Finite Group Databasebeta

← #27302296

Subgroups — #27302296

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283407C211yesmaximal, characteristic
2#20276480C211yesmaximal, characteristic, complemented
3#20276343C211yesmaximal, characteristic, complemented
4#20276325C211yesmaximal, characteristic, complemented
5#20276325C211yesmaximal, characteristic, complemented
6#20276319C211yesmaximal, characteristic, complemented
7C53×#2117C211yesmaximal, characteristic, complemented
8C22×C8×D53C2211yescharacteristic
9#16811130C411yescharacteristic
10C2×#13133157C2211yescharacteristic
11C2×#13133157C2211yescharacteristic
12#16810692—22no—
13C8×C22⋊D53—22no—
14#16810651C411yescharacteristic
15#16810590C411yescharacteristic
16#16810586C411yescharacteristic
17#16810578C2211yescharacteristic
18#16810443—22no—
19#16810407—22no—
20#16810402—22no—
21#16810402—22no—
22#16810400—22no—
23C2×#131×C53C2211yescharacteristic
24C2×M32×C53C2211yescharacteristic
25C2×M32×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×D53—21no—
35C2×C8×D53—21no—
36#13133198—21no—
37#13133193—21no—
38#13133188—21no—
39#13133185—21no—
40C4×C22⋊D53—22no—
41#13133176C2×C411yescharacteristic
42#13133157—21no—
43#13133157—21no—
44#13133157—21no—
45#13133157—21no—
46C2×#11719908—21no—
47C2×#11719908D412yescomplemented
48#13133146—22no—
49C22⋊#88322—22no—
50#13133129—22no—
51#13133127D412yes—
52C8×Dic53D412yes—
53C2×#41×C53C2×C411yescharacteristic
54#131×C53—21no—
55#131×C53—21no—
56#131×C53—21no—
57#131×C53—21no—
58C2×M16×C53—21no—
59C2×M16×C53C2×C411yescharacteristic
60C22×C8×C53—21no—
61C22×C8×C53C2311yescharacteristic
62M32×C53—21no—
63M32×C53—21no—
64M32×C53—21no—
65M32×C53—21no—
66C2×C16×C53—21no—
67C2×C16×C53D412yescomplemented
68C23×D53M1611yescharacteristic
69#41×C53—21no—
70#41×C53—21no—
71#41×C53—21no—
72#41×C53—21no—
73C2×Q8×C53M1611yescharacteristic
74C2×D4×C53—21no—
75C2×D4×C53M1611yescharacteristic
76C22×C4×C53—21no—
77C22×C4×C53C22×C411yescharacteristic
78C2×C22⋊D53—22no—
79C22×Dic53M1611yescharacteristic
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86M16×C53—21no—
87M16×C53—21no—
88M16×C53—41no—
89M16×C53—41no—
90C2×C8×C53—21no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53C2×D411yescharacteristic
94C2×C8×C53C2×D411yescharacteristic
95C2×C8×C53C2×D411yescharacteristic
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53#4111yescharacteristic
98C2×C8×C53#4111yescharacteristic
99C22⋊Dic53—22no—
100C22⋊Dic53—21no—
101#11719921—22no—
102C4⋊Dic53—21no—
103#11719919—22no—
104C4×Dic53C22⋊C412yes—
105#11719917—21no—
106#11719917—21no—
107C2×#88322—21no—
108C2×#88322—21no—
109C2×#88322—21no—
110C2×#88322C22⋊C412yes—
111C8×D53—41no—
112C8×D53—41no—
113C8×D53—41no—
114C8×D53—41no—
115C16×C53—22no—
116C16×C53—41no—
117#11719908—22no—
118#11719908—41no—
119C23×C53—21no—
120C23×C53C2×M1611yescharacteristic
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—42no—
125Q8×C53—21no—
126Q8×C53—21no—
127D4×C53—21no—
128D4×C53—21no—
129D4×C53—41no—
130D4×C53—41no—
131C2×C4×C53—21no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53C2×C22⋊C411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C2×C4×C53#13111yescharacteristic
139C2×C4×C53#13111yescharacteristic
140C22⋊D53—44no—
141C2×Dic53—21no—
142C2×Dic53—22no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C4×D53—41no—
146C4×D53—41no—
147C4×D53—41no—
148C4×D53—41no—
149C8×C53—41no—
150C8×C53#12111yescharacteristic
151C8×C53#12111yescharacteristic
152C8×C53#12411yescharacteristic
153C8×C53#12711yescharacteristic
154#88322—22no—
155#88322—41no—
156#13200—531nomaximal, Sylow
157C22×C53—41no—
158C22×C53—42no—
159C22×C53—41no—
160C22×C53#40711yescharacteristic
161C22×C53#43511yescharacteristic
162C22×C53#43511yescharacteristic
163C2×D53—22no—
164C2×D53—22no—
165C2×D53—22no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C4×C53—41no—
171C4×C53#39011yescharacteristic
172C4×C53#40811yescharacteristic
173C4×C53#43511yescharacteristic
174C4×C53#43511yescharacteristic
175Dic53—42no—
176Dic53—41no—
177#2929—531no—
178#2153—531no—
179#2117—531no—
180#2117D5311yescharacteristic, complemented
181#2112—531no—
182#2112—531no—
183#2110—531no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—81no—
188C2×C53#195511yescharacteristic
189D53—42no—
190C2×#131C2×D5311yescharacteristic
191C23×C8—531no—
192#525—531no—
193C2×M32—531no—
194C2×M32—531no—
195C2×M32C2×D5311yescharacteristic
196C2×M32C2×D5311yescharacteristic
197#435—1062no—
198C8×D4—1062no—
199#424—531no—
200#408—531no—
201#407—531no—
202#405—531no—
203#364—1062no—
204#350—22no—
205#350—1062no—
206C22⋊C16—1062no—
207C22⋊C16—1062no—
208#347—1062no—
209C53#1320011yescharacteristic, complemented, Sylow
210C2×#41C4×D5311yescharacteristic
211C23×C4—531no—
212#131—21no—
213#131—21no—
214#131—21no—
215#131—21no—
216C2×M16—21no—
217C2×M16—531no—
218C2×M16C4×D5311yescharacteristic
219C22×C8—21no—
220C22×C8—531no—
221C22×C8—1061no—
222C22×C8—1061no—
223C22×C8—1061no—
224C22×C8—1061no—
225C22×C8—1061no—
226C22×C8—1061no—
227C22×C8C22×D5311yescharacteristic
228#123—1061no—
229C22⋊Q8—1061no—
230#121—1061no—
231#120—1061no—
232C4×D4—1062no—
233#117—531no—
234M32—21no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32—1061no—
239M32—1061no—
240M32—1061no—
241M32—1061no—
242C2×C16—21no—
243C2×C16—532no—
244C2×C16—1061no—
245C2×C16C22⋊D5312yescomplemented
246C4⋊C8—1062no—
247C22⋊C8—1062no—
248C22⋊C8—1062no—
249#97—532no—
250C4×C8—532no—
251C24—531no—
252#41—21no—
253#41—21no—
254#41—21no—
255#41—21no—
256C2×Q8#1171991211yescharacteristic
257C2×D4—21no—
258C2×D4—1062no—
259C2×D4#1171991211yescharacteristic
260C22×C4—21no—
261C22×C4—531no—
262C22×C4—1061no—
263C22×C4—1061no—
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—1061no—
282C2×C8—2121no—
283C2×C8—2121no—
284C2×C8—2121no—
285C2×C8—2121no—
286C2×C8#1171994511yescharacteristic
287C2×C8D4×D5311yescharacteristic
288C2×C8D4×D5311yescharacteristic
289C2×C8D4×D5311yescharacteristic
290C2×C8#1171994711yescharacteristic
291C2×C8C2×C22⋊D5311yescharacteristic
292C4⋊C4—1062no—
293C4⋊C4—1061no—
294C22⋊C4—1062no—
295C22⋊C4—1062no—
296C22⋊C4—1061no—
297C42—532no—
298C16—22no—
299C16—41no—
300C16—1062no—
301C16—2121no—
302C23—21no—
303C23—1061no—
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—1062no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4C22⋊C4×D5311yescharacteristic
327C2×C4#1313317711yescharacteristic
328C2×C4C4×C22⋊D5311yescharacteristic
329C2×C4C2×#1171991211yescharacteristic
330C2×C4#1313322311yescharacteristic
331C2×C4#1313322511yescharacteristic
332C8—41no—
333C8—1062no—
334C8—2121no—
335C8#1313318911yescharacteristic
336C8#1313319011yescharacteristic
337C8#1313320311yescharacteristic
338C8#1313321311yescharacteristic
339C22—41no—
340C22—42no—
341C22—41no—
342C22—1062no—
343C22—1062no—
344C22—1062no—
345C22—2121no—
346C22—2121no—
347C22—2121no—
348C22—2121no—
349C22#1681058511yescharacteristic
350C22#1681069211yescharacteristic
351C22#1681069711yescharacteristic
352C4—41no—
353C4—2122no—
354C4—2121no—
355C4#1681052311yescharacteristic
356C4#1681058811yescharacteristic
357C4#1681069111yescharacteristic
358C4#1681069411yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—81no—
363C2—2122no—
364C2#2027551111yescharacteristic
subgroupG/H
centerC8#13133213
commutatorC22×C4×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2117C2
socleC2×C53#1955
prime radical#16811130C4
cosocleC22×C8×C53C23
hypercenter#2117D53
hypocenterC53#13200
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
2. #20276480C2semidirect
3. #20276343C2semidirect
4. #20276325C2semidirect
5. #20276325C2semidirect
6. #20276319C2semidirect
7. C53×#2117C2semidirect
47. C2×#11719908D4semidirect
67. C2×C16×C53D4semidirect
180. #2117D53semidirect
209. C53#13200semidirect
245. C2×C16C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202834072
#202764802
#202763432
#202763252
#202763192
C53×#21172
#1320053