Finite Group Databasebeta

← #27287892

Subgroups — #27287892

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810819C211yesmaximal, characteristic
2#20275520C211yesmaximal, characteristic, complemented
3#20275508C211yesmaximal, characteristic, complemented
4#20274712C211yesmaximal, characteristic, complemented
5C2×#16810373C211yesmaximal, characteristic, complemented
6C2×#16810366C211yesmaximal, characteristic, complemented
7C53×#1296C211yesmaximal, characteristic, complemented
8C22×D424—21no—
9C2×#13133214C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810819—22no—
12#16810819—22no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133139—21no—
15C2×#13133137—21no—
16C2×#13133137—21no—
17C2×#13133137—21no—
18C2×#13133129—21no—
19C2×#13133129—21no—
20C2×C4×#88322C2211yescharacteristic
21#16810385—21no—
22#16810373—22no—
23#16810366—22no—
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C53×#384C2211yescharacteristic
27C22×D212—21no—
28C22×D212—21no—
29C22×D212—21no—
30C22×D212D411yescharacteristic
31C22×C4×D53—21no—
32C2×D424—24no—
33C2×D424—42no—
34C2×D424—42no—
35C22×#88322—21no—
36C22×#88322D411yescharacteristic
37C22×#88322D411yescharacteristic
38#13133214—21no—
39#13133214—21no—
40#13133214—21no—
41#13133214D412yes—
42C4×D212—21no—
43C4×D212—21no—
44C4×D212—21no—
45C4×D212—21no—
46C2×C4⋊Dic53D411yescharacteristic
47C2×#11719921—21no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50#13133139—24no—
51#13133137—42no—
52#13133137—42no—
53#13133137—42no—
54#13133129—42no—
55#13133129—42no—
56C4×#88322—21no—
57C4×#88322—21no—
58C22×C8×C53—21no—
59C22×C8×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C4⋊C8×C53D412yes—
68C23×D53—21no—
69C23×D53—41no—
70C22×C4×C53—41no—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53#4111yescharacteristic
74C22×Dic53—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—22no—
80C2×D212—22no—
81C2×D212—21no—
82C2×D212—22no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—81no—
95C2×D212—81no—
96C2×D212SD1612yes—
97C2×C4×D53—41no—
98C2×C4×D53—41no—
99C2×C4×D53—41no—
100C2×C4×D53—41no—
101C2×C4×D53—41no—
102C2×C4×D53—41no—
103C2×C8×C53—22no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—81no—
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117C42×C53#4111yescharacteristic
118#11719921—41no—
119#11719921—41no—
120C4⋊Dic53—21no—
121C4⋊Dic53—21no—
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—21no—
125C2×#88322—21no—
126C2×#88322—21no—
127C2×#88322—41no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322D812yescomplemented
131D424—48no—
132C23×C53#12111yescharacteristic
133C22×D53—22no—
134C22×D53—41no—
135C22×D53—41no—
136C22×D53—41no—
137C22×D53—41no—
138C22×D53—41no—
139C22×D53—41no—
140C22×D53—42no—
141C22×D53—81no—
142C22×D53—81no—
143C22×D53—81no—
144C22×D53—81no—
145C22×D53—81no—
146C22×D53—81no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53—81no—
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12311yescharacteristic
156C2×C4×C53#12311yescharacteristic
157C2×C4×C53#12411yescharacteristic
158C2×C4×C53C2×D811yescharacteristic
159C2×C4×C53C2×SD1611yescharacteristic
160C2×C4×C53#13511yescharacteristic
161C2×C4×C53#13511yescharacteristic
162C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169D212—22no—
170D212—22no—
171D212—41no—
172D212—42no—
173D212—42no—
174D212—42no—
175D212—42no—
176D212—41no—
177D212—82no—
178D212—82no—
179D212—81no—
180D212—81no—
181D212—81no—
182D212—81no—
183C4×D53—81no—
184C4×D53—81no—
185C4×D53—81no—
186C4×D53—81no—
187C8×C53—42no—
188C8×C53—81no—
189#88322—22no—
190#88322—41no—
191#88322—81no—
192#10473—531nomaximal, Sylow
193C22×C53#39411yescharacteristic
194C22×C53#45911yescharacteristic
195C22×C53#46311yescharacteristic
196C22×C53#46511yescharacteristic
197C22×C53#46911yescharacteristic
198C22×C53#47111yescharacteristic
199C22×C53#47211yescharacteristic
200C2×D53—42no—
201C2×D53—42no—
202C2×D53—42no—
203C2×D53—81no—
204C2×D53—81no—
205C2×D53—81no—
206C2×D53—81no—
207C2×D53—82no—
208C2×D53—82no—
209C2×D53—82no—
210C2×D53—161no—
211C2×D53—161no—
212C2×D53—161no—
213C2×D53—161no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—161no—
217C4×C53#44911yescharacteristic
218C4×C53#45011yescharacteristic
219C4×C53#45911yescharacteristic
220C4×C53#46311yescharacteristic
221C4×C53#46611yescharacteristic
222C4×C53#46911yescharacteristic
223C4×C53#48111yescharacteristic
224C4×C53#48211yescharacteristic
225Dic53—81no—
226C2×#459—531no—
227#1957—531no—
228#1954—531no—
229#1802—531no—
230C2×#334—531no—
231C2×#331—531no—
232#1296D5311yescharacteristic, complemented
233C2×C53#166511yescharacteristic
234C2×C53#167711yescharacteristic
235C2×C53#168711yescharacteristic
236C2×C53#170111yescharacteristic
237C2×C53#203411yescharacteristic
238C2×C53#205111yescharacteristic
239C2×C53#205411yescharacteristic
240D53—82no—
241D53—162no—
242C22×D8—1061no—
243C2×#127—531no—
244C2×C4×D4—531no—
245#459—1062no—
246#459—1062no—
247C2×C4⋊C8—531no—
248C2×C4⋊C8C2×D5311yescharacteristic
249C2×C4⋊C8C2×D5311yescharacteristic
250C2×#102—1061no—
251C2×#102—1061no—
252C2×#102—1061no—
253C2×#102—1061no—
254C2×C22⋊C8—1061no—
255C2×C22⋊C8—1061no—
256C2×C4×C8—531no—
257#384C2×D5311yescharacteristic
258#340—1061no—
259#334—1062no—
260#331—1062no—
261C53#1047311yescharacteristic, complemented, Sylow
262C22×D4—531no—
263C22×D4—1061no—
264C22×D4—1061no—
265C22×D4—1061no—
266C23×C4—1061no—
267C2×D8—1064no—
268C2×D8—2122no—
269C2×D8—2122no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—531no—
273C22×C8—531no—
274C22×C8—1061no—
275#127—532no—
276#127—1061no—
277#127—1061no—
278#127—1061no—
279C4×D4—1061no—
280C4×D4—1061no—
281C4×D4—1061no—
282C4×D4—1061no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—531no—
287C2×C22⋊C4—1061no—
288C2×C42C22×D5311yescharacteristic
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—1061no—
293C4⋊C8—1061no—
294C4⋊C8D21212yes—
295#102—1064no—
296#102—2122no—
297#102—2122no—
298#102—2122no—
299C22⋊C8—2122no—
300C22⋊C8—2122no—
301C4×C8—1061no—
302C4×C8—1061no—
303C24—1061no—
304C24—2121no—
305C2×D4—532no—
306C2×D4—1061no—
307C2×D4—1062no—
308C2×D4—1062no—
309C2×D4—1061no—
310C2×D4—1062no—
311C2×D4—1061no—
312C2×D4—1061no—
313C2×D4—1061no—
314C2×D4—4241no—
315C2×D4—4241no—
316C2×D4—2121no—
317C2×D4—2121no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2121no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C2×D4—2121no—
327C22×C4—41no—
328C22×C4—1061no—
329C22×C4—2121no—
330C22×C4—2121no—
331C22×C4—2121no—
332C22×C4—2121no—
333C22×C4—2121no—
334C22×C4—2121no—
335C22×C4#1171994511yescharacteristic
336C22×C4D4×D5311yescharacteristic
337C22×C4D4×D5311yescharacteristic
338D8—2128no—
339C2×C8—22no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—532no—
346C2×C8—1061no—
347C2×C8—1061no—
348C2×C8—1061no—
349C2×C8—1061no—
350C2×C8—1061no—
351C2×C8—2121no—
352C2×C8—2121no—
353C2×C8—2121no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—81no—
359C4⋊C4—1061no—
360C4⋊C4—1061no—
361C22⋊C4—2121no—
362C22⋊C4—2121no—
363C42C2×D21211yescharacteristic
364C42#1171994511yescharacteristic
365C42D4×D5311yescharacteristic
366C42#1171994911yescharacteristic
367C23—1062no—
368C23—4241no—
369C23—4241no—
370C23—4241no—
371C23—4241no—
372C23—4241no—
373C23—4241no—
374C23—2122no—
375C23—2121no—
376C23—2121no—
377C23—2121no—
378C23—2121no—
379C23—2121no—
380C23—2121no—
381C23#1313319011yescharacteristic
382D4—1062no—
383D4—1062no—
384D4—4242no—
385D4—4242no—
386D4—4241no—
387D4—4241no—
388D4—4241no—
389D4—4241no—
390D4—2122no—
391D4—2121no—
392D4—2122no—
393D4—2122no—
394D4—2122no—
395D4—2121no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—4241no—
402C2×C4—4241no—
403C2×C4—4241no—
404C2×C4—4241no—
405C2×C4—2121no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4#1313318711yescharacteristic
409C2×C4#1313319011yescharacteristic
410C2×C4#1313319211yescharacteristic
411C2×C4#1313320211yescharacteristic
412C2×C4#1313320211yescharacteristic
413C2×C4#1313320511yescharacteristic
414C2×C4D8×D5311yescharacteristic
415C2×C4SD16×D5311yescharacteristic
416C2×C4#1313324011yescharacteristic
417C2×C4#1313324011yescharacteristic
418C2×C4#1313324111yescharacteristic
419C2×C4#1313324411yescharacteristic
420C2×C4#1313324411yescharacteristic
421C2×C4#1313324411yescharacteristic
422C8—42no—
423C8—81no—
424C8—1062no—
425C8—4241no—
426C8—2121no—
427C22—8481no—
428C22—8481no—
429C22—8481no—
430C22—8481no—
431C22—4242no—
432C22—4242no—
433C22—4242no—
434C22—4241no—
435C22—4241no—
436C22—4241no—
437C22—4241no—
438C22—2122no—
439C22—2122no—
440C22—2122no—
441C22#1681054111yescharacteristic
442C22#1681081711yescharacteristic
443C22#1681084111yescharacteristic
444C22#1681085511yescharacteristic
445C22#1681087511yescharacteristic
446C22#1681088911yescharacteristic
447C22#1681089211yescharacteristic
448C4—21no—
449C4—21no—
450C4—161no—
451C4—4241no—
452C4#1681077111yescharacteristic
453C4#1681077511yescharacteristic
454C4#1681081711yescharacteristic
455C4#1681084111yescharacteristic
456C4#1681086011yescharacteristic
457C4#1681087711yescharacteristic
458C4#1681094311yescharacteristic
459C4#1681095011yescharacteristic
460C2—8482no—
461C2—4242no—
462C2#2027405911yescharacteristic
463C2#2027412711yescharacteristic
464C2#2027419311yescharacteristic
465C2#2027427911yescharacteristic
466C2#2027590711yescharacteristic
467C2#2027601011yescharacteristic
468C2#2027602711yescharacteristic
subgroupG/H
centerC23#13133190
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53#121
prime radicalC2×#16810819C2
hypercenter#1296D53
hypocenterC53#10473
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #20275520C2semidirect
3. #20275508C2semidirect
4. #20274712C2semidirect
5. C2×#16810373C2semidirect
6. C2×#16810366C2semidirect
7. C53×#1296C2semidirect
130. C2×#88322D8semidirect
232. #1296D53semidirect
261. C53#10473semidirect

Maximal subgroups

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

subgroupindex
C2×#168108192
#202755202
#202755082
#202747122
C2×#168103732
C2×#168103662
C53×#12962
#1047353