Finite Group Databasebeta

← #27300962

Subgroups — #27300962

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283616C211yesmaximal, characteristic
2#20276344C211yesmaximal, characteristic, complemented
3C2×#16810407C211yesmaximal, characteristic, complemented
4#20276330C211yesmaximal, characteristic, complemented
5#20272805C211yesmaximal, characteristic, complemented
6#20272800C211yesmaximal, characteristic, complemented
7C53×#3407C211yesmaximal, characteristic, complemented
8#16811421C411yescharacteristic
9C2×#13133223C2211yescharacteristic
10C2×#13133157C2211yescharacteristic
11C22×#11719908C2211yescharacteristic
12#16810743—22no—
13#16810738—22no—
14#16810671C411yescharacteristic
15#16810640C411yescharacteristic
16#16810639C411yescharacteristic
17#16810578C2211yescharacteristic
18#16810408—22no—
19#16810407—22no—
20#16810402—22no—
21C2×#131×C53C2211yescharacteristic
22C53×#504—22no—
23C53×#504—22no—
24C2×M32×C53—22no—
25C2×M32×C53C2211yescharacteristic
26C2×M32×C53C2211yescharacteristic
27C2×#11719945C2×C411yescharacteristic
28#13133248—21no—
29#13133247C812yes—
30#13133243—21no—
31#13133242C812yes—
32#13133237—22no—
33#13133223—21no—
34#13133223—21no—
35#13133223—21no—
36#13133223—21no—
37C2×#11719917C2×C411yescharacteristic
38C2×#11719917C2×C411yescharacteristic
39C2×#11719912—21no—
40C2×C8×D53—21no—
41#13133176C2×C411yescharacteristic
42#13133157—21no—
43#13133157—21no—
44C2×#11719908—21no—
45C2×#11719908—21no—
46C2×#11719908—21no—
47C2×#11719908—21no—
48#13133152—22no—
49#13133144—22no—
50#13133143—22no—
51#13133127—21no—
52C8×Dic53—21no—
53C2×#41×C53C2×C411yescharacteristic
54#131×C53—21no—
55#131×C53—21no—
56#131×C53D412yes—
57#131×C53D412yes—
58C2×M16×C53—21no—
59C2×M16×C53C2×C411yescharacteristic
60C22×C8×C53—21no—
61C22×C8×C53C2311yescharacteristic
62M32×C53—24no—
63M32×C53—21no—
64M32×C53—21no—
65M32×C53—21no—
66M32×C53D412yes—
67C2×C16×C53—24no—
68C2×C16×C53—21no—
69C2×C16×C53D412yes—
70#41×C53—21no—
71#41×C53—21no—
72#41×C53C22⋊C412yes—
73#41×C53C2×C812yes—
74C2×Q8×C53M1611yescharacteristic
75C2×D4×C53—21no—
76C2×D4×C53M1611yescharacteristic
77C22×C4×C53—21no—
78C22×C4×C53C22×C411yescharacteristic
79C2×C22⋊D53—21no—
80#11719945—21no—
81#11719945—21no—
82#11719945—21no—
83#11719945—21no—
84C2×D212C2×C811yescharacteristic
85C2×C4×D53—21no—
86C2×Dic106C2×C811yescharacteristic
87M16×C53—21no—
88M16×C53—22no—
89M16×C53—22no—
90M16×C53C22⋊C412yes—
91C2×C8×C53—21no—
92C2×C8×C53—22no—
93C2×C8×C53—22no—
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—
101#11719925—22no—
102#11719924—22no—
103#11719923—22no—
104#11719922—22no—
105C4⋊Dic53—21no—
106C4×Dic53—21no—
107#11719917—21no—
108#11719917—21no—
109#11719917—21no—
110#11719917C2×C812yes—
111C2×#88322—21no—
112C2×#88322—21no—
113#11719912—41no—
114#11719912—41no—
115C8×D53—41no—
116C8×D53—41no—
117C16×C53—22no—
118C16×C53—22no—
119#11719908—41no—
120#11719908—41no—
121C23×C53—21no—
122C23×C53#13111yescharacteristic
123C22×D53—21no—
124Q8×C53—21no—
125Q8×C53C22⋊C812yes—
126D4×C53—22no—
127D4×C53—22no—
128D4×C53—21no—
129D4×C53C22⋊C812yes—
130C2×C4×C53—21no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53C2×C22⋊C411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53C22×C811yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C2×C4×C53#13111yescharacteristic
139C22⋊D53—41no—
140C22⋊D53—41no—
141C2×Dic53—21no—
142C2×Dic53—41no—
143D212—21no—
144D212—21no—
145C4×D53—41no—
146C4×D53—41no—
147Dic106—21no—
148Dic106—21no—
149C8×C53—22no—
150C8×C53#12011yescharacteristic
151C8×C53#12111yescharacteristic
152C8×C53#12111yescharacteristic
153C8×C53#12311yescharacteristic
154#88322—22no—
155#88322—41no—
156#12979—531nomaximal, Sylow
157C22×C53—22no—
158C22×C53—41no—
159C22×C53—81no—
160C22×C53#40811yescharacteristic
161C22×C53C8×D411yescharacteristic
162C22×C53#43511yescharacteristic
163C2×D53—42no—
164C2×D53—41no—
165C4×C53—22no—
166C4×C53#38611yescharacteristic
167C4×C53C2×C22⋊C811yescharacteristic
168C4×C53C8×D411yescharacteristic
169C4×C53#43511yescharacteristic
170Dic53—41no—
171Dic53—81no—
172#3407D5311yescharacteristic, complemented
173#2955—531no—
174#2117—531no—
175C2×#350—531no—
176#2113—531no—
177#1379—531no—
178#1378—531no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—42no—
183C2×C53#186911yescharacteristic
184D53—81no—
185#575—531no—
186C2×#131—531no—
187C2×#131C2×D5311yescharacteristic
188#504—22no—
189#504—22no—
190C2×M32—22no—
191C2×M32—531no—
192C2×M32C2×D5311yescharacteristic
193C2×M32C2×D5311yescharacteristic
194C22×C16—531no—
195#444—1062no—
196#443—1062no—
197#430—531no—
198#421—531no—
199#421—531no—
200#405—531no—
201#350—1062no—
202#350—1062no—
203C22⋊C16—1062no—
204C53#1297911yescharacteristic, complemented, Sylow
205C2×#41—531no—
206C2×#41C4×D5311yescharacteristic
207#137—532no—
208#137—1061no—
209AGL(1,ℤ/8)—532no—
210AGL(1,ℤ/8)—1061no—
211#135—1062no—
212#131—21no—
213#131—21no—
214#131—1061no—
215#131—1061no—
216#131—1061no—
217#131—1061no—
218#131D21212yes—
219#131C22⋊D5312yes—
220C2×M16—21no—
221C2×M16—531no—
222C2×M16—531no—
223C2×M16—1061no—
224C2×M16C4×D5311yescharacteristic
225C22×C8—21no—
226C22×C8—1061no—
227C22×C8C22×D5311yescharacteristic
228#117—531no—
229M32—24no—
230M32—21no—
231M32—21no—
232M32—21no—
233M32—1061no—
234M32—1061no—
235M32C22⋊D5312yes—
236C2×C16—24no—
237C2×C16—21no—
238C2×C16—1061no—
239C2×C16—1061no—
240C2×C16—1061no—
241C2×C16—1061no—
242C2×C16C22⋊D5312yes—
243#108—1062no—
244#104—1062no—
245#104—1062no—
246#97—1061no—
247C4×C8—1061no—
248#41—21no—
249#41—21no—
250#41—1061no—
251#41—1061no—
252#41—1061no—
253#41—1061no—
254#41C8×D5312yes—
255#41#1171992112yes—
256C2×Q8—531no—
257C2×Q8#1171991211yescharacteristic
258C2×D4—21no—
259C2×D4—531no—
260C2×D4—1061no—
261C2×D4#1171991211yescharacteristic
262C22×C4—21no—
263C22×C4—1061no—
264C22×C4C2×C4×D5311yescharacteristic
265Q16—1062no—
266SD16—1062no—
267SD16—1062no—
268D8—1062no—
269M16—21no—
270M16—22no—
271M16—22no—
272M16—532no—
273M16—1061no—
274M16—1061no—
275M16—1061no—
276M16—2121no—
277M16—2121no—
278M16#1171992112yes—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—1061no—
283C2×C8—1061no—
284C2×C8—2121no—
285C2×C8—2121no—
286C2×C8C2×D21211yescharacteristic
287C2×C8#1171994511yescharacteristic
288C2×C8#1171994511yescharacteristic
289C2×C8C2×C22⋊D5311yescharacteristic
290C2×C8C2×C22⋊D5311yescharacteristic
291C2×C8C2×C22⋊D5311yescharacteristic
292C4⋊C4—1061no—
293C22⋊C4—1061no—
294C42—1061no—
295C16—22no—
296C16—22no—
297C16—2121no—
298C16—2121no—
299C23—21no—
300C23—1061no—
301C23#1313322311yescharacteristic
302Q8—21no—
303Q8—1061no—
304Q8—1061no—
305Q8#1313312912yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4—1061no—
310D4—1061no—
311D4—2121no—
312D4—2121no—
313D4#1313312912yes—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—1061no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4C2×#1171992111yescharacteristic
322C2×C4C4×C22⋊D5311yescharacteristic
323C2×C4C4×C22⋊D5311yescharacteristic
324C2×C4C2×C8×D5311yescharacteristic
325C2×C4C2×#1171991211yescharacteristic
326C2×C4#1313322311yescharacteristic
327C8—22no—
328C8—1062no—
329C8—2121no—
330C8#1313318611yescharacteristic
331C8#1313319111yescharacteristic
332C8#1313319111yescharacteristic
333C8#1313320011yescharacteristic
334C22—22no—
335C22—41no—
336C22—81no—
337C22—2122no—
338C22—2121no—
339C22#1681058911yescharacteristic
340C22C8×C22⋊D5311yescharacteristic
341C22#1681069411yescharacteristic
342C4—22no—
343C4—4241no—
344C4—2121no—
345C4#1681050211yescharacteristic
346C4C2×#1313312911yescharacteristic
347C4C8×C22⋊D5311yescharacteristic
348C4#1681069411yescharacteristic
349C2—21no—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—4241no—
354C2#2027500711yescharacteristic
subgroupG/H
centerC8#13133186
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#3407C2
socleC2×C53#1869
prime radical#16811421C4
cosocleC22×C8×C53C23
hypercenter#3407D53
hypocenterC53#12979
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C22×C8 ◁ #3407

Complements

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

NH ≅ G/Ntype
2. #20276344C2semidirect
3. C2×#16810407C2semidirect
4. #20276330C2semidirect
5. #20272805C2semidirect
6. #20272800C2semidirect
7. C53×#3407C2semidirect
172. #3407D53semidirect
204. C53#12979semidirect

Maximal subgroups

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

subgroupindex
#202836162
#202763442
C2×#168104072
#202763302
#202728052
#202728002
C53×#34072
#1297953