Finite Group Databasebeta

← #27302294

Subgroups — #27302294

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#20283409C211yesmaximal, characteristic
2#20276480C211yesmaximal, characteristic, complemented
3#20276344C211yesmaximal, characteristic, complemented
4#20276342C211yesmaximal, characteristic, complemented
5#20276326C211yesmaximal, characteristic, complemented
6#20276319C211yesmaximal, characteristic, complemented
7C53×#2112C211yesmaximal, characteristic, complemented
8C2×#13133223C2211yescharacteristic
9#16811134C411yescharacteristic
10C2×#13133157C2211yescharacteristic
11C2×#13133157C2211yescharacteristic
12#16810694—22no—
13C8×C22⋊D53—22no—
14#16810649C411yescharacteristic
15#16810589C411yescharacteristic
16#16810587C411yescharacteristic
17#16810578C2211yescharacteristic
18#16810443—22no—
19#16810408—22no—
20#16810406—22no—
21C22⋊#11719908—22no—
22#16810400—22no—
23C23×C8×C53C2211yescharacteristic
24C2×M32×C53C2211yescharacteristic
25C2×M32×C53C2211yescharacteristic
26C53×C22⋊C16—22no—
27C2×#11719945C2×C411yescharacteristic
28#13133223—21no—
29#13133223—21no—
30#13133223—21no—
31#13133223—21no—
32C2×#11719917C2×C411yescharacteristic
33C2×#11719917C2×C411yescharacteristic
34C2×#11719912—21no—
35C2×C8×D53—21no—
36#13133200—21no—
37#13133196—21no—
38#13133191—21no—
39#13133186—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—
53C23×C4×C53C2×C411yescharacteristic
54C22×C8×C53—21no—
55C22×C8×C53—21no—
56C22×C8×C53—21no—
57C22×C8×C53—21no—
58C22×C8×C53—21no—
59C22×C8×C53—21no—
60C22×C8×C53C2×C411yescharacteristic
61C22×C8×C53C2311yescharacteristic
62M32×C53—21no—
63M32×C53—21no—
64M32×C53—21no—
65M32×C53—21no—
66C2×C16×C53—21no—
67C2×C16×C53D412yescomplemented
68C24×C53M1611yescharacteristic
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53M1611yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C2×C22⋊D53—22no—
78C2×C22⋊D53—21no—
79#11719945—21no—
80#11719945—21no—
81#11719945—21no—
82#11719945—21no—
83C2×D212M1611yescharacteristic
84C2×C4×D53—21no—
85C2×Dic106M1611yescharacteristic
86C2×C8×C53—21no—
87C2×C8×C53—21no—
88C2×C8×C53—21no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
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—
107#11719917—21no—
108#11719917—21no—
109C2×#88322—21no—
110C2×#88322C22⋊C412yes—
111#11719912—41no—
112#11719912—41no—
113C8×D53—41no—
114C8×D53—41no—
115C16×C53—22no—
116C16×C53—41no—
117#11719908—22no—
118#11719908—41no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—42no—
123C23×C53C2×M1611yescharacteristic
124C22×D53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53C2×C22⋊C411yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53C2×M1611yescharacteristic
136C2×C4×C53#13111yescharacteristic
137C2×C4×C53#13111yescharacteristic
138C22⋊D53—44no—
139C22⋊D53—41no—
140C22⋊D53—41no—
141C2×Dic53—21no—
142C2×Dic53—22no—
143D212—21no—
144D212—21no—
145C4×D53—41no—
146C4×D53—41no—
147Dic106—21no—
148Dic106—21no—
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—22no—
158C22×C53—22no—
159C22×C53—22no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53#40711yescharacteristic
166C22×C53#43511yescharacteristic
167C22×C53#43511yescharacteristic
168C2×D53—42no—
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#2117—531no—
181#2112—531no—
182#2112D5311yescharacteristic, complemented
183#2110—531no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—42no—
188C2×C53#195511yescharacteristic
189D53—81no—
190C2×#131—531no—
191C23×C8C2×D5311yescharacteristic
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—1062no—
205#350—1062no—
206C22⋊C16—22no—
207C22⋊C16—1062no—
208#347—1062no—
209C53#1320011yescharacteristic, complemented, Sylow
210C2×#41—531no—
211C23×C4C4×D5311yescharacteristic
212#131—1061no—
213#131—1061no—
214#131—1061no—
215#131—1061no—
216C2×M16—531no—
217C2×M16—531no—
218C2×M16—1061no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—1061no—
226C22×C8C4×D5311yescharacteristic
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#1171991211yescharacteristic
252#41—1061no—
253#41—1061no—
254#41—1061no—
255#41—1061no—
256C2×Q8—531no—
257C2×D4—531no—
258C2×D4—1062no—
259C2×D4—1061no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—1061no—
267C22×C4#1171991211yescharacteristic
268C22×C4C2×C4×D5311yescharacteristic
269M16—1061no—
270M16—1061no—
271M16—1061no—
272M16—1061no—
273M16—2121no—
274M16—2121no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—532no—
283C2×C8—1061no—
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—21no—
304C23—21no—
305C23—42no—
306C23—1061no—
307C23C2×#1171991211yescharacteristic
308Q8—1061no—
309Q8—1061no—
310D4—1061no—
311D4—1061no—
312D4—2124no—
313D4—2121no—
314D4—2121no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—1062no—
323C2×C4—1061no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4C22⋊C4×D5311yescharacteristic
327C2×C4#1313317711yescharacteristic
328C2×C4C4×C22⋊D5311yescharacteristic
329C2×C4M16×D5311yescharacteristic
330C2×C4#1313322311yescharacteristic
331C2×C4#1313322511yescharacteristic
332C8—41no—
333C8—1062no—
334C8—2121no—
335C8#1313318911yescharacteristic
336C8#1313319011yescharacteristic
337C8#1313320311yescharacteristic
338C8#1313321311yescharacteristic
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—2122no—
348C22—2121no—
349C22#1681058511yescharacteristic
350C22#1681069111yescharacteristic
351C22#1681069411yescharacteristic
352C4—41no—
353C4—2122no—
354C4—2121no—
355C4#1681052311yescharacteristic
356C4#1681058811yescharacteristic
357C4#1681069211yescharacteristic
358C4#1681069711yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—4241no—
364C2#2027551111yescharacteristic
subgroupG/H
centerC8#13133213
commutatorC22×C4×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2112C2
socleC2×C53#1955
prime radical#16811134C4
cosocleC22×C8×C53C23
hypercenter#2112D53
hypocenterC53#13200
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. #20276480C2semidirect
3. #20276344C2semidirect
4. #20276342C2semidirect
5. #20276326C2semidirect
6. #20276319C2semidirect
7. C53×#2112C2semidirect
47. C2×#11719908D4semidirect
67. C2×C16×C53D4semidirect
182. #2112D53semidirect
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
#202834092
#202764802
#202763442
#202763422
#202763262
#202763192
C53×#21122
#1320053