2×C4⋊C4C4×D5311yescharacteristic
232C2×C22⋊C4—1061no—
233C2×C42—21no—
234C2×C42C22×D5311yescharacteristic
235#107—1062no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—1061no—
239C4⋊C8—1061no—
240#103—1062no—
241#102—1062no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—532no—
245C22⋊C8—1064no—
246C22⋊C8—1061no—
247#97—1061no—
248#97—1061no—
249C4×C8—21no—
250C4×C8—1061no—
251C4×C8—1061no—
252C4×C8C22⋊D5312yes—
253C24—1061no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8#1171991212yescomplemented
260C2×D4—532no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4—1061no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—1061no—
269C22×C4—1062no—
270C22×C4—1062no—
271C22×C4—4242no—
272C22×C4C2×C4×D5311yescharacteristic
273C22×C4D4×D5311yescharacteristic
274C22×C4#1171994711yescharacteristic
275SD16—1068no—
276M16—4244no—
277C2×C8—22no—
278C2×C8—41no—
279C2×C8—532no—
280C2×C8—1061no—
281C2×C8—1061no—
282C2×C8—1061no—
283C2×C8—1062no—
284C2×C8—1061no—
285C2×C8—1061no—
286C2×C8—2121no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—532no—
292C4⋊C4—1061no—
293C22⋊C4—1062no—
294C22⋊C4—1062no—
295C42—41no—
296C42—41no—
297C42#1171994511yescharacteristic
298C42D4×D5311yescharacteristic
299C42D4×D5311yescharacteristic
300C42C2×C22⋊D5311yescharacteristic
301C23—1062no—
302C23—4241no—
303C23—4241no—
304C23—2121no—
305C23—2121no—
306C23#1313317711yescharacteristic
307Q8—22no—
308Q8—22no—
309Q8—41no—
310Q8—41no—
311D4—1062no—
312D4—1062no—
313D4—2121no—
314D4—2121no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—1062no—
323C2×C4—4242no—
324C2×C4—4241no—
325C2×C4—2124no—
326C2×C4C22⋊C4×D5311yescharacteristic
327C2×C4C4×C22⋊D5311yescharacteristic
328C2×C4#1313318511yescharacteristic
329C2×C4#1313318811yescharacteristic
330C2×C4#1313319011yescharacteristic
331C2×C4#1313319911yescharacteristic
332C2×C4M16×D5311yescharacteristic
333C2×C4C2×#1171991211yescharacteristic
334C2×C4#1313322311yescharacteristic
335C2×C4#1313322511yescharacteristic
336C8—42no—
337C8—81no—
338C8—1062no—
339C8—4241no—
340C8—2122no—
341C8—2121no—
342C22—21no—
343C22—21no—
344C22—4242no—
345C22—4242no—
346C22—2122no—
347C22—2122no—
348C22—2122no—
349C22#1681049911yescharacteristic
350C22#1681069111yescharacteristic
351C22#1681069811yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—81no—
358C4—2122no—
359C4#1681058511yescharacteristic
360C4#1681058811yescharacteristic
361C4#1681069411yescharacteristic
362C4#1681069911yescharacteristic
363C2—21no—
364C2—21no—
365C2—4242no—
366C2#2027500911yescharacteristic
367C2#2027518511yescharacteristic
368C2#2027519911yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1479C2
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1479D53
hypocenterC53#9091
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
4. #20273039C2semidirect
5. #20272559C2semidirect
6. #20272545C2semidirect
7. C53×#1479C2semidirect
74. C2×Q8×C53M16semidirect
114. C2×#88322M16semidirect
181. #1479D53semidirect
211. C53#9091semidirect
259. C2×Q8#11719912semidirect

Maximal subgroups

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

subgroupindex
#202837552
C2×#168107212
#202731072
#202730392
#202725592
#202725452
C53×#14792
#909153