Finite Group Databasebeta

← #27285209

Subgroups — #27285209

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810997C211yesmaximal, characteristic
2#20279714C211yesmaximal, characteristic
3#20275664C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic
5C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C2×C53×#477C211yesmaximal, characteristic, complemented
8C22×#11719924—22no—
9C2×#13133214C2211yescharacteristic
10C2×#13133206—21no—
11#16810997—22no—
12#16810997—22no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×C4×#88322C2211yescharacteristic
15#16810540—22no—
16#16810525—22no—
17#16810524—21no—
18C4×C4⋊Dic53C2211yescharacteristic
19#16810385—22no—
20#16810366—22no—
21Q8⋊#88322—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#477—22no—
25C53×#477—22no—
26C2×#107×C53—21no—
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×#103×C53—21no—
29C22×D212—21no—
30C22×D212—21no—
31C22×D212D412yes—
32C2×#11719924—28no—
33C2×#11719924—48no—
34C22×#88322—21no—
35C22×#88322—21no—
36#13133214—21no—
37#13133214—21no—
38#13133214—21no—
39#13133214—21no—
40#13133206—42no—
41#13133206—42no—
42C2×C4⋊Dic53D412yes—
43C2×#11719921—24no—
44C2×#11719921—24no—
45C2×C4×Dic53—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4×#88322—21no—
49C4×#88322D412yes—
50#13133124—21no—
51C22×Q8×C53—21no—
52C22×Q8×C53D411yescharacteristic
53C22×C8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53D412yes—
58C4×Q8×C53—21no—
59C4×Q8×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C42×C53—21no—
67C2×C42×C53C2311yescharacteristic
68#107×C53—24no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53D412yes—
71#103×C53—24no—
72C23×D53—22no—
73C2×Q8×C53—21no—
74C2×Q8×C53—21no—
75C2×Q8×C53—22no—
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—44no—
80C2×Q8×C53—41no—
81C2×Q8×C53SD1612yescomplemented
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—41no—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×Dic53—22no—
89C2×D212—22no—
90C2×D212—22no—
91C2×D212—44no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—42no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—42no—
102C2×C8×C53—22no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C42×C53—41no—
114C42×C53—41no—
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53#4111yescharacteristic
119#11719924—416no—
120#11719921—48no—
121#11719921—84no—
122C4⋊Dic53—22no—
123C4⋊Dic53—22no—
124C4×Dic53—41no—
125C4×Dic53—41no—
126C2×#88322—24no—
127C2×#88322—22no—
128C2×#88322—41no—
129C2×#88322—42no—
130C23×C53#12011yescharacteristic
131C22×D53—42no—
132C22×D53—42no—
133C22×D53—42no—
134C22×D53—44no—
135C22×D53—44no—
136Q8×C53—22no—
137Q8×C53—22no—
138Q8×C53—44no—
139Q8×C53—41no—
140Q8×C53—41no—
141Q8×C53—82no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—82no—
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12711yescharacteristic
156C2×C4×C53C2×SD1611yescharacteristic
157C2×C4×C53#13511yescharacteristic
158C2×C4×C53#13512yes—
159C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C53#13711yescharacteristic
161C2×C4×C53#13712yes—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165D212—44no—
166D212—82no—
167D212—81no—
168D212—81no—
169D212—82no—
170D212—81no—
171D212—81no—
172D212—81no—
173D212—81no—
174C8×C53—42no—
175#88322—24no—
176#88322—81no—
177#10055—531nomaximal, Sylow
178C22×C53#39211yescharacteristic
179C22×C53#45011yescharacteristic
180C22×C53#45211yescharacteristic
181C22×C53#45212yes—
182C22×C53#45312yes—
183C2×D53—84no—
184C2×D53—84no—
185C2×D53—82no—
186C2×D53—82no—
187C2×D53—82no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—42no—
191C4×C53—81no—
192C4×C53—82no—
193C4×C53#44811yescharacteristic
194C4×C53#44911yescharacteristic
195C4×C53#46412yes—
196C4×C53#46511yescharacteristic
197C4×C53#46811yescharacteristic
198C4×C53#49511yescharacteristic
199C4×C53#49711yescharacteristic
200Dic53—82no—
201C2×#492—531no—
202C2×#477D5311yescharacteristic, complemented
203#2548—531no—
204#1987—531no—
205C2×#331—531no—
206C2×Q8⋊C8—531no—
207#1279—531no—
208C2×C53#163012yes—
209C2×C53#163112yes—
210C2×C53#200011yescharacteristic
211C2×C53#200211yescharacteristic
212C2×C53#201511yescharacteristic
213D53—162no—
214C22×SD16—1062no—
215C2×C4⋊Q8C2×D5311yescharacteristic
216C2×#127—531no—
217C2×#124—1061no—
218C2×C4×Q8C2×D5311yescharacteristic
219#492—1062no—
220#492—1062no—
221#477—22no—
222#477—22no—
223C2×#107—21no—
224C2×C4⋊C8—531no—
225C2×C4⋊C8C2×D5311yescharacteristic
226C2×#103—21no—
227C2×C4×C8—531no—
228#394—1062no—
229#390—1062no—
230#390—1061no—
231C4×C4⋊C4—531no—
232#340—1062no—
233#331—1062no—
234Q8⋊C8—1062no—
235C53#1005511yescharacteristic, complemented, Sylow
236C22×Q8—21no—
237C22×Q8C22⋊D5311yescharacteristic
238C22×D4—532no—
239C22×D4—1061no—
240C22×D4—1061no—
241C2×SD16—1068no—
242C2×SD16—2128no—
243C22×C8—21no—
244C22×C8—1061no—
245C22×C8—1061no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8C22⋊D5312yes—
250#127—1061no—
251#127—1061no—
252#127—1061no—
253#127—1061no—
254#124—2122no—
255#124—2122no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—532no—
264C2×C4⋊C4C22⋊D5311yescharacteristic
265C2×C22⋊C4—1064no—
266C2×C22⋊C4—1064no—
267C2×C42—21no—
268C2×C42—1061no—
269C2×C42C22×D5311yescharacteristic
270#107—24no—
271C4⋊C8—21no—
272C4⋊C8—1061no—
273C4⋊C8—1061no—
274C4⋊C8C22⋊D5312yes—
275#103—24no—
276C4×C8—532no—
277C4×C8—1061no—
278#95—1061no—
279C24—1062no—
280C2×Q8—21no—
281C2×Q8—22no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—44no—
287C2×Q8—41no—
288C2×Q8#1171992412yescomplemented
289C2×D4—1062no—
290C2×D4—1062no—
291C2×D4—2124no—
292C2×D4—2121no—
293C2×D4—2121no—
294C2×D4—2121no—
295C2×D4—2121no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4—2122no—
299C2×D4—2121no—
300C2×D4—2121no—
301C2×D4—2122no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—1062no—
306C22×C4D4×D5311yescharacteristic
307C22×C4D4×D5311yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309SD16—21216no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—1064no—
314C2×C8—1062no—
315C2×C8—2121no—
316C2×C8—2122no—
317C4⋊C4—22no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—1062no—
326C4⋊C4—1062no—
327C22⋊C4—4244no—
328C22⋊C4—2128no—
329C42—41no—
330C42—41no—
331C42—2121no—
332C42—2121no—
333C42D4×D5311yescharacteristic
334C42#1171994711yescharacteristic
335C42C2×C22⋊D5311yescharacteristic
336C42C2×C22⋊D5311yescharacteristic
337C23—2122no—
338C23—2122no—
339C23—2124no—
340C23—2122no—
341C23—2124no—
342C23#1313318511yescharacteristic
343Q8—22no—
344Q8—22no—
345Q8—44no—
346Q8—41no—
347Q8—41no—
348Q8—82no—
349D4—4242no—
350D4—4241no—
351D4—4241no—
352D4—4241no—
353D4—4241no—
354D4—4242no—
355D4—4241no—
356D4—4241no—
357D4—2124no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—2122no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4#1313318511yescharacteristic
370C2×C4#1313318611yescharacteristic
371C2×C4#1313318811yescharacteristic
372C2×C4#1313318911yescharacteristic
373C2×C4#1313318911yescharacteristic
374C2×C4#1313321311yescharacteristic
375C2×C4C2×#1171992411yescharacteristic
376C2×C4#1313323711yescharacteristic
377C2×C4#1313324012yes—
378C2×C4#1313324211yescharacteristic
379C2×C4#1313324811yescharacteristic
380C2×C4#1313324912yes—
381C8—42no—
382C8—1064no—
383C8—4241no—
384C22—4242no—
385C22—4242no—
386C22—4242no—
387C22—4244no—
388C22—4244no—
389C22#1681053111yescharacteristic
390C22#1681077311yescharacteristic
391C22#1681078311yescharacteristic
392C22#1681078412yes—
393C22#1681078912yes—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—82no—
399C4—4242no—
400C4#1681076211yescharacteristic
401C4#1681076811yescharacteristic
402C4#1681084712yes—
403C4#1681085411yescharacteristic
404C4#1681087111yescharacteristic
405C4#1681100811yescharacteristic
406C4#1681101611yescharacteristic
407C2—8482no—
408C2#2027383512yes—
409C2#2027384012yes—
410C2#2027571611yescharacteristic
411C2#2027572711yescharacteristic
412C2#2027580611yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#477C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenterC2×#477D53
hypocenterC53#10055
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #20275664C2semidirect
5. C2×Q8⋊#88322C2semidirect
6. #20272554C2semidirect
7. C2×C53×#477C2semidirect
81. C2×Q8×C53SD16semidirect
202. C2×#477D53semidirect
235. C53#10055semidirect
288. C2×Q8#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168109972
#202797142
#202756642
C2×#168103662
C2×Q8⋊#883222
#202725542
C2×C53×#4772
#1005553