Finite Group Databasebeta

← #27280643

Subgroups — #27280643

Order 13568.

492 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285658C211yesmaximal, characteristic
2#20283662C211yesmaximal, characteristic, complemented
3#20275555C211yesmaximal, characteristic
4#20273403C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic
6#20272611C211yesmaximal, characteristic, complemented
7C53×#1283C211yesmaximal, characteristic, complemented
8#16811180C2211yescharacteristic
9C2×#13133214C2211yescharacteristic
10#16810996—21no—
11#16810970—21no—
12#16810824—22no—
13#16810776C414yes—
14#16810753—21no—
15#16810751—22no—
16C8×Dic106—21no—
17#16810651—21no—
18#16810647C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133133—21no—
21C2×#13133133—21no—
22#16810581—21no—
23#16810371—21no—
24#16810370—21no—
25#16810366—21no—
26#16810366—21no—
27C53×#431C2211yescharacteristic
28C2×C4⋊C8×C53C2211yescharacteristic
29C4×M16×C53C2211yescharacteristic
30C22×D212—21no—
31C22×D212—21no—
32C22×D212C2×C412yes—
33C2×#11719913—24no—
34C2×#11719917—21no—
35C22×#88322—21no—
36#13133217D411yescharacteristic
37#13133217D411yescharacteristic
38#13133214—21no—
39#13133214—21no—
40#13133214—21no—
41#13133214—21no—
42#13133209—21no—
43#13133196—21no—
44#13133196C2×C412yes—
45C4×Dic106—21no—
46C4×Dic106D412yes—
47#13133176—21no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322D411yescharacteristic
52C4⋊#88322D411yescharacteristic
53#13133146—22no—
54#13133146—22no—
55#13133139—24no—
56#13133133—42no—
57#13133133—42no—
58#13133127—22no—
59C8×Dic53—22no—
60C2×M16×C53D411yescharacteristic
61C2×M16×C53D411yescharacteristic
62C22×C8×C53—21no—
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C22⋊C8×C53C2×C412yes—
68#97×C53—21no—
69C4×C8×C53—21no—
70C4×C8×C53—21no—
71C23×D53—22no—
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53#4111yescharacteristic
75C2×D212—22no—
76C2×D212—22no—
77C2×D212—41no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—82no—
86C2×D212—81no—
87C2×D212—81no—
88C2×Dic106—21no—
89C2×Dic106C22⋊C412yes—
90M16×C53—21no—
91M16×C53—21no—
92M16×C53—21no—
93M16×C53SD1612yescomplemented
94C2×C8×C53—21no—
95C2×C8×C53—22no—
96C2×C8×C53—21no—
97C2×C8×C53—41no—
98C2×C8×C53—81no—
99C42×C53C2×D411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53#4111yescharacteristic
103C22⋊Dic53—22no—
104C4⋊Dic53—21no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53C22⋊C412yes—
108#11719919—22no—
109#11719919—22no—
110C4×Dic53—22no—
111#11719917—41no—
112#11719917—41no—
113C2×#88322—21no—
114C2×#88322—21no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118#11719913—84no—
119C23×C53C4×D411yescharacteristic
120C22×D53—42no—
121C22×D53—42no—
122C22×D53—42no—
123C22×D53—82no—
124C22×D53—82no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—41no—
128C2×C4×C53C2×C22⋊C411yescharacteristic
129C2×C4×C53C4×D411yescharacteristic
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53#12111yescharacteristic
132C2×C4×C53#12111yescharacteristic
133C2×C4×C53#12311yescharacteristic
134C2×C4×C53C2×SD1611yescharacteristic
135C2×C4×C53#13511yescharacteristic
136C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C53#13711yescharacteristic
138C2×Dic53—22no—
139C2×Dic53—22no—
140D212—44no—
141D212—82no—
142D212—82no—
143D212—81no—
144D212—81no—
145D212—81no—
146D212—81no—
147D212—81no—
148D212—81no—
149Dic106—41no—
150Dic106—41no—
151Dic106—42no—
152C8×C53—22no—
153C8×C53—42no—
154C8×C53—41no—
155C8×C53—81no—
156#88322—41no—
157#88322—41no—
158#88322—81no—
159#9274—531nomaximal, Sylow
160C22×C53—21no—
161C22×C53—21no—
162C22×C53#38611yescharacteristic
163C22×C53C4×SD1611yescharacteristic
164C22×C53#44011yescharacteristic
165C2×D53—82no—
166C2×D53—82no—
167C2×D53—82no—
168C2×D53—162no—
169C2×D53—162no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53#44811yescharacteristic
174C4×C53#44911yescharacteristic
175C4×C53#46011yescharacteristic
176C4×C53#46411yescharacteristic
177C4×C53#47111yescharacteristic
178C4×C53#47211yescharacteristic
179Dic53—42no—
180Dic53—42no—
181#3208—531no—
182#2962—531no—
183#1965—531no—
184#1550—531no—
185C2×#331—531no—
186#1301—531no—
187#1283D5311yescharacteristic, complemented
188C2×C53—21no—
189C2×C53—21no—
190C2×C53#188011yescharacteristic
191C2×C53#190711yescharacteristic
192C2×C53#191411yescharacteristic
193D53—162no—
194#533—531no—
195C2×#127—531no—
196#492—1061no—
197#486—1061no—
198#460—1062no—
199#450—534no—
200#446—1061no—
201#446—1062no—
202C8×Q8—1061no—
203#431C2×D5311yescharacteristic
204#424—1061no—
205#423—531no—
206C2×C4⋊C8—531no—
207C2×C4⋊C8C2×D5311yescharacteristic
208C2×#100—1061no—
209C2×#100—1061no—
210#405—1061no—
211C4×M16C2×D5311yescharacteristic
212#333—1061no—
213#332—1061no—
214#331—1061no—
215#331—1061no—
216C53#927411yescharacteristic, complemented, Sylow
217C22×D4—532no—
218C22×D4—1061no—
219C22×D4—1061no—
220C2×SD16—1064no—
221C2×M16—1061no—
222C2×M16D21211yescharacteristic
223C2×M16D21211yescharacteristic
224C22×C8—21no—
225C22×C8—1061no—
226C4⋊Q8—531no—
227C4⋊Q8—531no—
228#127—1061no—
229#127—1061no—
230#127—1061no—
231#127—1061no—
232#125—1061no—
233C22⋊Q8—532no—
234C22⋊Q8—1061no—
235C4×Q8—532no—
236C4×Q8—1061no—
237#117—1061no—
238C2×C42C22×D5311yescharacteristic
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—531no—
243C4⋊C8—531no—
244C4⋊C8—1062no—
245C4⋊C8—1062no—
246C4⋊C8—1061no—
247C4⋊C8—1061no—
248C4⋊C8—1061no—
249#102—1064no—
250#100—2122no—
251#100—2122no—
252C22⋊C8C4×D5312yes—
253#97—21no—
254#97—1062no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—1062no—
258C24—1062no—
259C2×Q8—532no—
260C2×Q8—1061no—
261C2×D4—1062no—
262C2×D4—1062no—
263C2×D4—4241no—
264C2×D4—4241no—
265C2×D4—4242no—
266C2×D4—2121no—
267C2×D4—2121no—
268C2×D4—2121no—
269C2×D4—2121no—
270C2×D4—2121no—
271C2×D4—2121no—
272C2×D4—2121no—
273C2×D4—2121no—
274C22×C4C2×C4×D5311yescharacteristic
275C22×C4C2×D21211yescharacteristic
276C22×C4#1171994511yescharacteristic
277SD16—4244no—
278M16—21no—
279M16—21no—
280M16—21no—
281M16—2121no—
282M16—2121no—
283M16#1171991312yescomplemented
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—41no—
288C2×C8—81no—
289C2×C8—1061no—
290C2×C8—1061no—
291C2×C8—2121no—
292C2×C8—2121no—
293C2×C8—2121no—
294C4⋊C4—532no—
295C4⋊C4—1062no—
296C4⋊C4—1061no—
297C4⋊C4—1061no—
298C4⋊C4—1061no—
299C4⋊C4—1062no—
300C22⋊C4—1062no—
301C42—1062no—
302C42D4×D5311yescharacteristic
303C42D4×D5311yescharacteristic
304C42D4×D5311yescharacteristic
305C42#1171994911yescharacteristic
306C23—4242no—
307C23—4242no—
308C23—2122no—
309C23—2122no—
310C23—2122no—
311C23C4×D21211yescharacteristic
312Q8—2122no—
313Q8—2121no—
314Q8—2121no—
315D4—4242no—
316D4—4242no—
317D4—4241no—
318D4—4241no—
319D4—4241no—
320D4—4241no—
321D4—4241no—
322D4—4241no—
323D4—2124no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—1062no—
328C2×C4—1062no—
329C2×C4C22⋊C4×D5311yescharacteristic
330C2×C4#1313317911yescharacteristic
331C2×C4#1313318711yescharacteristic
332C2×C4#1313319011yescharacteristic
333C2×C4#1313319211yescharacteristic
334C2×C4#1313320211yescharacteristic
335C2×C4C2×#1171991311yescharacteristic
336C2×C4#1313323911yescharacteristic
337C2×C4#1313324511yescharacteristic
338C2×C4#1313325011yescharacteristic
339C8—22no—
340C8—42no—
341C8—41no—
342C8—81no—
343C8—4241no—
344C8—2121no—
345C8—2121no—
346C22—21no—
347C22—21no—
348C22—8482no—
349C22—8482no—
350C22—4242no—
351C22—4242no—
352C22—4242no—
353C22#1681050111yescharacteristic
354C22C4×#1171991311yescharacteristic
355C22#1681072511yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—2122no—
360C4—2122no—
361C4#1681076411yescharacteristic
362C4#1681077011yescharacteristic
363C4#1681082411yescharacteristic
364C4#1681084611yescharacteristic
365C4#1681088911yescharacteristic
366C4#1681089211yescharacteristic
367C2—21no—
368C2—21no—
369C2—8482no—
370C2#2027506911yescharacteristic
371C2#2027522411yescharacteristic
372C2#2027526111yescharacteristic
subgroupG/H
centerC22#16810501
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1283C2
socleC22×C53#386
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1283D53
hypocenterC53#9274
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
2. #20283662C2semidirect
4. #20273403C2semidirect
6. #20272611C2semidirect
7. C53×#1283C2semidirect
93. M16×C53SD16semidirect
187. #1283D53semidirect
216. C53#9274semidirect
283. M16#11719913semidirect

Maximal subgroups

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

subgroupindex
#202856582
#202836622
#202755552
#202734032
C2×#168103662
#202726112
C53×#12832
#927453