Finite Group Databasebeta

← #27537557

Subgroups — #27537557

Order 13568.

632 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811064C211yesmaximal, characteristic
2#20283633C211yesmaximal, characteristic
3#20276604C218yesmaximal, complemented
4#20276487C211yesmaximal, characteristic, complemented
5C2×#16810439C211yesmaximal, characteristic, complemented
6#20276450C211yesmaximal, characteristic
7#20276320C211yesmaximal, characteristic, complemented
8C53×#2950C211yesmaximal, characteristic, complemented
9C2×#13133239C2211yescharacteristic
10#16811389—21no—
11#16811064C414yescomplemented
12#16811064C2214yes—
13C2×#13133163C412yescomplemented
14C2×#13133160C412yescomplemented
15C2×#13133157C2211yescharacteristic
16C2×#13133157C2211yescharacteristic
17#16810746C2214yes—
18#16810741—22no—
19C2×#13133153C2211yescharacteristic
20#16810641—21no—
21#16810445C2214yescomplemented
22#16810439C2214yescomplemented
23#16810432C2214yes—
24#16810401C2214yescomplemented
25C22×Q16×C53C2211yescharacteristic
26C53×#519—21no—
27C53×#441C2214yescomplemented
28C4×Q16×C53—22no—
29C53×#427C2211yescharacteristic
30C53×#419—21no—
31C53×#405C2211yescharacteristic
32C2×#11719945—21no—
33#13133239—21no—
34#13133239—21no—
35#13133239C2×C412yes—
36#13133239C2312yes—
37C2×Dic212C2×C411yescharacteristic
38C2×#11719913—21no—
39C2×D424C2×C411yescharacteristic
40#13133225—42no—
41#13133223—24no—
42C2×#11719917—21no—
43M16×D53—42no—
44#13133163C2×C418yes—
45#13133160C2×C418yes—
46#13133157—21no—
47#13133157C2×C412yes—
48#13133157C2312yes—
49#13133157C2312yes—
50C2×#11719908—21no—
51C2×#11719908C2×C412yes—
52#13133153—21no—
53#13133153C2312yes—
54#13133145—24no—
55C22×Q8×C53—21no—
56C2×Q16×C53—21no—
57C2×Q16×C53—21no—
58C2×Q16×C53—21no—
59C2×Q16×C53C2×C412yes—
60C2×Q16×C53C2×C412yes—
61C2×Q16×C53C2312yes—
62C2×M16×C53D411yescharacteristic
63C22×C8×C53C2311yescharacteristic
64C4×Q8×C53—24no—
65C4×Q8×C53—42no—
66#117×C53—21no—
67#117×C53—42no—
68#117×C53D411yescharacteristic
69#107×C53—21no—
70#106×C53C2312yes—
71#103×C53—24no—
72#97×C53C2312yes—
73C4×C8×C53D412yes—
74C2×Q8×C53—22no—
75C2×Q8×C53—22no—
76C2×Q8×C53—22no—
77C2×Q8×C53—41no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C22×C4×C53—41no—
81C22×C4×C53C2×D411yescharacteristic
82C2×C22⋊D53—41no—
83#11719945—22no—
84#11719945—22no—
85#11719945—41no—
86#11719945—41no—
87C2×D212—21no—
88C2×C4×D53—41no—
89C2×Dic106—21no—
90Q16×C53—22no—
91Q16×C53—22no—
92Q16×C53—22no—
93Q16×C53C22×C414yes—
94M16×C53—21no—
95M16×C53—21no—
96C2×C8×C53C22×C411yescharacteristic
97C2×C8×C53C22×C411yescharacteristic
98C2×C8×C53C2×D412yes—
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53#4111yescharacteristic
101C2×C8×C53#4111yescharacteristic
102C2×C8×C53C2411yescharacteristic
103C4⋊C4×C53—21no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—44no—
106C22⋊C4×C53—21no—
107C22⋊C4×C53—41no—
108C42×C53—44no—
109C42×C53—41no—
110C42×C53C2×D412yes—
111#11719917—22no—
112#11719917—41no—
113C2×#88322—41no—
114Dic212—21no—
115Dic212C22×C412yes—
116D424—21no—
117D424C22×C412yes—
118#11719913—22no—
119#11719913—22no—
120#11719912—44no—
121C8×D53—44no—
122#11719908—22no—
123#11719908C22×C414yes—
124C23×C53AGL(1,ℤ/8)11yescharacteristic
125C22×D53—41no—
126Q8×C53—24no—
127Q8×C53—42no—
128Q8×C53—42no—
129Q8×C53—42no—
130C2×C4×C53—22no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C53C22×D411yescharacteristic
141C22⋊D53—42no—
142C22⋊D53—42no—
143C2×Dic53—41no—
144D212—22no—
145D212—41no—
146C4×D53—42no—
147C4×D53—42no—
148Dic106—22no—
149Dic106—41no—
150C8×C53—22no—
151C8×C53C4×D411yescharacteristic
152C8×C53C4×D411yescharacteristic
153C8×C53C23×C411yescharacteristic
154C8×C53C2×#4111yescharacteristic
155#88322—42no—
156#33470—531nomaximal, Sylow
157C22×C53—41no—
158C22×C53#44211yescharacteristic
159C22×C53#44211yescharacteristic
160C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
161C2×D53—42no—
162C2×D53—161no—
163C4×C53—42no—
164C4×C53—44no—
165C4×C53—82no—
166C4×C53#44211yescharacteristic
167C4×C53#44211yescharacteristic
168C4×C53C2×C4×D411yescharacteristic
169C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
170Dic53—42no—
171C2×#510—531no—
172#2957—531no—
173#2950D5311yescharacteristic, complemented
174#2180—538no—
175#2155—531no—
176C2×#362—531no—
177#2145—531no—
178#2110—531no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53C2×#44211yescharacteristic
183D53—82no—
184C2×#135—531no—
185C22×Q16C2×D5311yescharacteristic
186#568—1061no—
187#519—21no—
188#510—534no—
189#510—534no—
190C2×Q32—532no—
191C2×SD32—532no—
192C2×M32—531no—
193C2×M32—531no—
194#444—534no—
195#443—1062no—
196#441C2×D5314yescomplemented
197C4×Q16—22no—
198C2×#108—531no—
199#427C2×D5311yescharacteristic
200#421—1061no—
201#419—21no—
202#405C2×D5311yescharacteristic
203#365—534no—
204#362—534no—
205#360—534no—
206#347—534no—
207C53#3347011yescharacteristic, complemented, Sylow
208C2×#41—1061no—
209C22×Q8—21no—
210#135—532no—
211#135—532no—
212#135—1061no—
213#135—1061no—
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16—21no—
217C2×Q16—531no—
218C2×Q16C4×D5312yes—
219C2×Q16C4×D5312yes—
220C2×Q16C22×D5312yes—
221C2×SD16—1061no—
222C2×D8—531no—
223#131—1064no—
224#131—2122no—
225C2×M16—1061no—
226C2×M16—2122no—
227C2×M16C22⋊D5311yescharacteristic
228C22×C8C22×D5311yescharacteristic
229C4×Q8—24no—
230C4×Q8—42no—
231#117—21no—
232#117—42no—
233#117C22⋊D5311yescharacteristic
234Q32—538no—
235SD32—538no—
236M32—532no—
237M32—532no—
238M32—532no—
239M32—1061no—
240C2×C16—532no—
241C2×C16—1061no—
242#108—532no—
243#108—1061no—
244#107—21no—
245#106C22×D5312yes—
246#104—1064no—
247#103—24no—
248#97C22×D5312yes—
249C4×C8C22⋊D5312yes—
250#41—1062no—
251#41—1062no—
252#41—2121no—
253#41—2121no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—1061no—
261C2×D4—1061no—
262C2×D4—2121no—
263C22×C4—41no—
264C22×C4—2121no—
265C22×C4C2×C22⋊D5311yescharacteristic
266Q16—22no—
267Q16—22no—
268Q16—22no—
269Q16—532no—
270Q16—1061no—
271Q16C2×C4×D5314yes—
272SD16—1062no—
273SD16—1062no—
274D8—532no—
275D8—1061no—
276M16—21no—
277M16—21no—
278M16—1062no—
279M16—2124no—
280M16—2121no—
281C2×C8—2124no—
282C2×C8—2121no—
283C2×C8C2×C4×D5311yescharacteristic
284C2×C8C2×C4×D5311yescharacteristic
285C2×C8#1171994511yescharacteristic
286C2×C8#1171994511yescharacteristic
287C2×C8C2×C22⋊D5312yes—
288C2×C8C2×C22⋊D5311yescharacteristic
289C2×C8C23×D5311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—44no—
293C22⋊C4—21no—
294C22⋊C4—41no—
295C42—44no—
296C42—41no—
297C42C2×C22⋊D5312yes—
298C16—534no—
299C16—1062no—
300C23—2121no—
301C23#1313324311yescharacteristic
302Q8—24no—
303Q8—42no—
304Q8—42no—
305Q8—42no—
306Q8—1062no—
307Q8—2121no—
308D4—1062no—
309D4—2122no—
310D4—2122no—
311D4—2121no—
312C2×C4—22no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—2121no—
318C2×C4—2122no—
319C2×C4—2122no—
320C2×C4C4×C22⋊D5311yescharacteristic
321C2×C4C4×C22⋊D5311yescharacteristic
322C2×C4#1313324211yescharacteristic
323C2×C4#1313324211yescharacteristic
324C2×C4#1313324311yescharacteristic
325C2×C4C22×C22⋊D5311yescharacteristic
326C8—22no—
327C8—2122no—
328C8C4×C22⋊D5311yescharacteristic
329C8C4×C22⋊D5311yescharacteristic
330C8C22×C4×D5311yescharacteristic
331C8C2×#1171994511yescharacteristic
332C22—41no—
333C22—8481no—
334C22—2122no—
335C22#1681073311yescharacteristic
336C22#1681073311yescharacteristic
337C22C2×#1313324311yescharacteristic
338C4—42no—
339C4—44no—
340C4—82no—
341C4—2122no—
342C4#1681073311yescharacteristic
343C4#1681073311yescharacteristic
344C4C2×C4×C22⋊D5311yescharacteristic
345C4C2×#1313324211yescharacteristic
346C2—21no—
347C2—21no—
348C2—21no—
349C2—4242no—
350C2C2×#1681073311yescharacteristic
subgroupG/H
centerC2C2×#16810733
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2950C2
socleC2×C53C2×#442
prime radicalC2×#13133239C22
cosocleC2×C8×C53C24
hypercenter#2950D53
hypocenterC53#33470
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2950

Complements

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

NH ≅ G/Ntype
3. #20276604C2semidirect
4. #20276487C2semidirect
5. C2×#16810439C2semidirect
7. #20276320C2semidirect
8. C53×#2950C2semidirect
11. #16811064C4semidirect
13. C2×#13133163C4semidirect
14. C2×#13133160C4semidirect
21. #16810445C22semidirect
22. #16810439C22semidirect
24. #16810401C22semidirect
27. C53×#441C22semidirect
173. #2950D53semidirect
196. #441C2×D53semidirect
207. C53#33470semidirect

Maximal subgroups

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

subgroupindex
C2×#168110642
#202836332
#202766042
#202764872
C2×#168104392
#202764502
#202763202
C53×#29502
#3347053