Finite Group Databasebeta

← #27300988

Subgroups — #27300988

Order 13568.

488 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
1C22×#13133156C211yesmaximal, characteristic, complemented
2#20283407C211yesmaximal, characteristic
3C2×#16810407C211yesmaximal, characteristic, complemented
4#20276325C211yesmaximal, characteristic, complemented
5#20272805C211yesmaximal, characteristic, complemented
6#20272803C211yesmaximal, characteristic, complemented
7C2×C53×#350C211yesmaximal, characteristic, complemented
8C22×C8×D53C2211yescharacteristic
9#16811130C411yescharacteristic
10C2×#13133157C2211yescharacteristic
11C2×#13133156—22no—
12C2×#13133156—22no—
13C2×#13133156—22no—
14C22×#11719908C2211yescharacteristic
15#16810692—22no—
16C8×C22⋊D53—22no—
17#16810651C411yescharacteristic
18#16810590C411yescharacteristic
19#16810586C411yescharacteristic
20#16810578C2211yescharacteristic
21#16810407—22no—
22#16810402—22no—
23C2×#131×C53C2211yescharacteristic
24C2×M32×C53C2211yescharacteristic
25C22×C16×C53C2211yescharacteristic
26C53×#350—22no—
27C22×C4×D53C2×C411yescharacteristic
28C2×#11719917C2×C411yescharacteristic
29C22×#88322C2×C411yescharacteristic
30C2×C8×D53—21no—
31C2×C8×D53—21no—
32C2×C8×D53—21no—
33C2×C8×D53—21no—
34C2×C8×D53D412yes—
35C2×C8×D53D412yes—
36#13133198—21no—
37#13133193C812yes—
38#13133188C812yes—
39#13133185—21no—
40C4×C22⋊D53—22no—
41#13133176C2×C411yescharacteristic
42#13133157—21no—
43#13133157—21no—
44#13133156—28no—
45C2×#11719908—21no—
46C2×#11719908—21no—
47C2×#11719908—21no—
48C2×#11719908D412yescomplemented
49#13133146—22no—
50C22⋊#88322—22no—
51#13133129—22no—
52#13133127—21no—
53C8×Dic53—21no—
54C2×#41×C53C2×C411yescharacteristic
55#131×C53—21no—
56#131×C53—21no—
57#131×C53—21no—
58#131×C53—21no—
59C2×M16×C53—21no—
60C2×M16×C53C2×C411yescharacteristic
61C22×C8×C53—21no—
62C22×C8×C53C2311yescharacteristic
63M32×C53—21no—
64M32×C53—21no—
65C2×C16×C53—21no—
66C2×C16×C53—21no—
67C2×C16×C53—21no—
68C2×C16×C53D412yescomplemented
69C23×D53M1611yescharacteristic
70#41×C53—21no—
71#41×C53—21no—
72#41×C53—21no—
73#41×C53—21no—
74C2×Q8×C53C2×C811yescharacteristic
75C2×D4×C53—21no—
76C2×D4×C53C2×C811yescharacteristic
77C22×C4×C53—21no—
78C22×C4×C53C22×C411yescharacteristic
79C2×C22⋊D53—22no—
80C22×Dic53M1611yescharacteristic
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53C22⋊C412yes—
86C2×C4×D53C2×C812yes—
87M16×C53—21no—
88M16×C53—21no—
89M16×C53—41no—
90M16×C53—41no—
91C2×C8×C53—21no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53C2×D411yescharacteristic
95C2×C8×C53C2×D411yescharacteristic
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53#4111yescharacteristic
99C2×C8×C53#4111yescharacteristic
100C22⋊Dic53—21no—
101C22⋊Dic53—22no—
102#11719921—22no—
103C4⋊Dic53C2×C812yes—
104#11719919—22no—
105C4×Dic53—21no—
106#11719917—21no—
107#11719917—21no—
108C2×#88322—21no—
109C2×#88322—21no—
110C2×#88322—21no—
111C2×#88322C22⋊C412yes—
112C8×D53—22no—
113C8×D53—22no—
114C8×D53—22no—
115C8×D53—22no—
116C16×C53—22no—
117C16×C53—41no—
118#11719908—22no—
119#11719908—41no—
120C23×C53—21no—
121C23×C53C2×M1611yescharacteristic
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—42no—
125C22×D53C22⋊C812yes—
126Q8×C53—21no—
127Q8×C53—21no—
128D4×C53—21no—
129D4×C53—21no—
130D4×C53—41no—
131D4×C53—41no—
132C2×C4×C53—21no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53C2×C22⋊C411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53C22×C811yescharacteristic
139C2×C4×C53#13111yescharacteristic
140C2×C4×C53#13111yescharacteristic
141C22⋊D53—44no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—22no—
145C2×Dic53C22⋊C812yes—
146C4×D53—22no—
147C4×D53—22no—
148C4×D53—22no—
149C4×D53—22no—
150C8×C53—41no—
151C8×C53#12011yescharacteristic
152C8×C53#12111yescharacteristic
153C8×C53#12111yescharacteristic
154C8×C53#12311yescharacteristic
155#88322—22no—
156#88322—41no—
157#12984—531nomaximal, Sylow
158C22×C53—22no—
159C22×C53—42no—
160C22×C53—41no—
161C22×C53C2×C22⋊C811yescharacteristic
162C22×C53#43511yescharacteristic
163C22×C53#43511yescharacteristic
164C2×D53—22no—
165C2×D53—22no—
166C2×D53—22no—
167C2×D53—42no—
168C2×D53—42no—
169C4×C53—41no—
170C4×C53#38611yescharacteristic
171C4×C53#40811yescharacteristic
172C4×C53C8×D411yescharacteristic
173C4×C53C8×D411yescharacteristic
174Dic53—22no—
175Dic53—42no—
176C22×M32—531no—
177#2929—531no—
178C2×#350—531no—
179C2×#350D5311yescharacteristic, complemented
180#2112—531no—
181#1379—531no—
182#1379—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—81no—
186C2×C53#186911yescharacteristic
187C2×C53#211712yes—
188D53—42no—
189C2×#131C2×D5311yescharacteristic
190C23×C8—531no—
191#525—531no—
192C2×M32—531no—
193C2×M32—1062no—
194C2×M32—1062no—
195C2×M32—1062no—
196C2×M32C2×D5311yescharacteristic
197C22×C16—531no—
198C22×C16C2×D5311yescharacteristic
199#435—1062no—
200C8×D4—1062no—
201#424—531no—
202#408—531no—
203#407—531no—
204#405—531no—
205#350—22no—
206#350—1062no—
207C22⋊C16—1062no—
208C53#1298411yescharacteristic, complemented, Sylow
209C2×#41C4×D5311yescharacteristic
210C23×C4—531no—
211#131—21no—
212#131—21no—
213#131—21no—
214#131—21no—
215C2×M16—21no—
216C2×M16—531no—
217C2×M16C4×D5311yescharacteristic
218C22×C8—21no—
219C22×C8—531no—
220C22×C8—532no—
221C22×C8—532no—
222C22×C8—1061no—
223C22×C8—1061no—
224C22×C8—1061no—
225C22×C8—1061no—
226C22×C8C22×D5311yescharacteristic
227#123—1061no—
228C22⋊Q8—532no—
229#121—532no—
230#120—1061no—
231C4×D4—1062no—
232#117—531no—
233M32—21no—
234M32—21no—
235M32—1068no—
236M32—1061no—
237M32—1061no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—532no—
242C2×C16—1061no—
243C2×C16—1061no—
244C2×C16—1061no—
245C2×C16C22⋊D5312yescomplemented
246C4⋊C8—1062no—
247C22⋊C8—1062no—
248C22⋊C8—1062no—
249#97—1061no—
250C4×C8—1061no—
251C24—531no—
252#41—21no—
253#41—21no—
254#41—21no—
255#41—21no—
256C2×Q8C8×D5311yescharacteristic
257C2×D4—21no—
258C2×D4—1062no—
259C2×D4C8×D5311yescharacteristic
260C22×C4—21no—
261C22×C4—531no—
262C22×C4—532no—
263C22×C4—532no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4C2×C4×D5311yescharacteristic
269M16—21no—
270M16—21no—
271M16—41no—
272M16—41no—
273M16—1061no—
274M16—1061no—
275C2×C8—21no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—532no—
279C2×C8—1061no—
280C2×C8—1061no—
281C2×C8—1062no—
282C2×C8—1062no—
283C2×C8—1062no—
284C2×C8—1062no—
285C2×C8—1061no—
286C2×C8#1171994511yescharacteristic
287C2×C8D4×D5311yescharacteristic
288C2×C8D4×D5311yescharacteristic
289C2×C8D4×D5311yescharacteristic
290C2×C8#1171994711yescharacteristic
291C2×C8C2×C22⋊D5311yescharacteristic
292C4⋊C4—532no—
293C4⋊C4—1062no—
294C22⋊C4—1062no—
295C22⋊C4—1062no—
296C22⋊C4—1061no—
297C42—1061no—
298C16—22no—
299C16—41no—
300C16—1062no—
301C16—2121no—
302C23—21no—
303C23—532no—
304C23—1061no—
305C23—1061no—
306C23—2122no—
307C23M16×D5311yescharacteristic
308Q8—21no—
309Q8—21no—
310D4—21no—
311D4—21no—
312D4—41no—
313D4—41no—
314D4—2124no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—532no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1062no—
322C2×C4—1062no—
323C2×C4—1062no—
324C2×C4—1062no—
325C2×C4—1062no—
326C2×C4C22⋊C4×D5311yescharacteristic
327C2×C4#1313317711yescharacteristic
328C2×C4C4×C22⋊D5311yescharacteristic
329C2×C4C2×C8×D5311yescharacteristic
330C2×C4#1313322311yescharacteristic
331C2×C4#1313322511yescharacteristic
332C8—41no—
333C8—1062no—
334C8—2121no—
335C8#1313318511yescharacteristic
336C8#1313318811yescharacteristic
337C8#1313319011yescharacteristic
338C8#1313319911yescharacteristic
339C22—22no—
340C22—41no—
341C22—42no—
342C22—1062no—
343C22—1062no—
344C22—1062no—
345C22—2122no—
346C22—2122no—
347C22C22⋊C8×D5311yescharacteristic
348C22#1681069211yescharacteristic
349C22#1681069711yescharacteristic
350C4—41no—
351C4—1062no—
352C4—2122no—
353C4#1681049911yescharacteristic
354C4#1681058811yescharacteristic
355C4#1681068611yescharacteristic
356C4C8×C22⋊D5311yescharacteristic
357C2—21no—
358C2—21no—
359C2—81no—
360C2—2122no—
361C2#2027500411yescharacteristic
362C2#2027634112yes—
subgroupG/H
centerC2×C8D4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC2×C53×#350C2
socleC22×C53C2×C22⋊C8
prime radical#16811130C4
cosocleC22×C8×C53C23
hypercenterC2×#350D53
hypocenterC53#12984
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
1. C22×#13133156C2semidirect
3. C2×#16810407C2semidirect
4. #20276325C2semidirect
5. #20272805C2semidirect
6. #20272803C2semidirect
7. C2×C53×#350C2semidirect
48. C2×#11719908D4semidirect
68. C2×C16×C53D4semidirect
179. C2×#350D53semidirect
208. C53#12984semidirect
245. C2×C16C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C22×#131331562
#202834072
C2×#168104072
#202763252
#202728052
#202728032
C2×C53×#3502
#1298453