Finite Group Databasebeta

← #27279292

Subgroups — #27279292

Order 13568.

476 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284719C211yesmaximal, characteristic
2#20275513C211yesmaximal, characteristic
3#20273372C211yesmaximal, characteristic, complemented
4#20273139C211yesmaximal, characteristic
5#20272610C211yesmaximal, characteristic, complemented
6#20272560C211yesmaximal, characteristic, complemented
7C53×#2961C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9#16810951C412yes—
10#16810910—21no—
11#16810825—21no—
12#16810773C412yes—
13C4×#11719924—22no—
14#16810676C2211yescharacteristic
15#16810676C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133129—21no—
18C2×#13133129—21no—
19#16810372—21no—
20#16810369—21no—
21#16810350—22no—
22C53×#533C2211yescharacteristic
23C53×#446—22no—
24C53×#446—22no—
25C53×#424—21no—
26C53×#423C2211yescharacteristic
27C53×#405—21no—
28C2×#97×C53C2211yescharacteristic
29C22×D212C2×C411yescharacteristic
30C22×C4×D53—21no—
31C2×#11719924—22no—
32C22×#88322—21no—
33C4×D212—21no—
34C4×D212—21no—
35C4×D212—21no—
36C4×D212—21no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719921—21no—
39#13133148—22no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44#13133141—22no—
45#13133137—22no—
46C22⋊#88322—21no—
47C22⋊#88322C2×C412yes—
48#13133129—42no—
49#13133129—42no—
50C4×#88322—21no—
51C4×#88322—21no—
52C2×M16×C53—21no—
53C22×C8×C53D411yescharacteristic
54C22×C8×C53D411yescharacteristic
55C4⋊Q8×C53D411yescharacteristic
56C4⋊Q8×C53D411yescharacteristic
57#125×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53C2×C412yes—
60C4×Q8×C53—21no—
61C4×Q8×C53D412yes—
62#117×C53—21no—
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53—24no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53D412yes—
67#97×C53—22no—
68#97×C53—21no—
69#97×C53—21no—
70C4×C8×C53—22no—
71C23×D53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53C22⋊C412yes—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53#4111yescharacteristic
77C22×Dic53—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—42no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87M16×C53—41no—
88M16×C53—41no—
89C2×C8×C53—21no—
90C2×C8×C53—21no—
91C2×C8×C53—22no—
92C2×C8×C53—21no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53SD1612yescomplemented
96C4⋊C4×C53—22no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53C22⋊C412yes—
102C22⋊C4×C53—22no—
103C42×C53—22no—
104C42×C53C2×D411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53#4111yescharacteristic
108#11719924—44no—
109#11719921—41no—
110#11719921—41no—
111C4⋊Dic53—21no—
112C4⋊Dic53—21no—
113C2×#88322—22no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C23×C53C4×D411yescharacteristic
119C22×D53—22no—
120C22×D53—41no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—41no—
126Q8×C53—42no—
127Q8×C53—81no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53C2×C22⋊C411yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×C4×C53#12011yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12311yescharacteristic
138C2×C4×C53C2×SD1611yescharacteristic
139C2×C4×C53#13511yescharacteristic
140C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C53#13711yescharacteristic
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145D212—22no—
146D212—22no—
147D212—41no—
148D212—41no—
149C4×D53—81no—
150C4×D53—81no—
151C4×D53—81no—
152C4×D53—81no—
153C8×C53—22no—
154C8×C53—41no—
155C8×C53—42no—
156#88322—42no—
157#88322—81no—
158#88322—81no—
159#9050—531nomaximal, Sylow
160C22×C53—21no—
161C22×C53—21no—
162C22×C53#38611yescharacteristic
163C22×C53C4×SD1611yescharacteristic
164C22×C53#44011yescharacteristic
165C2×D53—42no—
166C2×D53—42no—
167C2×D53—42no—
168C2×D53—81no—
169C2×D53—81no—
170C2×D53—81no—
171C2×D53—81no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—42no—
177C4×C53—42no—
178C4×C53#44811yescharacteristic
179C4×C53#44911yescharacteristic
180C4×C53#46011yescharacteristic
181C4×C53#46411yescharacteristic
182Dic53—81no—
183#3101—531no—
184#2961D5311yescharacteristic, complemented
185#1955—531no—
186#1542—531no—
187#1486—531no—
188#1301—531no—
189#1281—531no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53#188011yescharacteristic
193C2×C53#188111yescharacteristic
194C2×C53#189111yescharacteristic
195D53—82no—
196#533C2×D5311yescharacteristic
197C2×C4×D4—531no—
198#483—532no—
199#476—1061no—
200#461—1061no—
201#450—532no—
202#446—22no—
203#446—22no—
204C4×SD16—1062no—
205#431—531no—
206#431—531no—
207#424—21no—
208#423C2×D5311yescharacteristic
209C2×C4⋊C8—531no—
210C2×C22⋊C8—1061no—
211C2×C22⋊C8—1061no—
212#405—21no—
213C2×#97C2×D5311yescharacteristic
214#333—1061no—
215#332—1061no—
216D4⋊C8—1062no—
217C53#905011yescharacteristic, complemented, Sylow
218C22×D4—531no—
219C23×C4—1061no—
220C2×SD16—1062no—
221C2×M16—21no—
222C22×C8—1061no—
223C22×C8C22⋊D5311yescharacteristic
224C22×C8C22⋊D5311yescharacteristic
225C4⋊Q8D21211yescharacteristic
226C4⋊Q8D21211yescharacteristic
227#125—21no—
228C22⋊Q8—21no—
229C22⋊Q8C4×D5312yes—
230C4×Q8—21no—
231C4×Q8C22⋊D5312yes—
232C4×D4—1061no—
233C4×D4—1061no—
234C4×D4—1061no—
235C4×D4—1061no—
236#117—21no—
237C2×C4⋊C4—531no—
238C2×C22⋊C4—1061no—
239C2×C42C22×D5311yescharacteristic
240#106—1062no—
241C4⋊C8—24no—
242C4⋊C8—21no—
243C4⋊C8—1061no—
244C4⋊C8—1061no—
245C4⋊C8—1061no—
246C4⋊C8—1061no—
247C4⋊C8C22⋊D5312yes—
248#103—1062no—
249#102—1062no—
250C22⋊C8—532no—
251C22⋊C8—1061no—
252C22⋊C8—2122no—
253C22⋊C8—2122no—
254#97—22no—
255#97—21no—
256#97—21no—
257C4×C8—22no—
258C4×C8—1061no—
259C4×C8—1061no—
260C24—1061no—
261C2×Q8—21no—
262C2×Q8#1171992112yes—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4—2122no—
266C22×C4—1061no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4—2121no—
270C22×C4—2121no—
271C22×C4—2121no—
272C22×C4—2121no—
273C22×C4C2×C4×D5311yescharacteristic
274C22×C4#1171994511yescharacteristic
275C22×C4C2×C22⋊D5311yescharacteristic
276SD16—2124no—
277M16—41no—
278M16—41no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—1062no—
286C2×C8—2121no—
287C2×C8—2121no—
288C2×C8—2121no—
289C2×C8—2121no—
290C2×C8#1171992412yescomplemented
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—1061no—
297C4⋊C4—1061no—
298C4⋊C4#1171992112yes—
299C22⋊C4—22no—
300C22⋊C4—2121no—
301C22⋊C4—2121no—
302C42—22no—
303C42C2×D21211yescharacteristic
304C42#1171994511yescharacteristic
305C42C2×C22⋊D5311yescharacteristic
306C42C2×C22⋊D5311yescharacteristic
307C23—1062no—
308C23—2121no—
309C23—2121no—
310C23—2121no—
311C23—2121no—
312C23—2121no—
313C23—2121no—
314C23C4×C22⋊D5311yescharacteristic
315Q8—42no—
316Q8—81no—
317D4—1062no—
318D4—1062no—
319D4—2121no—
320D4—2121no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—4241no—
326C2×C4—4241no—
327C2×C4—4241no—
328C2×C4—4241no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4C2×#1171992111yescharacteristic
333C2×C4C4×C22⋊D5311yescharacteristic
334C2×C4#1313318611yescharacteristic
335C2×C4#1313319111yescharacteristic
336C2×C4#1313319111yescharacteristic
337C2×C4#1313320011yescharacteristic
338C2×C4C2×#1171992411yescharacteristic
339C2×C4#1313323711yescharacteristic
340C2×C4#1313324211yescharacteristic
341C2×C4#1313324811yescharacteristic
342C8—22no—
343C8—41no—
344C8—42no—
345C8—4241no—
346C8—4241no—
347C8—2122no—
348C22—21no—
349C22—21no—
350C22—4241no—
351C22—4241no—
352C22—4241no—
353C22—4241no—
354C22—2122no—
355C22—2122no—
356C22—2122no—
357C22#1681050211yescharacteristic
358C22C4×#1171992411yescharacteristic
359C22#1681072111yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—42no—
366C4—4241no—
367C4#1681076211yescharacteristic
368C4#1681076811yescharacteristic
369C4#1681082311yescharacteristic
370C4#1681084511yescharacteristic
371C2—21no—
372C2—21no—
373C2—4242no—
374C2#2027507011yescharacteristic
375C2#2027507711yescharacteristic
376C2#2027513311yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2961C2
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#2961D53
hypocenterC53#9050
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2961

Complements

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

NH ≅ G/Ntype
3. #20273372C2semidirect
5. #20272610C2semidirect
6. #20272560C2semidirect
7. C53×#2961C2semidirect
95. C2×C8×C53SD16semidirect
184. #2961D53semidirect
217. C53#9050semidirect
290. C2×C8#11719924semidirect

Maximal subgroups

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

subgroupindex
#202847192
#202755132
#202733722
#202731392
#202726102
#202725602
C53×#29612
#905053