Finite Group Databasebeta

← #27278678

Subgroups — #27278678

Order 13568.

584 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719924C211yesmaximal, characteristic
2C2×#16810700C211yesmaximal, characteristic
3#20275652C211yesmaximal, characteristic, complemented
4C2×#16810352C211yesmaximal, characteristic
5#20272549C211yesmaximal, characteristic, complemented
6#20272542C211yesmaximal, characteristic, complemented
7C2×C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C22×#11719924C411yescharacteristic
9C22×#11719912—21no—
10C2×C4×D212C2211yescharacteristic
11C4×#11719924—22no—
12C4×#11719924—22no—
13#16810700—22no—
14#16810700—22no—
15C2×#13133148C411yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133141C411yescharacteristic
18C2×#13133137C411yescharacteristic
19C2×#13133129—21no—
20C2×C4×#88322C2211yescharacteristic
21C2×C4×#88322C2211yescharacteristic
22#16810378—21no—
23#16810378—21no—
24#16810352—22no—
25C2×C4×Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4×C8×C53C2211yescharacteristic
28C53×Q8⋊C8—22no—
29C22×D212C2×C411yescharacteristic
30C22×C4×D53—21no—
31C2×#11719924—22no—
32C2×#11719924—22no—
33C2×#11719924—22no—
34C2×#11719912—24no—
35C2×#11719912—42no—
36C2×#11719912—42no—
37C22×#88322—21no—
38C22×#88322C2×C411yescharacteristic
39C22×#88322C2×C411yescharacteristic
40C22×#88322D411yescharacteristic
41C22×#88322D411yescharacteristic
42C4×D212—21no—
43C4×D212—21no—
44C4×D212D412yes—
45C4×D212D412yes—
46C2×C4⋊Dic53C2×C411yescharacteristic
47C2×#11719921—21no—
48#13133148—22no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51#13133141—22no—
52#13133137—22no—
53#13133129—24no—
54C4×#88322—21no—
55C4×#88322—21no—
56C4×#88322—21no—
57C4×#88322—21no—
58C22×Q8×C53C2×C411yescharacteristic
59C22×C8×C53—21no—
60C22×C8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×Q8×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53C2×C411yescharacteristic
67C2×C42×C53—21no—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—21no—
70C4⋊C8×C53D412yes—
71C4×C8×C53—21no—
72C4×C8×C53—21no—
73C23×D53—21no—
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53M1612yescomplemented
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53#4111yescharacteristic
85C22×Dic53—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212—21no—
90C2×D212—21no—
91C2×D212C22⋊C412yes—
92C2×C4×D53—22no—
93C2×C4×D53—22no—
94C2×C4×D53—41no—
95C2×C4×D53—41no—
96C2×C4×D53—41no—
97C2×C4×D53—41no—
98C2×C8×C53—22no—
99C2×C8×C53—22no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C42×C53—41no—
109C42×C53—41no—
110C42×C53C2×D411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53#4111yescharacteristic
114#11719924—28no—
115#11719921—22no—
116#11719921—22no—
117C4⋊Dic53—21no—
118C4⋊Dic53C22⋊C412yes—
119C2×#88322—21no—
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—21no—
125C2×#88322—21no—
126C2×#88322—21no—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322M1612yescomplemented
133C2×#88322SD1612yescomplemented
134#11719912—48no—
135C23×C53C4×D411yescharacteristic
136C22×D53—22no—
137C22×D53—41no—
138C22×D53—41no—
139C22×D53—41no—
140C22×D53—41no—
141C22×D53—41no—
142C22×D53—41no—
143Q8×C53—22no—
144Q8×C53—22no—
145Q8×C53—41no—
146Q8×C53—41no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53C2×C22⋊C411yescharacteristic
156C2×C4×C53C4×D411yescharacteristic
157C2×C4×C53#12011yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12111yescharacteristic
160C2×C4×C53#12311yescharacteristic
161C2×C4×C53C2×M1611yescharacteristic
162C2×C4×C53C2×M1611yescharacteristic
163C2×C4×C53#13111yescharacteristic
164C2×C4×C53#13111yescharacteristic
165C2×C4×C53C2×SD1611yescharacteristic
166C2×C4×C53#13511yescharacteristic
167C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C53#13711yescharacteristic
169C2×Dic53—22no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172D212—22no—
173D212—22no—
174D212—41no—
175D212—41no—
176C4×D53—42no—
177C4×D53—42no—
178C4×D53—42no—
179C4×D53—42no—
180C8×C53—42no—
181C8×C53—42no—
182#88322—22no—
183#88322—22no—
184#88322—41no—
185#88322—41no—
186#88322—81no—
187#8949—531nomaximal, Sylow
188C22×C53#38611yescharacteristic
189C22×C53#43511yescharacteristic
190C22×C53#43611yescharacteristic
191C22×C53C4×SD1611yescharacteristic
192C22×C53#44011yescharacteristic
193C22×C53#44311yescharacteristic
194C22×C53#44411yescharacteristic
195C2×D53—42no—
196C2×D53—42no—
197C2×D53—42no—
198C2×D53—81no—
199C2×D53—81no—
200C2×D53—81no—
201C2×D53—81no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—42no—
205C4×C53—81no—
206C4×C53#40711yescharacteristic
207C4×C53#40811yescharacteristic
208C4×C53#43511yescharacteristic
209C4×C53#43611yescharacteristic
210C4×C53#45011yescharacteristic
211C4×C53#45211yescharacteristic
212C4×C53#46111yescharacteristic
213C4×C53#46311yescharacteristic
214Dic53—42no—
215C2×C4×SD16—531no—
216C2×#436—531no—
217#1984—531no—
218C2×Q8⋊C8D5311yescharacteristic, complemented
219C2×D4⋊C8—531no—
220#1277—531no—
221#1273—531no—
222C2×C53#158311yescharacteristic
223C2×C53#158811yescharacteristic
224C2×C53#159011yescharacteristic
225C2×C53#159211yescharacteristic
226C2×C53#187011yescharacteristic
227C2×C53#187611yescharacteristic
228C2×C53#187711yescharacteristic
229D53—82no—
230C22×SD16—531no—
231C22×M16—1061no—
232C2×C4×Q8C2×D5311yescharacteristic
233C2×C4×D4—531no—
234C4×SD16—1062no—
235C4×SD16—1062no—
236#436—1062no—
237#436—1062no—
238C2×#106—531no—
239C2×C4⋊C8—531no—
240C2×C4⋊C8C2×D5311yescharacteristic
241C2×#103—531no—
242C2×#102—531no—
243C2×C22⋊C8—1061no—
244C2×C4×C8—531no—
245C2×C4×C8—531no—
246C2×C4×C8C2×D5311yescharacteristic
247#336—1061no—
248#336—1061no—
249Q8⋊C8—22no—
250D4⋊C8—1062no—
251C53#894911yescharacteristic, complemented, Sylow
252C22×Q8C4×D5311yescharacteristic
253C22×D4—531no—
254C23×C4—1061no—
255C2×SD16—1062no—
256C2×SD16—1062no—
257C2×SD16—1062no—
258C2×M16—1064no—
259C2×M16—2122no—
260C2×M16—2122no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—531no—
264C22×C8—531no—
265C22×C8—531no—
266C22×C8—531no—
267C22×C8—1061no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×D4—532no—
273C4×D4—532no—
274C4×D4—1061no—
275C4×D4—1061no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4C4×D5311yescharacteristic
279C2×C22⋊C4—1061no—
280C2×C42—21no—
281C2×C42C22×D5311yescharacteristic
282#106—1062no—
283C4⋊C8—21no—
284C4⋊C8—1061no—
285C4⋊C8—1061no—
286C4⋊C8C22⋊D5312yes—
287#103—1062no—
288#102—1062no—
289C22⋊C8—1064no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—1061no—
293C4×C8—1061no—
294C4×C8—1061no—
295C4×C8—1061no—
296C24—1061no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8#1171991212yescomplemented
303C2×D4—532no—
304C2×D4—1061no—
305C2×D4—1061no—
306C2×D4—1061no—
307C2×D4—1061no—
308C2×D4—1061no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—1061no—
312C22×C4—1062no—
313C22×C4—1062no—
314C22×C4—2121no—
315C22×C4—2121no—
316C22×C4—2121no—
317C22×C4—2121no—
318C22×C4C2×C4×D5311yescharacteristic
319C22×C4D4×D5311yescharacteristic
320C22×C4#1171994711yescharacteristic
321SD16—1068no—
322M16—2128no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—532no—
330C2×C8—532no—
331C2×C8—1061no—
332C2×C8—1061no—
333C2×C8—1061no—
334C2×C8—1061no—
335C2×C8—1061no—
336C2×C8—1061no—
337C2×C8—1061no—
338C2×C8—1061no—
339C2×C8—1061no—
340C2×C8—1061no—
341C2×C8—2121no—
342C2×C8—2121no—
343C2×C8—2121no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—532no—
349C4⋊C4—1061no—
350C22⋊C4—1062no—
351C22⋊C4—1062no—
352C42—41no—
353C42—41no—
354C42#1171994511yescharacteristic
355C42D4×D5311yescharacteristic
356C42D4×D5311yescharacteristic
357C42C2×C22⋊D5311yescharacteristic
358C23—1062no—
359C23—2121no—
360C23—2121no—
361C23—2121no—
362C23—2121no—
363C23—2121no—
364C23—2121no—
365C23#1313317711yescharacteristic
366Q8—22no—
367Q8—22no—
368Q8—41no—
369Q8—41no—
370D4—1062no—
371D4—1062no—
372D4—2121no—
373D4—2121no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—1062no—
383C2×C4—2121no—
384C2×C4—2121no—
385C2×C4—2122no—
386C2×C4—2122no—
387C2×C4—2122no—
388C2×C4—2122no—
389C2×C4C22⋊C4×D5311yescharacteristic
390C2×C4C4×C22⋊D5311yescharacteristic
391C2×C4#1313318511yescharacteristic
392C2×C4#1313318811yescharacteristic
393C2×C4#1313319011yescharacteristic
394C2×C4#1313319911yescharacteristic
395C2×C4M16×D5311yescharacteristic
396C2×C4C2×#1171991211yescharacteristic
397C2×C4#1313322311yescharacteristic
398C2×C4#1313322511yescharacteristic
399C2×C4SD16×D5311yescharacteristic
400C2×C4#1313324011yescharacteristic
401C2×C4#1313324411yescharacteristic
402C2×C4#1313324911yescharacteristic
403C8—42no—
404C8—42no—
405C8—1062no—
406C8—1062no—
407C8—4241no—
408C8—2121no—
409C8—2121no—
410C22—4241no—
411C22—4241no—
412C22—4241no—
413C22—4241no—
414C22—2122no—
415C22—2122no—
416C22—2122no—
417C22#1681049911yescharacteristic
418C22#1681069111yescharacteristic
419C22#1681069811yescharacteristic
420C22#1681070811yescharacteristic
421C22#1681072611yescharacteristic
422C22#1681073711yescharacteristic
423C22#1681074211yescharacteristic
424C4—21no—
425C4—21no—
426C4—42no—
427C4—81no—
428C4—2122no—
429C4#1681058511yescharacteristic
430C4#1681058811yescharacteristic
431C4#1681069411yescharacteristic
432C4#1681069911yescharacteristic
433C4#1681077211yescharacteristic
434C4#1681078411yescharacteristic
435C4#1681082711yescharacteristic
436C4#1681084111yescharacteristic
437C2—4242no—
438C2#2027355511yescharacteristic
439C2#2027359011yescharacteristic
440C2#2027360211yescharacteristic
441C2#2027362011yescharacteristic
442C2#2027500911yescharacteristic
443C2#2027503911yescharacteristic
444C2#2027504611yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×Q8⋊C8C2
socleC23×C53C4×D4
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenterC2×Q8⋊C8D53
hypocenterC53#8949
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #20275652C2semidirect
5. #20272549C2semidirect
6. #20272542C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
79. C2×Q8×C53M16semidirect
132. C2×#88322M16semidirect
133. C2×#88322SD16semidirect
218. C2×Q8⋊C8D53semidirect
251. C53#8949semidirect
302. C2×Q8#11719912semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199242
C2×#168107002
#202756522
C2×#168103522
#202725492
#202725422
C2×C53×Q8⋊C82
#894953