Finite Group Databasebeta

← #27279290

Subgroups — #27279290

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#20283656C211yesmaximal, characteristic, complemented
3#20275512C211yesmaximal, characteristic
4#20273141C211yesmaximal, characteristic
5#20272612C211yesmaximal, characteristic, complemented
6#20272559C211yesmaximal, characteristic, complemented
7C53×#1542C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9#16810951C412yes—
10#16810910—21no—
11#16810825—21no—
12#16810773C412yes—
13#16810754—22no—
14#16810754—22no—
15C4×#11719924—22no—
16#16810676C2211yescharacteristic
17#16810650—21no—
18#16810647C2211yescharacteristic
19C2×#13133129—21no—
20C2×#13133129—21no—
21#16810579—21no—
22C2×#13133128C2211yescharacteristic
23#16810352—22no—
24C53×#533C2211yescharacteristic
25C53×#431C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C53×#333—21no—
28C53×#332—21no—
29C22×D212C2×C411yescharacteristic
30C22×C4×D53—21no—
31C2×#11719924—22no—
32C2×#11719917—21no—
33C22×#88322D411yescharacteristic
34C22×#88322D411yescharacteristic
35C4×D212—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719921—21no—
41#13133148—22no—
42C4⋊#88322—24no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D412yes—
46#13133141—22no—
47#13133137—22no—
48C22⋊#88322C2×C412yes—
49#13133129—42no—
50#13133129—42no—
51#13133128—22no—
52#13133128—21no—
53#13133128—21no—
54C4×#88322—22no—
55C4×#88322—21no—
56C22×C8×C53—21no—
57C4⋊Q8×C53D411yescharacteristic
58C4⋊Q8×C53D411yescharacteristic
59#125×C53—21no—
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53C2×C412yes—
62C4×Q8×C53—21no—
63C4×Q8×C53D412yes—
64#117×C53—21no—
65C2×C42×C53C2311yescharacteristic
66C4⋊C8×C53—21no—
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69C22⋊C8×C53—21no—
70C4×C8×C53—21no—
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—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53C22⋊C412yes—
97C22⋊C4×C53—22no—
98C42×C53—22no—
99C42×C53C2×D411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53#4111yescharacteristic
103#11719924—44no—
104#11719921—41no—
105#11719921—41no—
106C4⋊Dic53—21no—
107C4⋊Dic53—21no—
108#11719917—41no—
109#11719917—41no—
110C2×#88322—21no—
111C2×#88322—21no—
112C2×#88322—22no—
113C2×#88322—22no—
114C2×#88322—21no—
115C2×#88322—21no—
116C2×#88322—21no—
117C2×#88322SD1612yescomplemented
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—81no—
154C8×C53—81no—
155#88322—22no—
156#88322—42no—
157#88322—41no—
158#88322—42no—
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#2961—531no—
185#1955—531no—
186#1542D5311yescharacteristic, complemented
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—1062no—
203#446—1062no—
204C4×SD16—1062no—
205#431—531no—
206#431C2×D5311yescharacteristic
207#424—1061no—
208#423—531no—
209C2×C4⋊C8C2×D5311yescharacteristic
210C2×C22⋊C8—1061no—
211C2×C22⋊C8—1061no—
212#405—1061no—
213C2×#97—531no—
214#333—21no—
215#332—21no—
216D4⋊C8—1062no—
217C53#905011yescharacteristic, complemented, Sylow
218C22×D4—531no—
219C23×C4—1061no—
220C2×SD16—1062no—
221C2×M16—1061no—
222C22×C8—21no—
223C22×C8—531no—
224C22×C8—531no—
225C4⋊Q8C22⋊D5311yescharacteristic
226C4⋊Q8C22⋊D5311yescharacteristic
227#125—21no—
228C22⋊Q8—21no—
229C22⋊Q8C4×D5312yes—
230C4×Q8—21no—
231C4×Q8D21212yes—
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—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—532no—
245C4⋊C8—1064no—
246C4⋊C8—1061no—
247C4⋊C8—1061no—
248#103—1062no—
249#102—1062no—
250C22⋊C8—21no—
251C22⋊C8—532no—
252C22⋊C8—2122no—
253C22⋊C8—2122no—
254#97—1062no—
255#97—1061no—
256#97—1061no—
257C4×C8—21no—
258C4×C8—1062no—
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×C4D4×D5311yescharacteristic
275C22×C4#1171994711yescharacteristic
276SD16—2124no—
277M16—2121no—
278M16—2121no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—532no—
284C2×C8—1061no—
285C2×C8—1061no—
286C2×C8—1062no—
287C2×C8—1062no—
288C2×C8—1061no—
289C2×C8—1061no—
290C2×C8—1061no—
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
304C42D4×D5311yescharacteristic
305C42#1171994711yescharacteristic
306C42C2×C22⋊D5311yescharacteristic
307C23—1062no—
308C23—2121no—
309C23—2121no—
310C23—2121no—
311C23—2121no—
312C23—2121no—
313C23—2121no—
314C23#1313317711yescharacteristic
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×C4#1313317711yescharacteristic
334C2×C4#1313318711yescharacteristic
335C2×C4#1313318911yescharacteristic
336C2×C4#1313318911yescharacteristic
337C2×C4#1313320111yescharacteristic
338C2×C4SD16×D5311yescharacteristic
339C2×C4#1313324011yescharacteristic
340C2×C4#1313324411yescharacteristic
341C2×C4#1313324911yescharacteristic
342C8—81no—
343C8—81no—
344C8—1062no—
345C8—2122no—
346C8—2121no—
347C8—2122no—
348C22—21no—
349C22—21no—
350C22—4241no—
351C22—4241no—
352C22—4241no—
353C22—4241no—
354C22—2122no—
355C22—2122no—
356C22—2122no—
357C22#1681050011yescharacteristic
358C22#1681070811yescharacteristic
359C22#1681072611yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—42no—
366C4—4241no—
367C4#1681076311yescharacteristic
368C4#1681077111yescharacteristic
369C4#1681082111yescharacteristic
370C4#1681084711yescharacteristic
371C2—21no—
372C2—21no—
373C2—4242no—
374C2#2027506811yescharacteristic
375C2#2027507511yescharacteristic
376C2#2027513111yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1542C2
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1542D53
hypocenterC53#9050
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
2. #20283656C2semidirect
5. #20272612C2semidirect
6. #20272559C2semidirect
7. C53×#1542C2semidirect
117. C2×#88322SD16semidirect
186. #1542D53semidirect
217. C53#9050semidirect

Maximal subgroups

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

subgroupindex
#202847192
#202836562
#202755122
#202731412
#202726122
#202725592
C53×#15422
#905053