Finite Group Databasebeta

← #27300642

Subgroups — #27300642

Order 13568.

420 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#108×D53C211yesmaximal, characteristic
2C2×#16810445C211yesmaximal, characteristic, complemented
3#20276325C211yesmaximal, characteristic
4#20274737C211yesmaximal, characteristic
5#20272815C211yesmaximal, characteristic, complemented
6#20272814C211yesmaximal, characteristic, complemented
7C2×C53×#365C211yesmaximal, characteristic, complemented
8C2×M16×D53—21no—
9C22×C8×D53C2211yescharacteristic
10C2×#13133157C2211yescharacteristic
11C2×#13133153C2211yescharacteristic
12C2×#13133152—21no—
13#108×D53—22no—
14#108×D53—22no—
15C2×#13133149C2211yescharacteristic
16#16810480—21no—
17#16810445—22no—
18#16810402C412yescomplemented
19#16810402C412yescomplemented
20#16810385—21no—
21C2×M32×C53C2211yescharacteristic
22C2×#108×C53C2211yescharacteristic
23C2×#106×C53C2211yescharacteristic
24C53×#365—22no—
25C22×C4×D53D411yescharacteristic
26C2×#11719917—21no—
27C2×#11719912—41no—
28M16×D53—24no—
29M16×D53—42no—
30C22×#88322D411yescharacteristic
31C2×C8×D53—21no—
32C2×C8×D53—21no—
33C2×C8×D53—21no—
34C2×C8×D53—21no—
35C2×C8×D53—41no—
36C2×C8×D53C2×C411yescharacteristic
37C2×C8×D53C2×C411yescharacteristic
38C2×C8×D53Q811yescharacteristic
39C2×C8×D53Q811yescharacteristic
40C2×C4⋊Dic53—21no—
41C2×#11719921—41no—
42#13133157—21no—
43#13133157—21no—
44C2×#11719908C2×C412yes—
45#13133153—21no—
46#13133153D412yes—
47#13133152—24no—
48#13133149—21no—
49#13133149—21no—
50#13133124—41no—
51C2×M16×C53—21no—
52C22×C8×C53C2311yescharacteristic
53C2×C4⋊C4×C53—21no—
54M32×C53—21no—
55M32×C53—21no—
56C2×C16×C53C2×C412yes—
57#108×C53—21no—
58#108×C53D412yes—
59#106×C53—21no—
60#106×C53—21no—
61C23×D53SD1611yescharacteristic
62C22×C4×C53—41no—
63C22×C4×C53C2×D411yescharacteristic
64C22×Dic53—41no—
65C22×Dic53SD1611yescharacteristic
66C2×C4×D53—21no—
67C2×C4×D53—21no—
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×C4×D53C4⋊C411yescharacteristic
71C2×C4×D53C4⋊C411yescharacteristic
72C2×C4×D53SD1611yescharacteristic
73C2×C4×D53SD1611yescharacteristic
74M16×C53—22no—
75M16×C53—41no—
76C2×C8×C53—41no—
77C2×C8×C53C22×C411yescharacteristic
78C2×C8×C53C2×D411yescharacteristic
79C2×C8×C53C2×D411yescharacteristic
80C2×C8×C53C2×Q811yescharacteristic
81C2×C8×C53#4111yescharacteristic
82C2×C8×C53#4111yescharacteristic
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85#11719921—82no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88#11719917—22no—
89#11719917—41no—
90C2×#88322—21no—
91C2×#88322—21no—
92C2×#88322—41no—
93C2×#88322C4⋊C411yescharacteristic
94C2×#88322C4⋊C411yescharacteristic
95#11719912—44no—
96C8×D53—21no—
97C8×D53—21no—
98C8×D53—21no—
99C8×D53—21no—
100C8×D53—21no—
101C8×D53—21no—
102C8×D53—21no—
103C8×D53—21no—
104C8×D53—44no—
105C16×C53—22no—
106#11719908—22no—
107C23×C53C2×SD1611yescharacteristic
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—22no—
111C22×D53—41no—
112C22×D53#10611yescharacteristic
113C22×D53#10611yescharacteristic
114C2×C4×C53—81no—
115C2×C4×C53—81no—
116C2×C4×C53—81no—
117C2×C4×C53C2×C4⋊C411yescharacteristic
118C2×C4×C53#12011yescharacteristic
119C2×C4×C53#12311yescharacteristic
120C2×C4×C53C2×SD1611yescharacteristic
121C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C53#13711yescharacteristic
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—81no—
126C2×Dic53—81no—
127C2×Dic53—81no—
128C2×Dic53#10611yescharacteristic
129C2×Dic53#10611yescharacteristic
130C4×D53—21no—
131C4×D53—21no—
132C4×D53—21no—
133C4×D53—21no—
134C4×D53—21no—
135C4×D53—21no—
136C4×D53—21no—
137C4×D53—21no—
138C8×C53—42no—
139C8×C53C4×D411yescharacteristic
140C8×C53C4×D411yescharacteristic
141C8×C53C22⋊Q811yescharacteristic
142C8×C53C22⋊Q811yescharacteristic
143#88322—21no—
144#88322—21no—
145#88322—42no—
146#12928—531nomaximal, Sylow
147C22×C53—21no—
148C22×C53—21no—
149C22×C53C2×#10611yescharacteristic
150C22×C53#44811yescharacteristic
151C22×C53#48111yescharacteristic
152C2×D53—21no—
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—42no—
158C2×D53—42no—
159C2×D53#36512yescomplemented
160C4×C53—161no—
161C4×C53#38511yescharacteristic
162C4×C53#42811yescharacteristic
163C4×C53#45011yescharacteristic
164C4×C53#48311yescharacteristic
165Dic53—21no—
166Dic53—21no—
167Dic53—161no—
168C22×#108—531no—
169C2×#365—531no—
170C2×#365D5311yescharacteristic, complemented
171#2112—531no—
172#1807—531no—
173#1383—531no—
174#1383—531no—
175C2×C53—21no—
176C2×C53—21no—
177C2×C53#185311yescharacteristic
178C2×C53C2×#36511yescharacteristic
179C2×C53#215511yescharacteristic
180D53—22no—
181D53—41no—
182C22×M16—1061no—
183C23×C8—531no—
184C2×M32—531no—
185C2×M32C2×D5311yescharacteristic
186C2×#108—531no—
187C2×#108—1062no—
188C2×#108—1061no—
189C2×#108—1062no—
190C2×#108C2×D5311yescharacteristic
191C2×#106—531no—
192C2×#106C2×D5311yescharacteristic
193#380—1061no—
194#365—22no—
195#365—1062no—
196C22⋊C16—532no—
197C22⋊C16—532no—
198#340—1061no—
199C53#1292811yescharacteristic, complemented, Sylow
200C23×C4—531no—
201C2×M16—21no—
202C2×M16—1064no—
203C2×M16—1061no—
204C2×M16—2121no—
205C2×M16—2122no—
206C22×C8—531no—
207C22×C8—531no—
208C22×C8—531no—
209C22×C8—531no—
210C22×C8—531no—
211C22×C8—1061no—
212C22×C8—1061no—
213C22×C8—1061no—
214C22×C8—1061no—
215C22×C8—2121no—
216C22×C8C22×D5311yescharacteristic
217C2×C4⋊C4—21no—
218C2×C4⋊C4—1061no—
219C2×C22⋊C4—2121no—
220M32—21no—
221M32—21no—
222M32—1061no—
223M32—1061no—
224C2×C16—532no—
225C2×C16C4×D5312yes—
226#108—21no—
227#108—532no—
228#108—1064no—
229#108—1061no—
230#108C22⋊D5312yes—
231#106—21no—
232#106—21no—
233#106—1061no—
234#106—1061no—
235#95—2121no—
236C24—531no—
237C22×C4—41no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4—2121no—
248C22×C4D4×D5311yescharacteristic
249M16—22no—
250M16—41no—
251M16—1062no—
252M16—2124no—
253M16—2121no—
254C2×C8—41no—
255C2×C8—531no—
256C2×C8—531no—
257C2×C8—1061no—
258C2×C8—1061no—
259C2×C8—1061no—
260C2×C8—1061no—
261C2×C8—1061no—
262C2×C8—1061no—
263C2×C8—1061no—
264C2×C8—1061no—
265C2×C8—1061no—
266C2×C8—1061no—
267C2×C8—2124no—
268C2×C8—2121no—
269C2×C8C2×C4×D5311yescharacteristic
270C2×C8#1171994511yescharacteristic
271C2×C8D4×D5311yescharacteristic
272C2×C8Q8×D5311yescharacteristic
273C2×C8#1171994911yescharacteristic
274C2×C8C2×C22⋊D5311yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—2121no—
278C4⋊C4—2121no—
279C22⋊C4—4242no—
280C16—22no—
281C16—1062no—
282C23—531no—
283C23—531no—
284C23—1062no—
285C23—1061no—
286C23—1061no—
287C23—2121no—
288C23SD16×D5311yescharacteristic
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—531no—
293C2×C4—531no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—4241no—
305C2×C4—4241no—
306C2×C4—4241no—
307C2×C4C4⋊C4×D5311yescharacteristic
308C2×C4#1313318511yescharacteristic
309C2×C4#1313320211yescharacteristic
310C2×C4SD16×D5311yescharacteristic
311C2×C4#1313324411yescharacteristic
312C2×C4#1313324611yescharacteristic
313C8—42no—
314C8—1061no—
315C8—1061no—
316C8—2122no—
317C8C4×C22⋊D5311yescharacteristic
318C8#1313317911yescharacteristic
319C8#1313319311yescharacteristic
320C8#1313319511yescharacteristic
321C22—21no—
322C22—21no—
323C22—532no—
324C22—1061no—
325C22—1061no—
326C22—1061no—
327C22—1061no—
328C22—1061no—
329C22—2122no—
330C22—2122no—
331C22#106×D5311yescharacteristic
332C22#1681076011yescharacteristic
333C22#1681094311yescharacteristic
334C4—161no—
335C4—8481no—
336C4—1061no—
337C4—1061no—
338C4#1681049411yescharacteristic
339C4#1681066211yescharacteristic
340C4#1681077211yescharacteristic
341C4#1681095511yescharacteristic
342C2—21no—
343C2—21no—
344C2—1062no—
345C2—2121no—
346C2#2027490611yescharacteristic
347C2#365×D5311yescharacteristic
348C2#2027648611yescharacteristic
subgroupG/H
centerC22#106×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#365C2
socleC22×C53C2×#106
prime radicalC23×D53SD16
cosocleC22×C8×C53C23
hypercenterC2×#365D53
hypocenterC53#12928
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#365

Complements

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

NH ≅ G/Ntype
2. C2×#16810445C2semidirect
5. #20272815C2semidirect
6. #20272814C2semidirect
7. C2×C53×#365C2semidirect
18. #16810402C4semidirect
19. #16810402C4semidirect
159. C2×D53#365semidirect
170. C2×#365D53semidirect
199. C53#12928semidirect

Maximal subgroups

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

subgroupindex
C2×#108×D532
C2×#168104452
#202763252
#202747372
#202728152
#202728142
C2×C53×#3652
#1292853