Finite Group Databasebeta

← #27285325

Subgroups — #27285325

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810970C211yesmaximal, characteristic
2C2×#16810908C211yesmaximal, characteristic, complemented
3#20279714C211yesmaximal, characteristic
4#20275637C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic
6#20272555C211yesmaximal, characteristic, complemented
7C2×C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×#13133214C2211yescharacteristic
10C2×#13133206—21no—
11#16810970—22no—
12#16810970—22no—
13#16810908—22no—
14#16810908—22no—
15C2×#13133148—21no—
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133142—22no—
18C2×#13133141—21no—
19C2×#13133139—22no—
20#16810540—22no—
21#16810525—22no—
22#16810524—21no—
23C4×C4⋊Dic53C2211yescharacteristic
24#16810366—22no—
25C2×C4×Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4×C8×C53C2211yescharacteristic
28C53×Q8⋊C8—22no—
29C22×Dic106—21no—
30C22×D212—21no—
31C22×D212—21no—
32C22×D212D412yes—
33C22×#88322—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217—21no—
37#13133217D412yes—
38#13133214—21no—
39#13133214—21no—
40#13133214—21no—
41#13133214—21no—
42#13133206—42no—
43#13133206—42no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53D412yes—
47C2×#11719921—24no—
48C2×#11719921—24no—
49C2×C4×Dic53—21no—
50#13133148—24no—
51C4⋊#88322—21no—
52C4⋊#88322D412yes—
53#13133142—44no—
54#13133141—24no—
55#13133139—28no—
56#13133124—21no—
57C22×Q8×C53D411yescharacteristic
58C22×C8×C53—21no—
59C22×C8×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C42×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4×C8×C53—21no—
71C4×C8×C53D412yes—
72C23×D53—22no—
73C2×Q8×C53—21no—
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53SD1612yes—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—22no—
85C22×Dic53—41no—
86C2×D212—22no—
87C2×D212—22no—
88C2×D212—44no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—42no—
96C2×D212—41no—
97C2×D212—42no—
98C2×D212—41no—
99C2×Dic106—22no—
100C2×Dic106—42no—
101C2×Dic106—42no—
102C2×Dic106—41no—
103C2×C8×C53—22no—
104C2×C8×C53—42no—
105C2×C8×C53—41no—
106C2×C8×C53—42no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C42×C53—41no—
112C42×C53—41no—
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117#11719921—48no—
118#11719921—84no—
119C4⋊Dic53—22no—
120C4⋊Dic53—22no—
121C4⋊Dic53—22no—
122C4⋊Dic53—22no—
123C4⋊Dic53—41no—
124C4⋊Dic53—41no—
125C4×Dic53—41no—
126C4×Dic53—41no—
127C2×#88322—22no—
128C2×#88322—41no—
129C2×#88322—41no—
130C23×C53#12011yescharacteristic
131C22×D53—42no—
132C22×D53—44no—
133C22×D53—42no—
134C22×D53—42no—
135C22×D53—44no—
136Q8×C53—22no—
137Q8×C53—22no—
138Q8×C53—41no—
139Q8×C53—41no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53#12011yescharacteristic
147C2×C4×C53#12011yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12711yescharacteristic
152C2×C4×C53C2×SD1611yescharacteristic
153C2×C4×C53#13511yescharacteristic
154C2×C4×C53#13512yes—
155C2×C4×C53AGL(1,ℤ/8)12yes—
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C53#13711yescharacteristic
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—82no—
163D212—44no—
164D212—82no—
165D212—81no—
166D212—81no—
167D212—82no—
168D212—81no—
169D212—81no—
170D212—81no—
171D212—81no—
172Dic106—44no—
173Dic106—81no—
174Dic106—81no—
175C8×C53—42no—
176C8×C53—81no—
177#88322—42no—
178#10071—531nomaximal, Sylow
179C22×C53#39211yescharacteristic
180C22×C53#44912yes—
181C22×C53#45011yescharacteristic
182C22×C53#45211yescharacteristic
183C22×C53#45512yes—
184C2×D53—84no—
185C2×D53—84no—
186C2×D53—82no—
187C2×D53—82no—
188C2×D53—82no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—42no—
192C4×C53—81no—
193C4×C53#44811yescharacteristic
194C4×C53#44911yescharacteristic
195C4×C53#46312yes—
196C4×C53#46511yescharacteristic
197C4×C53#46811yescharacteristic
198C4×C53#49511yescharacteristic
199C4×C53#49611yescharacteristic
200Dic53—82no—
201Dic53—82no—
202C2×#486—531no—
203C2×#475—531no—
204#2548—531no—
205#1981—531no—
206C2×#331—531no—
207C2×Q8⋊C8D5311yescharacteristic, complemented
208#1279—531no—
209C2×C53#164012yes—
210C2×C53#164212yes—
211C2×C53#200011yescharacteristic
212C2×C53#200411yescharacteristic
213C2×C53#201211yescharacteristic
214D53—162no—
215C2×C4⋊Q8—531no—
216C2×#127—531no—
217C2×#124—1061no—
218C2×C4×Q8C2×D5311yescharacteristic
219#486—1062no—
220#486—1062no—
221#475—1062no—
222#475—1062no—
223C2×#106—1061no—
224C2×C4⋊C8—531no—
225C2×C4⋊C8C2×D5311yescharacteristic
226C2×#103—1061no—
227C2×#103—1062no—
228C2×#102—1062no—
229C2×C4×C8C2×D5311yescharacteristic
230#394—1062no—
231#390—1062no—
232#390—1061no—
233C4×C4⋊C4—531no—
234#331—1062no—
235Q8⋊C8—22no—
236C53#1007111yescharacteristic, complemented, Sylow
237C22×Q8—1061no—
238C22×Q8D21211yescharacteristic
239C22×D4—532no—
240C22×D4—1061no—
241C22×D4—1061no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—1061no—
245C4⋊Q8—532no—
246C4⋊Q8—1061no—
247C4⋊Q8—1061no—
248C4⋊Q8—1061no—
249#127—1061no—
250#127—1061no—
251#127—1061no—
252#127—1061no—
253#124—2122no—
254#124—2122no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—532no—
261C2×C4⋊C4—1061no—
262C2×C4⋊C4—1061no—
263C2×C4⋊C4D21211yescharacteristic
264C2×C22⋊C4—1064no—
265C2×C22⋊C4—1064no—
266C2×C42—21no—
267C2×C42—1061no—
268C2×C42C22×D5311yescharacteristic
269#106—1064no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—532no—
273C4⋊C8—1061no—
274#103—1064no—
275#103—2124no—
276#102—1068no—
277C4×C8—21no—
278C4×C8C22⋊D5312yes—
279#95—1061no—
280C24—1062no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—1062no—
287C2×Q8—2122no—
288C2×Q8—2122no—
289C2×Q8—2121no—
290C2×Q8#1171991312yes—
291C2×D4—1062no—
292C2×D4—1062no—
293C2×D4—2124no—
294C2×D4—2121no—
295C2×D4—2121no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4—2121no—
299C2×D4—2121no—
300C2×D4—2122no—
301C2×D4—2122no—
302C2×D4—2121no—
303C2×D4—2121no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—1062no—
307C22×C4—2121no—
308C22×C4C2×D21211yescharacteristic
309C22×C4D4×D5311yescharacteristic
310C22×C4D4×D5311yescharacteristic
311C2×C8—22no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—1062no—
316C2×C8—2121no—
317C2×C8—2121no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—1062no—
323C4⋊C4—1062no—
324C4⋊C4—1062no—
325C4⋊C4—1062no—
326C4⋊C4—2121no—
327C4⋊C4—2121no—
328C22⋊C4—4244no—
329C22⋊C4—2128no—
330C42—41no—
331C42—41no—
332C42—2121no—
333C42—2121no—
334C42#1171994511yescharacteristic
335C42D4×D5311yescharacteristic
336C42D4×D5311yescharacteristic
337C42C2×C22⋊D5311yescharacteristic
338C23—2124no—
339C23—2122no—
340C23—2122no—
341C23—2122no—
342C23—2124no—
343C23#1313318711yescharacteristic
344Q8—22no—
345Q8—22no—
346Q8—41no—
347Q8—41no—
348Q8—4241no—
349Q8—4241no—
350Q8—2124no—
351D4—4242no—
352D4—4241no—
353D4—4241no—
354D4—4241no—
355D4—4241no—
356D4—4242no—
357D4—4241no—
358D4—4241no—
359D4—2124no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—4242no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4#1313318511yescharacteristic
372C2×C4#1313318711yescharacteristic
373C2×C4#1313319011yescharacteristic
374C2×C4#1313319011yescharacteristic
375C2×C4#1313319111yescharacteristic
376C2×C4#1313321311yescharacteristic
377C2×C4C2×#1171991311yescharacteristic
378C2×C4#1313323911yescharacteristic
379C2×C4#1313324012yes—
380C2×C4#1313324412yes—
381C2×C4#1313324511yescharacteristic
382C2×C4#1313325011yescharacteristic
383C8—42no—
384C8—81no—
385C8—2122no—
386C22—4242no—
387C22—4242no—
388C22—4242no—
389C22—4244no—
390C22—4244no—
391C22#1681053211yescharacteristic
392C22#1681077112yes—
393C22#1681077611yescharacteristic
394C22#1681078711yescharacteristic
395C22#1681080212yes—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—81no—
400C4—4242no—
401C4—4242no—
402C4#1681076411yescharacteristic
403C4#1681077011yescharacteristic
404C4#1681084112yes—
405C4#1681085211yescharacteristic
406C4#1681086911yescharacteristic
407C4#1681100811yescharacteristic
408C4#1681100911yescharacteristic
409C2—8482no—
410C2#2027390412yes—
411C2#2027391812yes—
412C2#2027572011yescharacteristic
413C2#2027574311yescharacteristic
414C2#2027579311yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×Q8⋊C8C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenterC2×Q8⋊C8D53
hypocenterC53#10071
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16810908C2semidirect
4. #20275637C2semidirect
6. #20272555C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
207. C2×Q8⋊C8D53semidirect
236. C53#10071semidirect

Maximal subgroups

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

subgroupindex
C2×#168109702
C2×#168109082
#202797142
#202756372
C2×#168103662
#202725552
C2×C53×Q8⋊C82
#1007153