Finite Group Databasebeta

← #27302299

Subgroups — #27302299

Order 13568.

420 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283416C211yesmaximal, characteristic
2#20276480C211yesmaximal, characteristic, complemented
3#20276343C211yesmaximal, characteristic, complemented
4C2×#16810404C211yesmaximal, characteristic, complemented
5C2×#16810404C211yesmaximal, characteristic, complemented
6#20276319C211yesmaximal, characteristic, complemented
7C53×#2117C211yesmaximal, characteristic, complemented
8C22×#11719912C2211yescharacteristic
9#16811130C411yescharacteristic
10C2×#13133157C2211yescharacteristic
11C2×#13133157C2211yescharacteristic
12#16810699—22no—
13#16810694—22no—
14#16810651C411yescharacteristic
15#16810590C411yescharacteristic
16#16810586C411yescharacteristic
17#16810578C2211yescharacteristic
18#16810443—22no—
19#16810407—22no—
20#16810404—22no—
21#16810404—22no—
22#16810400—22no—
23C2×#131×C53C2211yescharacteristic
24C2×M32×C53C2211yescharacteristic
25C2×M32×C53C2211yescharacteristic
26C53×#350—22no—
27C22×C4×D53C2×C411yescharacteristic
28C2×#11719917C2×C411yescharacteristic
29C2×#11719912—21no—
30C2×#11719912—21no—
31C2×#11719912—21no—
32C2×#11719912—21no—
33C2×#11719912—21no—
34C2×#11719912—21no—
35C22×#88322C2×C411yescharacteristic
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#13133157D412yescomplemented
46C2×#11719908—21no—
47C2×#11719908—21no—
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×C53D412yescomplemented
66C2×C16×C53—21no—
67C2×C16×C53—21no—
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—
111#11719912—41no—
112#11719912—41no—
113#11719912—41no—
114#11719912—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—81no—
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—82no—
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#13201—531nomaximal, Sylow
157C22×C53—41no—
158C22×C53—42no—
159C22×C53—41no—
160C22×C53#40711yescharacteristic
161C22×C53#43511yescharacteristic
162C22×C53#43511yescharacteristic
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—81no—
167C2×D53—81no—
168C4×C53—41no—
169C4×C53#39011yescharacteristic
170C4×C53#40811yescharacteristic
171C4×C53#43511yescharacteristic
172C4×C53#43511yescharacteristic
173Dic53—42no—
174Dic53—41no—
175#2930—531no—
176#2153—531no—
177#2117—531no—
178#2117D5311yescharacteristic, complemented
179C2×#349—531no—
180C2×#349—531no—
181#2110—531no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—81no—
186C2×C53#195511yescharacteristic
187D53—81no—
188C2×#131C2×D5311yescharacteristic
189C22×M16—531no—
190#525—531no—
191C2×M32—531no—
192C2×M32—531no—
193C2×M32C2×D5311yescharacteristic
194C2×M32C2×D5311yescharacteristic
195#436—1062no—
196#435—1062no—
197#424—531no—
198#408—531no—
199#407—531no—
200#405—531no—
201#364—1062no—
202#350—22no—
203#350—1062no—
204#349—1062no—
205#349—1062no—
206#347—1062no—
207C53#1320111yescharacteristic, complemented, Sylow
208C2×#41C4×D5311yescharacteristic
209C23×C4—531no—
210#131—21no—
211#131—21no—
212#131—21no—
213#131—21no—
214C2×M16—21no—
215C2×M16—531no—
216C2×M16—1061no—
217C2×M16—1061no—
218C2×M16—1061no—
219C2×M16—1061no—
220C2×M16—1061no—
221C2×M16—1061no—
222C2×M16C4×D5311yescharacteristic
223C22×C8—21no—
224C22×C8—531no—
225C22×C8C22×D5311yescharacteristic
226#123—1061no—
227C22⋊Q8—1061no—
228#121—1061no—
229#120—1061no—
230C4×D4—1062no—
231#117—531no—
232M32—21no—
233M32—21no—
234M32—21no—
235M32—532no—
236M32—1061no—
237M32—1061no—
238M32—1061no—
239M32C22⋊D5312yescomplemented
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—1061no—
243C2×C16—1061no—
244C4⋊C8—1062no—
245C22⋊C8—1062no—
246C22⋊C8—1062no—
247#97—532no—
248C4×C8—532no—
249C24—531no—
250#41—21no—
251#41—21no—
252#41—21no—
253#41—21no—
254C2×Q8#1171991211yescharacteristic
255C2×D4—21no—
256C2×D4—1062no—
257C2×D4#1171991211yescharacteristic
258C22×C4—21no—
259C22×C4—531no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4C2×C4×D5311yescharacteristic
267M16—21no—
268M16—21no—
269M16—41no—
270M16—41no—
271M16—1061no—
272M16—1061no—
273M16—2121no—
274M16—2121no—
275M16—2121no—
276M16—2121no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—532no—
281C2×C8—1061no—
282C2×C8—1061no—
283C2×C8—1061no—
284C2×C8#1171994511yescharacteristic
285C2×C8D4×D5311yescharacteristic
286C2×C8D4×D5311yescharacteristic
287C2×C8D4×D5311yescharacteristic
288C2×C8#1171994711yescharacteristic
289C2×C8C2×C22⋊D5311yescharacteristic
290C4⋊C4—1062no—
291C4⋊C4—1061no—
292C22⋊C4—1062no—
293C22⋊C4—1062no—
294C22⋊C4—1061no—
295C42—532no—
296C16—22no—
297C16—41no—
298C16—1062no—
299C16—2121no—
300C23—21no—
301C23—1061no—
302C23—1061no—
303C23—1061no—
304C23—4241no—
305C23M16×D5311yescharacteristic
306Q8—21no—
307Q8—21no—
308D4—21no—
309D4—21no—
310D4—41no—
311D4—41no—
312D4—4242no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—1062no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4C22⋊C4×D5311yescharacteristic
325C2×C4#1313317711yescharacteristic
326C2×C4C4×C22⋊D5311yescharacteristic
327C2×C4C2×#1171991211yescharacteristic
328C2×C4#1313322311yescharacteristic
329C2×C4#1313322511yescharacteristic
330C8—41no—
331C8—1062no—
332C8—2121no—
333C8#1313318911yescharacteristic
334C8#1313319011yescharacteristic
335C8#1313320311yescharacteristic
336C8#1313321311yescharacteristic
337C22—41no—
338C22—42no—
339C22—41no—
340C22—4241no—
341C22—4241no—
342C22—2121no—
343C22—2121no—
344C22—2121no—
345C22#1681058511yescharacteristic
346C22#1681069211yescharacteristic
347C22#1681069711yescharacteristic
348C4—41no—
349C4—2122no—
350C4—2121no—
351C4#1681052311yescharacteristic
352C4#1681058811yescharacteristic
353C4#1681069111yescharacteristic
354C4#1681069411yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—81no—
359C2—4241no—
360C2#2027551111yescharacteristic
subgroupG/H
centerC4#16810523
commutatorC22×C4×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2117C2
socleC2×C53#1955
prime radical#16811130C4
cosocleC22×C8×C53C23
hypercenter#2117D53
hypocenterC53#13201
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
2. #20276480C2semidirect
3. #20276343C2semidirect
4. C2×#16810404C2semidirect
5. C2×#16810404C2semidirect
6. #20276319C2semidirect
7. C53×#2117C2semidirect
45. #13133157D4semidirect
65. M32×C53D4semidirect
178. #2117D53semidirect
207. C53#13201semidirect
239. M32C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202834162
#202764802
#202763432
C2×#168104042
#202763192
C53×#21172
#1320153