Finite Group Databasebeta

← #27278680

Subgroups — #27278680

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×#11719913C211yesmaximal, characteristic
2C2×#16810702C211yesmaximal, characteristic
3#20275651C211yesmaximal, characteristic, complemented
4C2×#16810355C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic
6#20272550C211yesmaximal, characteristic, complemented
7C53×#1273C211yesmaximal, characteristic, complemented
8C22×#11719913C411yescharacteristic
9C22×#11719912—21no—
10C2×C4×Dic106C2211yescharacteristic
11C2×C4×D212C2211yescharacteristic
12C4×#11719913—22no—
13C4×#11719913—22no—
14#16810702—22no—
15#16810702—22no—
16C2×#13133149C411yescharacteristic
17C2×C4⋊#88322C2211yescharacteristic
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133142C411yescharacteristic
20C2×#13133139C411yescharacteristic
21C2×#13133129—21no—
22#16810377—21no—
23#16810377—21no—
24#16810355—22no—
25#16810352—22no—
26C2×C4×C8×C53C2211yescharacteristic
27C2×C4×C8×C53C2211yescharacteristic
28C2×C4×C8×C53C2211yescharacteristic
29C22×Dic106C2×C411yescharacteristic
30C22×D212C2×C411yescharacteristic
31C22×C4×D53—21no—
32C2×#11719913—22no—
33C2×#11719913—22no—
34C2×#11719913—22no—
35C2×#11719912—24no—
36C2×#11719912—42no—
37C2×#11719912—42no—
38C22×#88322—21no—
39C22×#88322—21no—
40C4×Dic106—21no—
41C4×Dic106—21no—
42C4×Dic106—21no—
43C4×Dic106—21no—
44C4×D212—21no—
45C4×D212—21no—
46C4×D212D412yes—
47C4×D212D412yes—
48C2×C4⋊Dic53C2×C411yescharacteristic
49C2×C4⋊Dic53C2×C411yescharacteristic
50C2×#11719919—21no—
51C2×#11719921—21no—
52C2×C4×Dic53—21no—
53#13133149—22no—
54C4⋊#88322—21no—
55C4⋊#88322—21no—
56C4⋊#88322—21no—
57C4⋊#88322D412yes—
58#13133142—22no—
59#13133139—22no—
60#13133129—24no—
61C22×C8×C53—21no—
62C22×C8×C53C2×C411yescharacteristic
63C22×C8×C53C2×C411yescharacteristic
64C22×C8×C53D411yescharacteristic
65C22×C8×C53D411yescharacteristic
66C2×C42×C53C2311yescharacteristic
67C4×C8×C53—21no—
68C4×C8×C53—21no—
69C4×C8×C53—21no—
70C4×C8×C53—21no—
71C4×C8×C53—21no—
72C4×C8×C53—21no—
73C23×D53—21no—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53#4111yescharacteristic
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212C22⋊C412yes—
86C2×C4×D53—22no—
87C2×C4×D53—22no—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×Dic106—21no—
93C2×Dic106—21no—
94C2×Dic106—21no—
95C2×Dic106—21no—
96C2×Dic106—21no—
97C2×Dic106M1612yescomplemented
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—21no—
101C2×C8×C53—21no—
102C2×C8×C53—22no—
103C2×C8×C53—21no—
104C2×C8×C53—21no—
105C2×C8×C53—21no—
106C2×C8×C53—21no—
107C2×C8×C53—21no—
108C2×C8×C53—21no—
109C2×C8×C53—41no—
110C2×C8×C53—41no—
111C2×C8×C53M1612yescomplemented
112C2×C8×C53SD1612yescomplemented
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117#11719921—22no—
118#11719921—22no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121C4⋊Dic53—21no—
122C4⋊Dic53C22⋊C412yes—
123#11719919—41no—
124#11719919—41no—
125C4×Dic53—41no—
126C4×Dic53—41no—
127C2×#88322—22no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—41no—
133#11719913—28no—
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—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53C2×C22⋊C411yescharacteristic
146C2×C4×C53C4×D411yescharacteristic
147C2×C4×C53#12011yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12311yescharacteristic
151C2×C4×C53C2×M1611yescharacteristic
152C2×C4×C53C2×M1611yescharacteristic
153C2×C4×C53#13111yescharacteristic
154C2×C4×C53#13111yescharacteristic
155C2×C4×C53C2×SD1611yescharacteristic
156C2×C4×C53#13511yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53#13711yescharacteristic
159C2×Dic53—22no—
160C2×Dic53—22no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168D212—22no—
169D212—22no—
170D212—41no—
171D212—41no—
172C4×D53—42no—
173C4×D53—42no—
174C4×D53—42no—
175C4×D53—42no—
176Dic106—22no—
177Dic106—22no—
178Dic106—41no—
179Dic106—41no—
180C8×C53—22no—
181C8×C53—22no—
182C8×C53—42no—
183C8×C53—41no—
184C8×C53—41no—
185#88322—42no—
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#40711yescharacteristic
205C4×C53#40811yescharacteristic
206C4×C53#43511yescharacteristic
207C4×C53#43611yescharacteristic
208C4×C53#45011yescharacteristic
209C4×C53#45211yescharacteristic
210C4×C53#46111yescharacteristic
211C4×C53#46311yescharacteristic
212Dic53—42no—
213Dic53—42no—
214Dic53—81no—
215C2×C4×SD16—531no—
216C2×#436—531no—
217#1984—531no—
218C2×Q8⋊C8—531no—
219C2×D4⋊C8—531no—
220#1277—531no—
221#1273D5311yescharacteristic, complemented
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×Q8—531no—
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⋊C8—531no—
241C2×#103—531no—
242C2×#102—531no—
243C2×C22⋊C8—1061no—
244C2×C4×C8C2×D5311yescharacteristic
245C2×C4×C8C2×D5311yescharacteristic
246C2×C4×C8C2×D5311yescharacteristic
247#336—1061no—
248#336—1061no—
249Q8⋊C8—1062no—
250D4⋊C8—1062no—
251C53#894911yescharacteristic, complemented, Sylow
252C22×Q8—531no—
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—1061no—
263C22×C8—1061no—
264C22×C8C4×D5311yescharacteristic
265C22×C8C4×D5311yescharacteristic
266C22×C8D21211yescharacteristic
267C22×C8D21211yescharacteristic
268C4×Q8—1061no—
269C4×Q8—1061no—
270C4×Q8—1061no—
271C4×Q8—1061no—
272C4×D4—532no—
273C4×D4—532no—
274C4×D4—1061no—
275C4×D4—1061no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4—1061no—
279C2×C22⋊C4—1061no—
280C2×C42—1061no—
281C2×C42C22×D5311yescharacteristic
282#106—1062no—
283C4⋊C8—532no—
284C4⋊C8—1061no—
285C4⋊C8—1061no—
286C4⋊C8—1061no—
287#103—1062no—
288#102—1062no—
289C22⋊C8—1064no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C24—1061no—
297C2×Q8—532no—
298C2×Q8—1061no—
299C2×Q8—1061no—
300C2×Q8—1061no—
301C2×Q8—1061no—
302C2×Q8—1061no—
303C2×D4—532no—
304C2×D4—1061no—
305C2×D4—1061no—
306C2×D4—1061no—
307C2×D4—1061no—
308C2×D4—1061no—
309C22×C4—1061no—
310C22×C4—1061no—
311C22×C4—1062no—
312C22×C4—1062no—
313C22×C4—1061no—
314C22×C4—2121no—
315C22×C4—2121no—
316C22×C4—2121no—
317C22×C4—2121no—
318C22×C4C2×C4×D5311yescharacteristic
319C22×C4C2×D21211yescharacteristic
320C22×C4#1171994511yescharacteristic
321SD16—1068no—
322M16—2128no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—1062no—
337C2×C8—2121no—
338C2×C8—2121no—
339C2×C8—2121no—
340C2×C8—2121no—
341C2×C8—2121no—
342C2×C8#1171991212yescomplemented
343C2×C8#1171991312yescomplemented
344C4⋊C4—532no—
345C4⋊C4—1061no—
346C4⋊C4—1061no—
347C4⋊C4—1061no—
348C4⋊C4—2121no—
349C4⋊C4—2121no—
350C22⋊C4—1062no—
351C22⋊C4—1062no—
352C42—2121no—
353C42—2121no—
354C42D4×D5311yescharacteristic
355C42D4×D5311yescharacteristic
356C42D4×D5311yescharacteristic
357C42#1171994911yescharacteristic
358C23—1062no—
359C23—2121no—
360C23—2121no—
361C23—2121no—
362C23—2121no—
363C23—2121no—
364C23—2121no—
365C23C4×D21211yescharacteristic
366Q8—1062no—
367Q8—1062no—
368Q8—2121no—
369Q8—2121no—
370D4—1062no—
371D4—1062no—
372D4—2121no—
373D4—2121no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—1062no—
377C2×C4—1062no—
378C2×C4—2121no—
379C2×C4—2121no—
380C2×C4—2121no—
381C2×C4—2121no—
382C2×C4—2121no—
383C2×C4—2121no—
384C2×C4—2121no—
385C2×C4—2122no—
386C2×C4—2122no—
387C2×C4—2122no—
388C2×C4—2122no—
389C2×C4C22⋊C4×D5311yescharacteristic
390C2×C4#1313317911yescharacteristic
391C2×C4#1313318711yescharacteristic
392C2×C4#1313319011yescharacteristic
393C2×C4#1313319211yescharacteristic
394C2×C4#1313320211yescharacteristic
395C2×C4M16×D5311yescharacteristic
396C2×C4C2×#1171991211yescharacteristic
397C2×C4#1313322311yescharacteristic
398C2×C4#1313322511yescharacteristic
399C2×C4C2×#1171991311yescharacteristic
400C2×C4#1313323911yescharacteristic
401C2×C4#1313324511yescharacteristic
402C2×C4#1313325011yescharacteristic
403C8—22no—
404C8—22no—
405C8—42no—
406C8—41no—
407C8—41no—
408C8—4241no—
409C8—2122no—
410C22—4241no—
411C22—4241no—
412C22—4241no—
413C22—4241no—
414C22—2122no—
415C22—2122no—
416C22—2122no—
417C22#1681050111yescharacteristic
418C22#1681069611yescharacteristic
419C22#1681070211yescharacteristic
420C22C4×#1171991311yescharacteristic
421C22#1681072511yescharacteristic
422C22#1681074111yescharacteristic
423C22#1681074611yescharacteristic
424C4—21no—
425C4—21no—
426C4—4241no—
427C4—2122no—
428C4—2122no—
429C4#1681058511yescharacteristic
430C4#1681058811yescharacteristic
431C4#1681069311yescharacteristic
432C4#1681070011yescharacteristic
433C4#1681077611yescharacteristic
434C4#1681078711yescharacteristic
435C4#1681082911yescharacteristic
436C4#1681083911yescharacteristic
437C2—4242no—
438C2#2027356011yescharacteristic
439C2#2027359511yescharacteristic
440C2#2027360711yescharacteristic
441C2#2027361911yescharacteristic
442C2#2027501111yescharacteristic
443C2#2027504111yescharacteristic
444C2#2027504811yescharacteristic
subgroupG/H
centerC22×C4C2×D212
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1273C2
socleC23×C53C4×D4
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1273D53
hypocenterC53#8949
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1273

Complements

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

NH ≅ G/Ntype
3. #20275651C2semidirect
4. C2×#16810355C2semidirect
6. #20272550C2semidirect
7. C53×#1273C2semidirect
97. C2×Dic106M16semidirect
111. C2×C8×C53M16semidirect
112. C2×C8×C53SD16semidirect
221. #1273D53semidirect
251. C53#8949semidirect
342. C2×C8#11719912semidirect
343. C2×C8#11719913semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199132
C2×#168107022
#202756512
C2×#168103552
C2×#168103522
#202725502
C53×#12732
#894953