Finite Group Databasebeta

← #27388383

Subgroups — #27388383

Order 13568.

862 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810705C211yesmaximal, characteristic
2#20277354C212yesmaximal, complemented
3#20275591C211yesmaximal, characteristic
4#20275589C211yesmaximal, characteristic
5#20275454C211yesmaximal, characteristic, complemented
6#20275399C212yesmaximal, complemented
7#20275397C212yesmaximal, complemented
8#106×Dic53C211yesmaximal, characteristic, complemented
9#20274539C212yesmaximal, complemented
10#20274403C211yesmaximal, characteristic, complemented
11C53×#1943C211yesmaximal, characteristic, complemented
12C22×D424C2211yescharacteristic
13C2×#13133179C2212yes—
14#16810705C418yescomplemented
15C2×#13133149C2211yescharacteristic
16C2×#13133139C2212yes—
17C2×#13133137C2212yes—
18C2×#13133137C2212yes—
19C2×#13133127C2211yescharacteristic
20C2×#13133127C2211yescharacteristic
21C2×C4×#88322C2211yescharacteristic
22C2×C8×Dic53C2211yescharacteristic
23C2×C8×Dic53C2211yescharacteristic
24#16810525C2212yes—
25#16810524C2212yes—
26#16810522C2212yes—
27#16810515—22no—
28#16810513—22no—
29#16810485—22no—
30#16810480—22no—
31C4⋊C4×Dic53C2212yes—
32#16810469C2211yescharacteristic
33#16810385C2212yes—
34#16810385C2212yes—
35C2×#107×C53C2211yescharacteristic
36C2×#106×C53C2211yescharacteristic
37C2×#97×C53C2211yescharacteristic
38C53×#389C2212yes—
39C53×#340C2212yes—
40C22×D212C2312yes—
41C22×C4×D53—22no—
42C2×D424—22no—
43C2×D424—22no—
44C2×D424C2×C414yes—
45C22×#88322C2311yescharacteristic
46C22×#88322C2311yescharacteristic
47#13133179—24no—
48#13133179C2×C418yes—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53C2312yes—
51C2×#11719921—22no—
52C2×#11719921—22no—
53C2×#11719921—22no—
54C2×#11719921—22no—
55C2×#11719921—22no—
56C2×C4×Dic53—22no—
57C2×C4×Dic53C2311yescharacteristic
58C2×C4×Dic53C2311yescharacteristic
59#13133149—22no—
60#13133139—24no—
61#13133137—24no—
62#13133137C2×C418yes—
63#13133127—22no—
64#13133127—22no—
65C4×#88322—22no—
66C8×Dic53—22no—
67C8×Dic53C2×C414yes—
68#13133124—22no—
69#13133124—22no—
70#13133124D412yes—
71#13133124D412yes—
72C22×C8×C53C2311yescharacteristic
73C22×C8×C53C2311yescharacteristic
74C2×C4⋊C4×C53—22no—
75C2×C4⋊C4×C53C2312yes—
76C2×C4⋊C4×C53C2312yes—
77C2×C42×C53C2311yescharacteristic
78#107×C53C2×C414yes—
79#106×C53—22no—
80#97×C53—22no—
81#95×C53—22no—
82C23×D53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—22no—
85C22×C4×C53C2×D412yes—
86C22×C4×C53C2411yescharacteristic
87C22×Dic53—22no—
88C22×Dic53C2×D412yes—
89C22×Dic53C2×D411yescharacteristic
90C22×Dic53C2×D411yescharacteristic
91C2×D212—22no—
92C2×D212—22no—
93C2×D212—22no—
94C2×D212—22no—
95C2×D212—22no—
96C2×D212C22×C414yes—
97C2×C4×D53—28no—
98C2×C4×D53—82no—
99C2×C4×D53—82no—
100C2×C8×C53—21no—
101C2×C8×C53—21no—
102C2×C8×C53C22×C412yes—
103C2×C8×C53#4112yes—
104C2×C8×C53#4112yes—
105C2×C8×C53#4112yes—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—82no—
110C4⋊C4×C53C22×C414yes—
111C42×C53—21no—
112C42×C53—21no—
113#11719921—28no—
114#11719921—82no—
115#11719921—82no—
116#11719921—82no—
117#11719921—82no—
118C4⋊Dic53—22no—
119C4⋊Dic53—22no—
120C4⋊Dic53—82no—
121C4×Dic53—21no—
122C4×Dic53—21no—
123C4×Dic53—82no—
124C4×Dic53C22×C412yes—
125C4×Dic53#4112yes—
126C2×#88322—21no—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322C22×C412yes—
131C2×#88322#4112yes—
132D424—28no—
133C23×C53C22×D411yescharacteristic
134C22×D53—24no—
135C22×D53—42no—
136C22×D53—42no—
137C22×D53—42no—
138C22×D53—42no—
139C22×D53—42no—
140C22×D53—42no—
141C2×C4×C53—24no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53C23×C411yescharacteristic
151C2×C4×C53C2×#4111yescharacteristic
152C2×C4×C53C2×#4111yescharacteristic
153C2×C4×C53C2×#4111yescharacteristic
154C2×C4×C53C2×#4111yescharacteristic
155C2×C4×C53C2×#4111yescharacteristic
156C2×Dic53—22no—
157C2×Dic53—22no—
158C2×Dic53—22no—
159C2×Dic53—22no—
160C2×Dic53—22no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53C4×D412yes—
165C2×Dic53C4×D412yes—
166D212—24no—
167D212—24no—
168D212—42no—
169D212—42no—
170C4×D53—88no—
171C8×C53—22no—
172C8×C53#11714yes—
173#88322—22no—
174#88322—22no—
175#21284—531nomaximal, Sylow
176C22×C53C2×C4×D411yescharacteristic
177C22×C53#53211yescharacteristic
178C22×C53#53211yescharacteristic
179C22×C53#57612yes—
180C22×C53#57712yes—
181C2×D53—44no—
182C2×D53—44no—
183C2×D53—44no—
184C2×D53—82no—
185C2×D53—82no—
186C2×D53—82no—
187C2×D53—82no—
188C4×C53—42no—
189C4×C53—44no—
190C4×C53—82no—
191C4×C53C2×#11711yescharacteristic
192C4×C53C4×#4111yescharacteristic
193C4×C53#52911yescharacteristic
194C4×C53#52911yescharacteristic
195Dic53—22no—
196Dic53—22no—
197Dic53—42no—
198Dic53—82no—
199C2×C4×D8—531no—
200#2319—532no—
201#1972—531no—
202#1972—531no—
203#1943—531no—
204#1943D5311yescharacteristic, complemented
205#1934—532no—
206#1934—532no—
207C4×#106—531no—
208#1762—532no—
209#1726—531no—
210C2×C53#229311yescharacteristic
211C2×C53#294811yescharacteristic
212C2×C53#295211yescharacteristic
213C2×C53#317912yes—
214C2×C53#317912yes—
215D53—84no—
216C22×D8—531no—
217C2×C4×D4—532no—
218C4×D8—538no—
219C2×#107C2×D5311yescharacteristic
220C2×#106—531no—
221C2×#106C2×D5311yescharacteristic
222C2×#102—532no—
223C2×#102—532no—
224C2×#102—532no—
225C2×#97—531no—
226C2×#97—531no—
227C2×#97C2×D5311yescharacteristic
228C2×C4×C8—531no—
229C2×C4×C8—531no—
230C2×C4×C8—531no—
231#390—532no—
232#390—532no—
233#389—532no—
234#389C2×D5312yes—
235#388—1062no—
236#388—1062no—
237#381—1062no—
238#380—1062no—
239C4×C4⋊C4—532no—
240#376—531no—
241#340—532no—
242#340—532no—
243#340C2×D5312yes—
244C53#2128411yescharacteristic, complemented, Sylow
245C22×D4—532no—
246C23×C4—1062no—
247C2×D8—534no—
248C2×D8—1062no—
249C2×D8—1062no—
250C22×C8—531no—
251C22×C8—531no—
252C22×C8C22×D5311yescharacteristic
253C22×C8C22×D5311yescharacteristic
254C4×D4—538no—
255C4×D4—1064no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—532no—
258C2×C4⋊C4—1062no—
259C2×C4⋊C4C22×D5312yes—
260C2×C4⋊C4C22×D5312yes—
261C2×C22⋊C4—1062no—
262C2×C22⋊C4—1062no—
263C2×C22⋊C4—1062no—
264C2×C22⋊C4—1062no—
265C2×C22⋊C4—1062no—
266C2×C42—531no—
267C2×C42—531no—
268C2×C42—1062no—
269C2×C42C22×D5311yescharacteristic
270#107C4×D5314yes—
271#106—22no—
272#106—1062no—
273#102—538no—
274#102—1064no—
275#102—1064no—
276#97—22no—
277#97—1062no—
278#97—1062no—
279C4×C8—534no—
280C4×C8—1062no—
281C4×C8—1062no—
282#95—22no—
283#95—532no—
284#95—532no—
285#95—1062no—
286#95—1062no—
287C24—1062no—
288C2×D4—534no—
289C2×D4—1062no—
290C2×D4—1062no—
291C2×D4—1062no—
292C2×D4—1062no—
293C2×D4—1062no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—531no—
297C22×C4—531no—
298C22×C4—532no—
299C22×C4—1068no—
300C22×C4—1062no—
301C22×C4—4242no—
302C22×C4—4242no—
303C22×C4D4×D5312yes—
304C22×C4C23×D5311yescharacteristic
305D8—1068no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—532no—
309C2×C8—532no—
310C2×C8—1061no—
311C2×C8—1061no—
312C2×C8—1061no—
313C2×C8—1061no—
314C2×C8C2×C4×D5312yes—
315C2×C8#1171994712yes—
316C2×C8#1171994912yes—
317C2×C8#1171994912yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—82no—
322C4⋊C4—1062no—
323C4⋊C4—1062no—
324C4⋊C4—4242no—
325C4⋊C4C2×C4×D5314yes—
326C22⋊C4—1068no—
327C22⋊C4—4242no—
328C22⋊C4—4242no—
329C22⋊C4—4242no—
330C22⋊C4—4242no—
331C42—21no—
332C42—21no—
333C42—532no—
334C42—532no—
335C42—1061no—
336C42—1061no—
337C42—4242no—
338C23—1064no—
339C23—2122no—
340C23—2122no—
341C23—2122no—
342C23—2122no—
343C23—2122no—
344C23—2122no—
345C23C2×D4×D5311yescharacteristic
346D4—1064no—
347D4—1064no—
348D4—2122no—
349D4—2122no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—532no—
360C2×C4—532no—
361C2×C4—1062no—
362C2×C4—1062no—
363C2×C4—1062no—
364C2×C4—1062no—
365C2×C4—1062no—
366C2×C4—4248no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4C22×C4×D5311yescharacteristic
371C2×C4#41×D5311yescharacteristic
372C2×C4#41×D5311yescharacteristic
373C2×C4C2×#1171994711yescharacteristic
374C2×C4C2×#1171994911yescharacteristic
375C2×C4C2×#1171994911yescharacteristic
376C8—22no—
377C8—1062no—
378C8—1062no—
379C8#1313317514yes—
380C22—4242no—
381C22—4242no—
382C22—4242no—
383C22—4242no—
384C22—2124no—
385C22—2124no—
386C22—2124no—
387C22C4×D4×D5311yescharacteristic
388C22#1681116811yescharacteristic
389C22#1681117411yescharacteristic
390C22#1681142712yes—
391C22#1681143212yes—
392C4—44no—
393C4—42no—
394C4—82no—
395C4—1062no—
396C4—1062no—
397C4—4242no—
398C4—2122no—
399C4C2×#1313317511yescharacteristic
400C4C4×#1171994911yescharacteristic
401C4#1681115811yescharacteristic
402C4#1681115911yescharacteristic
403C2—4244no—
404C2#2027717311yescharacteristic
405C2#2028356411yescharacteristic
406C2#2028359511yescharacteristic
407C2#2028539112yes—
408C2#2028540312yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1943C2
socleC23×C53C22×D4
prime radicalC22×D424C22
cosocleC22×C4×C53C24
hypercenter#1943D53
hypocenterC53#21284
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1943

Complements

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

NH ≅ G/Ntype
2. #20277354C2semidirect
5. #20275454C2semidirect
6. #20275399C2semidirect
7. #20275397C2semidirect
8. #106×Dic53C2semidirect
9. #20274539C2semidirect
10. #20274403C2semidirect
11. C53×#1943C2semidirect
14. #16810705C4semidirect
204. #1943D53semidirect
244. C53#21284semidirect

Maximal subgroups

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

subgroupindex
C2×#168107052
#202773542
#202755912
#202755892
#202754542
#202753992
#202753972
#106×Dic532
#202745392
#202744032
C53×#19432
#2128453